Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Exponential stability of the semigroup associated with a thermoelastic system

Authors: Zhuangyi Liu and Song Mu Zheng
Journal: Quart. Appl. Math. 51 (1993), 535-545
MSC: Primary 35Q72; Secondary 34G10, 47N20, 73B30, 93C25
DOI: https://doi.org/10.1090/qam/1233528
MathSciNet review: MR1233528
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper it is proved that the semigroup associated with the one-dimensional thermoelastic system with Dirichlet boundary conditions is an exponentially stable $ {C_0}$-semigroup of contraction on the space $ H_0^1 \times {L^2} \times {L^2}$. The technique of the proof is completely different from the usual energy method. It is shown that the exponential decay in $ D\left( A \right)$ recently obtained by Revira is a consequence of our main result. An important application of our main result to the Linear-Quadratic-Gaussian optimal control problem is also discussed.

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  • [Ba] A. V. Balakrishnan, Applied Functional Analysis, second ed., Springer-Verlag, New York, 1981 MR 612793
  • [BLM] J. A. Burns, Z. Y. Liu, and R. E. Miller, Approximations of thermoelastic and viscoelastic control systems, Numer. Funct. Anal. Optim. 12, 79-136 (1991) MR 1125046
  • [CP] R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear System Theory, Springer-Verlag, New York, 1978 MR 516812
  • [D] C. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational. Mech. Anal. 29, 241-271 (1968) MR 0233539
  • [G] J. S. Gibson, The Riccati integral equations for optimal control problems on Hilbert spaces, SIAM J. Control Optim. 17, 537-565 (1979) MR 534423
  • [GA] J. S. Gibson and A. Adamian, Approximation theory for LQG optimal control of flexible structures, SIAM J. Control Optim. 29, No. 1, 1-37 (1991) MR 1088217
  • [GRT] J. S. Gibson, I. G. Rosen, and G. Tao, Approximation in control of thermoelastic systems, SIAM J. Control Optim. 30, No. 5, 1163-1189 (1992) MR 1178657
  • [H] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl. 167, 429-442 (1992) MR 1168599
  • [Hu] F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations 1, 43-56 (1985) MR 834231
  • [PA] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983 MR 710486
  • [RS] R. Racke and Y. Shibata, Global smooth solutions and asymptotic stability in one-dimensional nonlinear thermoelasticity, Arch. Rational Mech. Anal. 116, 1-34 (1992) MR 1130241
  • [RSZ] R. Racke, Y. Shibata, and S. Zheng, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity, preprint of Bonn Univ., 1991 MR 1247439
  • [R] J. E. M. Rivera, Energy decay rate in linear thermoelasticity, Funkcial. Ekvac. (to appear) MR 1172418
  • [S] M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity, Arch. Rational. Mech. Anal. 76, 97-133 (1981) MR 629700

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DOI: https://doi.org/10.1090/qam/1233528
Article copyright: © Copyright 1993 American Mathematical Society

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