A closed-form expression parameterizing the solutions of the extended symplectic difference equation over a finite time interval is given under the mild assumption of modulus-controllability. This representation is expressed in terms of the strongly unmixed solution of a discrete ARE and of an algebraic Stein equation.The most important application of this result is a generalized version of the finite-horizon LQ regulator: In particular our framework enables different kind of boundary conditions to be treated in a unified fashion, without resorting to the Riccati difference equation for the computation of the optimal control function.
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