Memoryless Approach to the LQ and LQG Problems with Variable Input Delay

This note studies the LQ and LQG problems for linear time invariant systems with a single time-varying input delay and instantaneous (memoryless) state feedback. We extend the memoryless state feedback solution proposed in [1] in two directions. We prove that in the deterministic case a memoryless state feedback can be in general optimal only up to a certain delay, for which we provide a sufficient, and sometimes strict, bound. Moreover, we show that this memoryless control is optimal also in the case of time-varying delays and that the quadratic cost functional has the same value as in the case without delay. For time varying delays the control law requires that the relationship between time points in which the input is generated and applied is known and invertible even if the delay function needs not to be differentiable or even continuous. Finally, we prove that the cost functional is bounded also in the stochastic case for the same delay interval as in the deterministic case, but with a larger cost than the delay-less LQG solution.