LaSalle’s invariance principle

E695937

mathematical theorem result in dynamical systems theory

LaSalle’s invariance principle is a fundamental result in dynamical systems theory that extends Lyapunov’s direct method by characterizing the asymptotic behavior of trajectories through invariant sets where a Lyapunov function’s derivative vanishes.

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Statements (48)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in dynamical systems theory ⓘ
appliedIn	biological systems modeling ⓘ engineering ⓘ mechanical systems ⓘ power systems stability ⓘ robotics ⓘ
appliesTo	autonomous dynamical systems ⓘ continuous-time systems ⓘ nonlinear differential equations ⓘ ordinary differential equations ⓘ
assumes	existence of a Lyapunov function with nonpositive derivative ⓘ forward completeness of solutions on the considered set ⓘ
characterizes	asymptotic stability ⓘ limit behavior of trajectories ⓘ
concerns	invariance of sets under system flow ⓘ long-term behavior of solutions ⓘ
concludes	solutions converge to invariant sets where Lyapunov derivative vanishes ⓘ trajectories approach the largest invariant set in the region ⓘ
conditionOn	existence of largest invariant set where Lyapunov derivative is zero ⓘ negative semidefinite derivative of Lyapunov function ⓘ nonincreasing Lyapunov function along trajectories ⓘ
field	control theory ⓘ dynamical systems ⓘ nonlinear systems ⓘ stability theory ⓘ
generalizes	Lyapunov’s direct method NERFINISHED ⓘ
hasVariant	LaSalle–Yoshizawa theorem NERFINISHED ⓘ discrete-time LaSalle invariance principle ⓘ
implies	asymptotic stability of equilibrium under suitable conditions ⓘ convergence to equilibrium when the largest invariant set is an equilibrium ⓘ
namedAfter	Joseph P. LaSalle NERFINISHED ⓘ
relatedTo	Barbashin–Krasovskii theorem NERFINISHED ⓘ Lyapunov stability theory NERFINISHED ⓘ invariant set theorems ⓘ
timeDomain	time-continuous dynamics ⓘ
typicalAssumptionOnSet	compact positively invariant set ⓘ trajectories remain in a bounded region ⓘ
usedFor	analysis of limit sets ⓘ convergence analysis in adaptive control ⓘ convergence analysis in observer design ⓘ global asymptotic stability proofs ⓘ proving asymptotic stability without strict Lyapunov decrease ⓘ stability analysis of nonlinear control systems ⓘ
usesConcept	Lyapunov function NERFINISHED ⓘ asymptotic behavior of trajectories ⓘ invariant set ⓘ omega-limit set ⓘ

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Lyapunov stability theory → relatedTo → LaSalle’s invariance principle ⓘ