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, 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 $0\leq\nu\leq1,$ 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 and new even for these fractional delay systems.