Numerical Computation of Control Lyapunov Functions in the Sense of Generalized Gradients

Abstract : The existence of a control Lyapunov function with the weak infinitesimal decrease via the Dini or the proximal subdifferential and the lower Hamiltonian characterizes asymp-totic controllability of nonlinear control systems and differential inclusions. We study the class of nonlinear differential inclusions with a right-hand side formed by the convex hull of active C 2 functions which are defined on subregions of the domain. For a simplicial triangulation we parametrize a control Lyapunov function (clf) for nonlinear control systems by a continuous, piecewise affine (CPA) function via its values at the nodes and demand a suitable negative upper bound in the weak decrease condition on all vertices of all simplices. Applying estimates of the proximal subdifferential via active gradients we can set up a mixed integer linear problem (MILP) with inequalities at the nodes of the triangulation which can be solved to obtain a CPA function. The computed function is a clf for the nonlinear control system. We compare this novel approach with the one applied to compute Lyapunov functions for strongly asymptotically stable differential inclusions and give a first numerical example.