Geodesic
Optimal control for a controlled ill-posed wave equation without requiring the Slater hypothesis
Ural mathematical journal, Tome 6 (2020) no. 1, pp. 84-94 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we investigate the problem of optimal control for an ill-posed wave equation without using the extra hypothesis of Slater i.e. the set of admissible controls has a non-empty interior. Firstly, by a controllability approach, we make the ill-posed wave equation a well-posed equation with some incomplete data initial condition. The missing data requires us to use the no-regret control notion introduced by Lions to control distributed systems with incomplete data. After approximating the no-regret control by a low-regret control sequence, we characterize the optimal control by a singular optimality system.
Keywords: Ill-posed wave equation, No-regret control, Incomplete data, Null-controllability.
Mots-clés : Carleman estimates 
@article{UMJ_2020_6_1_a6,
     author = {Abdelhak Hafdallah},
     title = {Optimal control for a controlled ill-posed wave equation without requiring the {Slater} hypothesis},
     journal = {Ural mathematical journal},
     pages = {84--94},
     year = {2020},
     volume = {6},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UMJ_2020_6_1_a6/}
}
TY  - JOUR
AU  - Abdelhak Hafdallah
TI  - Optimal control for a controlled ill-posed wave equation without requiring the Slater hypothesis
JO  - Ural mathematical journal
PY  - 2020
SP  - 84
EP  - 94
VL  - 6
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/UMJ_2020_6_1_a6/
LA  - en
ID  - UMJ_2020_6_1_a6
ER  - 
%0 Journal Article
%A Abdelhak Hafdallah
%T Optimal control for a controlled ill-posed wave equation without requiring the Slater hypothesis
%J Ural mathematical journal
%D 2020
%P 84-94
%V 6
%N 1
%U http://geodesic.mathdoc.fr/item/UMJ_2020_6_1_a6/
%G en
%F UMJ_2020_6_1_a6
Abdelhak Hafdallah. Optimal control for a controlled ill-posed wave equation without requiring the Slater hypothesis. Ural mathematical journal, Tome 6 (2020) no. 1, pp. 84-94. http://geodesic.mathdoc.fr/item/UMJ_2020_6_1_a6/

[1] Baleanu D., Joseph C., Mophou G., “Low-regret control for a fractional wave equation with incomplete data”, Adv. Difference Equ., 2016:240 (2016), 1–20 | DOI | MR

[2] Baudouin L., De Buhan M., Ervedoza S., “Global Carleman estimates for waves and applications”, Comm. Partial Differential Equations, 38:5 (2013), 823–859 | DOI | MR | Zbl

[3] Berhail A., Omrane A., “Optimal control of the ill-posed Cauchy elliptic problem”, Int. J. Differ. Equ., 2018 (2015), 468918, 1–9 | DOI | MR

[4] Ciprian G., Gal N. J., “A spectral approach to ill-posed problems for wave equations”, Ann. Mat. Pura Appl., 187:4 (2008), 705–717 | DOI | MR | Zbl

[5] A. Hafdallah, A. Ayadi, C. Laouar, “No-regret optimal control characterization for an illposed wave equation”, Int. J. Math. Trends Tech., 45:3 (2017), 283–287 | DOI

[6] Hafdallah A., Ayadi A., “Optimal control of electromagnetic wave displacement with an unknown velocity of propagation”, Published online: 2018, Int. J. Control, 92:11 (2019), 2693–2700 | DOI | MR | Zbl

[7] Hafdallah A., Ayadi A., “Optimal control of a thermoelastic body with missing initial conditions”, Published online: 2018, Int. J. Control, 93:7 (2020), 1570–1576 | DOI | MR

[8] Hafdallah A., “On the optimal control of linear systems depending upon a parameter and with missing data”, Nonlinear Stud., 27:2 (2020), 457–469 | MR

[9] Jacob B., Omrane A., “Optimal control for age-structured population dynamics of incomplete data”, J. Math. Anal. Appl., 370:1 (2010), 42–48 | DOI | MR | Zbl

[10] Lions J. L., Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren Math. Wiss., 170, Springer-Verlag, Berlin, Heidelberg, 1971, 400 pp. | MR | Zbl

[11] Lions J. L., Control of Distributed Singular Systems, Gauthier-Villars, Paris, 1985, 552 pp. | MR

[12] Lions J. L., “Contrôle á moindres regrets des systémes distribués”, C. R. Acad. Sci. Paris. Sér. I, Math., 315:12 (1992), 1253–1257 | MR | Zbl

[13] Lions J. L., No-Regret and Low-Regret Control, in Environment, Economics and Their Mathematical Models, Masson, Paris, 1994

[14] Nakoulima O., Omrane A., Velin J., “On the Pareto control and no-regret control for distributed systems with incomplete data”, SIAM J. Control Optim., 42:4 (2003), 1167–1184 | DOI | MR | Zbl

[15] Savage L. J., The Foundations of Statistics, 2nd, Dover Publ., New York, 1972, 310 pp. | MR | Zbl