2019/05/31 by Antil, Harbir, Verma, Deepanshu, Warma, Mahamadi
#35S15 #49J20 #49K20 #65N30 #65R20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.1906.00032
This paper introduces the notion of state constraints for optimal control problems governed by fractional elliptic PDEs of order s ∈ (0,1). There are several mathematical tools that are developed during the process to study this problem, for instance, the characterization of the dual of the fractional order Sobolev spaces and well-posedness of fractional PDEs with measure-valued datum. These tools are widely applicable. We show well-posedness of the optimal control problem and derive the first order optimality conditions. Notice that the adjoint equation is a fractional PDE with measure as the right-hand-side datum. We use the characterization of the fractional order dual spaces to study the regularity of the state and adjoint equations. We emphasize that the classical case (s=1) was considered by E. Casas in \citeECasas1986a but almost none of the existing results are applicable to our fractional case.