Feedback stabilization methods for the numerical solution of ordinary differential equations

Iasson Karafyllis; Lars Grüne

doi:10.3934/dcdsb.2011.16.283

Article Contents

2011, Volume 16, Issue 1: 283-317. Doi: 10.3934/dcdsb.2011.16.283

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Feedback stabilization methods for the numerical solution of ordinary differential equations

Iasson Karafyllis^1, and
Lars Grüne^2,

1.
Department of Environmental Engineering, Technical University of Crete, 73100 Chania
2.
Mathematisches Institute, Universität Bayreuth, 95440 Bayreuth

Received: April 30, 2010

Revised: July 31, 2010

Published: June 2011

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Abstract

In this work we study the problem of step size selection for numerical schemes, which guarantees that the numerical solution presents the same qualitative behavior as the original system of ordinary differential equations. We apply tools from nonlinear control theory, specifically Lyapunov function and small-gain based feedback stabilization methods for systems with a globally asymptotically stable equilibrium point. Proceeding this way, we derive conditions under which the step size selection problem is solvable (including a nonlinear generalization of the well-known A-stability property for the implicit Euler scheme) as well as step size selection strategies for several applications.

Keywords:

Asymptotic stability of numerical approximations,
feedback stabilization,
nonlinear differential equations.

Mathematics Subject Classification: Primary: 65L07, 34D20; Secondary: 65L06, 93D15.

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