A Unified Approach to the Finite-Horizon Linear Quadratic Optimal Control Problem**

Under the mild assumption of sign-controllability, a closed-form expression parameterizing all the solutions of the Hamiltonian differential equation over a finite time interval is presented in terms of a strongly unmixed solution of an algebraic Riccati equation (ARE) and of the solution of an algebraic Lyapunov equation. This result is employed for the solution of a generalized version of the finite-horizon linear quadratic (LQ) problem, encompassing the case of fixed end-point. Furthermore, it is shown how this method can be applied to the H2 preview decoupling problem.