Optimal control problem for an elliptic equation which has exactly two solutions (Q2847228)
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scientific article; zbMATH DE number 6205325
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Optimal control problem for an elliptic equation which has exactly two solutions |
scientific article; zbMATH DE number 6205325 |
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Optimal control problem for an elliptic equation which has exactly two solutions (English)
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4 September 2013
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optimal control
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semilinear elliptic equations
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Pontryagin's maximum principle
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relaxed controls
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This article deals with an optimal control problem associated with an elliptic boundary value problem that has two positive solutions. Both the control domain and the cost functional may be nonconvex. The control problem is studied by means of the penalization method. Since it is possible that the penalized problem has no solution, the relaxation theory is used. In this way the space of admissible controls is extended and both the control system and the cost functional become convex. A uniform estimate of the state variable with respect to the control one is established in order to show that the minimum of the original problem is equal to the minimum of the relaxed problem. Finally, necessary conditions of optimality are derived by a limit process.
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