Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks

Zengyun Wang; Jinde Cao; Zuowei Cai; Lihong Huang

doi:10.3934/dcdsb.2020200

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May 2021, 26(5): 2677-2692. doi: 10.3934/dcdsb.2020200

Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks

Zengyun Wang ^a,b,c, , Jinde Cao ^2,3,, , Zuowei Cai ^d, and Lihong Huang ^e,

a.	Department of Mathematics Hunan First Normal University, Changsha, Hunan 410205, China
b.	School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China
c.	ool of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China c Jiangsu Provincial Key Laboratory of Networked Collective Intelligence Southeast University, Nanjing, Jiangsu 210096, China
d.	Department of Information Technology, Hunan Women's University Changsha, Hunan 410002, China
e.	School of Mathematics and Statistics, Changsha University of Science and Technology Changsha, Hunan 410114, China

^* Corresponding author: Jinde Cao

Received December 2019 Revised March 2020 Published May 2021 Early access June 2020

Fund Project: This work was supported in part by NSF of China(No.11601143, 61833005), Jiangsu Provincial Key Laboratory of Networked Collective Intelligence (No.BM2017002), China Postdoctoral Science Foundation (No.2018M632207) and Teaching Reform Project of Ordinary Colleges and Universities in Hunan Province (No. 844)

In this article, we present several results on Finite-Time Stability (FTS) of impulsive differential inclusion. In order to investigate the FTS problem, a new concept of Finite-Time Stable Function Pair (FTSFP) is proposed. By virtue of average impulsive interval and FTSFP, two unified criteria on FTS of impulsive differential inclusion are obtained, which are effective for both the destabilizing impulses and the stabilizing impulses. In addition, the settling-time depends not only on the initial value, but also on the information of impulsive sequence. As an extension, a delay-independent FTS result of impulsive delayed differential inclusion is presented. Finally, the obtained results are applied to study the FTS of discontinuous impulsive neural networks.

Keywords: Finite-time stability, impulsive differential inclusion, finite-time stable function pair, improved Lyapunov method, discontinuous impulsive neural networks.

Mathematics Subject Classification: Primary: 39A30, 34K34, 34K45; Secondary: 34A60.

Citation: Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

References:

[1]	N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations, 246 (2009), 3834-3863. doi: 10.1016/j.jde.2009.03.004. Google Scholar
[2]	J. Abderrahim nd E. Vilches, A differential equation approach to implicit sweeping processes, J. Differential Equations, 266 (2019), 5168-5184. doi: 10.1016/j.jde.2018.10.024. Google Scholar
[3]	W. Allegretto, D. Papini and M. Forti, Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks, IEEE Trans. Neural Netw., 21 (2010), 1110-1125. Google Scholar
[4]	F. Amato, G. De Tommasi and A. Pironti, Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems, Automatica J. IFAC, 49 (2013), 2546-2550. doi: 10.1016/j.automatica.2013.04.004. Google Scholar
[5]	R. Ambrosino, F. Calabrese, C. Cosentino and G. Tommasi, Sufficient conditions for finite-time stability of impulsive dynamical systems, IEEE Trans. Automat. Control, 54 (2009), 861-865. doi: 10.1109/TAC.2008.2010965. Google Scholar
[6]	J.-P. Aubin and A. Cellina., Differential Inclusions. Set-Valued Functions and Viability Theory, Grundlehren der Mathematischen Wissenschaften, 264. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. Google Scholar
[7]	G. Ballinger and X. Z. Liu, Existence and uniqueness results for impulsive delay differential equation, Dyn. Contin. Discrete Impuls. Syst., 5 (1999), 579-591. Google Scholar
[8]	J. Cao, G. Stamov, I. Stamova and S. Simeonov, Almost periodicity in impulsive fractional-order reaction-diffusion neural networks with time-varying delays, IEEE Trans. Cybern., (2020), http://dx.doi.org/10.1109/TCYB.2020.2967625. Google Scholar
[9]	G. Chen, Y. Yang and J. Li, Finite time stability of a class of hybrid dynamical systems, IET Control Theory Appl., 6 (2012), 8-13. doi: 10.1049/iet-cta.2010.0259. Google Scholar
[10]	G. Craciun, Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAM J. Appl. Algebra Geometry, 3 (2019), 87-106. doi: 10.1137/17M1129076. Google Scholar
[11]	S. Djebali, L. Gorniewicz and A. Ouahab, First-order perodic impulsive semilinear differential inclusions: Existence and structure of solution sets, Math. Comput. Modelling., 52 (2010), 683-714. doi: 10.1016/j.mcm.2010.04.016. Google Scholar
[12]	A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar
[13]	M. Forti and P. Nistri, Global convergence of neural networks with discontinuous neuron activations, IEEE Trans. Circuits Systems I Fund. Theory Appl., 50 (2003), 1421-1435. doi: 10.1109/TCSI.2003.818614. Google Scholar
[14]	M. Forti and D. Papini, Global exponential stability and global convergence in finite time of delayed neural network with infinite gain, IEEE Trans. Neural Netw., 16 (2005), 1449-1463. Google Scholar
[15]	H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Progr. Theoret. Phys., 69 (1983), 32-47. doi: 10.1143/PTP.69.32. Google Scholar
[16]	G. Haddad, Monotone viable trajectories for functional differential inclusions, J. Differential Equations, 41 (1981), 1-24. doi: 10.1016/0022-0396(81)90031-0. Google Scholar
[17]	G. Haddad, Topological propertyies of the sets of solutions for functional differntial inclusion, Nonlinear Anal., 39 (1981), 1349-1366. doi: 10.1016/0362-546X(81)90111-5. Google Scholar
[18]	J. P. Hespanha, D. Liberzon and A. R. Teel, Lyapuov conditions for input-to-state stability of impulsive systems, Automatica J. IFAC, 44 (2008), 2735-2744. doi: 10.1016/j.automatica.2008.03.021. Google Scholar
[19]	S. C. Hu, D. A. Kandilakis and N. S. Papageorgiou, Periodic solutions for nonconvex differential inclusions, Proc. Amer. Math. Soc., 127 (1999), 89-94. doi: 10.1090/S0002-9939-99-04338-5. Google Scholar
[20]	L. Huang, Z. Guo and J. Wang, Theory and Applications of Differential Equations with Discontinuous Right-hand Sides, Science Press, Beijing, 2011. Google Scholar
[21]	P. Hur, B. Duiser, S. Salapaka and E. Weckster, Measuring robustness of the postural control system to a mild impulsive perturbation, IEEE Trans Neur. Syst. Rehab. Engin., 18 (2010), 461-467. Google Scholar
[22]	X. D. Li, D. W. C. Ho and J. D. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368. doi: 10.1016/j.automatica.2018.10.024. Google Scholar
[23]	Y. C. Li and R. G. Sanfelice, Finite time stability of sets for hybrid dynamical systems, Automatica J. IFAC, 100 (2019), 200-211. doi: 10.1016/j.automatica.2018.10.016. Google Scholar
[24]	J. X. Liu, L. G. Wu, C. W. Wu, W. S. Luo and L. Franquelo, Event-triggering dissipative control of switched stochastic systems via sliding mode, Automatica J. IFAC, 103 (2019), 261-273. doi: 10.1016/j.automatica.2019.01.029. Google Scholar
[25]	K.-Z. Liu, X.-M. Sun, J. Liu and R. Andrew, Stability theorems for delayed differential inclusions, IEEE Trans. Autom. Control., 61 (2016), 3215-3220. doi: 10.1109/TAC.2015.2507782. Google Scholar
[26]	W. L. Lu and T. P. Chen, Almost periodic dynamics of a class of delayed neural networks with discontinuous activations, Neural Comput., 20 (2008), 1065-1090. doi: 10.1162/neco.2008.10-06-364. Google Scholar
[27]	J. Q Lu, D. W. C. Ho and J. D. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica J. IFAC, 46 (2010), 1215-1221. doi: 10.1016/j.automatica.2010.04.005. Google Scholar
[28]	E. Moulay and W. Perruquetti, Finite time stability of differential inclusions, IMA J. Math. Control Inform., 22 (2005), 465-475. doi: 10.1093/imamci/dni039. Google Scholar
[29]	E. Moulay and W. Perruquetti, Finite time stability and stabilization of a class of conitnuous systems, J. Math. Anal. Appl., 323 (2006), 1430-1443. doi: 10.1016/j.jmaa.2005.11.046. Google Scholar
[30]	E. Moulay, M. Dambrine, N. Yeganefar and W. Perruquetti, Finite time stability and stabilization of time-delayed systems, Systems Control Lett., 57 (2008), 561-566. doi: 10.1016/j.sysconle.2007.12.002. Google Scholar
[31]	J. Nygren and K. Pelckmans, A stability criterion for switching Lur'e systems with switching-path restrictions, Automatica J. IFAC, 96 (2018), 337-341. doi: 10.1016/j.automatica.2018.06.038. Google Scholar
[32]	B. E. Paden and S. S. Sastry, A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulator, IEEE Trans. Circuits Syst., 34 (1987), 73-82. doi: 10.1109/TCS.1987.1086038. Google Scholar
[33]	S. G. Peng, F. Q. Deng and Y. Zhang, A unified Razumikhin-type criteria on input-to-state stability of time-varying impulsive delayed system, Systems Control Lett., 216 (2018), 20-26. doi: 10.1016/j.sysconle.2018.04.002. Google Scholar
[34]	A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Auto. Contr., 57 (2012), 2106-2100. doi: 10.1109/TAC.2011.2179869. Google Scholar
[35]	A. Polyakov, D. Efimov and W. Perruquetti, Finite-time and fixed-time stabilization: Implicit Lyapunov function approach, Automatica J. IFAC, 51 (2015), 332-340. doi: 10.1016/j.automatica.2014.10.082. Google Scholar
[36]	S. T. Qin and X. P. Xue, Periodic solutions for nonlinear differential inclusions with multivalued perturbations, J. Math, Anal. Appl., 424 (2015), 988-1005. doi: 10.1016/j.jmaa.2014.11.057. Google Scholar
[37]	E. Serpelloni, M. Maggiore and C. Damaren, Bang-bang hybrid stabilization of perturbed double-integrators, Automatica J. IFAC, 69 (2016), 315-323. doi: 10.1016/j.automatica.2016.02.028. Google Scholar
[38]	S. Vaddi, K. Alfriend, S. Vadali and P. Sengupta., Formation establishment and reconfiguration using impulsive control, J. Guid Control. Dynam., 28 (2005), 262-268. Google Scholar
[39]	A. Vinodkumar and A. Anguraj, Existence of random impulsive abstract neutral non-autonomous differeential inclusions with delayes, Nonlinear Anal. Hybrid Syst., 5 (2011), 413-426. doi: 10.1016/j.nahs.2011.04.002. Google Scholar
[40]	X. T. Wu, Y. Tang and W. B. Zhang, Input-to-state stability of impulsive stochastic delayed systems under linear assumptions, Automatica J. IFAC, 66 (2016), 195-2014. doi: 10.1016/j.automatica.2016.01.002. Google Scholar
[41]	T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272. Springer-Verlag, Berlin, 2001. Google Scholar
[42]	B. Zhou, On asymptotic stability of linear time-varying systems, Automatica J. IFAC, 68 (2016), 266-276. doi: 10.1016/j.automatica.2015.12.030. Google Scholar

show all references

References:

[1]	N. Abada, M. Benchohra and H. Hammouche, Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions, J. Differential Equations, 246 (2009), 3834-3863. doi: 10.1016/j.jde.2009.03.004. Google Scholar
[2]	J. Abderrahim nd E. Vilches, A differential equation approach to implicit sweeping processes, J. Differential Equations, 266 (2019), 5168-5184. doi: 10.1016/j.jde.2018.10.024. Google Scholar
[3]	W. Allegretto, D. Papini and M. Forti, Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks, IEEE Trans. Neural Netw., 21 (2010), 1110-1125. Google Scholar
[4]	F. Amato, G. De Tommasi and A. Pironti, Necessary and sufficient conditions for finite-time stability of impulsive dynamical linear systems, Automatica J. IFAC, 49 (2013), 2546-2550. doi: 10.1016/j.automatica.2013.04.004. Google Scholar
[5]	R. Ambrosino, F. Calabrese, C. Cosentino and G. Tommasi, Sufficient conditions for finite-time stability of impulsive dynamical systems, IEEE Trans. Automat. Control, 54 (2009), 861-865. doi: 10.1109/TAC.2008.2010965. Google Scholar
[6]	J.-P. Aubin and A. Cellina., Differential Inclusions. Set-Valued Functions and Viability Theory, Grundlehren der Mathematischen Wissenschaften, 264. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. Google Scholar
[7]	G. Ballinger and X. Z. Liu, Existence and uniqueness results for impulsive delay differential equation, Dyn. Contin. Discrete Impuls. Syst., 5 (1999), 579-591. Google Scholar
[8]	J. Cao, G. Stamov, I. Stamova and S. Simeonov, Almost periodicity in impulsive fractional-order reaction-diffusion neural networks with time-varying delays, IEEE Trans. Cybern., (2020), http://dx.doi.org/10.1109/TCYB.2020.2967625. Google Scholar
[9]	G. Chen, Y. Yang and J. Li, Finite time stability of a class of hybrid dynamical systems, IET Control Theory Appl., 6 (2012), 8-13. doi: 10.1049/iet-cta.2010.0259. Google Scholar
[10]	G. Craciun, Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAM J. Appl. Algebra Geometry, 3 (2019), 87-106. doi: 10.1137/17M1129076. Google Scholar
[11]	S. Djebali, L. Gorniewicz and A. Ouahab, First-order perodic impulsive semilinear differential inclusions: Existence and structure of solution sets, Math. Comput. Modelling., 52 (2010), 683-714. doi: 10.1016/j.mcm.2010.04.016. Google Scholar
[12]	A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar
[13]	M. Forti and P. Nistri, Global convergence of neural networks with discontinuous neuron activations, IEEE Trans. Circuits Systems I Fund. Theory Appl., 50 (2003), 1421-1435. doi: 10.1109/TCSI.2003.818614. Google Scholar
[14]	M. Forti and D. Papini, Global exponential stability and global convergence in finite time of delayed neural network with infinite gain, IEEE Trans. Neural Netw., 16 (2005), 1449-1463. Google Scholar
[15]	H. Fujisaka and T. Yamada, Stability theory of synchronized motion in coupled-oscillator systems, Progr. Theoret. Phys., 69 (1983), 32-47. doi: 10.1143/PTP.69.32. Google Scholar
[16]	G. Haddad, Monotone viable trajectories for functional differential inclusions, J. Differential Equations, 41 (1981), 1-24. doi: 10.1016/0022-0396(81)90031-0. Google Scholar
[17]	G. Haddad, Topological propertyies of the sets of solutions for functional differntial inclusion, Nonlinear Anal., 39 (1981), 1349-1366. doi: 10.1016/0362-546X(81)90111-5. Google Scholar
[18]	J. P. Hespanha, D. Liberzon and A. R. Teel, Lyapuov conditions for input-to-state stability of impulsive systems, Automatica J. IFAC, 44 (2008), 2735-2744. doi: 10.1016/j.automatica.2008.03.021. Google Scholar
[19]	S. C. Hu, D. A. Kandilakis and N. S. Papageorgiou, Periodic solutions for nonconvex differential inclusions, Proc. Amer. Math. Soc., 127 (1999), 89-94. doi: 10.1090/S0002-9939-99-04338-5. Google Scholar
[20]	L. Huang, Z. Guo and J. Wang, Theory and Applications of Differential Equations with Discontinuous Right-hand Sides, Science Press, Beijing, 2011. Google Scholar
[21]	P. Hur, B. Duiser, S. Salapaka and E. Weckster, Measuring robustness of the postural control system to a mild impulsive perturbation, IEEE Trans Neur. Syst. Rehab. Engin., 18 (2010), 461-467. Google Scholar
[22]	X. D. Li, D. W. C. Ho and J. D. Cao, Finite-time stability and settling-time estimation of nonlinear impulsive systems, Automatica J. IFAC, 99 (2019), 361-368. doi: 10.1016/j.automatica.2018.10.024. Google Scholar
[23]	Y. C. Li and R. G. Sanfelice, Finite time stability of sets for hybrid dynamical systems, Automatica J. IFAC, 100 (2019), 200-211. doi: 10.1016/j.automatica.2018.10.016. Google Scholar
[24]	J. X. Liu, L. G. Wu, C. W. Wu, W. S. Luo and L. Franquelo, Event-triggering dissipative control of switched stochastic systems via sliding mode, Automatica J. IFAC, 103 (2019), 261-273. doi: 10.1016/j.automatica.2019.01.029. Google Scholar
[25]	K.-Z. Liu, X.-M. Sun, J. Liu and R. Andrew, Stability theorems for delayed differential inclusions, IEEE Trans. Autom. Control., 61 (2016), 3215-3220. doi: 10.1109/TAC.2015.2507782. Google Scholar
[26]	W. L. Lu and T. P. Chen, Almost periodic dynamics of a class of delayed neural networks with discontinuous activations, Neural Comput., 20 (2008), 1065-1090. doi: 10.1162/neco.2008.10-06-364. Google Scholar
[27]	J. Q Lu, D. W. C. Ho and J. D. Cao, A unified synchronization criterion for impulsive dynamical networks, Automatica J. IFAC, 46 (2010), 1215-1221. doi: 10.1016/j.automatica.2010.04.005. Google Scholar
[28]	E. Moulay and W. Perruquetti, Finite time stability of differential inclusions, IMA J. Math. Control Inform., 22 (2005), 465-475. doi: 10.1093/imamci/dni039. Google Scholar
[29]	E. Moulay and W. Perruquetti, Finite time stability and stabilization of a class of conitnuous systems, J. Math. Anal. Appl., 323 (2006), 1430-1443. doi: 10.1016/j.jmaa.2005.11.046. Google Scholar
[30]	E. Moulay, M. Dambrine, N. Yeganefar and W. Perruquetti, Finite time stability and stabilization of time-delayed systems, Systems Control Lett., 57 (2008), 561-566. doi: 10.1016/j.sysconle.2007.12.002. Google Scholar
[31]	J. Nygren and K. Pelckmans, A stability criterion for switching Lur'e systems with switching-path restrictions, Automatica J. IFAC, 96 (2018), 337-341. doi: 10.1016/j.automatica.2018.06.038. Google Scholar
[32]	B. E. Paden and S. S. Sastry, A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulator, IEEE Trans. Circuits Syst., 34 (1987), 73-82. doi: 10.1109/TCS.1987.1086038. Google Scholar
[33]	S. G. Peng, F. Q. Deng and Y. Zhang, A unified Razumikhin-type criteria on input-to-state stability of time-varying impulsive delayed system, Systems Control Lett., 216 (2018), 20-26. doi: 10.1016/j.sysconle.2018.04.002. Google Scholar
[34]	A. Polyakov, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans. Auto. Contr., 57 (2012), 2106-2100. doi: 10.1109/TAC.2011.2179869. Google Scholar
[35]	A. Polyakov, D. Efimov and W. Perruquetti, Finite-time and fixed-time stabilization: Implicit Lyapunov function approach, Automatica J. IFAC, 51 (2015), 332-340. doi: 10.1016/j.automatica.2014.10.082. Google Scholar
[36]	S. T. Qin and X. P. Xue, Periodic solutions for nonlinear differential inclusions with multivalued perturbations, J. Math, Anal. Appl., 424 (2015), 988-1005. doi: 10.1016/j.jmaa.2014.11.057. Google Scholar
[37]	E. Serpelloni, M. Maggiore and C. Damaren, Bang-bang hybrid stabilization of perturbed double-integrators, Automatica J. IFAC, 69 (2016), 315-323. doi: 10.1016/j.automatica.2016.02.028. Google Scholar
[38]	S. Vaddi, K. Alfriend, S. Vadali and P. Sengupta., Formation establishment and reconfiguration using impulsive control, J. Guid Control. Dynam., 28 (2005), 262-268. Google Scholar
[39]	A. Vinodkumar and A. Anguraj, Existence of random impulsive abstract neutral non-autonomous differeential inclusions with delayes, Nonlinear Anal. Hybrid Syst., 5 (2011), 413-426. doi: 10.1016/j.nahs.2011.04.002. Google Scholar
[40]	X. T. Wu, Y. Tang and W. B. Zhang, Input-to-state stability of impulsive stochastic delayed systems under linear assumptions, Automatica J. IFAC, 66 (2016), 195-2014. doi: 10.1016/j.automatica.2016.01.002. Google Scholar
[41]	T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272. Springer-Verlag, Berlin, 2001. Google Scholar
[42]	B. Zhou, On asymptotic stability of linear time-varying systems, Automatica J. IFAC, 68 (2016), 266-276. doi: 10.1016/j.automatica.2015.12.030. Google Scholar

Figure 1. The state trajectories of $ x_{i}(t) $ $ (i = 1,2) $ without impulsive effects in Example 1

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Figure 2. The trajectories of states $ x_{i}(t) $ $ (i = 1,2) $ with different impulsive sequences in Example 1

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