November  2020, 25(11): 4493-4513. doi: 10.3934/dcdsb.2020109

Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps

1. 

School of Mathematics, Southeast University, Nanjing 211189, China

2. 

Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 211189, China

* Corresponding author: Jinde Cao

Received  January 2019 Revised  November 2019 Published  November 2020 Early access  March 2020

Fund Project: This work was jointly supported by the Key Project of National Science Foundation of China under Grant No. 61833005

The main aim of this article is to examine almost sure exponential stabilization and suppression of nonlinear systems by periodically intermittent stochastic perturbation with jumps. On the one hand, some sufficient criteria ensure almost sure stabilization of the unstable deterministic system by applying exponential martingale inequality with jumps. On the other hand, sufficient conditions of destabilization are provided under which the system is stable by the well-known strong law of large numbers of local martingale and Poisson process. Both the sample Lyapunov exponents are closely related to the control period $ T $ and noise width $ \theta $. As for applications, the well-known Lorenz chaotic systems and nonlinear Liénard equation with jumps are discussed. Finally, two simulation examples demonstrating the effectiveness of the results are provided.

Citation: Wensheng Yin, Jinde Cao. Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4493-4513. doi: 10.3934/dcdsb.2020109
References:
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[2]

D. Applebaum, Extending stochastic resonance for neuron models to general Lévy noise, IEEE Trans. Neural Netw., 20 (2009), 1993-1995.  doi: 10.1109/TNN.2009.2033183.

[3]

D. Applebaum and M. Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Probab., 46 (2009), 1116-1129.  doi: 10.1239/jap/1261670692.

[4]

D. Applebaum and M. Siakalli, Stochastic stabilization of dynamical systems using Lévy noise, Stoch. Dyn., 10 (2010), 509-527.  doi: 10.1142/S0219493710003066.

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A. D. ApplebyX. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Tran. Automat. Control, 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.

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R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203485217.

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F. DengQ. Luo and X. Mao, Stochastic stabilization of hybrid differential equations, Automatica J. IFAC, 48 (2012), 2321-2328.  doi: 10.1016/j.automatica.2012.06.044.

[8]

Y. FengX. YangQ. Song and J. Cao, Synchronization of memristive neural networks with mixed delays via quantized intermittent control, Appl. Math. Comput., 339 (2018), 874-887.  doi: 10.1016/j.amc.2018.08.009.

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R. Fernholz and I. Karatzas, Relative arbitrage in volatility-stabilized markets, Ann. Financ., 1 (2005), 149-177.  doi: 10.1007/s10436-004-0011-6.

[10]

H. Gao and Y. Wang, Stochastic mutualism model under regime switching with Lévy jumps, Phys. A, 515 (2019), 355-375.  doi: 10.1016/j.physa.2018.09.189.

[11]

B. GuoY. WuY. Xiao and C. Zhang, Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control, Appl. Math. Comput., 331 (2018), 341-357.  doi: 10.1016/j.amc.2018.03.020.

[12]

R. Hasminskii, Stochastic Stability of Differential Equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. doi: 10.1007/978-3-642-23280-0.

[13]

N. Li and J. Cao, Intermittent control on switched networks via $\omega$-matrix measure method, Nonlinear Dynam., 77 (2014), 1363-1375.  doi: 10.1007/s11071-014-1385-2.

[14]

S. Li, J. Cao and Y. He, Pinning controllability scheme of directed complex delayed dynamical networks via periodically intermittent control, Discrete Dyn. Nat. Soc., (2016), 10 pp. doi: 10.1155/2016/1585928.

[15]

C. LiZ. Dong and R. Situ, Almost sure stability of linear stochastic differential equations with jumps, Probab. Theory Related Fields, 123 (2002), 121-155.  doi: 10.1007/s004400200198.

[16]

L. LiuY. Shen and F. Jiang, The almost sure asymptotic stability and $p$th moment asymptotic stability of nonlinear stochastic differential systems with polynomial growth, IEEE Trans. Automat. Control, 56 (2011), 1985-1990.  doi: 10.1109/TAC.2011.2146970.

[17]

L. Liu and Y. Shen, Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition, Automatica J. IFAC, 48 (2012), 619-624.  doi: 10.1016/j.automatica.2012.01.022.

[18]

X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.

[19]

X. MaoG. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica J. IFAC, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.

[20]

X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Tran. Automat. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.

[21]

L. Pan and J. Cao, Stochastic quasi-synchronization for delayed dynamical networks via intermittent control, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1332-1343.  doi: 10.1016/j.cnsns.2011.07.010.

[22]

A. Patel and B. Kosko, Stochastic resonance in continuous and spiking neuron models with Lévy noise, IEEE Trans. Neural Netw., 19 (2008), 1993-2008.  doi: 10.1109/TNN.2008.2005610.

[23]

M. Scheutzow, Stabilisation and destabilisation by noise in the plane, Stochastic Anal. Appl., 11 (1993), 97-113.  doi: 10.1080/07362999308809304.

[24]

M. Siakalli, Stability Properties of Stochastic Differential Equations Driven by Lévy Noise, Ph.D thesis, University of Sheffield, 2009.

[25]

Y. Wan and J. Cao, Distributed robust stabilization of linear multi-agent systems with intermittent control, J. Franklin Inst., 352 (2015), 4515-4527.  doi: 10.1016/j.jfranklin.2015.06.024.

[26]

P. WangY. Hong and H. Su, Stabilization of stochastic complex-valued coupled delayed systems with Markovian switching via periodically intermittent control, Nonlinear Anal. Hybrid Syst., 29 (2018), 395-413.  doi: 10.1016/j.nahs.2018.03.006.

[27]

F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control, 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108.

[28]

F. WuG. Yin and Z. Jin, Kolmogorov-type systems with regime-switching jump diffusion perturbations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2293-2319.  doi: 10.3934/dcdsb.2016048.

[29]

Y. XuX. WangH. Zhang and W. Xu, Stochastic stability for nonlinear systems driven by Lévy noise, Nonlinear Dynam., 68 (2012), 7-15.  doi: 10.1007/s11071-011-0199-8.

[30]

Y. Xu, H. Zhou and W. Li, Stabilization of stochastic delayed systems with Lévy noise on networks via periodically intermittent control, Internat. J. Control, (2018). doi: 10.1080/00207179.2018.1479538.

[31]

G. YinY. Talafha and F. Xi, Stochastic Liénard equations with random switching and two-time scales, Comm. Statist. Theory Methods, 43 (2014), 1533-1547.  doi: 10.1080/03610926.2012.741741.

[32]

Q. Zhu, Asymptotic stability in the $p$th moment for stochastic differential equations with Lévy noise, J. Math. Anal. Appl., 416 (2014), 126-142.  doi: 10.1016/j.jmaa.2014.02.016.

[33]

Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching, Internat. J. Control, 90 (2017), 1703-1712.  doi: 10.1080/00207179.2016.1219069.

[34]

Q. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Systems Control Lett., 118 (2018), 62-68.  doi: 10.1016/j.sysconle.2018.05.015.

[35]

B. ZhangF. DengS. Peng and S. Xie, Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system, J. Franklin Inst., 355 (2018), 3829-3852.  doi: 10.1016/j.jfranklin.2017.12.033.

[36]

S. ZhuK. SunS. Zhou and Y. Shi, Stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial, J. Franklin Inst., 354 (2017), 6550-6566.  doi: 10.1016/j.jfranklin.2017.08.007.

[37]

X. ZongT. Li and J. Zhang, Consensus conditions for continuous-time multi-agent systems with additive and multiplicative measurement noises, SIAM J. Control Optim., 56 (2018), 19-52.  doi: 10.1137/15M1019775.

[38]

X. ZongF. WuG. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.

show all references

References:
[1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2009.  doi: 10.1017/CBO9780511755323.
[2]

D. Applebaum, Extending stochastic resonance for neuron models to general Lévy noise, IEEE Trans. Neural Netw., 20 (2009), 1993-1995.  doi: 10.1109/TNN.2009.2033183.

[3]

D. Applebaum and M. Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Probab., 46 (2009), 1116-1129.  doi: 10.1239/jap/1261670692.

[4]

D. Applebaum and M. Siakalli, Stochastic stabilization of dynamical systems using Lévy noise, Stoch. Dyn., 10 (2010), 509-527.  doi: 10.1142/S0219493710003066.

[5]

A. D. ApplebyX. Mao and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Tran. Automat. Control, 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.

[6]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203485217.

[7]

F. DengQ. Luo and X. Mao, Stochastic stabilization of hybrid differential equations, Automatica J. IFAC, 48 (2012), 2321-2328.  doi: 10.1016/j.automatica.2012.06.044.

[8]

Y. FengX. YangQ. Song and J. Cao, Synchronization of memristive neural networks with mixed delays via quantized intermittent control, Appl. Math. Comput., 339 (2018), 874-887.  doi: 10.1016/j.amc.2018.08.009.

[9]

R. Fernholz and I. Karatzas, Relative arbitrage in volatility-stabilized markets, Ann. Financ., 1 (2005), 149-177.  doi: 10.1007/s10436-004-0011-6.

[10]

H. Gao and Y. Wang, Stochastic mutualism model under regime switching with Lévy jumps, Phys. A, 515 (2019), 355-375.  doi: 10.1016/j.physa.2018.09.189.

[11]

B. GuoY. WuY. Xiao and C. Zhang, Graph-theoretic approach to synchronizing stochastic coupled systems with time-varying delays on networks via periodically intermittent control, Appl. Math. Comput., 331 (2018), 341-357.  doi: 10.1016/j.amc.2018.03.020.

[12]

R. Hasminskii, Stochastic Stability of Differential Equations, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980. doi: 10.1007/978-3-642-23280-0.

[13]

N. Li and J. Cao, Intermittent control on switched networks via $\omega$-matrix measure method, Nonlinear Dynam., 77 (2014), 1363-1375.  doi: 10.1007/s11071-014-1385-2.

[14]

S. Li, J. Cao and Y. He, Pinning controllability scheme of directed complex delayed dynamical networks via periodically intermittent control, Discrete Dyn. Nat. Soc., (2016), 10 pp. doi: 10.1155/2016/1585928.

[15]

C. LiZ. Dong and R. Situ, Almost sure stability of linear stochastic differential equations with jumps, Probab. Theory Related Fields, 123 (2002), 121-155.  doi: 10.1007/s004400200198.

[16]

L. LiuY. Shen and F. Jiang, The almost sure asymptotic stability and $p$th moment asymptotic stability of nonlinear stochastic differential systems with polynomial growth, IEEE Trans. Automat. Control, 56 (2011), 1985-1990.  doi: 10.1109/TAC.2011.2146970.

[17]

L. Liu and Y. Shen, Noise suppresses explosive solutions of differential systems with coefficients satisfying the polynomial growth condition, Automatica J. IFAC, 48 (2012), 619-624.  doi: 10.1016/j.automatica.2012.01.022.

[18]

X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.

[19]

X. MaoG. Yin and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica J. IFAC, 43 (2007), 264-273.  doi: 10.1016/j.automatica.2006.09.006.

[20]

X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Tran. Automat. Control, 61 (2016), 1619-1624.  doi: 10.1109/TAC.2015.2471696.

[21]

L. Pan and J. Cao, Stochastic quasi-synchronization for delayed dynamical networks via intermittent control, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1332-1343.  doi: 10.1016/j.cnsns.2011.07.010.

[22]

A. Patel and B. Kosko, Stochastic resonance in continuous and spiking neuron models with Lévy noise, IEEE Trans. Neural Netw., 19 (2008), 1993-2008.  doi: 10.1109/TNN.2008.2005610.

[23]

M. Scheutzow, Stabilisation and destabilisation by noise in the plane, Stochastic Anal. Appl., 11 (1993), 97-113.  doi: 10.1080/07362999308809304.

[24]

M. Siakalli, Stability Properties of Stochastic Differential Equations Driven by Lévy Noise, Ph.D thesis, University of Sheffield, 2009.

[25]

Y. Wan and J. Cao, Distributed robust stabilization of linear multi-agent systems with intermittent control, J. Franklin Inst., 352 (2015), 4515-4527.  doi: 10.1016/j.jfranklin.2015.06.024.

[26]

P. WangY. Hong and H. Su, Stabilization of stochastic complex-valued coupled delayed systems with Markovian switching via periodically intermittent control, Nonlinear Anal. Hybrid Syst., 29 (2018), 395-413.  doi: 10.1016/j.nahs.2018.03.006.

[27]

F. Wu and S. Hu, Suppression and stabilisation of noise, Internat. J. Control, 82 (2009), 2150-2157.  doi: 10.1080/00207170902968108.

[28]

F. WuG. Yin and Z. Jin, Kolmogorov-type systems with regime-switching jump diffusion perturbations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2293-2319.  doi: 10.3934/dcdsb.2016048.

[29]

Y. XuX. WangH. Zhang and W. Xu, Stochastic stability for nonlinear systems driven by Lévy noise, Nonlinear Dynam., 68 (2012), 7-15.  doi: 10.1007/s11071-011-0199-8.

[30]

Y. Xu, H. Zhou and W. Li, Stabilization of stochastic delayed systems with Lévy noise on networks via periodically intermittent control, Internat. J. Control, (2018). doi: 10.1080/00207179.2018.1479538.

[31]

G. YinY. Talafha and F. Xi, Stochastic Liénard equations with random switching and two-time scales, Comm. Statist. Theory Methods, 43 (2014), 1533-1547.  doi: 10.1080/03610926.2012.741741.

[32]

Q. Zhu, Asymptotic stability in the $p$th moment for stochastic differential equations with Lévy noise, J. Math. Anal. Appl., 416 (2014), 126-142.  doi: 10.1016/j.jmaa.2014.02.016.

[33]

Q. Zhu, Razumikhin-type theorem for stochastic functional differential equations with Lévy noise and Markov switching, Internat. J. Control, 90 (2017), 1703-1712.  doi: 10.1080/00207179.2016.1219069.

[34]

Q. Zhu, Stability analysis of stochastic delay differential equations with Lévy noise, Systems Control Lett., 118 (2018), 62-68.  doi: 10.1016/j.sysconle.2018.05.015.

[35]

B. ZhangF. DengS. Peng and S. Xie, Stabilization and destabilization of nonlinear systems via intermittent stochastic noise with application to memristor-based system, J. Franklin Inst., 355 (2018), 3829-3852.  doi: 10.1016/j.jfranklin.2017.12.033.

[36]

S. ZhuK. SunS. Zhou and Y. Shi, Stochastic suppression and almost surely stabilization of non-autonomous hybrid system with a new general one-sided polynomial, J. Franklin Inst., 354 (2017), 6550-6566.  doi: 10.1016/j.jfranklin.2017.08.007.

[37]

X. ZongT. Li and J. Zhang, Consensus conditions for continuous-time multi-agent systems with additive and multiplicative measurement noises, SIAM J. Control Optim., 56 (2018), 19-52.  doi: 10.1137/15M1019775.

[38]

X. ZongF. WuG. Yin and Z. Jin, Almost sure and $p$th-moment stability and stabilization of regime-switching jump diffusion systems, SIAM J. Control Optim., 52 (2014), 2595-2622.  doi: 10.1137/14095251X.

Figure 1.  The state $ x(t) $ of the system (29)
Figure 2.  The state $ x(t) $ of system (30)
Figure 5.  The exponent dynamical behaviors of state $ x(t) $ of the system (34)
Figure 3.  The state $ x(t) $ of the system (32)
Figure 4.  The state $ x(t) $ of system (33)
Figure 6.  The state $ x(t) $ of system (34)
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