Memoryless approach to the LQ and LQG problems with variable input delay

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