  This note introduces a new analytic approach to the solution of a very
general class of finite-horizon optimal control problems formulated 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. Importantly, the results established in this paper hold
under assumptions that are weaker than the ones considered in the literature so
far. Indeed, this approach does not require neither the regularity of the
symplectic pencil, nor the modulus controllability of the underlying system. In
the development of the approach presented in this paper, several ancillary
results of independent interest on generalised Riccati equations and on the
eigenstructure of the extended symplectic pencil will also be presented.
