Stochastic differential system output control by the quadratic criterion. I. Dynamic programming optimal solution

The problem of optimal control for the Ito diffusion process and a controlled linear output is solved. The considered statement is close to the classical linear-quadratic Gaussian control (LQG control) problem. Differences consist in the fact that the state is described by the nonlinear differential Ito equation $dy_y = A_t(y_t) \,dt+\Sigma_t(y_t)\,dv_t$ and does not depend on the control $u_t$, optimization subject is controlled linear output $dz_t=a_ty_t\,dt +b_tz_t\,dt +c_t u_t\,dt +\sigma_t \,dw_t$. Additional generalizations are included in the quadratic quality criterion for the purpose of statement such problems as state tracking by output or a linear combination of state and output tracking by control. The method of dynamic programming is used for the solution. The assumption about Bellman function in the form $V_t(y,z)= \alpha_t z^2+\beta_t(y) z+\gamma_t(y)$ allows one to find it. Three differential equations for the coefficients $\alpha_t$, $\beta_t(y)$, and $\gamma_t(y)$ give the solution. These equations constitute the optimal solution of the problem under consideration.