A dual to Lyapunov's stability theorem

Abstract

Lyapunov's second theorem is a standard tool for stability analysis of ordinary differential equations. Here we introduce a theorem which can be viewed as a dual to Lyapunov's result. From existence of a scalar function satisfying certain inequalities it follows that “almost all trajectories” of the system tend to zero. The scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories. If the stationary density is bounded everywhere except at a singularity in the origin, then almost all trajectories tend towards the origin. The weaker notion of stability allows for applications also in situations where Lyapunov's theorem cannot be used. Moreover, the new criterion has a striking convexity property related to control synthesis.