Structural Invariant Subspaces of Singular Hamiltonian Systems and Nonrecursive Solutions of Finite-Horizon Optimal Control Problems

This note introduces an analytic, nonrecursive approach to the solution of finite-horizon optimal control problems formulated for discrete- time stabilizable systems. The procedure, which adapts to handle both the case where the final state is weighted by a generic quadratic function and the case where the final state is an admissible, sharply assigned one, provides the optimal control sequences, as well as the corresponding optimal state trajectories, in closed form, as functions of time, by exploiting an original characterization of a pair of structural invariant subspaces associated to the singular Hamiltonian system. The results hold on the fairly general assumptions which guarantee the existence and uniqueness of the stabilizing solution of the corresponding discrete algebraic Riccati equation and, as a consequence, solvability of an appropriately defined symmetric Stein equation. Some issues to be considered in the numerical implementation of the proposed approach are mentioned. The application of the suggested methodology to H
₂
optimal rejection with preview is also discussed.