On the problem of guidance of an autonomous conflict-controlled system onto a cylinder set
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ON THE PROBLEM OF GUIDANCE OF AN AUTONOMOUS CONFLICT-CONTROLLED SYSTEM ONTO A CYLINDER SET1
UDC 517.977.8
Yu. V. Averboukh
The problem of guidance of a conflict-controlled system onto a cylinder set is considered. This problem is compared with the problem of guidance of a transformed system onto the base of the cylinder at the last moment. It is shown that the problems considered are equivalent. Moreover, the sequences constructed by the programmed iteration method coincide for both problems. Keywords: differential game, maximal stable bridge, “at a moment” problem, programmed iteration method. INTRODUCTION This work is devoted to the investigation of the structure of the solution of a differential guidance game in the case when the goal set is of the form of a cylinder in the space of positions. Problems of this form are sometimes called “at a moment” guidance problems. The structure of a nonlinear guidance game problem is exhaustively characterized by the theorem on the alternative proved by N. N. Krasovskii and A. I. Subbotin [1, 2]; it asserts the existence of a saddle point in the class of corresponding positional strategies under conditions of information consistency. In the case when the condition of information consistency is not satisfied, a saddle point exists in the class of pairs “counterstrategy-strategy” [2] (the existence of a saddle point in pairs “strategy-counterstrategy” and “mixed strategy-mixed strategy” [2] is also established). The following form of an optimal strategy in the guidance problem [2] is well-known: a strategy (or a counterstrategy in the case when the condition of information consistency is not satisfied) is constructed by the method of extremal shift to some set; this set is a maximal u-stable bridge in the sense of N. N. Krasovskii. Thus, the solution of the guidance game problem is reduced to the problem of construction of the maximal u-stable bridge. If the problem is considered in the classes “mixed strategy-mixed strategy” and “strategy-counterstrategy,” then the corresponding optimal control can be obtained by the ~ method of extremal shift to the maximal u-stable and u * -stable bridge, respectively [2]. Along with guidance problems, game problems of minimization of a functional are considered in the theory of differential games. For these problems, based on the theorem on the alternative, N. N. Krasovskii and A. I. Subbotin established the existence of a function of price [2]. A concrete positional absorption set or a function of price can be constructed on the basis of the programmed iteration method proposed by A. G. Chentsov [3–6] (see also [7–9]). The following two important classes are often selected from the set of problems considered in the theory of differential games: “at the moment” problems and “at a moment” problems. A guidance game problem is called an “at the moment” problem if a system should approach a set in the phase space at the last moment and an “at a moment” problem if the system should approach a set in the phase space at any moment. In the latt
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