Almost sure properties of controlled diffusions and worst case properties of deterministic systems

Dipartimento di Matematica P. e A., Università di Padova, via Trieste 63, 35131 Padova, Italy; bardi@math.unipd.it; acesar@math.unipd.it

Received: 11 January 2006
Revised: 26 October 2006

Abstract

We compare a general controlled diffusion process with a deterministic system where a second controller drives the disturbance against the first controller. We show that the two models are equivalent with respect to two properties: the viability (or controlled invariance, or weak invariance) of closed smooth sets, and the existence of a smooth control Lyapunov function ensuring the stabilizability of the system at an equilibrium.