Stochastic Exponential Stability Research Articles

Stability and mean growth rate of stochastic Solow model driven by jump–diffusion process

This work focuses on the stability and mean growth rate of stochastic Solow growth model. The uncertainty in our model is generated by the technology part, which includes fluctuations on the technological accumulation and jump growth from new inventions and ideas. We introduce the criteria of stability, including stochastic stability and exponential stability, and the sufficient condition for the existence of balanced growth path. Furthermore, we precisely calculate the higher-order moment of effective capital–labor ratio and the mean growth rate of major economic quantities on the balanced growth path. Our analysis sheds light on the impact of various stochastic effects on economic quantities and provides insight into the stability of the stochastic Solow model with technological uncertainty.

LMIs-based exponential stabilization for interval delay systems via congruence transformation: Application in chaotic Lorenz system

This paper investigates the stochastic Takagi-Sugeno (T-S) fuzzy uncertainty-dependent state feedback sampled-data control, asymptotical stability and exponential stability analysis for a class of stochastic T-S fuzzy system with measurable uncertainties, mixed interval time varying delays and Gaussian white noise. Firstly, the T-S fuzzy model and stochastic Bernoulli theory are employed to approximate the system plant. Secondly, the stochastic T-S fuzzy uncertainty-dependent state feedback sampled-data controller is designed via uncertainty-dependent parameters and the hold times series of sampled-data. Thirdly, the Bernoulli probability distribution is employed to describe the random occurrence characteristic of multipath packet dropouts. Three classes of delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) are derived for the closed-loop system. The multiple integral fuzzy-basis-dependent Lyapunov functional and augmented state variable are employed to derive the delay-dependent stability conditions, then the closed-loop system is asymptotically stable and exponentially stable. The congruence transformation method and prescribed H-infinity performance functional are employed to derive the less conservative delay-dependent stability conditions, and the prescribed H-infinity performance is guaranteed. The relaxation matrix and congruence transformation method are employed to design the linear transformation equalities and congruence transformation matrices, then the controller gain matrix is determined and computation complexity of solving LMIs is reduced. Finally, the proposed method is applied in the chaotic Lorenz system to show the effectiveness and advantage of proposed method.

Stability of nonlinear positive semi-Markovian jump systems with mode-dependent delay subject to mode-dependent average dwell time

This article concentrates on the stability of nonlinear positive semi-Markovian jump systems (NPSMJSs) with mode-dependent delay, where the transition rate functions depend on both the sojourn-times and the average dwell time (ADT) switching. By resorting to linear programming (LP) approach, sufficient conditions for stochastic exponential stability of mode-dependent delay for NPSMJSs are developed. In addition, we obtain the $ L_{1}- $gian performance analysis. Finally, two numerical examples are proposed to illustrate the advantages of the theoretical results.

Event-generator-based H∞ control of fuzzy distributed parameter systems

This paper studies the adaptive event-based H∞ control problem for a class of T-S fuzzy distributed parameter system with parameter uncertain and actuator faults. To overcome the drawback of the period control scheme, an event-triggered control scheme with adaptive threshold is proposed to minimize the number of unnecessary sampled data transmission and to reduce the update frequency of the controller. Furthermore, a T-S fuzzy controller based on the adaptive event-triggered sampled data is designed to ensure the stochastic exponential stability of the distributed parameter system with H∞ disturbance attenuation performance. Based on a new Lyapunov functional and inequality technique, the stochastic exponential stability criterion of the closed-loop system is obtained, and the controller parameters are designed. The new developed inequality can reduce the conservatism of the stability criterion. Finally, the effectiveness of the theoretical calculation results is verified by numerical simulation, and the results are compared with the relevant literature in the simulation, showing that the methods and results in this paper are less conservative.

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

Stochastic global exponential stability of disease-free equilibrium of HIV/AIDS model

This paper investigates the dynamics behavior of the mathematical model of Human Immunodeficiency Virus (HIV) influenced by stochastic perturbations. This paper is involved in exploring the disease-free equilibrium’s stochastic global exponential stability with a basic reproductive number $$R_{0} < 1$$ . Necessary and sufficient conditions for stochastic global exponential stability of the disease-free equilibrium $$E^{0}$$ of the nonlinear HIV stochastic system are derived. This can be accomplished by the exponential analysis of the global mean square stability of trivial equilibrium of the corresponding linear system. Finally, we provide areas of stability of $$E^{0}$$ and numerical simulations to confirm the analytical results by using the fundamental Euler–Maruyama (EM) algorithm.

Robust stochastic stability for multi-group coupled models with Markovian switching

As one of the most important topics in control theory, robust stability for the systems plays a key role in many areas of science and industry. This paper discusses the robust stochastic stability a class of multi-group coupled stochastic models with Markovian switching. Based on discrete-time observations in the diffusion part with time-delay, the robust stochastic exponential stability criterion in mean square sense is established in the forms of Lyapunov-type and coefficient-type. An application to the robust stochastic stability for a class of coupled oscillators is proposed.

FRACTAL DIMENSION OF FRACTIONAL BROWNIAN MOTION BASED ON RANDOM SETS

The fractal dimension of fractional Brownian motion can effectively describe random sets, reflecting the regularity implicit in complex random sets. Data mining algorithms based on fractal theory usually follow the calculation of the fractal dimension of fractional Brownian motion. However, the existing fractal dimension calculation methods of fractal Brownian motion have high time complexity and space complexity, which greatly reduces the efficiency of the algorithm and makes it difficult for the algorithm to adapt to high-speed and massive data flow environments. Therefore, several existing fractal dimension calculation methods of fractional Brownian motion are summarized and analyzed, and a random method is proposed, which uses a fixed memory space to quickly estimate the associated dimension of the data stream. Finally, a comparison experiment with existing algorithms proves the effectiveness of this random algorithm. Second, in the sense of two different measures, based on the principle of stochastic comparison, the stability of the stochastic fuzzy differential equations is derived using the stability of the comparison equations, and the practical stability criterion of two measures according to probability is obtained. Then, the stochastic fuzzy differential equations are discussed. The definition of stochastic exponential stability is given and the stochastic exponential stability criterion is proved.

Robust H∞ filtering for polytopic uncertain stochastic systems under quantized sampled outputs

In this paper, the robust H∞ filtering of polytopic uncertain stochastic systems is designed under quantized sampled outputs. An novel parameters-dependent time-scheduled Lyapunov functional is constructed using segmentation technology and linear interpolation. Based on this new Lyapunov functional and convex combination method, sufficient stochastic exponential stability criteria of the filtering error dynamics are proposed and a prescribed H∞ performance level can be guaranteed by designing the stochastically stable filter. The better H∞ performance can be obtained with a larger interval dividing parameter. A numerical example is provided to illustrate the effectiveness of the proposed method.

Stability analysis of 2-D switched systems with multiplicative noise under arbitrary and restricted switching signals

ABSTRACTThis paper introduces a new class of discrete-time two-dimensional (2-D) switched systems with multiplicative noise. Firstly, we extend the definition of the asymptotic stability and establish a sufficient stochastically stability condition for this new model under arbitrary switching signal. Then, by introducing the average dwell time into this new model and combining with the Lyapunov function, we investigate the extended stochastic exponential stability of the 2-D switched systems with multiplicative noise for the restricted switching case. Moreover, some remarks and discussions are given to illustrate the significance of obtained results, which generalise and comprise some previous results the literature. Finally, two examples are provided to show the effectiveness of the theoretical results.

Stochastic stability of positive Markov jump linear systems with completely/partially known transition rates

This paper addresses the stochastic stability problem of positive Markov jump linear systems (PMJLSs). A necessary and sufficient condition of stochastic stability for PMJLSs is given which can be checked by solving linear programming feasibility problems due to the positivity property. A common co-positive Lyapunov function is constructed to solve the problem, and the equivalence among stochastic stability, 1-moment stability and exponential mean stability is proved afterwards. Considering the uncertain transition rates situation, PMJLSs with partially known transition rate matrix are investigated, and a necessary and sufficient condition for robust stochastic stability is proposed in linear programming form. Numerical examples are presented to show the effectiveness of the proposed results.

Stochastic Exponential Robust Stability of Delayed Complex-Valued Neural Networks With Markova Jumping Parameters

This paper deals with the problem on stochastic exponential robust stability for a class of complex-valued interval neural networks with Markova jumping parameters and mixed delays, including both time-varying delays and continuously distributed delays. By applying the M-matrix theory and coupling with the vector Lyapunov function method, some sufficient conditions are derived to guarantee the existence, uniqueness, and stochastic exponential robust stability of the equilibrium point of the addressed system. The obtained results not only are easy to judge the dynamical behavior of the addressed system, but also are with lower level conservatism in comparison with some existing results. Finally, two numerical examples with simulation results are given to illustrate the effectiveness of the proposed results.

Fuzzy-Model-Based Reliable Static Output Feedback $\mathscr{H}_{\infty }$ Control of Nonlinear Hyperbolic PDE Systems

This paper investigates the problem of output feedback robust $\mathscr{H}_{\infty }$ control for a class of nonlinear spatially distributed systems described by first-order hyperbolic partial differential equations (PDEs) with Markovian jumping actuator faults. The nonlinear hyperbolic PDE systems are first expressed by Takagi–Sugeno fuzzy models with parameter uncertainties, and then, the objective is to design a reliable distributed fuzzy static output feedback controller guaranteeing the stochastic exponential stability of the resulting closed-loop system with certain $\mathscr{H}_{\infty }$ disturbance attenuation performance. Based on a Markovian Lyapunov functional combined with some matrix inequality convexification techniques, two approaches are developed for reliable fuzzy static output feedback controller design of the underlying fuzzy PDE systems. It is shown that the controller gains can be obtained by solving a set of finite linear matrix inequalities based on the finite-difference method in space. Finally, two examples are presented to demonstrate the effectiveness of the proposed methods.

Reliable fuzzy output feedback control of nonlinear parabolic distributed parameter systems with sensor faults1

This paper studies the problem of output feedback control for a class of nonlinear spatially distributed systems described by parabolic partial differential equations (PDEs) with Markovian jumping sensor faults. The nonlinear parabolic distributed parameter systems (DPSs) are firstly expressed by T-S fuzzy models with parameter uncertainties and the aim is to design a reliable distributed fuzzy static output feedback (SOF) Proportional-spatial Derivative (P-sD) controller guaranteeing the robust stochastic exponential stability of the resulting closed-loop system. Based on a Markovian Lyapunov functional, a descriptor system approach is developed for reliable fuzzy SOF P-sD controller synthesis for the underlying nonlinear parabolic DPSs. It is shown that the controller gains can be obtained by solving a set of finite linear matrix inequalities. Finally, an example is given to illustrate the effectiveness of the proposed method.

LMI-Based Stability Criterion for Impulsive Delays Markovian Jumping Time-Delays Reaction-Diffusion BAM Neural Networks via Gronwall-Bellman-Type Impulsive Integral Inequality

Lyapunov stability theory, variational methods, Gronwall-Bellman-type inequalities theorem, and linear matrices inequality (LMI) technique are synthetically employed to obtain the LMI-based global stochastic exponential stability criterion for a class of time-delays Laplace diffusion stochastic equations with large impulsive range under Dirichlet boundary value, whose backgrounds of physics and engineering are the impulsive Markovian jumping time-delays reaction-diffusion BAM neural networks. As far as the authors know, it is the first time to derive the LMI-based criterion by way of Gronwall-Bellman-type inequalities, which can be easily and efficiently computed by computer Matlab LMI toolbox. And the obtained criterion improves the allowable upper bounds of impulse against those of some previous related literature. Moreover, a numerical example is presented to illustrate the effectiveness of the proposed methods.

Delay-Dependent exponential stability for nonlinear reaction-diffusion uncertain Cohen-Grossberg neural networks with partially known transition rates via Hardy-Poincaré inequality

In this paper, stochastic global exponential stability criteria for delayed impulsive Markovian jumping reaction-diffusion Cohen-Grossberg neural networks (CGNNs for short) are obtained by using a novel Lyapunov-Krasovskii functional approach, linear matrix inequalities (LMIs for short) technique, Ito formula, Poincare inequality and Hardy-Poincare inequality, where the CGNNs involve uncertain parameters, partially unknown Markovian transition rates, and even nonlinear p-Laplace diffusion (p > 1). It is worth mentioning that ellipsoid domains in ℝ m (m ≥ 3) can be considered in numerical simulations for the first time owing to the synthetic applications of Poincare inequality and Hardy-Poincare inequality. Moreover, the simulation numerical results show that even the corollaries of the obtained results are more feasible and effective than the main results of some recent related literatures in view of significant improvement in the allowable upper bounds of delays.

Delay-dependent exponential stability for Markovian jumping stochastic Cohen-Grossberg neural networks with p-Laplace diffusion and partially known transition rates via a differential inequality

In this paper, new stochastic global exponential stability criteria for delayed impulsive Markovian jumping p-Laplace diffusion Cohen-Grossberg neural networks (CGNNs) with partially unknown transition rates are derived based on a novel Lyapunov-Krasovskii functional approach, a differential inequality lemma and the linear matrix inequality (LMI) technique. The employed methods are different from those of previous related literature to some extent. Moreover, a numerical example is given to illustrate the effectiveness and less conservatism of the proposed method due to the significant improvement in the allowable upper bounds of time delays.

Stability analysis for impulsive stochastic fuzzy p-Laplace dynamic equations under Neumann or Dirichlet boundary condition

Under Neumann or Dirichlet boundary conditions, the stability of a class of delayed impulsive Markovian jumping stochastic fuzzy p-Laplace partial differential equations (PDEs) is considered. Thanks to some methods different from those of previous literature, the difficulties brought by fuzzy stochastic mathematical model and impulsive model have been overcome. By way of the Lyapunov-Krasovskii functional, Itô formula, Dynkin formula and a differential inequality, new LMI-based global stochastic exponential stability criteria for the above-mentioned PDEs are established. Some applications of the obtained results improve some existing results on neural networks. And some numerical examples are presented to illustrate the effectiveness of the proposed method due to the significant improvement in the allowable upper bounds of time delays.MSC:34D20, 34D23, 34B45, 34B37, 34K20.

Exponential Synchronization for Neutral Complex Dynamical Networks with Interval Mode-Dependent Delays and Sampled Data

The exponential synchronization and sampled-data controller problem for a class of neutral complex dynamical networks (NCDNs) with Markovian jump parameters, partially unknown transition rates and delays, is investigated in this paper. Both the discrete and neutral delays are considered to be interval mode dependent and time varying, while the sampling period is assumed to be time varying and bounded. Based on a new augmented stochastic Lyapunov functional, the delay-range-dependent and rate-dependent exponential stability conditions for the closed-loop error system are obtained by the Lyapunov-Krasovskii stability theory and reciprocally convex lemma. Then according to the proposed exponential stability conditions, the sampled-data synchronization controllers are designed in terms of the solution to linear matrix inequalities that can be solved effectively by using Matlab. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the proposed methods.

Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen–Grossberg neural networks with mixed delays

This paper is devoted to investigating the robust stochastic exponential stability for reaction-diffusion Cohen–Grossberg neural networks (RDCGNNs) with Markovian jumping parameters and mixed delays. The parameter uncertainties are assumed to be norm bounded. The delays are assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. Some criteria for delay-dependent robust exponential stability of RDCGNNs with Markovian jumping parameters are established in terms of linear matrix inequalities (LMIs), which can be easily checked by utilizing Matlab LMI toolbox. Numerical examples are provided to demonstrate the efficiency of the proposed results.