Feedback Stabilization Methods for the Numerical Solution of Systems of Ordinary Differential Equations
For numerical analysts, provides a control-theoretic framework for step size control to ensure qualitative fidelity, but is incremental as it applies existing methods to a known problem.
This work addresses step size selection for numerical ODE solvers to preserve qualitative behavior, using Lyapunov and Small-Gain feedback stabilization methods. Results enable control of global discretization error for systems with globally asymptotically stable equilibria.
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.