In this paper a finite horizon H?? optimal control problem is treated. Results guaranteeing the existence of a unique optimal control and the worst exogenous input are derived. A criterion for the evaluation of the infimal H?? norm is then given in terms of the least positive value for which a certain boundary value problem has a nontrivial solution. Once the infimal value is known, a noninfimal value can be selected and suboptimal H?? controllers can be synthesized.