The motivation of this paper stems from a family of optimal control problems wherein the control’s active duration is constrained within a predefined limit. The duration constraint can be perceived as an additional variable in the dynamics, and the relaxation of the naturally associated control problem is equivalent to a an $\mathbb{L}^{1}$-constraint. The paper provides a generalization of a preliminary work by the authors to encompass scenarios with non-convex dynamics. The relaxed problems are formulated through Linear Programming techniques and their qualitative properties, alongside the interrelations between different formulations, are investigated.