Ordinary differential equations Stability Theorem

LaSalle's invariance principle

Builds on Lyapunov's stability theorem Lyapunov function Asymptotic stability

Statement

Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be locally Lipschitz with $f(0) = 0$ , and let $V : \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable. Assume that $\dot{V}(x) = \nabla V(x) \cdot f(x) \le 0$ for all $x \in \mathbb{R}^n$ , that $V(0) = 0$ and $V(x) > 0$ for $x \neq 0$ , and that $V(x) \to \infty$ as $\|x\| \to \infty$ . Let $E = \{x \in \mathbb{R}^n : \dot{V}(x) = 0\}$ . Assume that the only solution $x(t)$ of $x' = f(x)$ with $x(t) \in E$ for all $t \ge 0$ is the solution $x(t) = 0$ . Then the origin is globally asymptotically stable for $x' = f(x)$ .

Remarks

The general principle: if $\dot{V} \le 0$ , then each solution that is bounded for $t \ge 0$ approaches the largest invariant subset of $E$ . The principle allows $\dot{V} = 0$ outside the origin, where Lyapunov's stability theorem needs a strict Lyapunov function. For the damped oscillator $x' = y$ , $y' = -x - c y$ with $c > 0$ , take $V = \frac{1}{2}(x^2 + y^2)$ . Then $E = \{y = 0\}$ , and a solution in $E$ has $y' = -x = 0$ . So the origin is globally asymptotically stable. A local version holds when the conditions on $V$ hold only near the origin.

History

The principle is named after Joseph LaSalle. It is also named after Nikolay Krasovskii and Evgenii Barbashin.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Lipschitz condition definition
Grönwall's inequality statement
Solution of an ordinary differential equation definition
Autonomous system definition
Initial value problem definition
Equilibrium point definition
Uniqueness of solutions for Lipschitz vector fields statement
Lyapunov function definition
Maximal solution definition
Flow of a vector field definition
Lyapunov stability definition
Asymptotic stability definition
Lyapunov's stability theorem statement
LaSalle's invariance principle statement

Tag	`0659J5X7B5`
URI	`m2://0659J5X7B5`
Aliases	`invariance-principle`, `krasovskii-lasalle-principle`
Kind	statement, theorem
Writing	W0 stub
MSC	34D20, 34D23
Status	This copy of the site has no build data. Run `m2 status 0659J5X7B5`.
Uses	Lyapunov's stability theorem, Lyapunov function, Asymptotic stability
Same as	Wikidata Q3922068 Wikipedia LaSalle's invariance principle

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