Lyapunov characterization of boundedness of reachability sets for infinite-dimensional systems

We prove a converse Lyapunov theorem for boundedness of reachability sets for a general class of control systems whose flow is Lipschitz continuous on compact intervals with respect to trajectory-dominated inputs. We show that this condition is satisfied by broad classes of semilinear evolution equations. For ordinary differential equations, as a consequence of our results, we obtain a converse Lyapunov theorem for forward completeness, without a priori restrictions on the magnitude of inputs. As an example, we rederive Wintner's classical result using our Lyapunov approach.