Consider a control system of the form $\dot{x}(t) = f(x(t), u(t))$ where $u(t)$ is the control input, $x \in \mathbb{R}^{n}$, $u \in \mathbb{R}^{m}$. Assume $f$ is Lipschitz continuous so that the existence and uniqueness of solutions holds.
Following are the definitions of attracting sets and invariant sets that I use:
- A set $S$ is (control) invariant if for any initial state $x(0) \in S$, there exists a control signal $u(\cdot)$ such that $x(t) \in S$ for all $t \geq 0$.
- A set $A$ is (weakly) attracting with basin of attraction $B$ if for any initial state $x(0) \in B$, there exists a control signal $u(\cdot)$ such that $x(t)$ converges to $A$ as $t \to +\infty$, that is $\lim_{t \to +\infty} \operatorname{dist} (x(t), A) = 0$.
It seems to me that attracting sets and invariant sets are closely related, in particular when $B = \mathbb{R}^{n}$ then in many cases, $A$ is also invariant. My questions are:
- Is there any example in which an attracting set is not invariant?
- Under what condition an attracting set is invariant?
I come from the control community and not many control textbooks mention these concepts. If there are any good (math) books that discuss these sets well, please let me know.