Approximate Controllability of Ψ-Hilfer Fractional Neutral Differential Equation with Infinite Delay
Abstract
:1. Introduction
2. Preliminaries
- (ci)
- is closed and ;
- (cii)
- is the resolvent set of containing and , we write
- 1.
- ;
- 2.
- , for ;
- 3.
- .
- (a)
- For any , and are bounded linear operators with
- (b)
- The operators, and , are strongly continuous for all ; thus, we write
- (c)
- If is a compact operator , then and are compact for all .
- (d)
- If and are the compact strongly continuous semigroup of bounded linear operators for , then and are continuous in the uniform operator topology.
- (H1)
- is the -semigroup, such that .
- (H2)
- For , , are continuous functions, and for each , and are strongly measurable.
- (H3)
- There exists an increasing function and , such that for all , and ∃ a constant , then
- (H4)
- There exists a constant , such that: for all .
- (H5)
- The function is continuous, and there exists for any is strongly measurable, there exists such that:
3. Approximate Controllability
4. Application
4.1. Application 1
4.2. Application 2
- Product modulator 1 accepts the input and produces the output .
- Product modulator 2 accepts the input and and gives out put .
- Product modulator 3 accepts the input and produces the output .
- Product modulator 4 accepts the input and -function, and obtains the output .
- Product modulator 5 accepts the input and , and produces the output .
- Product modulator 6 accepts the input and , and gives the output .
- Product modulator 7 accepts the input and , and gives the output over the period .
- The integrator executes the input and and produces the outputover the period of time .
- Product modulator 8 accepts the input and and gives the output .
- Product modulator 9 accepts and at time , and produces .
- The integrators execute the following value:, and produces the integral value over the period .Finally, we turn all outputs from the integrators to the summer network and the output of is obtained; it is bounded and approximately controllable.
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
Hilfer Fractional Derivative | |
Hilfer Fractional Differential | |
MNC | Measure of Noncompactness. |
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Varun Bose, C.S.; Udhayakumar, R.; Velmurugan, S.; Saradha, M.; Almarri, B. Approximate Controllability of Ψ-Hilfer Fractional Neutral Differential Equation with Infinite Delay. Fractal Fract. 2023, 7, 537. https://doi.org/10.3390/fractalfract7070537
Varun Bose CS, Udhayakumar R, Velmurugan S, Saradha M, Almarri B. Approximate Controllability of Ψ-Hilfer Fractional Neutral Differential Equation with Infinite Delay. Fractal and Fractional. 2023; 7(7):537. https://doi.org/10.3390/fractalfract7070537
Chicago/Turabian StyleVarun Bose, Chandrabose Sindhu, Ramalingam Udhayakumar, Subramanian Velmurugan, Madhrubootham Saradha, and Barakah Almarri. 2023. "Approximate Controllability of Ψ-Hilfer Fractional Neutral Differential Equation with Infinite Delay" Fractal and Fractional 7, no. 7: 537. https://doi.org/10.3390/fractalfract7070537