Finite Time Stability of Fractional Order Systems of Neutral Type
Abstract
:1. Introduction
- Knowing that, FTS of NFOTSs are proved in the literature based on the Gronwall inequality, see [23]. The novelty of our suggested work comes from the use of the fixed point theory to demonstrate the FTS of NFOTSs;
- A novel FTS result of FOS of neutral type is given;
- The theoretical contributions are confirmed and validated by two examples.
2. Basic Results
- S1
- if, and only if, ;
- S2
- for all ;
- S3
- for all .
- (a)
- with ;
- (b)
- is the unique fixed point of K in ;
- (c)
- If , then .
3. Stability Analysis
3.1. The Case
3.2. The Case
4. Examples
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Warrier, P.; Shah, P. Fractional Order Control of Power Electronic Converters in Industrial Drives and Renewable Energy Systems: A Review. IEEE Access 2021, 9, 58982–59009. [Google Scholar] [CrossRef]
- Afshari, A. Solution of fractional differential equations in quasi-b-metric and bmetric- like spaces. Adv. Differ. Equ. 2019, 2019, 285. [Google Scholar] [CrossRef] [Green Version]
- Afshari, A.; Gholamyan, H.; Zhai, C.B. New applications of concave operators to existence and uniqueness of solutions for fractional differential equations. Math. Commun. 2020, 25, 157–169. [Google Scholar]
- Afshari, A.; Sajjadmanesh, M.; Baleanu, D. Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives. Adv. Differ. Equ. 2020, 2020, 111. [Google Scholar] [CrossRef] [Green Version]
- Feng, Y.Y.; Yang, X.J.; Liu, J.G.; Chen, Z.Q. A new fractional Nishihara-type model with creep damage considering thermal effect. Eng. Fract. Mech. 2021, 242, 107451. [Google Scholar] [CrossRef]
- Ibrahim, R.W.; Baleanu, D. On quantum hybrid fractional conformable differential and integral operators in a complex domain. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 2021, 31, 514–531. [Google Scholar] [CrossRef]
- Jafari, H.; Jassim, H.K.; Baleanu, D.; Chu, Y. On the Approximate Solutions for a System of Coupled Korteweg De Vries Equations with Local Fractional Derivative. Fractals 2021, 29, 2140012. [Google Scholar] [CrossRef]
- Sakar, M.G. Numerical solution of neutral functional-differential equations with proportional delays. Int. J. Optim. Control. Theor. Appl. (IJOCTA) 2017, 7, 186–194. [Google Scholar] [CrossRef]
- Veeresha, P.; Yavuz, M.; Baishya, C. A computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators. Int. J. Optim. Control. Theor. Appl. (IJOCTA) 2021, 11, 52–67. [Google Scholar] [CrossRef]
- Vigya; Mahto, T.; Malik, H.; Mukherjee, V.; Alotaibi, M.A.; Almutairi, A. Renewable generation based hybrid power system control using fractional order-fuzzy controller. Energy Rep. 2021, 7, 641–653. [Google Scholar] [CrossRef]
- Zhang, K.; Wu, L. Using a fractional order grey seasonal model to predict the dissolved oxygen and pH in the Huaihe River. Water Sci. Technol. 2021, 83, 475–486. [Google Scholar] [CrossRef] [PubMed]
- Daoui, A.; Yamni, M.; Karmouni, H.; Sayyouri, M.; Qjidaa, H. Biomedical signals reconstruction and zero-watermarking using separable fractional order Charlier-Krawtchouk transformation and Sine Cosine Algorithm. Signal Process. 2021, 180, 107854. [Google Scholar] [CrossRef]
- Higazy, M.; Allehiany, F.M.; Mahmoud, E.E. Numerical study of fractional order COVID-19 pandemic transmission model in context of ABO blood group. Results Phys. 2021, 22, 103852. [Google Scholar] [CrossRef] [PubMed]
- Liu, D.; Zhao, S.; Luo, X.; Yuan, Y. Synchronization for fractional-order extended Hindmarsh-Rose neuronal models with magneto-acoustical stimulation input. Chaos Solitons Fractals 2021, 144, 110635. [Google Scholar] [CrossRef]
- Zhang, C.; Yang, H.; Jiang, B. Fault Estimation and Accommodation of Fractional-Order Nonlinear, Switched, and Interconnected Systems. IEEE Trans. Cybern. 2020, 52, 1443–1453. [Google Scholar] [CrossRef]
- Amiri, S.; Keyanpour, M.; Asaraii, A. Observer-based output feedback control design for a coupled system of fractional ordinary and reaction-diffusion equations. IMA J. Math. Control. Inf. 2021, 38, 90–124. [Google Scholar] [CrossRef]
- Feng, T.; Wang, Y.E.; Liu, L.; Wu, B. Observer-based event-triggered control for uncertain fractional-order systems. J. Frankl. Inst. 2020, 357, 9423–9441. [Google Scholar] [CrossRef]
- Edrisi-Tabriz, Y.; Lakestani, M.; Razzaghi, M. Study of B-spline collocation method for solving fractional optimal control problems. Trans. Inst. Meas. Control 2021, 43, 2425–2437. [Google Scholar] [CrossRef]
- Brandibur, O.; Kaslik, E. Stability analysis of multi-term fractional-differential equations with three fractional derivatives. J. Math. Anal. Appl. 2021, 495, 124751. [Google Scholar] [CrossRef]
- Ivanescu, M.; Popescu, N.; Popescu, D. Physical Significance Variable Control for a Class of Fractional-Order Systems. Circuits Syst. Signal Process. 2021, 40, 1525–1541. [Google Scholar] [CrossRef]
- Ben Makhlouf, A. A Novel Finite Time Stability Analysis of Nonlinear Fractional-Order Time Delay Systems: A Fixed Point Approach. Asian J. Control 2021. [Google Scholar] [CrossRef]
- Du, F.; Jia, B. Finite-time stability of a class of nonlinear fractional delay difference systems. Appl. Math. Lett. 2019, 98, 233–239. [Google Scholar] [CrossRef]
- Du, F.; Lu, J.G. Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities. Appl. Math. Comput. 2020, 375, 125079. [Google Scholar] [CrossRef]
- Lu, Q.; Zhu, Y.; Li, B. Finite-time stability in mean for Nabla Uncertain Fractional Order Linear Difference Systems. Chaos Solitons Fractals 2021, 29, 2150097. [Google Scholar] [CrossRef]
- Phat, V.N.; Thanh, N.T. New criteria for finite-time stability of nonlinear fractional-order delay systems: A Gronwall inequality approach. Appl. Math. Lett. 2018, 83, 169–175. [Google Scholar] [CrossRef]
- Thanh, N.T.; Phat, V.N. Switching law design for finite-time stability of singular fractional-order systems with delay. IET Control Theory Appl. 2019, 13, 1367–1373. [Google Scholar] [CrossRef]
- Thanh, N.T.; Phat, V.N.; Niamsup, T. New finite-time stability analysis of singular fractional differential equations with time-varying delay. Fract. Calc. Appl. Anal. 2020, 23, 504–519. [Google Scholar] [CrossRef]
- Wu, R.; Lu, Y.; Chen, L. Finite-time stability of fractional delayed neural networks. Neurocomputing 2015, 149, 700–707. [Google Scholar] [CrossRef]
- Wu, G.C.; Baleanu, D.; Zeng, S.D. Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion. Commun. Nonlinear Sci. Numer. Simul. 2018, 57, 299–308. [Google Scholar] [CrossRef]
- Ye, H.; Gao, J.; Ding, Y. A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 2007, 328, 1075–1081. [Google Scholar] [CrossRef] [Green Version]
- Podlubny, I. Fractional Differential Equations; Academic Press: New York, NY, USA, 1999. [Google Scholar]
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Ben Makhlouf, A.; Baleanu, D. Finite Time Stability of Fractional Order Systems of Neutral Type. Fractal Fract. 2022, 6, 289. https://doi.org/10.3390/fractalfract6060289
Ben Makhlouf A, Baleanu D. Finite Time Stability of Fractional Order Systems of Neutral Type. Fractal and Fractional. 2022; 6(6):289. https://doi.org/10.3390/fractalfract6060289
Chicago/Turabian StyleBen Makhlouf, Abdellatif, and Dumitru Baleanu. 2022. "Finite Time Stability of Fractional Order Systems of Neutral Type" Fractal and Fractional 6, no. 6: 289. https://doi.org/10.3390/fractalfract6060289