Author: Jane Kehoe CullumPublisher:
ISBN:
Category :
Languages : en
Pages : 202
Book Description
The first part is devoted to proving that for a control problem satisfying the proper differentiability hypotheses and in which the optimization is made over a set of trajectories of the associated differential system that are in a fixed closed subset A of E(n), if an optimal solution exists such that the optimal trajectory is on the boundary of A and such that in a neighborhood of this trajectory, the boundary of A is the C(2)- diffeomorphic image of an open set in E(n-1), then this optimal solution satisfies a modified version of Pontryagin's maximum principle. The proof presented is direct and uses only the constructions used in the proof of Pontryagin's principle. If a C(2)- diffeomorphism exists, it is proved that the problems considered by Gamkrelidze are included in the problems considered in this paper. The restriction made by Gamkrelidze that the controls be piecewise smooth is removed, and the condition that the control sets be regular is relaxed. In the second part, three types of approximations of sequences of trajectories paired with their controls to a trajectory and its control are defined. The first type involves only the convergence of the trajectories, the second and third types add the convergence of the corresponding controls in the weak L(2)-topology and the strong L(2)-topology respectively. Next penalty functions are introduced and the problems generated perturbed; it is proved that the preceding results still hold for this new family of problems. Finally, results involving controllability hypotheses are obtained, and a specialized theorem involving approximations of type three is proved.