Optimal Control of Partial Differential Equations
Optimal Control of Partial Differential Equations
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DOI：
10.1007/978-3-319-49631-3_1
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
E. Casas;M. Mateos
中科院分区：
文献类型：
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作者：
E. Casas;M. Mateos
在本章中，我们介绍了偏微分方程的最优控制。在解释了什么是最优控制问题以及分析这些问题的目的之后，我们将重点放在一个模型实例上。考虑一类由半线性椭圆方程控制的最优控制问题，该控制受有界约束。然后说明了证明解的存在性的方法；导出了一阶和二阶最优性条件；用离散问题逼近控制问题；为了证明离散化的收敛性并得到一些误差估计。最后给出了一种求解离散问题的数值算法，并给出了一些数值结果。虽然整个分析是针对椭圆控制问题进行的，但在分布控制下，还提出了一些其他控制问题，这表明了控制理论领域的范围和分析所需的各种数学方法。在这些问题中，我们考虑了演化方程、Neumann或Dirichlet边界控制和状态约束的情况。
In this chapter, we present an introduction to the optimal control of partial differential equations. After explaining what an optimal control problem is and the goals of the analysis of these problems, we focus the study on a model example. We consider an optimal control problem governed by a semilinear elliptic equation, the control being subject to bound constraints. Then we explain the methods to prove the existence of a solution; to derive the first and second order optimality conditions; to approximate the control problem by discrete problems; to prove the convergence of the discretization and to get some error estimates. Finally we present a numerical algorithm to solve the discrete problem and we provide some numerical results. Though the whole analysis is done for an elliptic control problem, with distributed controls, some other control problems are formulated, which show the scope of the field of control theory and the variety of mathematical methods necessary for the analysis. Among these problems, we consider the case of evolution equations, Neumann or Dirichlet boundary controls, and state constraints.