Controllability and Minimum-Energy Control of Fractional Differential Systems with Time-Varying State and Control Delays

This paper presents a unified framework for controllability and minimum-energy control of linear fractional differential systems with Caputo derivative order γ∈(0,1) and fully time-varying state and control delays. An explicit mild solution representation is derived using the fractional fundamental matrix, and a new controllability Gramian is introduced. Using analytic properties of the matrix-valued Mittag-Leffler function, we prove a fractional Kalman-type theorem showing that bounded time-varying delays do not change the algebraic controllability structure determined by (F,G,K). The minimum-energy control problem is solved in closed form through Hilbert space methods. Efficient numerical strategies and several examples—including delayed viscoelastic, neural, and robotic models—demonstrate practical applicability and computational feasibility.