Abstract
In this paper, we extend the notion of regional observability of the gradient for linear systems to a class of semilinear parabolic systems. To reconstruct the gradient in the subregion of the system evolution domain, we begin with the first approach which combines the extension of the HUM method and the fixed point techniques. The analytical case is then tackled using sectorial property of the considered dynamic operator and converted to a fixed point problem. The two approaches lead to algorithms which are successfully implemented numerically and illustrated with examples and simulations.
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L. Afifi, A. El Jai, and M. Merry. Detection and spy sensors in discrete distributed systems. Int. J. Systems Sci. 36 (2005), No. 12, 777–789.
M. Amouroux, A. El Jai, and E. Zerrik. Regional observability of distributed systems. Int. J. Systems Sci. 25 (1994), 301–313.
E. M. Aparron. On the calculation of radiant interchange between surfaces. In: Modern Developments in Heat Transfert, Academic Press, New York (1963), 181212.
A. Boutoulout, H. Bourray, and F. Z. El Alaoui. Regional boundary observability for semilinear systems: Approach and simulation. Int. J. Math. Anal. 4 (2010), No. 24, 1153–1173.
L. Cuniasse-Langhans. Evaluation par méthode inverse de la distribution des transferts de chaleur pariétaux le long dune plaque verticale en convection naturelle. Thése de Doctorat. Toulouse (1998).
F. Chuli and Q. Chunyu. Wavelet and error estimation of surface heat flux. J. Comput. Appl. Math. 150 (2003), 143–155.
R. F. Curtain and A. J. Pritchard. Infinite-Dimensional Linear Systems Theory. Springer-Verlag, Berlin (1978).
D. Henry. Geometry theory of semilinear parabolic systems. Lect. Notes Math., 840 (1981).
K. Kassara and A. El Jai. Algorithme pour la commande d’une classe de systèmes à paramètre répartis non linéaires. Rev. Mar. d’Aut. Trait. Signal 1 (1983), 3–24.
A. Khapalov. Local controllability for a swimming model. SIAM J. Control. Optim. 46 (2007), 655–682.
J. Klamka. Constained controllability of semilinear systems. Nonlin. Anal. 47 (2001), No. 5, 2939–2949.
J. L. Lions. Controlabilité exacte. Perturbation et Stabilisation des systèmes distribués. Masson (1988).
J. L. Lions. Optimal control of systems gouverned by partial differential equations. Springer-Verlag, New York (1971).
A. Pazy. Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York (1990).
E. Zerrik and H. Bourray. Gradient observability for diffusion systems. Int. J. App. Math Comp. Sci. 13 (2003), No. 2, 139–150.
E. Zerrik, H. Bourray, and A. El Jai. Regional flux reconstruction for parabolic systems. Int. J. Systems Sci. 34 (2003), 641–650.
_____ , Regional observability for semilinear distributed parabolic systems. J. Dynam. Control Systems 10 (2004), No. 3, 413–430.
E. Zuazua. Exact controllability for the semilinear wave equation. J. Math. Pures Appl. 69 (1990), 1–31.
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Boutoulout, A., Bourray, H. & El Alaoui, F.Z. Regional gradient observability for distributed semilinear parabolic systems. J Dyn Control Syst 18, 159–179 (2012). https://doi.org/10.1007/s10883-012-9138-3
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DOI: https://doi.org/10.1007/s10883-012-9138-3
Key words and phrases
- Distributed systems
- semilinear parabolic systems
- gradient reconstruction
- regional observability
- fixed point