Stabilization and asymptotic stability of the Caputo–Fabrizio fractional-order linear and semilinear evolution equations

Abstract

We discuss the asymptotic stability of the fractional-order linear and semilinear evolution equations involving a Caputo–Fabrizio fractional derivative for fractional-order α∈(0,1) with a non-singular kernel. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo–Fabrizio system admits only a constant solution and a new concept of a global solution for the Caputo–Fabrizio system is introduced. Laplace transform and Grönwall inequality are used to derive the local and global asymptotic stability conditions. By constructing a suitable linear feedback control and using our main results, we stabilize the Caputo–Fabrizio fractional-order linear and semilinear evolution equations. At the end, by using the stabilization result, we stabilize a fractional-order chaotic system to support the obtained results.