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Mittag-Leffler stability for a fractional Euler-Bernoulli problem. (English) Zbl 1485.35040

Summary: 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.

MSC:

35B35 Stability in context of PDEs
35R11 Fractional partial differential equations
35R10 Partial functional-differential equations
35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
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