Abstract
We extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions (mixed-integer optimal control problems, MIOCP). We improve a theorem (Sager et al. in Math Program 118(1): 109–149, 2009) that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements. Unlike in previous publications the new proof avoids the usage of the Krein-Milman theorem, which is undesirable as it only states the existence of a solution that may switch infinitely often. We present a constructive way to obtain an integer solution with a guaranteed bound on the performance loss in polynomial time. We prove that this bound depends linearly on the control discretization grid. A numerical benchmark example illustrates the procedure. As a byproduct, we obtain an estimate of the Hausdorff distance between reachable sets. We improve the approximation order to linear grid size h instead of the previously known result with order \({\sqrt{h}}\) (Häckl in Reachable sets, control sets and their computation, augsburger mathematisch-naturwissenschaftliche schriften. Dr. Bernd Wißner, Augsburg, 1996). We are able to include a Special Ordered Set condition which will allow for a transfer of the results to a more general, multi-dimensional and nonlinear case compared to the Theorems in Pietrus and Veliov in (Syst Control Lett 58:395–399, 2009).
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Sager, S., Bock, H.G. & Diehl, M. The integer approximation error in mixed-integer optimal control. Math. Program. 133, 1–23 (2012). https://doi.org/10.1007/s10107-010-0405-3
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DOI: https://doi.org/10.1007/s10107-010-0405-3