Control and numerical approximation of fractional diffusion equations
| dc.contributor.author | Biccari, Umberto | |
| dc.contributor.author | Warma, Mahamadi | |
| dc.contributor.author | Zuazua, Enrique | |
| dc.date.accessioned | 2025-01-23T14:15:49Z | |
| dc.date.available | 2025-01-23T14:15:49Z | |
| dc.date.issued | 2022 | |
| dc.date.updated | 2025-01-23T14:15:49Z | |
| dc.description.abstract | The aim of this chapter is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain Ω. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the L2 and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset ω⊂Ω. Secondly, we move our attention to the exterior control problem, in which the control region O⊂Ωc is located outside Ω. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on ∂Ω. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation. | en |
| dc.description.sponsorship | This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement NO: 694126-DyCon). The work of the three authors is supported by the Air Force Office of Scientific Research (AFOSR) under Award NO: FA9550-18-1-0242. The work of MW is supported by the US Army Research Office (ARO) under Award NO: W911NF-20-1-0115. The work of U.B. and E.Z. is supported by Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain) and by the Elkartek grant KK-2020/00091 CONVADP of the Basque government. The work of E.Z. is funded by the Alexander von Humboldt-Professorship program, the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex and the Transregio 154 Project ‘‘Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks’’, project C08, of the German DFG. | en |
| dc.identifier.citation | Biccari, U., Warma, M., & Zuazua, E. (2022). Control and numerical approximation of fractional diffusion equations. Handbook of Numerical Analysis, 23, 1-58. https://doi.org/10.1016/BS.HNA.2021.12.001 | |
| dc.identifier.doi | 10.1016/BS.HNA.2021.12.001 | |
| dc.identifier.isbn | 9780323850599 | |
| dc.identifier.issn | 1570-8659 | |
| dc.identifier.uri | http://hdl.handle.net/20.500.14454/2242 | |
| dc.language.iso | eng | |
| dc.publisher | Elsevier B.V. | |
| dc.rights | © 2022 Elsevier B.V. | |
| dc.subject.other | Exterior control | |
| dc.subject.other | Finite element method | |
| dc.subject.other | Fractional diffusion equation | |
| dc.subject.other | Fractional Laplacian | |
| dc.subject.other | Interior control | |
| dc.subject.other | Numerical approximation | |
| dc.subject.other | Simultaneous control | |
| dc.title | Control and numerical approximation of fractional diffusion equations | en |
| dc.type | research article | |
| dcterms.accessRights | metadata only access | |
| oaire.citation.endPage | 58 | |
| oaire.citation.startPage | 1 | |
| oaire.citation.title | Handbook of Numerical Analysis | |
| oaire.citation.volume | 23 |