Computation of non-monotonic Lyapunov functions for continuous-time systems

In this paper, we propose two methods to compute non-monotonic Lyapunov functions for continuous-time systems which are asymptotically stable. The first method is to solve a linear optimization problem on a compact and bounded set. The proposed linear programming based algorithm delivers a CPA1 non-monotonic Lyapunov function on a suitable triangulation covering the given compact and bounded set excluding a small neighbourhood of the equilibrium. It is shown that for every asymptotically stable system there exists a suitable triangulation such that the proposed algorithm terminates successfully. The second method is to verify a CPA function constructed based on the values of the norm of the state at all vertices of a suitable triangulation covering the given compact and bounded set is a non-monotonic Lyapunov function on the given set without a small neighbourhood of the equilibrium. It is further proved that if system is asymptotically stable then there exists a suitable triangulation such that the second way works. The comparison of the proposed two methods are discussed via three examples.