Necessary conditions for a nonclassical control problem with state constraints

We consider the problem of minimizing the cost h(x(T)) at the endpoint of a trajectory x subject to the finite dimensional dynamics (x)over dot is an element of- N-C(x) f(x, u), x(0) = x(0), where N-c denotes the normal cone to the convex set C. Such differential inclusion is termed, after Moreau, sweeping process. We label it as a "nonclassical" control problem with state constraints, because the right hand side is discontinuous with respect to the state, and the constraint x(t) is an element of C for all t is implicitly contained in the dynamics. We prove necessary optimality conditions in the form of Pontryagin Maximum Principle by requiring, essentially, that C is independent of time. If the reference trajectory is in the interior of C, necessary conditions coincide with the usual ones. In the general case, the adjoint vector is a BV function and a signed vector measure appears in the adjoint equation. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.