Optimal Control Problems Involving Nonlocal Operators

In this paper,we study two kinds of optimal control problems.One is an optimal control problem governed by a variational inequality involving a nonlocal operator,and the other is an optimal control problem governed by a partial differential equation involving a nonlocal operator.In order to study these optimal control problems,first,we need to study the existence and multiplicity of solutions,and other properties,such as regularity,positivity or sign-changing properties,of those solutions for general variational inequalities and partial differential equations with nonlocal operators.Then,we study the existence and regularity of solutions for the optimal control problems governed by variational inequalities or partial differential equations with nonlocal operators.Finally,we study the optimality conditions for the existence of optimal control.In Chapter 3,we are concerned with the existence of the least energy sign-changing solution for a partial differential equation involving the fractional p-Laplacian.We first establish a constraint manifold,and using the topology degree theory in nonlinear functional,we then prove that the energy functional corresponding to the equation admits a minimum point in the constraint manifold.Finally,we obtain by a quantitative deformation lemma that the minimum point is a least energy sign-changing solution.In Chapter 4,we consider the existence of positive solution for a partial differential equation involving the fractional p-Laplacian,whose nonlinear term does not satisfy the Ambrosetti-Rabinowitz condition.Using the mountain pass theorem,we obtain a solution for the equation.Moreover,by Stampacchia truncation method and previous regularity theory of fractional p-Laplacian,we improve the regularity of the solutions for the equation.Finally,we obtain that the equation admits a positive least energy solution.In Chapter 5,we study an optimal control problem governed by a Krichhoff type variational inequality.After establishing the existence of multiple solutions for the Krichhoff type variational inequality,we prove that the optimal control problem governed by Krichhoff type variational inequality has an optimal solution.In Chapter 6,we study the optimality conditions for an optimal control problem governed by a Krichhoff type equation whose nonlinear term without any monotonicity assumption.We remark that in general the uniqueness of solution for the above equation needs a monotonicity assumption,which will guarantee the differentiability of control-state mapping.The lack of monotonicity will brings us a great difficulty.In order to overcome this difficulty,we represent the solution set of the state equation with finite continuous differentiable mappings by using topological degree theory and implicit function theorem.Then we obtain the first-order optimality conditions of the optimal control problem.In Chapter 7,we investigate the approximate optimality condition for a weakly εefficient solution to an robust multi-objective optimization problem.Under the robust characteristic cone constraint qualification,we obtain approximate optimality conditions for a weakly ε-efficient solution to the multi-objective optimization problem in view of its associated minimax optimization problem.