Approximation Properties and Tight Bounds for Constrained Mixed-Integer Optimal Control

We extend recent work on solving mixed-integer nonlinear optimal control problems (MIOCPs) to the case of integer control functions subject to constraints that involve a pointwise coupling of the state with the integer controls. We extend a theorem due to [S. Sager, H. Bock, and M. Diehl, Math. Program. Ser. A, 133 (2012), pp. 1--23] to the case of MIOCPs with constraints on the integer control and show that the integrality gap vanishes in function space when the coarseness of the rounding grid is driven to zero even after adding constraints of this type. For the time-discretized problem, we extend a sum-up rounding (SUR) scheme due to [S. Sager, C. Reinelt, and H. Bock, Math. Program. Ser. A, 118 (2009), pp. 109--149] to the new problem class. Our scheme permits one to constructively obtain an $\varepsilon$-feasible and $\varepsilon$-optimal binary feasible control. We derive new, tight upper bounds on the integer control approximation error made by SUR. For unconstrained binary controls on equidistant grids, we reduce the approximation error bound from $\mathcal{O}(|\Omega|)$ to $\mathcal{O}(\log |\Omega|)$ asymptotically for $|\Omega| \to \infty$ and a fixed coarseness of the rounding grid, where $|\Omega|$ is the number of binary controls. For constrained binary controls, we show that the approximation problem is more difficult, and we give a proof of an approximation error bound of complexity $\mathcal{O}(|\Omega|)$. A numerical example compares our approach to a state-of-the-art mixed-integer nonlinear programming solver and illustrates the applicability of our results when solving MIOCPs using the direct and simultaneous approach.