Abstract
AbstractMixed‐integer optimal control problems can be reformulated by means of partial outer convexification, which introduces binary‐valued switching functions for the different realizations of a discrete‐valued control variable. They can be relaxed naturally by allowing them to take values in [0, 1]. Sum‐Up Rounding (SUR) algorithms approximate feasible switching functions of the relaxation with binary ones. If the controls are distributed in one dimension, the approximants are known to converge in the weak∗ topology of L∞. We show that this still holds true for controls that are distributed in more than one dimension if an appropriate grid refinement strategy that is coupled with a deliberate ordering of the grid cells is chosen. This condition is satisfied by the iterates of space‐filling curves, e.g. the Hilbert curve.
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