OPTIMAL CONTROL OF A NON-SMOOTH SEMILINEAR ELLIPTIC EQUATION

被引：59

作者：

Christof, Constantin ^{[1
]}

Meyer, Christian ^{[1
]}

Walther, Stephan ^{[1
]}

Clason, Christian ^{[2
]}

机构：

[1] TU Dortmund, Fac Math, Vogelpothsweg 87, D-44227 Dortmund, Germany

[2] Univ Duisburg Essen, Fac Math, Thea Leymann Str 9, D-45127 Essen, Germany

来源：

MATHEMATICAL CONTROL AND RELATED FIELDS | 2018年 / 8卷 / 01期

关键词：

Optimal control of PDEs; non-smooth optimization; Bouligand subdifferential; strong stationarity; semi-smooth Newton method; STRONG STATIONARITY; VARIATIONAL-INEQUALITIES; OPTIMIZATION PROBLEMS; OBSTACLE PROBLEM; MPECS;

D O I：

10.3934/mcrf.2018011

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.

引用

页码：247 / 276

页数：30

共 40 条

[1]

[Anonymous], 2006, GRUNDLEHREN MATH WIS

[2]

[Anonymous], 2010, Graduate studies in Mathematics

[3]

[Anonymous], 1990, Optimal control of nonsmooth distributed parameter systems

[4]

[Anonymous], 2002, NONCON OPTIM ITS APP

[5]

[Anonymous], 1990, OPTIMIZATION NONSMOO

[6]

[Anonymous], 1983, Elliptic Partial Equations of Second Order

[7]

Barbu V., 1984, OPTIMAL CONTROL VARI, V100, DOI DOI 10.1007/BF01442167

[8] Optimal control of problems governed by abstract elliptic variational inequalities with state constraints [J].

Bergounioux, M .

SIAM JOURNAL ON CONTROL AND OPTIMIZATION, 1998, 36 (01) :273-289

[9]

Brenner S., 2007, The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics

[10]

Cervinka M., 2008, THESIS

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