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

Open access
Date
2021Type
- Journal Article
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 Carathéodory 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. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000454287Publication status
publishedJournal / series
SIAM Journal on Control and OptimizationVolume
Pages / Article No.
Publisher
SIAMSubject
Discontinuous systems; Differential inclusions; projected gradient descentOrganisational unit
09478 - Dörfler, Florian / Dörfler, Florian
09481 - Hug, Gabriela / Hug, Gabriela
Funding
160573 - Plug-and-Play Control & Optimization in Microgrids (SNF)
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