Mittag–Leffler stability for a fractional Euler–Bernoulli problem

Abstract

• A fractional fourth order viscoelastic problem is treated. • The problem is an interpolation of a parabolic and a hyperbolic problem. • A fractional lower order term is proved to be an effective damping. • The problem can be stabilized by a viscoelastic memory term as well. • New functionals are introduced in the context of the multiplier technique. We investigate the stability of an Euler–Bernoulli type problem of fractional order. By adding a fractional term of lower-order, namely of order half of the order of the leading fractional derivative , the problem will generalize the well-known telegraph equation. It is shown that this term is capable of stabilizing the system to rest in a Mittag–Leffler manner. Moreover, we consider a much weaker dissipative term consisting of a memory term in the form of a convolution known as viscoelastic term. It is proved that we can still obtain Mittag–Leffler stability under a smallness condition on the involved kernels. The results rely heavily on some established properties of fractional derivatives and some newly introduced functionals.