Exponential Stabilization of a Flexible Structure: A Delayed Boundary Force Control Versus a Delayed Boundary Moment Control

• Boumediène Chentouf ^[1] ; Nejib Smaoui ^[1]
  1. [1] Kuwait University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 3, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • The main concern of this paper is to study the boundary stabilization problem of the disk-beam system. To do so, we assume that the boundary control is either of a force type control or a moment type control and is subject to the presence of a constant time-delay. First, we show that in both cases, the system is well-posed in an appropriate functional space. Next, the exponential stability property is established. Finally, the obtained outcomes are ascertained through numerical simulations.

• Referencias bibliográficas
  • Adams, R.: Sobolev Spaces. Academic Press, Cambridge (1975)
  • Baillieul, J., Levi, M.: Rotational elastic dynamics. Physica 27, 43–62 (1987)
  • Bloch, A.M., Titi, E.S.: On the dynamics of rotating elastic beams. In: Proceedings of the Conference on New Trends System Theory, Genoa,...
  • Brezis, H.: Functional analysis. In: Pibas, I. (ed.) Sobolev Spaces and Partial Differential Equations. Universitex, Springer, Berlin (2011)
  • Chentouf, B., Wang, J.M.: Optimal energy decay for a non-homogeneous flexible beam with a tip mass. J. Dyn. Control Syst. 13, 37–53 (2007)
  • Chentouf, B., Smaoui, N.: Exponential stabilization of a non-uniform rotating disk-beam system via a torque control and a finite memory type...
  • Chen, X., Chentouf, B., Wang, J.M.: Nondissipative torque and shear force controls of a rotating flexible structure. SIAM J. Control Optim....
  • Chentouf, B., Couchouron, J.F.: Nonlinear feedback stabilization of a rotating body-beam without damping. ESAIM COCV 4, 515–535 (1999)
  • Chentouf, B.: A simple approach to dynamic stabilization of a rotating body-beam. Appl. Math. Lett. 19, 97–107 (2006)
  • Chentouf, B.: Stabilization of memory type for a rotating disk-beam system. Applied Math. Comput. 258, 227–236 (2015)
  • Chentouf, B.: Stabilization of the rotating disk-beam system with a delay term in boundary feedback. Nonlinear Dyn. 78, 2249–2259 (2014)
  • Chentouf, B.: Stabilization of a nonlinear rotating flexible structure under the presence of time-dependent delay term in a dynamic boundary...
  • Chentouf, B.: Effect compensation of the presence of a time-dependent interior delay on the stabilization of the rotating disk-beam system....
  • Chentouf, B.: Compensation of the interior delay effect for a rotating disk-beam system. IMA J. Math. Control Inf. 33(4), 963–978 (2016)
  • Chentouf, B.: A minimal state approach to dynamic stabilization of the rotating disk-beam system with infinite memory. IEEE Trans. Autom....
  • Chentouf, B.: Exponential stabilization of the rotating disk-beam system with an interior infinite memory control: a minimal state framework....
  • Chentouf, B., Wang, J.M.: Stabilization and optimal decay rate for a non-homogeneous rotating body-beam with dynamic boundary controls. J....
  • Chentouf, B., Wang, J.M.: On the stabilization of the disk-beam system via torque and direct strain feedback controls. IEEE Trans. Autom....
  • Chentouf, B., Han, Z.: On the elimination of infinite memory effects on the stability of a nonlinear nonhomogeneous rotating body-beam system....
  • Chentouf, B., Mansouri, S.: Exponential stabilization of a flexible structure with a local interior control and under the presence of a boundary...
  • Geng, H., Han, Z.J., Wang, J., Xu, G.Q.: Stabilization of nonlinear rotating disk-beam system with localized thermal effect. Nonlinear Dyn....
  • Gibson, J.S.: A note on stabilization of infinite dimensional linear oscillators by compact linear feedback. SIAM J. Control Optim. 18, 311–316...
  • Han, Z., Chentouf, B., Geng, H.: Stabilization of a rotating disk-beam system with infinite memory via minimal state variable: a moment control...
  • Huang, F.L.: Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1, 43–56...
  • Huang, F.L.: Strong asymptotic stability of linear dynamical systems in Banach spaces. J. Diff. Equ. 104, 307–324 (1993)
  • Kato, T.: Perturbation Theory of Linear Operators. Springer-Verlag, New York (1976)
  • Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Hoboken (2002)
  • Laousy, H., Xu, C.Z., Sallet, G.: Boundary feedback stabilization of a rotating body-beam system. IEEE Trans. Autom. Control 41, 241–245 (1996)
  • Lee, S.W.R., Li, H.L.: Development and characterization of a rotary motor driven by anisotropic piezoelectric composite laminate. Smart Mater....
  • Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems. Chapman and Hall/CRC Res. Notes Math. vol. 398. Chapman and Hall (1999)
  • Morgül, O.: Orientation and stabilization of a flexible beam attached to a rigid body: planar motion. IEEE Trans. Autom. Control 36, 953–963...
  • Morgül, O.: Control and stabilization of a rotating flexible structure. Automatica 30, 351–356 (1994)
  • Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks....
  • Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983)
  • Prüss, J.: On the spectrum of -semigroups. Trans. Am. Math. Soc. 284, 847–857 (1984)
  • Shang, Y.F., Xu, G.Q., Chen, Y.L.: Stability analysis of Euler-Bernoulli beam with input delay in the boundary control. Asian J. Control 14,...
  • Triggiani, R.: Lack of uniform stabilization for noncontractive semigroups under compact perturbation. Proc. Am. Math. Soc. 105, 375–383 (1989)
  • Wang, H.C.: Orthonormal aspect on vibration modes for a class of non-uniform beams. J. Frankl. Inst. 287, 175–178 (1969)
  • Xu, G.Q., Yung, S.P., Li, L.K.: Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var. 12,...
  • Xu, C.Z., Baillieul, J.: Stabilizability and stabilization of a rotating body-beam system with torque control. IEEE Trans. Autom. Control...