A minimax control problem for sampled linear systems

A linear differential system is subject to a bounded control and a bounded disturbance. The controller receives the value of the state at a finite number of fixed sampling times. The cost is a convex or concave function of the state at a fixed final time. Given any control law, there is a maximum cost over all perturbations, the guaranteed performance for this control law. It is desired to find the minimum of this number over all control laws. Fenchel's theory of conjugate convex functions is used to set up an algorithm dual to the dynamic programming approach. This algorithm deals with the support functions of reachable sets, more easily determined than other descriptions of these sets. A discrete minimax principle is generally incorrect for problems of this class.