Relaxation of Hypotheses in LaSalle–Krasovskii-Type Invariance Results

The usual invariance results for asymptotic stability for continuous autonomous finite-dimensional dynamical systems involve a positive definite Lyapunov function whose time derivative along the system motions is negative semidefinite (along with certain invariance conditions). This is equivalent to requiring that along the system motions, the Lyapunov function is nonincreasing at all times with increasing time. In this paper we establish an invariance result for asymptotic stability for continuous and discontinuous nonautonomous finite-dimensional dynamical systems, which requires a positive definite Lyapunov function which when evaluated along the system motions is nonincreasing only on certain unbounded discrete time sets $E$ with increasing time. This allows that between the time instants determined by $E$, the Lyapunov function may increase (i.e., over some finite time intervals, the system may exhibit unstable behavior). We also show that the usual invariance theorem for asymptotic stability reduces to the invariance theorem for continuous dynamical systems presented herein. In addition, we establish a variant to the above result involving estimates of the asymptotic behavior of the system's motions. We apply our results to four examples. One of these involves the stabilization of conservative mechanical systems using energy dissipation intermittently.