Stabilization of the transmission wave/plate equation with variable coefficients

Stabilization of string–beam–string transmission systems with localized frictional damping and delay

“…Since then, the stability of the transmission problem has been gradually studied. Up to now, the stability of the transmission problem in two regions has been extensively studied by scholars [1, 7, 9, 11, 16, 18‐20, 24, 29, 30]. We mention for example, Li et al [19] considered a string–beam system with localized frictional damping, $$$$ \left\{\begin{array}{ll}{y}_{tt}\left(x,t\right)-{y}_{xx}\left(x,t\right)+\alpha {y}_t\left(x,t\right)=0,& \left(x,t\right)\in \left(0,1\right)\times \left(0,\infty \right),\\ {}{\theta}_{tt}\left(x,t\right)+{\theta}_{xx xx}\left(x,t\right)+\gamma {\theta}_t\left(x,t\right)=0,& \left(x,t\right)\in \left(1,2\right)\times \left(0,\infty \right).\end{array}\right.$$…”

Regularity and asymptotic behavior for a damped plate–membrane transmission problem

“…This was generalized in [26] to a model with curved middle surface by virtue of geometric multiplier method. In [16], stabilization of a damped wave / damped plate system with variable coefficients is studied by means of a Riemannian geometrical approach. For stability of coupled wave-plate systems within the same domain, we mention, e.g., [21].…”

Energy decay estimates for a two‐dimensional coupled wave‐plate system with localized frictional damping

“…[15] Since its potential applications in engineering, the coupled systems have received increasing attention. [2,3,9] For example, the polynomial decay of one-dimensional hyperbolic-parabolic coupled system is considered in [24], and the exponential decay of the energy for the solution of a string-beams network is treated in [2] by frequency domain method. For the damped wave equation coupled with a damped Kirchhoff plate equation in the two-dimensional Euclidean space, the exponential stability is established in [3] by multipliers technique.…”