Contractivity of Multi-Stage Runge-Kutta Dynamics

TL;DR

This paper provides conditions under which multi-stage Runge-Kutta methods preserve strong contractivity when discretizing continuous-time systems, ensuring stability and robustness in control, optimization, and learning algorithms.

Contribution

It establishes new criteria for preserving strong contractivity of Runge-Kutta methods, extending classical results to multiple norms and implicit schemes, with a novel analysis of well-definedness via auxiliary systems.

Findings

Explicit methods' preservation depends on Lipschitz bounds of stage mappings.

Implicit methods' preservation depends on algebraic coefficient conditions.

Auxiliary system analysis ensures implicit method solvability.

Abstract

Many control, optimization, and learning algorithms rely on discretizations of continuous-time contracting systems, where preservation of contractivity under numerical integration is key for stability, robustness, and reliable fixed-point computation. In this paper, we establish conditions under which multi-stage Runge-Kutta methods preserve strong contractivity when discretizing infinitesimally contractive continuous-time systems. For explicit Runge-Kutta methods, preservation conditions are derived by bounding Lipschitz constants of the associated composite stage mappings, leading to coefficient-dependent criteria. For implicit methods, the algebraic structure of the stage equations enables explicit conditions on the Runge-Kutta coefficients that guarantee preservation of strong contractivity. In the implicit case, these results extend classical guarantees, typically limited to weak contractivity in the Euclidean metric, to strong contractivity with respect to the $\ell_1$-, $\ell_2$-, and $\ell_\infty$-norms. In addition, we study well-definedness of implicit methods through an auxiliary continuous-time system associated with the stage equations. We show that strong infinitesimal contractivity of this auxiliary system is sufficient to guarantee unique solvability of the stage equations. This analysis generalizes standard well-definedness conditions and provides a dynamic implementation approach that avoids direct solution of the implicit algebraic equations.