• Previous Article
    Modeling the influence of human population and human population augmented pollution on rainfall
  • DCDS-B Home
  • This Issue
  • Next Article
    The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations
June  2022, 27(6): 2959-2978. doi: 10.3934/dcdsb.2021168

Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights

1. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

2. 

Department of Mathematics, Federal University of Bahia, Salvador, BA, Brazil

3. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

4. 

Faculty of Exact Sciences and Technology, Federal University of Pará, Manoel de Abreu Street, s/n, 68440-000, Abaetetuba, Pará, Brazil

5. 

Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, 36307-352, Minas Gerais, Brazil

* Corresponding authors: wjliu@nuist.edu.cn (W. Liu), jeremias@ufpa.br (M. Santos)

† These authors contributed equally to this work

Received  March 2021 Revised  April 2021 Published  June 2022 Early access  June 2021

This paper is concerned with system of magnetic effected piezoelectric beams with interior time-varying delay and time-dependent weights, in which the beam is clamped at the two side points subject to a single distributed state feedback controller with a time-varying delay. Under appropriate assumptions on the time-varying delay term and time-dependent weights, we obtain exponential stability estimates by using the multiplicative technique, and prove the equivalence between stabilization and observability.

Citation: Aowen Kong, Carlos Nonato, Wenjun Liu, Manoel Jeremias dos Santos, Carlos Raposo. Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 2959-2978. doi: 10.3934/dcdsb.2021168
References:
[1]

H. Y. S. Al-ZahraniJ. PalM. A. MiglioratoG. Tse and D. Yu, Piezoelectric field enhancement in III-V core-shell nanowires, Nano Energy, 14 (2015), 382-391.  doi: 10.1016/j.nanoen.2014.11.046.

[2]

V. Barros and C. Nonato and C. Raposo, Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights, Electronic Research Archive, 28 (2020), 205-220.  doi: 10.3934/era.2020014.

[3]

A. Benaissa, A. Benguessoum and S. A. Messaoudi, Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback, Electronic Journal of Qualitative Theory of Differential Equations, 2014 (2014), 13 pp. doi: 10.14232/ejqtde.2014.1.11.

[4]

A. Blanguernon and F. Léné and M. Bernadou, Active control of a beam using a piezoceramic element, Smart Materials and Structures, 8 (1999), 116-124.  doi: 10.1088/0964-1726/8/1/013.

[5]

W. G. Cady, Piezoelectricity: An Introduction to the Theory and Applications of Electrical Phenomena in Crystals, Dover Publications, New York, 1964.

[6]

M. Chen and W. Liu and W. Zhou, Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms, Advances in Nonlinear Analysis, 7 (2018), 547-569.  doi: 10.1515/anona-2016-0085.

[7]

D. Damjanovic, Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics, Reports on Progress in Physics, 61 (1999), 1267-1324.  doi: 10.1088/0034-4885/61/9/002.

[8]

G. Davi and A. Milazzo, Multidomain boundary integral formulation for piezoelectric materials fracture mechanics, International Journal of Solids and Structures, 38 (2001), 7065-7078.  doi: 10.1016/S0020-7683(00)00416-9.

[9]

J. M. DietlA. M. Wickenheiser and E. Garcia, A Timoshenko beam model for cantilevered piezoelectric energy harvesters, Smart Materials and Structures, 19 (2010), 547-569.  doi: 10.1088/0964-1726/19/5/055018.

[10]

B. Feng and X. G. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Applicable Analysis, 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139.

[11]

M. M. FreitasA. J. A. RamosA. Özer and D. S. Almeida Júnior, Long-time dynamics for a fractional piezoelectric system with magnetic effects and Fourier's law, Journal of Differential Equations, 280 (2021), 891-927.  doi: 10.1016/j.jde.2021.01.030.

[12]

C. Galassi, M. Dinescu, K. Uchino and M. Sayer, Piezoelectric materials: Advances in science, technology and applications, Nato Science Partnership Subseries 3, Springer, Berlin, 2000. doi: 10.1007/978-94-011-4094-2.

[13]

A. Haraux, Two remarks on hyperbolic dissipative problems, Research Notes in Mathematics Pitman, 122 (1985), 161-179. 

[14]

H. Kawai, The Piezoelectricity of poly (vinylidene Fluoride), Japanese Journal of Applied Physics, 8 (1969), 975-976.  doi: 10.1143/JJAP.8.975.

[15]

T. Kato, Linear and Quasi-Linear Equations of Evolution of Hyperbolic Type, Summer Sch., 72, Springer, Heidelberg, 2011,125–191. doi: 10.1007/978-3-642-11105-1_4.

[16]

M. Kirane and B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Communications on Pure and Applied Analysis, 10 (2011), 667-686.  doi: 10.3934/cpaa.2011.10.667.

[17]

G. Liu and L. Diao, Energy decay of the solution for a weak viscoelastic equation with a time-varying delay, Acta Applicandae Mathematicae, 155 (2018), 9-19.  doi: 10.1007/s10440-017-0142-1.

[18]

W. Liu and D. Chen and Z. Chen, Long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, Acta Mathematica Scientia, 41 (2021), 609-632.  doi: 10.1007/s10473-021-0220-3.

[19]

W. Liu and M. Chen, Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback, Continuum Mechanics and Thermodynamics, 29 (2017), 731-746.  doi: 10.1007/s00161-017-0556-z.

[20]

W. Liu and H. Zhuang, Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback, Discrete and Continuous Dynamical Systems-Series B, 26 (2021), 907-942.  doi: 10.3934/dcdsb.2020147.

[21]

S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a viscoelastic equation with boundary feedback, Topological Methods in Nonlinear Analysis, 51 (2018), 413-427.  doi: 10.12775/tmna.2017.066.

[22]

S. A. Messaoudi, A. Fareh and N. Doudi, Well posedness and exponential stability in a wave equation with a strong damping and a strong delay, Journal of Mathematical Physics, 57 (2016), 13pp. doi: 10.1063/1.4966551.

[23]

K. A. Morris and A. Özer, Strong stabilization of piezoelectric beams with magnetic effects, in 52nd IEEE Conference on Decision and Control, 2013, 3014–3019. doi: 10.1109/CDC.2013.6760341.

[24]

K. A. Morris and A. Özer, Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects, SIAM Journal on Control and Optimization, 52 (2014), 2371-2398.  doi: 10.1137/130918319.

[25]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electronic Journal of Differential Equations, 2011 (2011), 20pp.

[26]

S. Nicaise and J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete and Continuous Dynamical Systems-Series S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[27]

C. Nonato, M. J. {Dos Santos}, C. Raposo, Dynamics of Timoshenko system with time-varying weight and time-varying delay, Discrete and Continuous Dynamical Systems-Series B, in press. doi: 10.3934/dcdsb.2021053.

[28]

C. Nonato, C. Raposo and B. Feng, Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay, Asymptotic Analysis, in press. doi: 10.3233/ASY-201668.

[29]

R. L. Oliveira and H. P. Oquendo, Stability and instability results for coupled waves with delay term, Journal of Mathematical Physics, 61 (2020), 13pp. doi: 10.1063/1.5144987.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31]

G. Poulin-VittrantC. OshmanC. OpokuA. S. DahiyaN. CamaraD. AlquierHu eL. -P. T. H and M. Lethiecq, Fabrication and characterization of ZnO nanowire-based piezoelectric nanogenerators for low frequency mechanical energy harvesting, Physics Procedia, 70 (2015), 909-913.  doi: 10.1016/j.phpro.2015.08.188.

[32]

A. J. A. Ramos, M. M. Freitas, D. S. Almeida Jr., S. S. Jesus and T. R. S. Moura, Equivalence between exponential stabilization and boundary observability for piezoelectric beams with magnetic effect, Zeitschrift Für Angewandte Mathematik Und Physik, 70 (2019), 14pp. doi: 10.1007/s00033-019-1106-2.

[33]

A. J. A. Ramos, A. Özer, M. M. Freitas, D. S. Almeida Jr. and J. D. Martins, Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback, Zeitschrift Für Angewandte Mathematik Und Physik, 72 (2021), 26pp. doi: 10.1007/s00033-020-01457-8.

[34]

A. J. A. RamosC. S. L. Gon\c{c}alves and S. S. Corrêa Neto, Exponential stability and numerical treatment for piezoelectric beams with magnetic effect, ESAIM Mathematical Modelling and Numerical Analysis, 52 (2018), 255-274.  doi: 10.1051/m2an/2018004.

[35]

C. RaposoJ. A. D. ChuquipomaJ. A. J. Avila and M. L. Santos, Exponential decay and numerical solution for a Timoshenko system with delay term in the internal feedback, International Journal of Analysis and Applications, 3 (2013), 1-13. 

[36]

Z. SabbaghA. KhemmoudjM. Ferhat and M. Abdelli, Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with internal distributed delay, Rendiconti del Circolo Matematico di Palermo Series 2, 68 (2019), 477-498.  doi: 10.1007/s12215-018-0373-7.

[37]

B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback, Applied Mathematics and Computation, 217 (2010), 2857-2869.  doi: 10.1016/j.amc.2010.08.021.

[38]

P. Wang and J. Hao, Asymptotic stability of memory-type Euler-Bernoulli plate with variable coefficients and time delay, Journal of Systems Science and Complexity, 32 (2019), 1375-1392.  doi: 10.1007/s11424-018-7370-y.

[39]

H. J. Xiang and Z. F. Shi, Static analysis for multi-layered piezoelectric cantilevers, International Journal of Solids and Structures, 45 (2008), 113-128.  doi: 10.1016/j.ijsolstr.2007.07.022.

[40]

J. Yang, A Review of a few topics in piezoelectricity, Applied Mechanics Reviewes, 59 (2006), 335-345.  doi: 10.1115/1.2345378.

[41]

Y. Zheng, W. Liu and Y. Liu, Equivalence between internal observability and exponential stabilization for suspension bridge problem, Ricerche di Matematica, in press. doi: 10.1007/s11587-021-00566-4.

[42]

F. ZhuM. B. WardJ. F. Li and S. J. Milne, Core-shell grain structures and ferroelectric properties of Na$_{0.5}$K$_{0.5}$NbO$_3$-LiTaO$_3$-BiScO$_3$ piezoelectric ceramics, Data in Brief, 4 (2015), 34-39.  doi: 10.1016/j.dib.2015.04.002.

show all references

References:
[1]

H. Y. S. Al-ZahraniJ. PalM. A. MiglioratoG. Tse and D. Yu, Piezoelectric field enhancement in III-V core-shell nanowires, Nano Energy, 14 (2015), 382-391.  doi: 10.1016/j.nanoen.2014.11.046.

[2]

V. Barros and C. Nonato and C. Raposo, Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights, Electronic Research Archive, 28 (2020), 205-220.  doi: 10.3934/era.2020014.

[3]

A. Benaissa, A. Benguessoum and S. A. Messaoudi, Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback, Electronic Journal of Qualitative Theory of Differential Equations, 2014 (2014), 13 pp. doi: 10.14232/ejqtde.2014.1.11.

[4]

A. Blanguernon and F. Léné and M. Bernadou, Active control of a beam using a piezoceramic element, Smart Materials and Structures, 8 (1999), 116-124.  doi: 10.1088/0964-1726/8/1/013.

[5]

W. G. Cady, Piezoelectricity: An Introduction to the Theory and Applications of Electrical Phenomena in Crystals, Dover Publications, New York, 1964.

[6]

M. Chen and W. Liu and W. Zhou, Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms, Advances in Nonlinear Analysis, 7 (2018), 547-569.  doi: 10.1515/anona-2016-0085.

[7]

D. Damjanovic, Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics, Reports on Progress in Physics, 61 (1999), 1267-1324.  doi: 10.1088/0034-4885/61/9/002.

[8]

G. Davi and A. Milazzo, Multidomain boundary integral formulation for piezoelectric materials fracture mechanics, International Journal of Solids and Structures, 38 (2001), 7065-7078.  doi: 10.1016/S0020-7683(00)00416-9.

[9]

J. M. DietlA. M. Wickenheiser and E. Garcia, A Timoshenko beam model for cantilevered piezoelectric energy harvesters, Smart Materials and Structures, 19 (2010), 547-569.  doi: 10.1088/0964-1726/19/5/055018.

[10]

B. Feng and X. G. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Applicable Analysis, 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139.

[11]

M. M. FreitasA. J. A. RamosA. Özer and D. S. Almeida Júnior, Long-time dynamics for a fractional piezoelectric system with magnetic effects and Fourier's law, Journal of Differential Equations, 280 (2021), 891-927.  doi: 10.1016/j.jde.2021.01.030.

[12]

C. Galassi, M. Dinescu, K. Uchino and M. Sayer, Piezoelectric materials: Advances in science, technology and applications, Nato Science Partnership Subseries 3, Springer, Berlin, 2000. doi: 10.1007/978-94-011-4094-2.

[13]

A. Haraux, Two remarks on hyperbolic dissipative problems, Research Notes in Mathematics Pitman, 122 (1985), 161-179. 

[14]

H. Kawai, The Piezoelectricity of poly (vinylidene Fluoride), Japanese Journal of Applied Physics, 8 (1969), 975-976.  doi: 10.1143/JJAP.8.975.

[15]

T. Kato, Linear and Quasi-Linear Equations of Evolution of Hyperbolic Type, Summer Sch., 72, Springer, Heidelberg, 2011,125–191. doi: 10.1007/978-3-642-11105-1_4.

[16]

M. Kirane and B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Communications on Pure and Applied Analysis, 10 (2011), 667-686.  doi: 10.3934/cpaa.2011.10.667.

[17]

G. Liu and L. Diao, Energy decay of the solution for a weak viscoelastic equation with a time-varying delay, Acta Applicandae Mathematicae, 155 (2018), 9-19.  doi: 10.1007/s10440-017-0142-1.

[18]

W. Liu and D. Chen and Z. Chen, Long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, Acta Mathematica Scientia, 41 (2021), 609-632.  doi: 10.1007/s10473-021-0220-3.

[19]

W. Liu and M. Chen, Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback, Continuum Mechanics and Thermodynamics, 29 (2017), 731-746.  doi: 10.1007/s00161-017-0556-z.

[20]

W. Liu and H. Zhuang, Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback, Discrete and Continuous Dynamical Systems-Series B, 26 (2021), 907-942.  doi: 10.3934/dcdsb.2020147.

[21]

S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a viscoelastic equation with boundary feedback, Topological Methods in Nonlinear Analysis, 51 (2018), 413-427.  doi: 10.12775/tmna.2017.066.

[22]

S. A. Messaoudi, A. Fareh and N. Doudi, Well posedness and exponential stability in a wave equation with a strong damping and a strong delay, Journal of Mathematical Physics, 57 (2016), 13pp. doi: 10.1063/1.4966551.

[23]

K. A. Morris and A. Özer, Strong stabilization of piezoelectric beams with magnetic effects, in 52nd IEEE Conference on Decision and Control, 2013, 3014–3019. doi: 10.1109/CDC.2013.6760341.

[24]

K. A. Morris and A. Özer, Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects, SIAM Journal on Control and Optimization, 52 (2014), 2371-2398.  doi: 10.1137/130918319.

[25]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electronic Journal of Differential Equations, 2011 (2011), 20pp.

[26]

S. Nicaise and J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete and Continuous Dynamical Systems-Series S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[27]

C. Nonato, M. J. {Dos Santos}, C. Raposo, Dynamics of Timoshenko system with time-varying weight and time-varying delay, Discrete and Continuous Dynamical Systems-Series B, in press. doi: 10.3934/dcdsb.2021053.

[28]

C. Nonato, C. Raposo and B. Feng, Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay, Asymptotic Analysis, in press. doi: 10.3233/ASY-201668.

[29]

R. L. Oliveira and H. P. Oquendo, Stability and instability results for coupled waves with delay term, Journal of Mathematical Physics, 61 (2020), 13pp. doi: 10.1063/1.5144987.

[30]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[31]

G. Poulin-VittrantC. OshmanC. OpokuA. S. DahiyaN. CamaraD. AlquierHu eL. -P. T. H and M. Lethiecq, Fabrication and characterization of ZnO nanowire-based piezoelectric nanogenerators for low frequency mechanical energy harvesting, Physics Procedia, 70 (2015), 909-913.  doi: 10.1016/j.phpro.2015.08.188.

[32]

A. J. A. Ramos, M. M. Freitas, D. S. Almeida Jr., S. S. Jesus and T. R. S. Moura, Equivalence between exponential stabilization and boundary observability for piezoelectric beams with magnetic effect, Zeitschrift Für Angewandte Mathematik Und Physik, 70 (2019), 14pp. doi: 10.1007/s00033-019-1106-2.

[33]

A. J. A. Ramos, A. Özer, M. M. Freitas, D. S. Almeida Jr. and J. D. Martins, Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback, Zeitschrift Für Angewandte Mathematik Und Physik, 72 (2021), 26pp. doi: 10.1007/s00033-020-01457-8.

[34]

A. J. A. RamosC. S. L. Gon\c{c}alves and S. S. Corrêa Neto, Exponential stability and numerical treatment for piezoelectric beams with magnetic effect, ESAIM Mathematical Modelling and Numerical Analysis, 52 (2018), 255-274.  doi: 10.1051/m2an/2018004.

[35]

C. RaposoJ. A. D. ChuquipomaJ. A. J. Avila and M. L. Santos, Exponential decay and numerical solution for a Timoshenko system with delay term in the internal feedback, International Journal of Analysis and Applications, 3 (2013), 1-13. 

[36]

Z. SabbaghA. KhemmoudjM. Ferhat and M. Abdelli, Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with internal distributed delay, Rendiconti del Circolo Matematico di Palermo Series 2, 68 (2019), 477-498.  doi: 10.1007/s12215-018-0373-7.

[37]

B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback, Applied Mathematics and Computation, 217 (2010), 2857-2869.  doi: 10.1016/j.amc.2010.08.021.

[38]

P. Wang and J. Hao, Asymptotic stability of memory-type Euler-Bernoulli plate with variable coefficients and time delay, Journal of Systems Science and Complexity, 32 (2019), 1375-1392.  doi: 10.1007/s11424-018-7370-y.

[39]

H. J. Xiang and Z. F. Shi, Static analysis for multi-layered piezoelectric cantilevers, International Journal of Solids and Structures, 45 (2008), 113-128.  doi: 10.1016/j.ijsolstr.2007.07.022.

[40]

J. Yang, A Review of a few topics in piezoelectricity, Applied Mechanics Reviewes, 59 (2006), 335-345.  doi: 10.1115/1.2345378.

[41]

Y. Zheng, W. Liu and Y. Liu, Equivalence between internal observability and exponential stabilization for suspension bridge problem, Ricerche di Matematica, in press. doi: 10.1007/s11587-021-00566-4.

[42]

F. ZhuM. B. WardJ. F. Li and S. J. Milne, Core-shell grain structures and ferroelectric properties of Na$_{0.5}$K$_{0.5}$NbO$_3$-LiTaO$_3$-BiScO$_3$ piezoelectric ceramics, Data in Brief, 4 (2015), 34-39.  doi: 10.1016/j.dib.2015.04.002.

[1]

Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Jamille L.L. Almeida. Dynamics of piezoelectric beams with magnetic effects and delay term. Evolution Equations and Control Theory, 2022, 11 (2) : 583-603. doi: 10.3934/eect.2021015

[2]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273

[3]

Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024

[4]

Baowei Feng, Carlos Alberto Raposo, Carlos Alberto Nonato, Abdelaziz Soufyane. Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022011

[5]

Mounir Afilal, Abdelaziz Soufyane, Mauro de Lima Santos. Piezoelectric beams with magnetic effect and localized damping. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021056

[6]

Eugenii Shustin. Exponential decay of oscillations in a multidimensional delay differential system. Conference Publications, 2003, 2003 (Special) : 809-816. doi: 10.3934/proc.2003.2003.809

[7]

Vilmos Komornik, Anna Chiara Lai, Paola Loreti. Simultaneous observability of infinitely many strings and beams. Networks and Heterogeneous Media, 2020, 15 (4) : 633-652. doi: 10.3934/nhm.2020017

[8]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773

[9]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221

[10]

Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407

[11]

Vilmos Komornik, Bernadette Miara. Cross-like internal observability of rectangular membranes. Evolution Equations and Control Theory, 2014, 3 (1) : 135-146. doi: 10.3934/eect.2014.3.135

[12]

Zhong-Jie Han, Gen-Qi Xu. Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Networks and Heterogeneous Media, 2011, 6 (2) : 297-327. doi: 10.3934/nhm.2011.6.297

[13]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169

[14]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. On the exponential stabilization of a thermo piezoelectric/piezomagnetic system. Evolution Equations and Control Theory, 2012, 1 (2) : 315-336. doi: 10.3934/eect.2012.1.315

[15]

Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure and Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667

[16]

Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051

[17]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997

[18]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693

[19]

Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010

[20]

Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure and Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (413)
  • HTML views (442)
  • Cited by (0)

[Back to Top]