This article investigates the uniform null controllability problem for a system of coupled parabolic equations with a periodic oscillating coefficient. Our approach combines spectral analysis and Carleman estimates. First, we analyze the spectral properties of an elliptic operator with an oscillating coefficient to control the low frequencies. Then the system is allowed to evolve freely to achieve the required decay. In the third step, we establish a Carleman estimate that leads to a suitable observability result. By combining the three steps, we prove the uniform null controllability of the system, which is then used to homogenize the associated coupled parabolic system.
Ammar KhodjaF.BenabdallahA.DupaixC.González-BurgosM. (2009). A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems. Difference Equations and Applications, 1, 427–457.
2.
Ammar-KhodjaF.BenabdallahA.DupaixC.González-BurgosM. (2009). A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems. Journal of Evolution Equations, 9, 267–291.
3.
Ammar-KhodjaF.BenabdallahA.González-BurgosM.de TeresaL. (2011). Recent results on the controllability of linear coupled parabolic problems: A survey. Mathematical Control and Related Fields, 1, 267–306.
4.
BoyerF. (2022). Controllability of linear parabolic equations and systems. Master. France. 2022. hal-02470625v4.
5.
CardanobileS.MugnoloD. (2007). Analysis of a FitzHugh–Nagumo–Rall model of a neuronal network. Mathematical Methods in the Applied Sciences, 30.18, 2281–2308.
6.
CastroC.ZuazuaE. (2000). Low frequency asymptotic analysis of a string with rapidly oscillating density. SIAM Journal on Applied Mathematics, 60, 1205–1233.
7.
CioranescuD.DonatoP. (1999). An introduction to homogenization. Oxford University Press.
8.
CoronJ.-M. (2007). Control and nonlinearity. American Mathematical Society.
9.
CoronJ.-M.GuerreroS. (2009). Null controllability of the n-dimensional stokes system with scalar controls. Journal of Differential Equations, 246, 2908–2921.
10.
CoronJ.-M.GuerreroS.RosierL. (2010). Null controllability of a parabolic system with a cubic coupling term. SIAM Journal on Control and Optimization, 48, 5629–5653.
11.
DoleckiS.RussellD. L. (1977). A general theory of observation and control. SIAM Journal on Control and Optimization, 15, 185–220.
12.
DonatoP.JoseE. C. (2015). Asymptotic behavior of the approximate controls for parabolic equations with interfacial contact resistance. ESAIM: Control, Optimisation and Calculus of Variations, 21, 138–164.
13.
DonatoP.JoseE. C.OnofreiD. (2021). On the approximate controllability of parabolic problems with non-smooth coefficients. Asymptotic Analysis, 122, 395–402.
14.
DonatoP.NabilA. (2001). Approximate controllability of linear parabolic equations in perforated domains. ESAIM: Control, Optimisation and Calculus of Variations, 6, 21–38.
15.
EvansL. C. (1997). Partial differential equations (Vol. 19). American Mathematical Society.
16.
FaellaL.MonsurròS.PerugiaC. (2019). Exact controllability for evolutionary imperfect transmission problems. Journal de Mathématiques Pures et Appliquées, 122, 235–271.
17.
FattoriniO. H.RussellD. L. (1974). Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations. Quarterly of Applied Mathematics, 32.1, 45–69.
18.
FursikovA. V.ImanuvilovO. Y. (1996). Controllability of evolution equations. Lecture notes series, (Vol. 34). Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul.
19.
González-BurgosM.de TeresaL. (2010). Controllability results for cascade systems of coupled parabolic PDEs by one control force. Portugaliae Mathematica, 67, 91–113.
20.
LeivaH. (2005). Controllability of a system of parabolic equations with non-diagonal diffusion matrix. IMA Journal of Mathematical Control and Information, 22, 187–199.
21.
LionsJ. L. (1988). Exact controllability, stabilizability, and perturbations for distributed systems. SIAM Review, 30, 1–68.
22.
LópezA.ZuazuaE. (2002). Uniform null-controllability for the one-dimensional heat equation with rapidly oscillating periodic density. Annales de l’Institut Henri Poincaré C: Analyse Non Linéaire, 19, 543–580.
23.
PedregalP.PeriagoF. (2006). Some remarks on homogenization and exact boundary controllability for the one-dimensional wave equation. Quarterly of Applied Mathematics, 64, 529–546.
24.
TakahashiT.de TeresaL.Wu-ZhangY. (2024). A Kalman condition for the controllability of a coupled system of Stokes equations. Journal of Evolution Equations, 24, 25.
25.
TebouL. (2012). Uniform null controllability of a parabolic equation with rapidly oscillating periodic coefficients. Asymptotic Analysis, 80, 149–170.
26.
TucsnakM.WeissG. (2009). Observation and control for operator semigroups. Birkhäuser advanced texts: Basler Lehrbücher. [Birkhäuser advanced texts: Basel textbooks]. Birkhäuser Verlag.
27.
ZuazuaE. (1994). Approximate controllability for linear parabolic equations with rapidly oscillating coefficients. Control and Cybernetics, 23(4), 793–801.
