Dynamic optimization using adaptive direct multiple shooting

• Wavelet-based grid is applied to direct multiple shooting.
• Adaptive multiple shooting is used to solve dynamic optimization (or optimal control) problems.
• Adaptive multiple shooting, automatically generates a problem-dependent parameterization of the control profiles.
• We demonstrate the superiority of adaptive grid refinement over an equidistant discretization for two examples.

We present a wavelet-based grid refinement approach for direct multiple shooting applied to dynamic optimization problems. The algorithm, named adaptive multiple shooting, automatically generates a problem-dependent parameterization of the control profiles: Starting from an initially coarse parameterization, the control grid is refined iteratively using a wavelet analysis of the previously obtained optimal solution. Additional grid points are only inserted where required and redundant grid points are eliminated. Hence, the algorithm minimizes the number of grid points required to obtain accurate optimal control trajectories.First, we demonstrate the superiority of adaptive grid refinement over an equidistant discretization for the Williams–Otto semi-batch reactor employing multiple and single shooting. Here, the accuracy is checked using an optimal solution obtained by an indirect optimization approach. Second, we successfully demonstrate the efficiency of adaptive grid refinement compared to an equidistant discretization employing multiple shooting to a dynamically unstable HIPS (high impact polystyrene) polymerization reactor.