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Controllability of Hilfer Fractional Langevin Dynamical System with Impulse in an Abstract Weighted Space

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Abstract

The foremost goal of this paper is to study the sufficient conditions for controllability of Hilfer fractional Langevin dynamical system with impulse. The main results are obtained by using the generalized fractional calculus and fixed point theory. Finally, a pair of examples are equipped to demonstrate the importance of the obtained theoretical result. The homotopy perturbation method (HPM) is successfully used in the numerical example.

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Acknowledgements

The authors would like to specially thank the editor and the anonymous referees for their valuable suggestions that led to the improvement of the article.

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Authors and Affiliations

  1. Department of Mathematics, PSG College of Technology, Coimbatore, 641004, TamilNadu, India

    B. Radhakrishnan & T. Sathya

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  1. B. Radhakrishnan
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  2. T. Sathya
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Correspondence to B. Radhakrishnan.

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Appendix A

Appendix A

1.1 Analysis for HPM using Hilfer fractional derivative

To gain a numerical solution for FDEs, the homotopy perturbation method is one of the most beneficial methods in the literature for solving nonlinear fractional differential equations. This method was first offered by He [9]. The foremost benefit of applying HPM has accelerated the convergence to the solutions, and the results are promptly received after few iterations. On this basis, example 5.2 is solved by using the homotopy perturbation method. The following systematic procedure is used to solve the nonlinear fractional differential equations.

Step 1: Consider the nonlinear operator

$$\begin{aligned} N({\mathscr {Y}})= D^{\alpha , \beta }_{0^+} {\mathscr {Y}} (\eta ) -{\mathscr {E}}_{1}({\mathscr {Y}})-\displaystyle \int _{0}^{1}F_{1} (x,\eta ){\mathscr {E}}_{2}({\mathscr {Y}}(\eta ))\mathrm{d}\eta ,\ \ \ \eta \in (0,1], \end{aligned}$$
(A.1)

where \(D^{\alpha , \beta }_{0^+} {\mathscr {Y}} (\eta )\) is the Hilfer fractional derivative and \({\mathscr {Y}}\) is a sufficiently smooth function in [0, 1].

Step 2: Define the homotopy \({\mathbb {H}}({\mathscr {Y}},P)\), \(P\in [0,1]\), such that it satisfies

$$\begin{aligned} {\mathbb {H}}({\mathscr {Y}},0)=L({\mathscr {Y}})=0, \ \ {\mathbb {H}}({\mathscr {Y}},1)=N({\mathscr {Y}})=0. \end{aligned}$$

In general, \(L({\mathscr {Y}})\) is considered as a linear operator, and nonlinear operator \(N({\mathscr {Y}})\) is obtained from the given problem. Now, choose a homotopy by using the convex combination of \(L({\mathscr {Y}})\) and \(N({\mathscr {Y}})\) as

$$\begin{aligned} {\mathbb {H}}({\mathscr {Y}},P)=(1-P)L({\mathscr {Y}}) +P\big [D^{\alpha , \beta }_{0^+} {\mathscr {Y}} (\eta ) -{\mathscr {E}}_{1}({\mathscr {Y}})-\displaystyle \int _{0}^{1}F_{1} (x,\eta ){\mathscr {E}}_{2}({\mathscr {Y}}(\eta ))\mathrm{d}\eta \big ]=0. \end{aligned}$$
(A.2)

The homotopy perturbation parameter P satisfies \(0\le P\le 1\), and it can be increased monotonically from zero to one in such a way that the solution of the problem \(L({\mathscr {Y}})=0\) is continuously deformed to the solution of the original problem \(N({\mathscr {Y}})=0.\)

Step 3 : The solution of (A.1) is assumed as a series of P

$$\begin{aligned} {\mathscr {Y}}={\mathscr {Y}}_{0}+P{\mathscr {Y}}_{1}+P^{2}{\mathscr {Y}}_{2}+\cdots \ \ . \end{aligned}$$
(A.3)

Substituting (A.3) into (A.2) and collecting the same powers of P,  the series equations can be obtained in the following form:

$$\begin{aligned} P^{0}: D^{\alpha , \beta }_{0^+} {\mathscr {Y}}_{0} (\eta )= & {} 0, \\ P^{1}: D^{\alpha , \beta }_{0^+} {\mathscr {Y}}_{1} (\eta )= & {} {\mathscr {E}}_{1}({\mathscr {Y}})-\displaystyle \int _{0}^{1}F_{1} (x,\eta ){\mathscr {E}}_{2}({\mathscr {Y}}(\eta ))\mathrm{d}\eta , \ldots \ \ \ . \end{aligned}$$

Now, applying Hilfer fractional derivative on these set of equations, the approximate solution \({\mathscr {Y}}(\eta )\) is obtained in the form of series like \({\mathscr {Y}}(\eta )=\displaystyle \sum _{n=0}^{\infty }{\mathscr {Y}}_{n}(\eta ).\)

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Radhakrishnan, B., Sathya, T. Controllability of Hilfer Fractional Langevin Dynamical System with Impulse in an Abstract Weighted Space. J Optim Theory Appl 195, 265–281 (2022). https://doi.org/10.1007/s10957-022-02081-4

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