Game approach problems for dynamic processes with impulse controls
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GAME APPROACH PROBLEMS FOR DYNAMIC PROCESSES WITH IMPULSE CONTROLS A. N. Khimicha† and K. A. Chikriia‡
UDC 518.9
This paper deals with game problems with impulse and geometric constraints on the players’ controls. To analyze conflict-controlled processes with discontinuous trajectories, the method of resolving functions is used. This makes it possible to derive sufficient conditions of the game termination in a finite guaranteed time. Keywords: impulsive system, Dirac delta function, conflict-controlled process, multivalued mapping, Minkowski functional, Aumann integral, Pontryagin condition, Cauchy formula. INTRODUCTION Discrete-continuous or hybrid systems have appeared in scientific practice due to the development of the methods of digital automatic control in continuous systems. Such processes usually have continuous and discrete (impulsive) parts, whose concurrent operation generates new properties of the system. Elements operating in impulsive mode significantly change characteristics of the system. Impulse control, as an action, causes an instantaneous change of the state of the system and, therefore, a discontinuity of its trajectory as a function of time. It appeared natural in mathematical models to replace these rapid changes by abrupt ones. A natural extension of purely impulsive controls is measure controls [28], and differential equations with measures [4] are an adequate mathematical tool to describe the corresponding trajectories. Impulsive processes are analyzed, for example, in [4–6, 22, 24, 29–31], and various applications of impulsive controlled systems to flight dynamics, economic analysis, data processing, queuing systems, etc. are described in extensive literature (see, e.g., [25, 27]). The present paper studies game dynamic problems for processes with impulse controls. The study is based on the principle of a guaranteed result, which is implemented by the method of resolving functions [8–10, 12, 13]. This method is closely related to Minkowski functionals [15, 21] and requires inverse mappings, so-called inverse Minkowski functionals, to be introduced [8, 26]. The essence of the method is to use known parameters to generate a conflict-controlled process of numerical functions related to the above-mentioned functionals and integrally characterizing how the trajectory is close to the terminal set. Note that the theory of dynamic games includes alternative highly-functional methods for the optimization of conflict-controlled processes [1–3, 7, 11, 16–18]. An advantage of the method of resolving functions [8] is that it provides a complete substantiation of the classical parallel-approach rule and allows efficient use of the modern technique of multivalued mappings and their selectors [14, 15, 20, 21, 23] to substantiate game structures and obtain substantial results on their basis. The present paper sequentially considers three cases of impulse control of the pursuer, evader, and both players. Using generalized functions (Dirac delta functions in our case) as controls of one of the players ma
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