In this work, we investigate a temporal decomposition approach to long-horizon dynamic optimization problems. The problems are discrete-time, linear, time-dependent and with box constraints on the control variables. We prove that an overlapping domains temporal decomposition, while inexact, approaches the solution of the long-horizon dynamic optimization problem exponentially fast in the size of the overlap. The resulting subproblems share no solution information and thus can be computed independently in parallel. Our findings are demonstrated with a small, synthetic production cost model with real demand data.
Receding horizon control has been a widespread method in industrial control engineering as well as an extensively studied subject in control theory. In this work, we consider a lag L receding horizon strategy that applies the initial L optimal controls from each quadratic program to each receding horizon. We investigate a discrete-time and time-varying linear-quadratic optimal control problem that includes a nonzero reference trajectory and constraints on both state and control. We prove that, under boundedness and controllability conditions, the solution obtained by the receding horizon strategy converges to the solution of the full problem interval exponentially fast in the length of the receding horizon for some lag L. The exponential rate of convergence provides a systematic way of choosing the receding horizon length given a desired accuracy level. We illustrate our theoretical findings using a small, synthetic production cost model with real demand data.
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