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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 67, 2018 - Issue 10
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Articles

Optimal control problem for cancer invasion parabolic system with nonlinear diffusion

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Pages 1819-1836 | Received 26 Sep 2017, Accepted 02 Jun 2018, Published online: 22 Jun 2018

References

  • Araujo RP, McElwain DLS. A history of the study of solid tumour growth: the contribution of mathematical modelling. Bull Math Biol. 2004;66:1039–1091. doi: 10.1016/j.bulm.2003.11.002
  • Bellomo N, Li N, Maini PK. On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math Models Methods Appl Sci. 2008;18:593–646. doi: 10.1142/S0218202508002796
  • Gatenby RA, Gawlinski ET. A reaction-diffusion model of cancer invasion. Cancer Res. 1996;56:5745–5753.
  • Shangerganesh L, Balachandran K. Solvability of reaction-diffusion model with variable exponents. Math Methods Appl Sci. 2014;37:1436–1448. doi: 10.1002/mma.2905
  • Shangerganesh L, Barani Balan N, Balachandran K. Existence and uniqueness of solutions of degenerate chemotaxis system. Taiwanese J Math. 2014;18:1605–1622. doi: 10.11650/tjm.18.2014.3080
  • de Araujo ALA, de Magalhes PMD. Existence of solutions and optimal control for a model of tissue invasion by solid tumours. J Math Anal Appl. 2015;421:842–877. doi: 10.1016/j.jmaa.2014.07.038
  • Swan GW. Role of optimal control theory in cancer chemotherapy. Math Biosci. 1990;101:237–284. doi: 10.1016/0025-5564(90)90021-P
  • Chiu M, Yu J-L. An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems. Math Biosci Eng. 2007;4:187–203. doi: 10.3934/mbe.2007.4.187
  • Ryu S-U, Yagi A. Optimal control of Keller-Segel equations. J Math Anal Appl. 2001;256:45–66. doi: 10.1006/jmaa.2000.7254
  • Ryu S-U. Optimal control for a parabolic system modelling chemotaxis. Trends Math Inf Center Math Sci. 2003;6:45–49.
  • Ainseba B, Bendahmane M, Ruiz-Baier R. Analysis of an optimal control problem for the tridomain model in cardiac electrophysiology. J Math Anal Appl. 2012;388:231–247. doi: 10.1016/j.jmaa.2011.11.069
  • Gnanavel S, Barani Balan N, Balachandran K. Simultaneous identification of parameters and initial datum of reaction diffusion system by optimization method. Appl Math Model. 2013;37:8251–8263. doi: 10.1016/j.apm.2013.03.052
  • Sakthivel K, Gnanavel S, Barani Balan N, et al. Inverse problem for the reaction diffusion system by optimization method. Appl Math Model. 2011;35:571–579. doi: 10.1016/j.apm.2010.07.024
  • Xiang H, Liu B. Solving the inverse problem of an SIS epidemic reaction-diffusion model by optimal control methods. Comput Math Appl. 2015;70:805–819. doi: 10.1016/j.camwa.2015.05.025
  • Liu X. Insensitizing controls for a class of quasilinear parabolic equations. J Differ Equ. 2012;253:1287–1316. doi: 10.1016/j.jde.2012.05.018
  • Morales-Rodrigo C. Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours. Math Comput Model Dyn Syst. 2008;47:604–613. doi: 10.1016/j.mcm.2007.02.031
  • Swan GW, Vincent TL. Optimal control analysis in the chemotherapy of IgG multiple myeloma. Bull Math Biol. 1977;39:317–337. doi: 10.1007/BF02462912
  • Chakrabarty SP, Hanson FB. Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method. Math Biosci. 2009;219:129–141. doi: 10.1016/j.mbs.2009.03.005
  • Coldman AJ, Murray JM. Optimal control for a stochastic model of cancer chemotherapy. Math Biosci. 2000;168:187–200. doi: 10.1016/S0025-5564(00)00045-6
  • Knopoff DA, Fernndez DR, Torres GA, et al. Adjoint method for a tumor growth PDE-constrained optimization problem. Comput Math Appl. 2013;66:1104–1119. doi: 10.1016/j.camwa.2013.05.028
  • Ledzewicz U, Naghnaeian M, Schattler H. Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics. J Math Biol. 2012;64:557–577. doi: 10.1007/s00285-011-0424-6
  • Quiroga AAI, Fernndez D, Torres GA, et al. Adjoint method for a tumor invasion PDE-constrained optimization problem in 2D using adaptive finite element method. Appl Math Comput. 2015;270:358–368.
  • Bendahmane M, Karlsen KH. Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue. Netw Heterog Media. 2006;1:185–218. doi: 10.3934/nhm.2006.1.185
  • Bendahmane M, Langlais M. A reaction-diffusion system with cross-diffusion modeling the spread of an epidemic disease. J Evol Equ. 2010;10:883–904. doi: 10.1007/s00028-010-0074-y
  • Amann H. Quasilinear evolution equations and parabolic systems. Trans Amer Math Soc. 1986;293:191–227. doi: 10.1090/S0002-9947-1986-0814920-4
  • Amann H. Quasilinear parabolic systems under nonlinear boundary conditions. Arch Ration Mech Anal. 1986;92:153–192. doi: 10.1007/BF00251255
  • Boldrini JL, Fernndez-Cara E, Rojas-Medar MA. A mathematical analysis of an optimal control problem for a generalized Boussinesq model for viscous incompressible flows. Campinas: UNICAMP; 2005.

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