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We introduce the delayed Mittag-Leffler type matrix functions, delayedfractional cosine, delayed fractional sine and use the Laplace transform toobtain an analytical solution to the IVP for a Hilfer type fractional lineartime-delay system $D_{0,t}^{\mu,\nu}z\left( t\right) +Az\left( t\right) +\Omegaz\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, westudy Ulam-Hyers stability of the Hilfer type fractional linear time-delaysystem. Obtained results extend those for Caputo and Riemann-Liouville typefractional linear time-delay systems and new even for these fractional delaysystems.

Analytical solution of the fractional linear time-delay systems and their Ulam-Hyers stability