A non-coercive Lyapunov framework for stability of distributed parameter systems

We show that the existence of a non-coercive Lyapunov function is sufficient for uniform global asymptotic stability (UGAS) of infinite-dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied. These additional assumptions cannot be dropped in general. However, for the special classes of systems as evolution equations in Banach spaces with Lipschitz continuous nonlinearities, delay systems, or linear switched systems on Banach spaces they become much simpler to check. Moreover, for time-delay systems we prove an alternative non-coercive Lyapunov theorem, which is close in spirit to the Lyapunov-Krasovskii method.