This paper is focused on studying an important concept of the system analysis, which is the regional enlarged observability or constrained observability of the gradient for distributed parabolic systems evolving in the spatial domain Ω We will explore an approach based on the Hilbert Uniqueness Method (HUM), which can reconstruct the initial gradient state between two prescribed functions f1 and f2 only in a critical subregion ω of Ω without the knowledge of the state. Finally, the obtained results are illustrated by numerical simulations.
References
[1]
EL Jai, A. and Pritchard, A.J. (1988) Sensors and Actuators in Distributed Systems Analysis. Ellis Horwood Series in Applied Mathematics, John Wiley & Sons, Hoboken, New Jersey.
[2]
Curtain, R.F. and Pritchard, A.J. (1978) Infinite Dimensional Linear Systems Theory. Lecture Notes in Control and Information Sciences, 8, Springer, New York.
https://doi.org/10.1007/BFb0006761
[3]
Curtain, R.F. and Zwart, H. (1995) An Introduction to Infinite Dimensional Linear Systems Theory. Texts in Applied Mathematics, 21, Springer Verlag, New York.
https://doi.org/10.1007/978-1-4612-4224-6
[4]
Amouroux, M., EL Jai, A. and Zerrik, E. (1994) Regional Observability of Distributed Systems. International Journal of Systems Science, 25, 301-313.
https://doi.org/10.1080/00207729408928961
[5]
EL Jai, A., Simon, M.C. and Zerrik, E. (1993) Regional Observability and Sensors Structures. Sensors and Actuators Journal, 39, 95-102.
https://doi.org/10.1016/0924-4247(93)80204-T
[6]
Zerrik, E., Badraoui, L. and El Jai, A. (1999) Sensors and Regional Boundary State Reconstruction of Parabolic Systems. Sensors and Actuators Journal, 75, 102-117.
https://doi.org/10.1016/S0924-4247(98)00293-3
[7]
Zerrik, E., Bourray, H. and Boutoulout, A. (2002) Regional Boundary Observability: A Numerical Approach. International Journal of Applied Mathematics and Computer Science, 12, 143-151.
[8]
Zerrik, E. and Bourray, H. (2003) Gradient Observability for Diffusion Systems. Int. Journal of Applied Mathematics and Computing, 13, 139-150.
[9]
Boutoulout, A., Bourray, H. and Baddi, M. (2011) Constrained Observability for Parabolic Systems. International Journal of Mathematical Analysis, 5, 1695-1710.
[10]
Boutoulout, A., Bourray, H. and Baddi, M. (2011) Regional Observability with Constraints of the Gradient. International Journal of Pure and Applied Mathematical, 73, 235-253.
[11]
Lions, J.L. (1988) Contrôlabilité exacte perturbations et stabilisation des systèmes distribués, Tome 1, contrôlabilité exacte. Masson, Paris.
[12]
Lions, J.L. (1989) Sur la contrôlabilité exacte élargie. Progress in Nonlinear Differential Equations and Their Applications, 1, 703-727.
[13]
Balakrichnan, A. (1976) Applied Function Analysis. Springer-Verlag, Berlin.
[14]
Pritchard, A.J. and Wirth, A. (1978) Unbounded Control and Observation Systems and Their Duality. SIAM Journal on Control and Optimization, 16, 535-545.
https://doi.org/10.1137/0316036
[15]
Reed, M. and Simon, B. (1972) Methods of Mathematical Physics 1: Functional Analysis. Academic Press, New York.
[16]
Lions, J.L. and Magenes, E. (1968) Problèmes aux limites non homogènes et applications. Vol. 1 and 2, Dunod, Paris.
