Open Access
Issue |
ESAIM: COCV
Volume 29, 2023
|
|
---|---|---|
Article Number | 27 | |
Number of page(s) | 34 | |
DOI | https://doi.org/10.1051/cocv/2023009 | |
Published online | 26 April 2023 |
- R.A. Adams and J.J. Fournier, Sobolev spaces, Second edition. Vol. 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam (2003). [Google Scholar]
- V. Alekseev, V. Tikhomirov and S. Fomin, Optimal control, contemp. Soviet Math., Consultants Bureau, New York (1987). [CrossRef] [Google Scholar]
- V. Barbu, Boundary controllability of phase transition region of a two-phase Stefan problem. Syst. Control Lett. 150 (2021), Paper No. 104896. [CrossRef] [Google Scholar]
- L.C. Berselli and M. Ůžička, Global regularity for systems with p-structure depending on the symmetric gradient. Adv. Nonlinear Anal. 9 (2019) 176–192. [Google Scholar]
- L. Bonifacius and I. Neitzel, Second order optimality conditions for optimal control of quasi-linear parabolic equations. Math. Control Related Fields 8 (2018) 1–34. [CrossRef] [MathSciNet] [Google Scholar]
- F. Boyer, On the penalised hum approach and its applications to the numerical approximation of null-controls for parabolic problems. Congrès National d’Analyse Numérique, 15-58, ESAIM Proc., 41, EDP Sci., Les Ulis (2013). [Google Scholar]
- F. Boyer, F. Hubert and J. Le Rousseau, Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications. SIAM J. Control Optim. 48 (2010) 5357–5397. [CrossRef] [MathSciNet] [Google Scholar]
- C. Carthel, R. Glowinski and J.-L. Lions, On exact and approximate boundary controllabilities for the heat equation: a numerical approach. J. Optim. Theory Appl. 82 (1994) 429–484. [CrossRef] [MathSciNet] [Google Scholar]
- E. Casas and K. Chrysafinos, Analysis and optimal control of some quasi-linear parabolic equations. Math. Control Related Fields 8 (2018) 607. [CrossRef] [MathSciNet] [Google Scholar]
- E. Casas, L.A. Fernández and J. Yong, Optimal control of quasi-linear parabolic equations. Proc. Royal Soc. Edinb. Sect. A 125 (1995) 545–565. [CrossRef] [Google Scholar]
- F.W. Chaves-Silva and S. Guerrero, A uniform controllability result for the Keller-Segel system. Asymp. Anal. 92 (2015) 313–338. [Google Scholar]
- F.W. Chaves-Silva and S. Guerrero, A controllability result for a chemotaxis-fluid model. J. Differ. Equ. 262 (2017) 4863–4905. [CrossRef] [Google Scholar]
- H.R. Clark, E. Fernández-Cara, J. Limaco and L.A. Medeiros, Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities. Appl. Math. Comput. 223 (2013) 483–505. [CrossRef] [MathSciNet] [Google Scholar]
- J. Crank, The mathematics of diffusion, Second edition. Clarendon Press, Oxford (1975). [Google Scholar]
- P. De Carvalho, J. Limaco, D. Menezes and Y. Thamsten, Local null controllability of a class of non-Newtonian incompressible viscous fluids. Evolut. Equ. Control Theory (to appear). [Google Scholar]
- E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur. (3) 3 (1957) 25–43. [Google Scholar]
- Q. Du and M.D. Gunzburger, Analysis of a Ladyzhenskaya model for incompressible viscous flow. J. Math. Anal. Appl. 155 (1991) 21–45. [CrossRef] [MathSciNet] [Google Scholar]
- M. Duprez and P. Lissy, Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control. To appear arXiv:1909.02831 (2019). [Google Scholar]
- C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A: Math. 125 (1995) 31–61. [CrossRef] [Google Scholar]
- E. Fernández-Cara and A. Münch, Numerical exact controllability of the 1D heat equation: duality and Carleman weights. J. Optim. Theory Appl. 163 (2014) 253–285. [CrossRef] [MathSciNet] [Google Scholar]
- E. Fernández-Cara, J. Límaco and S. De Menezes, Theoretical and numerical local null controllability of a Ladyzhenskaya-Smagorinsky model of turbulence. J. Math. Fluid Mech. 17 (2015) 669–698. [CrossRef] [MathSciNet] [Google Scholar]
- E. Fernández-Cara, J. Limaco and I. Marín-Gayte, Theoretical and numerical local null controllability of a quasi-linear parabolic equation in dimensions 2 and 3. J. Franklin Inst. 358 (2021) 2846–2871. [CrossRef] [MathSciNet] [Google Scholar]
- E. Fernández-Cara and A. Münch, Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods. Math. Control Relat. Fields 2 (2012) 217–246. [Google Scholar]
- E. Fernández-Cara and A. Münch, Strong convergent approximations of null controls for the 1D heat equation. SeMA J 61 (2013) 49–78. [Google Scholar]
- E. Fernández-Cara, A. Münch and D.A. Souza, On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations. J. Sci. Comput. 70 (2017) 819–858. [CrossRef] [MathSciNet] [Google Scholar]
- E. Fernández-Cara, D. Nina-Huamán, M.R. Núñez-Chávez and F.B. Vieira, On the theoretical and numerical control of a one-dimensional nonlinear parabolic partial differential equation. J. Optim. Theory Appl. 175 (2017) 652–682. [CrossRef] [MathSciNet] [Google Scholar]
- E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. l’Inst. Henri Poincare (C) Non Linear Anal. 17 (2000) 583–616. [CrossRef] [Google Scholar]
- A.V. Fursikov and O.Yu. Imanuvilov, Controllability of evolution equations. Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). [Google Scholar]
- F. Hecht, The mesh adapting software: BAMG. INRIA report (1998), vol. 250, p. 252, http://wwwrocq.inria.fr/gamma/cdrom/www/bamg/eng.htm. [Google Scholar]
- F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. [CrossRef] [MathSciNet] [Google Scholar]
- T. Hillen and K.J. Painter, A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58 (2009) 183. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- E.F. Keller and L.A. Segel, Initiation of slime mold aggregation viewed as an instability. J Theor. Biol. 26 (1970) 399–415. [CrossRef] [Google Scholar]
- S. Koga and M. Krstic, Single-boundary control of the two-phase Stefan system. Syst. Control Lett. 55 (2020) 597–609. [Google Scholar]
- S. Labbé and E. Trélat, Uniform controllability of semidiscrete approximations of parabolic control systems. Syst. Control Lett. 55 (2006) 597–609. [CrossRef] [Google Scholar]
- I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems. Appl. Math. Optim. 23 (1991) 109–154. [CrossRef] [MathSciNet] [Google Scholar]
- G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur. Comm. PDE 20 (1995) 335–356. [CrossRef] [Google Scholar]
- J. Límaco, M. Clark, A. Marinho, S.B. de Menezes and A.T. Louredo, Null controllability of some reaction-diffusion systems with only one control force in moving domains. Chin. Ann. Math. Ser. B 37 (2016) 29–52. [CrossRef] [Google Scholar]
- X. Liu and X. Zhang, Local controllability of multidimensional quasi-linear parabolic equations. SIAM J. Control Optim. 50 (2012) 2046–2064. [CrossRef] [MathSciNet] [Google Scholar]
- J. Málek, J. Nečas and M. Růžička, On the non-newtonian incompressible fluids. Math. Models Methods Appl. Sci. 3 (1993) 35–63. [CrossRef] [Google Scholar]
- MATLAB, version 7.10.0 (R2010a). The MathWorks Inc., Natick, Massachusetts (2010) [Google Scholar]
- J. Moser, On Harnack’s theorem for elliptic differential equations. Commun. Pure Appl. Math. 14 (1961) 577–591. [CrossRef] [Google Scholar]
- J. Moser, A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17 (1964) 101–134. [CrossRef] [Google Scholar]
- A. Münch and E. Zuazua, Numerical approximation of null controls for the heat equation: ill-posedness and remedies. Inverse Probl. 26 (2010) 085018. [CrossRef] [Google Scholar]
- J. Nash, Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80 (1958) 931–954. [CrossRef] [Google Scholar]
- L. Pan and J. Yong, Optimal control for quasi-linear retarded parabolic systems. ANZIAM J. 42 (2001) 532–551. [CrossRef] [MathSciNet] [Google Scholar]
- P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Analysis and Mach. Intell. 12 (1990) 629–639. [CrossRef] [Google Scholar]
- O. Pironneau, F. Hecht, A. Le Hyaric and J. Morice, FreeFem++. (2020), http://www.freefem.org/ff++/index.htm [Google Scholar]
- R.D. Skeel and M. Berzins, A Method for the Spatial Discretization of Parabolic Equations in One Space Variable. SIAM J. Sci. Stat. Comput. 11 (1990) 1–32. [CrossRef] [Google Scholar]
- M. Teixeira, M. Rincon and I.-S. Liu, Numerical analysis of quenching-heat conduction in metallic materials. Appl. Math. Model. 33 (2009) 2464–2473. [CrossRef] [MathSciNet] [Google Scholar]
- Z. Wang, On chemotaxis models with cell population interactions. Math. Modell. Natl. Phenomena 5 (2010) 173–190. [CrossRef] [EDP Sciences] [Google Scholar]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.