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