Riesz Basis Property and Exponential Stability for One-Dimensional Thermoelastic System with Variable Coefficients

Guo, Bao-Zhu; Ren, Han-Jing

ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 98

In this paper, we study Riesz basis property and stability for a nonuniform thermoelastic system with Dirichlet-Dirichlet boundary condition, where the heat subsystem is considered as a control to the whole coupled system. By means of the matrix operator pencil method, we obtain the asymptotic expressions of the eigenpairs, which are exactly coincident to the constant coefficients case. We then show that there exists a sequence of generalized eigenfunctions of the system, which forms a Riesz basis for the state space and the spectrum determined growth condition is therefore proved. As a consequence, the exponential stability of the system is concluded.

Classification : 35Q79, 37L15, 47B06
Keywords: Thermoelastic system, variable coefficient, Riesz basis

[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces. Academic Press, Amsterdam, second edition (2003).

[2] B. T. Bilalov, Bases of exponentials, cosines, and sines formed by eigenfunctions of differential operators. Differ. Equ. 39 (2003) 652–657.

[3] A. Day, Heat Conduction within Linear Thermoelasticity. Springer-Verlag, New York (1985).

[4] B. Z. Guo and S. P. Yung, Asymptotic behavior of the eigenfrequency of a one-dimensional linear thermoelastic system. J. Math. Anal. Appl. 213 (1997) 406–421.

[5] B. Z. Guo, Further results for a one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary conditions. ANZIAM J. 43 (2002) 449–462.

[6] B. Z. Guo and J. C. Chen, The first real eigenvalue of a one-dimensional linear thermoelastic system. Comput. Math. Appl. 38 (1999) 249–256.

[7] B. Z. Guo, J. M. Wang and G. D. Zhang, Spectral analysis of a wave equation with Kelvin-Voigt damping. ZAMM Z. Angew. Math. Mech. 90 (2010) 323–342.

[8] B. Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM J. Control Optim. 39 (2001) 1736–1747.

[9] B. Z. Guo and J. M. Wang, Control of Wave and Beam PDEs – The Riesz Basis Approach. Springer-Verlag, Cham (2019).

[10] B. Z. Guo and G. D. Zhang, On spectrum and Riesz basis property for one-dimensional wave equation with Boltzmann damping. ESAIM: COCV 18 (2012) 889–913.

[11] B. Z. Guo and K. Y. Chan, Riesz basis generation, eigenvalues distribution, and exponential stability for a Euler-Bernoulli beam with joint feedback control. Rev. Mat. Complut. 14 (2001) 205–229.

[12] B. Z. Guo, Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients. SIAM J. Control Optim. 40 (2002) 1905–1923.

[13] Z. J. Han and G. Q. Xu, Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping. Math. Methods Appl. Sci. 38 (2015) 94–112.

[14] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, Berlin, second edition (1976).

[15] Z. H. Luo, B. Z. Guo and O. Morgül, Stability and Stabilization of Infinite-Dimensional Systems with Applications. Springer-Verlag, London (1999).

[16] K. S. Liu and Z. Y. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems. Z. Angew. Math. Phys. 48 (1997) 885–904.

[17] Z. Y. Liu and S. M. Zheng, Exponential stability of the semigroup associated with a thermoelastic system. Quart. Appl. Math. 51 (1993) 535–545.

[18] M. A. Naimark, Linear Differential Operators, Part I: Elementary Theory of Linear Differential Operators. Frederick Ungar Publishing Co., New York (1967).

[19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).

[20] M. Renardy, On the type of certain C₀−semigroups. Comm. Partial Differ. Equ. 18 (1993) 1299–1307.

[21] M. A. Shubov, Asymptotic and spectral analysis of the spatially nonhomogeneous Timoshenko beam model. Math. Nachr. 241 (2002) 125–162.

[22] C. Tretter, Spectral problems for systems of differential equations y′ + A₀y = λA₁y with λ-polynomial boundary conditions. Math. Nachr. 214 (2000) 129–172.

[23] C. Tretter, Boundary eigenvalue problems for differential equations Nη = λPη and λ-polynomial boundary conditions. J. Differ. Equ. 170 (2001) 408–471.

[24] J. M. Wang, G. Q. Xu and S. P. Yung, Riesz basis property, exponential stability of variable coefficient Euler-Bernoulli beams with indefinite damping. IMA J. Appl. Math. 70 (2005) 459–477.

[25] J. M. Wang and B. Z. Guo, On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type. J. Franklin Inst. 344 (2007) 75–96.

[26] J. M. Wang, G. Q. Xu and S. P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls. SIAM J. Control Optim. 44 (2005) 1575–1597.