On some special cases in the problem of stochastic differential system output control by the quadratic criterion

A general study of the optimal control problem for the Ito diffusion process and linear controlled output with the quadratic quality criterion was carried out in the author's previous works (coauthored by A. I. Stefanovich). An analysis of the available results allows to single out some models that are of a particular nature in relation to the general setting but have special practical significance. This article examines two such particular models. The first model is determined by the assumption of linear drift in the equation of state while maintaining nonlinear diffusion. It is shown that such a model provides linearity to the optimal control and the absence of the need to solve a parabolic equation for its implementation. But in this case, the quadratic Bellman function does not appear in the problem, the corresponding expression, as in the general case, is described by the solution of a parabolic equation and retains a meaningful stochastic interpretation expressed by the Feynman–Katz formula. The second model implements the assumption about the dependence of disturbances in the equations of state and output. The modified dynamic programming equation is solved in the same way as in the general case considered in previous works, including and within the framework of a combined model involving both cases presented. This model will be especially useful in the problems with incomplete information, when the assumption of the presence of complete information about the state and output will be replaced by a description of the observation system, in which the output is interpreted as indirect observations of the state. A numerical example, studied in detail in the author's previous works (coauthored by A. I. Stefanovich), is briefly discussed, since it turns out that it satisfies the assumption of linear drift in the equation of state and, accordingly, the previously obtained approximate solutions can be refined.