Inequalities for Subgradients of a Value Functional in Differential Games for Time-Delay Systems

Corresponding Member of the RAS N. Yu. Lukoyanov^a^,^b^,* and A. R. Plaksin^a^,^b^,**

^aKrasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620990 Russia

^bUral Federal University, Yekaterinburg, 620002 Russia

Correspondence to: *e-mail: nyul@imm.uran.ru
Correspondence to: **e-mail: a.r.plaksin@gmail.com

Received 14 June, 2019

Abstract—A two-person zero-sum positional differential game is considered in which the motion of a dynamical system is described by a nonlinear delay equation and the initial motion history is determined by a piecewise continuous function. Inequalities for directional derivatives of the value functional and inequalities for its sub- and supergradients are obtained.

Keywords: differential games, time-delay systems, Hamilton–Jacobi equations, invariant derivatives, subdifferentials, minimax solutions, viscosity solutions

DOI: 10.1134/S1064562420010238