We focus on optimal control problems where the cost to be minimized is the total time a trajectory of a control system spends inside a given set K, known as “the minimal time crisis”. The distintctive aspect of this problem is that it involves a discontinuous functional with respect to the state variable, a fact that prevents the straightforward application of the classical maximum principle. Here, we propose a multiprocess approach to define necessary conditions of optimality for this problem. In contrast to previous literature, we allow the state to live on the boundary of set K.