Stability analysis of fully dynamic piezoelectric beams with internal fractional delay

Abstract

In this paper, we are interested in the stability analysis of one-dimensional fully dynamic piezoelectric beam systems with internal fractional delay under different feedback controls (boundary and internal). By introducing two new equations to deal with fractional delay, equivalent new systems are obtained. Based on classical semigroup theory, we prove the well-posedness of the related systems. In the analysis of their stability, different research methods are adopted in this paper, aiming to reach conclusions in a more concise way. One of the most critical is some estimations which are caused by the existence of fractional delay terms. When the feedback control acts on the boundary, we give the non-trivial conditions of the delay parameter and the control parameter, then the exponential decay of the system with boundary feedback is obtained by using Lyapunov functional theory. When the feedback control operates in the internal domain, the delay parameter is less than the control parameter, the exponential stability of system with internal feedback is obtained by estimating of the resolvent operator norm. This is the first study of piezoelectric beam system with fractional delay and magnetic effect, which has certain significance in its development.