Sufficient Conditions For Stability Research Articles

GLOBAL STABILITY OF AN SEIS EPIDEMIC MODEL WITH BEDDINGTON-DEANGELIS INFECTION RATE

In this paper, the global dynamic behaviors of an SEIS epidemic model with Beddington-DeAngelis infection rate are investigated. This model takes into account constant recruitment, death from the disease, and the phenomenon of latency. Under some hypotheses, it is shown that the global dynamics is determined by the basic reproduction number $R_0$. If $R_0$ is less than unity, the disease-free equilibrium is both locally and globally stable and the disease dies out. If $R_0$ is greater than unity, sufficient conditions for the global stability of the endemic equilibrium are obtained by the geometric approach of Li and Muldowney. Some numerical simulations are also presented to confirm the analytical results.

Asymptotic stability of fractional order switching nonlinear system based on short memory principle

AbstractFractional order derivatives have memory effects and are widely used in real world applications. However, they require large storage space and lead to low computational efficiency. Therefore, fractional order systems based on the short memory principle have gradually attracted the scholars' attention. In this paper, the asymptotic stability of Caputo fractional order switching nonlinear systems is investigated based on the Markov process and short memory principle. Firstly, a model of Caputo fractional order Markovian switching nonlinear systems (CFMNSs) based on the short memory principle is constructed so that the lower bound initial time and the corresponding initial state values are updated synchronously with switching. Secondly, the stability of the system is investigated based on the probabilistic analysis method and stochastic multi‐Lyapunov functions and the sufficient conditions for the asymptotic stability of the system are given. Using a similar method, we also study the asymptotic stability of CFMNs with variable fractional order. Finally, the simulation results show that the proposed stability scheme is effective and reasonable.

Graph-based necessary and sufficient conditions for exponential stability of switched positive systems with marginally stable subsystems

This paper derives necessary and sufficient (N&S) conditions for the exponential stability of discrete-time switched positive linear systems under transfer-restricted switching. The transfer-restricted switching property is characterized by a switching digraph, and the structural properties for subsystems are characterized by a novel class of state component digraphs. Combining with the two properties, some joint path conditions involving the sum and product matrices of multiple subsystems respectively are presented as N&S conditions based on weak common linear co-positive Lyapunov functions (weak-CLCLFs). The presented conditions allow all subsystems to be marginally stable rather than asymptotically stable. Finally, a simulation example is provided to show the effectiveness and superiority of the proposed method.

Leader-Following Consensus of Linear Multiagent Systems With Aperiodically Sampled Outputs: A Distributed Impulsive-Observer-Based Approach.

This article studies the leader-following consensus problem for a class of linear multiagent systems over a directed graph with aperiodically sampled outputs. First, a novel distributed impulsive-observer-based consensus protocol is designed. This protocol requires only the output measurements at sporadic time instants for the observer and control gain design. Second, by using time-varying Lyapunov function techniques, sufficient conditions for exponential stability of a class of linear impulsive systems are established; subsequently, these stability results are applied for the distributed impulsive observer design. Different from the existing related works, the impulsive observer gain designed in this work is decoupled from the graph properties. As a result, once the impulsive observer gain is designed for one network topology, it can be directly used for other network topologies, as long as the graph properties and the dynamics of local agents satisfy certain conditions. Furthermore, the resilience of the designed protocol is tested under denial of service (DoS) attacks. It is shown that the protocol is robust with respect to low-frequency DoS attacks occurring in the observer communication network. Finally, two examples illustrating the validity and effectiveness of the proposed protocol are included.

Uniform exponential stabilization for distributed discrete bilinear systems with time delay

In this paper we investigate the feedback stabilization problem for distributed discrete bilinear systems with time delay. Sufficient and necessary conditions for uniform exponential stabilization are given. Furthermore, under weaker conditions, the weak stabilization is achieved for the considered systems. Applications and illustrative simulations are presented.

Fault-Tolerant Time-Varying Formation Trajectory Tracking Control for Multi-Agent Systems with Time Delays and Semi-Markov Switching Topologies

The fault-tolerant time-varying formation (TVF) trajectory tracking control problem is investigated in this paper for uncertain multi-agent systems (MASs) with external disturbances subject to time delays under semi-Markov switching topologies. Firstly, based on the characteristics of actuator faults, a failure distribution model is established, which can better describe the occurrence of the failures in practice. Secondly, switching the network topologies is assumed to follow a semi-Markov stochastic process that depends on the sojourn time. Subsequently, a novel distributed state-feedback control protocol with time-varying delays is proposed to ensure that the MASs can maintain a desired formation configuration. To reduce the impact of disturbances imposed on the system, the H∞ performance index is introduced to enhance the robustness of the controller. Furthermore, by constructing an advanced Lyapunov–Krasovskii (LK) functional and utilizing the reciprocally convex combination theory, the TVF control problem can be transformed into an asymptotic stability issue, achieving the purpose of decoupling and reducing conservatism. Furthermore, sufficient conditions for system stability are obtained through linear matrix inequalities (LMIs). Eventually, the availability and superiority of the theoretical results are validated by three simulation examples.

On numerical stability of continued fractions

The paper considers the numerical stability of the backward recurrence algorithm (BR-algorithm) for computing approximants of the continued fraction with complex elements. The new method establishes sufficient conditions for the numerical stability of this algorithm and the error bounds of the calculation of the $n$th approximant of the continued fraction with complex elements. It follows from the obtained conditions that the numerical stability of the algorithm depends not only on the rounding errors of the elements and errors of machine operations but also on the value sets and the element sets of the continued fraction. The obtained results were used to study the numerical stability of the BR-algorithm for computing the approximants of the continued fraction expansion of the ratio of Horn's confluent functions $\mathrm{H}_7$. Bidisc and bicardioid regions are established, which guarantee the numerical stability of the BR-algorithm. The obtained result is applied to the study of the numerical stability of computing approximants of the continued fraction expansion of the ratio of Horn's confluent function $\mathrm{H}_7$ with complex parameters. In addition, the analysis of the relative errors arising from the computation of approximants using the backward recurrence algorithm, the forward recurrence algorithm, and Lenz's algorithm is given. The method for studying the numerical stability of the BR-algorithm proposed in the paper can be used to study the numerical stability of the branched continued fraction expansions and numerical branched continued fractions with elements in angular and parabolic domains.

Dynamics of a non-autonomous delay mosquito population suppression model with Wolbachia-infected male mosquitoes

In this paper, we develop a non-autonomous delay differential equation model for mosquito population suppression. After establishing the positiveness and boundedness of the solutions, we study the dynamical behaviours of the model with or without Wolbachia-infected male mosquitoes. More specifically, for the model without infected male mosquitoes, we analyse the asymptotic stability of the equilibria and demonstrate that the model undergo Hopf bifurcations under certain conditions. For the model incorporating infected male mosquitoes, we derive sufficient conditions for the global asymptotic stability of the origin. Numerical examples are provided to illustrate and support our theoretical findings.

Stability analysis of a seabed under combined action of waves and seismic loading

The stability analysis of a seabed under wave and seismic loading is conducted within the framework of yield design theory using the pseudo-static method. The strength properties of the constitutive material are modeled by means of a purely cohesive condition. The loading associated with water waves is modeled through the linear wave theory, whereas the inertial forces induced by the passage of seismic waves is addressed in the framework of pseudo-static method. The material cohesion that increases linearly with dept is adopted in the analysis. A sufficient stability condition is obtained through the static approach of limit analysis, while a necessary condition is determined from the kinematic approach. The effects of seafloor surface inclination, as well as seismic intensity, are analyzed through pseudo-static coefficients. The determination of the limit load obtained through the pseudo-static approach is based on existing literature, particularly when seismic coefficients are not considered.

Asymptotic Stability of Delayed Boolean Networks With Random Data Dropouts.

In real networks, communication constraints often prevent the full exchange of information between nodes, which is inevitable. This brief investigates the problem of time delay and randomly missing data in Boolean networks (BNs). A Bernoulli random variable is assigned to each node to characterize the probability of data packet dropout. Time delay and missing data are modeled by independent random variables. A novel data-sending rule that incorporates both communication constraints is proposed. An augmented system, comprising current states, delayed information, and successfully transmitted data, is established for theoretical analysis. Using the semitensor product (STP), the necessary and sufficient condition for asymptotic stability of delayed BNs with random data dropouts is derived. The convergence rate is also obtained.

An explicit discrete-time dynamic vehicle model with assured numerical stability

Numerical stability is of great significance for discrete-time dynamic vehicle model. Among the unstable factors, low-speed singularity stands out as one of the most challenging issues, which arises from that the denominator of tire side angle term only contains the vehicle longitudinal speed. Consequently, for the common low-speed and stop-start driving scenarios, the calculated tire slip angle will approach infinity, which will further lead to the numerical explosion of other vehicle states. In response to this critical challenge, we propose a discrete-time dynamic vehicle model that effectively mitigates the low-speed singularity issue, ensuring numerical stability and maintaining the explicit form – highly favoured by model-based control algorithms. To validate the numerical stability of our model, we conduct a rigorous theoretical analysis, establishing sufficient conditions for stability, and conduct extensive empirical validation tests across a wide spectrum of speeds. Subsequent to the validation process, we conduct comprehensive simulations comparing our proposed model with both kinematic models and existing dynamic models discretised through the forward Euler method. The results demonstrate that our proposed model shows better comprehensive performance in terms of both the accuracy and numerical stability. Finally, the real vehicle experiments are carried out to support that our proposed model can closely aligns to the real vehicle trajectories showcasing its practicality and ease of use. Notably, our work stands as the pioneering endeavour in introducing an explicit discrete-time dynamic vehicle model suitable for common urban driving scenarios including low-speed and stop-start.

Dynamics of a Memory‐Based Diffusion Model With Maturation Delay and Spatial Heterogeneity

ABSTRACTIn this paper, we consider a single memory‐based diffusion population model with maturation delay, spatial heterogeneity, and Neumann boundary condition. When the integral of the intrinsic growth rate over the domain is nonnegative, we obtain sufficient conditions for the local stability of the positive steady state and the critical values of maturation delay for the associated Hopf bifurcation. When the integral of the intrinsic growth rate over the domain is negative, considering that the characteristic equation involves a non‐self‐adjoint operator and two delays, we utilize a geometric method to determine all bifurcation points in terms of memory and maturation delays. The impact of spatial heterogeneity on the distribution of solutions is also examined via numerical simulations. It is found that the core area of high population density is coincident with the source area of growth rate. This suggests the importance of spatial heterogeneity in shaping the distribution and dynamics of the species.

On the Global Existence of Solution of the Comparison System and Vector Lyapunov Asymptotic Eventual Stability for Nonlinear Impulsive Differential Systems

In this paper, the existence of maximal solution of the comparison system as well as the asymptotic eventual stability of nonlinear impulsive differential equations with fixed moments of impulse is examined using the vector Lyapunov functions which is generalized by a class of piecewise continuous Lyapunov functions. It was distinctly established that the maximal solution of the comparison system majorizes the vector form of the Lyapunov functions. The novelty in the use of the vector Lypunov functions lies in the fact that the "restrictions" encountered by the scalar Lyapunov function is safely handled especially for large scale dynamical systems, since it involves splitting the Lyapunov functions into components so that each of the components can easily describe the behavior of the solution state. Together with comparison results, sufficient conditions for eventual stability and asymptotic are presented.

An aperiodically intermittent control for finite-time and fixed-time synchronization of stochastic FCNN with switching parameters

The main objective of this paper is to design an aperiodically intermittent control to ensure the finite and fixed time synchronization of fuzzy cellular neural networks (FCNNs) involving switching parameters with threshold properties, time-dependent discrete, continuous-type delays, and stochastic disturbances during transmission. Cellular neural networks (CNNs), with their grid-like structure, excel in image-processing tasks. Incorporating the fuzziness in the CNNs may handle the uncertainties and incomplete information in an effective manner. Furthermore, this paper introduces the concept of user-controlled FCNNs, which retain the dynamic properties of conventional FCNNs while incorporating significant factors that could potentially degrade the performance of the considered model. Due to randomness, FCNNs are framed as a set of stochastic differential equations with memristors. Generally, a memristor functions as a non-volatile memory device, reserving past data and utilizing it in the event of transmission failure. Further, the synchronization process is an effective approach that helps to ensure the dynamical behaviors of two different models; hence, the paper formulates the synchronization problem for the FCNN model with control (response) and without control input (drive). In terms of control design, this paper also considers the aperiodically intermittent control (APIC) scheme that can drive the user-controlled FCNN solution trajectories onto conventional FCNN solution trajectories within a prescribed time. Additionally, the APIC scheme incorporates an adaptive mechanism that effectively manages switching behaviors and uncertainties. Furthermore, this study proposes a theoretical framework in the form of sufficient stability conditions that helps to ensure the synchronization of the drive-response model via the global asymptotical stability of the stochastic error model derived from the drive-response model under the APIC scheme in the mean square. The proposed stochastic FCNN model is numerically simulated, and the corresponding outcomes are graphically illustrated.

Global exponential stability of nonlinear time-delay system under impulsive control with distributed delay

This paper investigates the global exponential stability of nonlinear time-delay system (NTDS) under impulsive control with distributed delay. By establishing a connection between the distributed delay in impulsive control and the number of impulsive times, a novel Razumikhin-type impulsive inequality is formulated. Using this inequality and the concept of the average impulsive interval, some delay-independent sufficient conditions of the global exponential stability of NTDS are derived. This indicates that the upper bounds of the impulsive interval and state time delay are no longer constrained, and the mutual constraint between the distributed delay in impulse and state time delay is removed, which greatly improves the results of recent studies. Finally, two numerical examples are presented to showcase the validity and practicality of the results.

Lyapunov-like conditions for prescribed-time stability of perturbed impulsive systems

The present work deals with the problem of prescribed-time control of non-linear impulsive systems consisting of external perturbations. Lyapunov-like sufficient conditions for prescribed-time and practical prescribed-time stability are provided for vanishing and non-vanishing perturbations, respectively. Depending on the user's requirements, some sequences of stabilizing impulses are constructed in this regard. It is shown that the systems consisting of vanishing-type disturbances can attain prescribed-time stabilization at the origin. On the other hand, in the presence of non-vanishing disturbances, which can consist of bounded or unbounded disturbances, the trajectories of the system can enter a stable region only within the prescribed time. Moreover, the stable region is independent of the impulsive strength and the prescribed-time. The efficacy of the proposed results is provided through various examples and their numerical simulations.

An active queue management method for multiple bottleneck networks based on distributed average tracking

The active queue management (AQM) is a key technology for the information infrastructure and the Internet. The AQM for multiple bottleneck networks is especially challenging than that for single-bottleneck networks because the stability analysis of multiple bottleneck networks is more complex. In this paper, the active queue management problem for multiple bottleneck networks is considered. A novel random early detection algorithm is designed based on distributed average tracking technology, which is used to estimate the global queue length and improve the network's performance. Sufficient conditions for global asymptotic stability of the closed-loop active queue management system are presented, and some simulation results with NS-2 are given. Compared with the RED algorithm, simulation results show that the presented algorithm has smaller oscillations at routers, which highlights its effectiveness.

STABILITY OF SOLUTIONS TO SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONSOF NEUTRAL TYPE WITH INFINITE DISTRIBUTED DELAY

A class of systems of linear differential equations of neutral type with infinite distributed delay and periodic coefficients is considered. Using the Lyapunov-Krasovskii functional, sufficient conditions for exponential stability of the zero solution are obtained, estimates of solutions characterizing exponential decrease at infinity are established, conditions for perturbations of the coefficients of the system, under which exponential stability is preserved, are specified.

On the Global Existence of Solution and Lyapunov Asymptotic Practical Stability for Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle

This paper examines the existence of maximal solution of the comparison differential system and also establishes sufficient conditions for the asymptotic practical stability of the trivial solution of a nonlinear impulsive Caputo fractional differential equations with fixed moments of impulse using the vector Lyapunov functions. First, it was discovered that the vector form of the Lyapunov function was majorized by the maximal solution of the comparison system. From the results obtained, it was also established that the main system is asymptotically practically stable in the sense of Lyapunov.

Minimum-Time Control of a Linear System With Input Saturation: A Practical Approach

Abstract A simple algorithm is presented to find near-minimum-time control inputs for a controllable continuous-time linear time-invariant system with any number of states and inputs. The well-known digital deadbeat control algorithm is modified to satisfy input constraints, and the design issue is the selection of sampling period to minimize the time required to reach the desired final state. During each sampling period, the input required to place all the poles at the origin of z-plane (deadbeat control) via state feedback is computed. If the infinity norm of the deadbeat input exceeds the input constraint, the input is found by placing the poles as close to the origin as possible. A sufficient condition for the system stability is developed. Numerical examples illustrate that this algorithm can lead to true time-optimal control or near time-optimal control.