Viscosity Solutions of Hamilton--Jacobi--Bellman--Isaacs Equations for Time-Delay Systems
Anton Romanovich Plaksin · SIAM Journal on Control and Optimization · 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.