Regularity and stability of feedback relaxed controls
University of Oxford（牛津大学）
Christoph Reisinger, Yufei Zhang
备注：
Additional comments have been included, such that the importance of stable feedback controls for reinforcement learning. The manuscript will be published in SIAM Journal on Control and Optimization
This paper proposes a relaxed control regularization with general exploration rewards to design robust feedback controls for multi-dimensional continuous-time stochastic exit time problems. We establish that the regularized control problem admits a Hölder continuous feedback control, and demonstrate that both the value function and the feedback control of the regularized control problem are Lipschitz stable with respect to parameter perturbations. Moreover, we show that a pre-computed feedback relaxed control has a robust performance in a perturbed system, and derive a first-order sensitivity equation for both the value function and optimal feedback relaxed control. These stability results provide a theoretical justification for recent reinforcement learning heuristics that including an exploration reward in the optimization objective leads to more robust decision making. We finally prove first-order monotone convergence of the value functions for relaxed control problems with vanishing exploration parameters, which subsequently enables us to construct the pure exploitation strategy of the original control problem based on the feedback relaxed controls.
R-regularity of set-valued mappings under the relaxed constant positive linear dependence constraint qualification with applications to parametric and bilevel optimization
A Relaxed Quadratic-Program-based Framework for Trajectory Tracking of Unicycle Robots with Singularity Avoidance
Department of Mechanical；Industrial Engineering at the New Jersey Institute of Technology (NJIT), NJ, USA；School of Engineering, Infrastructure；Sustainability De Montfort University, Leicester, LE1 9BH, UK
Uniform stabilization of Boussinesq systems in critical $\mathbf{L}^q$-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls
我们考虑定义在一个充分光滑有界区域上的d维Boussinesq系统，具有齐次边界条件，并且受到外部源的影响，假设引起不稳定.流体方程和热方程的初始条件都是低正则性的。然后，我们寻求一致稳定这样的Boussinesq系统在一个不稳定的平衡对附近，在相应的低正则性空间的临界设置，通过显式构造，反馈控制，这是本地化的一个任意小的内部子域。此外，它们的数量将是最少的，并且具有减小的尺寸：更准确地说，它们将具有用于流体部件的尺寸$(d-1)$和用于热部件的尺寸$1$。所得到的适定性和稳定化的空间对于流体速度分量（接近于$\mathbf{L}^3(Ω$）对于$ d = 3 $）和对于热分量$ q > d $的空间$L^q(Ω$）是合适的、紧的Besov空间。因此，本文可以看作是\cite{LPT.1}的一个推广，在\cite{LPT.1}中，在相同的Besov条件下，通过对Navier-Stokes方程采用有限维反馈控制，获得了相同的内部局部一致镇定结果。
Deterministic Mean Field Games on Networks and Related Optimal Control Problems
Université de Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions（巴黎西岱大学与索邦大学，法国国家科学研究中心，雅克-路易·利翁斯实验室）；Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova（帕多瓦大学，“图利奥·莱维-奇维塔”数学系）；Univ Rennes, CNRS, IRMAR - UMR 6625（雷恩大学，法国国家科学研究中心，IRMAR（UMR 6625））
Impedance control of a cable-driven SEA with mixed $H_2/H_\infty$ synthesis
Institute of Robotics and Automatic Information Systems, Nankai University（机器人与自动信息系统研究所，南开大学）；Tianjin Key Laboratory of Intelligent Robotics, Nankai University（天津智能机器人重点实验室，南开大学）
