Controllability and Necessary Conditions in Unilateral Problems without Differentiability Assumptions

We study the attainable set and derive necessary conditions for relaxed, original and strictly original minimum in control problems defined by ordinary differential equations with unilateral restrictions. The functions defining the problem are assumed to be Lipschitz-continuous in their dependence on the state variables except for the unilateral restriction where continuous differentiability is also required. We define an extremal control as one satisfying a generalized Pontryagin maximum principle, with set-valued “derivate containers” replacing nonexistent derivatives. We prove that a nonextremal control (either original or relaxed) yields an interior point of the attainable set generated by original controls, and that, in normal problems, a minimizing original solution must also be a minimizing relaxed solution. The proofs are carried out with the help of an inverse function theorem for Lipschitz-continuous functions that is formulated in terms of derivate containers.