Reconstruction of dynamically distorted signals based on the theory of optimal control of solutions for Sobolev type equations in the spaces of stochastic processes

This paper investigates the solvability of the optimal control problem for solutions of stochastic Sobolev type equations. It is shown that the optimal dynamic measurement problem can be considered as an optimal control problem. To do this, the mathematical model of dynamic measurements is reduced to a stochastic Sobolev type equation of the first order in the spaces of stochastic processes. The article presents theorems on the existence of a unique classical and strong solutions of the Sobolev type equation with initial condition of Showalter-Sidorov in the spaces of stochastic processes. The theorem of the unique solvability of the optimal control problem for such equation is proved. The abstract results obtained for Sobolev type equation are applied to the problem of restoring a dynamically distorted signal as an optimal dynamic measurement.