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Suppose that I have some dynamical system

$$\dot x = f(x)$$ $f$ is locally Lipschitz, etc. I know that $f(x) = 0$ whenever $x \in \Gamma$, where $\Gamma$ is some closed (and possibly bounded) set in $\mathbb{R}^n$, could be a small disc, a line, a plane. In other words, not just an equilibrium.

How can dynamical system theory deal with this case?

Normally the approach is through Lyapunov theorem, but it is usually with respect to a single point.

Is it possible to extend the Lyapunov theorem by defining a Lyapunov function $V$ such that it is $V(\Gamma) = 0$ and $V(x) > 0$ elsewhere and proceed as usual by showing $\dot V(\Gamma) = 0$ and $\dot V(x) < 0$ elsewhere?

Is this possible? Are there any catch for using the above? I have went through several books but could not find this extension.

Curaçao Hajek
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    The answer depends on what "How can dynamical system theory deal with this case?" means. But indeed functions of the type that you mention are commonly used. – John B Mar 15 '21 at 22:34

1 Answers1

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You can use https://en.wikipedia.org/wiki/LaSalle%27s_invariance_principle, though I don't like the statement off of Wikipedia. The following comes from Nonlinear Systems, Third Edition by Khalil.

Consider the autonomous system

\begin{align} \dot{x} = f(x) \qquad (1) \end{align}

where $f: D\to \mathbb{R}^{n}$ is a locally Lipschitz map from a domain $D\subset \mathbb{R}^{n}$ into $\mathbb{R}^{n}$.

Theorem 4.4 in Nonlinear Systems, Third Edition by Khalil:

Let $\Omega \subset D$ be a compact set that is positively invariant with respect to $(1)$. Let $V: D \to \mathbb{R}$ be a continuously differentiable function such that $\dot{V} \leq 0$ in $\Omega$. Let $E$ be the set of all points in $\Omega$ where $\dot{V}=0$. Let $M$ be the largest invariant set in $E$. Then every solution starting in $\Omega$ approaches $M$ as $t\to\infty$.

Note that this theorem is more general than the extension you mention, $V$ doesn't need to be positive definite. As long as you can show that $\Omega$ is a compact set and positively invariant. If $V$ is positive definite then $\Omega$ can be obtained as the sublevel set of $V$, i.e., $\Omega := \{ x\ |\ V(x) \leq c\}$ for some $c$ and showing that $\dot{V}\leq 0$ for all elements in $\Omega$. If you can find a Lyapunov function such that your set $\Gamma = M$ then you can use LaSalles to show that you converge to the set $\Gamma$.

dgadjov
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  • Thanks. But my question lingers. I just want to extend the Lyapunov theorem a bit to a set version. I do not want to use LaSalle principle. Also what I do not like about LaSalle is that it does not tell you how this convergence looks like or what happens when you converge to this set. Does the ODE solution keep evolving within this set? Does it come to a stop? If so, where? – Curaçao Hajek Mar 16 '21 at 06:26
  • I don't think I understand. LaSalle is the way you "extend the Lyapunov theorem a bit to a set version". Your comment that "do not like about LaSalle is that it does not tell you how this convergence looks like or what happens when you converge to this set", will have the same issues with your method because they both rely on $V$. After using LaSalles you can often reduce the dynamics of $\dot{x}=f(x)$ on $M$ by using the constraint that $\dot{V}=0$. In these reduced dynamics, the ode might continue evolving or stop at some element. You have to investigate these reduced dynamics. – dgadjov Mar 16 '21 at 19:28