# American Institute of Mathematical Sciences

October  2015, 20(8): 2477-2495. doi: 10.3934/dcdsb.2015.20.2477

## 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, China 2 Lehrstuhl für Angewandte Mathematik, Universität Bayreuth, 95440 Bayreuth, Germany, Germany 3 School of Science and Engineering, Reykjavik University, Menntavegi 1, Reykjavik, IS-101 4 Fakultät für Informatik und Mathematik, Universität Passau, 94030 Passau, Germany

Received  June 2014 Revised  March 2015 Published  August 2015

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.
Citation: Huijuan Li, Robert Baier, Lars Grüne, Sigurdur F. Hafstein, Fabian R. Wirth. Computation of local ISS Lyapunov functions with low gains via linear programming. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2477-2495. doi: 10.3934/dcdsb.2015.20.2477
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