Numerical computation of control Lyapunov functions in the sense of generalized gradients

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Baier, Robert ; Hafstein, Sigurdur Freyr:
Numerical computation of control Lyapunov functions in the sense of generalized gradients.
In: MTNS 2014 : Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems. - Groningen : University of Groningen Press , 2014 . - S. 1173-1180
ISBN 978-90-367-6321-9

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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 asymptotic 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 C² [$C^2$] 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.