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 systemD0,tμ,νzt+Azt+Ωzt−h=ftof order1<μ<2and type0≤ν≤1, with nonpermutable matricesAandΩ. 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.