The Problem of Approach of Controlled Objects in Dynamic Game Problems with a Terminal Payoff Function
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THE PROBLEM OF APPROACH OF CONTROLLED OBJECTS IN DYNAMIC GAME PROBLEMS WITH A TERMINAL PAYOFF FUNCTION
UDC 517.977
J. S. Rappoport
Abstract. To solve the problem of convergence of controlled objects in dynamic game problems with the terminal payoff function, the author proposes a method that systematically uses the Fenchel–Moreau ideas as applied to the general scheme of the method of resolving functions. The essence of the method is that the resolving function can be expressed in terms of the function conjugate to payoff function and, using the involution of the conjugation operator for a convex closed function, a guaranteed estimate of the terminal value of the payoff function is obtained, which can be presented in terms of the payoff value at the initial instant of time and integral of the resolving function. The concepts of upper and lower resolving functions of two types are introduced and sufficient conditions for a guaranteed result in a differential game with a terminal payoff function are obtained for the case where the Pontryagin condition does not hold. Two schemes of the method of resolving functions are considered, the corresponding control strategies are generated, and guaranteed times are compared. The results are illustrated by a model example. Keywords: terminal payoff function, quasilinear differential game, multi-valued mapping, measurable selector, stroboscopic strategy, resolving function. INTRODUCTION In the paper, we will consider the problem of approach of controlled objects in dynamic game problems with terminal payoff function on the basis of the method of resolving functions [1]. We will introduce the concepts of upper and lower resolving functions of two types and will obtain sufficient conditions for the guaranteed result in the differential game with terminal payoff function in the case where the Pontryagin condition is not satisfied. We will propose two schemes of the method of resolving functions, generate appropriate control strategies, and compare the guaranteed times. The results will be illustrated by a modeling example. The paper continues the studies from [1, 2], is related to the publications [3–22], extends the class of solvable game problems of approach of controlled objects, and reveals new capabilities of applying convex analysis to the theory of conflict-controlled processes. THE GENERAL SCHEME OF THE METHOD. RESOLVING FUNCTIONS OF THE FIRST TYPE Let us consider a conflict-controlled process whose evolution can be described by the equality t
z ( t ) = g ( t ) + ò W ( t, t) j ( u( t), u ( t)) dt, t ³ 0 .
(1)
0
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2020, pp. 157–173. Original article submitted February 18, 2020. 820
1060-0396/20/5605-0820 ©2020 Springer Science+Business Media, LLC
Here, z ( t ) Î R n , function g ( t ) , g : R + ® R n , is Lebesgue measurable [8] and is bounded for t > 0 , the matrix function
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