【作者】 栾姝；

【导师】 高夯；

【作者基本信息】 东北师范大学 ， 应用数学， 2009， 博士

【摘要】 本文主要研究了几类多解椭圆型方程的最优控制问题．首先，在凸性条件下考虑了半线性多解椭圆型方程的最优边界控制问题，证明了变分不等式．其次，在非凸情形下首次采用松弛控制方法研究了具有两点边值条件的多解常微分方程以及一类只有两个解的半线性椭圆型方程的最优控制问题，得到了Pontryagin’s最大值原理．最后讨论了一类半线性椭圆方程的最优控制的存在性及不存在性问题．全文共分为三部分内容：在第一部分即本文的第二章和第三章中，我们分别讨论了带有线性边界条件以及非线性边界条件的半线性椭圆型方程的最优边界控制问题．在凸性条件下，通过构造相应的惩罚问题并取极限，我们以变分不等式的形式给出了原始问题最优对所满足的必要条件．在第二部分即本文的第四章中，我们考虑了一类具有两点边值条件的多解常微分方程的最优控制问题．由于控制区域及指标泛函可能非凸，那么相应的惩罚问题可能无解．在这样的情形下，我们引入了新的工具即松弛控制理论．通过构造松弛惩罚问题再取极限，从而得到了原问题最优对所满足的Pontryagin’s最大值原理．第三部分包含了本文的第五章，本章分为两节．第一节，在非凸情形下我们致力于一类只有两个解的半线性椭圆型方程的最优控制问题，通过对状态方程解性质的分析并建立适当的能量估计，我们证明了松弛控制方法对本问题的适用性并得到了最优对的Pontryagin’s最大值原理．在第二节，采用松弛控制方法我们证明了一类最优控制问题的存在性及不存在性结果．更多还原

【Abstract】 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.更多还原