LaSalle's invariance principle

LaSalle's invariance principle (also known as the invariance principle,Barbashin-Krasovskii-LaSalle principle, or Krasovskii-LaSalle principle ) is a criterion for the asymptotic stability of an autonomous (possibly nonlinear) dynamical system.

Suppose a system is represented as

where ${\displaystyle \mathbf {x} }$ is the vector of variables, with

If a ${\displaystyle C^{1}}$ function ${\displaystyle V(\mathbf {x} )}$ can be found such that

then the set of accumulation points of any trajectory is contained in ${\displaystyle {\mathcal {I}}}$ where ${\displaystyle {\mathcal {I}}}$ is the union of complete trajectories contained entirely in the set ${\displaystyle \{\mathbf {x} :{\dot {V}}(\mathbf {x} )=0\}}$.

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