Finite horizon H ^∞ control problems with terminal penalties

The authors generalize the standard H ^∞ control problem to the finite horizon case with two (possibly singular) terminal penalties at the initial and final times. The major objective of the generalization is to increase flexibility of H _∞ controls; the terminal penalties correspond to treating an intrinsic issue of finite horizon cases within the framework of H ^∞ control problems. The authors give a complete solution, a necessary and sufficient condition, and a parametrization to the finite horizon H ^∞ control problem. The solution is a natural extension of the Riccati equation solution; in the special case when all the terminal penalties vanish, the solution is reduced to the existing one to the finite horizon standard H ^∞ control problem. The present approach to the problem is based on completing the square argument of a particular quadratic form.