We consider a stochastic optimal feedback control problem for a single-degree-of-freedom vibrational system, where uncertainty is described by two independent noises. The first of them is induced by the control actions and called internal, whereas the second one acts externally. The drift vector also depends on the control function. The set of pointwise control constraints is assumed to be bounded. The minimization functional is taken as the mean system response energy. The Cauchy problem for the corresponding Hamilton–Jacobi–Bellman (HJB) equation without the control constraints is first investigated. This allows us to find the sought-for feedback control strategy in a specific domain of the space of state and time variables. Then a proper extension to the remaining parts of the space is constructed, and the optimality of the resulting global feedback control strategy is proved. The obtained control law is compared with the dry friction and saturated viscous friction control laws.