Necessary conditions for a nonclassical control problem with state constraints * *The second author is partially supported by Padova University project PRAT2015 “Control of dynamics with active constraints” and by Istituto Nazionale di Alta Matematica

We consider the problem of minimizing the cost h(x(T )) at the endpoint of a trajectory subject to the finite dimensional dynamics the x ̇ ∈ −N_C (x) + f(x, u), x(0) = x_0 , where N_C denotes the normal cone to the convex set C. Such inclusion is termed, after J-J. Moreau, the 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) ∈ 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.