Optimal Control Problems For Multisolution Elliptic Type Equations

This dissertation investigates mainly the optimal control problems for severalclasses of multi-solution elliptic type equations.Firstly,under the assumption of con-vexity,we consider an optimal boundary control problem for semilinear multi-solutionelliptic type equations and prove the varional inequality.Secondly,in the non-convexcase,using the relaxed controls we study an optimal control problem for a multi-solutionordinary differential equation with two-point boundary value condition and that for aclass of semilinear elliptic type equations which admit exactly two solutions.We obtainthe Pontryagin's maximum principle.Finally we discuss the existence and nonexistenceof an optimal control problem governed by a class of semilinear elliptic equations.This dissertation consists of three parts.In the first part,i.e.Chapter 2 and Chapter 3 of this dissertation,we discussoptimal boundary control problems for a semilinear elliptic type equation with linearboundary condition and nonlinear boundary condition,respectively.Under the convexcase,by constructing the corresponding penalizing problem and passing to the limit,we give the necessary conditions for the optimal pairs of the original problem in theform of variational inequality.In the second part,i.e.Chapter 4 of this dissertation,we consider an optimalcontrol problem for a class of multi-solution ordinary differential equations with two-point boundary value condition.Since the control domain and the objective functionalmay be non-convex,the corresponding penalizing problem may have no solution.Inthis case,we introduce a new tool,i.e.relaxed control theory.By constructing relaxedpenalizing problem and then passing to the limit,we obtain the Pontryagin's maximumprinciple for the optimal pairs of the original problem.The third part consists of Chapter 5 and this chapter consists of two sections.Inthe first section,under the non-convex case we are devoted to investigate an optimalcontrol problem for a class of semilinear elliptic type equations which admit exactly twosolutions.By analyzing the property of a soltuion of the state equation and establishing the proper energy estimate,we prove the relaxed control method is fit for our problemand gain the Pontryagin's maximum principle.In the second section,by using relaxedcontrols we prove the existence and nonexistence results of a class of optimal controlproblems.