Stochastic differential system output control by the quadratic criterion. III. Optimal control properties analysis

The investigation of the optimal control problem for the Ito diffusion process and linear controlled output with a quadratic quality criterion is continued. The properties of the optimal solution defined by the Bellman function of the form $V_t(y,z)=\alpha_tz^2+\beta_t(y)z+\gamma_t(y)$, whose coefficients $\beta_t(y)$ and $\gamma_t(y)$ are described by linear parabolic equations, are studied. For these coefficients, alternative equivalent descriptions are defined in the form of stochastic differential equations and a theoretical-to-probabilistic representation of their solutions, known as the Kolmogorov equation. It is shown that the obtained differential representation is equivalent to the Feynman–Kac integral formula. In the future, the obtained description of the coefficients and, as a result, the solutions of the original control problem can be used to implement an alternative numerical method for calculating them as a result of computer simulation of the solution of a stochastic differential equation.