The Value and Optimal Strategies in a Positional Differential Game for a Neutral-Type System

• Published in: Proceedings of the Steklov Institute of Mathematics Vol. 327; no. Suppl 1; pp. S112 - S123
• Main Authors: Gomoyunov, M. I., Lukoyanov, N. Yu
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Moscow Pleiades Publishing 01.12.2024; Springer Nature B.V
• Subjects: Closed loops; Differential equations; Differential games; Dynamical systems; Feedback control; Hamilton-Jacobi equation; Mathematics; Mathematics and Statistics; Minimax technique; Saddle points; game value; optimal strategies; neutral-type equation; path-dependent Hamilton–Jacobi equation; coinvariant derivatives; minimax solution; differential game
• ISSN: 0081-5438; 1531-8605
• DOI: 10.1134/S0081543824070083

On a finite time interval, a differential game for the minimax–maximin of a given cost functional is considered. In this game, the motion of a conflict-controlled dynamical system is described by functional differential equations of neutral type in Hale’s form. Under assumptions more general than those considered previously, a theorem on the existence of the value and saddle point of the game in classes of players’ closed-loop control strategies with memory of the motion history is proved. The proof involves the technique of the corresponding path-dependent Hamilton–Jacobi equations with coinvariant derivatives and the theory of minimax (generalized) solutions of such equations. In order to construct optimal strategies, which constitute a saddle point of the game, a recent result on the existence and uniqueness of a suitable minimax solution and a special Lyapunov–Krasovskii functional are used.