Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 2477 - 2495, Volume 20, Issue 8, October 2015      doi:10.3934/dcdsb.2015.20.2477

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Huijuan Li - School of Mathematics and Physics, Chinese University of Geosciences (Wuhan), 430074, Wuhan, China (email)
Robert Baier - Lehrstuhl für Angewandte Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany (email)
Lars Grüne - Lehrstuhl für Angewandte Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany (email)
Sigurdur F. Hafstein - School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101, Iceland (email)
Fabian R. Wirth - Fakultät für Informatik und Mathematik, Universität Passau, 94030 Passau, Germany (email)

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:  Nonlinear systems, local input-to-state stability, local ISS Lyapunov function, robust Lyapunov function, linear programming.
Mathematics Subject Classification:  Primary: 37B25, 93D09, 93D30; Secondary: 34D20, 90C05.

Received: June 2014;      Revised: March 2015;      Available Online: August 2015.