The Extended Symplectic Pencil and the Finite-Horizon LQ Problem With Two-Sided Boundary Conditions

This technical note introduces a new approach to the solution of a very general class of finite-horizon optimal control problems for discrete-time systems. This approach provides a parametric expression for the optimal control sequences, as well as the corresponding optimal state trajectories, by exploiting a new decomposition of the so-called extended symplectic pencil. This decomposition provides an original strategy for a more direct solution of the problem with no need of the system-theoretic hypotheses (including regularity of the symplectic pencil) that have always been assumed in the literature so far.