In this chapter, we present the normal maximum principle, complemented by sensitivity relations and an initial—time transversality condition. We prove the existence of accompanying co-state variables, associated measures, and a measurable selection that together satisfy the adjoint equations and optimality conditions. When the trajectory lies strictly within the admissible set, the standard maximum-principle relations hold. If instead the trajectory reaches the boundary of the constraint set, we introduce modified adjoint equations and boundary-adjusted optimality conditions.