A parametrization of the solutions of the finite-horizon LQ problem with general cost and boundary conditions

A generalization of the finite-horizon linear quadratic regulator problem is proposed for LTI continuous-time controllable systems. In particular, a formulation of the linear quadratic (LQ) problem is considered, with affine constraints on the initial and the terminal states and with general quadratic costs in the initial and terminal states. The solution presented is simple and attractive from a computational point of view, and is based on the solutions of an algebraic Riccati equation and of a Lyapunov equation, that enable all the solutions of the Hamiltonian differential equation to be parametrized in closed form.