CPA Lyapunov functions for time-periodic systems in one spatial dimension

Continuous piecewise affine (CPA) functions have been successfully used to construct Lyapunov functions for autonomous ordinary differential equations with an exponentially stable equilibrium. However, for time-periodic systems this method cannot be used. In this paper, we develop a method to compute a Lyapunov function for a time-periodic ordinary differential equation with a (known) exponentially stable periodic solution. While the usual approach with a triangulation works away from the periodic solution, a new approach is required in a neighborhood of the periodic solution. We derive sufficient conditions for a Lyapunov function and, moreover, we prove a converse theorem, showing that these conditions can always be satisfied if the triangulation is sufficiently fine. Finally, we formulate the construction problem as a linear programming problem and apply it to two examples. This paper deals with the case of one spatial dimension, but we believe that the method can be generalised to any dimension, which will be done in a subsequent paper.