OPTIMAL CONTROL OF SEMILINEAR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS WITH NON-LIPSCHITZIAN NONLINEARITIES

被引:0
作者
Christof, Constantin [1 ]
机构
[1] Univ Duisburg Essen, Fac Math, Thea Leymann Str 9, D-45127 Essen, Germany
关键词
Optimal control; nonsmooth optimization; optimality condition; KKT-; system; non-Lipschitzian nonlinearity; semilinear partial differential equation; porous media flow; FINITE-ELEMENT APPROXIMATION; NONSMOOTH;
D O I
10.3934/mcrf.2024063
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We study optimal control problems that are governed by semilinear elliptic partial differential equations that involve non-Lipschitzian nonlinearities. It is shown that, for a certain class of such PDEs, the solution map is Fr & eacute;chet differentiable even though the differential operator contains a nondifferentiable term. We exploit this effect to establish first-order necessary optimality conditions for minimizers of the considered control problems. The resulting KKT-conditions take the form of coupled PDE-systems that are posed in non-Muckenhoupt weighted Sobolev spaces and raise interesting questions regarding the regularity of optimal controls, the derivation of second-order optimality conditions, and the analysis of finite element discretizations.
引用
收藏
页码:1049 / 1065
页数:17
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