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Optimal Control of a Passive Particle Advected by a Point Vortex

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Advanced Research in Technologies, Information, Innovation and Sustainability (ARTIIS 2021)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1485))

Abstract

The objective of this work is to develop a mathematical framework for modeling, control and optimization of the movement of a passive particle advected by a point vortex. Dynamic equations are rewritten in polar coordinates, where the control acts only on the radial coordinate. The optimal control found is explicitly time dependent. This framework should provide a sound basis for the design and control of new advanced engineering systems arising in many important classes of applications, some of which encompass underwater gliders and mechanical fishes.

The research effort has been focused in applying necessary conditions of optimality for some class of flow driven dynamic control systems, by using the vortex methods. The control problem of moving a passive particle between two given points driven by this class of flow in a prescribed time and minimizing the energy of the process has been solved by using the maximum principle.

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References

  1. Clarke, F., Ledyaev, Y., Stern, R., Wolenski, P.: Nonsmooth Analysis and Control Theory. Springer, New York (1998). https://doi.org/10.1007/b97650

    Book  MATH  Google Scholar 

  2. Lions, J.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)

    Book  Google Scholar 

  3. Protas, B.: Vortex dynamics models in flow control problems. Nonlinearity 21(9), R203 (2008)

    Article  MathSciNet  Google Scholar 

  4. Mahmoudian, N., Geisbert, J., Woolsey, C.: Dynamics and control of underwater gliders I: steady motions, Virginia Center for Autonomous Systems, Technical report no. VaCAS-2007-01 (2009)

    Google Scholar 

  5. Mahmoudian, N., Woolsey, C.: Dynamics and control of underwater gliders II: motion planning and control, Virginia Center for Autonomous Systems, Technical report no. VaCAS-2010-02 (2010)

    Google Scholar 

  6. Liu, J., Hu, H.: Biological inspiration: from carangiform fish to multijoint robotic fish. J. Bionic Eng. 7, 35–48 (2010)

    Article  Google Scholar 

  7. Pereira, F.L., Grilo, T., Gama, S.: Optimal multi-process control of a two vortex driven particle in the plane. IFAC-PapersOnLine 50(1), 2193–2198 (2017)

    Article  Google Scholar 

  8. Pereira, F.L., Grilo, T., Gama, S.: Optimal power consumption motion control of a fish-like vehicle in a vortices vector field. In: OCEANS 2017, Aberdeen, pp. 1–4. IEEE (2017)

    Google Scholar 

  9. Aref, H.: Integrable, chaotic, and turbulent vortex motion in two-dimensional flows. Annu. Rev. Fluid Mech. 15(1), 345–389 (1983)

    Article  MathSciNet  Google Scholar 

  10. Babiano, A., Boffetta, G., Provenzale, A., Vulpiani, A.: Chaotic advection in point vortex models and two-dimensional turbulence. Phys. Fluids 6(7), 2465–2474 (1994)

    Article  MathSciNet  Google Scholar 

  11. Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, New York (1992)

    Google Scholar 

  12. Newton, P.K.: The N-Vortex Problem - Analytical Techniques. Springer, New York (2001). https://doi.org/10.1007/978-1-4684-9290-3

    Book  MATH  Google Scholar 

  13. Chorin, A.J.: Vorticity and Turbulence. Springer, New York (1994). https://doi.org/10.1007/978-1-4419-8728-0

  14. Gama, S., Milheiro-Oliveira, P.: Statistical properties of passive tracers in a positive four-point vortex model. Phys. Rev. E 62(1), 1424 (2000)

    Article  Google Scholar 

  15. Pontryagin, L., Boltyanskiy, V., Gamkrelidze, R., Mishchenko, E.: Mathematical theory of optimal processes. Interscience Publish, New York (1962)

    Google Scholar 

  16. Arutyunov, A.V., Karamzin, D.Y., Pereira, F.L.: The maximum principle for optimal control problems with state constraints by RV Gamkrelidze: revisited. J. Optim. Theory Appl. 149(3), 474–493 (2011)

    Article  MathSciNet  Google Scholar 

  17. Levi, M.: Classical Mechanics with Calculus of Variations and Optimal Control: An Intuitive Introduction, vol. 69. American Mathematical Society, Providence (2014)

    Google Scholar 

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Acknowledgments

This work was partially supported by CMUP (UID/MAT/00144/2019), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020; by SYSTEC - POCI-01-0145-FEDER-006933/SYSTEC funded by ERDF - COMPETE2020 - FCT/MEC - PT2020; by project MAGIC - POCI-01-0145- FEDER-032485, funded by ERDF NORTE 2020; and by project SNAP - reference NORTE-01-0145-FEDER-000085, co-financed by the European Regional Development Fund (ERDF), through the North Portugal Regional Operational Programme (NORTE2020), under the PORTUGAL 2020 Partnership Agreement.

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Authors and Affiliations

  1. CMUP - Department of Mathematics, FCUP, University of Porto, Porto, Portugal

    G. Marques & S. Gama

  2. CFisUC - Department of Physics, FCTUC, University of Coimbra, Coimbra, Portugal

    T. Grilo

  3. SYSTEC - Department of Electrical and Computer Engineering, FEUP, University of Porto, Porto, Portugal

    F. L. Pereira

Authors
  1. G. Marques
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  2. T. Grilo
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  3. S. Gama
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  4. F. L. Pereira
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Corresponding author

Correspondence to S. Gama .

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Editors and Affiliations

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

    Teresa Guarda

  2. Universidade do Minho, Guimarães, Portugal

    Filipe Portela

  3. Universidade do Minho, Guimarães, Portugal

    Manuel Filipe Santos

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Marques, G., Grilo, T., Gama, S., Pereira, F.L. (2021). Optimal Control of a Passive Particle Advected by a Point Vortex. In: Guarda, T., Portela, F., Santos, M.F. (eds) Advanced Research in Technologies, Information, Innovation and Sustainability. ARTIIS 2021. Communications in Computer and Information Science, vol 1485. Springer, Cham. https://doi.org/10.1007/978-3-030-90241-4_39

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  • DOI: https://doi.org/10.1007/978-3-030-90241-4_39

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-90240-7

  • Online ISBN: 978-3-030-90241-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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