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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 and 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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[20] L. Huang, Z. Guo and J. Wang, Theory and Applications of Differential Equations with Discontinuous Right-hand Sides, Science Press, Beijing, 2011. 
[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[41]

T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272. Springer-Verlag, Berlin, 2001.

[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.

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.

[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.

[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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[20] L. Huang, Z. Guo and J. Wang, Theory and Applications of Differential Equations with Discontinuous Right-hand Sides, Science Press, Beijing, 2011. 
[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[41]

T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, 272. Springer-Verlag, Berlin, 2001.

[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.

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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