In this paper we consider the heat equation with Neumann, Robin and mixed boundary conditions (with coefficients on the boundary which depend on the space variable). The main results concern the behaviour of the cost of the null controllability with respect to the diffusivity when the control acts in the interior. First, we prove that if we almost have Dirichlet boundary conditions in the part of the boundary in which the flux of the transport enters, the cost of the controllability decays for a time T sufficiently large. Next, we show some examples of Neumann and mixed boundary conditions in which for any time T > 0 the cost explodes exponentially as the diffusivity vanishes. Finally, we study the cost of the problem with Neumann boundary conditions when the control is localized in the whole domain.
Keywords: Heat equation, singular limits, spectral decomposition, transport equation, uniform controllability
@article{COCV_2021__27_1_A108_0,
author = {B\'arcena-Petisco, Jon Asier},
title = {Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021103},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021103/}
}
TY - JOUR AU - Bárcena-Petisco, Jon Asier TI - Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021103/ DO - 10.1051/cocv/2021103 LA - en ID - COCV_2021__27_1_A108_0 ER -
%0 Journal Article %A Bárcena-Petisco, Jon Asier %T Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021103/ %R 10.1051/cocv/2021103 %G en %F COCV_2021__27_1_A108_0
Bárcena-Petisco, Jon Asier. Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 106. doi: 10.1051/cocv/2021103
[1] , and , Vol. 343 of Fourier analysis and nonlinear partial differential equations. Springer Science & Business Media (2011).
[2] , Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit. SIAM J. Control Optim. 58 (2020) 1597–1625.
[3] , , , and , Control of hyperbolic and parabolic equations on networks and singular limits. Preprint hal-03233211 (2021).
[4] and , Boundary null-controllability of coupled parabolic systems with Robin conditions. Evol. Equ. Control The. 10 (2021) 61–102.
[5] and , Quantitative biomedical optics: theory, methods, and applications. Cambridge University Press (2016).
[6] and , On the non-uniform null controllability of a linear KdV equation. Asymptotic Anal. 94 (2015) 33–69.
[7] and , Uniform null controllability of a linear KdV equation using two controls. J. Math. Anal. Appl. 457 (2018) 922–943.
[8] and , On the cost of null controllability of a fourth-order parabolic equation. J. Differ. Equ. 261 (2016) 6485–6520.
[9] and , An explicit time for the uniform null controllability of a linear Korteweg-de Vriesequation (2021).
[10] and , Spectral inequality and optimal cost of controllability for the Stokes system. ESAIM: COCV 22 (2016) 1137–1162.
[11] and , Controllability and observability of an artificial advection–diffusion problem. Math. Control Signal 24 (2012) 265–294.
[12] and , On the cost of null-control of an artificial advection-diffusion problem. ESAIM: COCV 19 (2013) 1209–1224.
[13] , Control and Nonlinearity. Number 136. American Mathematical Soc. (2007).
[14] and , Singular optimal control: a linear 1-D parabolic–hyperbolic example. Asymptotic Anal. 44 (2005) 237–257.
[15] and , Sharp observability estimates for heat equations. Arch. Ratl. Mech. An. 202 (2011) 975–1017.
[16] , Partial Differential Equation. American Mathematical Society (2010), second edition.
[17] , , and , Null controllability of the heat equation with boundary Fourier conditions: the linear case. ESAIM: COCV 12 (2006) 442–465.
[18] and , Global carleman inequalities for parabolic systems and applications to controllability. SIAM J. Control. Optim. 45 (2006) 1395–1446.
[19] and , Controllability of evolution equations. Number 34. Seoul National University (1996).
[20] , A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit. J. Funct. Anal. 258 (2010) 852–868.
[21] and , On the uniform controllability of the Burgers equation. SIAM J. Control Optim. 46 (2007) 1211–1238.
[22] and , Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit. Asymptotic Anal. 60 (2008) 61–100.
[23] and , Uniform controllability of a transport equation in zero diffusion–dispersion limit. Math. Mod. Meth. Appl. S. 19 (2009) 1567–1601.
[24] and , Singular optimal control for a transport-diffusion equation. Commun. Part. Diff. Eq. 32 (2007) 1813–1836.
[25] , 100 years of Weyl’s law. B. Math. Sci. 6 (2016) 379–452.
[26] , Uniform controllability of a transport equation in zero fourth order equation-dispersion limit. Preprint hal-03080969 (2020).
[27] , and , Vol. 23 of Linear and quasi-linear equations of parabolic type. American Mathematical Soc. (1988).
[28] and , On uniform observability of gradient flows in the vanishing viscosity limit. J. de l’École polytechnique–Math. 8 (2021) 439–506.
[29] and , Contrôle exact de l’équation de la chaleur. Commun. Part. Diff. Eq. 20 (1995) 335–356.
[30] , Contrôlabilité exacte, perturbations et stabilisation de systems distribués, tome 1, RMA 8 (1988).
[31] and , A generique uniqueness result for the Stokes system and its control theoretical consequences. Part. Differ. Equ. Appl. 177 (1996) 221–235.
[32] , A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation. C. R. Math. Acad. Sci. Paris 350 (2012) 591–595.
[33] , An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection–diffusion equation in the vanishing viscosity limit. Syst. Control Lett. 69 (2014) 98–102.
[34] , Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport–diffusion equation. J. Differ. Equ. 259 (2015) 5331–5352.
[35] and , Uniform null controllability of a fourth-order parabolic equation with a transport term. J. Math. Anal. Appl. 498 (2021) 124979.
[36] , Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. Siam Rev. 20 (1978) 639–739.
Cité par Sources :
