<p>We study the quadratic regulator problem on a finite time horizon for the wave equation with high internal damping controlled on the boundary by square integrable controls. The approach in this paper transforms the wave equation with high internal damping to an equation with persistent memory controlled on the boundary. One of the results of this paper is the introduction of a state space which is an extended Hilbert space, so a time dependent Hilbert space. We prove that the unique optimal control can be represented as a feedback control via a Riccati operator which solves a suitable version of the Riccati equation. Both the feedback operator and the Riccati equation act on such time dependent space. The derivation of these main results requires a very precise analysis of the properties of the derivatives of the value function and we find an explicit form for the derivative of the Riccati operator.</p>