In the paper, we consider a path-dependent Hamilton-Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We study generalized solutions of the considered Hamilton-Jacobi equation both in the minimax and in the viscosity sense. A minimax solution is defined as a functional which epigraph and subgraph satisfy certain conditions of weak invariance, while a viscosity solution is defined in terms of a pair of inequalities for coinvariant sub- and super-gradients. We prove that these two notions are equivalent, which is the main result of the paper. As a corollary, we obtain comparison and uniqueness results for viscosity solutions of a Cauchy problem for the considered Hamilton-Jacobi equation and a right-end boundary condition. The proof is based on a certain property of the coinvariant subdifferential. To establish this property, we develop a technique going back to the proofs of multidirectional mean-value inequalities. In particular, the absence of the local compactness property of the underlying continuous function space is overcome by using Borwein-Preiss variational principle with an appropriate guage-type functional.
Minimax solutions of path-dependent Hamilton--Jacobi equations under weakened assumptions with application to differential games
Krasovskii Institute of Mathematics and Mechanics Ural Branch of Russian Academy of Sciences（俄罗斯科学院乌拉尔分支克拉索夫斯基数学力学研究所）；Ural Federal University（乌拉尔联邦大学）
Viscosity and minimax solutions for path-dependent Hamilton-Jacobi equations in infinite dimensions and related differential games
University of Michigan（密歇根大学）；N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences（俄罗斯科学院乌拉尔分校 N.N.克拉索夫斯基数学力学研究所）；Ural Federal University（乌拉尔联邦大学）；University of Central Florida（中佛罗里达大学）
Policy iteration for Hamilton-Jacobi-Isaacs equations with control constraints and comparison with Hamilton-Jacobi-Bellman equations
Rajiv Gandhi Institute of Petroleum Technology（拉吉夫·甘地石油技术学院）；University of Graz（格拉茨大学）；Radon Institute for Computational and Applied Mathematics (RICAM)（雷登计算与应用数学研究所）
