Optimal Controls for Problems with a Restricted State Space

In this paper we derive conditions necessary for a curve to minimize a functional of the Bolza form, subject to differential equation side conditions and the restriction that the trajectory of an admissible curve is constrained to lie inside a closed set S with a smooth boundary. These results are obtained as limits of conditions derived by E. J. McShane for curves whose trajectories are contained in a neighborhood of S. This limit procedure verifies the continuity of optimal solutions as a “soft” boundary of S “hardens” until it allows no penetration at all.