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<p>This paper addresses the problems of global well-posedness and asymptotic stabilization for a viscoelastic Euler-Bernoulli beam model incorporating distributed damping and a nonlinear feedback with time delay acting at the right boundary as a shear force. The nonlinearity is merely assumed to be Lipschitz continuous and to vanish at the origin, without satisfying boundedness or monotonicity as in saturation or cone-bounded cases. By a change of variables to handle the delay, the system is reformulated as an evolutionary coupled beam-transport system in a suitable Hilbert space. We then show that the corresponding nonlinear operator generates a contraction semigroup, ensuring the global well-posedness of the system. Furthermore, by designing an appropriate Lyapunov functional and applying LaSalle’s invariance principle in infinite dimension, we establish the global asymptotic stability of the system. This work extends existing results by effectively addressing the complex interaction between a distributed damping mechanism, a broad class of nonlinearities, and a boundary time delay. A numerical simulation validating the proposed approach is provided.</p>

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Well-Posedness and Global Stabilization of an Euler-Bernoulli Beam with Distributed Damping and Nonlinear Boundary Delay

  • 期刊论文

  • 发表日期：

    2026年6月1日

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  • Nizar Mahfoudhi,
  • Habib Ayadi

摘要

This paper addresses the problems of global well-posedness and asymptotic stabilization for a viscoelastic Euler-Bernoulli beam model incorporating distributed damping and a nonlinear feedback with time delay acting at the right boundary as a shear force. The nonlinearity is merely assumed to be Lipschitz continuous and to vanish at the origin, without satisfying boundedness or monotonicity as in saturation or cone-bounded cases. By a change of variables to handle the delay, the system is reformulated as an evolutionary coupled beam-transport system in a suitable Hilbert space. We then show that the corresponding nonlinear operator generates a contraction semigroup, ensuring the global well-posedness of the system. Furthermore, by designing an appropriate Lyapunov functional and applying LaSalle’s invariance principle in infinite dimension, we establish the global asymptotic stability of the system. This work extends existing results by effectively addressing the complex interaction between a distributed damping mechanism, a broad class of nonlinearities, and a boundary time delay. A numerical simulation validating the proposed approach is provided.