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Minimization with differential inequality constraints applied to complete Lyapunov functions
Author(s) -
Peter Giesl,
Carlos Argáez,
Sigurður Hafstein,
Holger Wendland
Publication year - 2021
Publication title -
mathematics of computation
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.95
H-Index - 103
eISSN - 1088-6842
pISSN - 0025-5718
DOI - 10.1090/mcom/3629
Subject(s) - mathematics , lyapunov function , lyapunov redesign , discretization , control lyapunov function , lyapunov equation , lyapunov optimization , dynamical systems theory , mathematical optimization , nonlinear system , mathematical analysis , physics , quantum mechanics
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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