Problem Of Exponential Stability Research Articles

New Algebraic Criteria for Global Exponential Periodicity and Stability of Memristive Neural Networks with Variable Delays

This paper concentrates on the problem of global exponential periodicity and stability of memristive neural networks with variable delays. By constructing the appropriate Lyapunov functionals and utilizing some inequality techniques, new algebraic criteria are proposed to guarantee the existence and global exponential stability of periodic solution of the considered system. In addition, the proposed theoretical results not only expand and complement the earlier publications, but also are easy to be checked with the parameters of system itself. A numerical example is given to demonstrate the effectiveness of our results.

Exponential Pointwise Stabilization of Semilinear Parabolic Distributed Parameter Systems via the Takagi–Sugeno Fuzzy PDE Model

This paper deals with the problem of exponential stabilization for nonlinear parabolic distributed parameter systems using the Takagi–Sugeno (T–S) fuzzy partial differential equation (PDE) model, where a finite number of actuators are active only at some specified points of the spatial domain (these actuators are referred to as pointwise actuators). Three cases of state feedback are respectively considered in this study as follows: full state feedback, piecewise state feedback, and collocated pointwise state feedback. It is initially assumed that a T–S fuzzy PDE model obtained via the sector nonlinearity approach is employed to accurately represent the semilinear parabolic PDE system. Based on the obtained T–S fuzzy PDE model, Lyapunov-based design methodologies of fuzzy feedback control laws are subsequently derived for the above three state feedback cases by using the vector-valued Wirtinger's inequality to guarantee locally exponential pointwise stabilization of the semilinear PDE system, and presented in terms of standard linear matrix inequalities (LMIs). Moreover, the favorable property offered by sharing all the same premises in the T–S fuzzy PDE models and fuzzy controllers is not applicable for the case of collocated pointwise state feedback. A parameterized LMI is introduced for this case to enhance the stabilization ability of the fuzzy controller. Finally, the merit and effectiveness of the proposed design methods are demonstrated by numerical simulation results of two examples.

LMI-based results on exponential stability of BAM-type neural networks with leakage and both time-varying delays: A non-fragile state estimation approach

In this epigrammatic, the problem of exponential stability for BAM-type neural networks (BAMNNs) with non-fragile state estimator is investigated under time-varying delays. The delays in discrete and distributed terms are assumed to be time-varying, which means that the lower and upper bounds can be derived. Without involving the time-delays or the activation functions, the non-fragile estimators are constructed in terms of simple linear formation and also the implementation of state estimators are uncomplicated. In addition, the non-fragile estimators are reduced the possible implementation errors in neural networks. For consequence, reason of energy saving, the non-fragile estimators are designed with neural networks. By fabricating a suitable LKF (Lyapunov–Krasovskii functional) and enroling some analysis techniques, a novel sufficient conditions for exponential stability of the designated neural networks are derived in terms of Linear Matrix Inequalities (LMIs), which can be easily assessed by MATLAB LMI Control toolbox. Accordingly, the research proposed here, is advanced and less conservative than the previous one exists in the literature. Finally, two numerical examples with simulations and comparative studies are performed to substantiate the advantage and validity of our theoretical findings.

Exponential Stabilization of Time‐varying Delayed Complex‐valued Memristor‐based Neural Networks Via Impulsive Control

AbstractThe exponential stabilization problem for a class of time‐varying delayed complex‐valued memristor‐based neural networks via discontinuous impulsive control is investigated in this paper. Firstly, the time‐varying delayed complex‐valued memristor‐based neural networks is translated to real‐valued memristor‐based neural networks. Secondly, an impulsive control law is constructed to guarantee exponential stabilization of the complex‐valued memristor‐based neural networks with time‐varying delays. Thirdly, by constructing an appropriate Lyapunov Krasovskii functional and adopting linear matrix inequality (LMI) technique, some sufficient exponential stabilization criteria are derived. Finally, an illustrative example is provided to illustrate the effectiveness of the obtained theoretical results.

Extended robust global exponential stability for uncertain switched memristor-based neural networks with time-varying delays

This paper is concerned with the problem of global exponential stability for uncertain memristive-based neural networks (UMNNs) with time-varying delays and switching parameters subject to unstable subsystems. Different from most of the existing papers, the considered uncertain switched MNNs with discrete-delays are modeled as switched neural networks (SNNs) with uncertain time-varying parameters. Based on multiple Lyapunov–Krasovskii functional (MLF) approach, average dwell time (ADT) technique and mode-dependent average dwell time (MDADT) method, some LMIs-based stability criteria are derived to design the switching signal and guarantee the exponential stability of the considered uncertain switched neural networks. By exploring the mode-dependent property of each subsystem, all the subsystems are categorized into stable and unstable ones. The concerned SNNs with both stable and unstable subsystems are more general and applicable than the existing models of SNNs only view all subsystems being stable, thus getting less conservatism criteria. The proposed sufficient conditions can be simplified into the forms of LMIs for conveniently using Matlab LMI toolbox. Finally, two numerical examples are exploited to demonstrate the effectiveness and applicability of the proposed theoretical results.

Exponential Stability of Two-Dimensional Homogeneous Monotone Systems With Bounded Directional Delays

One-dimensional (1-D) monotone systems have received considerable attention recently due to their wide applicability and interesting mathematical properties. One of these special properties is that, for LTI monotone systems, exponential stability is insensitive to time-delays. Some extensions to 1-D nonlinear monotone systems based on conditions of homogeneity have also been reported. In this paper, we study the problem of exponential stability of discrete-time 2-D nonlinear monotone systems described by the Roesser model with time-varying delays. Specifically, based on the monotone property and homogeneity of the associated vector fields, necessary and sufficient delay-independent exponential stability conditions are derived. The magnitudes of delays are also taken into deriving an explicit estimation of the exponential decay rate which correlates the impact of delays on the system performance. Two examples are given to demonstrate the effectiveness of the obtained results.

Mean-square exponential stability for the hysteretic Hopfield neural networks with stochastic disturbances

ABSTRACTIn this paper, the exponential stability problem is considered for a class of hysteretic Hopfield neural networks with stochastic disturbances. The hysteretic nonlinearities are characterized by a Lipschitz-type constraint where the internal parameters of the hysteretic function are reflected. By resorting to Lyapunov function approach and stochastic analysis, a sufficient condition has been obtained under which the underlying hysteretic Hopfield neural network is exponentially stable in the mean square. The obtained condition is expressed in terms of linear matrix inequalities (LMIs) which can be easily checked via the Matlab toolbox. Finally, an illustrative example is provided to show the effectiveness of the results derived in this paper.

On exponential stabilisation for descriptor time-delay systems with nonlinear input via SMC

The problem of exponential stabilisation for a class of uncertain descriptor time-delay systems (DTS) is investigated, where both nonlinear input and external disturbance are involved. A novel sliding mode control (SMC) scheme is developed to achieve a stable uncertain system. A delay-free linear sliding surface is first introduced for control design of the DTS. A new adaptive controller is designed to ensure the state trajectories globally to converge onto the sliding surface. Furthermore, the considered sliding mode dynamics (SMDs) are derived via equivalent control of the SMC, once the system is in the sliding mode, the proposed criterion can guarantee the SMDs to be exponentially admissible and passive. Finally, the effectiveness of the proposed scheme is verified via an example.

On exponential stabilisation for descriptor time-delay systems with nonlinear input via SMC

The problem of exponential stabilisation for a class of uncertain descriptor time-delay systems (DTS) is investigated, where both nonlinear input and external disturbance are involved. A novel sliding mode control (SMC) scheme is developed to achieve a stable uncertain system. A delay-free linear sliding surface is first introduced for control design of the DTS. A new adaptive controller is designed to ensure the state trajectories globally to converge onto the sliding surface. Furthermore, the considered sliding mode dynamics (SMDs) are derived via equivalent control of the SMC, once the system is in the sliding mode, the proposed criterion can guarantee the SMDs to be exponentially admissible and passive. Finally, the effectiveness of the proposed scheme is verified via an example.

Delay-Dependent Exponential Stabilization for Linear Distributed Parameter Systems With Time-Varying Delay

This paper extends the framework of Lyapunov–Krasovskii functional to address the problem of exponential stabilization for a class of linearly distributed parameter systems (DPSs) with continuous differentiable time-varying delay and a spatiotemporal control input, where the system model is described by parabolic partial differential-difference equations (PDdEs) subject to homogeneous Neumann or Dirichlet boundary conditions. By constructing an appropriate Lyapunov–Krasovskii functional candidate and using some inequality techniques (e.g., spatial integral form of Jensen's inequalities and vector-valued Wirtinger's inequalities), some delay-dependent exponential stabilization conditions are derived, and presented in terms of standard linear matrix inequalities (LMIs). These stabilization conditions are applicable to both slow-varying and fast-varying time delay cases. The detailed and rigorous proof of the closed-loop exponential stability is also provided in this paper. Moreover, the main results of this paper are reduced to the constant time delay case and extended to the stochastic time-varying delay case, and also extended to address the problem of exponential stabilization for linear parabolic PDdE systems with a temporal control input. The numerical simulation results of two examples show the effectiveness and merit of the main results.

Exponential stability for the neutral-type singular neural network with time-varying delays

This paper deals with the problem of global exponential stability for neutral type singular networks with time-varying delays. Delay-dependent criterion is proposed to guarantee the exponential stability of neutral type singular networks via linear matrix inequality (LMI) approach. Improved global exponential stability condition is derived by employing a new Lyapunov–Krasovskii functional and a rarely integral inequality. The developed result is less conservative than previous published ones in the literature, which is illustrated by representative numerical examples.

Global exponential stability of memristive Cohen–Grossberg neural networks with mixed delays and impulse time window

This paper addresses the problem of global exponential stability for a class of memristive Cohen–Grossberg with mixed time delays and impulsive time window. Based on the memristor theory, impulse control theory, and mathematical induction method, stability criteria for the considered memristive Cohen–Grossberg neural networks are derived by employing appropriate Lyapunov functions. Compared with existing impulse control schemes, the impulse instants are extended to some bounded time intervals, we will show that the neural networks can still be stable under reasonable assumptions. Furthermore, the conditions obtained in this paper are easy to be checked, and they can be applied to improve previous results concerning stability for memristive neural networks. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.

Robust Stability of Switched Interconnected Systems With Time-Varying Delays

The problem of robust exponential stability for a class of switched nonlinear dynamical systems with uncertainties and time-varying delays is investigated. On the assumption that each isolated subsystem of the interconnected system can be exponentially stabilized and the corresponding Lyapunov functions are available, using M-matrix property, the differential inequalities with time-varying delays are constructed. By the stability analysis of the differential inequalities, the sufficient conditions to ensure the robust exponential stability of the switched interconnected systems (SIS) under arbitrary switching are obtained. The proposed method, which neither requires the individual subsystems to share a common Lyapunov function (CLF), nor needs to know the values of individual Lyapunov functions at each switching time, would provide a new mentality for studying stability of arbitrary switching. In addition, by resorting to average dwell time approach, conditions for guaranteeing the robust exponential stability of SIS under constrained switching are derived. The proposed criteria are explicit, and they are convenient for practical applications. Finally, two numerical examples are given to illustrate the validity and correctness of the proposed theories.

Exponential stability and [formula omitted]-gain analysis for positive time-delay Markovian jump systems with switching transition rates subject to average dwell time

The paper deals with the problems of exponential stability and L1-gain analysis for positive time-delay Markovian jump systems (MJSs) with switching transition rates. Another set of useful regime-switching model is given, which extends fixed transition rates to time-varying transition rates. By resorting to the linear co-positive Lyapunov function and average dwell time, sufficient conditions for exponential stability are proposed in terms of standard linear programming. Based on the obtained results, L1-gain performance is analyzed. Finally, an example is proposed to illustrate the validity of the main results.

Stabilization of Variable Coefficients Euler-Bernoulli Beam with Viscous Damping under a Force Control in Rotation and Velocity Rotation

This paper investigates the problem of exponential stability for a damped Euler-Bernoulli beam with variable coefficients clamped at one end and subjected to a force control in rotation and velocity rotation. We adopt the Riesz basis approach for show that the closed-loop system is a Riesz spectral system. Therefore, the exponential stability and the spectrum-determined growth condition are obtained.

Exponential stabilisation of stochastic memristive neural networks under intermittent adaptive control

This study focuses on the exponential stabilisation problem for a general class of memristive neural networks subjected to both stochastic disturbance and time-varying delays under periodically intermittent adaptive control. The stochastic disturbances are described as Brownian motions in the considered networks. An adaptive updated rule and a periodically intermittent adaptive control strategy are designed for the exponential stabilisation of memristive neural networks subjected to both stochastic disturbance and time-varying delays. Then, by adopting the adaptive control technique, differential inclusion theory and set-valued maps, and by building a new Lyapunov–Krasovskii functional, many novel sufficient conditions are proposed to guarantee exponential stabilisation for stochastic memristive neural networks. Different from existing results on stabilisation of stochastic memristive neural networks, the obtained criteria in this study are directly derived according to the parameters of networks. Finally, an example is carried out to demonstrate the validity of the theoretic results.

Exponential stability with respect to part of the variables for a class of nonlinear stochastic systems with Markovian switchings

This paper deals with the exponential stability problem with respect to part of the variables of a class of nonlinear stochastic systems for three classes of Markovian switching processes. The methodologies of stability of stochastic hybrid systems and stability with respect to part of the variables based on Lyapunov methods are combined to find sufficient conditions of the exponential yp-stability for a class of nonlinear stochastic systems. Moreover, the detailed analysis and criteria of exponential mean square y-stability based on LMI methodology are also given for the case of linear systems.

Stability and [formula omitted]-gain analysis for impulsive delay systems: An impulse-time-dependent discretized Lyapunov functional method

The problems of exponential stability and L2-gain for a class of time-delay systems with impulsive effects are studied. The main tool used is the construction of an impulse-time-dependent complete Lyapunov functional. By dividing the impulse interval and delay interval into several segments, the matrix functions of this functional are chosen to be continuous piecewise linear. Moreover, an impulse-time-dependent weighting factor is introduced to coordinate the dynamical behavior of the nondelayed and integral terms of this functional along the trajectories of the system. By applying this functional, delay-dependent sufficient conditions for exponential stability and L2-gain are derived in terms of linear matrix inequalities. As by-products, new delay-independent sufficient conditions for the same problems are also derived. The efficiency of the proposed results is illustrated by numerical examples.

Exponential stability of impulsive systems with random delays under sampled‐data control

In this study, the exponential stability problem of impulsive system is investigated via sampled-data control in the presence of random time-varying delays and non-linear perturbations. In particular, the time delays are considered to be randomly time varying and they obey the Bernoulli distributions. The authors' main attention is focused on the design of a sampled-data controller to ensure an exponential stability for the closed-loop system. By extending the first- and second-order reciprocal convex approach, an efficient method called third-order reciprocal convex technique is used to manipulate the main results. Through the construction of a suitable Lyapunov–Krasovskii functional combined with input delay approach and Briat lemma, several delay-dependent sufficient conditions for the concerned system are derived in the form of linear matrix inequalities which can be readily solved by utilising the valid software packages. Some numerical examples are given to illustrate the effectiveness of the developed control technique.

Robust control for switched systems subject to input saturation and parametric uncertainties

This paper studies the exponential stability problem for switched systems with input saturation and parametric uncertainties. On the basis of the gain scheduling scheme (GSS) and the low gain feedback (LGF), a robust controller is constructed. The main contributions of this paper are increasing the state convergence speed of the closed-loop system by the introduced parameters and estimating the maximized region of attraction by the designed outermost ellipsoids. The corresponding conditions of the designed controller are expressed as the linear matrix inequalities (LMIs). The constructed controller is easily computed by solving the optimization problem. The numerical simulation is performed to show the feasibility and the effectiveness of the obtained results.