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**Computation of local ISS Lyapunov functions with low gains via linear programming.**
*(English)*
Zbl 1366.37146

Summary: 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.

### MSC:

37M99 | Approximation methods and numerical treatment of dynamical systems |

93D09 | Robust stability |

93D30 | Lyapunov and storage functions |

90C05 | Linear programming |

93D25 | Input-output approaches in control theory |

90C90 | Applications of mathematical programming |

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\textit{H. Li} et al., Discrete Contin. Dyn. Syst., Ser. B 20, No. 8, 2477--2495 (2015; Zbl 1366.37146)

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