Lyapunov Functions and Duality for Convex Processes

The paper studies convex Lyapunov functions for differential and difference inclusions with right-hand sides given by convex processes, that is, by set-valued mappings the graphs of which are convex cones. Convex conjugacy between weak Lyapunov functions for such inclusions and Lyapunov functions for adjoint inclusions is established. Asymptotic stability concepts are compared and the existence of convex Lyapunov functions for classes of convex processes is shown. Relevance of the results for the study of asymptotic controllability or stabilizability and of detectability for linear control systems with conical control and state constraints is underlined.