# zbMATH — the first resource for mathematics

Controllability of a parabolic system with a diffusive interface. (English) Zbl 1319.35078
Summary: We consider a linear parabolic transmission problem across an interface of codimension one in a bounded domain or on a Riemannian manifold, where the transmission conditions involve an additional parabolic operator on the interface. This system is an idealization of a three-layer model in which the central layer has a small thickness $$\delta$$. We prove a Carleman estimate in the neighborhood of the interface for an associated elliptic operator by means of partial estimates in several microlocal regions. In turn, from the Carleman estimate, we obtain a spectral inequality that yields the null controllability of the parabolic system. These results are uniform with respect to the small parameter $$\delta$$.
##### MSC:
 35K51 Initial-boundary value problems for second-order parabolic systems 93B05 Controllability 93C20 Control/observation systems governed by partial differential equations 35Q93 PDEs in connection with control and optimization
Full Text:
##### References:
 [1] M. Bellassoued, Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization, Asymptotic Anal. 35 (2003), 257-279. · Zbl 1137.35388 [2] A. Benabdallah, Y. Dermenjian, and J. Le Rousseau, Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem, J. Math. Anal. Appl. 336 (2007), 865-887. · Zbl 1189.35349 [3] —, On the controllability of linear parabolic equations with an arbitrary control location for stratified media, C. R. Acad. Sci. Paris, Ser I. 344 (2007), 357-362. · Zbl 1115.35055 [4] —, Carleman estimates for stratified media, J. Funct. Anal. 260 (2011), 3645-3677. · Zbl 1218.35238 [5] A. Benabdallah, P. Gaitan, and J. Le Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim. 46 (2007), 1849-1881. · Zbl 1155.35485 [6] A. Doubova, A. Osses, and J.-P. Puel, Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients, ESAIM Control Optim. Calc. Var. 8 (2002), 621-661. · Zbl 1092.93006 [7] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963. · Zbl 0321.35001 [8] —, The Analysis of Linear Partial Differential Operators, vol. III, Springer-Verlag, 1985, Second printing 1994. [9] —, The Analysis of Linear Partial Differential Operators, vol. IV, Springer-Verlag, 1985. [10] D. Jerison and G. Lebeau, Harmonic analysis and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Mathematics, ch. Nodal sets of sums of eigenfunctions, pp. 223-239, The University of Chicago Press, Chicago, 1999. · Zbl 0946.35055 [11] H. Koch and E. Zuazua, A hybrid system of pde’s arising in multi-structure interaction: coupling of wave equations in $$n$$ and $$n-1$$ space dimensions, Recent trends in partial differential equations, Contemp. Math. 409 (2006), 55-77. · Zbl 1108.35110 [12] J. Le Rousseau, Carleman estimates and controllability results for the one-dimensional heat equation with $$BV$$ coefficients, J. Differential Equations 233 (2007), 417-447. · Zbl 1128.35020 [13] J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., DOI:10.1051/cocv/2011168 (2011). · Zbl 1262.35206 [14] J. Le Rousseau and N. Lerner, Carleman estimates for anisotropic elliptic operators with jumps at an interface, Preprint (2011). · Zbl 1319.47038 [15] J. Le Rousseau and L. Robbiano, Carleman estimate for elliptic operators with coefficents with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations, Arch. Rational Mech. Anal. 105 (2010), 953-990. · Zbl 1202.35336 [16] —, Local and global Carleman estimates for parabolic operators with coefficients with jumps at interfaces, Invent. Math. 183 (2011), 245-336. · Zbl 1218.35054 [17] M. Léautaud, Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems, J. Funct. Anal. 258 (2010), 2739-2778. · Zbl 1185.35153 [18] G. Lebeau, Équation des ondes amorties, Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., vol. 19, Kluwer Acad. Publ., Dordrecht, 1996, pp. 73-109. · Zbl 0863.58068 [19] G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm. Partial Differential Equations 20 (1995), 335-356. · Zbl 0819.35071 [20] —, Stabilisation de l’équation des ondes par le bord, Duke Math. J. 86 (1997), 465-491. [21] G. Lebeau and E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Rational Mech. Anal. 141 (1998), 297-329. · Zbl 1064.93501 [22] V. Lescarret and E. Zuazua, Numerical scheme for waves in multi-dimensional media: convergence in asymmetric spaces, Preprint (2010). · Zbl 1311.65112 [23] L. Miller, On the controllability of anomalous diffusions generated by the fractional laplacian, Mathematics of Control, Signals, and Systems 3 (2006), 260-271. · Zbl 1105.93015 [24] —, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 4, 1465-1485. · Zbl 1219.93017 [25] G. Tenenbaum and M. Tucsnak, On the null-controllability of diffusion equations, ESAIM, Control Optim. Calc. Var., DOI: 10.1051/cocv/2010035 (2011). · Zbl 1236.93025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.