On a stochastic optimality of the feedback control in the LQG-problem

We show that the optimal feedback control $\widehat u$ in the standard nonhomogeneous LQG-problem with infinite horizon has the following property. There is a constant $b_*$ such that, whatever $b> b_*$ is, the deficiency process of optimal control with respect to any possible control $u$, i.e., the difference $J_T(\widehat u\hspace*{0.2pt})- J_T(u)$ between the optimal cost process $J_T(\widehat u\hspace*{0.2pt})$ and the cost process corresponding to control $u$, is majorated at infinity by a deterministic function $b\log T$. In other words, $b\log T$ is an upper function for any deficiency process. This result, combined with an example of an LQG-regulator where, for certain $b>0$, the function $b\log T$ is not an upper function for certain deficiency processes, gives an answer to the long-standing open problem about the best possible rate function for sensitive probabilistic criteria. Our setting covers the optimal tracking problem.