Output optimal control of infinite-dimensional hyperbolic bilinear systems
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Optimal control, Tome 178 (2020), pp. 77-90.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we examine a regional optimal control problem for a class of infinite-dimensional hyperbolic bilinear systems evolving on a spatial domain $\Omega$. We characterize an optimal control that minimizes a cost functional, which is composed of the gap between the desired state and final state using optimality conditions. The approach is successfully illustrated by simulations.
Keywords: hyperbolic system, infinite-dimensional bilinear system, distributed optimal control, regional controllability.
@article{INTO_2020_178_a5,
     author = {E. Zerrik and A. El Kabouss and R. Larhrissi},
     title = {Output optimal control of infinite-dimensional hyperbolic bilinear systems},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {77--90},
     publisher = {mathdoc},
     volume = {178},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_178_a5/}
}
TY  - JOUR
AU  - E. Zerrik
AU  - A. El Kabouss
AU  - R. Larhrissi
TI  - Output optimal control of infinite-dimensional hyperbolic bilinear systems
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 77
EP  - 90
VL  - 178
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_178_a5/
LA  - ru
ID  - INTO_2020_178_a5
ER  - 
%0 Journal Article
%A E. Zerrik
%A A. El Kabouss
%A R. Larhrissi
%T Output optimal control of infinite-dimensional hyperbolic bilinear systems
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 77-90
%V 178
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_178_a5/
%G ru
%F INTO_2020_178_a5
E. Zerrik; A. El Kabouss; R. Larhrissi. Output optimal control of infinite-dimensional hyperbolic bilinear systems. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Optimal control, Tome 178 (2020), pp. 77-90. http://geodesic.mathdoc.fr/item/INTO_2020_178_a5/

[1] Adams R. A., Sobolev Spaces, Academic Press, New York–San Francisco–London, 1975 | MR | Zbl

[2] Ball J. M., Marsden J. E. Slemrod M., “Controllability for distributed bilinear systems”, SIAM J. Control Optim., 20:4 (1982), 575–597 | DOI | MR | Zbl

[3] Beauchard K., “Local controllability and noncontrollability for a 1d wave equation with bilinear control”, J. Differ. Equations., 250:4 (2011), 2064–2098 | DOI | MR | Zbl

[4] Beauchard K., “Local controllability of a one-dimensional beam equation”, SIAM J. Control Optim., 47:3 (2008), 1219–1273 | DOI | MR | Zbl

[5] Brezis H., Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983 | MR

[6] El Jai A., Zerrik E., Afiri L., Systems Theory : Modeling, Analysis and Control, Presse Univ. Perpignan, 2009

[7] El Jai A., Zerrik E., Pritchard A. , “Regional controllability of distributed parameter systems”, Int. J. Control., 62:6 (1995), 1351–1365 | DOI | MR | Zbl

[8] Evans L. C., Partial Differential Equations, Am. Math. Soc., Providence, Rhode Island, 1998 | Zbl

[9] Ferreira S. C. Jr., Martins M. L., Vilela M. J., “A reaction-diffusion model for the growth of avascular tumor”, Phys. Rev., 65:2 (2002), 021907 | MR

[10] Jacques S., “Compact sets in the space $L^p(0,T;B)$”, Ann. Mat. Pura Appl., 146:6 (1987), 65–96 | MR | Zbl

[11] Khapalov A. Y., Controllability of Partial Differential Equations Governed by Multiplicative Controls, Springer-Verlag, 2010 | MR | Zbl

[12] Liang M., “Bilinear optimal control for a wave equation”, Math. Models Meth. Appl. Sci., 9:1 (1999), 45–68 | DOI | MR | Zbl

[13] Ouzahra M., “Controllability of the wave equation with bilinear controls”, Eur. J. Control., 20:2 (2014), 57–63 | DOI | MR | Zbl

[14] Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin–Heidelberg–New York–Tokyo, 1983 | MR | Zbl

[15] Zerrik E., El Kabouss A., “Regional optimal control of a class of bilinear systems”, IMA J. Math. Control Inform., 34:4 (2017), 1157–1175 | MR | Zbl

[16] Zerrik E., El Kabouss A., “Regional optimal control of a class of infinite-dimensional bilinear systems”, Int. J. Control., 90:7 (2017), 1495–1504 | DOI | MR | Zbl

[17] Zerrik E., Larhrissi R., “Regional target control of the wave equation”, Int. J. Syst. Sci., 32:10 (2001), 1233–1242 | DOI | MR | Zbl

[18] Zerrik E., Larhrissi R., “Regional boundary controllability of hyperbolic systems. Numerical approach”, J. Dynam. Control syst., 8:3 (2002), 293–311 | DOI | MR | Zbl

[19] Zerrik E., Ould sidi M., “\textquotedblleft Regional controllability for infinite-dimensional bilinear systems: Approach and simulations”, Int. J. Control., 84:12 (2011), 2108–2116 | DOI | MR | Zbl

[20] Ztot K., Zerrik E., Bourray H., “Regional control problem for distributed bilinear systems: Approach and simulations”, Int. J. Appl. Math. Comput. Sci., 21:3 (2011), 499–580 | DOI | MR