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

This work develops an energy-based reachability framework for linear fractional-order dynamical systems governed by Caputo derivatives of order α∈(0,1) in the presence of time-dependent delays acting on both the state and control channels. By combining a controllability Gramian formulation with a delay-independent algebraic characterization, explicit quantitative descriptions of reachability under finite energy constraints are obtained. It is shown that the set of terminal states attainable with bounded control energy admits a geometric characterization in terms of a Gramian-induced ellipsoidal region centered at the uncontrolled terminal state. In addition, the minimum eigenvalue of the controllability Gramian is identified as an energy-based controllability margin that provides certified reachability guarantees. Stability and sensitivity properties of the associated minimum-energy control law with respect to perturbations in the terminal target are also established. The theoretical developments are supported by implementable numerical procedures and illustrative examples that demonstrate the computation of the controllability Gramian, its spectral characteristics, and the resulting minimum-energy control inputs.