Stabilitiy Analysis using LaSalle’s Principle

We consider autonomous systems of the form \[ \dot{x}=f(x) \] with a vector field $f:\mathbb{R}^n\to\mathbb{R}^n$. A set $\Omega\subseteq\mathbb{R}^n$ is called positively invariant if \[ \forall t\geq 0: \quad x(0)\in\Omega\quad\implies\quad x(t)\in\Omega. \]

Let $\Omega\subsetneq\mathbb{R}^n$ be a compact, positively invariant set. A smooth map $V:\mathbb{R}^n\to[0,\infty)$ is called a Lyapunov-type function if \[ \forall x\in\Omega: \quad L_fV(x) \leq 0, \] where $L_fV(x)=V^\prime(x)\cdot f(x)$ denotes the Lie derivative of $V$ along the vector field $f$.

The positive limit set $M^+\subseteq\Omega$ is contained in the set \[ M_1= \left\{ x\in\Omega:\;L_fV(x)=0 \right\} \subseteq M^+. \] It can be shown that all solution starting in $\Omega$ converge to the set \[ M_{\infty}= \left\{ x\in\Omega:\;L_fV(x)=0 \land L_f^2V(x)=0 \land \ldots \right\}. \] For polynomial systems, this set is an algebraic variety. The associated polynomial ideal can be computed with a finite number of steps.


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