Feedback Stabilization Methods for the Numerical Solution of Systems of Ordinary Differential Equations

TL;DR

This paper introduces feedback stabilization techniques from nonlinear control theory to improve step size selection in numerical solutions of ODEs, ensuring qualitative behavior preservation and controlling discretization errors.

Contribution

It applies Lyapunov-based and Small-Gain feedback methods to numerical ODE schemes, a novel approach for stability and error control in this context.

Findings

Effective stabilization of numerical solutions with feedback methods

Preservation of qualitative behavior in ODE numerical schemes

Potential for improved global error control

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 exploited and numerous illustrating applications are presented for systems with a globally asymptotically stable equilibrium point. The obtained results can be used for the control of the global discretization error as well.