• Ganesh C Gorain

      Articles written in Proceedings – Mathematical Sciences

    • Boundary stabilization of a hybrid Euler—Bernoulli beam

      Ganesh C Gorain Sujit K Bose

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      We consider a problem of boundary stabilization of small flexural vibrations of a flexible structure modeled by an Euler-Bernoulli beam which is held by a rigid hub at one end and totally free at the other. The hub dynamics leads to a hybrid system of equations. By incorporating a condition of small rate of change of the deflection with respect tox as well ast, over the length of the beam, for appropriate initial conditions, uniform exponential decay of energy is established when a viscous boundary damping is present at the hub end.

    • Uniform stability of damped nonlinear vibrations of an elastic string

      Ganesh C Gorain Sujit K Bose

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      Here we are concerned about uniform stability of damped nonlinear transverse vibrations of an elastic string fixed at its two ends. The vibrations governed by nonlinear integro-differential equation of Kirchoff type, is shown to possess energy uniformly bounded by exponentially decaying function of time. The result is achieved by considering an energy-like Lyapunov functional for the system.

    • Stabilization for the Vibrations Modeled by the `Standard Linear Model' of Viscoelasticity

      Ganesh C Gorain

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      We study the stabilization of vibrations of a flexible structure modeled by the `standard linear model’ of viscoelasticity in a bounded domain in $\mathbb{R}^n$ with a smooth boundary. We prove that amplitude of the vibrations remains bounded in the sense of a suitable norm in a space $\mathbb{X}$, defined explicitly in (22) subject to a restriction on the uncertain disturbing forces on $\mathbb{X}$. We also estimate the total energy of the system over time interval $[0,T]$ for any $T>0$, with a tolerance level of the disturbances. Finally, when the input disturbances are insignificant, uniform exponential stabilization is obtained and an explicit form for the energy decay rate is derived. These results are achieved by a direct method under undamped mixed boundary conditions.

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