STABILITY ANALYSIS OF LARGE-SCALE SYSTEMS WITH MULTIPLE DELAYS

This paper is concerned with the stability analysis of a class of linear timeinvariant large-scale systems with multiple delays. By using the properties of matrices, a sufficient condition of absolute stability is first derived for low order systems, described by differential-difference equations. Then a comparison theorem is presented for differential-difference inequalities. Finally, a sufficient condition of absolute stability for large-scale systems with multiple delays is established by using the aggregation technique based on vector Lyapunov functions. A numerical example is also given to illustrate the applicability of the stability criterion obtained in this paper.     

关于运用Liapunov函数的滞后型系统部分稳定性的注释

Sufficient conditions for the stability with respect to part of the functional differential equation variables are given. These conditions utilize Lyapunov functions to determine the uniform stability and uniform asymptotic stability of functional differential equations. These conditions for the partial stability develop the Razumikhin theorems on uniform stability and uniform asymptotic stability of functional differential equations. An example is presented which demonstrates these results and gives insight into the new stability conditions.     

STABILIZED FINITE ELEMENT METHODS FOR THE REISSNER-MINDLIN PLATE

A new stabilized finite element method which is different from Hughes and Franco's (1988) is presented for the Reissner-Mindlin plate model. The least square mesh-dependent residual form of the shear constitute equation is added to the Partial Projection scheme to enhance the stability. Using piecewise polynomials of order k≥1 for the rotations, of order k+1 for the displacement and of order k-1 for the shear, the kth order error-estimates are obtained. Besides, our computing scheme can be also applied to some lower order elements. All error-estimates are obtained independent of the plate thickness, and the stability parameter is an arbitrary positive constant.     

The Partial Stability for Linear Discrete Systems

In this paper, we study the partial stability of linear discrete systems by means of Liapunov's functions of quadratic form. We obtain a necessary and sufficient condition for the system being stable with respect to part of variables and generalize Liapunov's equation to the partial stability of linear discrete systems. A method of constructing Liapunov's function of quadratic form for the stability of the systems is given.     

截断分层B样条基的Lp稳定性

The truncated hierarchical B-spline basis has been proposed for adaptive data fitting and has already drawn a lot of attention in theory and applications.However the stability with respect to the L_p-norm,1≤p∞,is not clear.In this paper,we consider the L_p stability of the truncated hierarchical B-spline basis,since the L_p stability is useful for curve and surface fitting,especially for least squares fitting.We prove that this basis is weakly L_p stable.This means that the associated constants to be considered in the stability analysis are at most of polynomial growth in the number of the hierarchy depth.     

STABILITY OF THEORETICAL SOLUTION AND NUMERICAL SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS WITH PIECEWISE DELAYS

This paper is concerned with the stability of theoretical solution and numerical solutionof a class of nonlinear differential equations with piecewise delays.At first,a sufficientcondition for the stability of theoretical solution of these problems is given,then numericalstability and asymptotical stability are discussed for a class of multistep methods whenapplied to these problems.     

Exponential Stability of Traveling Pulse Solutions of a Singularly Perturbed System of Integral Differential Equations Arising From Excitatory Neuronal Networks

We establish the exponential stability of fast traveling pulse solutions to nonlinear singularly per-turbed systems of integral differential equations arising from neuronal networks.It has been proved that expo-nential stability of these orbits is equivalent to linear stability.Let (?) be the linear differential operator obtainedby linearizing the nonlinear system about its fast pulse,and let σ((?)) be the spectrum of (?).The linearizedstability criterion says that if max{Reλ:λ∈σ((?)),λ≠0}(?)-D,for some positive constant D,and λ=0 is asimple eigenvalue of (?)(ε),then the stability follows immediately (see [13] and [37]).Therefore,to establish theexponential stability of the fast pulse,it suffices to investigate the spectrum of the operator (?).It is relativelyeasy to find the continuous spectrum,but it is very difficult to find the isolated spectrum.The real part ofthe continuous spectrum has a uniformly negative upper bound,hence it causes no threat to the stability.Itremains to see if the isolated spectrum is safe.Eigenvalue functions (see [14] and [35,36]) have been a powerful tool to study the isolated spectrum of the as-sociated linear differential operators because the zeros of the eigenvalue functions coincide with the eigenvaluesof the operators.There have been some known methods to define eigenvalue functions for nonlinear systems ofreaction diffusion equations and for nonlinear dispersive wave equations.But for integral differential equations,we have to use different ideas to construct eigenvalue functions.We will use the method of variation of param-eters to construct the eigenvalue functions in the complex plane C.By analyzing the eigenvalue functions,wefind that there are no nonzero eigenvalues of (?) in {λ∈C:Reλ(?)-D} for the fast traveling pulse.Moreoverλ=0 is simple.This implies that the exponential stability of the fast orbits is true.     

A New Approach to Synchronization Analysis of Linearly Coupled Map Lattices

In this paper, a new approach to analyze synchronization of linearly coupled map lattices (LCMLs) is presented. A reference vector x(t) is introduced as the projection of the trajectory of the coupled system on the synchronization manifold. The stability analysis of the synchronization manifold can be regarded as investigating the difference between the trajectory and the projection. By this method, some criteria are given for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to the eigenvalue "0" of the coupling matrix play key roles in the stability of synchronization manifold for the coupled system. Moreover, it is revealed that the stability of synchronization manifold for the coupled system is different from the stability for dynamical system in usual sense. That is, the solution of the coupled system does not converge to a certain knowable s(t) satisfying s(t 1) = f(s(t)) but to the reference vector on the synchronization manifold, which in fact is a certain weighted average of each xi(t) for i = 1, ... ,m, but not a solution s(t) satisfying s(t 1) = f(s(t)).     

A Notion of Stochastic Input-to-State Stability and Its Application to Stability of Cascaded Stochastic Nonlinear Systems

In this paper, the property of practical input-to-state stability and its application to stability of cascaded nonlinear systems are investigated in the stochastic framework. Firstly, the notion of （practical） stochastic input-to-state stability with respect to a stochastic input is introduced, and then by the method of changing supply functions, （a） an （practical） SISS-Lyapunov function for the overall system is obtained from the corresponding Lyapunov functions for cascaded （practical） SISS subsystems.     

SAMPLED-DATA STATE ESTIMATION FOR NEURAL NETWORKS WITH ADDITIVE TIME–VARYING DELAYS

In this paper, we consider the problem of delay-dependent stability for state estimation of neural networks with two additive time–varying delay components via sampleddata control. By constructing a suitable Lyapunov–Krasovskii functional with triple and four integral terms and by using Jensen's inequality, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities(LMIs) to ensure the asymptotic stability of the equilibrium point of the considered neural networks. Instead of the continuous measurement,the sampled measurement is used to estimate the neuron states, and a sampled-data estimator is constructed. Due to the delay-dependent method, a significant source of conservativeness that could be further reduced lies in the calculation of the time-derivative of the Lyapunov functional. The relationship between the time-varying delay and its upper bound is taken into account when estimating the upper bound of the derivative of Lyapunov functional. As a result, some less conservative stability criteria are established for systems with two successive delay components. Finally, numerical example is given to show the superiority of proposed method.     

Delay-dependent exponential stability of cellular neural networks with time-varying delays

The global exponential stability of cellular neural networks (CNNs) with time-varying delays is analyzed. Two new sufficient conditions ensuring global exponential stability for delayed CNNs are obtained. The conditions presented here are related to the size of delay. The stability results improve the earlier publications. Two examples are given to demonstrate the effectiveness of the obtained results.     

Stability analyses of cellular neural networks with continuous time delay

The stability of cellular neural networks (CNNs) with continuous time delay is investigated in the present paper. A main theorem together with a few corollaries related to the existence and boundedness of the solution for this kind of CNNs, as well as its global asymptotic stability is derived. Some numerical simulation examples aimed at justifying the theoretical results are also given.     

Decentralized feedback control of uncertain cellular neural networks subject to time-varying delays and dead-zone input

This paper investigates the stabilization problem for uncertain cellular neural networks (CNNs) subject to time-varying delays and dead-zone input. On the basis of Lyapunov stability theory, a memoryless decentralized feedback control law is derived for guaranteeing global exponential stability of the system. The main results illustrate that the derived control law does not impose restriction on the derivative of the time-varying delays and can be applied to stabilizing the uncertain CNNs with time-varying delays and dead-zone input. An illustrative example is given to justify the validity and feasibility of the proposed control scheme.     

Existence and global stability of a periodic solution for a cellular neural network

Purpose of this study is to investigate the dynamical properties of Chua and Yang cellular neural networks (CNNs). Based on the continuation theorem of coincidence degree theory, a novel sufficient condition with respect to the existence of periodic solution for CNNs is derived. Moreover, a generalized Lyapunov–Krasovskii functional is designed to guarantee the global stability of the existed periodic solution. An illustrative example is given to verify the effectiveness and correctness of the proposed method, furthermore, random disturbance is added in the numerical simulation in order to verify the robustness of the proposed approach.     

Edge detection of noisy images based on cellular neural networks

This paper studies a technique employing both cellular neural networks (CNNs) and linear matrix inequality (LMI) for edge detection of noisy images. Our main work focuses on training templates of noise reduction and edge detection CNNs. Based on the Lyapunov stability theorem, we derive a criterion for global asymptotical stability of a unique equilibrium of the noise reduction CNN. Then we design an approach to train edge detection templates, and this approach can detect the edge precisely and efficiently, i.e., by only one iteration. Finally, we illustrate performance of the proposed methodology from the aspect of peak signal to noise ratio (PSNR) through computer simulations. Moreover, some comparisons are also given to prove that our method outperforms classical operators in gray image edge detection.     

Stability in cellular neural networks with a piecewise constant argument

In this paper, by using the concept of differential equations with piecewise constant arguments of generalized type [1], [2], [3] and [4], a model of cellular neural networks (CNNs) [5] and [6] is developed. The Lyapunov-Razumikhin technique is applied to find sufficient conditions for the uniform asymptotic stability of equilibria. Global exponential stability is investigated by means of Lyapunov functions. An example with numerical simulations is worked out to illustrate the results.     

A topological approach to the existence of solutions for nonlinear differential equations with piecewise constant argument

In this paper, we investigate qualitative behavior of nonlinear differential equations with piecewise constant argument (PCA). A topological approach of Wa?ewski-type which gives sufficient conditions to guarantee that the graph of at least one solution stays in a given domain is formulated. Moreover, our results are also suitable for a class of general system of discrete equations. By using a regular polyfacial set, we apply our developed topological approach to cellular neural networks (CNNs) with PCA. Some new results are attained to reveal dynamic behavior of CNNs with PCA and discrete-time CNNs. Finally, an illustrative example of CNNs with PCA shows usefulness and effectiveness of our results.     

Improved global exponential stability criteria of cellular neural networks with time-varying delays

In this paper, the existence and uniqueness of the equilibrium point and stability of the cellular neural networks (CNNs) with time-varying delays are analyzed and proved. Several global exponential stability conditions of the neural networks are obtained by the delay differential inequality and matrix measures approach. The obtained results are extensions of the earlier literature. The approach used in this paper is also suitable for delayed Hopfield neural networks and delayed bi-directional associative memory neural networks whose activation functions are often nondifferentiable or unbounded. Two simulation examples in comparison to previous results in literature are shown to check the theory in this paper.     

Existence and global exponential stability of almost periodic solution for CNNs with variable coefficients and delays

By using exponential dichotomy and the Banach fixed-point theory and combining Yang inequality, some sufficient conditions are derived ensuring existence and global exponential stability of almost periodic solution for CNNs with variable coefficients and delays. Without assuming the boundedness of signal function, these results obtained here can be expected to have highly important significance in designs and applications of the networks. We extend and improve previously known results.     

The Effect of Clone Template Parameters on the Spreading Speeds in CNNs

The aim of this paper is to investigate the effect of clone template parameters on the spreading speeds in cellular neural networks(CNNs). According to the property analysis of spreading speeds of monotone semiflows developed by Yu and Zhang [{\it European Journal of Applied Mathematics}, {\bf 31} (2020), 369-384], we investigate the sign of spreading speeds, continuity and limit cases with no propagation phenomena for CNNs with general output functions where each cell interacts with its 2-neighborhood cell.