On the problem of quasi-singular optimal controls in hyperbolic type stochastic systems

The proposed work examines the optimal control problem of a stochastic system, the dynamics of which are described by a stochastic second-order partial differential equation of hyperbolic type with Goursat boundary conditions. The system is controlled using measurable and constrained controls. The case is considered when two-parameter “white noise” enters the right side of a controlled system of second-order nonlinear hyperbolic equations. The goal of control is to minimize the mathematical expectation of the quality functional at the final point of the domain. Problems of this type arise, for example, when modeling a number of processes of drying, sorption, etc. in the presence of random influences such as standard two-parameter “white noise” on a plane. Using a modified version of the increment method, a formula for the increment of the second-order quality criterion for the quality functional is established, which allows us to obtain the necessary first-order optimality conditions of the type of the linearized Pontryagin maximum principle, as well as to study quasi-singular controls (i. e., the case of degeneration of the first-order optimality condition), in of the stochastic problem under consideration. Necessary conditions for first- and second-order optimality are established. In the end, based on the use of a special variation of control, a pointwise necessary condition for the optimality of quasi-special controls is obtained.