On a Differential Game in a Stochastic System
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a Differential Game in a Stochastic System L. A. Vlasenko1,∗ , A. G. Rutkas1,∗∗ , and A. A. Chikrii2,∗∗∗ Received April 5, 2019; revised May 15, 2019; accepted May 20, 2019
Abstract—We study the game problem of approach for a system whose dynamics is described by a stochastic differential equation in a Hilbert space. The main assumption on the equation is that the operator multiplying the system state generates a strongly continuous semigroup (a semigroup of class C0 ). Solutions of the equation are represented by a stochastic variation of constants formula. Using constraints on the support functionals of sets defined by the behavior of the pursuer and the evader, we obtain conditions for the approach of the system state to a cylindrical terminal set. The results are illustrated with a model example of a simple motion in a Hilbert space with random perturbations. Applications to distributed systems described by stochastic partial differential equations are considered. By taking into account a random external influence, we consider the heat propagation process with controlled distributed heat sources and sinks. Keywords: differential game, stochastic differential equation, Wiener process, generator of a strongly continuous semigroup, set-valued mapping, support functional, resolving functional, stochastic partial differential equation.
DOI: 10.1134/S0081543820040203 INTRODUCTION The Ural research school in the field of mathematical control theory and the theory of dynamic games was founded by Nikolai Nikolaevich Krasovskii and is a major research center well-known among specialists. Krasovskii’s works [1–5] and the fundamental methods he had developed became the basis for further progress in this area by his students and followers (see, for example, [6–8]). A number of mathematical concepts introduced into scholarly discourse by Ural scientists have become key to many studies. Terms such as positional differential games, stable bridges, saddle point in the small game, extremal aiming rule, extremal shift, alternative, program iterations, and guided control are widely used by specialists. One of the most important areas of studies of the Yekaterinburg research school is stochastic differential games. Sometimes a conflict between opposing parties is complicated by the influence of various kinds of random interference, which leads to the consideration of conflict-control random processes. A series of works by Krasovskii, Tret’yakov, and their students [9–13] is devoted to the study of such problems. A review of the theory of stochastic differential games can be found in the monograph [14]. Such games involve lumped systems with states described by stochastic differential equations in finite-dimensional spaces. However, in a number of areas of physics and technology, the dynamics 1 2
Kharkiv National University, Kharkiv, 61022 Ukraine Institute of Cybernetics, National Academy of Sciences of Ukraine, Kiev, 03680 Ukraine e-mail: ∗ [email protected], ∗∗ [email protected], ∗∗∗ [email protected]
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