Abstract
We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general data and locally Hadamard differentiable obstacle mappings, thereby extending in particular the results of our previous work which provided the first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area.
Highlights
Quasi-variational inequalities (QVIs) are generalisations of variational inequalities (VIs) where the constraint set in which the solution is sought depends on the unknown solution itself
We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs
We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general data and locally Hadamard differentiable obstacle mappings, thereby extending in particular the results of our previous work which provided the first differentiability result for QVIs in infinite dimensions
Summary
Quasi-variational inequalities (QVIs) are generalisations of variational inequalities (VIs) where the constraint set in which the solution is sought depends on the unknown solution itself. We develop in this paper theory addressing the matters of existence for (1), directional differentiability of Q and stationarity conditions for optimal control problems with QVI constraints of the form min 1 y − yd 2 + ν u 2.
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