Existence of Minimum Upcrossing Controllers

An optimal stochastic control problem that minimizes the probability that a signal upcrosses a level is solved by rewriting it as a one-parametric optimization problem over a set of LQG control problem solutions. Finding the optimal controller can be interpreted as finding an optimal costing transfer function. The existence of the optimal controller is investigated in a constructive way, and it is shown that it is equivalent to the existence of a controller with sufficiently small closed-loop variance of the controlled signal.