📂Dynamics

Proof of the LaSalle Invariance Principle

Principle

Buildup

Let’s consider a vector field given by the following differential equation for space $X$ and function $f : X \to X$. $$ \dot{x} = f(x) $$ Let’s call the compact positively invariant set under the flow $\phi_t \left( \cdot \right)$ as $\mathcal{M} \subset \mathbb{R}^{n}$.

When a Lyapunov function $V : \mathcal{M} \to \mathbb{R}$ is defined as in $\mathcal{M}$, consider the following two sets. $$ E := \left\{ x \in \mathcal{M} : V ' (x) = 0 \right\} $$ The set $M$ defined as follows for $E$ is called the Positively Invariant Part. $$ M:=\left\{ \text{The union of all trajectories that start in E and remain in E for all } t >0 \right\} $$

Lasalle Invariance Principle

For all $x \in \mathcal{M}$, when $t \to \infty$ then it is $\phi_{t} (x) \to M$.

Proof ¹

Strategy: Mainly uses the definition of Lyapunov function and properties of omega limit sets.

Definition of Lyapunov function: Let’s consider a vector field given by the following differential equation for space $X$ and function $f : X \to X$. $$ \dot{x} = f(x) $$ Given a point $x_{0} \in X$ in such an autonomous system, a scalar function $V \in C^{1} \left( \mathcal{N} (x_{0}) , \mathbb{R} \right)$ defined in the neighborhood $\mathcal{N} \left( x_{0} \right)$ of $x_{0}$ is called a Liapunov Function if it satisfies the following conditions:

• (i): If $V(x_{0}) = 0$ and $x \ne x_{0}$, then $V(x) > 0$
• (ii): In $x \in \mathcal{N} \left( x_{0} \right) \setminus \left\{ x_{0} \right\}$, it is $V ' (x) \le 0$

Properties of Omega Limit Sets: Assuming the whole space is the Euclidean space $X = \mathbb{R}^{n}$ and given a point $p \in \mathcal{M}$ of the compact positively invariant set $\mathcal{M}$ under the flow $\phi_{t} ( \cdot )$:

• [1]: $\omega (p) \ne \emptyset$
• [2]: $\omega (p)$ is a closed set.
• [3]: $\omega (p)$ is invariant under the flow. In other words, $\omega (p)$ is a union of orbits.
• [4]: $\omega (p)$ is a connected space.

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First, let’s show that in the omega limit set $\omega (x)$, $V$ becomes a constant function $V = \chi$. $$ \overline{x} \in \omega (x) \\ \chi = V \left( \overline{x} \right) $$ If we assume that, then $V$ does not increase according to the flow $\phi_{t}$. In other words, for $t_{i} \le t \le t_{i+1}$, $$ V \left( \phi_{t_{i}} (x) \right) \ge V \left( \phi_{t} (x) \right) \ge V \left( \phi_{t_{i+1}} (x) \right) $$ it is, and due to the continuity of Lyapunov function $V$, $\chi$ becomes the greatest lower bound, namely the infimum, of $\left\{ V \left( \phi_{t} (x) \right) : t \ge 0 \right\}$. Since the omega limit set $\omega (x)$ is invariant under the flow, $\phi_{t} ( \overline{x} )$ also becomes an omega limit point of $\phi_{t}(x)$. Since $\chi$ was the infimum of $\left\{ V \left( \phi_{t} (x) \right) : t \ge 0 \right\}$ as mentioned above, $$ V \left( \phi_{t} \left( \overline{x} \right) \right) = \chi $$ $\chi$ is constant, so in $\omega (x)$, it is $V’ = 0$, and according to the definition of $E$, it is $\omega (x) \subset E$. Meanwhile, since $\omega (x)$ is an invariant set and according to the definition of $M$, it is also $\omega (x) \subset M$. Thus, when $t \to \infty$, it is $\phi_{t} (x) \to M$.

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