Volume 27, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Number of page(s)||31|
|Published online||01 March 2021|
Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation*
Technische Universität München, Chair of Optimal Control, Center for Mathematical Sciences, M17,
2 Universität Konstanz, Department of Mathematics and Statistics, WG Numerical Optimization, Universitätsstraße 10, 78457 Konstanz, Germany.
** Corresponding author: firstname.lastname@example.org
Accepted: 19 September 2020
This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are – in a certain sense – exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures.
Mathematics Subject Classification: 35J20 / 49J52 / 49K20 / 58E17 / 90C29
Key words: Multiobjective optimal control / non-smooth optimization / first-order necessary optimality condition / strong stationarity / semilinear partial differential equation / Pareto front
This research has been partially funded by the German Research Foundation (DFG) through the Priority Programme SPP 1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”, Project P02 “Multiobjective Optimization of Non-smooth PDE-constrained Problems – Switches, State Constraints, and Model Order Reduction”. The first author gratefully acknowledges the support by the International Research Training Group IGDK 1754, funded by the German Research Foundation (DFG) and the Austrian Science Fund (FWF) under project number 188264188/GRK1754.
© EDP Sciences, SMAI 2021
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