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

Please always quote using this URN: urn:nbn:de:0297-zib-11933
  • The paper 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, 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 and the two notions of solutions of Casas and Alibert-Raymond, and show that they are the same. As side results, we reformulate the solution in the sense of Stampacchia, and prove the existence and uniqueness of solutions in in case of mixed boundary conditions.

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Author:Christian Meyer, Lucia Panizzi, Anton Schiela
Document Type:ZIB-Report
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Dxx Generalized solutions / 35D99 None of the above, but in this section
46-XX FUNCTIONAL ANALYSIS (For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx) / 46Nxx Miscellaneous applications of functional analysis [See also 47Nxx] / 46N10 Applications in optimization, convex analysis, mathematical programming, economics
49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Kxx Optimality conditions / 49K20 Problems involving partial differential equations
Date of first Publication:2010/12/22
Series (Serial Number):ZIB-Report (10-28)