pth Moment exponential stability of impulsive stochastic functional differential equations and application to control problems of NNs

This paper investigates the pth moment exponential stability of impulsive stochastic functional differential equations. Some sufficient conditions are obtained to ensure the pth moment exponential stability of the equilibrium solution by the Razumikhin method and Lyapunov functions. Based on these results, we further discuss the pth moment exponential stability of generalized impulsive delay stochastic differential equations and stochastic Hopfield neural networks with multiple time-varying delays from the impulsive control point of view. The results derived in this paper improve and generalize some recent works reported in the literature. Moreover, we see that impulses do contribute to the stability of stochastic functional differential equations. Finally, two numerical examples are provided to demonstrate the efficiency of the results obtained.     

Stability and L2-gain analysis for switched neutral systems with mixed time-varying delays

This paper considers the stability and L₂-gain for a class of switched neutral systems with time-varying discrete and neutral delays. Some new delay-dependent sufficient conditions for exponential stability and weighted L₂-gain are developed for a class of switching signals with average dwell time. These conditions are formulated in terms of linear matrix inequalities (LMIs) and are derived by employing free weighting matrices method. As a special case of such switching signals, we can obtain exponential stability and normal L₂-gain under arbitrary switching signals. Finally, two numerical examples are given to illustrate the effectiveness of the theoretical results.     

Noise suppresses exponential growth for neutral stochastic differential delay equations

Our aim in this paper is to investigate the polynomial growth for a class of neutral stochastic differential delay systems. We reveal that environmental noise will suppress the exponential growth if it is sufficiently strong. That is, the given system (neutral ordinary differential delay equation) whose solution grows exponentially and becomes a new system (neutral stochastic differential delay equation) whose solution will grow polynomially with probability one. We also provide two examples to illustrate the main result.     

The Razumikhin approach on general decay stability for neutral stochastic functional differential equations

The topic of the paper is both the pth moment and almost sure stability on a general decay rate for neutral stochastic functional differential equations, by applying the Razumikhin approach. This concept is extended to neutral stochastic differential delay equations. The results obtained in the paper are more general and they may be specialized on the exponential, polynomial or logarithmic stability. Moreover, some neutral stochastic functional differential equations which are not pth moment or almost surely exponentially stable, could be stable with respect to a certain lower decay rate. In that sense, some nontrivial examples are presented to justify and illustrate the usefulness of the theory. More precisely, one can say anything about both the pth moment and almost sure exponential stability, although the solutions are pth moment and almost surely polynomially or logarithmically stable.     

pth Moment exponential stability of impulsive stochastic functional differential equations with Markovian switching

In this paper, the pth moment exponential stability for a class of impulsive stochastic functional differential equations with Markovian switching is investigated. Based on the Lyapunov function, Dynkin formula and Razumikhin technique with stochastic version as well as stochastic analysis theory, many new sufficient conditions are derived to ensure the pth moment exponential stability of the trivial solution. The obtained results show that stochastic functional differential equations with/without Markovian switching may be pth moment exponentially stabilized by impulses. Moreover, our results generalize and improve some results obtained in the literature. Finally, a numerical example and its simulations are given to illustrate the theoretical results.     

马尔可夫调制的随机微分方程的稳定性

在现有文献的基础上，对一类马尔可夫调制的随机微分方程进行了研究，得到了其平凡解2阶均值指数稳定性和几乎必然指数稳定性的充分条件。对现有成果进行了改进。     

Exponential stabilization and sampled-date H∞ control for uncertain T–S fuzzy systems with time-varying delay

In this paper, the exponential stabilization problem of uncertain T–S fuzzy systems with time-varying delay is emulated by fuzzy sampled-data H_∞ control. Firstly, a novel suitable Lyapunov–Krasovskii function is constructed, which contains all the information about the sampling pattern. Secondly, a less conservative result is achieved by using an extended Jensen inequality, and purposefully using a compact free weighting matrix. In addition, according to the linear matrix inequality (LMI), some sampled-data H_∞ exponential stability sufficient conditions and controller design of T–S fuzzy systems are established. Finally, effectiveness gives some illustrative examples may be used to display the value of the current proposed method as well as a significant improvement.     

Existence and exponential stability of almost periodic solution for neutral delay BAM neural networks with time-varying delays in leakage terms

This paper is concerned with a class of neutral delay BAM neural networks with time-varying delays in leakage terms. Some sufficient conditions are established to ensure the existence and exponential stability for such class of neural networks by employing the exponential dichotomy of linear differential equations, fixed point theorems and differential inequality techniques. An example is provided to show the effectiveness of the theoretical results. The results of this paper are completely new and complementary to the previously known results.     

Attitude tracking control of a quadrotor UAV in the exponential coordinates

A new approach to control the attitude of a quadrotor UAV in terms of the exponential coordinates is developed in this paper. The exponential coordinate is a minimal representation of the rotation matrix, but it can avoid singularities. Since the quadrotor UAV can be considered as a rigid body aircraft, the analytic closed-form expressions of a rigid body's attitude kinematics are derived from differential of exponential on SO(3). Furthermore, based on the exponential expressions of attitude kinematics, the controller of a fully actuated rigid body is designed using trajectory linearization control method. The overall attitude controller contains two loops, which are designed according to the torque equation and the angular velocity equation respectively. In the numerical simulation, the proposed attitude controller is compared to a controller in the Euler angles, showing that singularities induced by Euler angles are avoided by using exponential coordinates. The robustness test of the attitude controller is also demonstrated in the simulation. The simulation results indicate that the proposed method can be applied to the attitude tracking control of an aerial robot especially when the robot needs to make aggressive maneuverings.     

Stability in mean of partial variables for stochastic reaction–diffusion systems with Markovian switching

This paper is denoted to investigating stability in mean of partial variables for stochastic reaction–diffusion equations with Markovian switching (SRDEMS). By transforming the integral of the trajectory with respect to spatial variables as the solution of the stochastic ordinary differential equations with Markovian switching (SODEMS) and using Itô formula, sufficient criteria on uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, exponential stability in mean of partial variables for SRDEMS are first derived. An example is presented to illustrate the effectiveness and efficiency of the obtained results.     

A sliding mode approach to H∞ synchronization of master–slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties

In this paper, a sliding-mode approach is proposed for exponential H_∞ synchronization problem of a class of master–slave time-delay systems with both discrete and distributed time-delays, norm-bounded nonlinear uncertainties and Markovian switching parameters. Using an appropriate Lyapunov–Krasovskii functional, some delay-dependent sufficient conditions and a synchronization law, which include the master–slave parameters are established for designing a delay-dependent mode-dependent sliding mode exponential H_∞ synchronization control law in terms of linear matrix inequalities. The controller guarantees the H_∞ synchronization of the two coupled master and slave systems regardless of their initial states. Two numerical examples are given to show the effectiveness of the method.     

L1-gain analysis and control of impulsive positive systems with interval uncertainty and time delay

This paper investigates L₁-gain analysis and control of impulsive positive systems (IPSs) with interval uncertainty and time delay. For different types of impulsive effect, by means of the Razumikhin techniques and Lyapunov function theory, conditions are developed for guaranteeing the robust exponential stability with L₁-gain performance. Then the positive stabilization with L₁-gain performance is also addressed for IPSs with interval uncertainty and time delay through the state feedback control. In addition, the way to explore the minimum L₁-gain is discussed. All the obtained conditions can be easily inspected by the linear programming (LP) method when some parameters are preset. Finally, simulations are provided to demonstrate the validity of the theoretical results.     

Stability and L1-gain analysis for switched positive T–S fuzzy systems under asynchronous switching

This paper is concerned with the exponential stability and L₁-gain analysis problem for switched positive T–S fuzzy systems under both time-varying delays and asynchronous switching. By permitting the increase of the designed multiple Lyapunov functionals during the running time of the activated subsystem, solvable conditions for the stability and L₁-gain are developed by adopting the mode-dependent average dwell time (MDADT) technique. The desired controllers guaranteeing the stability and the L₁-gain performance are designed based on the obtained solvable conditions. An example is given to demonstrate the effectiveness of the proposed methods.     

Exponential stability of non-linear neutral stochastic delay differential system with generalized delay-dependent impulsive points

This work aims to analyze the exponential stability of a non-linear impulsive neutral stochastic delay differential system. In this study, impulse perturbation is considered a delay-dependent state variable. The solution of the delay-dependent impulsive neutral stochastic delay differential system is associated with the solution of the system without impulses. First, we developed a relation connecting the solution of the neutral stochastic delay differential system without impulses and the solution of the corresponding system with impulses. Then, the conditions of the exponential stability of the proposed impulsive system are derived by determining the stability analysis of the respective system without impulse. The numerical approach for the neutral stochastic delay system without impulses is generated using the Euler-Maruyama method and adopted for the corresponding impulsive system. Finally, the achieved theoretical results are illustrated for applying the Malthusian single species neutral stochastic delay population model with immigration impulses.     

KP差分-微分方程组及其精确解(英文)

通过定义平移算子和差分算子 ,并利用Lax配对的方法 ,找到了KP差分 -微分方程组的正确形式 .定义了差分指数函数 .借助穿衣算子法 ,得到了KP差分 -微分方程组的精确解析解 .还讨论了KP差分 -微分方程组及其解的展开形式 .     

Spatiotemporal sampled-data fuzzy control for switched singularly perturbed PDE systems with average dwell-time switching mechanism

This paper investigates a spatiotemporal sampled-data fuzzy control strategy for switched singularly perturbed partial differential equation (PDE) systems, where the systems’ operation modes obey average dwell-time switching mechanism. To efficiently deal with nonlinear terms and guarantee the system stability for the considered systems, a spatiotemporal sampled-data fuzzy control scheme is developed. Furthermore, based on the fact that mode mismatch phenomena during switching and sampling, through formulating novel Lyapunov functionals (LFs) with the discontinuous terms and mode-dependent two-sided looped-functionals, which can fully utilize the state information of the sampling period, a new exponential stability criterion is provided for the target systems. Finally, an example is provided to prove the validity of the proposed control approach.     

Exponential stabilization for fuzzy sampled-data system based on a unified framework and its application

This article deals with the exponential stabilization problems of Takagi–Sugeno (T–S) fuzzy system under aperiodic sampling. The objective is to solve the H_∞, $L_2?L_{\infty },$ passive and dissipative stability and stabilization problems based on a unified performance index-extended dissipativity. Some new stability conditions consisting of both exponential stability and extended dissipativity criterion are first established. A sampling period dependent Lyapunov–Krasovskii function together with a novel efficient integral inequality, which has the advantage of reducing conservativeness, is adopted. On the basis of the stability conditions, a sampled-data controller that can not only exponentially stabilize the system but also guarantee the prescribed extended-dissipativity performance is then designed. Moreover, an exponential decay rate can be set in advance to achieve better system performance. Finally, a quarter-vehicle active suspension system with considering payload uncertainties and aperiodic sampling is provided for evaluating the validity and superiority of the extended dissipative control approach proposed in this article.     

Model-Free distributed optimal consensus control of nonlinear multi-agent systems: A graphical game approach

In this paper, the optimal consensus control problem of nonlinear multi-agent systems(MASs) with completely unknown dynamics is considered. The problem is formulated in a differential graphical game approach which can be solved by Hamilton-Jacobi (HJ) equations. The main difficulty in solving the HJ equations lies in the nonlinear coupling between equations. Based on the Adaptive Dynamic Programming (ADP) technique, an VI-PI mixed HDP algorithm is proposed to solve the HJ equations distributedly. With the PI step, a suitable iterative initial value can be obtained according to the initial policies. Then, VI steps are run to get the optimal solution with exponential convergence rate. Neural networks (NNs) are applied to approximate the value functions, which makes the data-driven end-to-end learning possible. A numerical simulation is conducted to show the effectiveness of the proposed algorithm.     

Robust simultaneous fault detection and control for a class of nonlinear stochastic switched delay systems under asynchronous switching

This paper concerns the simultaneous fault detection and control (SFDC) problem for a class of nonlinear stochastic switched systems with time-varying state delay and parameter uncertainties. The switching signal of detector/controller unit (DCU) is assumed to be with switching delay, which results in the asynchronous switching between the subsystems and DCU. By constructing a switching strategy depending on the state and switching delays, new sufficient conditions expressed by a set of linear matrix inequalities (LMIs) is derived to design DCU gains. This problem is formulated as an H_∞ optimization problem and both mean square exponential stability and fault detection of augmented system are considered. A numerical example is finally exploited to verify the effectiveness and potential of the achieved scheme.