Uniqueness criteria for the adjoint equation in state-constrained elliptic optimal control

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Meyer, Christian ; Panizzi, Lucia ; Schiela, Anton:
Uniqueness criteria for the adjoint equation in state-constrained elliptic optimal control.
In: Numerical Functional Analysis and Optimization. Bd. 32 (2011) Heft 9 . - S. 983-1007.
ISSN 1532-2467
DOI: https://doi.org/10.1080/01630563.2011.587074

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Abstract

The article considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control problems. By a counter example of Serrin [18], it is known that, in the presence of non-smooth data, a standard weak formulation does not ensure uniqueness for such equations. Therefore several notions of solution have been developed that guarantee uniqueness. In this note, we compare different definitions of solutions, namely the ones of Stampacchia [19] and Boccardo-Galouët [4] and the two notions of solutions of [2, 7], and show that they are equivalent. As side results, we reformulate the solution in the sense of [19], and prove the existence of solutions in the sense of [2, 4, 7] in case of mixed boundary conditions.