Existence and uniqueness of a hybrid system with variable coefficients

Volume 15, Issue 1, pp 67--78 http://dx.doi.org/10.22436/jnsa.015.01.06

Publication Date: August 20, 2021 Submission Date: May 02, 2021 Revision Date: June 16, 2021 Accteptance Date: July 06, 2021

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 789 Downloads
• 1703 Views

Authors

Y. S. A. Joresse - Universite Nangui Abrogoua d'Abobo, Adjame and UFR Sciences Fondamentales et Appliquees, Cote d'Ivoire. Y. Gozo - Universite Nangui Abrogoua d'Abobo, Adjame and UFR Sciences Fondamentales et Appliquees, Cote d'Ivoire. B. G. Jean-Marc - Universite Nangui Abrogoua d'Abobo, Adjame and UFR Sciences Fondamentales et Appliquees, Cote d'Ivoire. T. K. Augustin - Institut National Polytechnique Houphouet, Boigny de Yamoussoukro, Cote d'Ivoire.

Abstract

In this paper we consider a long flexible Euler-Bernoulli beam with boundary conditions imposed at the two ends, the resulting model being called hybrid system. The beam is hybrid in the sense that it holds both rigid and elastic motions. Our main result is to show the existence and uniqueness of the weak solution of close-loop system. The closed-loop system stability is shown through Lyapunov-based analysis.

Share and Cite

Keywords

• Beam equations
• variable coefficients
• stability
• existence
• uniqueness

MSC

•  32C37
•  34B05
•  35A15
•  35L10
•  78M10

References

• [1] A .P. G. Abro, J. M. Gossrin Bomisso, A. K. Toure, A. Coulibaly, {A Numerical Method by Finite Element Method (FEM) of an Euler-Bernoulli Beam to Variable Coefficients, Adv. Math. Sci. J., 9 (2020), 8485--8510
  • View Article
  • Google Scholar


• [2] H. T. Banks, I. G. Rosen, Computational methods for the identification of spatially varying stiffness and damping in beams, Control Theory Adv. Tech., 3 (1987), 1--32
  • View Article
  • MathSciNet
  • Google Scholar


• [3] B. J. S. Ben, C. Adama, Dissipative Numerical Method for an Euler-Bernoulli Beam with Rigid Body, Global J. Pure Appl. Math., 14 (2018), 989--1010
  • View Article
  • Google Scholar


• [4] G. J. M. Bomisso, A. Toure, G. Yoro, Dissipative Numerical Method for a Flexible Euler-Bernoulli Beam with a Force Control in Rotation and Velocity Rotation, J. Math. Res., 9 (2017), 30--48
  • View Article
  • Google Scholar


• [5] G. J. M. Bomisso, A. Toure, G. Yoro, Stabilization of Variable Coefficients Euler-Bernoulli Beam with Viscous Damping under a Force Control in Rotation and Velocity Rotation, J. Math. Res., 9 (2017), 1--13
  • View Article
  • Google Scholar


• [6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York (2011)
  • View Article
  • MathSciNet
  • Google Scholar


• [7] B. Z. Guo, On the boundary control of a hybrid system with variable coefficients, J. Optim. Theory Appl., 114 (2002), 373--395
  • View Article
  • MathSciNet
  • Google Scholar


• [8] J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1, Dunod, Paris (1968)
  • View Article
  • MathSciNet
  • Google Scholar


• [9] W. Littman, L. Markus, Exact Boundary Controllability of a Hybrid System of Elasticity, Arch. Rational Mech. Anal., 103 (1988), 193--236
  • View Article
  • MathSciNet
  • Google Scholar


• [10] W. Littman, L. Markus, Stabilization of a Hybrid System of Elasticity by Feedback Boundary Damping, Ann. Mat. Pura Appl. (4), 152 (1988), 281--330
  • View Article
  • MathSciNet
  • Google Scholar


• [11] Z.-H. Luo, Direct Strain Feedback Control of Flexible Robot Arms: New Theoretical and Experimental Results, IEEE Trans. Automat. Control, 38 (1993), 1610--1622
  • View Article
  • MathSciNet
  • Google Scholar


• [12] M. Miletic, Stability analysis and a dissipative FEM for an Euler-Bernoulli beam with tip body and passivity-based boundary control, Ph.D. Thesis (Vienna University of Technology), Vienna (2015)
  • View Article
  • Google Scholar


• [13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York (1983)
  • View Article
  • MathSciNet
  • Google Scholar


• [14] B. P. Rao, Uniform Stabilization of a Hybrid System of Elasticity, SIAM J. Control Optim., 33 (1995), 440--454
  • View Article
  • MathSciNet
  • Google Scholar


• [15] R. Temam, Infinite-dimensional Dynamical Systems In Mechanics And Physics, Springer Verlag, New York (1988)
  • View Article
  • MathSciNet
  • Google Scholar


• [16] J.-M. Wang, G.-Q. Xu, S.-P. Yung, Riesz basis property, exponential stability of variable coefficient Euler--Bernoulli beams with indefinite damping, IMA J. Appl. Math., 70 (2005), 459--477
  • View Article
  • MathSciNet
  • Google Scholar


• [17] K. Yosida, Functional Analysis, Springer, Berlin (1965)
  • MathSciNet
  • Google Scholar