2 00 5 Characterization of the solution to a constrained H ∞ optimal control problem ∗

This paper characterizes the solution to a finite horizon min-max optimal control problem where the system is linear and discrete-time with control and state constraints, and the cost quadratic; the disturbance is negatively costed, as in the standard H ∞ problem, and is constrained. The cost is minimized over control policies and maximized over disturbance sequences so that the solution yields a feedback control. It is shown that the value function is piecewise quadratic and the optimal control policy piecewise affine, being quadratic and affine, respectively, in polytopes that partition the domain of the value function .