An Infinite Horizon Linear Quadratic Problem with Unbounded Controls in Hilbert Space

An infinite horizon linear quadratic optimal control problem for analytic semigroup with unbounded control in Hilbert space is considered. The state weight operator is allowed to be indefinite while the control weight operator is coercive. Under the exponential stabilization condition, it is proved that any optimal control and its optimal trajectory are continuous. The positive real lemma as a necessary and sufficient condition for the unique solvability of this problem is established. The closed-loop synthesis of optimal control is given via the solution to the algebraic Riccati equation.