Optimal Finite-Dimensional Controller of the Stochastic Differential Object’s State by Its Output. I. Incomplete Precise Measurements

The well-known problem of synthesizing the optimal on the average and on given time interval of the inertial control law for a continuous stochastic object if only a part of its state variables are accurately measured is considered. Due to the practical unrealizability of its classical infinite-dimensional Stratonovich-Mortensen solution, it is proposed to limit ourselves to optimizing the structure of a finite-dimensional dynamic controller, whose order is chosen by the user. This finiteness allows using a truncated version of the a posteriori probability density that satisfies a deterministic partial differential integrodifferential equation. Using the Krotov extension principle, sufficient optimality conditions for the structural functions of the controller and the Lagrange–Pontryagin equation for finding their extremals are obtained. It is shown that in particular cases of the absence of measurements, complete measurements and taking into account only the values of incomplete measurements, the proposed controller turns out to be static (inertialess), and the relations for its synthesis coincide with the known ones. For a dynamic controller, algorithms for finding each of its structural functions are given.