Controllability of Prabhakar Fractional System of Integro-Differential Equations of Order η∈(1,2) with Nonlocal Conditions: Application to Viscoelastic Mechanical Systems

This paper advances a comprehensive controllability framework for Prabhakar fractional differential systems (PFDSs) of order η∈(1,2) with nonlocal initial conditions, where the second-order setting requires the joint specification of both an initial state and an initial velocity. Explicit solution representations for four structurally distinct classes of second-order Prabhakar systems are derived via the Laplace transform method and Neumann series expansions, revealing that the placement of the forcing term directly in the system or under the Prabhakar fractional integral operator produces fundamentally different convolution kernels. For linear integro-differential systems, necessary and sufficient controllability conditions are established through a Gramian rank criterion with an explicit norm-bounded control law, while for nonlinear systems, sufficient conditions are obtained via the Schauder fixed-point theorem under an asymptotic growth condition. Three numerical examples validate the theory: a three-dimensional linear system and a two-dimensional nonlinear integro-differential system achieve terminal errors of order 10−12 and 10−7, respectively, and a Prabhakar fractional mass–spring–damper system with viscoelastic hereditary damping demonstrates direct physical relevance, with all theoretical conditions verified and a terminal error of 7.42×10−5 confirming precise rest-position steering by the Gramian-based control law.