Qualitative Analysis of the Prabhakar-Caputo Type Fractional Delayed Equations

Abstract

The representation of an explicit solution to the Prabhakar fractional differential delayed system is studied employing the far-famed Laplace transform technique. Second, the existence uniqueness of the solution is debated together with the Ulam-Hyers stability of a semilinear Prabhakar fractional differential delayed system. Thirdly, the necessary and sufficient circumstances for the controllability of linear Prabhakar fractional differential delayed system are determined by describing the Gramian matrix. A sufficient circumstance for the relative controllability of a semilinear Prabhakar fractional differential delayed system is studied via the Krasnoselskii's fixed point theorem. Numerical examples are offered to verify the theoretical findings.