Exponential stability for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights

Abstract

<p style='text-indent:20px;'>In this paper, we consider a system of magnetic effected piezoelectric beams with interior time-varying delay and time-dependent weights, in which the beam is clamped at the two side points subject to a single distributed state feedback controller with a time-varying delay. By combining the semigroup theory with Kato's variable norm technique, we obtain the existence and uniqueness of solution. By imposing appropriate assumptions on the time-varying delay term and time-dependent weights, we introduce suitable perturbed Lyapunov functional to obtain exponential stability estimates.</p>

New general decay results for a fully dynamic piezoelectric beam system with fractional delays under memory boundary controls

“…In other words, even if the delay had a negative effect on the stability of the vibration of the system, the viscous damping was sufficient to offset this effect. For the corresponding time‐varying weight, see Kong et al [24]. Soufyane et al [25] considered the case of nonlinear time delays and obtained general energy decay results under appropriate assumptions of delay weights, which generalized previous conclusions.…”

Asymptotic behavior of the linearized compressible barotropic Navier‐Stokes system with a time varying delay term in the boundary or internal feedback

“…Now, using the variable norm theory of Kato [58][59][60], we will show the existence and uniqueness of the above abstract system with time dependent operator. Similar types of analysis has been done in many works; see [48,51,53,61] for instances. The main idea of the existence-uniqueness theory is to show that the triplet {, , ((0)} with  = {(t) ∶ t ∈ [0, T]} for some fixed T > 0 forms a constant domain system.…”

“…Now, using the variable norm theory of Kato [58–60], we will show the existence and uniqueness of the above abstract system with time dependent operator. Similar types of analysis has been done in many works; see [48, 51, 53, 61] for instances. The main idea of the existence‐uniqueness theory is to show that the triplet $$$ \left\{\mathcal{A},\mathcal{H},\mathcal{D}\right(\mathcal{A}(0)\Big\} $$$ with $$$ \mathcal{A}=\left\{\mathcal{A}(t):t\in \left[0,T\right]\right\} $$$ for some fixed $$$ T>0 $$$ forms a constant domain system.…”

Long-time dynamics for a piezoelectric system with time-varying delay and memory

“…Recently, the control of partial differential equations with time-varying delays has become a hot research topic [29,28,41].…”