Feedback stabilization methods for the numerical solution of ordinary differential equations

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.