Conditions For Asymptotic Stability Research Articles

Quantized Event-Triggered Synchronization of Discrete-Time Chaotic Neural Networks With Stochastic Deception Attack

This article focuses on the event-triggered synchronization of delayed discrete-time chaotic neural networks with quantized effect and stochastic deception attack. First, for alleviating the network communication and communication burden, an event-triggered mechanism and a logarithmic quantizer are employed, separately. Second, for integrating the impact of event-triggered scheme, quantization, and cyberattack in a unified framework, a synchronization error model is introduced. Third, based on the Lyapunov–Krasvovskii functional (LKF), some sufficient conditions are established to guarantee the synchronization of drive system and response system. Furthermore, the co-design controller and homologous event-triggered parameters are also derived according to the presented asymptotic stability condition. Finally, the availability of the proposed method is verified by some numerical examples.

Stable nonlinear model predictive control with a changing economic criterion

This paper proposes a novel stable economic model predictive control (EMPC) strategy for constrained nonlinear systems with changing economic criteria. The traditional EMPC may lead to infeasibility and even instability of the closed-loop system when the criterion has been changed. Firstly, a generalised terminal constraint is introduced to ensure the recursive feasibility of the economic optimisation problem, which enforces the terminal states to converge to an arbitrary equilibrium state rather than a predetermined fixed one at the initial time. Then the stability constraint is constructed by solving an auxiliary optimisation problem online, which is the key to guaranteeing the asymptotic stability of the closed-loop system. Finally, sufficient conditions for feasibility and asymptotic stability are derived in the context of changing economic criteria. The main feature of the proposed strategy is that it does not need to know the changes of economic criterion in advance, and its properties and effectiveness are exemplified by simulations on a nonlinear chemical process example.

Bifurcations in a Plant-Pollinator Model with Multiple Delays

The plant-pollinator model is a common model widely researched by scholars in population dynamics. In fact, its complex dynamical behaviors are universally and simply expressed as a class of delay differential-difference equations. In this paper, based on several early plant-pollinator models, we consider a plant-pollinator model with two combined delays to further describe the mutual constraints between the two populations under different time delays and qualitatively analyze its stability and Hopf bifurcation. Specifically, by selecting different combinations of two delays as branch parameters and analyzing in detail the distribution of roots of the corresponding characteristic transcendental equation, we investigate the local stability of the positive equilibrium point of equations, derive the sufficient conditions of asymptotic stability, and demonstrate the Hopf bifurcation for the system. Under the condition that two delays are not equal, some explicit formulas for determining the direction of Hopf bifurcation and some conditions for the stability of periodic solutions of bifurcation are obtained for delay differential equations by using the theory of norm form and the theorem of center manifold. In the end, some examples are presented and corresponding computer numerical simulations are taken to demonstrate and support effectiveness of our theoretical predictions.

Asymptotic behavior of fractional-order nonlinear systems with two different derivatives

This paper addresses the asymptotic behavior of systems described by nonlinear differential equations with two fractional derivatives. Using the Mittag–Leffler function, the Laplace transform, and the generalized Gronwall inequality, a sufficient asymptotic stability condition is derived for such systems. Numerical examples illustrate the theoretical results.

Data‐driven control for networked systems with multiple packet dropouts

AbstractThis paper investigates data‐driven control for a class of networked control systems with multiple packet dropouts and unknown system parameters. Here, multiple packet dropouts occur randomly in the controller‐to‐actuator (C/A) and sensor‐to‐controller (S/C) channels, where successive packet dropouts are limited by known upper bounds. By introducing the summation inequality, a model‐based mean‐square asymptotic stability condition is established for the closed‐loop networked control system with the potential to migrate to a data‐driven solution. Then, the data‐driven mean‐square asymptotic stability condition for the closed‐loop system is presented by merging the model‐based stability condition with noisy data‐parameterized representations. On this basis, the data‐driven controller gain design approaches for combining unknown and known input gain matrices are presented in turn. Finally, two simulation examples are provided to show the applicability of the developed approaches.

A Switching Strategy for Stabilization of Discrete-Time Switched Positive Time-Varying Delay Systems with All Modes Being Unstable and Application to Uncertain Data

The stability problem of switched systems plays an essential role in the study of long-term behavior. In fact, systems containing both time delay and uncertainty terms may lead to performance degradation of those systems. Therefore, we are interested in the robust stability for discrete-time switched positive time-varying delay systems with interval uncertainties in the case of all modes being unstable. Based on the proposed time-scheduled multiple co-positive Lyapunov–Krasovskii functional of each mode, new sufficient conditions for the global uniform asymptotic stability of the systems are derived. An effective time-dependent switching law utilized in this work is mode-dependent dwell time. In addition, the robust stability criteria in an asymptotic sense are formulated for the systems without time-varying delay. Compared with the existing related works, our results are less conservative and more general than some previous research. Finally, two numerical examples are provided to illustrate the effectiveness and correctness of the developed theoretical results.

A predator–prey model with additive Allee effect and intraspecific competition on predator involving Atangana–Baleanu–Caputo derivative

A mathematical model of an interaction between two populations namely prey and predator is studied based on a Gause-type predator–prey model involving the additive Allee effect and intraspecific competition on the predator. A famous fractional operator called Atangana–Baleanu–Caputo fractional derivative (ABC) is employed to integrate the impact of the memory effect on the dynamical behavior of the model. The existence, uniqueness, non-negativity, and boundedness of the solution are given to confirm the biological feasibility and validity of the model. Three types of equilibrium points are identified on the origin, axial, and interior including their existence conditions. The Lyapunov direct method for the ABC model is used to investigate the asymptotic stability condition for each equilibrium point. The numerical simulations are provided to demonstrate the impact of several biological parameters on the dynamics of the solutions. The emergence of transcritical, saddle–node, and backward bifurcations driven by the Allee constant are provided which resulted in the appearance of bistability condition. A Hopf bifurcation as well as the evolution of the limit-cycle also occurs as the impact of the memory effect. Each analytical and numerical result is biologically interpreted to show the way that the density of both populations always balances in their ecosystem.

Global asymptotic stability of switched boolean networks with missing data

This paper investigates the global asymptotic stability of a switched Boolean network (SBN) with missing data. Due to the physical constraints of the communication media, data loss often occurs during the transmission. In this paper, the data loss is described as a Bernoulli distribution sequence, and the switching signal sequence is determined by a logical system. Using a semi-tensor product, the dynamics of a SBN with missing data are converted to an algebraic form. Based on this, a necessary and sufficient condition for global asymptotic stability of SBNs with missing data and auxiliary theorems are provided. Two illustrative examples are presented to show the applicability of the developed theorems.

Obstructions to Asymptotic Stabilization

Necessary conditions for asymptotic stability and stabilizability of subsets for dynamical and control systems are obtained. The main necessary condition is homotopical and is in turn used to obtain a homological one. A certain extension is ruled out. Questions are posed.

Stochastic asymptotic stability for stochastic inertial Cohen-Grossberg neural networks with time-varying delay

This paper studies stochastic asymptotic stability for stochastic inertial Cohen-Grossberg neural networks with time-varying delay. Firstly, the second-order differential equation is converted into the first-order differential equation by appropriate variable substitution. Secondly, the existence of the equilibrium point is derived by using homeomorphic mapping, finite increment formula of Lagrange mean value theorem and linear matrix inequality. The sufficient conditions for the stochastic asymptotic stability of the equilibrium point of the system are derived by defining the appropriate operator, and constructing the appropriate positive Lyapunov function and positive-definite matrix. Thirdly, a numerical example illustrates the correctness of these theorems.

Asymptotic stability of non‐instantaneous impulsive systems and T‐S fuzzy non‐instantaneous impulsive control for nonlinear systems

AbstractThis paper studies the asymptotic stability for a class of non‐instantaneous impulsive systems and Takagi‐Sugeno (T‐S) fuzzy non‐instantaneous impulsive control for linear and nonlinear systems. First, a class of concrete comparison system for a more accurate and universal nonlinear non‐instantaneous impulsive model is constructed. Then, the less conservative sufficient conditions of Lyapunov asymptotic stability are studied by impulsive differential inequalities and generalized comparison principle. On the other hand, an improved non‐instantaneous impulsive control scheme with T‐S fuzzy model is also presented for the asymptotic stability of the nonlinear systems. Finally, simulation results on controlling a linear and a nonlinear system are presented to illustrate the rationality and effectiveness of the obtained results.

Spatiotemporal Tipping Induced by Turing Instability and Hopf Bifurcation in a Population Ecosystem Model with the Fear Factor

Population ecosystems can display the tipping points at which extinctions of species happens. To predict the appearance of tipping points and to understand their evolution mechanism are of uttermost importance for ecological balance. Using techniques from bifurcation theory, we can predict the emergence of tipping points based on a spatiotemporal predator-prey system having a fear effect before an instability is encountered. In the case of no time delay, the tipping induced by Turing instability is studied. The conditions for Turing instability and local asymptotic stability of coexisting equilibrium points are given. It is ascertained that the introduction of diffusion causes the ecosystem to change from stable to unstable, and then the tipping occurs. Then, we investigate another tipping due to Hopf bifurcation. The delay-dependent stability criterion and Hopf bifurcation condition are derived, and the onset of Hopf bifurcations (tipping point) is also determined. In order to further probe into the mechanism of tipping evolution, explicit formulae are derived to ascertain the stability of bifurcated oscillations and the direction of bifurcation via the center manifold reduction. It is revealed that many tipping points may exist in ecological competition systems, and the tipping occurs many times as the fear delay increases. Finally, several simulation examples are provided to substantiate the analytical results.

Delay-dependent and order-dependent asymptotic stability conditions for Riemann–Liouville fractional-order systems with time delays

Delay-dependent and order-dependent asymptotic stability conditions for Riemann–Liouville fractional-order systems with time delays

Stability Analysis and Control Strategy of a Generalized Food Chain Model

The stability analysis and control strategy for a generalized food chain model is considered in this paper. Firstly, we investigate the existence of the positive equilibrium and give the conditions of local asymptotic stability. Meanwhile, we design a feedback controller and prove the stability of the proposed model with the controller by using the Lyapunov function. Due to the addition of control, the unstable food chain system can become stable. At last, the analysis results are verified by numerical simulations. Our results may help to maintain the stability of the biological systems and avoid ecosystem imbalance.

Computationally Efficient Stability-Based Nonlinear Model Predictive Control Design for Quadrotor Aerial Vehicles

This article presents the design of a stability-guaranteed nonlinear model predictive controller for quadrotor-type microaerial vehicles to operate robustly on fast trajectories. The basic controller structure operates without having to use terminal conditions in the optimization problem. As a result, the controller is computationally less demanding and provides more stable closed-loop performance than traditional nonlinear predictive control schemes. This article presents a detailed stability analysis without terminal costs or terminal constraints and proves the asymptotic stability and necessary conditions for the recursive feasibility of the system. This article derives the growth-bound sequence that enables obtaining the shortest possible prediction horizon for stability. The proposed analysis provides the necessary conditions to implement the controller while using the shortest stabilizing prediction horizon compared to major traditional predictive control schemes reported in the literature. This particular feature enables the proposed controller to perform fast optimization and, hence, the capability to implement fast trajectories using feedback regularization. In order to demonstrate the validity of this new proposed control scheme, first, several MATLAB simulations are conducted to demonstrate the improved performance of the controller especially when the quadrotor vehicle follows fast trajectories. Real-time lab experiments are also conducted to validate the performance of the proposed scheme for point stabilization (hovering) and trajectory tracking problems. The results show that the proposed scheme can stabilize the system with a relatively short prediction horizon, a fast convergence rate, and a small tracking error.

Influence of dissipative asymmetry on the of rotation stability in a resisting medium of a asymmetric rigid body under the action of a constant moment in inertial reference frame

Assuming that the center of mass of an asymmetric rigid body is located on the third main axis of inertia of a rigid body, the influence of dissipative asymmetry on the stability of uniform rotation in a medium with resistance of a dynamically asymmetric rigid body is estimated. A rigid body rotates around a fixed point, is under the action of gravity, dissipative moment and constant moment in an inertial frame of reference. The stability conditions are represented by a system of three inequalities. The first and second inequalities have the first degree with respect to the dissipative asymmetry, and the third inequality has the third degree. The third inequality is the most difficult to study. Analytical studies of the influence of small and large dissipative asymmetries, restoring, overturning and constant moments on the stability of rotation of a rigid body are carried out. Conditions for asymptotic stability are obtained for sufficiently small values of the dissipative asymmetry and conditions for instability for sufficiently large values of the asymmetry. The stability conditions are written down to the second order of smallness with respect to the constant moment and the first - with respect to the restoring or overturning moments. Stability conditions for the rotation of a rigid body around the center of mass are studied.

Stability Criterion and Sharp Estimates for the “Super-Twisting” Algorithm

A new method for proving necessary and sufficient conditions for the global asymptotic stability of the “super-twisting” algorithm is given. The new method is based on obtaining a complete analytical solution of the system for the “worst-case” perturbation and permits one to obtain a criterion in a simpler, completely real form as well as to find estimates for the worst-case (majorizing) trajectory.

Controller design for discrete-time nonlinear feed-forward cascades with application to bacterial quorum sensing

This paper investigates the semi-global practical asymptotic stability (SPA stability) of a class of nonlinear feed-forward cascade systems. In particular, using general theories presented on the stabilization of sampled-data systems, a SPA stabilizing controller has been designed and the essential conditions for the semi-global asymptotic stability of this class of nonlinear systems have been presented. In doing so, using the approximated discrete-time model of a general form of feed-forward cascade systems in conjugation with the idea of cross-term constructed Lyapunov function, sampled data stabilizing conditions for the discretized system have been investigated and subsequently, the proper SPA stabilizing controller has been derived. To illustrate the effectiveness of the proposed scheme, the designed controller is applied on three examples. First, the framework has been applied to a nonlinear mathematical example and then to the well-known ball and beam system. In the end, the Quorum Sensing mechanism has been investigated as a novel application that extends the use of this set of frameworks to biological systems.

Exponential Stability of Nonlinear Time-Varying Delay Differential Equations via Lyapunov–Razumikhin Technique

In this article, some new sufficient conditions for the exponential stability of nonlinear time-varying delay differential equations are given. An extension of the classical asymptotical stability theorem in terms of a Lyapunov–Razumikhin function is obtained. The condition of non-positivity of the time derivative of a Razumikhin function is weakened. Additionally, the resulting sufficient asymptotic stability conditions allow us to guarantee uniform exponential stability and evaluate the exponential convergence rate of the system solutions. The effectiveness of the results is demonstrated by some examples.

Asymptotic stability for nonlinear time-varying systems on time scales

This paper is concerned with nonlinear time-varying systems on time scales, which can cover not only trivial continuous and discrete dynamical systems, but also some other nontrivial cases. Some less conservative sufficient conditions for asymptotic stability of nonlinear systems are derived. Compared with the traditional stability results, the negative definiteness of the time derivatives of Lyapunov functions is weakened, that is, the time derivatives of related Lyapunov functions in this paper are allowed to be indefinite. Two examples about trivial and nontrivial systems are included to illustrate the effectiveness of the results obtained.