An approximating approach to an optimal control problem for an elliptic variational inequality on a mixed boundary. (English) Zbl 1493.49010
Summary: In this article, an optimal control problem (OCP) for a system governed by a class of double obstacle elliptic variational inequalities on a mixed boundary is proposed. An approximating optimal control problem \((\mathrm{OCP})_\varepsilon\) is then defined to approximate the original problem (OCP) by a penalty method. Under some assumptions, the existence result for the problem \((\mathrm{OCP})_\varepsilon\) is proved. The sequence of solutions to the problem \((\mathrm{OCP})_\varepsilon\) converges to a solution of the original problem (OCP) as the penalty parameter tends to zero. Next, the necessary optimality conditions are obtained for the solution of the problem \((\mathrm{OCP})_\varepsilon\) with a given concrete penalty function. At last, an algorithm is used to solve a given concrete optimal control problem. Numerical experimental results are presented to show that the approximation method is effective and practical.
MSC:
| 49J40 | Variational inequalities |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
Keywords:
double obstacle elliptic variational inequality; mixed boundary optimal control problem; penalty approximation method; the necessary optimality conditionsReferences:
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