Feedback Integrators: Non-Asymptotic Invariance for One-Step Methods and Gain Selection under Euler Discretization

Abstract

For dynamical systems evolving on a manifold and admitting first integrals, standard one-step numerical methods generally cause the discrete trajectory to drift off the manifold and the numerical values of the first integrals to deviate from their prescribed values. Feedback integrators address this by extending the dynamics to an ambient Euclidean space and adding a feedback term that drives the numerical trajectory toward the set satisfying both the manifold constraint and the prescribed values of the first integrals. Existing theory, however, has two limitations: it remains asymptotic, guaranteeing only eventual entrance into an attractor containing the desired set, and it does not explain how the feedback gain should be chosen. In this paper, we first close the former gap for general one-step methods by proving positive invariance of arbitrarily small sublevel neighborhoods of the feedback Lyapunov function for sufficiently small step sizes. We then specialize to Euler discretization and analyze how the feedback gain enters the Taylor-based error bound. In this setting, we characterize a range of scaled gains that guarantee positive invariance for sufficiently small step sizes and identify the scaling that minimizes the Taylor-based upper bound. We further propose adaptive gain-selection rules under Euler discretization, including both stepwise and periodically updated variants, and establish corresponding boundedness guarantees for the resulting discrete trajectories. These results identify Euler discretization as the first setting in which gain selection for feedback integrators closes in explicit form, whereas extensions to general higher-order one-step methods remain genuinely method-dependent. Numerical experiments on free rigid body motion in $\operatorname{SO}(3)$, the Kepler problem, and a perturbed Kepler problem with rotational symmetry support the analysis.