Advertisement

Regional Optimal Control Problem Governed by Distributed Bi-linear Hyperbolic Systems

  • Rabie Zine
  • Maawiya Ould Sidi
Regular Papers Control Theory and Applications
  • 21 Downloads

Abstract

This paper considers the regional bi-linear control problem of an important class of hyperbolic systems. The objective is to bring the state solutions at time T close to a desired observations w d only on a sub-region ω along the spatial domain Ω. We prove the existence of solution by minimizing sequence method. The adjoint system of this problem is introduced and used to characterize the optimal control. A numerical approach is developed and illustrated successfully by simulations.

Keywords

Adjoint systems bi-linear optimal control regional analysis simulations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. E. Bradly and S. Lenhart, “Bilinear optimal control of a Kirchhoff plate to a desired profile,” Journal of Optimal Control Applications & Methods, vol. 18, pp. 217–226, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Y. Khapalov, Controllability of Partial Differential Equations Governed by Multiplicative Controls, Springer-Verlag, Berlin, Heidelberg, 2010.CrossRefzbMATHGoogle Scholar
  3. [3]
    H. Hermes, “Local controllability of observables in finite and infinite dimensional nonlinear control systems,” Appl. Math. Optim., vol. 5, pp. 117–125, 1979.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. Ball, J. E. Marsden, and M. Slemrod, “Controllability for distributed bilinear systems,” SIAM J. on Control and Opt., vol. 40, pp. 575–597, 1982. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A. Y. Khapalov, “On bilinear controllability of the parabolic equation with the reaction diffusion term satisfying Newton’s law,” J. Comput. Appl. Math., vol. 21, pp. 1–23, 2002.MathSciNetzbMATHGoogle Scholar
  6. [6]
    A. Y. Khapalov, “Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: a qualitative approach,” Siam J. Control Optim., vol. 41, pp. 1886–1990, 2003. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    K. Beauchard, “Local controllability and noncontrollability for a 1D wave equation with bilinear control,” Journal of Differential Equations, vol. 250, pp. 2064–2098, 2001. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Ouzahra, “Controllability of the wave equation with bilinear controls,” Europpean Journal of Control, vol. 20, pp. 57–63, 2014. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I. E. Harraki and A. Boutoulout, “Controllability of the wave equation via multiplicative controls,” IMA Journal of Mathematical Control and Information, 2016. dwn055. Doi: 10.1093/imamci/dnw055.Google Scholar
  10. [10]
    E. Zerrik and F. Ghafrani, “Regional constrained controllability problem: approaches and simulations,” International Journal of Control, Automation and Systems, vol. 7, no. 2, pp. 297–304, 2009. [click]CrossRefzbMATHGoogle Scholar
  11. [11]
    M. E. Bradly, S. Lenhart, and J. Yong, “Bilinear optimal control of the velocity term in a Kirchhoff plate equation,” Journal of Mathematical Analysis and Applications, vol. 238, pp. 451–467, 1999. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Addou and A. Benbrik, “Existence and uniqueness of optimal control for a distributed-parameter bi-linear system,” Journal of Dynamical and Control Systems, vol. 8, pp. 141–152, 2002. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. E. Jai, A. J. Pritchard, M. C. Simon, and E. Zerrik, “Regional controllability of distributed systems,” International Journal of Control, vol.62, pp. 1351–1365, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    E. Zerrik and M. Ould Sidi, “Regional controllability of linear and semi linear hyperbolic systems,” Int. Journal of Math. Analysis, vol. 4, no. 44, pp. 2167–2198, 2010.MathSciNetzbMATHGoogle Scholar
  15. [15]
    E. Zerrik and M. O. Sidi, “An output controllability of bilinear distributed system,” International Review of Automatic Control, vol. 3, pp. 466–473, 2010.Google Scholar
  16. [16]
    E. Zerrik and M. O. Sidi, “Regional controllability for infinite dimensional distributed bilinear systems,” Int. Journal of Control, vol. 84, pp. 2108–2116, 2011. [click]CrossRefzbMATHGoogle Scholar
  17. [17]
    K. Ztot, E. Zerrik, and H. Bourray, “Regional control problem for distributed bilinear systems,” Int. J. Appl. Math. Comput. Sci., vol. 21, pp. 499–508, 2011. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    E. Zerrik and M. O. Sidi, “Constrained regional control problem for distributed bilinear systems,” IET Cont. Theory. Appl., vol. 7, pp. 1914–1921, 2013. [click]MathSciNetCrossRefGoogle Scholar
  19. [19]
    E. Zerrik and A. E. Kabouss, “Regional optimal control of a class of bilinear systems,” IMA Journal of Mathematical Control and Information, 2016. doi: 10.1093/imamci/dnw015.Google Scholar
  20. [20]
    R. Zine and M. O. Sidi, “Regional optimal control problem with minimum energy for a class of bilinear distributed systems,” IMA J. Math. Control Info., 2017. doi: 10.1093/imamci/dnx022.Google Scholar
  21. [21]
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New-York, 1983.CrossRefzbMATHGoogle Scholar
  22. [22]
    H. Brezis, Analyse Fonctionnelle: Théorie et Application, Masson, 1983.zbMATHGoogle Scholar
  23. [23]
    J. L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, vol. 1 and 2. Dunod, 1968.zbMATHGoogle Scholar
  24. [24]
    J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués, Tome 1, Contrôlabilité Exacte, Masson, 1988.zbMATHGoogle Scholar
  25. [25]
    H. T. Banks and Y. Wang, “Damage detection and characterization in smart material structures,” International Series of Numerical Mathematics, vol. 118, pp. 21–43, 1994.MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences and Humanities in AflajPrince Sattam Bin Abdulaziz UniversityAl KharjKingdom of Saudi Arabia
  2. 2.Department of Mathematics, College of ScienceJouf UniversitySakakahKingdom of Saudi Arabia

Personalised recommendations