SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 635-668, January 2021.
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.

中文翻译：

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不规则，非欧氏域上的投影动力系统用于非线性优化

SIAM控制与优化杂志，第59卷，第1期，第635-668页，2021年1月。
连续时间投影动力系统是不连续动力系统的基本类别，其轨迹通过投影​​向外的矢量场而保持在可行域内。在对控制系统中的物理饱和度和运动约束建模以及研究基于投影的数值优化算法时，它们是必不可少的。受基于物理系统来强制执行非线性硬约束的基于反馈的连续时间优化方案的新兴应用的推动，我们研究了这些动力学在一般局部欧几里得集上的基本性质。除其他外，我们建议使用Krasovskii解，显示它们在低正则黎曼流形的非凸，不规则子集上的存在，并研究它们与常规Carathéodory解的关系。此外，我们建立了唯一性的条件，从而引入了适合非平坦域的近似正则性的广义定义。最后，我们使用这些结果来研究投影梯度流的稳定性和收敛性，以此作为我们框架的说明性应用。我们为主要结果提供了简单的反例，以说明我们本已虚弱的假设的必要性。