A general existence theorem for a multi-valued control problem

Behrouz Emamizadeh; Yichen Liu; Mohsen Zivari-Rezapour

doi:10.5802/crmath.568

Article de recherche - Équations aux dérivées partielles, Théorie du contrôle

A general existence theorem for a multi-valued control problem

Behrouz Emamizadeh ^1,² ; Yichen Liu ³ ; Mohsen Zivari-Rezapour ⁴

¹ Department of Mathematical Sciences, University of Nottingham Ningbo China, Ningbo, China
² Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
³ Department of Applied Mathematics, School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University, Suzhou, China
⁴ Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 195-202.

Résumé

In this paper, we propose a general existence theorem for a multi-valued control problem. The proof of the theorem is based on a decomposition result of the weak $^{☆}$ closure of the set containing all the multi-valued controls and the bathtub principle. We also obtain the optimality condition for the optimal control.

Reçu le : 2023-06-01
Révisé le : 2023-09-05
Accepté le : 2023-09-05
Publié le : 2024-03-07

DOI : 10.5802/crmath.568

Classification : 49J20, 49K20, 35J20
Mots clés : multi-valued control, existence, uniqueness, decomposition, optimality condition

Affiliations des auteurs :

Behrouz Emamizadeh ^{1, 2} ; Yichen Liu ³ ; Mohsen Zivari-Rezapour ⁴

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Behrouz Emamizadeh and Yichen Liu and Mohsen Zivari-Rezapour},
     title = {A general existence theorem for a multi-valued control problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {195--202},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.568},
     language = {en},
}

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Behrouz Emamizadeh; Yichen Liu; Mohsen Zivari-Rezapour. A general existence theorem for a multi-valued control problem. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 195-202. doi : 10.5802/crmath.568. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.568/

Bibliographie
Cité par

[1] Claudia Anedda; Fabrizio Cuccu Optimal location of resources and Steiner symmetry in a population dynamics model in heterogeneous environments, Ann. Fenn. Math., Volume 47 (2022) no. 1, pp. 305-324 | DOI | MR | Zbl

[2] Geoffrey R. Burton Rearrangements of functions, maximization of convex functionals, and vortex rings, Math. Ann., Volume 276 (1987) no. 2, pp. 225-253 | DOI | MR | Zbl

[3] Geoffrey R. Burton Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 6 (1989) no. 4, pp. 295-319 | DOI | Numdam | MR | Zbl

[4] Geoffrey R. Burton; John B. McLeod Maximisation and minimisation on classes of rearrangements, Proc. R. Soc. Edinb., Sect. A, Math., Volume 119 (1991) no. 3-4, pp. 287-300 | DOI | MR | Zbl

[5] Sagun Chanillo; Daniel Grieser; Masaki Imai; Kazuhiro Kurata; Isamu Ohnishi Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Commun. Math. Phys., Volume 214 (2000) no. 2, pp. 315-337 | DOI | MR | Zbl

[6] Steven J. Cox; Joyce R. McLaughlin Extremal eigenvalue problems for composite membranes. I, II, Appl. Math. Optim., Volume 22 (1990) no. 2, p. 153-167, 169–187 | MR | Zbl

[7] Fabrizio Cuccu; Behrouz Emamizadeh; Giovanni Porru Optimization of the first eigenvalue in problems involving the $p$ -Laplacian, Proc. Am. Math. Soc., Volume 137 (2009) no. 5, pp. 1677-1687 | DOI | MR | Zbl

[8] Behrouz Emamizadeh; Yichen Liu Constrained and unconstrained rearrangement minimization problems related to the $p$ -Laplace operator, Isr. J. Math., Volume 206 (2015) no. 1, pp. 281-298 | DOI | MR | Zbl

[9] Behrouz Emamizadeh; Yichen Liu Bang-bang and multiple valued optimal solutions of control problems related to quasi-linear elliptic equations, SIAM J. Control Optim., Volume 58 (2020) no. 2, pp. 1103-1117 | DOI | MR | Zbl

[10] David Gilbarg; Neil S. Trudinger Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001 | DOI

[11] Antoine Henrot; Hervé Maillot Optimization of the shape and the location of the actuators in an internal control problem, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), Volume 4 (2001) no. 3, pp. 737-757 | MR | Zbl

[12] Antoine Henrot; Michel Pierre Shape variation and optimization. A geometrical analysis, EMS Tracts in Mathematics, 28, European Mathematical Society, 2018, xi+366 pages | DOI

[13] Kazuhiro Kurata; Junping Shi Optimal spatial harvesting strategy and symmetry-breaking, Appl. Math. Optim., Volume 58 (2008) no. 1, pp. 89-110 | DOI | MR | Zbl

[14] Elliott H. Lieb; Michael Loss Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001

[15] Yichen Liu; Behrouz Emamizadeh Rearrangement minimization problems with indefinite external forces, Nonlinear Anal., Theory Methods Appl., Volume 145 (2016), pp. 162-175 | MR | Zbl

[16] Yichen Liu; Behrouz Emamizadeh Converse symmetry and intermediate energy values in rearrangement optimization problems, SIAM J. Control Optim., Volume 55 (2017) no. 3, pp. 2088-2107 | MR | Zbl

[17] Monica Marras Optimization in problems involving the $p$ -Laplacian, Electron. J. Differ. Equ. (2010), 02, 10 pages | MR | Zbl

[18] Monica Marras; Giovanni Porru; Stella Vernier-Piro Optimization problems for eigenvalues of $p$ -Laplace equations, J. Math. Anal. Appl., Volume 398 (2013) no. 2, pp. 766-775 | DOI | MR | Zbl

[19] Olivier Pironneau Optimal shape design for elliptic systems, Springer Series in Computational Physics, Springer, 1984, xii+168 pages | DOI

[20] Mohsen Zivari-Rezapour; Yichen Liu; Behrouz Emamizadeh A formula for the sum of $n$ weak $^{☆}$ closed sets in $L^{\infty}$ , C. R. Math. Acad. Sci. Paris, Volume 361 (2023), pp. 1635-1639 | MR

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