Neumann-Boundary Stabilization of the Wave Equation with Internal Damping Control and Applications

This paper is devoted to the Neumann boundary stabilization of a non-homogeneous ndimensional                      wave equation subject to static or dynamic boundary conditions. Using a linear feedback law involving only an internal term, we prove the well-posedness of the considered systems and provide a simple method to obtain an asymptotic convergence result for the solutions. The method consists of proposing a new energy norm, andapplying the semigroup theory and LaSalle's principle. Finally, the method presented in this work is also applied to several distributed parameter systems such as the Petrovsky system, coupled wave-wave equations and elastic system

On a Kelvin-Voigt Viscoelastic Wave Equation with Strong Delay

An initial-boundary value problem for a viscoelastic wave equation subject to a strong time-localized delay in a Kelvin & Voigt-type material law is considered. Transforming the equation to an abstract Cauchy problem on the extended phase space, a global well-posedness theory is established using the operator semigroup theory both in Sobolev-valued $C^{0}$- and BV-spaces. Under appropriate assumptions on the coefficients, a global exponential decay rate is obtained and the stability region in the parameter space is further explored using the Lyapunov's indirect method. The singular limit $\tau \to 0$ is further studied with the aid of the energy method. Finally, a numerical example from a real-world application in biomechanics is presented.Comment: 34 pages, 4 figures, 1 set of Matlab code

On a Kelvin-Voigt viscoelasticwave equation with strong delay

An initial-boundary value problem for a viscoelastic wave equation subject to a strong timelocalized delay in a Kelvin & Voigt-type material law is considered. Transforming the equation to an abstract Cauchy problem on the extended phase space, a global well-posedness theory is established using the operator semigroup theory both in Sobolev-valued C0- and BV-spaces. Under appropriate assumptions on the coefficients, a global exponential decay rate is obtained and the stability region in the parameter space is further explored using the Lyapunov’s indirect method. The singular limit τ -> 0 is further studied with the aid of the energy method. Finally, a numerical example from a real-world application in biomechanics is presented

A new approach of stabilization of nondissipative distributed systems

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