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

Abstract

In this paper, we investigate the general stability results of a fully dynamic piezoelectric beam system with internal fractional delays under memory boundary controls. Firstly, by introducing two new equations and using two Volterra's inverse operators to deal with fractional delay terms and boundary memory terms, respectively, a new equivalent system is obtained. Secondly, by combining Faedo–Galerkin approximations with the compactness argument, we prove the local existence and uniqueness of weak solution. Finally, under a broader class of kernel functions, by constructing Lyapunov functionals and new equations and applying some technical lemmas, we establish the optimal and explicit energy decay results for two cases, namely, initial values and without imposing initial values equal to 0, that is, . It is worth pointing out that we focus more on the analysis of , because the stability results of are special cases of . This is the first time to analyze the stability of strong coupling problem by combining internal fractional delay and boundary viscoelastic damping. The obtained general decay results are not necessarily exponential or polynomial, which generalize the relevant results in the existing literature.