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Research Papers

Finite Difference Computational Method for Trajectory Controllability of a Delayed Damped System Governed by Fractional Differential Equation

[+] Author and Article Information
P. Muthukumar

Department of Mathematics,
The Gandhigram Rural
Institute-Deemed University,
Dindigul,
Gandhigram 624302, Tamil Nadu, India
e-mail: pmuthukumargri@gmail.com

B. Ganesh Priya

Department of Mathematics,
The Gandhigram Rural
Institute-Deemed University,
Dindigul,
Gandhigram 624302, Tamil Nadu, India
e-mail: primath85@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 30, 2017; final manuscript received May 26, 2017; published online July 12, 2017. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 12(5), 051028 (Jul 12, 2017) (8 pages) Paper No: CND-17-1049; doi: 10.1115/1.4037076 History: Received January 30, 2017; Revised May 26, 2017

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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Figures

Grahic Jump Location
Fig. 1

The trajectory of the steering control function (16) for the system (15) with α = 1.95, β = 0.95, τ = 5, δ = 2, μ = 1, q = −3, γ = 0.1, and h = 0.1

Grahic Jump Location
Fig. 2

Approximate numerical trajectory of the state function of fractional damped system (15) with described trajectory z(t) and α = 1.95, β = 0.95, τ = 5, δ = 2, μ = 1, q = −3, γ = 0.1, and h = 0.1

Grahic Jump Location
Fig. 3

Trajectory of the steering control function for the fractional delayed damped Mathieu system (19) and α = 1.96, β = 0.8, τ = 2π, δ = 65, ε = 0.05, c = 1.5, d = −35, and h = 0.1

Grahic Jump Location
Fig. 4

Approximate numerical trajectory of the state function for the fractional delayed damped Mathieu system (19) with described trajectory (20) and α = 1.96, β = 0.8, τ = 2π, δ = 65, ε = 0.05, c = 1.5, d = −35, and h = 0.1

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