Asymmetrically constrained min-max arising from a problem of optimal control in the presence of uncertainty

This paper solves a particular min-max problem where the minimizing variables not only influence the performance index but also constrain the domain of action of the maximizing variables. The problem is approached as minimization of a supremal value function. It is shown here that if a weaker form of the classical sensitivity theorem of non-linear programming holds, then directional derivatives, and in some cases even ordinary derivatives, of the supremal value function exist. The existence of ordinary derivatives is especially useful for computation purposes, in that case the min-max becomes a stationary point under equality constraints with respect to both players separately. The problem originates from a new approach for control design of a dynamic uncertain system when a bound in norm of the approximation error is guaranteed.