Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback (Q2026453)

This paper is about the dynamic stability of the piezoelectric beams in one-dimensional space \(x\). The authors discuss the following ``strongly coupled differential equation: \[ \begin{matrix} \rho v_{tt} - \alpha v_{xx} + \gamma \beta p_{xx} +d_1u_1(x,t) = 0,\\ \mu p_{tt} -\beta p_{xx} +\gamma\beta v_{xx} + d_2 u_2(x,t) = 0, & (x,t) \in (0,L) \times (0,\infty)\\ v(0,t) = p(0,t) = 0\\ \alpha v_x(L,t) - \gamma\beta p_x(L,t) = d_3 u_3(t)\\ \beta p_x (L,t)- \gamma\beta v_x(L,t) = d_4 u_4(t), & t\geq 0 \end{matrix} \] where \(v\) and \(p\) describe the longitudinal vibration profile and the total charge accumulated at the electrodes of a clamped-free piezoelectric beam of length \(L\). Here, \(\alpha=\alpha_1+\gamma^2\beta\) and \(\rho,\alpha,\gamma,\mu,\beta>0\) denote the mass density per unit volume, elastic stiffness, piezoelectric coefficient, magnetic permeability, impermittivity coefficient of the beam, respectively, and \(u_1(t),u_2(t),u_3(t),u_4(t)\) are distributed mechanical and magnetic controllers, and boundary strain and voltage controllers.'' The authors discuss the existence and uniqueness of the solutions of BVP in section 2 by a change of variable to account for the delay feedback term. In section 3, the authors examine ``the asymptotic behavior of solutions of the dissipative piezoelectric beam \(\ldots\) with full magnetic effects \(\ldots\) with delay term in the internal state feedback.'' The core of the paper is Theorem 3.2 which states the energy of the piezo system decays exponentially as \(t\rightarrow \infty\). There are a total of 8 theorems and lemmas and zero figures/diagrams in the paper.