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We study the minimisation of a cost functional which measures the misfit onthe boundary of a domain between a component of the solution to a certainparametric elliptic PDE system and a prediction of the values of this solution.We pose this problem as a PDE-constrained minimisation problem for a supremalcost functional in ${\mathrm{L}}^\infty$, where except for the PDE constraintthere is also a unilateral constraint on the parameter. We utiliseapproximation by PDE-constrained minimisation problems in ${\mathrm{L}}^p$ as$p\to\infty$ and the generalised Kuhn-Tucker theory to derive the relevantvariational inequalities in ${\mathrm{L}}^p$ and ${\mathrm{L}}^\infty$. Theseresults are motivated by the mathematical modelling of the novel bio-medicalimaging method of Fluorescent Optical Tomography.

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