Optimal Control Problems Governed By Kirchhoff-type Equations

Kirchhoff-type equation has paid attention to engineers because it is more accurate description on the position change of string vibration.At the same time,the mathematical researchers have paid extensive attention to this problem because Kirchhoff-type equation contains non-local nonlinear terms;it brings essential difficulties to the qualitative theoretical research.On the other hand,as,optimal control theory is an important part of modern control theory.Optimal control problems governed by Kirchhoff-type equation can describe choosing a control strategy which can suppress the negative vibration or make the vibration state nearest to the target with the least energy consumption.Therefore,the research on Kirchhoff-type equation and its corresponding optimal control problem is of great significance.In this thesis,we deal with two kinds of optimal control problems.One is optimal internal control problems governed by the Kirchhoff-type equation with Schrodinger potential and the Kirchhoff-type equation with nonlinear nonhomogeneous respectively.Another is the optimal control problem on boundary governed by Kirchhoff-type equation with Schrodinger potential.The target functional is quadratic form.Theories of existence for solutions of controlled system are established.The sufficient conditions for the existence of optimal controls are given,and the necessary conditions for optimal controls are derived.The thesis is organized as follows:In Chapter 3,the Kirchhoff-type equation with Schrodinger potential and its corresponding optimal control problem are studied.Firstly,the existence and uniqueness of the solution for the controlled system are proved by variational method and proof by contradiction.Secondly,the existence of optimal control is proved by Sobolev embedding theorem and Mazur’s theorem.Finally,basis on the proof of continuous differentiability of control-state operator,the optimality conditions are derived by using cone theory and Dubovitskii-Milyutin formalism.Furthermore,the formalized optimality conditions are obtained by expressing variational inequalities in several cases.In Chapter 4,an optimal control problem governed by Kirchhoff-type equation with nonlinear non-homogeneous term is considered.The existence of solution for the controlled equation is proven.The existence theory of optimal control is given,and several equivalent optimality conditions are derived.The existence of solutions of adjoint equation are proved by Nehari manifolds,and some conditions are given to guarantee the existence of unique solution of adjoint equation.The optimal control problem on boundary governed by Kirchhoff-type equation with Schrodinger potential and the third boundary value condition is studied in Chapter 5.Because of the nonhomogeneous third boundary condition,the variational method fails in showing that the existence of solutions for controlled systems.By establishing auxiliary equations with parameters,combining operator theory,proof by contradiction and Lax-Milgram theorem,we prove the existence of the solution for state equation.Secondly,the existence of optimal control is proved and several equivalent optimality conditions are derived.Finally,the existence of solution for the adjoint equation is given by the solution of auxiliary equation and the geometric analysis method.We should mention that the difficulties caused by the non-local terms of Kirchhoff-type equations.In showing the existence of solutions for controlled systems and optimal controls,we use algebraic and geometric analysis methods,combined with the idea of variational method to overcome these difficulties.Meanwhile,the cone theory is applied to the derivation of optimality condition,which avoids the embarrassment that the solutions for state equation and adjoint system is not unique.The results in this thesis will enrich and develop optimal control theory.