Gradient Observability for Semilinear Hyperbolic Systems: Sectorial Approach
Affiliation(s)
TSI Team, MACS Laboratory, Department of Mathematics & Computer, Faculty of Sciences, Moulay Ismail University, Meknes, Morocco.
TSI Team, MACS Laboratory, Department of Mathematics & Computer, Faculty of Sciences, Moulay Ismail University, Meknes, Morocco.
ABSTRACT
The aim of this work is to study the notion of the gradient observability on a subregion ω of the evolution domain Ω for a class of semilinear hyperbolic systems. We show, under some hypothesis, that the gradient reconstruction is achieved following sectorial approach combined with fixed point techniques. The obtained results lead to an algorithm which can be implemented numerically.
KEYWORDS
Distributed Systems, Hyperbolic Systems, Gradient Reconstruction, Regional Observability, Fixed Point, Sectorial Operator
Distributed Systems, Hyperbolic Systems, Gradient Reconstruction, Regional Observability, Fixed Point, Sectorial Operator
Cite this paper
Khazari, A. and Boutoulout, A. (2014) Gradient Observability for Semilinear Hyperbolic Systems: Sectorial Approach. Intelligent Control and Automation, 5, 170-181. doi: 10.4236/ica.2014.53019.
Khazari, A. and Boutoulout, A. (2014) Gradient Observability for Semilinear Hyperbolic Systems: Sectorial Approach. Intelligent Control and Automation, 5, 170-181. doi: 10.4236/ica.2014.53019.
References
[1] El Jai, A., Simon, M.C. and Zerrik, E. (1993) Regional Observability and Sensor Structures. Sensors and Actuators Journal, 39, 95-102.
http://dx.doi.org/10.1016/0924-4247(93)80204-T [2] Zerrik, E., Bourray, H. and El Jai, A. (2004) Regional Observability for Semilinear Distributed Parabolic Systems. Journal of Dynamical and Control Systems, 10, 413-430.
http://dx.doi.org/10.1023/B:JODS.0000034438.72863.ca [3] Boutoulout, A., Bourray, H., El Alaoui, F.Z. and Benhadid, S. (2014) Regional Observability for Distributed Semi-Linear Hyperbolic Systems. International Journal of Control, 87, 898-910. [4] Zerrik, E., Bourray, H. and Benhadid, S. (2007) Sensors and Regional Observability of the Wave Equation. Sensors and Actuators Journal, 138, 313-328.
http://dx.doi.org/10.1016/j.sna.2007.05.017 [5] Boutoulout, A., Bourray, H. and El Alaoui, F.Z. (2013) Boundary Gradient Observability for Semilinear Parabolic Systems: Sectorial Approach. Mathematical Sciences Letters, 2, 45-54.
http://dx.doi.org/10.12785/msl/020106 [6] Zeidler, E. (1990) Nonlinear Functional Analysis and Its Applications II/A Linear Applied Functional Analysis. Springer, Berlin. [7] Boutoulout, A., Bourray, H. and Khazari, A. (2013) Gradient Observability for Hyperbolic System. International Review of Automatic Control (I.RE.A.CO), 6, 263-274. [8] Boutoulout, A., Bourray, H. and El Alaoui, F.Z. (2012) Regional Gradient Observability for Distributed Semilinear Parabolic Systems. Journal of Dynamical and Control Systems, 18, 159-179.
http://dx.doi.org/10.1007/s10883-012-9138-3 [9] Zuazua, E. (1993) Exact Controllability for Semilinear Wave Equations in One Space Dimension. Annalesde l’Institut Henri Poincaré: Analyse Non Linéaire, 10, 109-129. [10] Zuazua, E. (1990) Exact Controllability for the Semilinear Wave Equations. Journal de Mathématiques Pures et Appliquées, 59, 1-31. [11] Lions, J.L. (1988) Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome1, Masson, Paris. [12] Lions, J.L. and Magenes, E. (1968) Problèmes aux limites non homogènes et applications. Vols. 1 et 2, Dunod, Paris. [13] Henry, D. (1981) Geometric Theory of Semilinear Parabolic Systems. Lecture Notes in Mathematics 840, Springer-Verlag, Berlin Heidelberg, New York. [14] Kassara, K. and El Jai, A. (1983) Algorithme pour la commande d’une classe de systèmes à paramètre répartis non linéaires. Revue. Marocaine. D’automatique. et de Traitement de Signal, 1, 3-24.
[1] El Jai, A., Simon, M.C. and Zerrik, E. (1993) Regional Observability and Sensor Structures. Sensors and Actuators Journal, 39, 95-102.
http://dx.doi.org/10.1016/0924-4247(93)80204-T [2] Zerrik, E., Bourray, H. and El Jai, A. (2004) Regional Observability for Semilinear Distributed Parabolic Systems. Journal of Dynamical and Control Systems, 10, 413-430.
http://dx.doi.org/10.1023/B:JODS.0000034438.72863.ca [3] Boutoulout, A., Bourray, H., El Alaoui, F.Z. and Benhadid, S. (2014) Regional Observability for Distributed Semi-Linear Hyperbolic Systems. International Journal of Control, 87, 898-910. [4] Zerrik, E., Bourray, H. and Benhadid, S. (2007) Sensors and Regional Observability of the Wave Equation. Sensors and Actuators Journal, 138, 313-328.
http://dx.doi.org/10.1016/j.sna.2007.05.017 [5] Boutoulout, A., Bourray, H. and El Alaoui, F.Z. (2013) Boundary Gradient Observability for Semilinear Parabolic Systems: Sectorial Approach. Mathematical Sciences Letters, 2, 45-54.
http://dx.doi.org/10.12785/msl/020106 [6] Zeidler, E. (1990) Nonlinear Functional Analysis and Its Applications II/A Linear Applied Functional Analysis. Springer, Berlin. [7] Boutoulout, A., Bourray, H. and Khazari, A. (2013) Gradient Observability for Hyperbolic System. International Review of Automatic Control (I.RE.A.CO), 6, 263-274. [8] Boutoulout, A., Bourray, H. and El Alaoui, F.Z. (2012) Regional Gradient Observability for Distributed Semilinear Parabolic Systems. Journal of Dynamical and Control Systems, 18, 159-179.
http://dx.doi.org/10.1007/s10883-012-9138-3 [9] Zuazua, E. (1993) Exact Controllability for Semilinear Wave Equations in One Space Dimension. Annalesde l’Institut Henri Poincaré: Analyse Non Linéaire, 10, 109-129. [10] Zuazua, E. (1990) Exact Controllability for the Semilinear Wave Equations. Journal de Mathématiques Pures et Appliquées, 59, 1-31. [11] Lions, J.L. (1988) Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome1, Masson, Paris. [12] Lions, J.L. and Magenes, E. (1968) Problèmes aux limites non homogènes et applications. Vols. 1 et 2, Dunod, Paris. [13] Henry, D. (1981) Geometric Theory of Semilinear Parabolic Systems. Lecture Notes in Mathematics 840, Springer-Verlag, Berlin Heidelberg, New York. [14] Kassara, K. and El Jai, A. (1983) Algorithme pour la commande d’une classe de systèmes à paramètre répartis non linéaires. Revue. Marocaine. D’automatique. et de Traitement de Signal, 1, 3-24.