# CPA Lyapunov Functions: Switched Systems vs. Differential Inclusions

### Sigurdur Hafstein

#### Abstract

We present an algorithm that uses linear programming to parameterize continuous and piecewise affine Lyapunov functions for switched systems. The novel feature of the algorithm is, that it can compute Lyapunov functions for switched system with a strongly asymptotically stable equilibrium, for which the equilibrium of the corresponding differential inclusion is merely weakly asymptotically stable. For the differential inclusion no such Lyapunov function exists. This is achieved by removing constraints from a linear programming problem of an earlier algorithm to compute Lyapunov functions, that are not necessary to assert strong stability for the switched system. We demonstrate the benefits of this new algorithm sing Artsteinâ€™s circles as an example.

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