L.A. Vlasenko, A.G. Rutkas, A.A. Chikrii. On a differential game in a stochastic system ... P. 45-61

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 is the generator of a strongly continuous semigroup (a semigroup of class $C_0$). Solutions of the equation are represented by a stochastic formula of variation of constants. 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 leaks.

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

Received April 5, 2019

Revised May 15, 2019

Accepted May 20, 2019

Larisa Andreevna Vlasenko, Dr. Tech. Sci., Prof., Kharkov National University of Radio Electronics, Kharkov, 61166 Ukraine, e-mail: lara@rutrus.com

Anatolii Georgievich Rutkas, Dr. Phys.-Math. Sci., Prof., Kharkov National University of Radio Electronics, Kharkov, 61166 Ukraine, e-mail: anatoly@rutrus.com

Arkadii Alekseevich Chikrii, Dr. Phys.-Math. Sci., Prof., Ukranian NAS Academician, Glushkov Institute of Cybernetics, Kiev, 03187 Ukraine, e-mail: chik@insyg.kiev.ua

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Cite this article as: L.A. Vlasenko, A.G. Rutkas, A.A. Chikrii. On a differential game in a stochastic system, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 3, pp. 45–61.