LaSalle’s Invariance Principles Of Differential Inclusions And Its Applications

In this paper we first develop the strong LaSalle’s invariance principle of diferentialinclusions and give a direct and simpler proof of it. Then we make some further discus-sions on this topic. More specifically, in a particular but important case we address therobustness of strong LaSalle sets and discuss the case when the system under considera-tion has a weak Lyapunov function. The structures of the sets of weak-strong selectionsand control selections are also investigated, and a weak LaSalle’s invariance principle isestablished. As an application, we demonstrate how LaSalle’s invariance principles ofdiferential inclusions can be used to design feedback controls for nonsmooth dynami-cal systems by considering m-dimensional oscillators with frictions. Finally, the strongLaSalle’s invariance principle is applied to studying the asymptotic behavior of a classof nonsmooth Lagrange mechanical system.