Optimal control and directional differentiability for elliptic quasi-variational inequalities. (English) Zbl 07563231
Summary: 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.
MSC:
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 49J21 | Existence theories for optimal control problems involving relations other than differential equations |
| 49J40 | Variational inequalities |
| 49K21 | Optimality conditions for problems involving relations other than differential equations |
| 46G05 | Derivatives of functions in infinite-dimensional spaces |
Keywords:
quasi-variational inequality; obstacle problem; directional differentiability; sensitivity analysis; optimal control; stationarity conditionsReferences:
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