External optimal control of fractional parabolic PDEs
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 20.

In [Antil et al. Inverse Probl. 35 (2019) 084003.] we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been developed to handle the parabolic problem. We tackle the Dirichlet, Neumann and Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the control/source either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control/source in the exterior. We introduce the notions of weak and very-weak solutions to the fractional parabolic Dirichlet problem. We present an approach on how to approximate the fractional parabolic Dirichlet solutions by the fractional parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.

Reçu le :
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/cocv/2020005
Classification : 49J20, 49K20, 35S15, 65R20, 65N30
Mots-clés : Parabolic PDEs, fractional Laplacian, weak and very-weak solutions, Dirichlet, Neumann, Robin external control problems 
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     author = {Antil, Harbir and Verma, Deepanshu and Warma, Mahamadi},
     title = {External optimal control of fractional parabolic {PDEs}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2020005},
     mrnumber = {4065621},
     zbl = {1444.35144},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2020005/}
}
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Antil, Harbir; Verma, Deepanshu; Warma, Mahamadi. External optimal control of fractional parabolic PDEs. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 20. doi : 10.1051/cocv/2020005. http://www.numdam.org/articles/10.1051/cocv/2020005/

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The first and second authors are partially supported by NSF grants DMS-1818772, DMS-1913004 and the Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036. The third author is partially supported by the Air Force Office of Scientific Research under Award NO: FA9550-18-1-0242.