Rachıd LARHRISSI, Mustapha BENOUDİ

Regional enlarged controllability of a fractional derivative of an output linear system

This new research aims to extend the topic of the enlarged controllability of a fractional output linear system. Thus, we characterize the optimal control by two methods, ensuring that the Riemann-Liouville fractional derivative of the final state of the considered system lies between two given functions on a subregion of the evolution domain. Firstly, we transform the considered problem into the saddle point using the Lagrangian multiplier approach. Then, in the second one, we provide the technique of the subdifferential, which allows us to present the cost-explicit formula of the minimum energy control. Moreover, we construct an algorithm of Uzawa type to illustrate the theoretical results obtained through numerical simulations.

PDF

___

Oldham, K., & Spanier, J. (1974). The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier.
El Jai A., & El Yacoubi, S. (1993). On the number of actuators in parabolic system. International Journal of Applied Mathematics and Computer Science, 3(4), 673–686.
Zerrik, E. (1994). Controlabilite et observabilite regionales d’une classe de systemes distribues. PhD thesis Perpignan.
El Jai, A., Simon, C., & Zerrik, E., Pritchard, J. (1995). Regional controllability of distributed parameter systems. International Journal of Control, 62(6), 1351–1365.
Zerrik, E., Boutoulout, A., & El Jai A. (2002). Actuators and regional boundary controllability of parabolic systems. International Journal of Systems Science, 31(1), 73–82.
Zerrik, E., Boutoulout, A., & Bourray, H. (2001). Boundary strategic actuators. Sensors and Actuators A: Physical, 94(3), 197–203.
Zerrik, E., & Ghafrani, F. (2002). Minimum energy control subject to output constraints: numerical approach. IEE Proceedings-Control Theory and Applications, 149(1), 105–110.
Zerrik, E., & Ghafrani, F. (2003). Regional gradient-constrained control problem. Approaches and simulations. Journal of dynamical and control systems, 9(4), 585–599.
Zerrik, E., Boutoulout, A., & Kamal, A. (1996). Regional gradient controllability of parabolic systems. International Journal of Applied Mathematics and Computer Science, 9(4), 767–787.
Benoudi, M., & Larhrissi, R. (2023). Fractional controllability of linear hyperbolic systems. International Journal of Dynamics and Control, 11(3), 1375-1385.
Xie, J., & Dubljevic, S. (2019). Discrete-time Kalman filter design for linear infinite-dimensional systems. Processes, 7(7), 451.
Pazy, A. (1983). Semi-Groups of Linear Operators and Applications to Partial Differential Equations. Springer Verlag, New York.
Podlubny, I., & Chen, Q. (2007). Adjoint fractional differential expressions and operators. International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 4806, 1385–1390.
Miller, S., & Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. Wiley.
Ioffe, A. D. (2012). On the theory of sub differentials. Advances in Nonlinear Analysis, 1, 47-120.
Labrousse, J-Ph., M'bekhta, M. (1992). Les opérateurs points de continuité pour la conorme et l’inverse de Moore-Penrose. Houston Journal of Mathematics, 18(1), 7-23.
Yosida, K. (1980). Functional analysis. Springer Verlag, Berlin-Heidelberg, New York.
Fortin, M., & Glowinski, R. (2000). Augmented Lagrangian methods: applications to the numerical solution of boundary value problems. Elsevier, North-Holland-Amsterdam, New York, Oxford