Dependencies

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 equilibrium \(x_{\mathrm{eq}}\) is globally asymptotically stable (GAS) when it is Lyapunov stable (definition 30) and every trajectory \(\varphi \) satisfies \(\varphi (t) \to x_{\mathrm{eq}}\) as \(t \to \infty \).

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 classical GAS Lyapunov certificate: a \(C^{1}\) map \(V : \mathbb {R}^{n} \to \mathbb {R}\) with \(V(x_{\mathrm{eq}}) = 0\), \(V {\gt} 0\) elsewhere, \(\dot{V} {\lt} 0\) on \(\mathbb {R}^{n} \setminus \{ x_{\mathrm{eq}}\} \), and radially unbounded (\(V(x) \to \infty \) as \(\| x\| \to \infty \)). Implies definition 38 via theorem 35.

A class \(\mathcal{K}\) function on \([0,a)\) is a continuous strictly increasing map \(\alpha : [0,a) \to [0,b)\) with \(\alpha (0) = 0\). The structure records both \(\alpha \) and its inverse \(\alpha ^{-1} : [0,b) \to [0,a)\).

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 \(r \to \infty \).

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.

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 10 has real content.

A point \(x_{\mathrm{eq}} \in \mathbb {R}^{n}\) is an equilibrium of \(\dot{x} = f(x)\) when \(f(x_{\mathrm{eq}}) = 0\).

A function \(x : [t_0, t_1] \to E\) is an integral solution of \(\dot{x} = F(t,x)\) with initial value \(x_0\) when

A function \(V : \mathbb {R}^{n} \to \mathbb {R}\) is a local Lyapunov function on an open domain \(D \ni x_{\mathrm{eq}}\) when \(V\) is smooth, \(V(x_{\mathrm{eq}}) = 0\), \(V {\gt} 0\) on \(D \setminus \{ x_{\mathrm{eq}}\} \), and the Lie derivative satisfies \(\dot{V}(x) = \nabla V(x) \cdot f(x) \le 0\) for all \(x \in D\).

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 8 has real content.

A set \(S \subseteq \mathbb {R}^{n}\) is positively invariant for \(\dot{x} = f(x)\) when every trajectory \(\varphi \) starting in \(S\) remains in \(S\) for all future time:

A strict local Lyapunov function on \(D\) strengthens definition 36: the Lie derivative satisfies \(\dot{V}(x) {\lt} 0\) for all \(x \in D \setminus \{ x_{\mathrm{eq}}\} \), and there exists \(c {\gt} 0\) such that the compact sublevel set \(\Omega _{c}(V) \subseteq D\) (definition 33).

A global strict Lyapunov function for \(\dot{x} = f(x)\) at \(x_{\mathrm{eq}}\) is a \(C^{1}\) map \(V : \mathbb {R}^{n} \to \mathbb {R}\) with \(V(x_{\mathrm{eq}}) = 0\), \(V {\gt} 0\) everywhere else, \(\dot{V}(x) {\lt} 0\) on \(\mathbb {R}^{n} \setminus \{ x_{\mathrm{eq}}\} \), and all sublevel sets \(\Omega _{c}(V)\) compact (coercivity).

A trajectory of the autonomous ODE \(\dot{x} = f(x)\) is a globally defined map \(\varphi : \mathbb {R} \to \mathbb {R}^{n}\) satisfying

The equilibrium \(x_{\mathrm{eq}}\) is locally asymptotically stable (LAS) when it is Lyapunov stable (definition 30) and there exists \(c {\gt} 0\) such that every trajectory \(\varphi \) with \(\| \varphi (0) - x_{\mathrm{eq}}\| {\lt} c\) satisfies \(\varphi (t) \to x_{\mathrm{eq}}\) as \(t \to \infty \).

The equilibrium \(x_{\mathrm{eq}}\) is Lyapunov stable when

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 sublevel set of \(V : \mathbb {R}^{n} \to \mathbb {R}\) at level \(c \in \mathbb {R}\) is

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 12 reads as content.

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

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

If \(V : \mathbb {R}^{n} \to \mathbb {R}\) is continuous and radially unbounded (\(V(x) \to \infty \) as \(\| x\| \to \infty \)), then every sublevel set \(\Omega _{c}(V)\) (definition 33) is compact.

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 8.

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

Osgood construction. Given a class \(\mathcal{K}\) function \(\alpha \) satisfying the Osgood condition \(\int _0^{\varepsilon } \frac{1}{\alpha (x)}\, dx = +\infty \), the function \(\sigma (r, s) = \eta ^{-1}(\eta (r) + s)\), where \(\eta (y) = -\int _{\mathrm{base}}^{y} \frac{1}{\alpha (x)}\, dx\), is a class \(\mathcal{KL}\) function (definition 48).

Comparison Lemma 3.4. If \(u\) solves \(\dot{u} = f(t,u)\) with \(u(t_0) = u_0\), and \(v\) is continuous with \(D^{+}v(t) \le f(t,v(t))\) and \(v(t_0) \le u_0\), then \(v(t) \le u(t)\) for all \(t \in [t_0, t_1]\).

Theorem 3.5 (Continuous dependence on parameters). 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 \(t \in [t_0, t_1]\).

Gronwall–Bellman inequality. 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

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

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.

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 2 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.

Krasovskii’s theorem. If \(V \in C^1\) is positive definite, \(\dot{V} \le 0\) on \(\mathbb {R}^n\), radially unbounded, and every trajectory has \(\omega \)-limit set \(\subseteq \{ x_{\mathrm{eq}}\} \), then \(x_{\mathrm{eq}}\) is globally asymptotically stable (definition 32).

LaSalle’s invariance principle. Let \(\Omega \) be compact and positively invariant for \(\dot{x} = f(x)\), \(V \in C^{1}\) with \(\dot{V}(x) \le 0\) on \(\Omega \), and \(M \subseteq \Omega \) closed. If the \(\omega \)-limit set of any trajectory \(\varphi \) starting in \(\Omega \) satisfies \(\omega (\varphi ) \subseteq M\), then \(\varphi (t) \to M\) as \(t \to \infty \).

Barbashin’s theorem. If \(V \in C^1\) is positive definite on \(D\), \(\dot{V} \le 0\) on \(D\), there exists a compact sublevel set \(\Omega _c \subseteq D\), and every trajectory starting in \(\Omega _c\) has \(\omega \)-limit set \(\subseteq \{ x_{\mathrm{eq}}\} \), then \(x_{\mathrm{eq}}\) is locally asymptotically stable (definition 31).

Lyapunov’s global asymptotic stability theorem. If \(V\) is a global strict Lyapunov function (definition 38) and \(f\) is continuous, then \(x_{\mathrm{eq}}\) is globally asymptotically stable (definition 32).

Class K sandwich bounds. For any continuous positive-definite function \(V : \mathbb {R}^{n} \to \mathbb {R}\) on \(\overline{B}(0,r)\), there exist class \(\mathcal{K}\) functions \(\alpha _1, \alpha _2\) such that

Corollary. If \(V\) is a radially unbounded strict Lyapunov function (definition 39) and \(f\) is continuous, then \(x_{\mathrm{eq}}\) is globally asymptotically stable.

Lyapunov’s local asymptotic stability theorem. If \(V\) is a strict local Lyapunov function (definition 37) and \(f\) is continuous, then \(x_{\mathrm{eq}}\) is locally asymptotically stable (definition 31).

Lyapunov’s stability theorem. If \(V\) is a local Lyapunov function (definition 36) for \(\dot{x} = f(x)\) on a domain \(D \ni x_{\mathrm{eq}}\), then \(x_{\mathrm{eq}}\) is Lyapunov stable (definition 30).

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 finite-dimensional system \((A, C)\) is observable in the sense of definition 2 if and only if its unobservable subspace is trivial: