Sufficient Conditions For Finite-time Stability Research Articles

Finite-time stabilization of nonlinear time-varying systems involving impulsive action

This article investigates the finite-time stabilization of time-varying impulsive systems by time-varying feedback control. Moreover, the stabilizing and destabilizing effects of impulses on the stability of the system are considered, where the stabilizing impulses are used to regulate the unstable continuous dynamics. In addition, it can be found that the local and global characteristics of the finite-time stability may be affected by the stabilizing impulses. By establishing the relationship among the time-varying gain functions, the system structure and impulsive time sequences, some sufficient conditions for finite-time stability of time-varying systems involving impulsive action are proposed. Furthermore, based on the results mentioned above, a new class of finite-time controllers is designed for time-varying affine systems with external noises. The effectiveness of the proposed results is illustrated by three examples.

Finite-Time Stabilization and Optimal Feedback Control for Nonlinear Discrete-Time Systems

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Sufficient conditions for finite-time stability have recently been developed in the literature for discrete-time dynamical systems. In this article, we build on these results to develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finite-time stability and finite-time stabilization for nonlinear discrete-time controlled dynamical systems. Finite-time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that satisfies a difference inequality involving fractional powers and a minimum operator. This Lyapunov function can clearly be seen to be the solution to a difference equation that corresponds to a steady-state form of the Bellman equation, and hence, guaranteeing both finite-time stability and optimality. Finally, a numerical example is presented to demonstrate the efficacy of the proposed finite-time discrete stabilization framework.

Formation of a nontrivial finite-time stable attractor in a class of polyhedral sweeping processes with periodic input

We consider a differential inclusion known as a polyhedral sweeping process. The general sweeping process was introduced by J.-J. Moreau as a modeling framework for quasistatic deformations of elastoplastic bodies, and a polyhedral sweeping process is typically used to model stresses in a network of elastoplastic springs. Krejčí’s theorem states that a sweeping process with periodic input has a global attractor which consists of periodic solutions, and all such periodic solutions follow the same trajectory up to a parallel translation. We show that in the case of polyhedral sweeping process with periodic input the attractor has to be a convex polyhedron χ of a fixed shape. We provide examples of elastoplastic spring models leading to structurally stable situations where χ is a one- or two- dimensional polyhedron. In general, an attractor of a polyhedral sweeping process may be either exponentially stable or finite-time stable and the main result of the paper consists of sufficient conditions for finite-time stability of the attractor, with upper estimates for the settling time. The results have implications for the shakedown theory.

Finite-Time Stability for Caputo Nabla Fractional-Order Switched Linear Systems

In this paper, we address the finite-time stability problem of Caputo nabla fractional-order switched linear systems with α∈(0,1). Firstly, the monotonicity of the discrete Mittag-Leffler function is proposed. Secondly, under the per-designed switching rules, the form of the solution for Caputo nabla fractional-order switched linear systems is obtained by using the discrete unit step function. On the above basis, some sufficient conditions of finite-time stability for Caputo nabla fractional-order switched linear systems are proposed, according to the discrete Grönwall inequality and the monotonicity of the discrete Mittag-Leffler function. Finally, simulation verification is carried out via three numerical examples.

Robust Finite-time H∞ Control for Discrete-time Nonlinear Uncertain Singular Systems with Time-delay

This paper deals with the problems of finite-time H∞ control for a class of discrete-time nonlinear singular systems subject to uncertainties and external disturbance. Firstly, the finite-time stability problem is discussed for discrete-time nonlinear uncertain singular systems with time-delay. The sufficient conditions of finite-time stability of discrete-time nonlinear singular systems are established. Then, by using the Lyapunov functional method, a criterion is established to ensure that the closed-loop system with external disturbance is finite-time bounded. We design the controller gain matrix. Finally, we provide a numerical example to illustrate the effectiveness of the proposed results.

New conditions for finite‐time stability of impulsive dynamical systems via piecewise quadratic functions

In this paper, the use of time-varying piecewise quadratic functions is investigated to characterize the finite-time stability of state-dependent impulsive dynamical linear systems. Finite-time stability defines the behavior of a dynamic system over a bounded time interval. More precisely, a system is said to be finite-time stable if, given a set of initial conditions, its state vector does not exit a predefined domain for a certain finite interval of time. This paper presents new sufficient conditions for finite-time stability based on time-varying piecewise quadratic functions. These conditions can be reformulated as a set of Linear Matrix Inequalities that can be efficiently solved through convex optimization solvers. Different numerical analysis are included in order to prove that the presented conditions are able to improve the results presented so far in the literature.

Finite time stability for nonsingular impulsive first order delay differential systems

This primer article focuses on the representation of solutions and finite-time stability of impulsive first-order delay differential systems. We define delayed matrix function with impulses and use variation of parameters to obtain a representation of solutions of linear systems with impulse effects. The famous classical Grownwall inequalities and properties of delayed matrix exponential with impulses are used to develop sufficient conditions for finite-time stability. In the end, we provide some examples to support the results.

Further results on finite-time stability of neutral nonlinear multi-term fractional order time-varying delay systems

In this paper, the finite-time stability for nonlinear neutral multi-term fractional order systems with time-varying input and state delays is investigated. By use of the generalized Gronwall inequality and extended form of the generalized Gronwall inequality, new sufficient conditions for finite-time stability of such systems are obtained. Finally, numerical examples are given to illustrate the effectiveness and applicability of the proposed theoretical results.

Finite-time stabilization of nonlinear systems via impulsive control with state-dependent delay

This paper focuses on the issue of finite-time stability for a general form of nonlinear systems subject to state-dependent delayed impulsive controller. Based on the Lyapunov theory and the impulsive control theory, sufficient conditions for finite-time stability (FTS) and finite-time contractive stability (FTCS) are obtained. Additionally, we apply theoretical results to finite-time synchronization of chaotic systems and design the effective state-dependent delayed impulsive controllers in terms of techniques of linear matrix inequality (LMI). Finally, we present two numerical examples of finite-time synchronization of cellular neural networks and Chua’s circuit to verify the effectiveness of our results.

A survey on non-instantaneous impulsive fuzzy differential equations involving the generalized Caputo fractional derivative in the short memory case

In this paper, the existence results of the solution and the finite-time stability (FTS) are focused for fractional fuzzy differential equations (FFDEs) involving non-instantaneous impulsive effects and perturbance parameters. In view of the definitions of short-memory fractional derivatives and a short-memory fractional modeling approach, the existence of a unique solution is proved through utilizing the fixed point theorem of a weakly contractive mapping on the partially ordered fuzzy space, and the sufficient condition of FTS is obtained by using the generalized fractional Gronwall inequality. The fuzzy fractional derivatives used in this study include the concept of derivatives based on the generalized Hukuhara difference (gH-difference) and the granular difference (gr-difference). It is demonstrated that the obtained results based on the gr-difference can overcome restrictions associated with the previous approaches defined via the gH-difference concept. Finally, some numerical examples are provided to verify our main results.

Finite-time stability and optimal control of a stochastic reaction-diffusion model for Alzheimer’s disease with impulse and time-varying delay

With increased longevity in a nowadays society, Alzheimer’s disease (AD), an incurable neurodegenerative disease, becomes a serious threat to human health. To address the challenges to understanding and predicting AD development and treatment, we focus on the development of a stochastic reaction-diffusion model with time-varying delay and impulsive perturbations, which is driven by Le´vy jump process to model the in vivo progression of AD. Moreover, based on an bounded impulsive interval method, certain sufficient conditions of finite-time stability are provided. Further, from a cost-benefit perspective, an optimal control problem of AD is formulated to minimize the pathogenic proteins and control cost. Several illustrative examples are presented to demonstrate and verify theoretical results of this study.

Stability analysis of switched stochastic delay system with unstable subsystems

This paper investigates the stability of switched stochastic delay systems (SSDSs) with unstable subsystems. By using the multiple Lyapunov–Krasovskii functionals (MLKFs) method combined with the mode-dependent average dwell time of stable subsystems (MDADTSS) and the mode-dependent average dwell time of unstable subsystems (MDADTUS) method, SSDSs with unstable subsystems are analyzed, and several novel sufficient conditions for finite-time stability and pth moment exponential stability are given. Compared with the existing related works, our work considers the influences of random disturbance and delays on the stability of SSDSs. Finally, we provide two examples to illustrate the feasibility of this method.

Novel conditions for finite-time stability of neural networks with time varying delay

In this paper, the problem of finite-time stability of neural networks with time varying delay is considered by using improved reciprocally convex inequalities and some integral inequalities, combined with Wirtinger’s inequalities and some useful integral inequality. New sufficient conditions for finite-time stability are established to guarantee finite-time stability for the system are given in terms of linear matrix inequalities(LMIs). Numerical examples are given to illustrate the effectiveness of the obtained result.

Finite-time stability analysis of fractional-order memristive fuzzy cellular neural networks with time delay and leakage term

In this paper, we investigate finite-time stability analysis of fractional-order memristive fuzzy cellular neural networks(MFFCNNs) with time delay and leakage term. MFFCNNs are formulated by virtue of differential inclusion and set-valued map theories. By using generalized Bernoulli inequality and Holder inequality, we derived some new sufficient conditions of finite-time stability for the proposed system. Finally two numerical examples are given to show the effectiveness of the proposed approach.

Finite-time stability and optimal control of an impulsive stochastic reaction-diffusion vegetation-water system driven by Lévy process with time-varying delay.

In this paper, a reaction-diffusion vegetation-water system with time-varying delay, impulse and Lévy jump is proposed. The existence and uniqueness of the positive solution are proved. Meanwhile, mainly through the principle of comparison, we obtain the sufficient conditions for finite-time stability which reflect the effect of time delay, diffusion, impulse, and noise. Besides, considering the planting, irrigation and other measures, we introduce control variable into the vegetation-water system. In order to save the costs of strategies, the optimal control is analyzed by using the minimum principle. Finally, numerical simulations are shown to illustrate the effectiveness of our theoretical results.

Finite-time boundedness of two-dimensional positive continuous-discrete systems in Roesser model

This paper is concerned with the finite-time boundedness of two dimensional (2-D) positive continuous-discrete systems in Roesser model. By constructing an appropriate co-positive type Lyapunov function, sufficient conditions of finite-time stability for the nominal 2-D positive continuous-discrete system are established. Sufficient conditions of finite-time boundedness for the addressed system with external disturbances are also proposed. The proposed results are then extended to uncertain cases, where the interval and polytopic uncertainties are considered respectively. Finally, three examples are provided to illustrate the effectiveness of the proposed results.

Observer-based finite-time control of linear fractional-order systems with interval time-varying delay

In this paper, the problem of finite-time observer-based control for linear fractional-order systems with interval time-varying delay is studied. The delay is assumed to vary within an interval with known lower and upper bounds. A new proposition on estimating Caputo derivatives of quadratic functions is given. Based on the proposed result, delay-dependent sufficient conditions for finite-time stability and for designing state feedback controllers and observer gains for observer-based control problem are established in terms of a tractable linear matrix inequality and Mittag–Leffler function. A numerical example with simulation is given to demonstrate the effectiveness of the proposed method.

Finite-Time Control Analysis of Nonlinear Fractional-Order Systems Subject to Disturbances

This paper deals with finite-time control problem for nonlinear fractional-order systems with order $$0<\alpha <1$$ . We first derive sufficient conditions for finite-time stabilization based on Caputo derivative calculus and Lyapunov-like function method. Then, by introducing a new type of the cost control function, we study guaranteed cost control problem for such systems. In terms of linear matrix inequalities, an explicit expression for state and output feedback controllers is given to make the closed-loop system finite-time stable and to guarantee an adequate cost level of the performance. The proposed method is applied to analyze the finite-time control problem for a class of linear uncertain FOSs. Finally, numerical examples are given to illustrate the validity and effectiveness of the proposed results.

Finite-time stability of spring-mass system with unilateral impact constraints and frictions

In this paper, a spring-mass system with impacts and frictions is formulated by the impulsive differential system. An energy-like Lyapunov function and an auxiliary step function are constructed to analyze the finite-time stability of such impact system with a time-varying external force and sliding friction as well as air resistance. We establish the sufficient conditions of finite-time stability for three cases of the spring-mass system, and present numerical simulations for each case to verify the validity of the theoretical results.

Finite-Time Stability and Stabilization of Discrete-Time Switching Markov Jump Linear System

Switching Markov jump linear system (SMJLS), a special hybrid system, has attracted a lot of studies recently. SMJLS is governed by stochastic and deterministic commutations. This paper focuses on the switching strategy which stabilizes the SMJLS in a finite time interval in order to further expand the existing results and investigate new aspects of such systems. Several sufficient conditions for finite-time stability of discrete-time SMJLS are provided, and the numerical problems in these sufficient conditions are solved by solving linear matrix inequalities (LMIs). Finally, numerical examples are given to show the feasibility and effectiveness of the results.