Asymptotics of Rounding Algorithms for Integer Control Problems in the Presence of Convexified Pointwise Mixed Constraints

The analysis of computationally cheap algorithms to approximate the infimum of optimal control problems with discrete‐valued controls has been improved in recent literature. The overall method is often referred to as combinatorial integral approximation decomposition and the algorithms are often referred to as rounding algorithms. The studied algorithms have in common that they seek to minimize or at least bound a quantity called integrated control deviation. The algorithms operate on a grid (often called rounding grid) and the integrated control deviation vanishes when the grid constant of the grid on tends to zero. Assuming a suitable spatial coherence property of the grid refinement implies weak ∗ convergence of the computed controls in L ∞ . When the integer optimal control problem features additional pointwise mixed state‐control constraints, the aforementioned rounding algorithms cannot be applied directly. There exist examples for which the observed infeasibility of the constraint after applying the algorithms does not vanish in an L ∞ ‐sense when the grid constant is driven to zero. While modifications of the algorithms exist that resolve this problem, the asymptotics of the infeasibility without those modifications have not been analyzed so far. This work closes this gap in the existing literature and we show that the infeasibility vanishes in a pointwise almost everywhere sense.