Variational Control Principle Based on Free Energy Minimization

Abstract

This paper introduces a variational control principle grounded in the minimization of free energy on Riemannian manifolds. Unlike classical approaches that treat stability, constraints, and adaptation as separate problems, the proposed framework integrates them into a single variational principle. Control laws, including adaptive parameter updates, are derived as natural gradient flows. This geometric structure inherently enforces hard constraints via barrier potentials, eliminating the need for discontinuous projection operators, while an entropy term provides regularization and robustness. Rigorous analysis establishes the passivity and input-to-state stability of the resulting closed-loop systems. The proposed framework seamlessly bridges the gap between adaptive control and direct state-feedback design. Numerical examples illustrate the method’s geometric anti-windup properties, robustness to disturbances, and the explicit role of entropic smoothing in improving numerical conditioning.