The paper deals with a Bolza optimal control problem for a dynamical system which motion is described by a delay differential equation under an initial condition defined by a piecewise continuous function. For the value functional in this problem, the Cauchy problem for the Hamilton-Jacobi-Bellman equation with coinvariant derivatives is considered. Minimax and viscosity solutions of this problem are studied. It is proved that both of these solutions exist, are unique and coincide with the value functional.

中文翻译：

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时滞系统的 Hamilton-Jacobi-Bellman 方程的极小极大值和粘度解

本文研究了动态系统的 Bolza 最优控制问题，该动态系统在分段连续函数定义的初始条件下用延迟微分方程描述运动。对于此问题中的值泛函，考虑了具有协变导数的 Hamilton-Jacobi-Bellman 方程的 Cauchy 问题。研究了这个问题的极大极小解和粘度解。证明这两个解都存在，是唯一的，并且与值泛函一致。