Maximum Principle in the Optimal Control Problems for Systems with Integral Boundary Conditions and Its Extension

Abstract

The optimal control problem with integral boundary condition is considered. The sufficient condition is established for existence and uniqueness of the solution for a class of integral boundary value problems for fixed admissible controls. First-order necessary condition for optimality is obtained in the traditional form of the maximum principle. The second-order variations of the functional are calculated. Using the variations of the controls, various optimality conditions of second order are obtained.

Pontryagin’s Maximum Principle for the Optimal Control Problems with Multipoint Boundary Conditions

“…Taking into account equalities (24) and (25) in (23), we get the final form for the increment of the functional…”

“…Recently, the optimal control problems with nonlocal conditions are intensively investigated. In the papers [13,[20][21][22][23][24] the necessary optimality conditions for optimal control problems described by the systems of ordinary differential equations with nonlocal conditions were obtained. In these papers the nonlocal conditions contain two-point and integral boundary conditions.…”

Necessary optimality condition for the singular controls in an optimal control problem with nonlocal conditions

“…In the third type the necessary condition degenerates in the classical sense. Actually, the last case of singularity is investigated in [9].…”

“…In spite of the theory of necessary optimality conditions of higher order, while the theory of singular controls has been developed completely for the control problems described by the systems with local conditions [3,5,6,12,13], the study of optimal problems with nonlocal conditions has recently started [1,2,4,[7][8][9][10][11]14]. Now we shall note some earlier works in this direction.…”

“…The present paper is a continuation of the paper [9]. In this paper, we consider an optimal control problem for an ordinary differential equation with integral boundary conditions that generalizes the Cauchy problem.…”