Analytical Solution of the Fractional Linear Time-Delay Systems and their Ulam-Hyers Stability

Abstract

We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, and delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay system D0,t mu,nu zt+Azt+omega zt-h=ft of order 1 < 2 and type 0 & LE;nu & LE;1, with nonpermutable matrices A and omega. Moreover, we study Ulam-Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann-Liouville type fractional linear time-delay systems with permutable matrices and new even for these fractional delay systems.