Optimal control with limited controls

We consider a linear discrete-time optimal control problem where the control is limited in terms of the number of times it can be applied. We assume an additive quadratic performance criterion that does not penalize the control directly, and show that for a scalar plant driven by an independent additive Gaussian noise the optimal control is piecewise linear based on some offline computed thresholds on the optimal estimate of the plant state which can be generated by a Kalman filter