Lyapunov Theory

Khali의 Nonlinear-Systems 일부를 정리한 내용입니다.

Theorem 1. Let $f(x,t)$ be piecewise continuous in $t$ and locally Lipschitz in $x$ for all $t\geq t_0$ and all $x$ in a domain $D\subset R^n$. Let $W$ be a compact subset of $D$, $x_0 \in W$, and suppose it is known that every solution of 

$$\dot{x}=f(x,t), \quad x(t_0)=x_0$$

lies entirely in $W$. Then , there is a unique solution that is defined for all $t\geq t_0$.

Proof)  $W$가 compact set이므로 $W$에서 $f(x)$ lipschitz하므로, unique solution이 존재한다.

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Theorem 2 Let $x=0$ be an equilibrium point for $\dot{x}=f(x)$ and $D\subset R^n$ be a domain containing $x=0$. Let $V:D\rightarrow R$ be a continuously differentiable function such that

$$ V(0)=0 \quad\text{and}\quad V(x)>0\quad\text{in}\quad D-\{0\}$$

$$\dot{V}(x) \leq 0 \quad\text{in}\quad D$$

Then $x=0$ is stable. Moreover if 

$$\dot{V}(x) <0 \quad\text{in}\quad D-\{0\}$$

then $x=0$ is asymptotically stable.

$\dot{V}(x)$가 negative or non-positive 에 따라 stable과 asymptotic stability 가 정해진다.

Proof)

Given $\epsilon>0$, $r\in (0,\epsilon]$, take

$$B_r = \{ x\in R^n \mid ||x||\leq r\} \subset D$$

Let $\alpha = \min_{||x||=r}V(x)$. $x=0$에서만 $V(x)=0$이므로 $\alpha>0$이다. $\beta\in (0,\alpha)$인 

$$ \Omega_{\beta}=\{ x\in B_r | V(x) \leq \beta \}$$

$\Omega$가 $B_r$의 interior라는 것은 boundary point에서 만나는 점이 없다는 것으로으로 쉽게 증명할 수 있다.

또한 $\dot{V}(x(t))\leq 0$이므로 $V(x(t))\leq V(x(0))\leq \beta \quad \forall{t}\geq0$이다. 

또한 $\Omega$는 compact set이므로, $x(0)\in \Omega_{\beta}$이면 $x(t)$가 $\Omega$를 떠나지 않으므로 Theorem 1에 의해 unique solution이 정의된다.

$V(x)$가 continuous하므로 $||x||<\delta, \quad V(x)<\beta$인 $\delta$를 찾을 수 있다.

따라서 stability가 증명되었다.

관계를 정리해보면 $B_{\delta} \subset \Omega_{\beta} \subset B_r$

asymptotic stability를 증명하기 위해서는 $V(x(t))\rightarrow 0$임을 보이면 된다. $V(x(t))$의 continuity 로부터 $x(t)\rightarrow 0$임을 알 수 있다. 

Contradiction을 사용해 증명을 하는데 $V(x(t))\rightarrow c > 0$라고 하자.

$V(x(t))>c$이므로 $x(t) \notin B_d \subset \Omega_c$인 $d$를 fix 할 수 있다.

$x(t) \in B_r$인 것과 $ -\gamma = \max_{ d \leq ||x|| \leq } \dot{V}(x)$를 활용한다.

$V(x(t)) = V(x(0)) + \int^t_0 \dot{V}(x(w))dw \leq V(x(0)) - \gamma t$ 인데 이는 unbounded 되어있으므로 contradiction이다. 따라서 asymptotic stability가 증명되었다.

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Theorem 4.2 ( Global Stability ) Let $x=0$ be an equilibrium point for $\dot{x}=f(x)$. Let $V:R^n\rightarrow R$ be a continuously differentiable function such that

$$ V(0)=0\quad\text{and}\quad V(x)>0,\quad\forall{x} \neq 0$$

$$||x||\rightarrow \infty \Rightarrow V(x) \rightarrow \infty$$

$$\dot{V}(x)<0 \quad \forall{x} \neq 0$$

Proof)

Let $c=V(p)$. By radial unboundedness, there exists $r$ such that $V(x)>c$ whenver $||x||>r$.

Thus $\Omega_c \subset B_r$ hence, $\Omega_c$ is bounded.

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Theorem 4.2 ( Chetaev's Theorem )

 Let $x=0$ be an equilibrium point for $\dot{x}=f(x)$. Let $V:D\rightarrow R$ be a continuously differentiable function such that $V(0)=0$ and $V(x_0)>0$ for some $x_0$ with arbitrary small $||x_0||$. Define a set $U=\{x\in B_r \mid V(x)>0 \} $ and suppose that $\dot{V}(x)>0$ in $U$. Then $x=0$ is unstable.

Proof)

The point $x_0$ is in the interior of $U$ and $V(x_0)=a>0$. The trajectory starting at $x(0)=x_0$ must leave the set $U$. Since $\dot{V}(x)>0$ in $U$, $V(x(t))\geq a$.

Let

$$\gamma=\min \{ \dot{V} (x) \mid x\in U \quad\text{and}\quad V(x) \geq a\}$$

Since 

$$V(x(t))=V(x_0) + \int^t_0 \dot{V}(x(s))ds \geq a + \int^t_0 \gamma ds = a + \gamma t$$

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Theorem 4.3 ( LaSalle's Theorem) Let $\Omega \subset D$ be a compact set that is positively invariant with respect to $\dot{x}=f(x)$ . Let $V:D\rightarrow R$ be a continuously differentiable function such that $\dot{V}\leq 0$ in $\Omega$. Let $E$ be the set of all points in $\Omega$ where $\dot{V}(x)=0$. Let $M$ be the largest invariant set in $E$. Then every solution starting in $\Omega$ approaches $M$ as $t\rightarrow\infty$

여기서 $V(x)$는 꼭 positive definite 할 필요가 없다.

Theorem 4.4 Let $x=0$ be an equilibrium point for $\dot{x}=f(x)$. Let $V:D\rightarrow R$ be a continuously differentiable positive definite function on a domain $D$ containing the origin $x=0$, such that $\dot{V}(x) \leq 0$ in $D$. Let $S=\{x\in D\mid \dot{V}(x)=0\}$ and suppose that no solution can stay identically in $S$, other than the trivial solution.  Then, the origin is asymptotically stable.