On Time Extension in Differential Games with Impulse Controls
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ON TIME EXTENSION IN DIFFERENTIAL GAMES WITH IMPULSE CONTROLS
G. Ts. Chikrii
UDC 517.977
Abstract. In the development of ideas of B. N. Pshenichnyi, the paper considers a linear differential game of approach with impulse controls. A research technique is proposed, which is based on time extension and oriented to the case where the classical Pontryagin condition does not hold. Sufficient conditions for the finiteness of the guaranteed approach time are obtained. An illustrative example is given. Keywords: time extension function, differential game, impulse control, set-valued mapping, Minkowski’ geometric difference, Pontryagin’s condition. INTRODUCTION This paper is devoted to the cherished memory of my teacher, an outstanding mathematician and cybernetician, academician of the NAS of Ukraine, Boris Nikolaevich Pshenichnyi, who would have been 80 in April, 2017 [1, 2]. He is a founder of the Ukrainian school in necessary extremum conditions and theory of differential games. B. N. Pshenichnyi possessed highest analytical skills and exclusive intuition, he developed a number of efficient methods [3, 4], which made him world famous. Problems that Pshenichnyi formulated to his students then quite often became scientific directions. One of them is related to the effect of information delay and time extension principle, as well as their use to solve problems of approach with complete information that cannot be solved by direct application of well-known methods [5–11]. Within the framework of this direction, we will consider the linear game problem about bringing the trajectory of a conflict-controlled process to terminal set in case where it has cylindrical form [12] and opposing parties use impulse controls [13]. The problem becomes more complicated since the classical Pontryagin condition generally does not hold. Processes with impulse control occur, in particular, in space research when considering problems such as traveling salesman problem, related to target assignment and control of moving objects [14]. We propose a problem solution method based on time extension function, which allows obtaining sufficient conditions of game termination in a finite guaranteed time. To illustrate the results, we will use a model game example with the dynamics of simple pendulums. STATEMENT OF THE GAME PROBLEM OF APPROACH Let us consider a controlled system whose dynamics in the Euclidean space R n is described by the linear differential equation (1) z& = Az - u + u , z (0) = z 0 , where A is a quadratic matrix of order n and u = u( t ) and u = u ( t ) are vectors of control of the pursuer and evader, respectively. 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, 2017, pp. 58–66. Original article submitted April 12, 2017. 704
1060-0396/17/5305-0704 ©2017 Springer Science+Business Media New York
The game is considered from the standpoint of the pursuer whose purpose is, by choosing their
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