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 and 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 nonflat 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 counterexamples for our main results to illustrate the necessity of our already weak assumptions.


  1. differential inclusions
  2. optimization
  3. manifolds

MSC codes

  1. 34A60
  2. 34A12
  3. 49J24
  4. 49K24

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Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 635 - 668
ISSN (online): 1095-7138


Submitted: 28 November 2018
Accepted: 27 October 2020
Published online: 22 February 2021


  1. differential inclusions
  2. optimization
  3. manifolds

MSC codes

  1. 34A60
  2. 34A12
  3. 49J24
  4. 49K24



Funding Information

ETH Zürich Foundation https://doi.org/10.13039/501100012652

Funding Information

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung https://doi.org/10.13039/501100001711 : 160573

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