The method of the calculus of variations and the maximum principle are preposed for the design of `LQR' controllers with the worst case disturbance rejection for a linear time-varying (LTV) plant on finite horizon. The disturbance is bounded by either the windowed L-2-norm or the windowed L-infinity-norm, or both. In the case of the windowed L-2-normed disturbance, uncertain but norm bounded initial condition is also considered. Certain necessary and sufficient condtions for the existence of a linear controller are derived with the proof of the solution existence and uniqueness. The results are extended to the steady state ones for the linear time-invariant (LTIV) plant on the infinite horizon. A comparison to H-infinity control with transients is also presented. In the case of the windowed L-infinity-normed or both normed disturbances, the solution for the worst case disturbance is of switching (or bang-bang) type.