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This paper addresses the analysis of a boundary feedback system involving anon-homogeneous Euler-Bernoulli beam governed by the equation$m(x)u_{tt}+\mu(x)u_{t}$$+\left(r(x)u_{xx}\right)_{xx}=0$, subject to theinitial $u(x,0)=u_0(x)$, $u_t(x,0)=v_0(x)$ and boundary conditions $u(0,t)=0$,$\left (-r(x)u_{xx}(x,t)\right )_{x=0}=-k^{-}_r u_{x}(0,t)-k^{-}_au_{xt}(0,t)$, $u(\ell,t)=0$, $\left (-r(x)u_{xx}(x,t)\right )_{x=\ell}=-k^{+}_ru_{x}(\ell,t)-k^{+}_a u_{xt}(\ell,t)$, with boundary control at both endsresulting from the rotation and angular velocity. The approach proposed in thisstudy relies on the utilization of regular weak solutions, energy identity, anda physically motivated Lyapunov function. By imposing natural assumptionsconcerning physical parameters and other inputs, which ensure the existence ofa regular weak solution, we successfully derive a uniform exponential decayestimate for the system's energy. The decay rate constant featured in thisestimate is solely dependent on the physical and geometric properties of thebeam. These properties encompass crucial parameters such as the viscousexternal damping coefficient $\mu(x)$, as well as the boundary springs$k^{-}_r,k^+_r $ and dampers $k^{-}_a,k^+_a$. To illustrate the practicaleffectiveness of our theoretical findings, numerical examples are provided.These examples serve to demonstrate the applicability and relevance of ourderived results in real-world scenarios.

Exponential stability of Euler-Bernoulli beam under boundary controls in rotation and angular velocity