Problem Of Exponential Stability Research Articles

Lyapunov conditions for exponential stability of nonlinear delay systems via impulsive control involving stabilizing delays

The problem of global exponential stability (GES) for nonlinear delay impulsive systems is investigated in this paper. By extending the traditional comparison principle, delay effects on continuous and discrete dynamics of the system are estimated, based on which, the internal relationship between delays, parameters of impulsive control, and continuous dynamics of the system is revealed. Then some sufficient criteria are obtained for GES, which quantitatively shows the beneficial influences of delays in impulses on the system performance. Finally, two numerical examples are given to illustrate the effectiveness of the proposed result.

Exponential stability of nonlinear systems via event-triggered impulsive control based on partial states

In the framework of partial unknown states, this paper studies the exponential stability problem of nonlinear systems. A novel event-triggered mechanism (ETM) just involving partial known states is designed to guarantee the exponential stability of considered system, where forced impulse with great freedom degree is introduced in ETM. What's more, the information of known states of the system is fetched and applied to supplement the unknown part at impulse instants. With the help of matrix analysis technique, some linear matrix inequalities (LMI) based sufficient conditions are presented from the perspective of the quantity of known and unknown states. Besides, our proposed conditions establish the relationship of event-triggered parameters, impulse interval, and impulse strength. Finally, expository examples and their numerical modeling are appeared to prove the availability of main results.

On the exponential stability of Beck's Problem on a star-shaped graph

We deal with the as yet unresolved exponential stability problem for Beck's Problem on a metric star graph with three identical edges. The edges are stretched Euler–Bernoulli beams which are simply supported with respect to the outer vertices. At the inner vertex we have viscoelastic damping acting on the slopes of the edges. We carry out a complete spectral analysis of the system operator associated with the abstract spectral problem in Hilbert space. Within this framework it is shown that the eigenvectors have the property of forming a Riesz (i.e. an unconditional) basis, which makes it possible to directly deduce the exponential stability of the corresponding C0-semigroup using spectral information for the system operator alone. A physically interesting conclusion is that the particular choice of vertex conditions ensures the exponential stability even when the elasticity acting on the slopes of the edges is absent.

Exponential stability of time‐delay systems with flexible delayed impulse

AbstractThe problem of exponential stability for time‐delay systems with flexible delayed impulse control is investigated. Unlike the conventional Razumikhin‐type inequality, variable parameter based on the flexible impulsive gain is introduced. Sufficient conditions for exponential stability are developed for a class of time‐delay systems, for which the size of delay is not limited by impulsive intervals. A flexible impulse control scheme is developed by utilizing the flexible impulsive gain and time‐varying delays. When the system is disturbed by the impulse interferences, both the impulse gain and impulse delay can be adjusted accordingly to make the system exponential stable. The effectiveness of the present results is illustrated by two numerical examples and an application.

Exponential stability and boundedness of nonlinear perturbed systems by unbounded perturbation terms

We study the exponential stability and boundedness problem for perturbed nonlinear time-varying systems using Lyapunov Functions with indefinite derivatives. As the limiting function for the perturbation term, we use different forms and give stability and boundedness conditions in terms of the coefficients in these bounds. Contrary to most of the available conditions, we allow the coefficients to be unbounded. But instead, we put forward a condition that requires a series produced by coefficients to be limited and exponentially decaying. We perform our results on Linear time-varying systems and generalize many of the available results.

Exponential stabilization of conformable fractional bilinear systems with multiple inputs

In this paper, we consider a class of conformable fractional bilinear systems with multiple inputs. We study the exponential stabilization problem for these systems by using the quadratic Lyapunov functions and some additional hypothesis on the unit sphere. A numerical example is given to illustrate the efficiency of the obtained result.

Relaxed Exponential Stabilization for Coupled Memristive Neural Networks With Connection Fault and Multiple Delays via Optimized Elastic Event-Triggered Mechanism.

This article investigates the problem of relaxed exponential stabilization for coupled memristive neural networks (CMNNs) with connection fault and multiple delays via an optimized elastic event-triggered mechanism (OEEM). The connection fault of the two or some nodes can result in the connection fault of other nodes and cause iterative faults in the CMNNs. Therefore, the method of backup resources is considered to improve the fault-tolerant capability and survivability of the CMNNs. In order to improve the robustness of the event-triggered mechanism and enhance the ability of the event-triggered mechanism to process noise signals, the time-varying bounded noise threshold matrices, time-varying decreased exponential threshold functions, and adaptive functions are simultaneously introduced to design the OEEM. In addition, the appropriate Lyapunov-Krasovskii functionals (LKFs) with some improved delay-product-type terms are constructed, and the relaxed exponential stabilization and globally uniformly ultimately bounded (GUUB) conditions are derived for the CMNNs with connection fault and multiple delays by means of some inequality processing techniques. Finally, two numerical examples are provided to illustrate the effectiveness of the results.

Fuzzy Secure Event-Triggered Control for Networked Nonlinear Systems Under DoS and Deception Attacks

This article investigates the resilient dynamic event-triggered control issue of interval type-2 T-S fuzzy systems (IT2 T-S FSs) under denial-of-service (DoS) and deception attacks. A novel resilient dynamic event-triggering mechanism (DETM) and the corresponding fuzzy dynamic event-triggered controller are proposed to solve the stochastically exponential stability (SES) problem of concerned IT2 T-S FSs in the presence of DoS and deception attacks. Meanwhile, an aperiodic-sampling-based piecewise loop Lyapunov functional is constructed. It is shown that the resulting closed-loop IT2 T-S FSs can achieve SES, and under the piecewise loop Lyapunov functional, some relaxed conditions are derived. In addition, the derived results can apply to IT2 T-S FSs subject to DoS attacks, static event-triggered control, or dynamic periodic-sampling-based event-triggered control. Finally, the effectiveness of the achieved results is verified by a numerical simulation on one-link manipulator.

Practical exponential stability of hybrid impulsive stochastic functional differential systems with delayed impulses

SummaryThis paper is concerned with the problem of practical exponential stability for hybrid impulsive stochastic functional differential systems with delayed impulses, which comprise three classes of systems: the systems with unstable continuous stochastic dynamics and stable discrete dynamics, the systems with stable continuous stochastic dynamics and unstable discrete dynamics, and the systems where both the continuous stochastic dynamics and the discrete dynamics are stable. By using the Lyapunov‐Razumikhin approach, several new sufficient conditions for the practical exponential stability are established for each class of systems. It shows that the stabilizing and destabilizing delayed impulses that satisfy some conditions on their frequency and amplitude can stabilize the systems with unstable continuous stochastic dynamics in the practical exponential stability sense and ensure the practical exponential stability of the systems with stable continuous stochastic dynamics, respectively. Conversely, if the continuous stochastic dynamics are practically exponentially stable and the delayed impulses are stabilizing, then the systems can be practically exponentially stable regardless of the restrictions on the delayed impulses frequency and amplitude. Finally, two numerical examples are presented to illustrate the efficiency of the results.

Global exponential stabilization of the linearized Korteweg-de Vries equation with a state delay

Abstract In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval $[0,2\pi ]$ and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of $L^{2}(0,2\pi )$-norm by the use of an appropriate Lyapunov–Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result.

Adaptive memory-based event-triggering resilient LFC for power system under DoS attack

Considering denial-of-service (DoS) attacks in the communication network, this work develops the adaptive memory-based event triggering resilient LFC approach for multi-area power systems with wind power. First of all, the impact of each DoS attack is evaluated by the amount of continual information loss it causes. Meanwhile, an adaptive memory event triggering mechanism (AMETM) is proposed, where the dynamic event triggering parameter and the triggering mechanism are all related to the historical transmitted states. Then, based on the proposed AMETM and DoS attacks, a sampling-data LFC model is established. In order to construct the LFC scheme with the desired faster state response, the exponential decay index is introduced as a performance index. Next the exponential stability problem of LFC sampling model is transformed into the asymptotic stability problem of polynomial uncertain system. Secondly, the exponential stability criterion and the memory controllers design method are obtained by bilateral closed-loop Lyapunov–Krasovskii functional (LKF) and LMI technology, which ensure the proposed memory control scheme can quickly stabilize the LFC system under a longer DoS attack. Lastly, a one-area typical LFC system is presented to verify the superiority and effectiveness of the proposed AMETM. The resilient control with fast state response and DoS attack is studied by two-areas LFC systems with wind power.

Local exponential stabilization of a coupled ODE-Fisher’s PDE system

In this paper, we investigate the problem of exponential stabilization of a coupled control system consisting of a linear ordinary differential equation (ODE) and a nonlinear partial differential equation (PDE). The nonlinear PDE is the one-dimensional Fisher’s equation defined on a bounded domain. The control input for the whole system acts at the right boundary of the PDE domain, whereas the left boundary injects a Dirichlet term in the ODE subsystem and the coupling is given as a Robin condition at the left boundary. Based on the infinite dimensional backstepping method for the linear part, a state feedback boundary controller is explicitly derived. Under this controller, we show that the considered coupled ODE-Fisher’s PDE system is globally well-posed in the Hilbert space Rn×L2(0,1). Moreover, using a well-adapted strict Lyapunov functional, we establish its local exponential stabilization around its zero equilibrium in the topology of Rn×L2(0,1)-norm. A simulation result that reflects the effectiveness of the proposed approach is given.

Practical Stability for Conformable Time-Delay Systems

This article investigates the practical exponential stability and design problems of conformable time-delay systems. Sufficient conditions that confirm the practical exponential stability and design of the proposed class of systems are given by utilizing an adequate Lyapunov–Krasovskii functional (L-KF). These conditions are expressed in the form of linear matrix inequalities (LMI) which could be solved by using solvers in LMI Toolbox of MATLAB. Two numerical examples are given to illustrate the applicability of the proposed results.

Exponential stability of linear discrete systems with multiple delays by degenerated Lyapunov–Krasovskii functionals

The problem of exponential stability of delayed discrete systems with multiple delays x(n+1)=(I+A)x(n)+∑i=1sBix(n−i),n=0,1,…is studied, where x=(x1x2…xm)T is an unknown vector, m and s are fixed positive integers, A, Bi are square constant matrices and I is a unit matrix. A new degenerated Lyapunov–Krasovskii functional is used to derive sufficient conditions for exponential stability and to derive an exponential estimate of the norm of solutions. Though often used in the study of stability, the assumption that the spectral radius of the matrix of linear terms is less than 1 is not applied here. The criterion derived is illustrated by an example and compared with previously known results.

Exponential output feedback control for a class of strict-feedback nonlinear systems with different time-delay states

In this paper, the output feedback control scheme is developed to investigate the exponential stabilisation problem for a class of strict-feedback nonlinear systems subject to different time-delay states. First of all, the memoryless full-order observer and output feedback controller which depend on the same designed parameter are provided respectively. And then, by tactfully introducing the suitable Lyapunov–Krasovskii functional and selecting the applicable parameter to be designed, the exponential stability of the closed-loop system can be achieved. Finally, a numerical example is used to illustrate the effectiveness of the proposed control scheme.

Generalized stability criteria of switched linear systems with time-varying delays

In this paper, the global exponential stability problem of the switched linear system with time-varying delays is considered. We establish a new stability criterion for the switched linear time-delay system under the Φ-dependent average dwell time technique. By adopting different grouping methods for subsystems, the corresponding stability criteria based on average dwell time and mode-dependent dwell time techniques can be unified to the new results. Finally, an example is given to illustrate the effectiveness of the obtained results.

Robust exponential stabilization of stochastic coupled T‐S fuzzy complex networks subject to state‐dependent impulsive control

AbstractThe problem of robust exponential stabilization for T‐S fuzzy stochastic dynamical networks (T‐S FSDNs) based on the state‐dependent impulsive controller is investigated in the paper. Using B‐equivalence approach, some sufficient conditions are provided such that every solution of considered systems intersects each impulsive surface exactly once. Then, combining with the inequality technique and comparison principle, some meaningful sufficient criteria are obtained to ensure the mean square stability of T‐S fuzzy stochastic dynamical networks. Compared to related results, the results obtained in the paper are low conservatism and easy to verify. Finally, two numerical simulations based on Rssler's system and Lorenz system show the effectiveness of the obtained theoretical results.

New delay-range-dependent exponential stability criterion and $ H_\infty $ performance for neutral-type nonlinear system with mixed time-varying delays

<abstract><p>For a neutral system with mixed discrete, neutral and distributed interval time-varying delays and nonlinear uncertainties, the problem of exponential stability is investigated in this paper based on the $ H_\infty $ performance condition. The uncertainties are nonlinear time-varying parameter perturbations. By introducing a decomposition matrix technique, using Jensen's integral inequality, Peng-Park's integral inequality, Leibniz-Newton formula and Wirtinger-based integral inequality, utilization of a zero equation and the appropriate Lyapunov-Krasovskii functional, new delay-range-dependent sufficient conditions for the $ H_\infty $ performance with exponential stability of the system are presented in terms of linear matrix inequalities. Moreover, we present numerical examples that demonstrate exponential stability of the neutral system with mixed time-varying delays, and nonlinear uncertainties to show the advantages of our method.</p></abstract>

Impulsive control for stationary oscillation of nonlinear delay systems and applications

<abstract><p>In this paper, the problem of existence-uniqueness and global exponential stability of periodic solution (i.e., stationary oscillation) for a class of nonlinear delay systems with impulses was studied. Some new sufficient conditions ensuring the existence of stationary oscillation for the addressed equations were derived by using the inequality technique that has been reported in recent publications. Our proposed method, which is different with the existing results in the literature, shows that nonlinear delay systems may admit a stationary oscillation using proper impulsive control strategies even if it was originally unstable or divergent. As an application, we considered the single species logarithmic population model and established a new criterion to guarantee the existence of positive stationary oscillation. Some numerical examples and their computer simulations were also given at the end of this paper to show the effectiveness of our development control method.</p></abstract>

Global exponential stability conditions for quaternion-valued neural networks with leakage, transmission and distribution delays

<abstract><p>This paper studies the global exponential stability problem of quaternion-valued neural networks (QVNNs) with leakage, transmission, and distribution delays. To address this issue, a direct method based on system solutions is proposed to ensure the global exponential stability of the considered network models. In addition, this method does not need to construct any Lyapunov-Krasovskii functional, which greatly reduces the amount of computation. Finally, a numerical example is given to demonstrate the effectiveness of the proposed results.</p></abstract>