Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization

TL;DR

This paper develops a theoretical framework for analyzing projected dynamical systems on irregular, non-Euclidean domains, focusing on existence, uniqueness, and stability of solutions relevant to nonlinear optimization and control.

Contribution

It introduces Krasovskii solutions on nonconvex, low-regularity manifolds, establishes conditions for their existence and uniqueness, and extends stability analysis to these complex domains.

Findings

Krasovskii solutions exist on irregular, nonconvex sets.

Conditions for solution uniqueness are established.

Stability and convergence of projected gradient flows are analyzed.

Abstract

Continuous-time projected dynamical systems are an elementary class of discontinuous dynamical systems with trajectories that remain in a feasible domain by means of projecting outward-pointing vector fields. They are essential when modeling physical saturation in control systems, constraints of motion, as well as studying projection-based numerical optimization algorithms. Motivated by the emerging application of feedback-based continuous-time optimization schemes that rely on the physical system to enforce nonlinear hard constraints, we study the fundamental properties of these dynamics on general locally-Euclidean sets. Among others, we propose the use of Krasovskii solutions, show their existence on nonconvex, irregular subsets of low-regularity Riemannian manifolds, and investigate how they relate to conventional Carath\'eodory solutions. Furthermore, we establish conditions for uniqueness, thereby introducing a generalized definition of prox-regularity which is suitable for non-flat domains. Finally, we use these results to study the stability and convergence of projected gradient flows as an illustrative application of our framework. We provide simple counter-examples for our main results to illustrate the necessity of our already weak assumptions.