Structural invariants of the singular Hamiltonian system and non-iterative solution of finite-horizon optimal control problems

A non-iterative solution for a class of discrete-time finite-horizon linear quadratic optimal control problems is obtained through the characterization of a pair of structural invariants of the singular Hamiltonian system associated to the H2 optimal control problem stated for the generic, discrete-time quadruple (A,B,C,D). On the assumption that the final state weighting function in the performance index is represented by a quadratic surface, it is shown that the optimal cost is a function of the initial state with the same structure. Optimal control laws and state trajectories are analytically expressed as functions of the initial state as well. The results hold under rather extensive conditions: those that guarantee the existence and uniqueness of the stabilizing solution of the corresponding discrete algebraic Riccati equation.