Lyapunov functions and duality for convex processes (Q2862472)

This paper concerns convex processes, i.e. differential inclusions (resp. difference inclusions) \(\dot{x} \in F(x)\) (resp. \(x^+ \in F(x)\)), whose graphs are convex cones. The author studies the relationship between the stability properties of these processes and the stability properties of the adjoint differential inclusions (resp. adjoint difference inclusions) \(\dot{y} \in F^*(y)\) (resp. \(y^+ \in F^*(y)\)). Convex conjugacy between Lyapunov functions for such inclusions and Lyapunov functions for adjoint inclusions is established. It is shown that convex Lyapunov functions and their convex conjugates can be used to deduce the implication between certain asymptotic stability properties of a convex process and other asymptotic stability properties of the adjoint process. Conditions ensuring the existence of convex Lyapunov functions for some convex processes are also given.