Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping

Kouassi Ayo Ayébié Hermith; Bouallagui Zied; Diop Fatou N’diaye; Touré Kidjégbo Augustin

doi:doi:10.11648/j.acm.20251404.12

Research Article |

| Peer-Reviewed

Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping

Kouassi Ayo Ayébié Hermith^*, Bouallagui Zied, Diop Fatou N’diaye, Touré Kidjégbo Augustin

Published in Applied and Computational Mathematics (Volume 14, Issue 4)

Received: 6 May 2025 Accepted: 9 June 2025 Published: 5 August 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The transportation of masses or objects by cranes, necessary in the construction sector to increase productivity, often causes transverse vibrations of the system, which degrades production or process performance, and can even cause breakdowns. Actively controlling or mitigating these vibrations is becoming essential in many fields, such as the construction of satellite panels, etc. Therefore, many efforts have been made in recent years to find effective ways to eliminate these unwanted vibrations. Thus, the vibration behavior of a crane on a construction site, transporting a mass represented by beams, is translated into an equation using the theory of Euler-Bernoulli beam equations. These vibration effects are thus model led in a very reduced manner by a nonhomogeneous Euler-Bernoulli beam fixed at one end and subjected to a point mass at the free end. In this research article, we have generalized Wang’s results. We began by defining some fundamental properties of the closed-loop system and then analyzed its spectrum. Using the theory of perturbed problems, we obtained the basic Riesz property. The spectrum-determined growth condition and exponential stability were also derived. Moreover, when the damping was undefined, we provided a condition on the negative value of the damping without destroying the exponential stability of the system.

Published in	Applied and Computational Mathematics (Volume 14, Issue 4)
DOI	10.11648/j.acm.20251404.12
Page(s)	193-209
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Beam Equation, Semigroup Theory, Asymptotic Analysis, Riesz Basis, Exponential Stability

References

[1]	R. Adams, “Sobolev spaces,” Academic Press, New York, 1975.
[2]	O. Baysala, A. Hasanov and A. Kawano, “Exponential stability of damped Euler-Bernoulli beam controlled by boundary springs and dampers,” arXiv: 2307.06170v1 [math.AP] 12 Jul 2023.
[3]	A. Benaddi, B. P. Rao, “Energy decay rate of wave equations with indefinite damping,” J. Diff. Eqns, Vol. 161, issue 2, 1 March 2000, pp. 337-357. https://doi.org/10.1006/jdeq.2000.3714
[4]	G. D. Birkhoff, “On the asymptotic character of the solutions of certain linear differential equations containing a parameter,” Trans. Amer. Math. Soc., Vol. 9, No. 2 Apr., 1908, pp. 219-231. https://doi.org/10.2307/1988652
[5]	G. D. Birkhoff, “Boundary value and expansion problems of ordinary linear differential equations,” Trans. Amer. Math. Soc. Vol. 9, No. 4 (Oct., 1908), pp. 373-395. https://doi.org/10.2307/1988661
[6]	S. K. Bose, “On initial conditions for a boundary stabilized hybrid Euler-Bernoulli beam,” Proc.Indian Acad. Sci. (Math. Sci.), Vol. 111, No. 3, August 2001, pp. 365-370. https://doi.org/10.1007/BF02829602
[7]	Z. Bouallagui and M. Jellouli, “Stability of Euler-Bernoulli beam equation on a star-shaped network with indefinite damping,” Applicable Analysis, 20 Feb 2025, pp. 1-23. https://doi.org/10.1080/00036811.2025.2468518
[8]	H. Brezis, “Analyse fonctionnelle: Théorie et applications,” Masson. 1983.
[9]	R. F. Curtain and H. J. Zwart, “An Introduction to Infinite Dimensional Linera System Theory,” New York: Springer. Vol 38, iss. 3 1996. https://doi.org/10.1137/1038096
[10]	B. N. Data, S. Elhay, Y. M. Ram and D. R. Sarkissian, “Partial eigenstructure assignment for the quadratic pencil,” Journal of Sound and Vib, Vol. 230, Iss. 1, 10 February 2000, pp. 101-110. https://doi.org/10.1006/jsvi.1999.2620
[11]	B. N. Data, Y. M. Ram and D. R. Sarkissian, “Spectrum modification for gyroscopic system,” Z. Angew. Math. Mech. Vol. 82 Iss. 3, March 2002, pp. 191-200. https://doi.org/10.1002/1521-4001(200203)82:3<191::AID-ZAMM191>3.0.CO;2-L
[12]	A. Hasanov, “Exponential stability of Euler-Bernoulli beam under boundary controls in rotation and angular velocity,” arXiv: 2307.01518v1 [math.OC] 4 Jul 2023.
[13]	B. Z. Guo, “On the boundary control for a hybrid system with variable coefficients,” J. Optim: Theor. Appl., Vol. 114, No. 2, August 2002, pp. 373-395. https://doi.org/10.1023/A:1016039819069
[14]	B. Z. Guo, “Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equation with variable coefficients,” SIAM J. Control Optim., Vol. 40, Iss. 6, 2002, pp. 1905-1923. https://doi.org/10.1137/S0363012900372519
[15]	A. Haraux “Semi-Linear Hyperbolic Problems in Bounded Domain,” Harwood Academic and Breach New York, Vol. 3 January 1987.
[16]	A. A. H. Kouassi, F. N. Diop, S. A. J. Yapi, and K. A. Touré “Stability of variable coefficients Rayleigh beams with indefinite damping under force control” Far East Journal of Dynamical Systems. Vol. 38, No 1, 2025, pp. 127-157. https://doi.org/10.17654/0972111825006
[17]	V. Komornik, “Exact Contrllability and Stabilization (the Multiplier Method),” Masson, Paris: Wiley. 1995.
[18]	S. Li, Y. Wang, Z. Liang, J. Yu and G. Zhu, “Stabilization of Vibrating Beam with a Tip Mass Controlled by Combined Feedback Forces,” Vol. 256, Iss. 1, 1 April 2001, pp. 13-38. https://doi.org/10.1006/jmaa.2000.7217
[19]	B. R. Li, “The perturbation theory of a class of linear operators with applications,” Acta Math. Sin. (Chinese), Vol. 21, Iss. 3, 1078, pp. 206-222. https://doi.org/10.12386/A1978sxxb0023
[20]	K. M. Liew, Y. Xiang and S. Kitipornchai, “Research on thick plate vibration: a literature survey,” J. Sound Vib., Vol. 180, Iss. 1, 9 February 1995, pp. 163-176. https://doi.org/10.1006/jsvi.1995.0072
[21]	K. Liu, Z. Liu and B. Rao, “Exponential stability of an abstract non-dissipative linear system,” SIAM. J. Control Optim., Vol. 40, Iss. 1, July 2001, pp. 149-165. https://doi.org/10.1137/S0363012999364930
[22]	T. Mizusawa, “Vibration of rectangular mindlin plates with tapered thickness by the spline strip method,” Comput. Struct., Vol. 46, Iss. 3, 3 February 1993, pp. 451-463. https://doi.org/10.1016/0045-7949(93)90215-Y
[23]	A. My Driss and E. B. Abderrahman, “Stabilization of variable coefficients Euler-Bernoulli beam equation with a tip mass controlled by combined feedback forces,” Annals of the University of Craiova, Mathematics and Computer Science Series. Vol. 42 No: 1, 2015, pp. 238-248. https://doi.org/10.52846/ami.v42i1.754
[24]	M. A. Naimark, “Linear Differential Operators,” 1968, New York: Ugar.
[25]	A. Pazy, “Semigroup of Linear Operators and Applications to Partial Differential Equations,” 1983, New York: Springer.
[26]	A. A. Shkalikov, “Boundary value problems for ordinary differential equations with a parameter in the boundary conditions,” J. Sov. Math., Vol. 33, june 1986, pp. 1311-1342. https://doi.org/10.1007/BF01084754
[27]	A. K. Touré A. Coulibaly and H. A. A. Kouassi, “Riesz basis property and exponential stability of Euler-Bernoulli beams with variable coefficients and indefinite damping under a force control in position and velocity,” Electronic Journal of Differential Equations, Vol. 2015, February 2015 No: 54, pp 1-20.
[28]	K. A. Touré E. Mensah and M. Taha, “Riesz Basis Property and Exponential Stability of a Flexible Euler-Bernoulli Beam with a Force Control in Position and Velocity,” Far East Journal of Applied Mathematics, Vol. 69, No 1, 2012, pp 29-56.
[29]	C. Tretter, “On λ−Nonlinear Boundary Eigenvalue Problem,” Mathematical Research, Vol. 71, 1993, Berlin: Akademie.
[30]	J. Wang and S. P. Yung, “Stability of a nonuniform Rayleigh beam with indefinite damping,” Vol. 55, Iss. 10, October 2006, pp. 863-870. https://doi.org/10.1016/j.sysconle.2006.04.002
[31]	J. Wang, G. Q. Xu, and S. P. Yung, “Exponential stability of variable coefficients Rayleigh beams under boundary feedback control: a Riesz basis approach,” Syst. Control Lett., VOl. 51, Iss. 1 January 2004, pp. 33-50. https://doi.org/10.1016/S0167-6911(03)00205-6
[32]	J. Wang, “Riesz basis property of some infinite-dimensional control problems and its applications,” The HKU Scholars Hub The University of Hong Kong (thesis) 2004. http://dx.doi.org/10.5353/th-b3016350
[33]	G. Q. Xu and D. X. Feng, “The Riesz basis property of a Timoshenko beam with boundary feedback and application,” IMA J. Appl. Math., Vol. 67, Iss: 4, August 2002, pp. 357-370. https://doi.org/10.1093/imamat/67.4.357

Cite This Article

Plain Text BibTeX RIS

APA Style

Hermith, K. A. A., Zied, B., N’diaye, D. F., Augustin, T. K. (2025). Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping. Applied and Computational Mathematics, 14(4), 193-209. https://doi.org/10.11648/j.acm.20251404.12

Copy | Download

ACS Style

Hermith, K. A. A.; Zied, B.; N’diaye, D. F.; Augustin, T. K. Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping. Appl. Comput. Math. 2025, 14(4), 193-209. doi: 10.11648/j.acm.20251404.12

Copy | Download

AMA Style

Hermith KAA, Zied B, N’diaye DF, Augustin TK. Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping. Appl Comput Math. 2025;14(4):193-209. doi: 10.11648/j.acm.20251404.12

Copy | Download

@article{10.11648/j.acm.20251404.12,
  author = {Kouassi Ayo Ayébié Hermith and Bouallagui Zied and Diop Fatou N’diaye and Touré Kidjégbo Augustin},
  title = {Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {4},
  pages = {193-209},
  doi = {10.11648/j.acm.20251404.12},
  url = {https://doi.org/10.11648/j.acm.20251404.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251404.12},
  abstract = {The transportation of masses or objects by cranes, necessary in the construction sector to increase productivity, often causes transverse vibrations of the system, which degrades production or process performance, and can even cause breakdowns. Actively controlling or mitigating these vibrations is becoming essential in many fields, such as the construction of satellite panels, etc. Therefore, many efforts have been made in recent years to find effective ways to eliminate these unwanted vibrations. Thus, the vibration behavior of a crane on a construction site, transporting a mass represented by beams, is translated into an equation using the theory of Euler-Bernoulli beam equations. These vibration effects are thus model led in a very reduced manner by a nonhomogeneous Euler-Bernoulli beam fixed at one end and subjected to a point mass at the free end. In this research article, we have generalized Wang’s results. We began by defining some fundamental properties of the closed-loop system and then analyzed its spectrum. Using the theory of perturbed problems, we obtained the basic Riesz property. The spectrum-determined growth condition and exponential stability were also derived. Moreover, when the damping was undefined, we provided a condition on the negative value of the damping without destroying the exponential stability of the system.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping

AU  - Kouassi Ayo Ayébié Hermith
AU  - Bouallagui Zied
AU  - Diop Fatou N’diaye
AU  - Touré Kidjégbo Augustin
Y1  - 2025/08/05
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251404.12
DO  - 10.11648/j.acm.20251404.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 193
EP  - 209
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251404.12
AB  - The transportation of masses or objects by cranes, necessary in the construction sector to increase productivity, often causes transverse vibrations of the system, which degrades production or process performance, and can even cause breakdowns. Actively controlling or mitigating these vibrations is becoming essential in many fields, such as the construction of satellite panels, etc. Therefore, many efforts have been made in recent years to find effective ways to eliminate these unwanted vibrations. Thus, the vibration behavior of a crane on a construction site, transporting a mass represented by beams, is translated into an equation using the theory of Euler-Bernoulli beam equations. These vibration effects are thus model led in a very reduced manner by a nonhomogeneous Euler-Bernoulli beam fixed at one end and subjected to a point mass at the free end. In this research article, we have generalized Wang’s results. We began by defining some fundamental properties of the closed-loop system and then analyzed its spectrum. Using the theory of perturbed problems, we obtained the basic Riesz property. The spectrum-determined growth condition and exponential stability were also derived. Moreover, when the damping was undefined, we provided a condition on the negative value of the damping without destroying the exponential stability of the system.

VL  - 14
IS  - 4
ER  -

Copy | Download

Author Information

Kouassi Ayo Ayébié Hermith

National Polytechnic Institute Félix Houphouët-Boigny, Yamoussoukro, Côte d’Ivoire

Contact Email
Bouallagui Zied

Mathematics Laboratory of Avignon, Avignon University, Avignon, France

Contact Email
Diop Fatou N’diaye

National Polytechnic Institute Félix Houphouët-Boigny, Yamoussoukro, Côte d’Ivoire

Contact Email
Touré Kidjégbo Augustin

African Higher School of Information and Communication Technologies, Abidjan, Côte d’Ivoire

Contact Email

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Hermith, K. A. A., Zied, B., N’diaye, D. F., Augustin, T. K. (2025). Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping. Applied and Computational Mathematics, 14(4), 193-209. https://doi.org/10.11648/j.acm.20251404.12

Copy | Download

ACS Style

Hermith, K. A. A.; Zied, B.; N’diaye, D. F.; Augustin, T. K. Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping. Appl. Comput. Math. 2025, 14(4), 193-209. doi: 10.11648/j.acm.20251404.12

Copy | Download

AMA Style

Hermith KAA, Zied B, N’diaye DF, Augustin TK. Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping. Appl Comput Math. 2025;14(4):193-209. doi: 10.11648/j.acm.20251404.12

Copy | Download

@article{10.11648/j.acm.20251404.12,
  author = {Kouassi Ayo Ayébié Hermith and Bouallagui Zied and Diop Fatou N’diaye and Touré Kidjégbo Augustin},
  title = {Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {4},
  pages = {193-209},
  doi = {10.11648/j.acm.20251404.12},
  url = {https://doi.org/10.11648/j.acm.20251404.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251404.12},
  abstract = {The transportation of masses or objects by cranes, necessary in the construction sector to increase productivity, often causes transverse vibrations of the system, which degrades production or process performance, and can even cause breakdowns. Actively controlling or mitigating these vibrations is becoming essential in many fields, such as the construction of satellite panels, etc. Therefore, many efforts have been made in recent years to find effective ways to eliminate these unwanted vibrations. Thus, the vibration behavior of a crane on a construction site, transporting a mass represented by beams, is translated into an equation using the theory of Euler-Bernoulli beam equations. These vibration effects are thus model led in a very reduced manner by a nonhomogeneous Euler-Bernoulli beam fixed at one end and subjected to a point mass at the free end. In this research article, we have generalized Wang’s results. We began by defining some fundamental properties of the closed-loop system and then analyzed its spectrum. Using the theory of perturbed problems, we obtained the basic Riesz property. The spectrum-determined growth condition and exponential stability were also derived. Moreover, when the damping was undefined, we provided a condition on the negative value of the damping without destroying the exponential stability of the system.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping

AU  - Kouassi Ayo Ayébié Hermith
AU  - Bouallagui Zied
AU  - Diop Fatou N’diaye
AU  - Touré Kidjégbo Augustin
Y1  - 2025/08/05
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251404.12
DO  - 10.11648/j.acm.20251404.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 193
EP  - 209
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251404.12
AB  - The transportation of masses or objects by cranes, necessary in the construction sector to increase productivity, often causes transverse vibrations of the system, which degrades production or process performance, and can even cause breakdowns. Actively controlling or mitigating these vibrations is becoming essential in many fields, such as the construction of satellite panels, etc. Therefore, many efforts have been made in recent years to find effective ways to eliminate these unwanted vibrations. Thus, the vibration behavior of a crane on a construction site, transporting a mass represented by beams, is translated into an equation using the theory of Euler-Bernoulli beam equations. These vibration effects are thus model led in a very reduced manner by a nonhomogeneous Euler-Bernoulli beam fixed at one end and subjected to a point mass at the free end. In this research article, we have generalized Wang’s results. We began by defining some fundamental properties of the closed-loop system and then analyzed its spectrum. Using the theory of perturbed problems, we obtained the basic Riesz property. The spectrum-determined growth condition and exponential stability were also derived. Moreover, when the damping was undefined, we provided a condition on the negative value of the damping without destroying the exponential stability of the system.

VL  - 14
IS  - 4
ER  -

Copy | Download