Local Lyapunov Functions for periodic and finite-time ODEs

Lyapunov functions for general systems are difficult to construct. However, for autonomous linear systems with exponentially stable equilibrium, there is a classical way to construct a global Lyapunov function by solving a matrix equation. Consequently, the same function is a local Lyapunov function for a nonlinear system. In this paper, we generalise these results to time-periodic and, in particular, finitetime systems with an exponentially attractive zero solution. We show the existence of local Lyapunov functions for nonlinear systems. For finite-time systems, we consider a generalised notion of a Lyapunov function, which is not necessarily continuously differentiable, but just locally Lipschitz continuous; the derivative is then replaced by the Dini derivative. Dedicated to Jürgen Scheurle on the occasion of his 60th birthday.