Skip to main content
Log in

Regional averaged controllability for hyperbolic parameter dependent systems

  • Published:
Control Theory and Technology Aims and scope Submit manuscript

Abstract

The purpose of this paper is to extend the notion of regional controllability for hyperbolic parameter dependent systems. The key idea is the characterization of the averaged regional control with minimal energy. This control steers the state average (with respect to such a parameter) towards the desired state only on a given part of the system evolution domain. In this paper, we give the precis definition and the properties of this new concept. Then, we use an approach based on an extension of the Hilbert uniqueness method devoted to the calculation of the control in two different cases: zone control and pointwise control.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Curtain, H. Zwart. An Introduction to Infinite-dimensional Linear Systems Theory. Texts in Applied Mathematics. New York: Springer, 1995.

    Book  Google Scholar 

  2. A. El Jai. Actuators and controllability of distributed systems. Stabilization of Flexible Structures. Berlin/Heidelberg: Springer, 1990: 109–125.

    Chapter  Google Scholar 

  3. A. El Jai. Regional controllability of distributed parameter systems. International Journal of Control, 1995, 62(6): 1351–1365.

    Article  MathSciNet  Google Scholar 

  4. E. Zerrik, R. Larhrissi. Regional boundary controllability of hyperbolic systems. Numerical approach. Journal of Dynamical And Control Systems, 2002 8(3): 293–311.

    Article  MathSciNet  Google Scholar 

  5. E. Zerrik, A. Kamal, A. Boutoulout. Regional flux target with minimum energy. IEE Proceedings-Control Theory and Applications, 2002, 149(4): 349–356.

    Article  Google Scholar 

  6. E. Zerrik, M. Ould Sidi. Regional controllability of linear and semi linear hyperbolic systems. International Journal of Mathematical Analysis, 2010, 4(44): 2167–2198.

    MathSciNet  MATH  Google Scholar 

  7. E. Zuazua. Averaged control. Automatica, 2014, 50(12): 3077–3087.

    Article  MathSciNet  Google Scholar 

  8. J. Lohéac, E. Zuazua. Averaged controllability of parameter dependent conservative semigroups. Journal of Differential Equations, 2017, 262(3): 1540–1574.

    Article  MathSciNet  Google Scholar 

  9. Qi. Lu, E. Zuazua. Averaged controllability for random evolution partial differential equations. Journal de Mathématiques Pures et Appliquées, 2016, 105(3): 367–414.

    Article  MathSciNet  Google Scholar 

  10. A. Hafdallah, A. Ayadi. Optimal control of electromagnetic wave displacement with an unknown velocity of propagation. International Journal of Control, 2018, 92(11): 2693–2700.

    Article  MathSciNet  Google Scholar 

  11. J. L. Lions, E. Magenes. Problèmes Aux Limites Non Homogènes et Applications. Dunod, Paris, 1968.

    MATH  Google Scholar 

  12. J. L. Lions. Controlabilité exacte, perturbations et stabilisation de systemes distribués. Tome 1. Research in Applied Mathematics, Masson, 1988.

  13. A. El Jai, A. J. Pritchard. Capteurs et actionneurs dans l’analyse des systemes distribués. Research in Applied Mathematics, Masson, Paris, 1986.

    Google Scholar 

Download references

Acknowledgements

The authors wish to acknowledge the editor and anonymous reviewers for their insightful comments, which have improved the quality of this publication. The authors also acknowledge Prof. Carlos Castro, from Polytechnic University of Madrid (Spain), for a fruitful discussion and the referee for the remarks that have improved the final version of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mouna Abdelli.

Additional information

This work was supported by the Directorate-General for Scientific Research and Technological Development (DGRSDT).

Mouna ABDELLI received her M.Sc. degree in Applied Mathematics from Department of Mathematics, Tebessa University, Tebessa, Algeria in 2017. Since 2017 she has been pursuing her Ph.D. degree at Tebessa University. She is currently a researcher within LAMIS (the Laboratory of Mathematics Informatics and Systems). Her research interests include control theory and dynamic systems. E-mail: mouna.abdelli@univ-tebessa.dz.

Abdelhak HAFDALLAH is an associate professor in University Larbi Tebessi in Algeria. He graduated with a Ph.D. in Applied Mathematics from in 2018 at O.E.B. University, Algeria. He is currently a researcher within LAMIS working on optimal control problems with missing data, controllability and sentinel. E-mail: abdelhak.hafdallah@univtebessa.dz.

Meriem LOUAFI is a Ph.D. candidate at the Larbi Tebessi University. She is researching in Applied Mathematics, working exactly on control of some distributed systems with missing data. E-mail: meriem.louafi@univtebessa.dz.

Fayçal MERGHADI received his Ph.D. in Mathematics from Algeria in 2010 at Annaba university, Algeria. He has a comparative evaluation of his studies carried out outside Canada attesting to a Doctorate in Mathematics of functional analysis. He has taught for over 14 years at Tebessa University in Algeria. His area of research focuses on fixed point theory and control theory. Email: faycel.merghadi@univ-tebessa.dz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abdelli, M., Hafdallah, A., Merghadi, F. et al. Regional averaged controllability for hyperbolic parameter dependent systems. Control Theory Technol. 18, 307–314 (2020). https://doi.org/10.1007/s11768-020-0006-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11768-020-0006-5

Keywords

Navigation