On the Theory of Positional Differential Games for Neutral-Type Systems
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the Theory of Positional Differential Games for Neutral-Type Systems N. Yu. Lukoyanov1,2,∗ , and A. R. Plaksin1,2,∗∗ Received April 16, 2019; revised May 14, 2019; accepted May 20, 2019
Abstract—For a dynamical system whose motion is described by neutral-type differential equations in Hale’s form, we consider a minimax–maximin differential game with a quality index evaluating the motion history realized up to the terminal time. The control actions of the players are subject to geometric constraints. The game is formalized in classes of pure positional strategies with a memory of the motion history. It is proved that the game has a value and a saddle point. The proof is based on the choice of an appropriate Lyapunov– Krasovskii functional for the construction of control strategies by the method of an extremal shift to accompanying points. Keywords: neutral-type systems, control theory, differential games.
DOI: 10.1134/S0081543820040100 INTRODUCTION This paper is devoted to the development of the theory of positional differential games [1–3] for systems of neutral type. We consider a zero-sum differential game in which the motion of a dynamical system is described by differential equations of neutral type in Hale’s form [4]. There are geometric constraints on the control actions of the players. The quality of the control process is estimated in terms of the motion history of the system that has formed by the terminal time. The game is formalized in classes of pure positional strategies within the approach of [1–3]. The result of the paper is a theorem on the existence of a value and a saddle point in the differential game under consideration. Issues of the existence of a value and optimal strategies in positional differential games for neutral-type systems were studied earlier in [5–8]. Linear neutral-type systems were considered in [8]. Differential games for nonlinear systems formalized in classes of control strategies with a guide were the subject of [5,7]. The result closest to the present paper was established in [6], where a differential game in classes of pure positional strategies was considered for nonlinear neutraltype systems of a fairly general form. However, because of a special proof technique based on the constructions of two guides [9, 10], additional considerable constraints were imposed on the game in [6]: it was required that the functional defining the quality index and the functional under the derivative on the left-hand side of the motion equations should satisfy the Lipschitz condition, and 1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia 2 Ural Federal University, Yekaterinburg, 620002 Russia e-mail: ∗ [email protected], ∗∗ [email protected]
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the Lipschitz constant for the latter functional should be less than 1. These constraints are removed in the present paper, whereas the form of the system is slightly less general as compared to [5–7] but still quite typical. This result was obtained u
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