Internal stabilization of interconnected heat-wave equations
DOI:
https://doi.org/10.58997/ejde.2023.03Abstract
This article concerns the internal stabilization problem of 1-D interconnected heat-wave equations, where information exchange and the two actuators occur at the adjacent side of the two equations. By designing an inverse back-stepping transformation, the original system is converted into a dissipative target system. Moreover, we investigate the eigenvalues distribution and the corresponding eigenfunctions of the closed-loop system by an asymptotic analysis method. This shows that the spectrum of the system can be divided into two families: one distributed along the a line parallel to the left side of the imaginary axis and symmetric to the real axis, and the other on the left half real axis. Then we work on the properties of the resolvent operator and we verify that the root subspace is complete. Finally, we prove that the closed-loop system is exponentially stable.
For more information see https://ejde.math.txstate.edu/Volumes/2023/03/abstr.html
References
I. Aksikas, A. Alizadeh Moghadam, J. F. Forbes; Optimal linear-quadratic control of coupled parabolic-hyperbolic PDEs., International Journal of Control 90 (2017), no. 10, 2152-2164.
P. Albano, D. Tataru; Carleman estimates and boundary observability for a coupled parabolic- hyperbolic system., Electronic Journal of Differential Equations 204 (2000), no. 22, 15.
G. Avalos, I. Lasiecka, R. Triggiani; Higher regularity of a coupled parabolic-hyperbolic fluid- structure interactive system., Georgian Mathematical Journal 15 (2008), no. 3, 403-437.
G. Avalos, R. Triggiani; Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lame PDE system of fluid-structure interaction., Discrete and Continuous Dynamical Systems. Series S 2 (2009), no. 3, 417-447.
C. Batty, L. Paunonen, D. Seifert; Optimal energy decay for the wave-heat system on a rectangular domain, SIAM Journal on Mathematical Analysis 51 (2019), no. 2, 808-819.
H. Chen, S. Wu; On existence of solutions for some hyperbolic-parabolic-type chemotaxis systems., IMA Journal of Applied Mathematics 72 (2007), no. 3, 331-347.
N. Dunford, J. T. Schwartz; Linear operators. part iii: Spectral operators., John Wiley & Sons, Inc., New York-London-Sydney, 1971.
A. Gerisch, M. Kotschote, R. Zacher; Well-posedness of a quasilinear hyperbolic-parabolic system arising in mathematical biology., Nonlinear Differential Equations and Applications 14 (2007), no. 5-6, 593-624.
J. J. Gu, J. M. Wang; Sliding-mode control of the Orr-Sommerfeld equation cascaded by both Squire equation and ODE in the presence of boundary disturbances., SIAM Journal on Control and Optimization 56 (2018), no. 2, 837-867.
B. Z. Guo, J. M. Wang; Control of wave and beam PDEs: The riesz basis approach, Springer, Switzerland, 2019.
J. Hao, Z. Liu, J. Yong; Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations., Journal of Differential Equations 259 (2015), no. 9, 4763-4798.
X. L. Jin, Y. Li, F. Zheng; Spectrum and stability of a 1-d heat-wave coupled system with dynamical boundary control, Mathematical Problems in Engineering (2019).
M. Krstic, B. Z. Guo, A. Balogh, A. Smyshlyaev; Output-feedback stabilization of an unstable wave equation., Automatica. A Journal of IFAC, the International Federation of Automatic Control 44 (2008), no. 1, 63-74.
M. Krstic, A. Smyshlyaev; Boundary control of PDEs: A course on backstepping designs, Society for Industrial and Applied Mathematics, Philadelphia, 2008.
B. Y. Levin; Lectures on entire functions., Providence, Rhode Island, 1996.
S. Li, P. F. Yao; Stabilization of the Euler-Bernoulli plate with variable coefficients by non- linear internal feedback, Automatica 50 (2014), no. 9, 2225-2233.
Z. H. Luo, B. Z. Guo,o. Morgul; Stability and stabilization of infinite dimensional systems with applications., Springer-Verlag London, Ltd., London, 1999.
B. Muha; A note on optimal regularity and regularizing effects of point mass coupling for a heat-wave system, Journal of Mathematical Analysis and Applications 425 (2015), no. 2, 1134-1147.
M. A. Naimark; Linear differential operators, part i: Elementary theory of linear differential operators., Frederick Ungar Publishing Co., New York, 1967.
A. Pazy; Semigroup of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
J. Rauch, X. Zhang, E. Zuazua; Polynomial decay for a hyperbolic-parabolic coupled system., Journal de Mathematiques Pures et Appliquees. Neuvi`eme Serie 84 (2005), no. 4, 407-470.
A. A. Shkalikov; Boundary value problems for ordinary differential equations with a parame- ter in the boundary conditions., Journal of Soviet Mathematics 33 (1986), no. 4, 1311-1342.
A. Smyshlyaev, E. Cerpa, M. Krstic; Boundary stabilization of a 1-d wave equation with in-domain antidamping., SIAM Journal on Control and Optimization 48 (2010), no. 6, 4014- 4031.
A. Smyshlyaev, M. Krstic; On control design for PDEs with space-dependent diffusivity or time-dependent reactivity., Automatica 41 (2005), 1601-1608.
L. L. Su, J. M. Wang, M. Krstic; Boundary feedback stabilization of a class of coupled hyperbolic equations with nonlocal terms., IEEE Transactions on Automatic Control 63 (2018), no. 8, 2633-2640.
J. M. Wang, M. Krstic; Stability of an interconnected system of Euler-Bernoulli beam and heat equation with boundary coupling., ESAIM: Control, Optimisation and Calculus of Variations 21 (2015), no. 4, 1029-1052.
K. Yang, J. M. Wang; Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay., ESAIM: Control, Optimisation and Calculus of Variations 25 (2019), no. 4, 33.
P. F. Yao; On the observability inequalities for exact controllability of wave equations with variable coefficients., SIAM Journal on Control and Optimization 37 (1999), no. 5, 1568- 1599.
R. M. Young; An introduction to nonharmonic fourier series., Academic Press, Inc., San Diego, CA, 2001.
X. Zhang, E. Zuazua; Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system., Journal of Differential Equations 204 (2004), no. 2, 380-438.
Y. Zhang, Z. Tan, M. B. Sun; Global existence and asymptotic behavior of smooth solutions to a coupled hyperbolic-parabolic system., Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal 14 (2013), no. 1, 465-482.
Downloads
Published
License
Copyright (c) 2023 Xiu-Fang Yu, Jun-Min Wang, Han-Wen Zhang
This work is licensed under a Creative Commons Attribution 4.0 International License.
