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Minimization with differential inequality constraints applied to complete Lyapunov functions


Authors: Peter Giesl, Carlos Argáez, Sigurdur Hafstein and Holger Wendland
Journal: Math. Comp. 90 (2021), 2137-2160
MSC (2020): Primary 37M22, 37B35, 90C20; Secondary 35A23, 46N20
DOI: https://doi.org/10.1090/mcom/3629
Published electronically: March 17, 2021
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Abstract:

Motivated by the desire to compute complete Lyapunov functions for nonlinear dynamical systems, we develop a general theory of discretizing a certain type of continuous minimization problems with differential inequality constraints. The resulting discretized problems are quadratic optimization problems, for which there exist efficient solution algorithms, and we show that their unique solutions converge strongly in appropriate Sobolev spaces to the unique solution of the original continuous problem. We develop the theory and present examples of our approach, where we compute complete Lyapunov functions for nonlinear dynamical systems.

A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is divided into the chain-recurrent set, where the complete Lyapunov function is constant along solutions, and the part characterizing the gradient-like flow, where the complete Lyapunov function is strictly decreasing along solutions. We propose a new method to compute a complete Lyapunov function as the solution of a quadratic minimization problem, for which no information about the chain-recurrent set is required. The solutions to the discretized problems, which can be solved using quadratic programming, converge to the complete Lyapunov function.


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Additional Information

Peter Giesl
Affiliation: Department of Mathematics, University of Sussex, Falmer, BN1 9QH, United Kingdom
MR Author ID: 725648
Email: p.a.giesl@sussex.ac.uk

Carlos Argáez
Affiliation: The Science Institute, University of Iceland, Dunhagi 5, 107 Reykjavik, Iceland
ORCID: 0000-0002-0455-8015
Email: carlos@hi.is

Sigurdur Hafstein
Affiliation: The Science Institute, University of Iceland, Dunhagi 5, 107 Reykjavik, Iceland
MR Author ID: 697182
ORCID: 0000-0003-0073-2765
Email: shafstein@hi.is

Holger Wendland
Affiliation: Chair of Applied and Numerical Analysis, Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany
MR Author ID: 602098
Email: holger.wendland@uni-bayreuth.de

Keywords: Dynamical system, differential inequality, complete Lyapunov function, quadratic programming, meshless collocation, convergence
Received by editor(s): May 22, 2020
Received by editor(s) in revised form: December 4, 2020
Published electronically: March 17, 2021
Additional Notes: The research in this paper was supported by the Icelandic Research Fund (Rannís) grant number 163074-052, Complete Lyapunov functions: Efficient numerical computation.
Article copyright: © Copyright 2021 American Mathematical Society