In this paper, the trajectory controllability (T-controllability) of a nonlinear fractional-order damped system with time delay is studied. Existence and uniqueness of solution are obtained by using the Banach fixed point theorem and Green's function. Necessary and sufficient conditions of trajectory controllable for the nonlinear system are formulated and proved under a predefined trajectory. Modified fractional finite difference method is applied to the system for numerical approximation of its solution. The applicability of this technique is demonstrated by numerical simulation of two scientific models such as neuromechanical interaction in human snoring and fractional delayed damped Mathieu equation.

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