A dual to Lyapunov's stability theorem

Lyapunov's second theorem is a standard tool for stability analysis of ordi nary differential equations. Here we introduce a theorem which can be viewe d as a dual to Lyapunov's result. From existence of a scalar function satis fying 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 th e state space and flows along the system trajectories. If the stationary de nsity is bounded everywhere except at a singularity in the origin, then alm ost all trajectories tend towards the origin. The weaker notion of stabilit y allows for applications also in situations where Lyapunov's theorem canno t be used. Moreover, the new criterion has a striking convexity property re lated to control synthesis. (C) 2001 Elsevier Science B.V. All rights reser ved.