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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 5
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Research Article

Exact controllability of a semilinear reaction–diffusion equation governed by a bilinear control

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Pages 1414-1438 | Received 28 Oct 2020, Accepted 16 Sep 2021, Published online: 14 Oct 2021

References

  • Díaz JI, Tello JI. Mathematical analysis, controllability and numerical simulation of a simple model of avascular tumor growth. Handbook of Numerical Analysis. 2004;12:189–230. DOI:10.1016/S1570-8659(03)12003-0.
  • Keener J, Sneyd J. Mathematical physiology-interdisciplinary. New York (NY): Springer; 1998. (Applied mathematics; 8).
  • Khapalov A. Controllability of partial differential equations governed by multiplicative controls. Springer-Verlag; 2010. (Lecture notes in mathematics; vol. 1995).
  • Cannarsa P, Khapalov A. Multiplicative controllability for reaction–diffusion equations with target states admitting finitely many changes of sign. Discrete Cont Dyn Syst B. 2010;14:1293–1311.
  • Cannarsa P, Floridia G, Khapalov A. Multiplicative controllability for semilinear reaction–diffusion equations with finitely many changes of sign. J Math Pures Appl. 2017;108:425–458.
  • Khapalov A. Global non-negative controllability of the semilinear parabolic equation governed by bilinear control. ESAIM Control Optim Calc Var. 2002;7:269–283.
  • Khapalov AY. Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: a qualitative approach. SIAM J Control Optim. 2003;41:1886–1900.
  • Ouzahra M. Approximate controllability of the semilinear reaction–diffusion equation governed by a multiplicative control. Discrete Cont Dyn Syst B. 2021.
  • Fernández-Cara E. Null controllability of the semilinear heat equation. ESAIM Control Optim Calc Var. 1997;2:87–103.
  • Fursikov AV, Imanuvilov OY. Controllability of evolution equations. Korea: Seoul National University; 1996. (Lecture notes series; Seoul 34).
  • Imanuvilov OY. Controllability of parabolic equations (Russian). Mat Sb. 1995;186(6):109–132, translated from Sb. Math. 1995;186(6):879–900.
  • Zuazua E, Fernández-Cara E. Null and approximate controllability for weakly blowing up semilinear heat equations. Ann Inst H Poincaré Anal Non Linéaire. 2000;17:583–616.
  • Lions JL. Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 1988;30(1):1–68.
  • Zuazua E. Exponential decay for the semilinear wave equation with locally distributed damping. Commun Partial Differ Equ. 1990;15(2):205–235.
  • Zuazua E. Exact controllability for semilinear wave equations in one space dimension. Ann Inst H Poincaré C Non Linear Anal. 1993;10(1):109–129.
  • Ball JM, Marsden JE, Slemrod M. Controllability for distributed bilinear systems. SIAM J Control Optim. 1982;20:575–597.
  • Jidou Khayar M, Ouzahra M. Partial controllability of the bilinear reaction–diffusion equation. Int J Dyn Control. 2018;8:197–204.
  • Ouzahra M, Tsouli A, Boutoulout A. Exact controllability of the heat equation with bilinear control. Math Methods Appl Sci. 2015;38:5074–5084.
  • Ouzahra M. Approximate and exact controllability of a reaction–diffusion equation governed by bilinear control. Eur J Control. 2016;32:32–38.
  • Renardy M, Rogers RC. An introduction to partial differential. New York (NY): Springer-Verlag; 2004. (Texts in applied mathematics; 13).
  • Pazy A. Semi-groups of linear operators and applications to partial differential equations. New York (NY): Springer Verlag; 1983.
  • Bensoussan A, Da Prato G, Delfour M, et al. Representation and control of infinite dimensional systems. 2nd ed. Boston: Birkhäuser Boston, Inc.; 2007.
  • Brezis H, Marcus M. Hardy's inequalities revisited. Annali della Scuola Normale Superiore di Pisa – Classe di Scienze. 1997;25:217–237.
  • Barbu V. Analysis and control of nonlinear infinite dimensional system. Vol. 190. Mathematics in science and engineering. Elsevier; 1993. p.1–476.
  • Li X, Yong J. Optimal control theory for infinite dimensional systems. Basel: Birkhauser; 1995.
  • Haraux A, Cazenave T. An introduction to semilinear evolution. Oxford: Clarendon Press; 1998.
  • Attouch H, Buttazzo G, Michaille G. Variational analysis in Sobolev and BV spaces applications to PDEs and optimization. Philadelphia (PA): Society for Industrial and Applied Mathematics; 2006.
  • Fernández-Cara E, Munch A. Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares, and Newton methods. Math Control Relat Fields. 2012;2:217–246.
  • Fernández-Cara E, Munch A. Numerical null controllability of the 1D heat equation: Carleman weights and duality; 2011. Hal-00687887v2.

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