抛物方程Dirichlet边界控制问题的控制稀疏性
The Sparsity of Control for the DirichletBoundary Control Problem is Limitedby the Parabolic Equation
DOI: 10.12677/PM.2023.1310315, PDF, 下载: 97  浏览: 137  国家自然科学基金支持
作者: 文小进, 罗贤兵*：贵州大学数学与统计学院，贵州 贵阳
关键词: 盒子约束； Dirichlet 边界；控制问题；转化解；稀疏性质；Box Constraints； Dirichlet Bounday； Optimal Control Problem； Transformed Solution； Sparse Properties
摘要: 本文考虑了一个凸区域上受限于线性抛物方程的 Dirichlet 边界稀疏最优控制问题，其中目标泛函由一个线性二次泛函 F(·,·)和一个作用于满足盒子约束的控制u上的凸， Lipschitz 连续和Frechet 不可微泛函 j(·)组成。首先，在转化意义下证明状方程转化解和控制问题解存在唯一，并给出最优性系统。然后，针对不同情形下的泛函 j(·) 给出其次微分及控制的稀疏性质。最后给出一个正则性结果。
Abstract: In this paper, we consider a Dirichlet boundary sparse optimal control problem limited by the linear parabolic equation on convex regions. The goal functional consists of a linear quadratic functional F(·,·) and a convex, Lipschitz continuous and FrWchet non-differentiable functional j(·) acting on the control u that satisfies the box constraint. Firstly, we proved the existence and uniqueness of the solution of the state equation in the transformation sense and the control problem, as well as the optimality system. Then, we give the subdifferential of functional j(·) and sparse properties of the control in different cases. Finally, a additional regularity result is given.
文章引用：文小进, 罗贤兵. 抛物方程Dirichlet边界控制问题的控制稀疏性[J]. 理论数学, 2023, 13(10): 3048-3060. https://doi.org/10.12677/PM.2023.1310315

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