A minimisation problem in $L^{\infty }$ with PDE and unilateral constraints

Katzourakis, Nikos

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 60

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We study the minimisation of a cost functional which measures the misfit on the boundary of a domain between a component of the solution to a certain parametric elliptic PDE system and a prediction of the values of this solution. We pose this problem as a PDE-constrained minimisation problem for a supremal cost functional in L$$, where except for the PDE constraint there is also a unilateral constraint on the parameter. We utilise approximation by PDE-constrained minimisation problems in L$$ as p →∞ and the generalised Kuhn-Tucker theory to derive the relevant variational inequalities in L$$ and L$$. These results are motivated by the mathematical modelling of the novel bio-medical imaging method of Fluorescent Optical Tomography.

Classification : 35Q93, 49K20, 49J40
Keywords: Absolute minimisers, calculus of variations in L^∞, PDE-constrained optimisation, generalised Kuhn–Tucker theory, Lagrange multipliers, fluorescent optical tomography, Robin boundary conditions

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