Employing the algebraic Riccati equation for the solution of the finite-horizon LQ problem

A new approach to the study of the finite-horizon linear quadratic regulator problem is presented for linear, time-invariant controllable systems. This simple and computationally attractive procedure is based on the solutions of an algebraic Riccati equation and of a Lyapunov equation, that enable all the state-costate functions that satisfy the Hamiltonian system to be parametrized in a closed form. In this way, it is possible to easily solve in a unified framework optimal control problems in which either both the initial and the terminal states are assigned, or one of them is weighted in the performance index. In this way the optimal state trajectory and input control law are efficiently obtained without resorting to the integration of a Riccati differential equation. The optimal value of the cost is also explicitly parametrized.