Feedback stabilization methods for the numerical solution of ordinary differential equations

Titelangaben

Karafyllis, Iasson ; Grüne, Lars:
Feedback stabilization methods for the numerical solution of ordinary differential equations.
In: Discrete and Continuous Dynamical Systems. Series B. Bd. 16 (2011) Heft 1 . - S. 283-317.
ISSN 1531-3492
DOI: https://doi.org/10.3934/dcdsb.2011.16.283

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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, by means of tools from nonlinear control theory. Lyapunov-based and Small-Gain feedback stabilization methods are presented 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.