Optimal control of semilinear elliptic partial differential equations with non-Lipschitzian nonlinearities

Constantin Christof

doi:10.3934/mcrf.2024063

Article Contents

2025, Volume 15, Issue 3: 1049-1065. Doi: 10.3934/mcrf.2024063

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Optimal control of semilinear elliptic partial differential equations with non-Lipschitzian nonlinearities

Constantin Christof^,

Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany

^*Corresponding author: Constantin Christof
^*Corresponding author: Constantin Christof

Received: June 2024

Revised: November 2024

Early access: December 2024

Published: September 2025

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Abstract

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 Fréchet 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.

Keywords:

Optimal control,
nonsmooth optimization,
optimality condition,
KKT-system,
non-Lipschitzian nonlinearity,
semilinear partial differential equation,
porous media flow.

Mathematics Subject Classification: Primary: 49K20, 49K40; Secondary: 49J20.

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References

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