Problem of optimal dynamic measurement with multiplicative effects in spaces of differentiable “noises”

The article deals with a model of optimal dynamic measurement with multiplicative influence, considered as an optimal control problem for a non-stationary Leontief-type system. The existence of a solution to this problem in a stochastic formulation is established. The main objective is to find a recoverable signal (control action) that brings the system state as close as possible to the observed indicators, given the presence of an additional input process modeling noise. Solutions to the system must be sought in spaces of random processes. To achieve this, the optimal control problem in spaces of differentiable “noises” is preliminarily analyzed. The linearity of the transducer model, described by a non-stationary Leontief-type system, allows the original system to be decomposed into deterministic and stochastic subsystems. Based on the results regarding the solvability of optimal control problems for each subsystem, a solution to the original problem is obtained.
The first part of the article presents the solvability conditions for a stochastic non-stationary Leontief-type system. The second part explores the optimal control problem in the stochastic case and derives estimates for the minimized functionals using results previously obtained for the deterministic counterpart. In conclusion, an algorithm for studying the problem of optimal dynamic measurement with multiplicative influence in spaces of “noises” is presented.