Extension of discrete LQR-problem to symplectic systems*

In this paper we consider a discrete linear-quadratic regulator problem in the setting of discrete symplectic systems (S). We derive minimal conditions which guarantee the solvability of this problem. The matrices appearing in these conditions have close connection to the focal point definition of conjoined bases of (S). We show that the optimal solution of this problem has a feedback form and that it is constructed from a generalized discrete Riccati equation. Several examples are provided illustrating this theory. The results of this paper extend the results obtained earlier by the authors for the special case of discrete linear Hamiltonian systems.