On the iterated logarithm law in a control problem

We consider the limit behavior of some functional of a Wiener process and an arbitrary absolutely continuous adapted function. This kind of functional arises in the problem of control of the linear regulator with quadratic criteria disturbed by white noise. The results obtained make it possible to prove that one cannot improve the estimate of the quality of the control from [T. Belkina, Yu. Kabanov, and E. Presman, Stochastic Linear-Quadratic Regulator. Optimality Almost Sure and in Probability, manuscript] and to clarify some phenomena connected with the iterated logarithm law.