FINITE-TIME STABILITY OF NON-INSTANTANEOUS IMPULSIVE SET DIFFERENTIAL EQUATIONS

Peiguang Wang; Mengyu Guo; Junyan Bao

doi:10.11948/20220244

2023 Volume 13 Issue 2

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Peiguang Wang, Mengyu Guo, Junyan Bao. FINITE-TIME STABILITY OF NON-INSTANTANEOUS IMPULSIVE SET DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 954-968. doi: 10.11948/20220244

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Peiguang Wang, Mengyu Guo, Junyan Bao. FINITE-TIME STABILITY OF NON-INSTANTANEOUS IMPULSIVE SET DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 954-968. doi: 10.11948/20220244

FINITE-TIME STABILITY OF NON-INSTANTANEOUS IMPULSIVE SET DIFFERENTIAL EQUATIONS

1.
School of Mathematics and Information Science, Hebei University, 071002, China

Corresponding author: Email address: jybao@hbu.edu.cn (J. Bao)
Fund Project: The authors were supported by National Natural Science Foundation of China(Nos. 12171135, 11771115)

Abstract

Abstract

In this paper, we investigate the finite-time stability of non-instant-aneous impulsive set differential equations. By using the generalized Gronwall inequality and a revised Lyapunov method, the finite-time stability criteria for such equations are obtained. Finally, an example is given to illustrate the validity of the results.
- Set differential equations /
- non-instantaneous impulses /
- finite-time stability
MSC: 34A37, 34D20

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References

[1]	R. Agarwal, S. Hristova and D. O'Regan, Non-instantaneous impulses in differential equations, Springer Nature, Cham, 2017. Google Scholar
[2]	R. Agarwal, S. Hristova and D. O'Regan, Lipschitz stability for non-instantaneous impulsive Caputo fractional differential equations with state dependent delays, Axioms, 2019, 8, 4. Google Scholar
[3]	R. Agarwal, D. O'Regan and S. Hristova, Noninstantaneous impulses in Caputo fractional differential equations and practical stability via Lyapunov functions, J. Franklin Inst., 2017, 354(7), 3097–3119. Google Scholar
[4]	R. Agarwal, D. O'Regan and S. Hristova, Stability by Lyapunov like functions of nonlinear differential equations with non-instantaneous impulses, Appl. Math. Comput., 2017, 53(1–2), 147–168. Google Scholar
[5]	F. Amato, M. Ariola and P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 2001, 37(9), 1459–1463. doi: 10.1016/S0005-1098(01)00087-5 CrossRef Google Scholar
[6]	C. Appala Naidu, D. Dhaigude and J. Devi, Finite-time control of discrete-time linear systems, IEEE Trans. Automat. Control, 2005, 50(5), 724–729. doi: 10.1109/TAC.2005.847042 CrossRef Google Scholar
[7]	D. Azzam-Laouir and W. Boukrouk, A delay second-order set-valued differential equation with Hukuhara derivatives, Nonlinear Anal., 2015, 36(6), 704–729. Google Scholar
[8]	D. Azzam-Laouir and W. Boukrouk, Second-order set-valued differential equations with boundary conditions, J. Fixed Point Theory Appl., 2015, 17(1), 99–121. Google Scholar
[9]	A. Brandão Lopes Pinto, F. De Blasi and F. Iervolino, Uniqueness and existence theorems for differential equations with compact convex valued solutions, Boll. Unione Mat. Ital., 1969, 3, 47–54. Google Scholar
[10]	J. Devi, Basic results in impulsive set differential equations, Nonlinear Stud., 2003, 10(3), 259–271. Google Scholar
[11]	A. Fernandez, S. Ali and A. Zada, On non-instantaneous impulsive fractional differential equations and their equivalent integral equations, Math. Methods Appl. Sci., 2021, 3, 1–10. Google Scholar
[12]	E. Hernández and D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc., 2013, 141(5), 1641–1649. Google Scholar
[13]	S. Hristova and K. Ivanova, Lipschitz stability of delay differential equations with non-instantaneous impulses, Dynam. Syst. Appl., 2019, 28, 167–181. Google Scholar
[14]	S. Hristova and R. Terzieva, Lipschitz stability of differential equations with non-instantaneous impulses, Adv. Difference Equ., 2016, 1, 322. Google Scholar
[15]	V. Lakshmikantham, T. Bhaskar and J. Devi, Theory of set differential equations in metric spaces, Cambridge Scientific Publisher, UK, 2006. Google Scholar
[16]	V. Lakshmikantham, S. Leela and A. Martynyuk, Stability analysis of nonlinear systems, Springer International Publishing, Cham, 2015. Google Scholar
[17]	Q. Li, D. Luo, Z. Luo and Q. Zhu, On the novel finite-time stability results for uncertain fractional delay differential equations involving noninstantaneous impulses, Math. Probl. Eng., 2019, 2019, 1–9. Google Scholar
[18]	X. Li, D. Ho and J. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica, 2019, 99, 361–368. Google Scholar
[19]	D. Luo and Z. Luo, Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses, Adv. Difference Equ., 2019, 2019, 155. Google Scholar
[20]	A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, Springer Nature, Cham, 2019. Google Scholar
[21]	F. McRae and J. Devi, Impulsive set differential equations with delay, Appl. Anal., 2005, 84(4), 329–341. Google Scholar
[22]	V. Millman and A. Myshkis, On the stability of motion in the presence of impulses, Sibirski Math, 1960, 1(2), 233–237. Google Scholar
[23]	V. Millman and A. Myshkis, Random impulses in linear dynamical systems, Approximante Methods for Solving Differential Equations, 1963, 1, 64–81. Google Scholar
[24]	O. Naifar, A. Nagy, A. Makhlouf et al., Finite-time stability of linear fractional-order time-delay systems, Internat. J. Robust Nonlinear Control, 2018, 1–8. Google Scholar
[25]	B. Pervaiz, A. Zada, S. Etemad and S. Rezapour, An analysis on the controllability and stability to some fractional delay dynamical systems on time scales with impulsive effects, Adv. Difference Equ., 2021, 2021(1), 1–36. Google Scholar
[26]	H. Waheed, A. Zada and J. Xu, Well-posedness and hyers-ulam results for a class of impulsive fractional evolution equations, Math. Methods Appl. Sci., 2020, 1–23. Google Scholar
[27]	P. Wang and J. Bao, Asymptotic stability of neutral set-valued functional differential equation by fixed point method, Discrete Dyn. Nat. Soc., 2020, 2020, 1–8. Google Scholar
[28]	S. Wang and Y. Tian, Variational methods to the fourth-order linear and nonlinear differential equations with non-instantaneous impulses, J. Appl. Anal. Comput., 2020, 10(6), 2521–2536. Google Scholar
[29]	Z. Wang, J. Cao, Z. Cai and M. Abdel-Aty, A novel Lyapunov theorem on finite/fixed-time stability of discontinuous impulsive systems, Chaos, 2020, 30(1), 013139. Google Scholar
[30]	L. Weiss and E. Infante, On the stability of systems defined over a finite time interval, Proc. Natl. Acad. Sci., 1965, 54(1), 44–48. Google Scholar
[31]	J. Xu, B. Pervaiz, A. Zada and S. Shah, Stability analysis of causal integral evolution impulsive systems on time scales, Adv. Difference Equ., 2021, 41B(3), 781–800. Google Scholar
[32]	C. Yakar and H. Talab, Stability of perturbed set differential equations involving causal operators in regard to their unperturbed ones considering difference in initial conditions, Adv. Math. Phys., 2021, 2021, 1–12. Google Scholar
[33]	A. Zada, B. Pervaiz, M. Subramanian and I. Popa, Finite time stability for nonsingular impulsive first order delay differential systems, Appl. Math. Comput., 2022, 421, 126943. Google Scholar