Specialized Adaptive Algorithms for Model Predictive Control of PDEs

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Model Predictive Control is a control method in which the solution of optimal control problems on infinite or indefinitely long horizons is split up into the successive solution of optimal control problems on relatively short finite time horizons. Only the first piece of each optimal control us then used in order to synthesize the resulting control function on the infinite horizon. Under suitable conditions, this method can be shown to produce approximately optimal control functions on the infinite horizon. Moreover, due to the successive re-optimization the resulting control is of a feedback type form which provides robustness against model errors and perturbations. Since only the first piece of each optimal control function is used, one can expect that for its numerical computation a high accuracy is only needed at the beginning of the optimization interval while a lower accuracy is sufficient towards its end. The main objective of the proposed project is the construction of numerical algorithms for model predictive control of parabolic partial differential equations which exploit this fact via goal oriented error estimation and adaptivity.The grids, constructed by our algorithm, will reflect the sensitivity of the computed optimal control at the beginning of the control interval with respect to perturbations of the dynamics. We expect that we will obtain a relatively fine discretization near the beginning of the control interval that blends into coarser and coarser discretizations towards the end of this interval. Of course, previous grids are reused as the time-horizon moves on. This will result in an efficient overall method for model predictive control of parabolic PDEs that obtains a near optimal infinite horizon performance with low computational effort.Our strategy is to first establish our method for ODEs and then move on to linear and non-linear parabolic PDEs. The algorithmic development will be combined with theoretical investigations on the sensitivity of the first piece of the optimal control with respect to perturbations of the dynamics. Particularly, we will derive conditions under which we can rigorously prove that the sensitivity of the optimal control decreases over time. This will create a deeper understanding of the newly created algorithms and identify classes of problems for which these algorithms can be applied.