On stochastic optimality for a linear controller with attenuating disturbances

For a linear stochastic control system with quadratic objective functional, we introduce various generalizations of the notions of optimality on average and stochastic optimality on an infinite time interval that take into account possible degeneration of the parameter of the disturbing process with time (attenuation of the disturbances) or the presence of a discount function in the objective functional. This lets us improve upon the quality estimate for a well known optimal control in this problem from the point of view of both asymptotic behavior of the functional's expectation and its asymptotic probabilistic properties. In particular, in the considered case we have found an improvement for the well known logarithmic upper bound on the optimal control for a family of defect processes.