On exponential stability of hybrid neutral stochastic differential delay equations with different structures

Abstract

This article discusses the problem of exponential stability for a class of hybrid neutral stochastic differential delay equations with highly nonlinear coefficients and different structures in different switching modes. In such systems, the coefficients will satisfy the local Lipschitz condition and suitable Khasminskii-types conditions. The set of switching states will be divided into two subsets. In different subsets, the coefficients will be dominated by polynomials with different degrees.By virtue of M -matrices and suitable Lyapunov functions dependent on coefficient structures and switching modes, some results including the existence-and-uniqueness, boundedness and exponential stability of the solution are proposed and proved.

‐stability of hybrid neutral stochastic differential equations with infinite delay

“…For instance, in Reference 17 we imposed more restrictive conditions on the general decay rate function , for example, see conditions (d) and (e) in theorems 3.3 and 3.5, but in this article we will just impose conditions (i) and (ii) in Theorem 2 which allow us to obtain stability results for decay functions which can grow, for instance, like the product of a polynomial by an exponential, simply a polynomial or a logarithm (see References 10,11,17‐19 and 14 for other types or results). In Reference 4, the class of ‐function is more restrictive than the ones in this article due to the condition , condition (iii) in definition 3.1 and condition (3.1) in assumption 3.3 (in Reference 4). Different from the previous results in Reference 16, taking into account the influence of the neutral term, the infinite delay, the new class of ‐function and the M‐matrix technique in the stability theory make our results more general.…”

“…Different from the previous results in Reference 16, taking into account the influence of the neutral term, the infinite delay, the new class of ‐function and the M‐matrix technique in the stability theory make our results more general.…”

“…For hybrid neutral SDDE with different structures, 16 studied p th moment asymptotic boundedness, p th moment and almost sure exponential stability of solutions.…”

Stability of hybrid pantograph stochastic functional differential equations

“…For example, [21] investigated the exponential stability of highly nonlinear neutral pantograph stochastic differential equations, [25] built Razumikhin-type theorems on neutral SFDEs, [12] studied stability of neutral SFDEswMS driven by G-Brownian motion, [6] analyzed asymptotic stability and boundedness of SFDESwMS. More related work can be seen in [5,8,11,19,20,17,23,26].…”

Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

“…Hybrid SDDEs (which mean SDDEs with Markovian switching) can be utilized to describe many physical systems undergoing random abrupt changes in their parameters and structure triggered by some phenomena such as abrupt environmental disturbances, component repairs or failures, and changing subsystem interconnections (see, e.g., [27,15,32,49,10,44,26]). An area of particular interest for hybrid SDDEs is the analysis of stochastic stability, we can refer to Mao and Yuan [27], Wu et al [32], Zhu and Zhang [49], and Hu et al [10]. Inspired by the aforementioned works, some scholars started to investigate the stability of hybrid SPDEs (which are a kind of hybrid SDDEs) (see, e.g., [17,28,47,48,40,31]).…”