Computation of local ISS Lyapunov functions with low gains via linear programming

Huijuan  Li; Robert  Baier; Lars  Grüne; Sigurdur F. Hafstein; Fabian R.  Wirth

doi:10.3934/dcdsb.2015.20.2477

Article Contents

2015, Volume 20, Issue 8: 2477-2495. Doi: 10.3934/dcdsb.2015.20.2477

This issue Previous Article Grid refinement in the construction of Lyapunov functions using radial basis functions Next Article Separable Lyapunov functions for monotone systems: Constructions and limitations

Computation of local ISS Lyapunov functions with low gains via linear programming

1.
School of Mathematics and Physics, Chinese University of Geosciences (Wuhan), 430074, Wuhan
2.
Lehrstuhl für Angewandte Mathematik, Universität Bayreuth, 95440 Bayreuth
3.
School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101
4.
Fakultät für Informatik und Mathematik, Universität Passau, 94030 Passau

Received: June 2014

Revised: March 2015

Published: October 2015

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Abstract

In this paper, we present a numerical algorithm for computing ISS Lyapunov functions for continuous-time systems which are input-to-state stable (ISS) on compact subsets of the state space. The algorithm relies on a linear programming problem and computes a continuous piecewise affine ISS Lyapunov function on a simplicial grid covering the given compact set excluding a small neighborhood of the origin. The objective of the linear programming problem is to minimize the gain. We show that for every ISS system with a locally Lipschitz right-hand side our algorithm is in principle able to deliver an ISS Lyapunov function. For $C^2$ right-hand sides a more efficient algorithm is proposed.

Keywords:

Mathematics Subject Classification: Primary: 37B25, 93D09, 93D30; Secondary: 34D20, 90C05.

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