We perform the asymptotic analysis of the scalar advection-diffusion equation , , , with respect to the diffusion coefficient ε. We use the matched asymptotic expansion method which allows to describe the boundary layers of the solution. We then use the asymptotics to discuss the controllability property of the solution for .
H.Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
2.
J.-M.Coron and S.Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymptot. Anal.44 (2005), 237–257.
3.
P.Deuring, R.Eymard and M.Mildner, -stability independent of diffusion for a finite element-finite volume discretization of a linear convection-diffusion equation, SIAM J. Numer. Anal.53 (2015), 508–526. doi:10.1137/140961146.
4.
A.V.Fursikov and O.Y.Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, Seoul National University Research Institute of Mathematics, Vol. 34, Global Analysis Research Center, Seoul, 1996.
5.
O.Glass, A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit, Journal of Functional Analysis258 (2010), 852–868. doi:10.1016/j.jfa.2009.06.035.
6.
S.Guerrero and G.Lebeau, Singular optimal control for a transport-diffusion equation, Comm. Partial Differential Equations32 (2007), 1813–1836. doi:10.1080/03605300701743756.
7.
J.Kevorkian and J.D.Cole, Multiple Scale and Singular Perturbation Methods, Applied Mathematical Sciences, Vol. 114, Springer-Verlag, New York, 1996.
8.
S.G.Krantz and H.R.Parks, A Primer of Real Analytic Functions, 2nd edn, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2002.
9.
G.Lebeau and L.Robbiano, Contrôle exact de l’équation de la chaleur, Comm. Partial Differential Equations20 (1995), 335–356. doi:10.1080/03605309508821097.
10.
J.-L.Lions, Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin-New York, 1973.
11.
J.-L.Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Vol. 9, Masson, Paris, 1988, Perturbations. [Perturbations].
12.
P.Lissy, 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, Comptes Rendus Mathematique350 (2012), 591–595. doi:10.1016/j.crma.2012.06.004.
13.
P.Lissy, 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. Differential Equations259 (2015), 5331–5352. doi:10.1016/j.jde.2015.06.031.
14.
F.Marbach, Small time global null controllability for a viscous Burgers’ equation despite the presence of a boundary layer, J. Math. Pures Appl.9(102) (2014), 364–384. doi:10.1016/j.matpur.2013.11.013.
15.
A.Münch, Numerical estimation of the cost of boundary controls for the equation with respect to ε, SEMA-SIMAI Springer series – Recent Advances in PDEs: Analysis, Numerics and controls, 2018, in honor of Prof. Enrique Fernandez-Cara’s 60 birthday, 17.
16.
Y.Ou and P.Zhu, The vanishing viscosity method for the sensitivity analysis of an optimal control problem of conservation laws in the presence of shocks, Nonlinear Anal. Real World Appl.14 (2013), 1947–1974. doi:10.1016/j.nonrwa.2013.02.001.
17.
J.Sanchez Hubert and E.Sánchez-Palencia, Vibration and Coupling of Continuous Systems, Springer-Verlag, Berlin, 1989, Asymptotic methods.
18.
M.Van Dyke, Perturbation Methods in Fluid Mechanics, annotated edn, The Parabolic Press, Stanford, Calif., 1975.
