# zbMATH — the first resource for mathematics

Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems. (English) Zbl 1185.35153
Summary: We consider elliptic operators $$A$$ on a bounded domain, that are compact perturbations of a selfadjoint operator. We first recall some spectral properties of such operators: localization of the spectrum and resolvent estimates. We then derive a spectral inequality that measures the norm of finite sums of root vectors of $$A$$ through an observation, with an exponential cost. Following the strategy of G. Lebeau and L. Robbiano [Commun. Partial Differ. Equations 20, No. 1–2, 335–356 (1995; Zbl 0819.35071)], we deduce the construction of a control for the non-selfadjoint parabolic problem $$\partial _tu+Au=Bg$$. In particular, the $$L^{2}$$ norm of the control that achieves the extinction of the lower modes of $$A$$ is estimated. Examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of $$A$$.

##### MSC:
 35P15 Estimates of eigenvalues in context of PDEs 35K15 Initial value problems for second-order parabolic equations 47F05 General theory of partial differential operators 47A10 Spectrum, resolvent 49J20 Existence theories for optimal control problems involving partial differential equations
Full Text:
##### References:
 [1] Ammar Khodja, F.; Benabdallah, A.; Dupaix, C., Null controllability of some reaction-diffusion systems with one control force, J. math. anal. appl., 320, 928-943, (2006) · Zbl 1157.93004 [2] Ammar Khodja, F.; Benabdallah, A.; Dupaix, C.; González-Burgos, M., Controllability for a class of reaction-diffusion systems: the generalized Kalman’s condition, C. R. acad. sci. Paris, ser I., 345, 543-548, (2007) · Zbl 1127.93011 [3] Alessandrini, G.; Escauriaza, L., Null-controllability of one-dimensional parabolic equations, ESAIM control optim. calc. var., 14, 284-293, (2008) · Zbl 1145.35337 [4] Agranovich, M.S., Summability of series in root vectors of non-selfadjoint elliptic operators, Funct. anal. appl., 10, 165-174, (1976) · Zbl 0362.35062 [5] Agranovich, M.S., On series with respect to root vectors of operators associated with forms having symmetric principal part, Funct. anal. appl., 28, 151-167, (1994) · Zbl 0819.47025 [6] Boyer, F.; Hubert, F.; Le Rousseau, J., Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations, (2008), preprint [7] Donnelly, H.; Fefferman, C., Nodal sets of eigenfunctions on Riemannian manifolds, Invent. math., 93, 161-183, (1988) · Zbl 0659.58047 [8] Donnelly, H.; Fefferman, C., Nodal sets of eigenfunctions: Riemannian manifolds with boundary, (), 251-262 [9] Dzhanlatyan, L.S., Basis properties of the system of root vectors for weak perturbations of a normal operator, Funct. anal. appl., 28, 204-207, (1994) · Zbl 0873.47011 [10] Erdélyi, A., Asymptotic expansions, (1956), Dover Publications, Inc. New York · Zbl 0070.29002 [11] Fursikov, A.; Imanuvilov, O.Yu., Controllability of evolution equations, Lecture notes ser., vol. 34, (1996), Seoul National University, Research Institute of Mathematics, Global Analysis Research Center Seoul · Zbl 0862.49004 [12] González-Burgos, M.; Pérez-García, R., Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. anal., 46, 123-162, (2006) · Zbl 1124.35026 [13] M. González-Burgos, L. de Teresa, Controllability results for cascade systems of m coupled parabolic pdes by one control force, Port. Math. (2009), in press [14] Gohberg, I.C.; Krein, M.G., Introduction to the theory of linear non-selfadjoint operators, Transl. math. monogr., vol. 18, (1969), Amer. Math. Soc. Providence, RI · Zbl 0181.13504 [15] Grigis, A.; Sjöstrand, J., Microlocal analysis for differential operators, (1994), Cambridge University Press Cambridge · Zbl 0804.35001 [16] Haase, M., The functional calculus for sectorial operators, (2006), Birkhäuser Verlag Basel · Zbl 1101.47010 [17] Hörmander, L., Linear partial differential operators, (1963), Springer-Verlag Berlin · Zbl 0171.06802 [18] Hörmander, L., The analysis of linear partial differential operators, vol. II, (1983), Springer-Verlag Berlin [19] Hörmander, L., The analysis of linear partial differential operators, vol. I, (1990), Springer-Verlag Berlin [20] Jerison, D.; Lebeau, G., Nodal sets of sums of eigenfunctions, (), 223-239 · Zbl 0946.35055 [21] Katsnel’son, V.E., Conditions under which systems of eigenvectors of some classes of operators form a basis, Funct. anal. appl., 1, 122-132, (1967) [22] Kato, T., Perturbation theory for linear operators, (1980), Springer-Verlag Berlin [23] O. Kavian, L. de Teresa, Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var. (2009), doi:10.1051/cocv/2008077 [24] Le Rousseau, J.; Lebeau, G., On Carleman estimates for elliptic and parabolic operators, applications to unique continuation and control of parabolic equations, (2009), preprint [25] Lebeau, G.; Robbiano, L., Contrôle exact de l’équation de la chaleur, Comm. partial differential equations, 20, 335-356, (1995) · Zbl 0819.35071 [26] Lebeau, G.; Zuazua, E., Null-controllability of a system of linear thermoelasticity, Arch. ration. mech. anal., 141, 297-329, (1998) · Zbl 1064.93501 [27] Markus, A.S., Introduction to the spectral theory of polynomial operator pencils, Transl. math. monogr., vol. 71, (1988), Amer. Math. Soc. Providence, RI · Zbl 0678.47005 [28] Miller, L., On the controllability of anomalous diffusions generated by the fractional Laplacian, Math. control signals systems, 3, 260-271, (2006) · Zbl 1105.93015 [29] Micu, S.; Zuazua, E., On the controllability of a fractional order parabolic equation, SIAM J. control optim., 44, 1950-1979, (2006) · Zbl 1116.93022 [30] Pazy, A., Semigroups of linear operators and applications to partial differential equations, (1983), Springer-Verlag New York · Zbl 0516.47023 [31] Russell, D.L., A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Stud. appl. math., 52, 189-221, (1973) · Zbl 0274.35041 [32] de Teresa, L., Insensitizing controls for a semilinear heat equation, Comm. partial differential equations, 25, 39-72, (2000) · Zbl 0942.35028 [33] Treves, F., Introduction to pseudodifferential operators and Fourier integral operators, vol. I, (1980), Plenum Press New York · Zbl 0453.47027 [34] Wang, G., $$L^\infty$$-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. control optim., 47, 1701-1720, (2008) · Zbl 1165.93016
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.