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

A R Safari,M F Mekhtiyev,Y A Sharifov

doi:10.1155/2013/946910

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.

Highlights

Boundary value problems with integral conditions constitute a very interesting and important class of boundary problems
A traditional form of the necessary optimality condition will follow from the increment formula (28) if we show that on the needle-shaped variation ũ(t) = uε(t) the state increment Δ εx(t) has the order ε
The optimal control problem is considered when the state of the system is described by the differential equations with integral boundary conditions

Summary

Introduction

Boundary value problems with integral conditions constitute a very interesting and important class of boundary problems. Note that optimal control problems with integral boundary condition are considered and first-order necessary conditions are obtained in [23–25]. We investigate an optimal control problem in which the state of the system is described by differential equations with integral boundary conditions. Note that this problem is a natural generalization of the Abstract and Applied Analysis. Under the solution of boundary value problem (1)–(3) corresponding to the fixed control parameter u(t), we understand the function x(t) : [0, T] → Rn that is absolutely continuous on [0, T]. For any C ∈ Rn and for each fixed admissible control, boundary value problem (1)–(3) has the unique solution that satisfies the following integral equation:.

First-Order Optimality Condition

Variations of the Functional and Derivation of Legendre-Clebsh Conditions

Conclusion