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

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 $D_{0,t}^{\mu ,\nu }z\left(t\right)+Az\left(t\right)+\Omega z\left(t-h\right)=f\left(t\right)$ of order $1<\mu <2$ 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.