This paper considers optimal control problems for an abstract linear system with a quadratic cost functional in a Hilbert space on a fixed time interval $[0,T],$ $ 0 < T < +\infty$. The controls are unbounded and the state weight operators are indefinite and not necessarily smoothing. Some new inequalities are established and applied to prove that the optimal control and the optimal trajectory are continuous. The equivalences among the solvability of the LQ problems, the two-point boundary value problem and the Fredholm integral equation are proved and the state feedback representation for the optimal control is also given in terms of the solution to the Fredholm integral equation. Finally, we derive the closed-loop synthesis of the optimal control via the solution to the Riccati integral equation which exists under some mild conditions. An explanatory example is given at the end. The results are complementary to those of [15, 1}] and .
"A linear quadratic problem with unbounded control in Hilbert spaces." Differential Integral Equations 13 (4-6) 529 - 566, 2000.