6 Stability of continuous-time systems
Stability notions for an equilibrium of an autonomous continuous-time system \(\dot x = f(t, x)\) (definition 65); a system with input is first unforced (definition 74). Each notion is defined once, for time-varying systems, with the initial time \(t_0 \ge 0\) explicit; a time-invariant system is a special case, and that its notions may be read with \(t_0 = 0\) is a theorem rather than a second definition.
Bounds — stability, uniform convergence times, exponential estimates — quantify over trajectories on segments \([t_0, t_1]\), so that a solution escaping in finite time counts against stability instead of being silently excluded. Limits quantify over trajectories on rays \([t_0, \infty )\).
6.1 Definitions
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 equilibrium is unstable when it is not stable (definition 275).
Reference: Khalil, Nonlinear Systems (3rd ed.), Definition 4.4.
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 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 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 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.
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 \(t \in [t_0 + T, t_1]\).
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 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 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.
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.
6.2 Implications
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).
Stability at \(t_0 = 0\) with tolerance \(\varepsilon \) gives a \(\delta \); the trajectory for that \(\delta \) contradicts it.
A uniformly stable equilibrium (definition 277) is stable (definition 275).
A globally asymptotically stable equilibrium (definition 280) is locally asymptotically stable (definition 279).
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.
Uniform stability gives stability; a uniform convergence time gives convergence of every trajectory on a ray, with the same radius \(c\) for every \(t_0\).
A globally uniformly asymptotically stable equilibrium (definition 283) is globally asymptotically stable (definition 280), given forward completeness (definition 77).
Every trajectory starts in some ball, and every ball has a uniform convergence time.
A locally exponentially stable equilibrium (definition 284) is uniformly stable (definition 277).
The estimate bounds the distance by \(k\) times the initial distance, so \(\delta = \min (c, \varepsilon / k)\) works for every \(t_0\).
A locally exponentially stable equilibrium (definition 284) is uniformly asymptotically stable (definition 282).
Uniform stability is lemma 291. From the ball of radius \(c\) the distance after time \(T\) is below \(k c\, e^{-\gamma T}\), which is below any \(\eta {\gt} 0\) for \(T\) large, independently of \(t_0\).
A globally exponentially stable equilibrium (definition 285) is locally exponentially stable (definition 284).
A globally exponentially stable equilibrium (definition 285) is globally uniformly asymptotically stable (definition 283).
The estimate bounds the distance by \(k\) times the initial distance, so \(\delta (\varepsilon ) = \varepsilon / k\) witnesses global uniform stability. For each radius \(c\) the estimate is locally exponential from that ball, which gives a uniform convergence time by lemma 292.
6.3 Comparison-function characterizations
The uniform notions are equivalent to comparison-function bounds on the trajectories: uniform stability to a class \(\mathcal{K}\) bound, uniform asymptotic stability to a class \(\mathcal{KL}\) bound. The hard direction is the decay in elapsed time. A uniform convergence time says that for each tolerance \(\eta \) there is a delay after which every trajectory from a ball is inside the \(\eta \)-ball. Read the other way round it bounds how far a trajectory can still be after a given elapsed time, but only through a function of \(\eta \) that may be discontinuous. The results below regularize it into a decay envelope, a function of elapsed time alone that is continuous, positive and vanishing at \(+\infty \): the time-decay half of a class \(\mathcal{KL}\) bound, whose other half is a uniform class \(\mathcal{K}\) bound.
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).
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.
Let \(\bar{T}(\eta , r)\) be the least delay after which every trajectory starting within \(r\) of \(x_{\mathrm{eq}}\) lies inside the \(\eta \)-ball; it is finite by hypothesis and antitone in \(\eta \), but possibly discontinuous. Smooth it by the sliding average
which equals \(\frac{2}{\eta }\int _{\eta /2}^{\eta }\bar{T}(s,r)\, ds + r/\eta \). The averaged term is antitone in \(\eta \) and the \(r/\eta \) term is strictly antitone, so \(W(r, \cdot )\) is continuous and strictly antitone on \((0,\infty )\); it dominates \(r/\eta \), hence tends to \(+\infty \) as \(\eta \to 0^{+}\), and \(\bar{T}(\cdot , r)\) vanishes beyond \(\alpha (r)\), hence \(W(r, \eta ) \to 0\) as \(\eta \to \infty \). Inverting it (lemma 243, lemma 244, lemma 245, lemma 246, lemma 247) gives \(U = \mathrm{invFunOn}\, W(r,\cdot )\, (0,\infty )\), singular class \(\mathcal{L}\). Finally \(W(r, U(s)) = s\) and \(W \ge \bar{T} + r/\eta \) give \(\bar{T}(U(s), r) {\lt} s\), so \(s\) is itself a valid convergence time to the \(U(s)\)-ball, which is the claimed bound.
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
\(U(r, s) {\gt} 0\) for all \(r {\gt} 0\) and \(s {\gt} 0\);
\(U(r, \cdot )\) antitone on \((0,\infty )\) for each \(r {\gt} 0\);
\(U(r, s) \to 0\) as \(s \to \infty \), for each \(r {\gt} 0\);
\(r \mapsto U(r+1, s)\) monotone on \([0,\infty )\) for each \(s {\gt} 0\);
\(\| x(t) - x_{\mathrm{eq}}\| \le U(r, t - t_{0})\) 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).
Every radius is finite, so at radius \(r\) the global hypotheses restrict to the local ones (a uniform convergence time from the \(r\)-ball, and the class \(\mathcal{K}_{\infty }\) bound read as a class \(\mathcal{K}\) bound on \([0, r+1)\)). Take \(U(r, \cdot )\) to be the envelope of lemma 297 at that radius, namely the inverse of \(W(r, \cdot )\). Conditions (1)–(3) and (5) are then that lemma’s conclusions, and (4) holds because \(W\) is monotone in \(r\) — a larger ball has larger optimal convergence times — so inverting reverses nothing.
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).
For \(c {\lt} 0\) no trajectory starts in the ball. Otherwise every deviation is at most \(\beta (c, t - t_0)\), by monotonicity of \(\beta \) in its first argument, and \(\beta (c, \cdot ) \to 0\).
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 argument of lemma 301 with the initial time fixed.
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.
Forward: the worst deviation \(\omega (r)\) reachable from the \(r\)-ball is nondecreasing, vanishes at \(0\) and is continuous there by uniform stability; a continuous strictly increasing majorant (lemma 269) is the class \(\mathcal{K}\) bound. Backward: \(\alpha \) is continuous at \(0\) with \(\alpha (0) = 0\), so a small radius \(\delta \) has \(\alpha (\delta ) {\lt} \varepsilon \).
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.
Forward: uniform stability gives a class \(\mathcal{K}\) bound \(\alpha \) (lemma 301) and, with the uniform convergence time, a decay envelope \(U\) (lemma 297); \(\beta = \min (\alpha , \sqrt{\alpha \, U})\) is class \(\mathcal{KL}\) (theorem 254) and bounds the trajectories. Backward: \(\beta (\cdot , 0)\) is continuous at \(0\), which gives uniform stability, and lemma 299 gives the convergence time.
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
zero at the origin: \(\beta (0, s) = 0\) for \(s \ge 0\);
jointly continuous on \([0,\infty ) \times [0,\infty )\);
strictly increasing in \(r\) for each fixed \(s \ge 0\);
radially unbounded: \(\beta (r, 0) \to \infty \) as \(r \to \infty \);
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.
Forward: global uniform stability gives a class \(\mathcal{K}_\infty \) bound, the convergence times from every ball a family of decay envelopes (lemma 298), and lemma 266 combines them. Backward: \(\alpha = \beta (\cdot , 0)\) is class \(\mathcal{K}_\infty \) and \(\delta = \alpha ^{-1}\) gives global uniform stability; from the \(r\)-ball every deviation is at most \(\beta (r, \cdot ) \to 0\).
The same characterizations hold with the bounds on solutions over \([t_0, \infty )\). A bound on every segment restricts to those solutions, so that direction needs nothing more. The converse needs every segment from the ball to extend to such a solution: forward completeness from some ball about \(x_{\mathrm{eq}}\) locally, and from every state globally.
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.
lemma 301, with the bound moved between segments and solutions on \([t_0, \infty )\). Forward: restrict a solution to \([t_0, t]\). Converse: shrink the radius to \(a' = \min (c, a/2)\), restricting \(\alpha \) (definition 228); a segment starting within \(a'\) extends by completeness to a solution on \([t_0, \infty )\), which the bound covers.
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.
lemma 302, with the bound moved between segments and solutions on \([t_0, \infty )\) as in lemma 304, shrinking the radius to \(\min (c, a)\) (definition 258).
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.
6.4 Lyapunov functions
A Lyapunov function is a certificate: its existence implies a stability notion. The certificates are time-varying, \(V(t, x)\), and named by what they certify. What separates them is which bounds hold, not whether time appears: a positive definite (definition 27) lower bound gives stability, and a positive definite upper bound as well (decrescence) is what makes the conclusion uniform. A time-invariant \(V(x)\) is its own lower and upper bound, so it is a uniform certificate as soon as it decreases along the system.
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 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.
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 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.
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).
Take \(W_1 = W_2 = V\). The derivative along the system of a function of \(x\) alone is its directional derivative along the vector field.
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).
As for lemma 311; \(-W_3 \le 0\) gives the non-strict decrease.
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).
Take \(W_3 = -\nabla V \cdot f\): continuous since \(V\) is \(C^1\) and \(f\) is continuous, zero at \(x_{\mathrm{eq}}\) since \(f(x_{\mathrm{eq}}) = 0\), and positive elsewhere on \(D\); apply lemma 312.
6.5 Lyapunov’s direct method
For a time-varying \(V\) the derivative along the system acquires an explicit time term, and positive definiteness of \(V(t, \cdot )\) on its own buys nothing uniform in \(t_0\): \(V\) could flatten as \(t \to \infty \). Sandwiching \(V\) between two positive-definite functions of the state alone is what restores uniformity. Only trajectories on segments \([t_0, t_1]\) are used, so no solution is assumed to exist for all time.
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.
Given \(\varepsilon \), work on a radius \(r \le \varepsilon \) whose closed ball lies in the region of the certificate. Bound \(W_1\) below and \(W_2\) above by class \(\mathcal{K}\) functions \(\alpha _1, \alpha _2\) on it (theorem 270), fix a level \(d\) below both ranges, and take \(\delta = \alpha _2^{-1}(d)\): starting within \(\delta \) of \(x_{\mathrm{eq}}\) forces \(V(t_0, x(t_0)) {\lt} d\). To leave the \(r\)-ball the trajectory would first cross a sphere on which \(W_1 {\gt} d\); up to that time \(V\) has not increased (lemma 56), so it is still below \(d\), a contradiction.
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.
Uniform stability is theorem 314. For convergence, take class \(\mathcal{K}\) bounds \(\alpha _1, \alpha _2, \alpha _3\) for \(W_1, W_2, W_3\) on a ball in the region of the certificate, and the trapping radius \(c\) of that proof. Along a trajectory from the \(c\)-ball, \(\dot V \le -W_3(x) \le -\alpha _3(\| x - x_{\mathrm{eq}}\| ) \le -\alpha _3(\alpha _2^{-1}(V))\), so \(v(t) = V(t, x(t))\) satisfies \(D^+ v \le -\alpha (v)\) with \(\alpha = \alpha _3 \circ \alpha _2^{-1}\) of class \(\mathcal{K}\). theorem 274 gives a class \(\mathcal{KL}\) \(\sigma \) with \(v(t) \le \sigma (v(t_0), t - t_0)\); unwinding the sandwich,
a class \(\mathcal{KL}\) bound. Conclude by lemma 299.
In the global theorem compact sublevel sets of \(V\) are the hypothesis; for a continuous \(V\) they follow from radial unboundedness.
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.
Closed: it is \(V^{-1}((-\infty , c])\), with \(V\) continuous. Bounded: radial unboundedness gives a compact \(K\) off which \(V {\gt} c\), so the sublevel set lies in \(K\). Compact: closed and bounded in a proper space.
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.
\(V\) does not increase along trajectories: \(\nabla V \cdot f \le 0\) everywhere, since \(\nabla V(x_{\mathrm{eq}}) = 0\) at the minimum. So a trajectory starting below every value \(V\) takes outside the \(\varepsilon \)-ball never leaves it. Take \(\delta (\varepsilon )\) the supremum of the radii \(\rho \le \varepsilon \) whose closed ball lies strictly below those values: it is positive, since \(V\) is bounded below by a positive constant outside the \(\varepsilon \)-ball and small near \(x_{\mathrm{eq}}\), and it tends to \(\infty \), since \(V\) is bounded on each ball and large far out. This is global uniform stability.
For convergence from the \(c\)-ball, trajectories stay in the compact set \(\{ V \le M\} \), \(M = \max _{\overline B(x_{\mathrm{eq}}, c)} V\), on which, outside the \(\delta (\eta )\)-ball, \(\nabla V \cdot f \le -\gamma {\lt} 0\). So \(V\) falls at rate \(\gamma \) while the trajectory is outside, and since \(V \ge 0\) it enters the \(\delta (\eta )\)-ball by time \(t_0 + M/\gamma + 1\); from then on it stays in the \(\eta \)-ball.
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.
The bounds give \(\dot V \le -(k_3/k_2) V\), so \(e^{(k_3/k_2)t} V(t, x(t))\) does not increase and \(V(t, x(t)) \le V(t_0, x(t_0))\, e^{-(k_3/k_2)(t - t_0)}\) while the trajectory stays in \(D\); the sandwich and an \(a\)-th root give the estimate. To stay in \(D\), start within \(r/(K + 1)\) of \(x_{\mathrm{eq}}\), where \(\overline{B}(x_{\mathrm{eq}}, r) \subseteq D\) and \(K = (k_2/k_1)^{1/a}\): at the first time the trajectory reached the sphere of radius \(r\) the estimate would put it strictly inside.
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.
The estimate of theorem 318 holds on every trajectory segment, since every segment stays in \(D = E\).
6.6 LaSalle’s invariance principle
When \(\nabla V \cdot f \le 0\) only, \(V\) may stall, and Lyapunov’s theorems give stability but not convergence. LaSalle’s principle recovers convergence for a time-invariant system: a trajectory confined to a compact set converges to its \(\omega \)-limit set, on which \(V\) is constant, and forward invariance of that set turns constancy of \(V\) into \(\nabla V \cdot f = 0\).
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 \(\psi \) be a trajectory on \([a, b]\) with \(p = \psi (a) \in \omega (\varphi )\), and fix \(t \in [a, b]\). Given \(\varepsilon {\gt} 0\), put \(C = e^{K(b - a)}\) and pick, by the definition of the \(\omega \)-limit set, arbitrarily large \(r\) with \(\| \varphi (r) - p\| {\lt} \varepsilon / C\). The time-shifted curve \(u(\cdot ) = \varphi (\cdot + r - a)\) solves the same time-invariant equation on \([a, b]\) with \(u(a) = \varphi (r)\), so Grönwall gives
Since \(r\) may be taken arbitrarily large, so may \(r + t - a\); hence \(\varphi \) is frequently within \(\varepsilon \) of \(\psi (t)\), i.e. \(\psi (t) \in \omega (\varphi )\).
Khalil argues instead with the flow map and the semigroup identity, which presumes a globally defined flow; comparing the two solutions directly needs only uniqueness, in its quantitative Grönwall form.
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.
\(V(\varphi (t))\) is nonincreasing and bounded below on the compact \(\Omega \), so it converges to some \(L\), and \(V \equiv L\) on \(\omega (\varphi )\) (lemma 62). Through \(y \in \omega (\varphi )\) take a solution \(\psi \) on \([0, 1]\) (Picard–Lindelöf, theorem 46); it stays in \(\omega (\varphi )\) by lemma 320, so \(V \circ \psi \equiv L\) and its derivative at \(0\), which is \(\nabla V(y) \cdot f(y)\), vanishes.
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.
The \(\omega \)-limit set lies in \(\Omega \), which is closed and contains the trajectory (lemma 63), and \(\nabla V \cdot f\) vanishes on it (lemma 321), so \(\omega (\varphi ) \subseteq E\). It is forward invariant (lemma 320), so \(\omega (\varphi ) \subseteq M\). A trajectory confined to a compact set converges to every set containing its \(\omega \)-limit set (lemma 64).
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.
Stability: \(V\) is a uniform Lyapunov function (lemma 311, theorem 314). Forward completeness: \(f\) is Lipschitz (theorem 80). Convergence: near \(x_{\mathrm{eq}}\), \(V {\lt} c\), so a trajectory starting there starts in \(\Omega _c\), and theorem 322 with \(M = \{ x_{\mathrm{eq}}\} \) applies.
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.
Stability and forward completeness as in Barbashin’s theorem. The sublevel set \(\{ V \le V(x(t_0))\} \) is compact and, since \(V\) does not increase along trajectories, forward invariant; apply theorem 322 with \(M = \{ x_{\mathrm{eq}}\} \).
6.7 Instability
Instability is the failure of stability, and the simplest witness is a trajectory that leaves a fixed neighbourhood from arbitrarily close initial states (lemma 286). Chetaev’s theorem produces such trajectories from a function that grows along the system near the equilibrium.
The geometric criterion uses a positive certificate with positive derivative in an open region and vanishing on its boundary, with these conditions local to a closed ball.
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.
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.
Choose a positive seed near the boundary point and continue a cutoff curve. Boundary vanishing retains it in the region. The Lie derivative has a positive minimum on its compact superlevel, forcing growth beyond the ball’s certificate ceiling.
The exponential criterion is its corollary on the full positive set \(D=\{ V{\gt}0\} \).
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.
Take \(D=\{ V{\gt}0\} \). The quadratic bound gives \(V(x_{\rm eq})=0\); the nearby positive seeds put \(x_{\rm eq}\) in its closure. Continuity makes \(V=0\) on \(\partial D\), and \(\dot V\ge 2\alpha V{\gt}0\) in \(D\) within the ball. Apply theorem 326.