Computation of Lyapunov functions for nonautonomous systems on finite time-intervals by linear programming

We present an algorithm for numerically computing Lyapunov functions for nonautonomous systems on finite time-intervals. The algorithm relies on a linear optimization problem and delivers a continuous and piecewise affine function on a compact set. The level-sets of such a Lyapunov function give concrete bounds on the time-evolution of the system on the time-interval and for time-periodic systems they deliver an ultimate bound on solutions. Four examples of computed finite-time Lyapunov functions are given.