On a Differential Game in a Stochastic System

L. A. Vlasenko ; A. G. Rutkas ; A. A. Chikrii

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 3, pp. 45-61

Résumé

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 $C_0$). 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.

Détail

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.

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