Stability analysis of neutral stochastic differential delay equations driven by Lévy noises

Abstract This paper mainly analyzes the well-posedness, and the stability analysis for the global solution of neutral stochastic differential delay equations (NSDDEs) driven by Levy noises. By using an integral lemma and a Lyapunov function approach, the existence and uniqueness theorem is proved. Then, by using the inequality technique and the stochastic analysis theory, the exponential stability in pth(p ≥ 2) moment of such equations is discussed. By using another integral lemma, and using the Baralat lemma as well as the stochastic analysis, the almost surely asymptotic stability is also studied. Finally, one example is given to check the effectiveness of the findings derived.

[1]  Jinde Cao,et al.  Fractional extended Kalman filtering for non-linear fractional system with Lévy noises , 2017 .

[2]  S. Jankovic,et al.  Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations , 2009 .

[3]  Qiong Wu.  Stability analysis for a class of nonlinear time-changed systems , 2016 .

[4]  Feiqi Deng,et al.  Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with Lévy noise , 2017 .

[5]  Xuerong Mao,et al.  Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching , 2008 .

[6]  D. Applebaum,et al.  Asymptotic Stability of Stochastic Differential Equations Driven by Lévy Noise , 2009, Journal of Applied Probability.

[8]  Quanxin Zhu,et al.  Stability analysis of stochastic delay differential equations with Lévy noise , 2018, Syst. Control. Lett..

[9]  Xuerong Mao,et al.  Delay-Dependent Exponential Stability of Neutral Stochastic Delay Systems , 2009, IEEE Transactions on Automatic Control.

[10]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[11]  Yi Shen,et al.  New criteria on exponential stability of neutral stochastic differential delay equations , 2006, Syst. Control. Lett..

[12]  C. Yuan,et al.  Stochastic Population Dynamics Driven by Levy Noise , 2011, 1105.1174.

[13]  Qun Liu,et al.  Analysis of a general stochastic non-autonomous logistic model with delays and Lévy jumps , 2016 .

[14]  C. Yuan,et al.  Stability in distribution of neutral stochastic differential delay equations with Markovian switching , 2009 .

[15]  Liangjian Hu,et al.  The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps , 2015, Appl. Math. Comput..

[17]  P. Shi,et al.  Exponential Stability for Neutral Stochastic Markov Systems With Time-Varying Delay and Its Applications , 2016, IEEE Transactions on Cybernetics.

[18]  D. Bahuguna,et al.  Approximate Controllability of Nonlocal Neutral Fractional Integro-Differential Equations with Finite Delay , 2016 .

[19]  Feiqi Deng,et al.  Stability equivalence between the neutral delayed stochastic differential equations and the Euler–Maruyama numerical scheme , 2018 .

[20]  X. Mao,et al.  Asymptotic properties of neutral stochastic differential delay equations , 2000 .

[21]  Gang George Yin,et al.  Almost Sure and pth-Moment Stability and Stabilization of Regime-Switching Jump Diffusion Systems , 2014, SIAM J. Control. Optim..

[22]  Wei Xing Zheng,et al.  Delay-Dependent Stochastic Stability and $H_{\infty} $-Control of Uncertain Neutral Stochastic Systems With Time Delay , 2009, IEEE Transactions on Automatic Control.

[23]  Huabin Chen,et al.  On the Asymptotic Behavior for Neutral Stochastic Differential Delay Equations , 2019, IEEE Transactions on Automatic Control.

[24]  Feiqi Deng,et al.  pth moment exponential stability of highly nonlinear neutral pantograph stochastic differential equations driven by Lévy noise , 2018, Appl. Math. Lett..

[25]  Yong He,et al.  pth moment exponential stability of neutral stochastic differential equations driven by Lévy noise , 2012, J. Frankl. Inst..

[26]  X. Mao.  Exponential stability in mean square of neutral stochastic differential functional equations , 1995 .

[27]  Wei Xing Zheng,et al.  A new result on stability analysis for stochastic neutral systems , 2010, Autom..

[28]  A. I. Matasov,et al.  Neutral Stochastic Differential Delay Equations with Markovian Switching , 2003 .

[29]  Peiguang Wang,et al.  pth moment asymptotic stability for neutral stochastic functional differential equations with Lévy processes , 2015, Appl. Math. Comput..

[30]  O. Barndorff-Nielsen,et al.  Lévy processes : theory and applications , 2001 .

[31]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[32]  Tusheng Zhang,et al.  Strong solutions for a stochastic model of two-dimensional second grade fluids driven by Lévy noise , 2019, Journal of Mathematical Analysis and Applications.

[33]  Feiqi Deng,et al.  Razumikhin-Type Theorems on Stability of Neutral Stochastic Functional Differential Equations , 2008, IEEE Transactions on Automatic Control.

[34]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Theory of martingales , 1989 .

[35]  Hieu Minh Trinh,et al.  Novel Criteria for Exponential Stability of Linear Neutral Time-Varying Differential Systems , 2016, IEEE Transactions on Automatic Control.

[36]  Peng Shi,et al.  Stability analysis for neutral stochastic delay systems with Markovian switching , 2017, Syst. Control. Lett..

[37]  Chenggui Yuan,et al.  Stabilization of Partial Differential Equations by Lévy Noise , 2011, 1104.2779.

[38]  Nikolaos Bekiaris-Liberis,et al.  Nonlinear Control Under Nonconstant Delays , 2013, Advances in design and control.

[39]  Khac Duc Do,et al.  Almost sure exponential stability of dynamical systems driven by Lévy processes and its application to control design for magnetic bearings , 2018, Int. J. Control.

[40]  K. D. Do,et al.  Stochastic control of drill-heads driven by Lévy processes , 2019, Autom..

[41]  Jianhua Huang,et al.  Large deviation principle for stochastic Boussinesq equations driven by Lévy noise , 2016 .

[42]  Chengming Huang,et al.  Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay , 2010, Syst. Control. Lett..

[43]  Yonggui Kao,et al.  A sliding mode approach to H∞ non-fragile observer-based control design for uncertain Markovian neutral-type stochastic systems , 2015, Autom..

[44]  Bruno Lombard,et al.  Stability of a critical nonlinear neutral delay differential equation , 2012 .

[45]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .

[46]  Feiqi Deng,et al.  Exponential mean-square stability of the θ-method for neutral stochastic delay differential equations with jumps , 2017, Int. J. Syst. Sci..

[47]  J. Menaldi,et al.  Almost sure asymptotic stabilization of differential equations with time-varying delay by Lévy noise , 2015 .