Non-monotonic Lyapunov functions for stability of nonlinear and switched systems: Theory and computation

Lyapunov’s direct method, which is based on the existence of a scalar function of the state that decreases monotonically along trajectories, still serves as the primary tool for establishing stability of nonlinear systems. Since the main challenge in stability analysis based on Lyapunov theory is always to find a suitable Lyapunov function, weakening the requirements of the Lyapunov function is of great interest. In this thesis, we relax the monotonicity requirement of Lyapunov’s theorem to enlarge the class of functions that can provide certificates of stability. Both the discrete time case and the continuous time case are covered. Throughout the thesis, special attention is given to techniques from convex optimization that allow for computationally tractable ways of searching for Lyapunov functions. Our theoretical contributions are therefore amenable to convex programming formulations. In the discrete time case, we propose two new sufficient conditions for globa