Dependencies

Consider the linear time-varying system

with \(A\) continuous, and write \(\Phi (t,t_{0})\) for its state transition matrix (definition 148).

The origin is an exponentially stable equilibrium (definition 284) of this system if and only if it is globally exponentially stable (definition 285).

For the discrete-time linear system \(x(t+1) = A(t)x(t)\), the origin is locally exponentially stable (definition 337) if and only if it is globally exponentially stable (definition 338).

For \(\alpha \) class \(\mathcal{K}\) on \([0, a)\) (definition 225), the decay system is the scalar time-invariant system

as a continuous-time system (definition 65) with no input. Its meaningful states are \(y \in [0, a)\).

A continuous-time linear time-invariant (LTI) system is the data of four matrices \(A \in \mathbb {K}^{X \times X}\), \(B \in \mathbb {K}^{X \times U}\), \(C \in \mathbb {K}^{Y \times X}\), \(D \in \mathbb {K}^{Y \times U}\), describing the LTV system (definition 114) with these constant coefficients, \(\dot x = Ax + Bu\), \(y = Cx + Du\).

The continuous-time system (definition 65) of the LTV system of \(s\) (definition 115, definition 126).

The LTV system (definition 114) whose coefficients are the constant maps \(t \mapsto A\), \(t \mapsto B\), \(t \mapsto C\), \(t \mapsto D\).

A continuous-time linear time-varying (LTV) system with states indexed by \(X\), inputs by \(U\), outputs by \(Y\) and coefficients in \(\mathbb {K}\) is the data of four matrix-valued maps on \(\mathbb {R}\),

describing the continuous-time system (definition 65) \(\dot x = A(t)x + B(t)u\), \(y = C(t)x + D(t)u\) (definition 115).

A Carathéodory trajectory (definition 89) of definition 115.

A continuous-time LTV system is time-invariant when each of its four coefficients takes the same value at every pair of times.

A trajectory (definition 84) for all time of definition 115.

A map \(x\) is a trajectory of \(s\) under the input \(u\) on a set \(I \subseteq \mathbb {R}\) when it is a trajectory (definition 83) of definition 115:

The output equation, \((t, z) \mapsto C(t)\, z + D(t)\, u(t)\): the output map (definition 82) of definition 115.

The continuous-time system (definition 65) with \(F(t, x, u) = A(t)x + B(t)u\) and \(h(t, x, u) = C(t)x + D(t)u\).

The right-hand side of the state equation, \((t, z) \mapsto A(t)\, z + B(t)\, u(t)\): the vector field (definition 81) of definition 115.

For real square matrices \(A\), \(P\) and \(Q\), the matrix \(P\) satisfies the continuous Lyapunov equation for \(A\) with forcing \(Q\) when

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.2.

A continuous-time system with state space \(E\), input space \(U\) and output space \(Y\) is the data of two maps

to be read as \(\dot x = F(t, x, u)\), \(y = h(t, x, u)\). The structure is only the data.

The continuous-time system \(\dot x = f(t, x)\), \(y = g(t, x)\), which has no input: its input space is the one-point type.

The equilibrium is globally asymptotically stable when it is stable (definition 275), the system is forward complete from every state at every \(t_0 \ge 0\) (definition 76), and every trajectory on \([t_0, \infty )\) satisfies \(x(t) \to x_{\mathrm{eq}}\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Section 4.1.

The equilibrium is globally exponentially stable when the estimate of definition 284 holds for every initial state, with no radius \(c\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.5.

The equilibrium is globally uniformly asymptotically stable when it is globally uniformly stable (definition 278) and every \(c {\gt} 0\) admits a uniform convergence time (definition 281).

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4.

The equilibrium is globally uniformly stable when it is uniformly stable (definition 277) with a choice of \(\delta (\varepsilon )\) satisfying \(\delta (\varepsilon ) \to \infty \) as \(\varepsilon \to \infty \). In particular every trajectory is bounded, so none escapes in finite time.

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4.

The trajectories of \(s\) have the uniform class \(\mathcal{K}\) bound \(\alpha : [0, a) \to [0, b)\) about \(x_{\mathrm{eq}}\) when

for every trajectory on a segment \([t_0, t_1]\) with \(t_0 \ge 0\) and \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} a\), and every \(t \in [t_0, t_1]\).

Reference: Khalil, Nonlinear Systems (3rd ed.), (4.19).

The trajectories of \(s\) have the uniform class \(\mathcal{K}_{\infty }\) bound \(\alpha \) about \(x_{\mathrm{eq}}\) when \(\| x(t) - x_{\mathrm{eq}}\| \le \alpha (\| x(t_0) - x_{\mathrm{eq}}\| )\) for every trajectory on a segment \([t_0, t_1]\) with \(t_0 \ge 0\) and every \(t \in [t_0, t_1]\), with no restriction on the initial state.

Reference: Khalil, Nonlinear Systems (3rd ed.), (4.20).

Trajectories starting within \(c\) of \(x_{\mathrm{eq}}\) have a uniform convergence time: for every \(\eta {\gt} 0\) there is \(T = T(\eta ) {\gt} 0\) such that every trajectory on a segment \([t_0, t_1]\), \(t_0 \ge 0\), with \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies \(\| x(t) - x_{\mathrm{eq}}\| {\lt} \eta \) for \(t \in [t_0 + T, t_1]\).

A continuous-time system has unique solutions under the input \(u\) when any two trajectories on a segment \([t_0, t_1]\) (definition 83) that agree at \(t_0\) agree on all of \([t_0, t_1]\).

A continuous-time system is autonomous when \(F(t, x, u)\) and \(h(t, x, u)\) do not depend on \(u\).

A Carathéodory trajectory (definition 89) on all of \(\mathbb {R}\).

A map \(x : \mathbb {R} \to E\), with \(E\) a Banach space, is a Carathéodory trajectory of \(s\) under the input \(u\) on a set \(I \subseteq \mathbb {R}\) when, for all \(a \le b\) in \(I\), the map \(\tau \mapsto F(\tau , x(\tau ), u(\tau ))\) is integrable on \([a, b]\) and

Unlike definition 83 it does not require \(x\) to be differentiable, so it admits the solutions produced by discontinuous or merely measurable inputs, which input-to-state stability quantifies over.

Let \(s\) be an input-free continuous-time system. A Chetaev certificate consists of an open region \(D\), a point \(x_{\rm eq}\in \partial D\), a radius \(\rho {\gt}0\), and a globally \(C^1\) function \(V\). For \(\| x-x_{\rm eq}\| \le \rho \), \(V(x){\gt}0\) in \(D\) and \(V(x)=0\) on \(\partial D\); in \(D\) the derivative \(\nabla V(x)\cdot F(t,x,())\) is strictly positive for every \(t\ge 0\).

Reference: Khalil, Nonlinear Systems, 3rd ed., Theorem 4.3, p. 125.

A differentiable \(V : \mathbb {R} \times E \to \mathbb {R}\) is an exponential Lyapunov function on a neighbourhood \(D\) of \(x_{\mathrm{eq}}\) when there are \(k_1, k_2, k_3, a {\gt} 0\) with, for all \(t \ge 0\) and \(x \in D\),

It certifies local exponential stability (definition 284).

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.10.

A continuous-time system is forward complete under the input \(u\) when it is forward complete (definition 76) from every state at every initial time.

A continuous-time system is forward complete from a set \(S\) at time \(t_0\) under the input \(u\) when every trajectory on a segment \([t_0, t_1]\) (definition 83) with \(x(t_0) \in S\) agrees on that segment with a trajectory on \([t_0, \infty )\).

A set \(S \subseteq E\) is forward invariant (or positively invariant) for \(s\) under the input \(u\) when every trajectory on a segment \([t_0, t_1]\) (definition 83) that starts in \(S\) stays in \(S\):

A differentiable \(V : \mathbb {R} \times E \to \mathbb {R}\) is a Lyapunov function on a neighbourhood \(D\) of \(x_{\mathrm{eq}}\) when, for all \(t \ge 0\) and \(x \in D\),

for some \(W_1\) positive definite on \(D\) (definition 27), with \(\dot V\) the derivative along the system (definition 88). It certifies stability (definition 275).

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.8.

A uniform Lyapunov function (definition 308) with \(\dot V(t, x) \le -W_3(x)\) for all \(t \ge 0\), \(x \in D\), for some \(W_3\) positive definite on \(D\). It certifies uniform asymptotic stability (definition 282).

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.9.

A continuous-time system is time-invariant when \(F(t, \cdot , \cdot )\) and \(h(t, \cdot , \cdot )\) are the same for every \(t\).

A trajectory (definition 83) defined on all of \(\mathbb {R}\).

A map \(x : \mathbb {R} \to E\) is a trajectory of \(s\) under the input \(u\) on a set \(I \subseteq \mathbb {R}\) when

the derivative being taken within \(I\). Stated on a set: a map can satisfy the equation on an interval without being defined for all time.

A Lyapunov function (definition 307) that is also decrescent: \(V(t, x) \le W_2(x)\) for all \(t \ge 0\), \(x \in D\), for some \(W_2\) positive definite on \(D\). The upper bound is what makes the conclusion uniform: it certifies uniform stability (definition 277).

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.8.

For \(V : \mathbb {R} \times E \to \mathbb {R}\) and an input value \(u \in U\), the derivative of \(V\) along the system \(\dot x = F(t, x, u)\) is

the Lie derivative of \(V\) along the field \(F(\cdot , \cdot , u)\) (definition 52).

The equilibrium is locally asymptotically stable when it is stable (definition 275) and for every \(t_0 \ge 0\) there is \(c = c(t_0) {\gt} 0\) such that the system is forward complete from the ball of radius \(c\) about \(x_{\mathrm{eq}}\) at \(t_0\) (definition 76) and every trajectory on \([t_0, \infty )\) starting in that ball satisfies \(x(t) \to x_{\mathrm{eq}}\). Forward completeness keeps solutions that escape in finite time from being silently excluded.

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4 (asymptotically stable).

The equilibrium is locally exponentially stable when there are \(c, k, \gamma {\gt} 0\) such that every trajectory on a segment \([t_0, t_1]\), \(t_0 \ge 0\), with \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies

The constants do not depend on \(t_0\): exponential stability is uniform, as in the reference.

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.5.

Under an input signal \(u\), the output equation is \((t, x) \mapsto h(t, x, u(t))\). An instantaneous relation: no derivative appears.

The equilibrium \(x_{\mathrm{eq}}\) of an autonomous continuous-time system \(\dot x = f(t, x)\) is stable when for every \(\varepsilon {\gt} 0\) and \(t_0 \ge 0\) there is \(\delta = \delta (\varepsilon , t_0) {\gt} 0\) such that every trajectory on a segment \([t_0, t_1]\) with \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} \delta \) satisfies \(\| x(t) - x_{\mathrm{eq}}\| {\lt} \varepsilon \) for all \(t \in [t_0, t_1]\).

Quantifying over segments rather than over solutions on \([t_0, \infty )\) means a solution that escapes in finite time counts against stability instead of being silently excluded.

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4 (stable).

The continuous-time system \(\dot x = f(x, u)\), \(y = g(x, u)\), whose equations do not depend on time.

The equilibrium is uniformly asymptotically stable when it is uniformly stable (definition 277) and there is \(c {\gt} 0\) from which convergence has a uniform convergence time (definition 281).

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4 (uniformly asymptotically stable).

The equilibrium is uniformly stable when \(\delta \) in definition 275 can be chosen independently of \(t_0\): \(\delta = \delta (\varepsilon )\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4 (uniformly stable).

The equilibrium is unstable when it is not stable (definition 275).

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4.

Under an input signal \(u : \mathbb {R} \to U\), the right-hand side of the state equation is the time-dependent vector field \((t, x) \mapsto F(t, x, u(t))\).

The autonomous system (definition 70) \(\dot x = F(t, x, 0)\), \(y = h(t, x, 0)\) obtained by holding the input at \(0\). Stability of a system with input means stability of this system.

The controllability matrix of a pair \((A, B)\) with \(A \in \mathbb {F}^{n \times n}\) and \(B \in \mathbb {F}^{n \times m}\) is the block-column matrix

Columns are indexed by \(\mathrm{Fin}\, n \times \mathrm{Fin}\, m\), so that \(A^{k}\) is available without casting \(k : \mathrm{Fin}\, n\) through \(\mathrm{Fin.val}\).

The upper right Dini derivative of \(f : \mathbb {R} \to \mathbb {R}\) at \(t\) is

When \(f\) is differentiable at \(t\) this equals \(f'(t)\).

The forced response of \(x(t+1) = A(t)x(t) + B(t)u(t)\) from the state \(x_0\) at time \(t_0\),

with \(\Phi \) the discrete state transition matrix (definition 204). It solves the recursion forward from \(t_0\) only.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5, Section 5.3.

The homogeneous response of \(x(t+1) = A(t)x(t)\) from the state \(x_0\) at time \(t_0\), \(x(t) = \Phi (t, t_0)\, x_0\): the forced response (definition 214) with the input switched off.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5, Section 5.3.

A discrete-time linear time-invariant (LTI) system is the data of four matrices \(A\), \(B\), \(C\), \(D\), describing the discrete-time LTV system (definition 131) with these constant coefficients, \(x(k+1) = Ax(k) + Bu(k)\), \(y(k) = Cx(k) + Du(k)\).

The discrete-time system (definition 94) of the LTV system of \(s\) (definition 132, definition 142).

The discrete-time LTV system (definition 131) whose coefficients are the constant maps \(k \mapsto A\), \(k \mapsto B\), \(k \mapsto C\), \(k \mapsto D\).

A discrete-time linear time-varying (LTV) system is the same data as definition 114 with the coefficients indexed by \(\mathbb {N}\) instead of \(\mathbb {R}\), describing the discrete-time system (definition 94) \(x(k+1) = A(k)x(k) + B(k)u(k)\), \(y(k) = C(k)x(k) + D(k)u(k)\) (definition 132).

The forward time domain \(\mathbb {N}\) rather than \(\mathbb {Z}\) matches the discrete solution theory, whose state transition matrix is an ordered product over \([t_0, t)\).

A discrete-time LTV system is time-invariant when each of its four coefficients takes the same value at every pair of times.

A trajectory (definition 106) for all time of definition 132.

A sequence \(x\) is a trajectory of \(s\) under the input \(u\) on a set \(I \subseteq \mathbb {N}\) when it is a trajectory (definition 105) of definition 132:

The output equation, \((k, z) \mapsto C(k)\, z + D(k)\, u(k)\): the output map (definition 104) of definition 132.

The discrete-time system (definition 94) with \(F(k, x, u) = A(k)x + B(k)u\) and \(h(k, x, u) = C(k)x + D(k)u\).

The right-hand side of the state equation, \((k, z) \mapsto A(k)\, z + B(k)\, u(k)\): the next-state map (definition 103) of definition 132.

For real square matrices \(A\), \(P\) and \(Q\), the matrix \(P\) satisfies the discrete-time Lyapunov equation for \(A\) with forcing \(Q\) when

Along \(x(t+1) = Ax(t)\) the quadratic form \(V(x) = x^{\mathsf T}Px\) then changes by \(V(Ax) - V(x) = -x^{\mathsf T}Qx\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.4.

A discrete-time system with state space \(E\), input space \(U\) and output space \(Y\) is the data of two maps

to be read as \(x^{+} = F(k, x, u)\), \(y = h(k, x, u)\). The same data as definition 65 with time in \(\mathbb {N}\); what differs is the reading, a shift rather than a derivative.

The discrete-time system \(x^{+} = f(k, x)\), \(y = g(k, x)\), which has no input: its input space is the one-point type.

The equilibrium is globally asymptotically stable when it is stable (definition 328) and every trajectory on \([k_0, \infty )\) satisfies \(x(k) \to x_{\mathrm{eq}}\).

The equilibrium is globally exponentially stable when the estimate of definition 337 holds for every initial state, with no radius \(c\).

The equilibrium is globally uniformly asymptotically stable when it is globally uniformly stable (definition 331) and every \(c {\gt} 0\) admits a uniform convergence time (definition 334).

The equilibrium is globally uniformly stable when it is uniformly stable (definition 330) with a choice of \(\delta (\varepsilon )\) satisfying \(\delta (\varepsilon ) \to \infty \) as \(\varepsilon \to \infty \).

Trajectories are globally uniformly ultimately bounded with ultimate bound \(b\) around \(x_c\) when definition 339 holds for every \(a {\gt} 0\), with no radius \(c\).

Trajectories starting within \(c\) of \(x_{\mathrm{eq}}\) have a uniform convergence time: for every \(\eta {\gt} 0\) there is \(T = T(\eta ) \in \mathbb {N}\) such that every trajectory on \([k_0, \infty )\) with \(\| x(k_0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies \(\| x(k) - x_{\mathrm{eq}}\| {\lt} \eta \) for \(k \ge k_0 + T\).

A discrete-time system is autonomous when \(F(k, x, u)\) and \(h(k, x, u)\) do not depend on \(u\).

A discrete-time system is time-invariant when \(F(k, \cdot , \cdot )\) and \(h(k, \cdot , \cdot )\) are the same for every \(k\).

A trajectory (definition 105) satisfying the recursion at every \(k \in \mathbb {N}\).

A sequence \(x : \mathbb {N} \to E\) is a trajectory of \(s\) under the input \(u\) on a set \(I \subseteq \mathbb {N}\) when

Membership constrains the step out of \(k\), so \(I = [k_0, \infty )\) is the forward recursion from \(k_0\), and the horizon \([k_0, k_1]\) corresponds to \(I = [k_0, k_1)\).

The equilibrium is locally asymptotically stable when it is stable (definition 328) and for every \(k_0\) there is \(c = c(k_0) {\gt} 0\) such that every trajectory on \([k_0, \infty )\) with \(\| x(k_0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies \(x(k) \to x_{\mathrm{eq}}\). Unlike definition 279 no forward completeness is asked for: discrete-time trajectories always exist.

The equilibrium is locally exponentially stable, or geometrically stable, when there are \(c, C {\gt} 0\) and \(0 \le \rho {\lt} 1\) such that every trajectory on \([k_0, \infty )\) with \(\| x(k_0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies

The counterpart of definition 284, with \(\rho = e^{-\gamma }\).

Reference: Jungers and van de Wouw (2026), Definition 2, estimate (6).

Under an input sequence \(u\), the output equation is \((k, x) \mapsto h(k, x, u(k))\).

The equilibrium \(x_{\mathrm{eq}}\) of an autonomous discrete-time system \(x^{+} = f(k, x)\) is stable when for every \(\varepsilon {\gt} 0\) and \(k_0 \in \mathbb {N}\) there is \(\delta = \delta (\varepsilon , k_0) {\gt} 0\) such that every trajectory on \([k_0, \infty )\) with \(\| x(k_0) - x_{\mathrm{eq}}\| {\lt} \delta \) satisfies \(\| x(k) - x_{\mathrm{eq}}\| {\lt} \varepsilon \) for all \(k \ge k_0\).

The counterpart of definition 275. Trajectories on rays suffice: a discrete-time trajectory exists from every state for all later times (lemma 109).

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4; Jungers and van de Wouw (2026), Definition 2.

The discrete-time system \(x^{+} = f(x, u)\), \(y = g(x, u)\), whose equations do not depend on time.

The equilibrium is uniformly asymptotically stable when it is uniformly stable (definition 330) and there is \(c {\gt} 0\) from which convergence has a uniform convergence time (definition 334).

The equilibrium is uniformly stable when \(\delta \) in definition 328 can be chosen independently of \(k_0\).

Trajectories are uniformly ultimately bounded with ultimate bound \(b\) around \(x_c\) when there is \(c {\gt} 0\) such that for every \(0 {\lt} a {\lt} c\) there is \(T = T(a, b) \in \mathbb {N}\) with

for every trajectory on \([k_0, \infty )\). The point \(x_c\) need not be an equilibrium.

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.6; Jungers and van de Wouw (2026), Definition 3.

The equilibrium is unstable when it is not stable (definition 328).

Under an input sequence \(u : \mathbb {N} \to U\), the right-hand side of the state equation is the time-dependent next-state map \((k, x) \mapsto F(k, x, u(k))\). The same expression as definition 81; what differs is what a trajectory does with it, a shift rather than a derivative.

The autonomous system (definition 99) \(x^{+} = F(k, x, 0)\), \(y = h(k, x, 0)\) obtained by holding the input at \(0\). Stability of a system with input means stability of this system.

The discrete-time state transition matrix \(\Phi (t,t_0)\):

Formally, the ordered product of \(A\) over \([t_0, t)\), which for \(t \le t_0\) is the empty product \(I\). The system runs forward only, so the \(t {\lt} t_0\) branch carries no meaning; it is there to make the definition total.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5, equation (5.11).

The forced response of \(\dot x = A(t)x + B(t)u(t)\) from the state \(x_0\) at time \(t_0\), given by the variation-of-constants formula

with \(\Phi \) the state transition matrix (definition 148).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5, Theorem 5.2.

The Hautus controllability matrix of \((A, B)\) at a complex number \(\mu \) is the block-column matrix

The Hautus observability matrix of \((A, C)\) at a complex number \(\mu \) is the block-row matrix

The homogeneous response of \(\dot x = A(t)x\) from the state \(x_0\) at time \(t_0\),

with \(\Phi \) the state transition matrix (definition 148). This is the forced response (definition 172) with the input switched off, and the object Lyapunov stability is stated about.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.1.

A class \(\mathcal{K}\) function on \([0,a)\) is a continuous strictly increasing map \(\alpha : [0,a) \to [0,b)\) with \(\alpha (0) = 0\). All class K functions have an inverse \(\alpha ^{-1} : [0,b) \to [0,a)\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.2 (class \(\mathcal{K}\)). Differences:

• Khalil writes the codomain as \([0,\infty )\); here the bound \(b\) is tracked in the type.

• The inverse is bundled as data rather than derived.

A class \(\mathcal{K}_{\infty }\) function is a continuous strictly increasing map \(\alpha : [0,\infty ) \to [0,\infty )\) with \(\alpha (0) = 0\) and \(\alpha (r) \to \infty \) as \(r \to \infty \).

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.2 (class \(\mathcal{K}_{\infty }\)). The inverse is bundled as data rather than derived.

A class \(\mathcal{KL}\) function on \([0,a) \times [0,\infty )\) is continuous, class \(\mathcal{K}\) in the first argument, and for each fixed \(r {\gt} 0\) is strictly decreasing and tends to \(0\) as \(s \to \infty \). It arises as the bound \(\| x(t)\| \le \beta (\| x_0\| , t)\) in asymptotic stability estimates.

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.3. Difference: Khalil says \(\beta (\cdot , s)\) is class \(\mathcal{K}\); here that is spelled out as separate fields, so no class \(\mathcal{K}\) instance is required for each \(s\).

A global class \(\mathcal{KL}\) function is a map \(\beta : [0,\infty ) \times [0,\infty ) \to \mathbb {R}\) that is class \(\mathcal{K}_{\infty }\) in the first argument (continuous, strictly increasing, zero at zero, radially unbounded) and, for each fixed \(r \ge 0\), is antitone and tends to \(0\) as \(s \to \infty \).

Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.3 with \(a = \infty \).

A class \(\mathcal{L}\) function on \([0,\infty )\) is continuous, strictly positive, and strictly decreasing, and satisfies \(\sigma (s) \to 0\) as \(s \to \infty \). It carries the time decay of an asymptotic stability estimate: the product of a class \(\mathcal{K}\) function of the initial deviation with a class \(\mathcal{L}\) function of elapsed time is a class \(\mathcal{KL}\) bound.

A singular class \(\mathcal{L}\) function on \((0,\infty )\) is continuous, strictly positive and antitone, with \(U(s) \to 0\) as \(s \to \infty \) and \(U(s) \to \infty \) as \(s \to 0^{+}\). It differs from class \(\mathcal{L}\) in two ways: the domain excludes \(0\), where the function is instead required to blow up, and the decrease is only required to be weak. It arises as the inverse of the sliding-window function in the class \(\mathcal{KL}\) characterization of asymptotic stability.

A linear system \((A, B)\) is controllable when every target state \(x \in \mathbb {F}^{n}\) is reachable from the origin in \(n\) steps: there exist input vectors \(u_{0}, u_{1}, \dots , u_{n-1} \in \mathbb {F}^{m}\) such that

This phrasing does not name the controllability matrix, so the bridge theorem 446 has real content.

A function \(x\), defined on the segment between \(t_0\) and \(t_1\), is an integral solution of \(\dot{x} = F(t,x)\) with initial value \(x_0\) when

for every \(t\) between \(t_0\) and \(t_1\).

A linear system \((A, C)\) is observable when the only state \(x \in \mathbb {F}^{n}\) for which

is the zero state. This phrasing does not mention \(\mathcal{O}(A, C)\), so the bridge theorem 444 has real content.

A function \(W\) is positive definite on \(S\) about \(x_0\) when it is continuous on \(S\), \(W(x_0) = 0\), and \(W(x) {\gt} 0\) for every \(x \in S \setminus \{ x_0\} \).

Addition of the four Kalman coordinate sectors defines a linear equivalence \(X_{c\bar o}\times X_{co}\times X_{\bar c\bar o}\times X_{\bar c o} \simeq \mathbb {C}^{n}\). Choosing sector bases therefore gives an adapted basis of the original state space.

A Kalman decomposition of \((A,B,C)\) records four coordinate sectors adapted to the reachable and unobservable subspaces. Their direct sums recover those subspaces and the full state space. The chosen complements are noncanonical and are not individually asserted to be invariant under \(A\).

References: Hespanha, Linear Systems Theory; Kailath, Linear Systems.

For \(V : \mathbb {R} \times E \to \mathbb {R}\) and a time-dependent vector field \(v : \mathbb {R} \times E \to E\), the Lie derivative of \(V\) along \(v\) is

the derivative of \(V\) at \((t, x)\) in the direction \((1, v(t, x))\): the Lie derivative of \(V\), as a function on \(\mathbb {R} \times E\), along the autonomous field \((t, x) \mapsto (1, v(t, x))\).

For a real matrix \(A\), its complexification \(A_{\mathbb C}\) is obtained by applying \(\mathbb R \hookrightarrow \mathbb C\) to each entry.

The observability matrix of a pair \((A, C)\) with \(A \in \mathbb {F}^{n \times n}\) and \(C \in \mathbb {F}^{p \times n}\) is the block-row matrix

Rows are indexed by \(\mathrm{Fin}\, n \times \mathrm{Fin}\, p\), so that \(A^{k}\) is available without casting \(k : \mathrm{Fin}\, n\) through \(\mathrm{Fin.val}\).

The \(\omega \)-limit set of \(\varphi : \tau \to X\), for \(\tau \) a preordered index set such as \(\mathbb {R}\) or \(\mathbb {N}\), is the set of cluster points of \(\varphi \) as \(t \to \infty \):

the points \(y\) such that \(\varphi (t)\) is frequently in every neighbourhood of \(y\) as \(t \to \infty \).

The \(k\)-th term of the Peano–Baker series for a time-varying state matrix \(A(t)\):

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5 (Peano-Baker series).

A real matrix \(P\) represents the quadratic form \(x \mapsto x^{\mathsf T}Px\) on Euclidean state space.

Reference: standard quadratic Lyapunov-function construction.

The reachable subspace of a pair \((A,B)\) is the column span of its controllability matrix:

Thus it is exactly the span of \(B,AB,\ldots ,A^{n-1}B\).

Reference: Hespanha, Linear Systems Theory.

A real square matrix \(A\) is Hurwitz when every eigenpair \((\mu , v)\) of its complexification, with \(v \ne 0\), satisfies \(\operatorname {Re}(\mu ) {\lt} 0\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.1.

A real square matrix \(A\) is Hurwitz with decay rate \(\alpha \) when every eigenpair \((\mu , v)\) of its complexification, with \(v \ne 0\), satisfies \(\operatorname {Re}(\mu ) {\lt} -\alpha \).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.1.

A real square matrix \(A\) is Schur when every eigenpair \((\mu , v)\) of its complexification, with \(v \ne 0\), satisfies \(|\mu | {\lt} 1\). The discrete-time counterpart of definition 364.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.3.

The state transition matrix \(\Phi (t,t_0)\), given by the Peano–Baker series

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5, Theorem 5.1 (Peano-Baker series).

The unobservable subspace of \((A, C)\) is the \(A\)-invariant subspace

Equivalently, \(\mathcal{N}(A, C) = \ker \mathcal{O}(A, C)\), but here it is phrased without naming \(\mathcal{O}\) so the bridge theorem 448 reads as content.

Let \(f\) be antitone and locally integrable on \((0,\infty )\), and let \(a, b\) be monotone on a set \(S\) with \(0 {\lt} a(\eta ) {\lt} b(\eta )\). Then the mean value

is antitone on \(S\).

Under the surjectivity conditions of lemma 241, \(W\bigl(\mathrm{invFunOn}\, W\, (0,\infty )\, s\bigr) = s\) for all \(s {\gt} 0\).

Let \(V\) be a finite-dimensional complex vector space, \(T\) an endomorphism, and \(B\) a bilinear form with \(B(x,Ty) + B(Tx,y) = c\, B(x,y)\). If no pair of eigenvalues of \(T\) sums to \(c\), then \(B = 0\).

This is the uniqueness half of solvability for the Sylvester-type equation: the associated operator has trivial kernel exactly when the spectrum is nonresonant.

Let \(T\) be an endomorphism of a complex vector space and \(B\) a bilinear form satisfying the Sylvester-type identity

If \(\xi + \nu \ne c\), then \(B\) vanishes on \(\ker (T - \xi )^{k} \times \ker (T - \nu )^{l}\) for all \(k, l\).

Class \(\mathcal{K}\) is closed under composition: if \(\alpha \) is class \(\mathcal{K}\) on \([0, a) \to [0, b)\) and \(\beta \) is class \(\mathcal{K}\) on \([0, b) \to [0, c)\), then \(\beta \circ \alpha \) is class \(\mathcal{K}\) on \([0, a) \to [0, c)\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, third bullet.

Let \(f : \mathbb {R} \to \mathbb {R}\) satisfy \(f(0) = 0\) and \(f(a) = b\) and be continuous and strictly increasing on \([0, a]\), with \(a, b {\gt} 0\). Then \(f\) is a class \(\mathcal{K}\).

If \(\alpha \) is class \(\mathcal{K}\) on \([0, a) \to [0, b)\) and \(0 {\lt} c {\lt} a\), then \(\alpha \) restricted to \([0, c)\) is class \(\mathcal{K}\) on \([0, c) \to [0, \alpha (c))\).

definition 228 with the codomain bound named: if \(\alpha (c) = e\), the restriction of \(\alpha \) to \([0, c)\) is class \(\mathcal{K}\) on \([0, c) \to [0, e)\).

Class \(\mathcal{K}\) is closed under inversion. The inverse of a class K function is also class K.

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, first bullet. Khalil’s domain \([0, \alpha (a))\) is the codomain bound \(b\) carried by the type.

Class \(\mathcal{K}_{\infty }\) is closed under pointwise addition: if \(\alpha \) and \(\beta \) are class \(\mathcal{K}_{\infty }\), so is \(\alpha + \beta \).

Class \(\mathcal{K}_{\infty }\) is closed under composition: if \(\alpha \) and \(\beta \) are class \(\mathcal{K}_{\infty }\), so is \(\beta \circ \alpha \), with inverse \(\alpha ^{-1} \circ \beta ^{-1}\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fourth bullet.

Class \(\mathcal{K}_{\infty }\) is closed under pointwise minimum: if \(\alpha \) and \(\beta \) are class \(\mathcal{K}_{\infty }\), so is \(\min (\alpha , \beta )\).

Let \(f : \mathbb {R} \to \mathbb {R}\) satisfy \(f(0) = 0\), be continuous and strictly increasing on \([0, \infty )\), and have \(f(r) \to \infty \) as \(r \to \infty \). Then \(f\) is a class \(\mathcal{K}_{\infty }\) function, its inverse being the inverse of \(f\) on \([0,\infty )\).

For every \(p {\gt} 0\), the map \(r \mapsto r^{p}\) is class \(\mathcal{K}_{\infty }\).

Class \(\mathcal{K}_{\infty }\) is closed under multiplication by a positive constant: if \(\alpha \) is class \(\mathcal{K}_{\infty }\) and \(c {\gt} 0\), so is \(c\alpha \).

Class \(\mathcal{K}_{\infty }\) is closed under inversion: if \(\alpha \) is class \(\mathcal{K}_{\infty }\), so is \(\alpha ^{-1}\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, second bullet.

Every class \(\mathcal{K}_{\infty }\) function \(\alpha \) restricts, for each \(a {\gt} 0\), to a class \(\mathcal{K}\) function on \([0, a) \to [0, \alpha (a))\).

Let \(\beta \) be class \(\mathcal{KL}\) on \([0,a)\) and \(\alpha \) be class \(\mathcal{K}\) on \([0,b)\), and suppose \(\beta (r, s) {\lt} b\) for every \(r \in [0,a)\) and \(s \ge 0\). Then \((r, s) \mapsto \alpha (\beta (r, s))\) is class \(\mathcal{KL}\) on \([0,a)\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet (the outer \(\alpha _1\)). The range hypothesis is additional: \(\alpha _1\) is defined only on \([0,b)\), so \(\beta \)’s values must land there, which Khalil leaves implicit.

If \(\beta \) is class \(\mathcal{KL}\) on \([0,a)\) and \(\alpha \) is class \(\mathcal{K}_{\infty }\), then \((r, s) \mapsto \alpha (\beta (r, s))\) is class \(\mathcal{KL}\) on \([0,a)\). This is definition 255 with the range hypothesis discharged by \(\alpha \) being defined on all of \([0,\infty )\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet, with \(\alpha _1\) of class \(\mathcal{K}_{\infty }\); no range hypothesis is then needed.

If \(\beta \) is class \(\mathcal{KL}\) on \([0,b)\) and \(\alpha \) is class \(\mathcal{K}\) on \([0,a) \to [0,b)\), then \((r, s) \mapsto \beta (\alpha (r), s)\) is class \(\mathcal{KL}\) on \([0,a)\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet (the inner \(\alpha _2\)).

If \(\beta \) is class \(\mathcal{KL}\) on \([0,a)\) and \(\alpha \) is class \(\mathcal{K}\) on \([0,a)\), then \((r, s) \mapsto \min \bigl(\beta (r,s), \alpha (r)\bigr)\) is class \(\mathcal{KL}\) on \([0,a)\).

Capping a class \(\mathcal{KL}\) bound by a function of the initial condition alone preserves the class. Its use is to recover a range condition: a bound obtained only as a majorant — from a smoothing construction, say — carries no upper control, and capping restores it without disturbing the bound, since the minimum still dominates whatever \(\beta \) dominated.

If \(\alpha \) is class \(\mathcal{K}\) on \([0,a)\) and \(\gamma \) is class \(\mathcal{L}\), then \(\beta (r, s) := \alpha (r)\gamma (s)\) is class \(\mathcal{KL}\) on \([0,a)\).

Let \(\alpha \) be class \(\mathcal{K}\) on \([0,a)\) and let \(U\) be a singular class \(\mathcal{L}\) function (positive, antitone, \(U(s) \to 0\) as \(s \to \infty \) and \(U(s) \to \infty \) as \(s \to 0^{+}\)). Then

is class \(\mathcal{KL}\) on \([0,a)\).

If \(\beta \) is class \(\mathcal{KL}\) on \([0, a) \times [0, \infty )\) and \(0 {\lt} a' \le a\), then \(\beta \) is class \(\mathcal{KL}\) on \([0, a') \times [0, \infty )\).

If \(\beta \) is global class \(\mathcal{KL}\) and \(\alpha \) is class \(\mathcal{K}_{\infty }\), then \((r, s) \mapsto \alpha (\beta (r, s))\) is global class \(\mathcal{KL}\). No range hypothesis is needed, since \(\alpha \) is defined on all of \([0,\infty )\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet (the outer \(\alpha _1\)), in the global case \(a = \infty \).

If \(\beta \) is global class \(\mathcal{KL}\) and \(\alpha \) is class \(\mathcal{K}_{\infty }\), then \((r, s) \mapsto \beta (\alpha (r), s)\) is global class \(\mathcal{KL}\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.2, fifth bullet (the inner \(\alpha _2\)), in the global case \(a = \infty \).

If \(\beta \) is global class \(\mathcal{KL}\) and \(\alpha \) is class \(\mathcal{K}_{\infty }\), then \((r, s) \mapsto \min \bigl(\beta (r, s), \alpha (r)\bigr)\) is global class \(\mathcal{KL}\) — capping a \(\mathcal{KL}\) bound by a function of the initial condition alone preserves the class.

If \(\alpha \) is class \(\mathcal{K}_{\infty }\) and \(\gamma \) is class \(\mathcal{L}\), then \(\beta (r, s) := \alpha (r)\gamma (s)\) is global class \(\mathcal{KL}\).

Let \(\alpha \) be class \(\mathcal{K}_{\infty }\) and let \(U\) be a family of functions such that

1. \(U(r, s) {\gt} 0\) for all \(r {\gt} 0\) and \(s {\gt} 0\);

2. \(U(r, \cdot )\) is antitone on \((0,\infty )\) for each \(r {\gt} 0\);

3. \(U(r, s) \to 0\) as \(s \to \infty \), for each \(r {\gt} 0\);

4. \(r \mapsto U(r+1, s)\) is monotone on \([0,\infty )\) for each \(s {\gt} 0\).

Then there is a global class \(\mathcal{KL}\) function \(\beta \) such that

1. \(\beta (r, 0) \to \infty \) as \(r \to \infty \);

2. \(\alpha (r) \le \beta (r, 0)\) for all \(r \ge 0\);

3. \(\min \bigl(\alpha (r), \sqrt{\alpha (r)U(r+1, s)}\bigr) \le \beta (r, s)\) for all \(r \ge 0\) and \(s {\gt} 0\).

Entrywise complexification commutes with the matrix exponential: for a real matrix \(A\),

where \(\overline{\phantom{A}}\) denotes applying \(\mathbb {R} \hookrightarrow \mathbb {C}\) entrywise. This is the bridge used to transfer spectral facts, which live over \(\mathbb {C}\), to norm bounds on a real exponential.

Let \(g\) be an autonomous field that is \(C^1\) at \(x_0\). Then there is a \(T {\gt} 0\) and a curve \(\varphi \) with \(\varphi (0) = x_0\) that is an integral curve of \(g\) on \([0, T]\): a nontrivial solution segment exists from every point at which the field is \(C^1\).

The anchor \(0\) is a construction choice rather than a restriction — by lemma 39 a segment from any other anchor follows.

Reference: the Picard–Lindelöf local existence theorem.

If \(V\) is \(C^1\) and \(v\) is continuous, then \((t, x) \mapsto L_v V(t, x)\) is continuous.

Each term \(P_k(t,t_0)\) of the Peano–Baker series is continuous in \(t\), for \(A\) continuous.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

Let \(f : \mathbb {R} \times \mathbb {R} \to \mathbb {R}\) be nondecreasing in its first argument, nonincreasing in its second, and continuous in each argument separately. Then \(f\) is jointly continuous.

The LTV system of an LTI system (definition 126) is time-invariant (definition 123).

The system of an LTI system (definition 129) is the time-invariant system (definition 66) with \(f(x, u) = Ax + Bu\) and \(g(x, u) = Cx + Du\): the two routes from the LTI data to a general system agree.

A time-invariant LTV system (definition 123) is the LTV system of some LTI system. The converse of lemma 127.

If an LTV system is time-invariant (definition 123), so is the system it describes (definition 67).

Consider the linear time-varying system

with \(A\) continuous, and write \(\Phi (t,t_{0})\) for its state transition matrix (definition 148).

The origin is a stable equilibrium (definition 275) of this system if and only if for every \(t_{0} \ge 0\) there is \(C\) with

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.1(1).

At \(z = 0\) the state equation (definition 116) contributes only the input term, \(A(t)\, 0 + B(t)u(t) = B(t)u(t)\). With the input switched off the origin is therefore an equilibrium of the system.

At zero input the state equation (definition 116) reduces to \((t, z) \mapsto A(t)\, z\).

For continuous \(A\), \(B\) and \(u\), the forced response (definition 172) is continuous on the closed segment between \(t_0\) and \(t_1\), endpoints included.

Let \(f\) be locally integrable on \((0,\infty )\), and let \(a, b\) be continuous on a set \(S\) with \(a(\eta ), b(\eta ) {\gt} 0\) in \(S\). Then

is continuous on \(S\).

If \(f\) is integrable on \([a,t]\) with \(a \le t\), then \(s \mapsto \int _{a}^{s} f(\tau )\, \mathrm{d}\tau \) is continuous on \([a,t]\).

\(\Phi (\cdot , t_0)\) is continuous on the segment between \(t_0\) and \(t_1\) (including both endpoints), for \(A\) continuous and bounded by \(M\) there.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

With the first argument held fixed, \(\tau \mapsto \Phi (t, \tau )\) is continuous on the segment between \(t\) and \(t_1\) — the mirror of lemma 153, which varies the first argument. Needed wherever \(\Phi (t, \cdot )\) is an integrand, as in variation of constants.

Suppose the trajectories of \(s\) have a uniform convergence time from the ball of radius \(c\) (definition 281) and a uniform class \(\mathcal{K}\) bound \(\alpha \) defined on \([0, a_\alpha )\) (definition 295). Let \(0 {\lt} a \le c\) with \(a {\lt} a_\alpha \). Then there is a singular class \(\mathcal{L}\) function \(U\) (definition 249) such that

for every trajectory on a segment \([t_0, t_1]\) with \(t_0 \ge 0\) and \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} a\), and every \(t \in (t_0, t_1]\).

The bound depends on the elapsed time alone: neither on \(t_{0}\), nor on the trajectory, nor on where in the \(a\)-ball it started.

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.5.

Suppose the trajectories of \(s\) have a uniform convergence time from every ball (definition 281) and a uniform class \(\mathcal{K}_{\infty }\) bound (definition 296). Then there is a family \(U : (0,\infty ) \times (0,\infty ) \to \mathbb {R}\) with

1. \(U(r, s) {\gt} 0\) for all \(r {\gt} 0\) and \(s {\gt} 0\);

2. \(U(r, \cdot )\) antitone on \((0,\infty )\) for each \(r {\gt} 0\);

3. \(U(r, s) \to 0\) as \(s \to \infty \), for each \(r {\gt} 0\);

4. \(r \mapsto U(r+1, s)\) monotone on \([0,\infty )\) for each \(s {\gt} 0\);

5. \(\| x(t) - x_{\mathrm{eq}}\| \le U(r, t - t_{0})\) for every \(r {\gt} 0\), every trajectory on a segment \([t_0, t_1]\) with \(t_0 \ge 0\) and \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} r\), and every \(t \in (t_0, t_1]\).

Conditions (1)–(4) are exactly the hypotheses of lemma 266; (5) is the decay bound. Unlike the local case (lemma 297) no single envelope can serve every initial state, so the radius remains a parameter.

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.5 (global case).

Let \(\dot x = f(x)\) with \(f\) Lipschitz, \(\Omega \) compact and forward invariant, \(V \in C^1\) with \(\nabla V \cdot f \le 0\) on \(\Omega \), and \(\varphi \) a trajectory on \([t_0, \infty )\) with \(\varphi (t_0) \in \Omega \). Then \(\nabla V(y) \cdot f(y) = 0\) for every \(y \in \omega (\varphi )\).

Reference: Khalil, Nonlinear Systems (3rd ed.), proof of Theorem 4.4.

A globally asymptotically stable equilibrium (definition 280) is locally asymptotically stable (definition 279).

A globally exponentially stable equilibrium (definition 285) is globally uniformly asymptotically stable (definition 283).

A globally exponentially stable equilibrium (definition 285) is locally exponentially stable (definition 284).

A globally uniformly asymptotically stable equilibrium (definition 283) is globally asymptotically stable (definition 280), given forward completeness (definition 77).

The equilibrium \(x_{\mathrm{eq}}\) is globally uniformly asymptotically stable (definition 283) if and only if there is a \(\beta : [0,\infty ) \times [0,\infty ) \to \mathbb {R}\) that is

1. zero at the origin: \(\beta (0, s) = 0\) for \(s \ge 0\);

2. jointly continuous on \([0,\infty ) \times [0,\infty )\);

3. strictly increasing in \(r\) for each fixed \(s \ge 0\);

4. radially unbounded: \(\beta (r, 0) \to \infty \) as \(r \to \infty \);

5. antitone in \(s\) for each fixed \(r \ge 0\), with \(\beta (r,s) \to 0\) as \(s \to \infty \);

such that \(\| x(t) - x_{\mathrm{eq}}\| \le \beta (\| x(t_0) - x_{\mathrm{eq}}\| , t - t_0)\) for every trajectory on a segment \([t_0, t_1]\) with \(t_0 \ge 0\) and every \(t \in [t_0, t_1]\), with no restriction on the initial state.

Condition (4) is what makes the characterization global: without it \(\beta \) is class \(\mathcal{KL}\) on a bounded radius and the statement collapses to the local one (lemma 302).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.5, global case.

Let the system be forward complete (definition 77). Then \(x_{\mathrm{eq}}\) is globally uniformly asymptotically stable if and only if there is a \(\beta \) with the properties of lemma 303 and \(\| x(t) - x_{\mathrm{eq}}\| \le \beta (\| x(t_0) - x_{\mathrm{eq}}\| , t - t_0)\) for every solution \(x\) on \([t_0, \infty )\) with \(t_0 \ge 0\), and every \(t \ge t_0\). The forward direction holds without completeness.

Let \(\beta \) be class \(\mathcal{KL}\) on \([0, a) \times [0, \infty )\) (definition 250), and suppose every trajectory on a segment \([t_0, t_1]\), \(t_0 \ge 0\), with \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} a\) satisfies \(\| x(t) - x_{\mathrm{eq}}\| \le \beta (\| x(t_0) - x_{\mathrm{eq}}\| , t - t_0)\) on the segment. Then for every \(c {\lt} a\), trajectories from the ball of radius \(c\) have a uniform convergence time (definition 281).

The system of definition 70 is autonomous (definition 71).

If a continuous-time system is autonomous (definition 71) and its input space is nonempty, there are \(f\) and \(g\) with \(F(t, x, u) = f(t, x)\) and \(h(t, x, u) = g(t, x)\) for every input \(u\).

A Carathéodory trajectory on \(I\) is one on every subset of \(I\); in particular one for all time is one on every set.

Let \(\dot x = f(x)\) with \(f\) Lipschitz, and let \(\varphi \) be a trajectory on \([t_0, \infty )\). Then its \(\omega \)-limit set \(\omega (\varphi )\) (definition 59) is forward invariant (definition 87).

Neither boundedness of \(\varphi \) nor compactness of an enclosing set is needed; those hypotheses of Khalil’s Lemma 4.1 are what make \(\omega (\varphi )\) nonempty and compact, not what make it invariant.

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.1.

Let \(V : E \to \mathbb {R}\) be differentiable and positive definite on a neighbourhood \(D\) of \(x_{\mathrm{eq}}\), with \(\nabla V(x) \cdot f(t, x) \le -W_3(x)\) for all \(t \ge 0\), \(x \in D\), for some \(W_3\) positive definite on \(D\). Then \((t, x) \mapsto V(x)\) is a strict uniform Lyapunov function (definition 309).

Let the state equation be \(\dot x = f(x)\) with \(f\) continuous and \(f(x_{\mathrm{eq}}) = 0\), and let \(V : E \to \mathbb {R}\) be \(C^1\) and positive definite on a neighbourhood \(D\) of \(x_{\mathrm{eq}}\) with \(\nabla V(x) \cdot f(x) {\lt} 0\) for \(x \in D \setminus \{ x_{\mathrm{eq}}\} \). Then \((t, x) \mapsto V(x)\) is a strict uniform Lyapunov function (definition 309).

A time-invariant continuous-time system (definition 67) is \(\operatorname {timeInvariant} f\, g\) for some \(f\) and \(g\). The converse of lemma 68.

The system of definition 66 is time-invariant (definition 67).

A trajectory for all time is a trajectory on every set.

definition 83 at \(I = \mathbb {R}\) is definition 84.

Let \(V : E \to \mathbb {R}\) be differentiable and positive definite on a neighbourhood \(D\) of \(x_{\mathrm{eq}}\) (definition 27), with \(\nabla V(x) \cdot f(t, x) \le 0\) for all \(t \ge 0\), \(x \in D\). Then \((t, x) \mapsto V(x)\) is a uniform Lyapunov function (definition 308).

A locally exponentially stable equilibrium (definition 284) is uniformly asymptotically stable (definition 282).

A locally exponentially stable equilibrium (definition 284) is uniformly stable (definition 277).

The equilibrium \(x_{\mathrm{eq}}\) is stable (definition 275) if and only if for each \(t_0 \ge 0\) there is a class \(\mathcal{K}\) function \(\alpha _{t_0}\) on \([0, a_{t_0})\) with \(\| x(t) - x_{\mathrm{eq}}\| \le \alpha _{t_0}(\| x(t_0) - x_{\mathrm{eq}}\| )\) for every trajectory on a segment \([t_0, t_1]\) with \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} a_{t_0}\), and every \(t \in [t_0, t_1]\). A single \(\alpha \) for all \(t_0\) is uniform stability (lemma 301).

The equilibrium \(x_{\mathrm{eq}}\) is uniformly asymptotically stable (definition 282) if and only if there are a class \(\mathcal{KL}\) function \(\beta \) on \([0, a) \times [0, \infty )\) (definition 250) with

for every trajectory on a segment \([t_0, t_1]\) with \(t_0 \ge 0\) and \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} a\), and every \(t \in [t_0, t_1]\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.5, second half.

Let the system be forward complete from some ball \(B(x_{\mathrm{eq}}, c)\) at every \(t_0 \ge 0\) (definition 76). Then \(x_{\mathrm{eq}}\) is uniformly asymptotically stable if and only if there is a class \(\mathcal{KL}\) function \(\beta \) on \([0, a) \times [0, \infty )\) with \(\| x(t) - x_{\mathrm{eq}}\| \le \beta (\| x(t_0) - x_{\mathrm{eq}}\| , t - t_0)\) for every solution \(x\) on \([t_0, \infty )\) with \(t_0 \ge 0\) and \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} a\), and every \(t \ge t_0\). The forward direction holds without completeness.

A uniformly asymptotically stable equilibrium (definition 282) of a forward-complete system (definition 77) is locally asymptotically stable (definition 279). Forward completeness is needed: bounds on solution segments do not by themselves make the segments extend.

The equilibrium \(x_{\mathrm{eq}}\) is uniformly stable (definition 277) if and only if the trajectories have a uniform class \(\mathcal{K}\) bound (definition 295): there is a class \(\mathcal{K}\) function \(\alpha \) on \([0, a)\) with

for every trajectory on a segment \([t_0, t_1]\) with \(t_0 \ge 0\) and \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} a\), and every \(t \in [t_0, t_1]\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.5, first half.

Let the system be forward complete from some ball \(B(x_{\mathrm{eq}}, c)\) at every \(t_0 \ge 0\) (definition 76). Then \(x_{\mathrm{eq}}\) is uniformly stable if and only if there is a class \(\mathcal{K}\) function \(\alpha \) on \([0, a)\) with \(\| x(t) - x_{\mathrm{eq}}\| \le \alpha (\| x(t_0) - x_{\mathrm{eq}}\| )\) for every solution \(x\) on \([t_0, \infty )\) with \(t_0 \ge 0\) and \(\| x(t_0) - x_{\mathrm{eq}}\| {\lt} a\), and every \(t \ge t_0\). The forward direction holds without completeness.

A uniformly stable equilibrium (definition 277) is stable (definition 275).

Fix \(\varepsilon {\gt} 0\). If for every \(\delta {\gt} 0\) some trajectory on a segment \([0, T]\) starts within \(\delta \) of \(x_{\mathrm{eq}}\) and reaches distance at least \(\varepsilon \) from it, then \(x_{\mathrm{eq}}\) is unstable (definition 276).

A map is a trajectory of the unforced system (definition 74) on \(I\) exactly when it is a trajectory of \(s\) under the zero input on \(I\).

Block-column entry shape: at row \(i\) and column \((k, j)\), the controllability matrix coincides with the \((i, j)\) entry of \(A^{k}\, B\):

If the forward difference quotients of \(f\) at \(t\) are bounded above and below, and \(g\) is differentiable at \(t\), then \(D^{+}(f+g)(t) = D^{+}f(t) + g'(t)\). Subadditivity becomes equality once one summand has a genuine derivative.

The upper right Dini derivative is subadditive: \(D^{+}(f+g)(t) \le D^{+}f(t) + D^{+}g(t)\), provided the forward difference quotients of \(f\) are bounded above and below and those of \(g\) are bounded above and cobounded below.

If the forward difference quotients of \(f\) at \(t\) are bounded above and below, then \(D^{+}\bigl(f + c\, \mathrm{id}\bigr)(t) = D^{+}f(t) + c\) — the case \(g(s) = cs\) of lemma 12, and the form the comparison arguments use.

For \(c {\gt} 0\), and forward difference quotients of \(f\) at \(t\) bounded above and cobounded below, \(D^{+}(cf)(t) = c\, D^{+}f(t)\). Positivity is essential: a negative constant reverses the order and turns the \(\limsup \) into a \(\liminf \).

Let the forward difference quotients of \(f\) at \(t\) be bounded above and cobounded below along \(h \to 0^{+}\). Then \(D^{+}f(t) \le L\) if and only if for every \(\varepsilon {\gt} 0\) the quotient \(\frac{f(t+h)-f(t)}{h}\) is eventually at most \(L + \varepsilon \) as \(h \to 0^{+}\).

If \(f\) has right derivative \(L\) at \(t\), then \(D^{+}f(t) = L\): the Dini derivative agrees with the classical one wherever the latter exists.

The forced response (definition 214) takes the value \(x_0\) at \(t = t_0\).

For \(t \ge t_0\) the forced response (definition 214) satisfies \(x(t+1) = A(t)x(t) + B(t)u(t)\).

The homogeneous response (definition 215) takes the value \(x_0\) at \(t = t_0\).

For \(t \ge t_0\) the homogeneous response (definition 215) satisfies \(x(t+1) = A(t)x(t)\).

The LTV system of a discrete-time LTI system (definition 142) is time-invariant (definition 139).

The system of a discrete-time LTI system (definition 145) is the time-invariant system (definition 95) with \(f(x, u) = Ax + Bu\) and \(g(x, u) = Cx + Du\): the two routes from the LTI data to a general system agree.

A time-invariant discrete-time LTV system (definition 139) is the LTV system of some LTI system. The converse of lemma 143.

If a discrete-time LTV system is time-invariant (definition 139), so is the system it describes (definition 96).

For the discrete-time linear system \(x(t+1) = A(t)x(t)\) with state transition matrix \(\Phi \) (definition 204), the origin is stable (definition 328) if and only if for every \(t_0\) there is \(C \ge 0\) with

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.2(1).

At \(z = 0\) the state equation (definition 133) contributes only the input term, \(A(k)\, 0 + B(k)u(k) = B(k)u(k)\). With the input switched off the origin is therefore an equilibrium of the system.

At zero input the state equation (definition 133) reduces to \((k, z) \mapsto A(k)\, z\).

If \(A^{\mathsf T}PA - P = -Q\) (definition 423), then for every \(x\),

For every input \(u\), initial time \(k_0\) and state \(x_0\) there is a trajectory (definition 105) on \([k_0, \infty )\) with \(x(k_0) = x_0\).

A globally asymptotically stable equilibrium (definition 333) is locally asymptotically stable (definition 332).

A globally exponentially stable equilibrium (definition 338) is globally uniformly asymptotically stable (definition 336).

A globally exponentially stable equilibrium (definition 338) is locally exponentially stable (definition 337).

A globally uniformly asymptotically stable equilibrium (definition 336) is globally asymptotically stable (definition 333).

The system of definition 99 is autonomous (definition 100).

If a discrete-time system is autonomous (definition 100) and its input space is nonempty, there are \(f\) and \(g\) with \(F(k, x, u) = f(k, x)\) and \(h(k, x, u) = g(k, x)\) for every input \(u\).

A time-invariant discrete-time system (definition 96) is \(\operatorname {timeInvariant} f\, g\) for some \(f\) and \(g\). The converse of lemma 97.

The system of definition 95 is time-invariant (definition 96).

A trajectory at every time is a trajectory on every set.

Two trajectories (definition 105) on \([k_0, \infty )\) under the same input with \(x(k_0) = y(k_0)\) satisfy \(x(k) = y(k)\) for every \(k \ge k_0\).

If \(F(k, x, u(k)) = f(x)\) for all \(k\) and \(x\), every trajectory (definition 105) on \([k_0, \infty )\) satisfies \(x(k_0 + n) = f^{n}(x(k_0))\) for every \(n \in \mathbb {N}\).

definition 105 at \(I = \mathbb {N}\) is definition 106.

A locally exponentially stable equilibrium (definition 337) is uniformly asymptotically stable (definition 335).

A locally exponentially stable equilibrium (definition 337) is uniformly stable (definition 330).

If \(x_{\mathrm{eq}}\) is stable (definition 328), then \(f(k, x_{\mathrm{eq}}) = x_{\mathrm{eq}}\) for every \(k\).

A uniformly asymptotically stable equilibrium (definition 335) is locally asymptotically stable (definition 332). Unlike lemma 289, no forward completeness is needed.

A uniformly stable equilibrium (definition 330) is stable (definition 328).

Fix \(\varepsilon {\gt} 0\) and \(k_0\). If for every \(\delta {\gt} 0\) some trajectory on \([k_0, \infty )\) starts within \(\delta \) of \(x_{\mathrm{eq}}\) and reaches distance at least \(\varepsilon \) from it, then \(x_{\mathrm{eq}}\) is unstable (definition 329).

A sequence is a trajectory of the unforced system (definition 112) on \(I\) exactly when it is a trajectory of \(s\) under the zero input on \(I\).

\(\Phi (t_0,t_0) = I\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5, Property P5.5.

For \(t \ge t_0\), \(\Phi (t+1, t_0) = A(t)\, \Phi (t, t_0)\). With lemma 205 this is the existence half of Hespanha’s P5.5: the discrete state transition matrix solves the matrix recursion.

The discrete variation-of-constants formula

takes the value \(x_0\) at \(t = t_0\): the forcing sum is over the empty range and \(\Phi (t_0,t_0) = I\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5, Section 5.3 (discrete variation of constants).

Let \(f\) be \(K\)-Lipschitz in the state variable, uniformly in time, and let \(\alpha , \beta \) be integral curves of \(f\) on \([t_0, t_1]\). Then

Uniqueness (lemma 45) is the case where the initial distance is zero; continuous dependence on the initial state is the statement read as a bound on the whole interval.

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 3.4, in the unperturbed case: two solutions of the same field, so the perturbation bound is zero.

Let \(g : X \to Y\) be continuous, \(Y\) Hausdorff, and \(g(\varphi (t)) \to L\) as \(t \to \infty \). Then \(g(y) = L\) for every \(y \in \omega (\varphi )\) (definition 59).

Let \(f\) be \(K\)-Lipschitz in the state variable, uniformly in time, and let \(\alpha , \beta \) be integral curves of \(f\) on \([t_0, t_1]\) with \(\alpha (t_0) = \beta (t_0)\). Then \(\alpha = \beta \) on \([t_0, t_1]\).

Reference: Khalil, Nonlinear Systems (3rd ed.), the uniqueness half of Theorem 3.2.

Let \(f\) be \(C^{1}\) with \(Df(x_{\mathrm{eq}}) = A\), let \(M\) be a matrix, and let \(c {\gt} 0\). Then there is \(r {\gt} 0\) such that for every \(x\) with \(\| x - x_{\mathrm{eq}}\| {\lt} r\),

where \(q_{M}\) is the centred quadratic form \(q_{M}(x) = \langle M(x - x_{\mathrm{eq}}),\, x - x_{\mathrm{eq}}\rangle \).

The bracket is the first-order remainder of \(f\) at \(x_{\mathrm{eq}}\). The point is that \(c\) is arbitrary: near enough to the equilibrium the remainder perturbs \(\dot{q}_{M}\) by less than any prescribed multiple of \(\| x - x_{\mathrm{eq}}\| ^{2}\), which is the order of \(\dot{q}_{M}\) itself. This is why the linearization decides stability.

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.7 (the quadratic-Lyapunov proof of Lyapunov’s indirect method).

A continuous curve on \([t_0,t_1]\), with \(t_0\le t_1\), starting in an open set \(D\) and ending outside \(D\) meets \(\partial D\) on that interval.

Let \(\alpha \) be a class \(\mathcal{K}\) function on \([0, a)\). Then there are \(b' {\gt} 0\), \(L {\gt} 0\) and a class \(\mathcal{K}\) function \(\beta : [0, a) \to [0, b')\) with \(\beta (x) \le \alpha (x)\) for \(x \in [0, a)\), \(\beta (x) \le L x\) for \(x \ge 0\), and \(\beta \) \(L\)-Lipschitz on all of \(\mathbb {R}\).

The range of \(\beta \) cannot in general be the range \([0, b)\) of \(\alpha \): \(\alpha (x) = 1 - \sqrt{1 - x}\) on \([0, 1)\) approaches \(1\) more slowly than any Lipschitz function can.

Let \(\omega : \mathbb {R} \to \mathbb {R}\) satisfy \(\omega (0) = 0\), be monotone on \([0, \infty )\), and be continuous at \(0\) from the right. Then there is a class \(\mathcal{K}_{\infty }\) function \(\alpha \) with \(\omega (r) \le \alpha (r)\) for all \(r \ge 0\).

Let \(\psi : \mathbb {R} \to \mathbb {R} \to \mathbb {R}\) satisfy, for all \(r, s \ge 0\):

1. \(\psi (r, s) \ge 0\);

2. \(\psi (0, s) = 0\);

3. \(r \mapsto \psi (r, s)\) is monotone on \([0,\infty )\);

4. \(s \mapsto \psi (r, s)\) is antitone on \([0,\infty )\);

5. \(\psi (r, s) \to 0\) as \(s \to \infty \);

6. \(r \mapsto \psi (r, 0)\) is continuous at \(0\) within \([0,\infty )\).

Then there is a global class \(\mathcal{KL}\) function \(\beta \) with \(\psi (r,s) \le \beta (r,s)\) for all \(r, s \ge 0\).

If a continuous curve on \([t_0,T]\) starts in an open set \(D\) and is outside \(D\) at \(t_1\in [t_0,T]\), some \(\tau \in [t_0,t_1]\) satisfies \(\varphi (\tau )\in \partial D\) and \(\varphi (t)\in D\) for \(t\in [t_0,\tau )\).

Let \(g\) be continuous on \([t_0, t_1]\) with \(g(t_0) \le 0 {\lt} g(t_1)\). Then \(g\) has a last root: there is \(a \in [t_0, t_1)\) with \(g(a) = 0\) and \(g(t) {\gt} 0\) for every \(t \in (a, t_1]\).

If every positive definite \(Q\) admits a positive definite solution of \(PM + M^{\mathsf T}P = -Q\) (definition 383), then some positive definite \(P\) makes \(-(PM + M^{\mathsf T}P)\) positive definite.

This is Hespanha’s Theorem 8.2, \((4) \Rightarrow (5)\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.2.

Let \(\mathbb {A}\) be a nontrivial complete normed algebra over \(\mathbb {C}\) and let \(a \in \mathbb {A}\) have spectral radius \(r(a) {\lt} 1\). Then there is an \(m {\gt} 0\) with \(\| a^m\| {\lt} 1\).

Reference: Rudin, Functional Analysis (Gelfand’s spectral-radius formula).

Let \(r {\gt} 0\) and let \(\psi : \mathbb {R} \to \mathbb {R}\) satisfy \(\psi (0) = 0\), \(\psi (s) {\gt} 0\) for \(s \in (0, r]\), and \(\psi \) nondecreasing on \([0, r]\). Then there are \(b {\gt} 0\) and a function \(f\) with \(f(0) = 0\), \(f(r) = b\), \(f\) continuous and strictly increasing on \([0, r]\), and \(f(s) \le \psi (s)\) for all \(s \in [0, r]\).

That is: a positive nondecreasing function admits a class \(\mathcal{K}\) minorant on any bounded interval. This is what turns a positive-definite \(V\) into the lower comparison bound of theorem 270.

Let \(r {\gt} 0\) and let \(\varphi \) satisfy \(\varphi (0) = 0\), be monotone on \([0, r]\), and be continuous at \(0\) from the right. Then there are \(b {\gt} 0\) and an \(f\) with \(f(0) = 0\), \(f(r) = b\), \(f\) continuous and strictly increasing on \([0, r]\), and \(\varphi (s) \le f(s)\) for all \(s \in [0, r]\).

Let \(\varphi : \mathbb {R} \to \mathbb {R}\) satisfy \(\varphi (0) = 0\), be monotone on \([0, \infty )\), and be continuous at \(0\) from the right. Then there is an \(f\) with \(f(0) = 0\), continuous and strictly increasing on \([0, \infty )\), \(f(s) \to \infty \) as \(s \to \infty \), and \(\varphi (s) \le f(s)\) for all \(s \ge 0\).

That is: a nondecreasing function admits a class \(\mathcal{K}_{\infty }\) majorant. The unbounded domain is what distinguishes this from lemma 269, which it implies.

Continuity at \(0\) is necessary: the unit step has no continuous majorant vanishing at \(0\).

Let \(M\) be a square matrix and let \((\mu , v)\) be an eigenpair of \(M\), so that \(Mv = \mu v\). Then

Reference: standard power-series functional calculus for the exponential.

For \(P\) solving \(PA + A^{\mathsf T}P = -I\) (definition 383), the derivative of the centered quadratic form (definition 384) along \(y \mapsto Ay\) is \(-\| x - x_{\mathrm{eq}}\| ^{2}\).

(Hespanha, equation (5.8)). With \(x\) as in theorem 169, the output \(y(t) = C(t)x(t) + D(t)u(t)\) splits into the homogeneous response and the forced response:

The forced response (definition 172) takes the value \(x_0\) at \(t = t_0\): the forcing integral is over the degenerate interval and \(\Phi (t_0,t_0) = I\). Together with theorem 169 this makes it a solution of the initial value problem, not merely of the differential equation.

For constant \(A\) and \(B\) the forced response (definition 172) is

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 6.

With \(u = 0\) the forced response (definition 172) is the homogeneous response (definition 173): \(\Phi (t,t_0)x_0 + \int _{t_0}^{t}\Phi (t,\tau )B(\tau )\cdot 0\, \mathrm{d}\tau = \Phi (t,t_0)x_0\).

Khalil, Lemma A.1 (the case \(\Lambda \) and \(\mu \ge 0\) both constant). Let \(\mu \ge 0\) be constant and let \(y\) be continuous with

Then \(y(t) \le C\, e^{\mu (t-a)}\) on \([a,b]\). This is the form the continuous-dependence estimates use, and the reason their bounds carry a factor \(e^{L(t-t_0)}\).

Khalil, Lemma A.1 (the case \(\Lambda (t) \equiv \Lambda \) constant). Let \(\mu \) be continuous and nonnegative on \([a,b]\), and let \(y\) be continuous with

Then \(y(t) \le C\, e^{\int _{a}^{t}\mu (\tau )\, \mathrm{d}\tau }\) on \([a,b]\).

For \(\mu \) continuous on \([a,b]\) and \(t \in (a,b)\),

If \(\alpha \) is an integral curve of \(f\) on \([t_0, t_1]\), then for every \(s \in [t_0, t_1)\) it has right derivative \(f(s, \alpha (s))\) at \(s\), i.e. \(\alpha \) has derivative \(f(s,\alpha (s))\) within \([s,\infty )\) at \(s\).

If \(f\) has right derivative \(L\) at \(t\), then \(\frac{f(t+h)-f(t)}{h} \to L\) as \(h \to 0^{+}\).

If \(x \ne 0\) lies in the maximal generalized eigenspace of \(T\) at \(\xi \), then \(\xi \) is an eigenvalue of \(T\).

For every \(\mu \in \mathbb {C}\) and \(v \in \mathbb {C}^{n}\),

The homogeneous response (definition 173) takes the value \(x_{0}\) at \(t = t_{0}\), since \(\Phi (t_{0},t_{0}) = I\).

\(\int _a^b C \, \mathrm{d}s = (b-a)\, C\) for a real constant \(C\).

For \(\mu \) continuous on \([a, t]\) and \(s \in [a, t]\),

If \(f\) is jointly continuous and \(z\) is continuous on the segment between \(t_0\) and \(t_1\), then \(s \mapsto f(s, z(s))\) is interval-integrable between them. This is the integrability side condition every integral-form solution argument needs.

Under the conditions of lemma 241 and with \(W\) strictly antitone, \(\mathrm{invFunOn}\, W\, (0,\infty )\) is continuous on \((0,\infty )\).

Under the surjectivity conditions of lemma 241, \(\mathrm{invFunOn}\, W\, (0,\infty )\, s {\gt} 0\) for all \(s {\gt} 0\).

Under the conditions of lemma 241 and with \(W\) strictly antitone, \(\mathrm{invFunOn}\, W\, (0,\infty )\, s \to +\infty \) as \(s \to 0^{+}\).

Under the conditions of lemma 241 and with \(W\) strictly antitone, \(\mathrm{invFunOn}\, W\, (0,\infty )\, s \to 0\) as \(s \to +\infty \).

Let \(E\) be a proper normed space. If \(V : E \to \mathbb {R}\) is continuous and radially unbounded (\(V(x) \to \infty \) as \(\| x\| \to \infty \)), then every sublevel set \(\{ x \mid V(x) \le c\} \) is compact.

A real square matrix \(A\) is Hurwitz (definition 364) if and only if it is Hurwitz with decay rate \(0\) (definition 365).

If \(A\) is Hurwitz with decay rate \(\alpha \) (definition 365) and \(\beta \le \alpha \), then \(A\) is Hurwitz with decay rate \(\beta \).

For \(\alpha {\lt} \gamma \), the \(1 \times 1\) matrix \((-\gamma )\) is Hurwitz with decay rate \(\alpha \) (definition 365).

Included as a concrete check on the sign convention of the definition.

If the index type is empty then every \(A\) is Hurwitz with every decay rate \(\alpha \) (definition 365).

This records the convention chosen for the degenerate case: the predicate quantifies over nonzero eigenvectors, and in dimension zero there are none.

Let \(V\) be differentiable and \(\gamma \) an integral curve of \(v\) on an interval \(s\) with \(L_v V(t, \gamma (t)) \le 0\) at every interior point \(t\) of \(s\). Then \(t \mapsto V(t, \gamma (t))\) is nonincreasing on \(s\).

If \(x\) is an integral curve of an autonomous field \(g\) on a set \(s\), then \(t \mapsto x(t + \Delta t)\) is an integral curve of the same \(g\) on \(s - \Delta t\), for every \(\Delta t\).

An autonomous field has no preferred time origin, so translating a solution translates its interval of definition and changes nothing else. For a time-varying field the translated solution would solve the translated equation instead.

Let \(V\) be differentiable and \(\gamma \) an integral curve of \(v\) on \(I\). Then for \(t \in I\), within \(I\),

(definition 52).

Let \(x\) be an integral curve of \(F\) on the segment between \(t_0\) and \(t_1\), with \(s \mapsto F(s, x(s))\) continuous there. Then \(x\) is an integral solution on that segment, anchored at its own value \(x(t_0)\):

Continuity of \(x\) itself is not assumed — it follows from differentiability.

Let \(g\) be a continuous autonomous field and let \(x\) be an integral curve of \(g\) on \([t_0,t_1]\), \(t_0 \le t_1\). Then \(x\) is an integral solution there, anchored at \(x(t_0)\).

This is the autonomous, forward-time specialization of lemma 37, and so the shape a trajectory of an autonomous system has: continuity of the field is the only hypothesis, the continuity of \(x\) along which \(F\) must be continuous being supplied by the curve itself.

Let \(V\) be differentiable, \(\gamma \) an integral curve of \(v\) on \([t_0, t_1]\), and \(c \in \mathbb {R}\) with \(L_v V(t, \gamma (t)) \le -c\, V(t, \gamma (t))\) for \(t \in (t_0, t_1)\). Then \(V(t, \gamma (t)) \le e^{-c(t - t_0)}\, V(t_0, \gamma (t_0))\) on \([t_0, t_1]\).

lemma 39 in the form the finite-segment predicates need: a solution segment of an autonomous field on \([t_0, t_1]\) becomes one on \([0, t_1 - t_0]\) under \(s \mapsto x(s + t_0)\).

This is why the autonomous stability predicates may fix the initial time at \(0\) without loss: any segment can be shifted there.

Given continuity of \(s \mapsto F(s, x(s))\) on the segment between \(t_0\) and \(t_1\), the two formulations agree: \(x\) is an integral solution anchored at \(x(t_0)\) if and only if it is an integral curve of \(F\) there. Continuity of \(F\) along \(x\) is exactly the price of the equivalence — without it the integral form is the weaker notion.

lemma 36 in forward time: for \(t_0 \le t_1\), the equivalence holds over \([t_0, t_1]\). This is the interval the finite-forward stability predicates are phrased over, so this is the form the stability track would use to reach the results of this chapter.

Let \(x\) be an integral solution of \(\dot x = F(t,x)\) on the segment between \(t_0\) and \(t_1\), with \(s \mapsto F(s, x(s))\) continuous there. Then \(x\) is an integral curve of \(F\) on that segment: it is differentiable, with \(\dot x(t) = F(t, x(t))\).

Let \(u\) be continuous on \([t_0, t_1]\) with \(u(t_0) = u_0\) and \(\dot u(t) = f(t, u(t))\) at every interior point, and let \(f\) be jointly continuous. Then \(u\) is an integral solution:

.

Let \(x\) be a continuous integral solution of \(\dot x = F(t,x)\) with initial value \(x_0\) on the segment between \(t_0\) and \(t_1\), with \(F\) jointly continuous. Then for any \(s, r\) on that segment, \(x\) is an integral solution on the segment between \(s\) and \(r\), with initial value \(x(s)\):

Leaving both endpoints free means the domain need not shrink — taking \(s = t_1\) and \(r = t_0\) re-anchors at one end of the original segment while retaining all of it.

Let \(x\) be an integral solution of \(\dot x = F(t,x)\) with initial value \(x_0\) on the segment between \(t_0\) and \(t_1\). Then \(\sigma \mapsto x(t_0 + t_1 - \sigma )\) is an integral solution, with the same initial value \(x_0\), of the reflected field

on the segment between \(t_1\) and \(t_0\).

The map \(\sigma \mapsto t_0 + t_1 - \sigma \) is an involution exchanging the two endpoints, so reflection turns a statement anchored at \(t_0\) into one anchored at \(t_1\) and nothing more. Its purpose is to let an argument that is inherently forward-marching — a Grönwall bootstrap, say — be applied unchanged in backward time.

For a constant matrix \(C\), \(Y(t) := \Phi (t,t_0)\, C\) satisfies the integral equation

for every \(t\) between \(t_0\) and \(t_1\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

\(x(t) := \Phi (t,t_0)\, x_0\) satisfies the integral equation

for every \(t\) between \(t_0\) and \(t_1\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

If \(A\) is Schur (definition 414) then so is \(A^{\mathsf T}\).

Under the same boundedness conditions as lemma 9, \(L \le D^{+}f(t)\) if and only if for every \(\varepsilon {\gt} 0\) the quotient \(\frac{f(t+h)-f(t)}{h}\) exceeds \(L - \varepsilon \) frequently as \(h \to 0^{+}\).

Let \(a {\lt} b\), let \(z(a) = v(a)\) and \(z(t) {\lt} v(t)\) for all \(t \in (a, b]\), let \(z\) have right derivative \(d_z\) at \(a\), and let the forward difference quotients of \(v\) at \(a\) be bounded above. Then \(d_z \le D^{+}v(a)\).

If \(V\) does not depend on \(t\) and is differentiable at \(x\), then the Lie derivative of \((t, x) \mapsto V(x) + ct\) along \(v\) is \(DV(x)\, v(t, x) + c\).

If \(V\) does not depend on \(t\) and is differentiable at \(x\), then \(L_v V(t, x) = DV(x)\, v(t, x)\).

Let \(l\) be a nontrivial filter, let \(f\) be bounded above and below along \(l\), and let \(g \to 0\) along \(l\). Then \(\limsup _{l}(f + g) = \limsup _{l} f\).

If a real matrix \(P\) is positive definite, so is its complexification \(P_{\mathbb C}\), as a Hermitian matrix over \(\mathbb C\).

\(y \in \omega (\varphi )\) (definition 59) if and only if for every neighbourhood \(U\) of \(y\) there are arbitrarily large \(t\) with \(\varphi (t) \in U\).

Let \(A\in \mathbb {F}^{n\times n}\) and \(B\in \mathbb {F}^{n\times m}\), and let \(\mathcal C(A,B)\in \mathbb {F}^{n\times (nm)}\) be their controllability matrix. For \(x\in \mathbb {F}^n\),

Here \(\mathcal R(A,B)=\operatorname {range}\mathcal C(A,B)\) is the reachable subspace.

For a square matrix \(M\) over a field \(K\), \(\mu \in \sigma (M)\) if and only if \(Mv = \mu v\) for some \(v \ne 0\).

Let \(W : \mathbb {R} \to \mathbb {R}\) be continuous on \((0,\infty )\) with \(W(\eta ) \to +\infty \) as \(\eta \to 0^+\) and \(W(\eta ) \to 0\) as \(\eta \to +\infty \). Then for every \(s {\gt} 0\) there exists \(c {\gt} 0\) with \(W(c) = s\).

For a square matrix \(A\) over a commutative ring and scalars \(\alpha , \mu \), a vector \(v\) satisfies \((A + \alpha I)v = (\mu + \alpha )v\) if and only if \(Av = \mu v\).

\((\mu I - A)v = 0\) if and only if \(Av = \mu v\).

If the forward difference quotients of \(f\) at \(t\) are bounded above and below, then \(-D^{+}(-f)(t) \le D^{+}f(t)\). The left-hand side is the lower right Dini derivative, so this is the expected inequality \(D_{+}f(t) \le D^{+}f(t)\) written in terms of \(D^{+}\) alone.

If \(\| u(s)\| \le C\) for all \(s \in [a,b]\) with \(a \le b\), then

Let \(u\) take values in a normed space and \(v\) be real, both integrable on \([a,b]\) with \(a \le b\). If \(\| u(s)\| \le L\, v(s)\) for all \(s \in [a,b]\), then

If \(\| A(s)\| \le M\) for all \(s\) between \(t_0\) and \(t_1\), then

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

The bound in lemma 150, uniformized to the worst case \(t = t_1\):

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

Block-row entry shape: at row \((k, i)\) and column \(j\), the observability matrix coincides with the \((i, j)\) entry of \(C\, A^{k}\):

For every state \(x \in \mathbb {F}^{n}\) and every \((k, i) \in \mathrm{Fin}\, n \times \mathrm{Fin}\, p\),

This is the definitional bridge that powers theorem 444.

definition 59 is the \(\omega \)-limit, in the sense of topological dynamics, of the one-member family \(\{ \varphi \} \).

If \(\Omega \) is closed and \(\varphi (t) \in \Omega \) for all large \(t\), then \(\omega (\varphi ) \subseteq \Omega \).

For a constant state matrix \(A\), the \(k\)-th Peano–Baker term collapses to

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 6, equation (6.1).

If \((M_i)\) is a summable family of real positive semidefinite matrices, then \(\sum _i M_i\) is positive semidefinite.

If \(Av = \mu v\) then \(A^{k}v = \mu ^{k}v\) for every \(k\).

The reachable subspace is \(A\)-invariant: \(A\mathcal R(A,B)\subseteq \mathcal R(A,B)\).

Reference: Hespanha, Linear Systems Theory.

If \(Av = \mu v\) then \((cA)v = (c\mu )v\).

If every eigenpair \((\mu , v)\) of the complexification of a real matrix \(A\), with \(v \ne 0\), has \(\operatorname {Re}(\mu ) {\lt} 0\), then \(r(e^{A_{\mathbb C}}) {\lt} 1\).

Reference: standard spectral mapping for the matrix exponential.

For every \(\tau \) on the segment between \(t_0\) and \(t\), \(\Phi (t, t_0)\, \Phi (t_0, \tau ) = \Phi (t, \tau )\): composition through a fixed base point. This differs from theorem 166 in which of the three times is required to lie between the others — here the two fixed integration endpoints are the outer ones and \(\tau \) is free, which is the shape variation of constants needs.

For every \(z\) on the segment between \(a\) and \(b\), \(\Phi (z, a) = \Phi (z, b)\, \Phi (b, a)\). Composition with the first argument free.

For a constant state matrix \(A\), the state transition matrix is the power series

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 6, equation (6.1).

If \(W\) is strictly antitone on \((0,\infty )\), then \(\mathrm{invFunOn}\, W\, (0,\infty )\) is strictly antitone on \((0,\infty )\).

The Peano–Baker series \(\sum _{k=0}^\infty P_k(t,t_0)\) converges absolutely, for \(A\) bounded between \(t_0\) and \(t\) (in either order).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

For \(\gamma {\gt} 0\) and any constants \(K, t_{0}\),

Let \(f\) be nonneg, antitone and locally integrable on \((0,\infty )\) with \(f(s) \to 0\) as \(s \to +\infty \), and let \(a(\eta ) {\lt} b(\eta )\) with \(a(\eta ) \to +\infty \) as \(\eta \to +\infty \). Then

The mean is squeezed between \(0\) and \(f(a(\eta ))\); no hypothesis relating \(a\) to \(b\) is needed.

If \(\varphi (t)\) lies in a compact set for all large \(t\) and \(\omega (\varphi ) \subseteq M\), then \(\varphi (t) \to M\): for every open \(W \supseteq M\), \(\varphi (t) \in W\) for all large \(t\).

Let \(E\) be a normed space, \(f : \mathbb {R} \to E\), and \(\gamma {\gt} 0\). If eventually

then \(f(t) \to 0\) as \(t \to \infty \).

Stated over an arbitrary normed space because it is applied to both matrix- and vector-valued quantities.

Let \(c \ge 0\) and let \(U : \mathbb {R} \to \mathbb {R}\) satisfy \(U(s) \to 0\) as \(s \to +\infty \). Then \(\min \bigl(c, \sqrt{c\, U(s)}\bigr) \to 0\) as \(s \to +\infty \).

For square matrices \(H, A\) over a commutative semiring, \((A^{\mathsf T}H)(x,y) = H(Ax, y)\) as bilinear forms.

Together with lemma 402 this is what turns the matrix equation \(HA + A^{\mathsf T}H = cH\) into the Sylvester-type identity on bilinear forms that lemma 401 consumes.

For square matrices \(H, A\) over a commutative semiring, \((HA)(x,y) = H(x, Ay)\) as bilinear forms.

If \(H\) is a symmetric real matrix then \(H(x,y) = H(y,x)\) as a bilinear form.

The unobservable subspace is closed under the action of \(A\): for every \(v \in \mathcal{N}(A, C)\), also \(A\, v \in \mathcal{N}(A, C)\).

Let \(\alpha \) be class \(\mathcal{K}\) on \([0,a)\) and at most linear near the origin, \(\alpha (x) \le Lx\) on \((0, \mathrm{base}]\). Then there is a class \(\mathcal{KL}\) function \(\sigma \) (definition 250) with \(\sigma (y_0, 0) = y_0\) such that for every \(y_0 \in [0, a)\) and every \(t_0\), \(t \mapsto \sigma (y_0, t - t_0)\) is a trajectory on \([t_0, \infty )\) (definition 83) of the decay system \(\dot y = -\alpha (y)\) (definition 271).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.4. Differences:

• Khalil assumes \(\alpha \) locally Lipschitz; here the hypothesis is the linear bound \(\alpha (x) \le Lx\) near the origin, which is what the construction uses.

• Uniqueness of the solution is not asserted.

Let \(\alpha \) be class \(\mathcal{K}\) on \([0,a)\). Then there is a class \(\mathcal{KL}\) function \(\sigma \) with \(\sigma (r,0) \le r\) such that: whenever \(v\) is continuous on \([t_0,t]\), takes values in \([0,a)\) there, has bounded forward difference quotients, and is a Dini subsolution of the decay system (definition 271),

it obeys \(v(t) \le \sigma \bigl(v(t_0),\, t - t_0\bigr)\).

A differential inequality with a class \(\mathcal{K}\) decay rate therefore yields a class \(\mathcal{KL}\) bound — which is what converts a Lyapunov decay estimate into an asymptotic stability statement. No hypothesis is placed on \(\alpha \) beyond being class \(\mathcal{K}\): the linear-growth condition the construction needs is obtained internally by passing to a Lipschitz minorant.

Khalil, Lemma 3.4. Let \(f\) be jointly continuous and \(L\)-Lipschitz in its state argument on \([t_0, t_1]\), and suppose

1. \(u\) is continuous on \([t_0,t_1]\) and solves \(\dot u = f(t,u)\) with \(u(t_0) = u_0\);

2. \(v\) is continuous on \([t_0,t_1]\) with \(v(t_0) \le u_0\), satisfies the Dini subsolution inequality \(D^{+}v(t) \le f(t, v(t))\), and has forward difference quotients bounded above at each \(t\);

3. for every \(\lambda {\gt} 0\) the perturbed equation \(\dot z = f(t,z) + \lambda \), \(z(t_0) = u_0\), has a solution on \([t_0,t_1]\).

Then \(v(t) \le u(t)\) for all \(t \in [t_0, t_1]\).

The perturbed solutions in (3) are taken as a hypothesis for generality, so that callers with solutions already in hand need not re-derive them. Under the hypotheses already listed they always exist: \(f + \lambda \) is Lipschitz exactly when \(f\) is, so theorem 47 discharges (3).

Khalil, Theorem 3.4. Let \(f\) be jointly continuous and \(L\)-Lipschitz in its state argument on the segment between \(t_0\) and \(t_1\), let \(y\) be an integral solution of \(\dot y = f(t,y)\) with \(y(t_0) = y_0\), and let \(z\) be an integral solution of the perturbed equation \(\dot z = f(t,z) + g(t,z)\) with \(z(t_0) = z_0\), where \(\| g(t,x)\| \le \mu \). Then for every \(t\) on that segment,

Two solutions may separate, but no faster than exponentially, at a rate set by the Lipschitz constant alone; the two error sources — a different starting point and a perturbed field — contribute additively. The bound is stated on the unordered segment, so it runs in either time direction.

The perturbation \(g\) is assumed only bounded and integrable along \(z\), never continuous, so \(z\) need not be differentiable anywhere. That is why the theorem is phrased over the integral formulation rather than the differential one.

Khalil, Theorem 3.5. If \(y\) solves \(\dot{y} = f(t,y)\) and \(z\) solves \(\dot{z} = f(t,z) + g(t,z)\) with \(\| g(t,x)\| \le \alpha \) and \(\| z_0 - y_0\| \le \alpha \), and \(\alpha (1 + 1/L)e^{L(t_1-t_0)} \le \varepsilon \), then \(\| y(t) - z(t)\| \le \varepsilon \) for all \(t \in [t_0, t_1]\).

Consider the time-invariant system \(\dot x = Ax\) with \(A\) a real square matrix.

The origin is an asymptotically stable equilibrium (definition 279) if and only if \(A\) is Hurwitz (definition 364), that is, every eigenvalue of \(A\) has strictly negative real part.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.1(2).

Consider the time-invariant system \(\dot x = Ax\) with \(A\) a real square matrix.

The origin is an exponentially stable equilibrium (definition 284) if and only if \(A\) is Hurwitz (definition 364).

Together with theorem 381 this says the two notions coincide for a time-invariant system — a genuine feature of constant \(A\), not of linearity: for time-varying \(A(t)\) they differ, as \(\dot x = -x/(1+t)\) shows.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.1(3).

Consider the linear time-varying system

with \(A\) continuous, and write \(\Phi (t,t_{0})\) for its state transition matrix (definition 148).

The origin is a globally exponentially stable equilibrium (definition 285) of this system if and only if there are \(k, \gamma {\gt} 0\) with

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.1(3).

If \(A\) is continuous, the unforced system (definition 74) of definition 115 is forward complete (definition 77).

Consider the linear time-varying system

with \(A\) continuous, and write \(\Phi (t,t_{0})\) for its state transition matrix (definition 148).

The origin is an asymptotically stable equilibrium (definition 279) of this system if and only if, for every \(t_{0} \ge 0\) and every \(x_{0}\), the homogeneous response \(t \mapsto \Phi (t, t_{0}) x_{0}\) is bounded on \([t_{0}, \infty )\) and tends to \(0\) as \(t \to \infty \).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.1(2).

Consider the linear time-varying system

with \(A\) continuous, and write \(\Phi (t,t_{0})\) for its state transition matrix (definition 148).

The origin is an exponentially stable equilibrium (definition 284) of this system if and only if there are \(k, \gamma {\gt} 0\) with

The condition is the one in theorem 360: the radius in the local definition carries no information here.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.1(3).

Consider the linear time-varying system

with \(A\) continuous, and write \(\Phi (t,t_{0})\) for its state transition matrix (definition 148).

The origin is a stable equilibrium (definition 275) of this system if and only if, for every \(t_{0} \ge 0\) and every initial condition \(x_{0}\), the homogeneous state response \(x(t) = \Phi (t, t_{0})\, x_{0}\) is bounded on \([t_{0}, \infty )\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.1(1).

Consider the linear time-varying system

with \(A\) continuous, and write \(\Phi (t,t_{0})\) for its state transition matrix (definition 148).

The origin is a uniformly stable equilibrium (definition 277) of this system if and only if there is a single \(C\) with

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.1(1), with the margin independent of \(t_{0}\).

Consider the linear time-varying system

with \(A\) continuous, and write \(\Phi (t,t_{0})\) for its state transition matrix (definition 148).

The origin is unstable (definition 276) for this system if and only if there are \(t_{0} \ge 0\) and \(x_{0}\) whose homogeneous response \(t \mapsto \Phi (t, t_{0}) x_{0}\) is unbounded on \([t_{0}, \infty )\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.1(4).

For an input-free system \(\dot x=f(x)\) with \(f\) globally \(C^1\), a Chetaev certificate at \(x_{\rm eq}\) (definition 325) implies instability (definition 276).

Reference: Khalil, Nonlinear Systems, 3rd ed., Theorem 4.3, p. 125.

Let \(\dot x = f(x)\) on \(\mathbb {R}^n\) with \(f\) Lipschitz, and \(V \in C^1\) positive definite on \(\mathbb {R}^n\), with compact sublevel sets (equivalently, radially unbounded) and \(\nabla V \cdot f \le 0\) everywhere. Suppose that for every \(c\), \(\{ x_{\mathrm{eq}}\} \) contains every forward invariant subset of \(\{ x : V(x) \le c,\ \nabla V(x) \cdot f(x) = 0\} \). Then \(x_{\mathrm{eq}}\) is globally asymptotically stable (definition 280).

Unlike Barbashin’s theorem (theorem 323) no forward invariant set is assumed: every sublevel set is compact, and \(\nabla V \cdot f \le 0\) keeps a trajectory in the one through its initial state.

Reference: Khalil, Nonlinear Systems (3rd ed.), Corollary 4.2.

Let the state equation be \(\dot x = f(x)\) with \(f\) continuous, on \(\mathbb {R}^n\). Let \(V : \mathbb {R}^n \to \mathbb {R}\) be \(C^1\), positive definite about \(x_{\mathrm{eq}}\) on all of \(\mathbb {R}^n\) (definition 27), with compact sublevel sets \(\{ V \le c\} \) (equivalently, radially unbounded) and \(\nabla V(x) \cdot f(x) {\lt} 0\) for \(x \ne x_{\mathrm{eq}}\). Then \(x_{\mathrm{eq}}\) is globally uniformly asymptotically stable (definition 283).

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.2.

If \(x \mapsto F(t, x, u(t))\) is \(K\)-Lipschitz for every \(t\), the system has unique solutions under \(u\) (definition 78).

Let \(t_0 \le t_1\) and let \(\tau \mapsto F(\tau , x(\tau ), u(\tau ))\) be continuous on \([t_0, t_1]\). Then \(x\) is a Carathéodory trajectory on \([t_0, t_1]\) (definition 89) if and only if it is a trajectory there (definition 83).

Let \(I \subseteq \mathbb {R}\) be an interval (order-connected) and let \(\tau \mapsto F(\tau , x(\tau ), u(\tau ))\) be continuous on \(I\). Then \(x\) is a Carathéodory trajectory on \(I\) (definition 89) if and only if it is a trajectory there (definition 83).

Order-connectedness cannot be dropped: on \(\{ 0\} \cup [1, 2]\) the integral form relates \(x(1)\) to \(x(0)\), the derivative form does not.

If \(V\) is an exponential Lyapunov function for \(x_{\mathrm{eq}}\) on the whole space (definition 310 with \(D = E\)), then \(x_{\mathrm{eq}}\) is globally exponentially stable (definition 285).

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.10, global case.

If \(V\) is an exponential Lyapunov function for \(x_{\mathrm{eq}}\) (definition 310), then \(x_{\mathrm{eq}}\) is locally exponentially stable (definition 284): near \(x_{\mathrm{eq}}\),

The state space is any normed space.

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.10.

If \((t, x) \mapsto F(t, x, u(t))\) is continuous and \(K\)-Lipschitz in \(x\) uniformly in \(t\), the system is forward complete under \(u\) (definition 77).

If \(V\) is a strict uniform Lyapunov function for \(x_{\mathrm{eq}}\) (definition 309) on a system with state space \(\mathbb {R}^n\), then \(x_{\mathrm{eq}}\) is uniformly asymptotically stable (definition 282).

A strictly negative \(\dot V\) alone would not do: \(W_3\) must be bounded away from zero on each annulus, which is exactly what positive definiteness of a function of \(x\) alone buys.

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.9. Khalil’s two “moreover” clauses are not part of this conclusion: the explicit class \(\mathcal{KL}\) bound, which the proof constructs, and the global case.

If \(V\) is a uniform Lyapunov function for \(x_{\mathrm{eq}}\) (definition 308) on a system with state space \(\mathbb {R}^n\), then \(x_{\mathrm{eq}}\) is uniformly stable (definition 277).

The sandwich \(W_1(x) \le V(t, x) \le W_2(x)\) is what makes the conclusion uniform in \(t_0\): \(W_1\) and \(W_2\) do not depend on \(t\), so neither does the \(\delta (\varepsilon )\) extracted from them.

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.8.

Let \(\dot x = f(x)\) with \(f\) Lipschitz, \(\Omega \) compact and forward invariant (definition 87), and \(V \in C^1\) with \(\nabla V \cdot f \le 0\) on \(\Omega \). Put

and let \(M\) contain every forward invariant subset of \(E\). Then every trajectory \(\varphi \) on \([t_0, \infty )\) with \(\varphi (t_0) \in \Omega \) satisfies \(\varphi (t) \to M\) as \(t \to \infty \).

Khalil takes \(M\) to be the largest invariant subset of \(E\). Here \(M\) must absorb every forward invariant subset, a larger family, so the hypothesis on \(M\) is correspondingly stronger: backward invariance of \(\omega (\varphi )\) would need solutions through its points to extend backwards in time.

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.4.

Let \(\dot x = f(x)\) on \(\mathbb {R}^n\) with \(f\) Lipschitz, and \(V \in C^1\) positive definite on a neighbourhood \(D\) of \(x_{\mathrm{eq}}\) (definition 27) with \(\nabla V \cdot f \le 0\) on \(D\). Suppose that for some \(c {\gt} 0\) the sublevel set \(\Omega _c = \{ V \le c\} \) is contained in \(D\), compact and forward invariant, and that \(\{ x_{\mathrm{eq}}\} \) contains every forward invariant subset of \(\{ x \in \Omega _c : \nabla V(x) \cdot f(x) = 0\} \). Then \(x_{\mathrm{eq}}\) is locally asymptotically stable (definition 279).

The hypothesis on \(E\) is the one a user of LaSalle discharges: solve \(\nabla V \cdot f = 0\) and check that no solution can stay there but the equilibrium.

Reference: Khalil, Nonlinear Systems (3rd ed.), Corollary 4.1.

For the time-invariant system \(x^{+} = Ax\), the origin is locally asymptotically stable (definition 332) if and only if \(A\) is Schur (definition 414): every eigenvalue of \(A\) has magnitude strictly smaller than \(1\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.3(2).

For the time-invariant system \(x^{+} = Ax\), the origin is locally exponentially stable (definition 337) if and only if \(A\) is Schur (definition 414).

Together with theorem 421 this says the two notions coincide for a time-invariant system.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.3(2).

For the time-invariant system \(x^{+} = Ax\) the following are equivalent.

1. The origin is asymptotically stable (definition 332).

2. The origin is exponentially stable (definition 337).

3. \(A\) is Schur (definition 414).

4. For every positive definite \(Q\) there is a positive definite \(P\) with \(A^{\mathsf T}PA - P = -Q\) (definition 423), and this \(P\) is the unique matrix solution.

5. There is a positive definite \(P\) with \(-(A^{\mathsf T}PA - P)\) positive definite.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.4.

For the discrete-time linear system \(x(t+1) = A(t)x(t)\), the origin is globally exponentially stable (definition 338) if and only if there are \(c {\gt} 0\) and \(0 \le \rho {\lt} 1\) with

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.2(3).

For the discrete-time linear system \(x(t+1) = A(t)x(t)\), the origin is locally asymptotically stable (definition 332) if and only if, for every \(t_0\) and every \(x_0\), the homogeneous response \(t \mapsto \Phi (t, t_0)x_0\) is bounded on \(t \ge t_0\) and tends to \(0\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.2(2).

For the discrete-time linear system \(x(t+1) = A(t)x(t)\), the origin is locally exponentially stable (definition 337) if and only if there are \(c {\gt} 0\) and \(0 \le \rho {\lt} 1\) with \(\| \Phi (t, t_0)\, x\| \le c\, \rho ^{t - t_0}\, \| x\| \) for all \(t \ge t_0\) and all \(x\): the condition of theorem 410.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.2(3).

For the discrete-time linear system \(x(t+1) = A(t)x(t)\), the origin is stable (definition 328) if and only if, for every \(t_0\) and every initial condition \(x_0\), the homogeneous response \(x(t) = \Phi (t, t_0)\, x_0\) (definition 215) is bounded on \(t \ge t_0\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.2(1).

For the discrete-time linear system \(x(t+1) = A(t)x(t)\), the origin is uniformly stable (definition 330) if and only if there is a single \(C\) with \(\| \Phi (t, t_0)\, x\| \le C\, \| x\| \) for all \(t \ge t_0\) and all \(x\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.2(1), with the margin independent of \(t_0\).

For the discrete-time linear system \(x(t+1) = A(t)x(t)\), the origin is unstable (definition 329) if and only if there are \(t_0\) and \(x_0\) whose homogeneous response \(t \mapsto \Phi (t, t_0)x_0\) is unbounded on \(t \ge t_0\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Definition 8.2(4).

Let \(E\) be proper, and let the system be \(x^{+} = f(x)\) with \(f\) continuous and \(f(x_{\mathrm{eq}}) = x_{\mathrm{eq}}\). Let \(V\) be positive definite on \(E\) about \(x_{\mathrm{eq}}\) (definition 27), with compact sublevel sets \(\{ V \le c\} \) (equivalently, for continuous \(V\), radially unbounded: lemma 316), and

Then \(x_{\mathrm{eq}}\) is globally asymptotically stable (definition 333).

Continuity of \(f\) is not among the hypotheses of the reference but cannot be dropped: with \(V(x) = |x|\) on \(\mathbb {R}\), the map \(f(x) = x/2\) for \(|x| \le 1\), \(f(x) = \operatorname {sign}(x)(1 + |x|)/2\) otherwise, has \(\Delta V {\lt} 0\) away from \(0\), yet the trajectory from \(2\) converges to \(1\).

Reference: Jungers and van de Wouw (2026), Theorem 2, row 3.

Let the system be \(x^{+} = f(x)\), let \(0 {\lt} c_1 \le c_2\), \(p {\gt} 0\) and \(0 {\lt} c_3 {\lt} 1\), and let \(V : E \to \mathbb {R}\) satisfy, for every \(x\),

Then \(x_{\mathrm{eq}}\) is globally exponentially stable (definition 338), with \(C = (c_2/c_1)^{1/p}\) and \(\rho = (1 - c_3)^{1/p}\). No continuity of \(f\) is needed.

Reference: Jungers and van de Wouw (2026), Theorem 2, row 4, on the whole space. They ask for \(p \ge 1\); \(p {\gt} 0\) suffices.

Let \(E\) be proper, and let the system be \(x^{+} = f(x)\) with \(f(x_{\mathrm{eq}}) = x_{\mathrm{eq}}\). Let \(\mathcal{D}\) be a neighbourhood of \(x_{\mathrm{eq}}\) on which \(f\) is continuous, and \(V\) positive definite on \(\mathcal{D}\) about \(x_{\mathrm{eq}}\) (definition 27), with

Then \(x_{\mathrm{eq}}\) is locally asymptotically stable (definition 332).

Continuity of \(f\) on \(\mathcal{D}\) is not among the hypotheses of the reference, but it cannot be dropped. With \(V(x) = |x|\) on \(\mathbb {R}\), let \(f\) be odd with \(f(0) = 0\) and \(f(x) = (2^{-n-1} + x)/2\) on each interval \((2^{-n-1}, 2^{-n}]\): then \(\Delta V {\lt} 0\) away from \(0\), but a trajectory starting in \((2^{-n-1}, 2^{-n}]\) converges to \(2^{-n-1}\), at every scale.

Reference: Jungers and van de Wouw (2026), Theorem 2, row 2.

Let the system be \(x^{+} = f(x)\) with \(f(x_{\mathrm{eq}}) = x_{\mathrm{eq}}\) and \(f\) continuous at \(x_{\mathrm{eq}}\), and let \(\mathcal{D}\) be a neighbourhood of \(x_{\mathrm{eq}}\). Let \(0 {\lt} c_1 \le c_2\), \(p {\gt} 0\) and \(0 {\lt} c_3 {\lt} 1\), and let \(V : E \to \mathbb {R}\) satisfy, for every \(x \in \mathcal{D}\),

Then \(x_{\mathrm{eq}}\) is locally exponentially stable (definition 337).

Continuity of \(f\) at \(x_{\mathrm{eq}}\) is not among the hypotheses of the reference, but without it a trajectory can leave \(\mathcal{D}\) in a single step.

Reference: Jungers and van de Wouw (2026), Theorem 2, row 4.

Let \(E\) be proper, and let the system be \(x^{+} = f(x)\) with \(f(x_{\mathrm{eq}}) = x_{\mathrm{eq}}\) and \(f\) continuous at \(x_{\mathrm{eq}}\). Let \(\mathcal{D}\) be a neighbourhood of \(x_{\mathrm{eq}}\) and \(V\) positive definite on \(\mathcal{D}\) about \(x_{\mathrm{eq}}\) (definition 27), with

Then \(x_{\mathrm{eq}}\) is uniformly stable (definition 330).

Continuity of \(f\) at \(x_{\mathrm{eq}}\) is not among the hypotheses of the reference, but it cannot be dropped: on \(\mathbb {R}\), \(V(x) = x^2/(1 + x^4)\) and \(f(x) = 1/x^2\) for \(0 {\lt} |x| {\lt} 1\) (\(f(x) = x\) otherwise) satisfy everything else, yet \(x_0 = 10^{-3}\) jumps to \(x_1 = 10^{6}\).

Reference: Jungers and van de Wouw (2026), Theorem 2, row 1.

Hespanha, P5.6. For each \(i\), the \(i\)-th column of \(\Phi (t,t_0)\) is the unique solution of \(z(t+1) = A(t)z(t)\) with \(z(t_0) = e_i\), for \(t \ge t_0\).

For a constant state matrix \(A\), the discrete state transition matrix (definition 204) is \(\Phi (t, t_0) = A^{t - t_0}\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5, Section 5.3.

If \(z(t_0) = x_0\) and \(z(t+1) = A(t)z(t)\) for all \(t \ge t_0\), then \(z(t) = \Phi (t,t_0)x_0\) for all \(t \ge t_0\): the vector form of theorem 207.

Hespanha, P5.7. For \(\tau \le s \le t\),

Unlike the continuous-time case there is no inverse: \(\Phi \) is a product of matrices that need not be nonsingular, so the ordering \(\tau \le s \le t\) is a genuine restriction here rather than an artifact.

Hespanha, P5.5. If \(Z(t_0) = I\) and \(Z(t+1) = A(t)Z(t)\) for all \(t \ge t_0\), then \(Z(t) = \Phi (t,t_0)\) for all \(t \ge t_0\).

For \(t \ge t_0\), the formula of lemma 211 satisfies the forced recursion

Any two solutions of \(z(t+1) = A(t)z(t) + B(t)u(t)\) with the same value at \(t_0\) agree for all \(t \ge t_0\).

If \(x\) is a trajectory of \(s\) under \(u\) on \([t_0, \infty )\) (definition 137), then \(x(t)\) is the forced response (definition 214) from \(x(t_0)\) at \(t_0\), for every \(t \ge t_0\).

If \(x\) is a trajectory of the unforced system on \([t_0, \infty )\) (definition 137), then \(x(t) = \Phi (t, t_0)\, x(t_0)\) for every \(t \ge t_0\).

Let \(A\) be continuous and let \(x\) solve \(\dot x = A(t)x\) on \([t_{0}, t_{1}]\). Then

Let \(s\) have continuous \(A\) and let \(\varphi \) be a trajectory of the unforced system on \([t_{0}, t_{1}]\) (definition 120). Then \(\varphi (t) = \Phi (t, t_{0})\, \varphi (t_{0})\) for every \(t \in [t_{0}, t_{1}]\).

Hespanha, P6.5. There are scalar functions \(\alpha _0, \dots , \alpha _{n-1} : \mathbb {R} \to \mathbb {R}\) with

The matrix exponential — an infinite series in \(A\) — is a finite polynomial in \(A\) of degree less than \(n\), at the cost of coefficients that are transcendental in \(t\).

Let \(\varphi \) be continuous on \([t_0, T]\) with \(\| \varphi (t_0) - x_{\mathrm{eq}}\| {\lt} \rho \), and suppose some \(t_1 \in [t_0, T]\) satisfies \(\rho \le \| \varphi (t_1) - x_{\mathrm{eq}}\| \). Then there is a first hitting time \(\tau \in [t_0, T]\) of the sphere of radius \(\rho \):

Let \(f : \mathbb {R} \to E \to E\) be jointly continuous and \(K\)-Lipschitz in the state variable, uniformly in time, on a Banach space \(E\). Then for every \(t_0 \le t_1\) and every \(x_0 \in E\) there is an \(\alpha \) with \(\alpha (t_0) = x_0\) solving \(\dot\alpha = f(t, \alpha )\) on all of \([t_0, t_1]\).

Global — rather than merely local — Lipschitz continuity is what makes the solution reach \(t_1\): it forces linear growth in the state, ruling out finite-time blowup.

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 3.2 (Global Existence and Uniqueness), proved in his Appendix C.1. Difference: Khalil assumes \(f\) piecewise continuous in \(t\); here it is jointly continuous.

Let \(f\) be jointly continuous and \(K\)-Lipschitz in the state variable, uniformly in time. Then for every \(t_0\) and \(x_0\) there is an \(\alpha \) with \(\alpha (t_0) = x_0\) solving \(\dot\alpha = f(t,\alpha )\) on all of \([t_0,\infty )\), and any other such solution agrees with it on \([t_0,\infty )\).

Uniqueness is agreement on \([t_0,\infty )\), not equality of functions: the defining condition says nothing about \(t {\lt} t_0\), so solutions may differ there.

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 3.2, whose \(t_1\) is arbitrary, extended to the forward ray.

Let \(g : \mathbb {R} \to \mathbb {R} \to \mathbb {R}\) be jointly continuous and \(L\)-Lipschitz in its state argument, uniformly in time, and let \(t_0 \le t_1\). Then for every \(x_0\) there is a \(z\), continuous on \([t_0, t_1]\), that is an integral solution of \(\dot z = g(t, z)\) with \(z(t_0) = x_0\) and has right derivative \(g(s, z(s))\) at each \(s \in [t_0, t_1)\).

This is the scalar specialization of theorem 42; it discharges the perturbed-solution hypothesis that theorem 51 takes for generality.

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 3.2, scalar case.

If \(A\) is Hurwitz (definition 364) then there are a positive definite \(P\) and a rate \(c {\gt} 0\) with \(PA + A^{\mathsf T}P \preceq -2cP\).

Suppose \(\| e^{Am}\| {\lt} 1\) for some integer \(m {\gt} 0\). Then for every positive definite \(Q\) there is a positive definite \(P\) with \(PA + A^{\mathsf T}P = -Q\) (definition 383), and \(P\) is the unique matrix solution.

Reference: Khalil, Nonlinear Systems.

Let \(A\) be constant. For each \(i\), the \(i\)-th column of \(e^{A(t-t_0)}\) is the unique continuous integral solution of \(\dot x = Ax\) with \(x(t_0) = e_i\), the \(i\)-th standard basis vector. Equivalently, the columns of the matrix exponential are the solutions from the standard basis.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 6, Property P6.2.

Hespanha, P6.3. For every \(t, \tau \in \mathbb {R}\),

Let \(A\) be constant and let \(z\) be a continuous integral solution of \(\dot x = Ax\), \(x(t_0) = x_0\), on \([t_0, t_1]\). Then \(z(t) = e^{A(t-t_0)}x_0\) for every \(t\) in that interval.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 6, Property P6.1.

Suppose the vector field \(f\) and certificate \(V\) are \(C^1\), \(V\) is quadratically bounded on a ball, its positive set accumulates at the base point, and its Lie derivative satisfies \(\dot V\geq 2\alpha V\) there for some \(\alpha {\gt}0\). Then the base point is unstable (definition 276) for any system whose state equation is \(\dot x = f(x)\).

Reference: Khalil, Nonlinear Systems, 3rd ed. (Prentice Hall, 2002), Theorem 4.3, p. 125, for the method; this quantitative variant is not its statement.

Let \(P\) solve \(PA + A^{\mathsf T}P = -Q\) (definition 383). Then the derivative of the centered quadratic form \(x \mapsto (x - x_{\mathrm{eq}})^{\mathsf T}P(x - x_{\mathrm{eq}})\) (definition 384) along the linear vector field \(y \mapsto Ay\) is

This is what makes a Lyapunov solution a Lyapunov function: the forcing \(Q\) is exactly the dissipation rate.

Let \(A\) be the Fréchet derivative of \(f\) at \(x_0\). For every \(\varepsilon {\gt} 0\) there exists \(\delta {\gt} 0\) such that

This is the \(\varepsilon \)–\(\delta \) form of \(f(x) - f(x_0) - A(x-x_0) = o(\| x-x_0\| )\).

Khalil, Lemma A.1. Let \(\Lambda , \mu : [a,b] \to \mathbb {R}\) be continuous with \(\mu \ge 0\), and let \(y : [a,b] \to \mathbb {R}\) be continuous satisfying

Then

Hespanha, Theorem 5.2. For continuous \(A\), \(B\) and \(u\), the function

satisfies \(\dot x(t) = A(t)x(t) + B(t)u(t)\) at every interior \(t\).

The state transition matrix solves the matrix ODE

for \(t\) strictly between \(t_0\) and \(t_1\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

For a constant matrix \(C\), \(Y(t) := \Phi (t,t_0)\, C\) solves

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

Hespanha, Theorem 5.1. \(x(t) := \Phi (t,t_0)\, x_0\) solves

If \(\mu \in \mathbb {C}\) and \(v \neq 0\) satisfy \(H_{A, C}(\mu ) \cdot v = 0\), then \((A, C)\) is not observable.

Let \(f : \mathbb {R}^n \to \mathbb {R}^n\) be \(C^1\), and let \(x_{\rm eq}\) be an equilibrium. If its Jacobian \(A\) at the equilibrium is Hurwitz, then \(x_{\rm eq}\) is locally exponentially stable (definition 284) for any system whose state equation is \(\dot x = f(x)\).

Reference: Khalil, Nonlinear Systems.

If a real matrix \(A\) is Hurwitz, then for every positive-definite \(Q\) there is a positive-definite matrix \(P\) satisfying \(PA+A^{\mathsf T}P=-Q\), and this \(P\) is the unique matrix solution.

omit [DecidableEq X] in Reference: Khalil, Nonlinear Systems.

A finite-dimensional system \((A, B)\) over \(\mathbb {C}\) is controllable if and only if for every \(\mu \in \mathbb {C}\) the Hautus matrix \(H^{\mathrm{ctrl}}_{A, B}(\mu )\) has full row rank:

A linear system \((A, B)\) is controllable if and only if the controllability matrix has full row rank,

where \(n\) is the state dimension.

If \(A\) is Hurwitz (definition 364) then there are \(k {\gt} 0\) and \(\gamma {\gt} 0\) with

If \(A\) is Hurwitz (definition 364) then there are \(k {\gt} 0\) and \(\gamma {\gt} 0\) with

This is the form the Lyapunov stability predicates consume.

If \(A\) is Hurwitz then there is \(c {\gt} 0\) such that every eigenvalue \(\mu \) of \(A\) satisfies \(\operatorname {Re}(\mu ) {\lt} -c\).

A real square matrix \(A\) is Hurwitz (definition 364) if and only if there are \(k {\gt} 0\) and \(\gamma {\gt} 0\) with \(\| e^{At}\| \le k\, e^{-\gamma t}\) for all \(t \ge 0\).

Recorded separately from theorem 379 because the two are what clauses (2) and (3) of Theorem 8.1 respectively need: bare convergence for asymptotic stability, an explicit rate for exponential stability.

A real square matrix \(A\) is Hurwitz (definition 364) if and only if there are a positive definite \(P\) and a rate \(c {\gt} 0\) with \(PA + A^{\mathsf T}P \preceq -2cP\).

This is clause 6 of theorem 396, which is not in Hespanha’s list. It is the rate-carrying refinement of theorem 394, and the shape every contraction and guaranteed-decay-rate condition is written in.

A real square matrix \(A\) is Hurwitz (definition 364) if and only if some positive definite \(P\) makes \(-(PA + A^{\mathsf T}P)\) positive definite.

This is Hespanha’s Theorem 8.2, \((3) \Leftrightarrow (5)\), and the form in which a Lyapunov certificate is actually supplied: one matrix, not a family.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.2.

A real square matrix \(A\) is Hurwitz (definition 364) if and only if for every positive definite \(Q\) there is a positive definite \(P\) with \(PA + A^{\mathsf T}P = -Q\) (definition 383), unique among all matrix solutions.

This is Hespanha’s Theorem 8.2, \((3) \Leftrightarrow (4)\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.2.

A real square matrix \(A\) is Hurwitz (definition 364) if and only if \(e^{At} \to 0\) as \(t \to \infty \).

If \(P\) and \(Q\) are positive definite and \(PM + M^{\mathsf T}P = -Q\), then every eigenvalue of \(M\) has strictly negative real part.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.2.

If \(P\) is positive definite, \(c {\gt} 0\), and \(PA + A^{\mathsf T}P \preceq -2cP\) — that is, \(-(PA + A^{\mathsf T}P) - 2cP\) is positive semidefinite — then \(A\) is Hurwitz (definition 364).

Along a trajectory this inequality gives \(\dot V \le -2cV\) for \(V = x^{\mathsf T}Px\), so it certifies not merely stability but a decay rate.

If \(P\) is positive definite and \(-(PM + M^{\mathsf T}P)\) is positive definite, then \(M\) is Hurwitz (definition 364).

This is Hespanha’s Theorem 8.2, \((5) \Rightarrow (3)\). It is stated through positive definiteness of the negated residual rather than through a matrix order relation, so that it needs no Loewner order.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.2.

If \(e^{At} \to 0\) as \(t \to \infty \) then \(A\) is Hurwitz (definition 364).

Stated from mere convergence rather than from an exponential bound, so that it serves clause (2) and clause (3) of Theorem 8.1 alike.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.1.

If \(e^{At}x \to 0\) as \(t \to \infty \) for every real state \(x\), then \(A\) is Hurwitz (definition 364).

This is the composite that both clauses of Theorem 8.1 reach for: the stability predicates speak about states, not about the matrix.

A real matrix \(A\) is Hurwitz with decay rate \(\alpha \) if and only if the spectrally shifted matrix \(A + \alpha I\) is Hurwitz.

Reference: João P. Hespanha, Linear Systems Theory (2nd ed.), continuous-time stability criterion. This spectral-shift corollary is proved directly from eigenpairs.

The forced response (definition 172) satisfies

for every \(t\) on the closed segment between \(t_0\) and \(t_1\).

This is theorem 169 upgraded to include the endpoints, which the differential form cannot reach: its window has \(t_0\) as an endpoint for every choice of \(t_1\), so \(t_0\) is never interior to it.

A finite-dimensional system \((A, C)\) over \(\mathbb {C}\) is observable if and only if for every \(\mu \in \mathbb {C}\) the Hautus matrix \(H_{A, C}(\mu )\) has trivial kernel:

A finite-dimensional linear system \((A, C)\) is observable in the sense of definition 438 if and only if the observability matrix \(\mathcal{O}(A, C)\) has trivial kernel under matrix-vector multiplication:

A finite-dimensional system \((A, C)\) is observable if and only if the observability matrix \(\mathcal{O}(A, C)\) has full column rank, i.e.

where \(n\) is the state dimension.

If \(A\) is Schur (definition 414) then there are \(C {\gt} 0\) and \(0 \le \rho {\lt} 1\) with

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.3.

If \(A\) is Schur (definition 414) then there are \(C {\gt} 0\) and \(0 \le \rho {\lt} 1\) with \(\| A^{k}x\| \le C\, \rho ^{k}\, \| x\| \) for every \(k\) and \(x\).

If \(A\) is Schur (definition 414), then for every positive definite \(Q\) there is a positive definite \(P\) with \(A^{\mathsf T}PA - P = -Q\) (definition 423), and \(P\) is the unique matrix solution.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.4.

A real square matrix \(A\) is Schur (definition 414) if and only if there are \(C {\gt} 0\) and \(0 \le \rho {\lt} 1\) with \(\| A^{k}\| \le C\rho ^{k}\) for all \(k\).

A real square matrix \(A\) is Schur (definition 414) if and only if some positive definite \(P\) makes \(-(A^{\mathsf T}PA - P)\) positive definite.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.4.

A real square matrix \(A\) is Schur (definition 414) if and only if for every positive definite \(Q\) there is a positive definite \(P\) with \(A^{\mathsf T}PA - P = -Q\) (definition 423), unique among all matrix solutions.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.4.

A real square matrix \(A\) is Schur (definition 414) if and only if \(A^{k} \to 0\) as \(k \to \infty \).

If \(P\) and \(Q\) are positive definite and \(A^{\mathsf T}PA - P = -Q\), then \(A\) is Schur (definition 414).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.4.

If \(P\) is positive definite and \(-(A^{\mathsf T}PA - P)\) is positive definite, then \(A\) is Schur (definition 414).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.4.

If \(A^{k} \to 0\) as \(k \to \infty \) then \(A\) is Schur (definition 414).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.3.

If \(A^{k}x \to 0\) as \(k \to \infty \) for every real state \(x\), then \(A\) is Schur (definition 414).

Let \(s\) be a continuous-time linear system with continuous coefficients \(A\) and \(B\), and let \(u\) be a continuous input. Then the forced response (definition 172) from any state \(x_0\) at any time \(t_0\) is a trajectory of \(s\) (definition 121) — it satisfies the state equation at every \(t \in \mathbb {R}\), with a two-sided derivative.

Restriction to an interval, in particular to the ray \([t_0,\infty )\) that the stability predicates quantify over, is lemma 86.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5, Theorem 5.2.

For a continuous-time linear system with continuous coefficients, the homogeneous response (definition 173) from any state \(x_0\) at any time \(t_0\) is a trajectory (definition 121) under zero input.

Definition 8.1 is stated about this map, so this is the form in which the solution theory reaches the stability theory.

For an LTI system (definition 125) with continuous input \(u\), the closed form

is a trajectory (definition 121) of that system.

For a discrete-time LTV system \(s\) and input \(u\), the forced response (definition 214) from \(x_0\) at \(t_0\) is a trajectory of \(s\) under \(u\) on \([t_0, \infty )\) (definition 137).

The homogeneous response (definition 215) from \(x_0\) at \(t_0\) is a trajectory of the unforced system on \([t_0, \infty )\) (definition 137).

The homogeneous response (definition 173) is a trajectory of the unforced system (definition 120) on \([t_{0}, \infty )\).

In a basis adapted to the four Kalman coordinate sectors, ordered as controllable-unobservable, controllable-observable, uncontrollable-unobservable, uncontrollable-observable, the system has the forced-zero pattern

Every displayed zero is asserted entrywise; no vanishing claim is made for the starred blocks. The chosen complement sectors are not individually asserted to be invariant under \(A\).

References: Hespanha, Linear Systems Theory; Kalman, “Mathematical Description of Linear Dynamical Systems” (1963); Kailath, Linear Systems.

Every finite-dimensional complex state-space system has four coordinate sectors adapted to its reachable and unobservable subspaces. The complements are noncanonical vector-space complements and are not individually asserted to be invariant under \(A\).

References: Hespanha, Linear Systems Theory; Kailath, Linear Systems.

Let \(W : \mathbb {R}^{n} \to \mathbb {R}\) be positive definite (definition 27) about \(x_{\mathrm{eq}}\) on a set containing \(\overline{B}(x_{\mathrm{eq}}, r)\). Then there are class \(\mathcal{K}\) functions \(\alpha _1, \alpha _2\) on \([0, r)\) with

The conclusion holds on the open ball: \(\alpha _1\) and \(\alpha _2\) are class \(\mathcal{K}\) on \([0, r)\), so neither is defined at \(\| x - x_{\mathrm{eq}}\| = r\).

This is what lets a Lyapunov argument be run entirely in terms of comparison functions: positive definiteness is a pointwise condition, and this converts it into monotone bounds in the single variable \(\| x - x_{\mathrm{eq}}\| \).

Reference: Khalil, Nonlinear Systems (3rd ed.), Lemma 4.3 (proved in his Appendix C.4). Differences:

• The bound holds on the open ball, with \(\alpha _1, \alpha _2\) class \(\mathcal{K}\) on \([0,r)\); Khalil states it on \([0,r]\).

• The radially unbounded \(\mathcal{K}_{\infty }\) case is not derived here.

For the time-invariant system \(\dot x = Ax\) the following are equivalent.

1. The origin is asymptotically stable (definition 279).

2. The origin is exponentially stable (definition 284).

3. \(A\) is Hurwitz (definition 364).

4. For every positive definite \(Q\) there is a positive definite \(P\) with \(PA + A^{\mathsf T}P = -Q\) (definition 383), and this \(P\) is the unique matrix solution.

5. There is a positive definite \(P\) with \(-(PA + A^{\mathsf T}P)\) positive definite.

6. There are a positive definite \(P\) and a rate \(c {\gt} 0\) with \(PA + A^{\mathsf T}P \preceq -2cP\).

Clause 6 is not in Hespanha’s list. It is the rate-carrying form of clause 5: along a trajectory it gives \(\dot V \le -2cV\) for \(V = x^{\mathsf T}Px\), hence \(V(t) \le V(0)e^{-2ct}\), and it is the shape every contraction and guaranteed-decay-rate condition is written in.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 8, Theorem 8.2.

If \((A, C)\) is not observable, then there exists \(\mu \in \mathbb {C}\) and a nonzero vector \(v \in \mathbb {C}^{n}\) such that

i.e. \(\mu \) is an eigenvalue of \(A\) with an eigenvector in \(\ker C\).

A complex eigenpair of a real matrix \(A\) with \(\operatorname {Re}\mu {\gt}0\) yields a Hermitian real matrix \(H\), a shift \(\alpha {\gt}0\), and a direction \(w\) such that \(w^{\mathsf T}Hw{\gt}0\) and

is positive definite.

Reference: Hahn, Stability of Motion; Khalil, Nonlinear Systems.

Let \(f:\mathbb R^n\to \mathbb R^n\) be \(C^1\) with \(f(x_{\rm eq})=0\). If the Jacobian at \(x_{\rm eq}\) has a complex eigenvalue with positive real part, then \(x_{\rm eq}\) is unstable (definition 276) for any system whose state equation is \(\dot x = f(x)\).

Reference: Khalil, Nonlinear Systems (Lyapunov’s indirect method).

A pair \((A,B)\) is controllable if and only if its reachable subspace is the whole state space.

Reference: Hespanha, Linear Systems Theory.

Hespanha, P5.2. For every fixed \(i\), the \(i\)-th column of \(\Phi (t,t_0)\) is the unique solution of \(\dot x = A(t)\, x\), \(x(t_0) = e_i\), where \(e_i\) is the \(i\)-th standard basis vector.

Hespanha, P5.3. For any \(s\) on the segment between \(\tau \) and \(t\),

The \(\tau \le s {\lt} t\) ordering in theorem 163 matches the textbook’s literal statement but is not a genuine requirement; the variation-of-constants proof needs the unordered form, where \(s = t_0\) is fixed and \(\tau \) ranges over the whole segment.

For a constant state matrix \(A\), the state transition matrix is the matrix exponential:

This is the bridge that lets every property already proved for \(\Phi \) in the time-varying setting — existence, uniqueness, the semigroup law, invertibility, variation of constants — be reused for LTI systems by specialization rather than reproved.

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 6, equation (6.2).

Hespanha, P5.4. For \(A\) continuous and bounded on the segment between \(\tau \) and \(t\), \(\Phi (t,\tau )\) is nonsingular and

Any solution \(Y\) of \(\dot Y = A(t)\, Y\), \(Y(t_0) = C\), for \(t\) between \(t_0\) and \(t_1\), coincides with \(\Phi (t,t_0)\, C\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

Hespanha, Theorem 5.1. Any solution \(z\) of \(\dot x = A(t)\, x\), \(x(t_0) = x_0\), for \(t\) between \(t_0\) and \(t_1\), coincides with \(x(t) := \Phi (t,t_0)\, x_0\).

\(\Phi (t_0,t_0) = I\).

Reference: Hespanha, Linear Systems Theory (2nd ed.), Chapter 5.

Hespanha, P5.3. For \(A\) continuous and bounded on \([\tau , t]\), and any \(\tau \le s {\lt} t\),

A finite-dimensional system \((A, C)\) is observable in the sense of definition 438 if and only if its unobservable subspace is trivial:

If \(A \in \mathbb {R}^{n \times n}\) has a complex eigenvalue \(\mu \) with \(\operatorname {Re}\mu {\gt} 0\), then the origin is unstable (definition 276) for any system on \(\mathbb {R}^n\) whose state equation is \(\dot x = A x\).

Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.7.

Hespanha, Theorem 5.2. Any two continuous integral solutions of the forced system \(\dot x = A(t)x + B(t)u(t)\) with the same initial value \(x_0\) agree on the segment between \(t_0\) and \(t_1\).