VISCOSITY SOLUTIONS OF HAMILTON-JACOBI-BELLMAN-ISAACS EQUATIONS FOR TIME-DELAY SYSTEMS

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.

Automated summary

This paper addresses a zero-sum differential game for a dynamical system described by a nonlinear delay differential equation under a initial condition defined by a piecewise continuous function. It derives the corresponding Cauchy problem for Hamilton-Jacobi-Bellman-Isaacs equation with coinvariant derivatives, and considers the definition of a viscosity solution for this problem. It proves that the differential game has a unique viscosity solution value, and obtains an infinitesimal description of the viscosity solution based on notions of sub- and superdifferentials corresponding to coinvariant derivatives.