Almost sure exponential stability of hybrid stochastic functional differential equations

This paper is concerned with the almost sure exponential stability of the n -dimensional nonlinear hybrid stochastic functional differential equation (SFDE) dx(t)=f(ψ1(xt,t),r(t),t)dt+g(ψ2(xt,t),r(t),t)dB(t), where xt={x(t+u):−τ≤u≤0} is a C([−τ,0];Rn)C([−τ,0];Rn)-valued process, B(t)B(t) is an m -dimensional Brownian motion while r(t) is a Markov chain. We show that if the corresponding hybrid stochastic differential equation (SDE) dy(t)=f(y(t),r(t),t)dt+g(y(t),r(t),t)dB(t) is almost surely exponentially stable, then there exists a positive number τ⁎ such that the SFDE is also almost surely exponentially stable as long as τ<τ⁎. We also describe a method to determine τ⁎ which can be computed numerically in practice

Explicit approximation of the invariant measure for SDDEs with the nonlinear diffusion term

To our knowledge, the existing measure approximation theory requires the diffusion term of the stochastic delay differential equations (SDDEs) to be globally Lipschitz continuous. Our work is to develop a new explicit numerical method for SDDEs with the nonlinear diffusion term and establish the measure approximation theory. Precisely, we construct a function-valued explicit truncated Euler-Maruyama segment process (TEMSP) and prove that it admits a unique ergodic numerical invariant measure. We also prove that the numerical invariant measure converges to the underlying one of SDDE in the Fortet-Mourier distance. Finally, we give an example and numerical simulations to support our theory.Comment: 31 pages, 2 figure

Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time observations of state and mode

Mao [10] recently initiated the study of the mean-square exponential stabilisation of continuous-time hybrid stochastic differential equations (SDEs) by the feedback controls based on the discrete-time observations of the state. However, the feedback controls still depend on the continuous-time observations of the mode. Of course this is perfectly fine if the mode of the system is obvious (i.e. fully observable at no cost). However, it could often be the case where the mode is not obvious and it costs to identify the current mode of the system. To reduce the control cost, it is reasonable we identify the mode at the discrete times when we make observations for the state. Hence the feedback control should be designed based on the discrete-time observations of both state and mode. The aim of this paper is to show how to design such a feedback control to stabilise a given hybrid SDE

Robust quantised control of hybrid stochastic systems based on discrete-time state and mode observations

In this paper, the problems of robust quantized feedback control are studied for hybrid stochastic systems based on discrete-time observations of state and mode. All of the existing results in this area design the quantized feedback control based on continuous observations of the state and mode for all time t ≥ 0 (see [23–25]). This is the first paper where we propose to use the quantized feedback control based on discrete-time observations of the state and mode. The key reason for this is to reduce the burden of communication by using not only the quantization (i.e. in the direction of state axis), but also discrete-time observations of state and mode (i.e. in the direction of time axis). Thus, the designed quantized feedback controllers have to be based on the discrete-time observations of state and mode. Clearly, the new quantized feedback controllers are more realistic and cost less in practice. Two examples with computer simulations will be provided to illustrate the effectiveness of the proposed control method

Noise expresses exponential growth under regime switching

Consider a given system under regime switching whose solution grows at most polynomially, and suppose that the system is subject to environmental noise in some regimes. Can the regime switching and the environmental noise work together to make the system change signicantly? The answer is yes. In this paper, we will show that the regime switching and the environmental noise will make the original system whose solution grows at most polynomially become a new system whose solution will grow exponentially. In other words, we reveal that the regime switching and the environmental noise will exppress the exponential growth

Hybrid stochastic functional differential equations with infinite delay : approximations and numerics

This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is established for a large class of highly nonlinear hybrid SFDEs with infinite delay. Our new theory makes it possible to numerically approximate the solution of the hybrid SFDE with infinite delay, via the numerical solution of the corresponding hybrid SFDE with finite delay

Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state

Although the mean square stabilisation of hybrid systems by feedback controls based on discretetime observations of state and mode has been studied by several authors since 2013 (see, e.g., [17,19,27,31]), the corresponding almost sure stabilisation problem has little been investigated. Recent Mao [18] is the first to study the almost sure stabilisation of a given unstable system x(t) = f(x(t)) by a linear discretetime stochastic feedback control Ax([t/τ]τ)dB(t) (namely the stochastically controlled system has the form dx(t) = f(x(t))dt + Ax([t/τ]τ)dB(t)), where B(t) is a scalar Brownian, τ > 0 and [t/τ] is the integer part of t/τ. In this paper, we will consider a much more general problem. That is, we will to study the almost sure stabilisation of a given unstable hybrid system x(t) = f(x(t), r(t)) by nonlinear discrete-time stochastic feedback control u(x([t/τ]τ), r([t/τ]τ))dB(t) (so the stochastically controlled system is a hybrid stochastic system of the form dx(t) = f(x(t), r(t))dt + u(x([t/τ]τ), r([t/τ]τ))dB(t)), where B(t) is a multi-dimensional Brownian motion and r(t) is a Markov chain

Advances in Research on Biogenic Amines in Wine

Biogenic amines (BA) are the general term for a class of bioactive low-molecular mass organic compounds containing amino groups. Excessive intake of BA will cause adverse physiological reactions and even endanger life. This article presents a systematic review of the sources, biosynthetic mechanism, microbial traceability, detection technologies, and influential factors of BA in wine, as well as the strategies used to control BA during winemaking. This article will provide theoretical guidance for efficient analysis and control of BA in wine, which will in turn promote the healthy and green development of the Chinese wine industry

Prevalence and associated factors of internet addiction among Chinese adolescents: association with childhood trauma

IntroductionInternet addiction (IA) is common among adolescents and may have severe consequences. This study aimed to investigate the prevalence and factors associated with IA among middle school students of Hunan Province, China. Relevance between IA and childhood trauma was also explored.MethodsOne thousand six hundred ten students were enrolled in this cross-sectional study. Data collected included demographics; internet addiction (revised-Chen internet addiction scale); childhood trauma (CTQ-SF); depression, anxiety, and stress symptoms (DASS-21); suicidal behaviors, as well as non-suicidal self-injury (NSSI). Cramer’s V analysis, univariable logistic regression and multivariable logistic regression were used for associations and identifying independent relevance of IA, respectively.ResultsThe prevalence of IA was 12.8%. Cramer’s V analysis showed that IA was associated with emotional abuse, emotional and physical neglect, NSSI, suicidal behaviors, stress, anxiety and depressive symptoms, physical disorder history. Regression analysis showed that IA was independently associated with emotional neglect (OR = 3.062, 95% CI: 2.083, 4.501, p &lt; 0.001); physical neglect (OR = 2.328; 95% CI: 1.590, 3.409, p &lt; 0.001); depressive symptoms (OR = 2.218, 95% CI: 1.467, 3.353, p &lt; 0.001) nationality (OR = 1.888, 95% CI: 1.034, 3.447, p = 0.006) and age (OR = 1.253, 95% CI: 1.066, 1.471, p = 0.006).DiscussionIA is common among middle school students. Attention should be paid to students with childhood trauma since they have a higher risk for IA, which may increase the risk for suicidal behaviors

Extinction and recurrence of multi-group SEIR epidemic

In this paper, we consider a class of multi-group SEIR epidemic models with stochastic perturbations. By the method of stochastic Lyapunov functions, we study their asymptotic behavior in terms of the intensity of the stochastic perturbations and the reproductive number R0R0. When the perturbations are sufficiently large, the exposed and infective components decay exponentially to zero whilst the susceptible components converge weakly to a class of explicit stationary distributions regardless of the magnitude of R0R0. An interesting result is that, if the perturbations are sufficiently small and R0≤1R0≤1, then the exposed, infective and susceptible components have similar behaviors, respectively, as in the case of large perturbations. When the perturbations are small and R0>1R0>1, we construct a new class of stochastic Lyapunov functions to show the ergodic property and the positive recurrence, and our results reveal some cycling phenomena of recurrent diseases. Computer simulations are carried out to illustrate our analytical results