论文标题
一种基于偶性的方法，用于求解线性抛物线控制约束最佳控制问题
A duality-based approach for solving linear parabolic control constrained optimal control problems
论文作者
论文摘要
本文涉及由线性抛物线方程控制的最佳控制问题，并受到控制变量的框约束。这种类型的问题在加热和冷却系统中具有重要的应用。通过应用Fenchel二元性方案，我们明确得出了双重问题，其中将原始问题中的控制约束嵌入到双重问题的目标功能中。证明了解决双重问题的解决方案的存在和独特性，并得出了一阶最佳条件。此外，我们讨论了原始问题解决方案与双重问题之间的鞍点属性。原始问题的解决方案可以通过双重问题的解决方案很容易获得。为了以数值方式解决双重问题，我们设计了两个可实施的方法：共轭梯度方法和半齿牛顿方法。解决了三个示例问题，数值结果表明所提出的方法是有效而准确的。
This paper is concerned with the optimal control problem governed by a linear parabolic equation and subjected to box constraints on control variables. This type of problem has important applications in heating and cooling systems. By applying the scheme of Fenchel duality, we derive the dual problem explicitly where the control constraints in primal problem are embedded in the dual problem's objective functional. The existence and uniqueness of the solution to the dual problem are proved and the first-order optimality conditions are also derived. In addition, we discuss the saddle point property between solution of the primal problem and the dual problem. The solution of primal problem can be readily obtained by the solution of the dual problem. To solve the dual problem numerically, we design two implementable method: conjugate gradient method and semismooth Newton method. Three example problems are solved, numerical results show that the proposed method is efficient and accurate.
