论文标题
在莱因哈特（Reinhardt）的猜想和最佳控制的正式基础上
On the Reinhardt Conjecture and Formal Foundations of Optimal Control
论文作者
论文摘要
我们描述了1934年雷纳特在平面中凸形域的凸堆积上的重新制定（以下是Hales（2017）），这是最佳控制理论中的问题。提出了该问题的几个结构性结果，包括其哈密顿结构和宽松的形式主义。提出了用于恒定控制的该问题的一般解决方案，并用于证明控制问题的极端物质被约束在状态空间的紧凑型域中。我们进一步描述了其奇异基因座附近的控制问题的结构，并证明我们在这种情况下恢复了多维富富最佳控制问题的泛素蛋白系统（具有二维控制）。我们展示了该系统如何在控制集是2个简单的限制磁盘的情况下，在有限的时间内执行无限数量的旋转旋转时，该系统如何允许对数螺旋轨迹。我们还描述了基础最佳控制中的形式化项目，即基于模型和无模型的强化学习理论。考虑了这些形式化的新颖的关键成分，即，考虑了纪念单元和收缩共同诱导，并讨论了一些应用。
We describe a reformulation (following Hales (2017)) of a 1934 conjecture of Reinhardt on pessimal packings of convex domains in the plane as a problem in optimal control theory. Several structural results of this problem including its Hamiltonian structure and Lax pair formalism are presented. General solutions of this problem for constant control are presented and are used to prove that the Pontryagin extremals of the control problem are constrained to lie in a compact domain of the state space. We further describe the structure of the control problem near its singular locus, and prove that we recover the Pontryagin system of the multi-dimensional Fuller optimal control problem (with two dimensional control) in this case. We show how this system admits logarithmic spiral trajectories when the control set is the circumscribing disk of the 2-simplex with the associated control performing an infinite number of rotations on the boundary of the disk in finite time. We also describe formalization projects in foundational optimal control viz., model-based and model-free Reinforcement Learning theory. Key ingredients which make these formalization novel viz., the Giry monad and contraction coinduction are considered and some applications are discussed.
