Abstract: Optimization problems with discrete variables are exposed to the conflict between being a powerful modeling tool and being hard to solve. Dynamical systems, as described by differential equations, for example, constraining the optimization, may need to solve for distributed discrete control variables. We present approximation arguments that replace the need for solving the optimization problem by the need for solving a sequence of relaxations and computing an appropriate rounding for each relaxation to regain a discrete control. We provide conditions on the rounding algorithms and their grid refinement strategies to prove approximation of the relaxed controls by the discrete controls in a weak sense. If the control-to-state mapping of the constraining dynamical systems is sufficiently regular, we obtain can approximate the infimal value of the mixed-integer optimal control problem arbitrarily close. We apply the arguments on different classes of mixed-integer optimization problems that are constrained by partial differential equations with discrete control inputs, which are distributed in the time and/or spatial domain. Furthermore, we apply the arguments to a signal reconstruction problem: a dynamical system that is not governed by a differential equation. The findings are illustrated computationally.