Asymptotic stabilization for Bresse transmission systems with fractional damping
DOI:
https://doi.org/10.58997/ejde.2023.87Keywords:
Bresse system; fractional damping; asymptotic stability; exponential decay; polynomial decay.Abstract
In this article, we study the asymptotic stability of Bresse transmission systems with two fractional dampings. The dissipation mechanism of control is given by the fractional damping term and acts on two equations. The relationship between the stability of the system, the fractional damping index \(\theta\in[0,1]\) and the different wave velocities is obtained. By using the semigroup method, we obtain the well-posedness of the system. We also prove that when the wave velocities are unequal or equal with \(\theta\neq 0\), the system is not exponential stable, and it is polynomial stable. In addition, the precise decay rate is obtained by the multiplier method and the frequency domain method. When the wave velocities are equal with \(\theta=0\), the system is exponential stable.
For more information see https://ejde.math.txstate.edu/Volumes/2023/87/abstr.html
References
M. Akil, H. Badawi, S. Nicaise, A. Wehbe; On the stability of Bresse system with one discontinuous local internal Kelvin-Voigt damping on the axial force, Z. Angew. Math. Phys., 72(3) (2021), 1-27.
M. O. Alves, L. H. Fatori, M. A. Jorge Silva, R. N. Monteiro; Stability and optimality of decay rate for a weakly dissipative Bresse system, Math. Methods Appl. Sci., 38(5) (2015), 898-908.
M. Afilal, A. Guesmia, A. Soufyane, M. Zahri; On the exponential and polynomial stability for a linear Bresse system, Math. Methods Appl. Sci., 43(5) (2020), 2626-2645.
G. Aguilera Contreras, J. E. Munoz Rivera; Bresse systems with localized Kelvin-Voigt dissipation, Electron. J. Differential Equations, (2021), 1-14.
F. Alabau Boussouira, J. E. Munoz Rivera, D. da S. Almeida JŽunior; Stability to weak dissipative Bresse system, J. Math. Anal. Appl., 374(2) (2011), 481-498.
M. Astudillo, H. P. Oquendo; Stability results for a Timoshenko system with a fractional operator in the memory, Appl. Math. Optim., 83(3) (2021), 1247-1275.
R. Bekhouche, A. Guesmia, S. Messaoudi; Uniform and weak stability of Bresse system with one infinite memory in the shear angle displacements, Arab. J. Math., 11(2) (2022), 155-1784.
A. Benaissa, A. Kasmi; Well-posedness and energy decay of solutions to a Bresse system with a boundary dissipation of fractional derivative type, Discrete Contin. Dyn. Syst. Ser. B, 23(10) (2018), 4361-4395.
T. Bentrcia, A. Mennouni; On the asymptotic stability of a Bresse system with two fractional damping terms: Theoretical and numerical analysis, Discrete Contin. Dyn. Syst. Ser. B, 28(1) (2023), 580-622.
A. Borichev, Y. Tomilov; Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347(2) (2010), 455-478.
V. R. Cabanillas, C. A. Raposo; Exponential stability for laminated beams with intermediate damping, Arch. Math. (Basel), 118(6) (2022), 625-635.
H. Dridi, A. Djebabla; Timoshenko system with fractional operator in the memory and spatial fractional thermal effect, Rend. Circ. Mat. Palermo (2), 70(1) (2021), 593-621.
K. J. Engel, R. Nagel; One-parameter semigroups for linear evolution equations, Grad. Texts in Math., 194 (2000).
T. El Arwadi, W. Youssef; On the stabilization of the Bresse beam with Kelvin-Voigt damping, Appl. Math. Optim., 83(3) (2021), 1831-1857.
L. H. Fatori, J. E. Munoz Rivera; Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75(6) (2010), 881-904.
L. H. Fatori, R. N. Monteiro; The optimal decay rate for a weak dissipative Bresse system, Appl. Math. Lett., 25(3) (2012), 600-604.
L. Gearhart; Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc., 236 (1978), 385-394.
K F. Graff; Wave motion in elastic solids, Dover Publications, New York (1991).
A. Guesmia; Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements, Mediterr. J. Math., 14(2) (2017), 1-19.
A. Guesmia; Non-exponential and polynomial stability results of a Bresse system with one infinite memory in the vertical displacement, Nonauton. Dyn. Syst., 4(1) (2017), 78-97.
A. Guesmia; The effect of the heat conduction of types I and III on the decay rate of the Bresse system via the longitudinal displacement, Arab. J. Math. (Springer), 8(1) (2019), 15-41.
A. Guesmia; Polynomial and non exponential stability of a weak dissipative Bresse system, preprint.
A. Guesmia, M. Kafini; Bresse system with infinite memories, Math. Methods Appl. Sci., 38(11) (2015), 2389-2402.
A. Guesmia, M. Kirane; Uniform and weak stability of Bresse system with two infinite memories, Z. Angew. Math. Phys., 67(5) (2016), 1-39.
S. G. Kreui; Linear differential equations in Banach space, American Mathematical Soc, (2011).
A .A. Keddi, T. A. Apalara, S. A. Messaoudi; Exponential and polynomial decay in a thermoelastic-Bresse system with second sound, Appl. Math. Optim., 77(2) (2018), 315-341.
Z. B. Kuang, Z. Y. Liu, L. Tebou; Optimal semigroup regularity for velocity coupled elastic systems: a degenerate fractional damping case, ESAIM Control Optim. Calc. Var., 28 (2022), 1-20.
J. E. Lagnese, G. Leugering, E. J. P. G. Schmidt; Modeling, analysis and control of dynamic elastic multi-link structures, Systems Control Found. Appl., (1994).
Z. Y. Liu, B. P. Rao; Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56(4) (2005), 630-644.
Z. Y. Liu, B. P. Rao; Energy decay rate of the thermoelastic Bresse system, Z. Angew. Math. Phys., 60(1) (2009), 54-69.
J. E. Munoz Rivera, H. D. FernŽandez Sare; Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339(1) (2008), 482-502.
H. P. Oquendo, C. R. da Luz; Asymptotic behavior for Timoshenko systems with fractional damping, Asymptot. Anal., 118 (1-2) (2020), 123-142.
H. P. Oquendo, F. M. S. SuŽarez; Exact decay rates for coupled plates with partial fractional damping, Z. Angew. Math. Phys., 70(3) (2019), 1-18.
A. Pazy; Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44 (1983).
J. Pršuss; On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284(2) (1984), 847- 857.
M. de L. Santos, A. Soufyane, D. da S. Almeida JŽunior; Asymptotic behavior to Bresse system with past history, Quart. Appl. Math., 73(1) (2015), 23-54.
A. Soufyane, B. Said-Houari; The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system, Evol. Equ. Control Theory, 3(4) (2014), 713-738.
G. F. Tyszka, M. R. Astudillo, H. P. Oquendo; Stabilization by memory effects: Kirchhoff plate versus Euler-Bernoulli plate, Nonlinear Anal. Real World Appl., 68(2) (2022), 1-24.
L. Tebou; Regularity and stability for a plate model involving fractional rotational forces and damping, Z. Angew. Math. Phys., 72(4) (2021), 1-13.
A. Wehbe, W. Youssef; Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks, J. Math. Phys., 51(10) (2010), 1-17.
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