Exponential Stability Research Articles

Propagation phenomena for a nonlocal reaction-diffusion model with bounded phenotypic traits

In this paper we study the stability of cylinder front waves and the propagation of solutions of a nonlocal Fisher-type model describing the propagation of a population with nonlocal competition among bounded and continuous phenotypic traits. By applying spectral analysis and separation of variables we prove the spectral and local exponential stability of the cylinder waves with the noncritical speeds in some exponentially weighted spaces. By combining the detailed analysis with the spectral expansion and the special construction of sub-supersolutions, we further prove the uniform boundedness of the solutions and the global asymptotic stability of the cylinder waves for more general nonnegative bounded initial data, and prove that the spreading speeds and the asymptotic behavior of the solutions are determined by the decay rates of the initial data. Our results also extend some classical results on the stability of planar waves for Fisher-KPP equation to the nonlocal Fisher model in multi-dimensional cylinder case.

On the local quadratic stability of T–S fuzzy systems in the vicinity of the origin

The main goal of this paper is to introduce new local stability conditions for continuous-time Takagi-Sugeno (T-S) fuzzy systems. These stability conditions are based on linear matrix inequalities (LMIs) in combination with quadratic Lyapunov functions. Moreover, they integrate information on the membership functions at the origin and effectively leverage the linear structure of the underlying nonlinear system in the vicinity of the origin. As a result, the proposed conditions are proved to be less conservative compared to existing methods using fuzzy Lyapunov functions. Moreover, we establish that the proposed methods offer necessary and sufficient conditions for the local exponential stability of T-S fuzzy systems. Discussions on the inherent limitations associated with fuzzy Lyapunov approaches are also given. To illustrate the theoretical results, we provide comprehensive examples that demonstrate the core concepts and validate the efficacy of the proposed conditions.

Mean-square exponential stabilization of mixed-autonomy traffic PDE system

Control of mixed-autonomy traffic where Human-driven Vehicles (HVs) and Autonomous Vehicles (AVs) coexist on the road has gained increasing attention over the recent decades. This paper addresses the boundary stabilization problem for mixed traffic on freeways. The traffic dynamics are described by uncertain coupled hyperbolic partial differential equations (PDEs) with Markov jumping parameters, which aim to address the distinctive driving strategies between AVs and HVs. Considering that the spacing policies of AVs vary in mixed traffic, the stochastic impact area of AVs is governed by a continuous Markov chain. The interactions between HVs and AVs such as overtaking or lane changing are mainly induced by impact areas. Using backstepping design, we develop a full-state feedback boundary control law to stabilize the deterministic system (nominal system). Applying Lyapunov analysis, we demonstrate that the nominal backstepping control law is able to stabilize the traffic system with Markov jumping parameters, provided the nominal parameters are sufficiently close to the stochastic ones on average. The mean-square exponential stability conditions are derived, and the results are validated by numerical simulations.

Bounded real lemma and H∞ control for BAM neural networks with unbounded time‐varying delays

AbstractThis article investigates the exponential control issue of bidirectional associative memory neural network (BAMNN) with unbounded time‐varying delays. A bounded real lemma (BRL) is first established via a direct method, which is on the basis of the solutions of BAMNN. Second, based on the obtained BRL, the state feedback controller is designed to guarantee the global exponential stability of the resulting closed‐loop BAMNN with an performance index. Since no Lyapunov–Krasovskii functionals is constructed in the proposed method, the computation burden and complexity are reduced. Lastly, the effectiveness of the theoretical results is illustrated through two numerical examples.

Boundary stabilisation of stochastic delay impulsive Korteweg–de Vries–Burgers equations

Boundary stabilisation is addressed for stochastic delay impulsive Korteweg–de Vries–Burgers (SDIKdVB) equations. Initially, a novel boundary controller is devised. Utilizing the Lyapunov–Krasovskii functional approach and the impulsive theory, a sufficient condition is derived for achieving mean-square exponential stabilisation (MSES) of SDIKdVB equations. Secondly, the scenario is investigated in which uncertainties exist in the system parameters, and a sufficient condition is presented to attain the robust MSES. Additionally, the investigation is conducted on the impact of the diffusion term, the dispersion term, impulsive strengths and impulsive intervals on MSES. Lastly, the effectiveness of the theoretic findings is demonstrated via practical applications in both biomechanics and physics.

Analysis of energy dissipation in hyperbolic problems influenced by internal and boundary control mechanisms

ABSTRACT This article primarily focuses on the rational stabilisation of the wave equation, supplied with a second-order dynamical boundary condition of hyperbolic type, while considering an additional internal damping mechanism within the specified ring. To achieve rational decay rates of the associated energy, it is imperative to exponentially stabilise a portion of the domain using a global Ingham's-type estimate. This paper will subsequently illustrate how this partially localised exponential stabilisation, combined with a Bessel analysis, leads to a rational decrease in the overall energy of the system considered.

Stability, Uniqueness and Existence of Solutions to McKean–Vlasov Stochastic Differential Equations in Arbitrary Moments

We deduce stability and pathwise uniqueness for a McKean–Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz continuous drift and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients, we provide unique strong solutions even if the drift fails to be of affine growth. The theory that we develop rests on Itô’s formula and leads to pth moment and pathwise exponential stability for p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document} with explicit Lyapunov exponents.

Mode-dependent observer-based secured hybrid-driven reliable control for semi-Markov jump cyber–physical systems with DoS attacks

This study focuses on the issue of secured observer based hybrid-triggered quantized reliable control design for semi-Markov jump cyber–physical systems (S-MJCPSs) with general uncertain transition rates, quantization, actuator faults, denial-of-service (DoS) attacks and disturbances. In particular, a mode-dependent observer is developed to estimate the unmeasurable states of the considered system. Further, to minimize unnecessary data transmissions within the network channel, we have implemented a hybrid-triggered mechanism in which, the swap process between the time-triggered and the event-triggered mechanisms is governed by a Bernoulli distribution. Moreover, an observer-based hybrid-triggered quantized reliable controller is devised by utilizing the state estimation information obtained from the observer, which assists the S-MJCPSs to achieve primary intent of this study. Subsequently, the control input signals are quantized before being transmitted into the actuator due to the limited capacity of the data transmission medium. Concurrently, periodic DoS jamming attacks are taken into consideration, which have a significant impact on control performances. A set of adequate criteria is established in the context of linear matrix inequalities by employing the mode-dependent time-varying piecewise Lyapunov-Krasovskii functional and the integral inequalities, guaranteeing that the undertaken system accomplishes the global exponential stability and meet a specific H∞ performance level. Following that, by means of the deduced adequate criteria, the values of the controller and observer gain are computed. At last, the efficacy and applicability of the derived theoretical findings are demonstrated through two practical examples.

Adjustable order fault estimation observer based fault-tolerant control for switched fuzzy stochastic systems

Based on fault estimation, fault-tolerant control problem of switching stochastic nonlinear systems with actuator fault and sensor fault are studied. The fuzzy models are invited to solve the nonlinear problems of the systems. The fault estimation observer is designed to estimate the system states, actuator and sensor faults. The order of observer can be adjusted, which can improve the design flexibility and realize a compromise between the estimated cost and estimated accuracy. Unlike some classical results, the output differential is not required in the observer, which increases the application range. Based on the observer, the fault-tolerant controller is proposed, besides, the sufficient condition is achieved to ensure that the closed-loop system is mean-squared exponential stability. Finally, the feasibility is verified by two simulations.

Exponential stability of second order time delay semilinear systems with bilinear control

ABSTRACT This paper explores the stabilisation of delayed bilinear hyperbolic systems, examining scenarios with delays in bilinear control and state. Our contributions are twofold: First, we establish existence results under specific conditions for these systems. Second, we design bilinear controls ensuring exponential stability under Lipschitz nonlinear perturbations. For control delay, stability is demonstrated using Lyapunov's abstract function. In contrast, for state delay, we show an equivalence between the stabilisation of the delayed system and the observability of the corresponding undamped system. Our theoretical findings are exemplified through practical models, emphasising their applicability in real-world scenarios.

Existence and exponential stability in pth moment of non-autonomous stochastic integro-differential equations

This study is concerned with the existence and exponential stability of solutions of non-autonomous neutral stochastic integro-differential equations with delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic integro-differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences. Finally, we provide an example to illustrate the results.

Delay-dependent exponential stability of highly nonlinear stochastic functional switched systems: an average dwell time approach

The delay-dependent exponential stability of stochastic functional switched systems (SFSSs) with highly nonlinear coefficients is investigated in this paper. By virtue of the mode-dependent average dwell time (MDADT) technique, the existence of global solution and qth moment exponential stability is studied without linear growth condition (LGC). Both of drift and diffusion coefficients have functional terms which can embody many existing delay cases, e.g. constant delay, time-varying delay and distributed delay, etc. By employing common/multiple Lyapunov function method, sufficient delay-dependent stability criteria of qth moment exponential stability are proposed. The MDADT condition reveals the positive role of switching signal in the delay-dependent stability analysis. Finally, an example is given to validate our theoretical results.

Asynchronous control for nonlinear switched stochastic delayed systems with weighted mode‐dependent average dwell time

AbstractThis article investigates the asynchronous control for a class of nonlinear switched stochastic delayed systems. In response to this issue, the weighted mode‐dependent average dwell time is first applied to switched stochastic systems as a more general switching signal. In addition, the asynchronous phenomenon is considered to address the mismatch in practical applications, where “mismatch” means the switching of the controllers has a delay to the switching of system modes. Then by selecting new multiple Lyapunov–Krasovskii functionals, sufficient conditions for mean‐square exponential stability and an performance are derived. Finally, numerical examples are provided to illustrate the effectiveness of the proposed methods.

New global exponential stability conditions for nonlinear delayed differential systems with three kinds of time-varying delays

For a class of nonlinear differential systems with heterogeneous time-varying delays, including distributed, leakage and transmission time-varying delays, a novel global exponential stability (GES) analysis method was developed. Based on the GES definition, some sufficient conditions and rigorous convergence analysis of nonlinear delayed differential systems are presented directly, which ensure all states to be globally exponentially convergent. The proposed analysis method not only avoids the construction of the Lyapunov–Krasovskii functional, but also uses some simple integral reduction techniques to determine the global exponential convergence rate. Furthermore, the main advantages and low calculation complexity are demonstrated through a theoretical comparison. Finally, three numerical examples are provided to verify the effectiveness of the theoretical results.

On the equivalence between the uniform exponential stability of a $$C_0$$-semigroup and the boundedness of the resolvent

On the equivalence between the uniform exponential stability of a $$C_0$$-semigroup and the boundedness of the resolvent

Parameter-dependent conditions for H ∞ stabilisation of polynomial parameter-varying system with delay

This paper considers the problems of achieving exponential stability and H ∞ stabilisation for polynomial parameter-varying system with time-varying delay. Unlike some existing linear parameter-varying results that treat system parameters as time invariant factors, this paper makes full use of the nonlinearity and time-varying characteristics of system through introducing a novel delay-and-parameter-dependent Lyapunov function. The main conclusions are discussed in three cases (system without input, system with disturbance input, system with disturbance input and control input). And the H ∞ stabilisation problem is solved by constructing a controller which is inserted with system parameters and their derivatives. The structures of Lyapunov function and controller lead to that the derived criteria for exponential stabilisation are expressed by state-and-parameter-dependent linear matrix inequalities, which are efficiently solved by sum-of-squares program. The effectiveness and superiority are verified through two numerical examples.

Stability of highly nonlinear stochastic systems with non-random switching signals and multiple delays

The situation of multiple delays often occur in hybrid stochastic systems. In this paper, multiple time-varying delays are considered for highly nonlinear switching stochastic systems. The function of time-varying delays is differentiable, and its derivative is a positive constant less than 1. By the Lyapunov functional method and the mode-dependent average dwell time approach, existence and uniqueness of solution, asymptotic stability and pth moment exponential stability for highly nonlinear switching stochastic multiple delays systems are obtained. Finally, an example is afforded to explicate the effectiveness our theoretical analysis results.

Dynamic event-triggered dual-channel quantized sliding mode control for uncertain networked control systems under DoS attacks

This paper introduces a dynamic event-triggered sliding mode control method to address the issue of quantized stability in uncertain networked control systems (NCSs) subjected to random Denial-of-Service (DoS) attacks. Firstly, a dynamic event-triggered mechanism is utilized to compensate for the damage caused by DoS attacks while effectively reducing network bandwidth pressure. Subsequently, a sliding mode controller is designed to ensure the robustness of uncertain NCSs with external disturbances. Additionally, we construct a novel control framework that considers network delays and incorporates quantizers for state and control signals, respectively. Based on this framework, both a sufficient condition for exponential stabilization of NCSs and a co-design method for dynamic event trigger and controller are given. Finally, the effectiveness of the proposed method is verified with two generalized simulation examples.

A Novel Coupled Memristive Izhikevich Neuron Model and Its Complex Dynamics

This paper proposes a novel, five-dimensional memristor synapse-coupled Izhikevich neuron model under electromagnetic induction. Firstly, we analyze the global exponential stability of the presented system by constructing an appropriate Lyapunov function. Furthermore, the Hamilton energy functions of the model and its corresponding error system are derived by using Helmholtz’s theorem. In addition, the influence of external current and system parameters on the dynamical behavior are investigated. The numerical simulation results indicate that the discharge pattern of excitatory and inhibitory neurons changes significantly when the amplitude and frequency of the external stimulus current are applied at different degrees. And the crucial dynamical behavior of the neuronal system is determined by the intensity of modulation of the induced current and the gain in the electromagnetic induction. Moreover, the amount of Hamilton energy released by the model could be evaluated during the conversion between the distinct dynamical behaviors.

On exponential and L2-exponential stability of continuous-time delay-difference systems

In this paper, the exponential stability (ES) and L2-exponential stability (L2-ES) of continuous-time delay-difference systems are studied. Firstly, the relationship between ES and L2-ES of the studied system is systematically presented. Secondly, a novel Lyapunov stability theorem is proposed to test both the ES and L2-ES with a guaranteed convergence rate of the system. Then, for a particular class of delay-difference systems with both point delays and distributed delays having exponential integral kernels, some stability criteria based on linear matrix inequalities (LMIs) are established by selecting suitable Lyapunov-Krasovskii functionals (LKFs) and using a delay decomposition technique. Finally, a numerical example is worked out to illustrate the effectiveness of the theoretical results.