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A Control Problem with Passive Particles Driven by Point Vortices on the Sphere

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Advanced Research in Technologies, Information, Innovation and Sustainability (ARTIIS 2022)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1675))

Abstract

The objective of this study is to control the motion of a passive particle advected by N point vortices in a sphere. The square of the \(L^2\) norm of control, necessary for the system to evolve from a starting point to an end point in an a priori fixed time, must be minimized. If the motion is generated by a single vortex (\(N = 1\)), we show that the system is controllable. The problem is also solved by a direct approach, where the control problem is transformed into a nonlinear optimization problem that is solved numerically. In the case of one (\(N = 1\)), two (\(N = 2\)), or three (\(N = 3\)) point vortices, the numerical results show the existence of near/quasi-optimal control.

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Acknowledgements

Carlos Balsa was partially supported by CeDRI which is financed by FCT within the Project Scope: UIDB/05757/2020.

Sílvio Gama was partially supported by (i) CMUP, member of LASI, which is financed by national funds through FCT - Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144/2020, and (ii) project SNAP NORTE- 01–0145–FEDER–000085, co-financed by the European Regional Development Fund (ERDF) through the North Portugal Regional Operational Programme (NORTE2020) under Portugal 2020 Partnership Agreement.

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Authors and Affiliations

  1. Research Centre in Digitalization and Intelligent Robotics (CeDRI), Instituto Politécnico de Bragança, 5300-253, Bragança, Portugal

    Carlos Balsa

  2. Centro de Matemática da Universidade do Porto, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre s/n, 4169-007, Porto, Portugal

    Sílvio Gama

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  1. Carlos Balsa
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Correspondence to Carlos Balsa .

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Editors and Affiliations

  1. Universidad Estatal Península de Santa, La Libertad, Ecuador

    Teresa Guarda

  2. University of Minho, Guimarães, Portugal

    Filipe Portela

  3. BITrum Research Group, Leon, Spain

    Maria Fernanda Augusto

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Balsa, C., Gama, S. (2022). A Control Problem with Passive Particles Driven by Point Vortices on the Sphere. In: Guarda, T., Portela, F., Augusto, M.F. (eds) Advanced Research in Technologies, Information, Innovation and Sustainability. ARTIIS 2022. Communications in Computer and Information Science, vol 1675. Springer, Cham. https://doi.org/10.1007/978-3-031-20319-0_11

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  • DOI: https://doi.org/10.1007/978-3-031-20319-0_11

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  • Online ISBN: 978-3-031-20319-0

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