SIAM Journal on Control and Optimization, Volume 59, Issue 3, Page 1951-1972, January 2021.
The paper deals with a zero-sum differential game for a dynamical system which motion is described by a nonlinear delay differential equation under an initial condition defined by a piecewise continuous function. The corresponding Cauchy problem for Hamilton--Jacobi--Bellman--Isaacs equation with coinvariant derivatives is derived, and the definition of a viscosity solution of this problem is considered. It is proved that the differential game has a value that is the unique viscosity solution. Moreover, based on notions of sub- and superdifferentials corresponding to coinvariant derivatives, the infinitesimal description of the viscosity solution is obtained. An example of applying these results is given.

中文翻译：

---

时滞系统的 Hamilton--Jacobi--Bellman--Isaacs 方程的粘度解

SIAM Journal on Control and Optimization，第 59 卷，第 3 期，第 1951-1972 页，2021
年1 月。该论文涉及动态系统的零和微分游戏，该动态系统的运动由在定义的初始条件下的非线性延迟微分方程描述由分段连续函数。导出了具有协变导数的Hamilton--Jacobi--Bellman--Isaacs方程对应的柯西问题，并考虑了该问题的粘度解的定义。证明了微分博弈具有唯一的粘度解的价值。此外，基于对应于共变导数的亚微分和超微分的概念，获得了粘度解的无穷小描述。给出了应用这些结果的示例。