Stability analysis in a second-order differential equation with delays

A second-order differential equation with finite discrete delays is considered. Local stability of the zero equilibrium is investigated, and we obtain some sufficient conditions for the zero equilibrium is stable or unstable. Moreover, it is found that there exist the local Hopf bifurcations of the system when the delay varies.     

Stability of some linear systems with delays

Asymptotic stability of a class of linear equations with arbitrary discrete and distributed delays is investigated. Both delay-independent and delay-dependent stability conditions are formulated in terms of existence of positive definite solutions to Riccati matrix equations. The approach of deriving various Riccati equations using the direct Lyapunov method is proposed     

具线性脉冲的微分系统的稳定性

研究了具有线性脉冲作用的常微分系统平衡态的稳定性问题.应用常数变易法和Lyapunov方法,证明了在文中假设的条件下,总存在线性脉冲控制使一般非自治常微分系统的平衡态是一致渐近稳定.将自治微分系统作为特殊情形,在更一般的条件下得到了相同结论.最后给出了自治微分系统和非自治微分系统的两个例子以说明所得结论.     

Stability criteria for linear systems with multiple time-varying delays

New delay-independent and deky-dependent stability criteria for linear systems with multiple time-varying delays are established by using the time-domain method. The results are derived based on a new-type stability theorem for general retarded dynamical systems and new analysis techniques developed in the author's previous work. Unlike some results in the literature, all of the established results do not depend on the derivative of time-varying dekys. Therefore, they are suitable for the case with very fast time-varying dekys. In addition, some remarks are also given to expkin the obtained results and to point out the limitations of the previous results in the literature.     

Stability of linear systems with interval time delays excluding zero

The stability of linear systems with multiple, time-invariant, independent and uncertain delays is investigated. Each delay is assumed to reside within a known interval excluding zero. A delay-free sufficient comparison system is formed by replacing the delay elements with parameter-dependent filters, satisfying certain properties. It is shown that robust stability of this finite dimensional parameter-dependent comparison system, guarantees stability of the original time-delay system. This result is novel in the sense that it does not require any a priori knowledge regarding stability of the time-delay system for some fixed delay. When the parameter-dependent filters are formed in a particular manner using Pade approximations, an upper bound on the degree-of-conservatism of the comparison system may be obtained, which is independent of the time-delay system considered. With this, it is shown that the conservatism of this comparison system may be made arbitrarily small. A linear matrix ineqaulity (LMI) formulation is presented for analysis of the stability of the parameter-dependent comparison system. In the single-delay case, an eigenvalue criterion is also available for stability analysis which incurs no additional conservatism.     

Stability crossing surfaces for linear time-delay systems with three delays

In this article, the stability of linear systems with time-delays is studied. The cases where the characteristic equations of the system include three delays are investigated. Using the geometrical relations in a normalised polynomial plane, a graphical method is presented to visualise the stability domains in the space of the time-delays. In this space, the points at which the characteristic equation has a zero on the imaginary axis (the border between stability and instability regions) are identified. These points form several surfaces called the ‘stability crossing surfaces’. This work extends the results of the previous works on the ‘stability crossing curves’ defined in the two-dimensional space (plane) of delays to a higher dimension and provides new geometric interpretations for the stability crossing conditions.     

Double-frequency oscillations of an impulse neuron equation with two delays

The dynamics of the generalized impulse neuron equation with two delays are studied. A local analysis of the loss of the stability of the nonzero equilibrium state is carried out. The phase reorganizations of the equation under study are numerically analyzed using the obtained asymptotic formulas.     

Stability of linear impulsive differential equations

The idea of the stability of linear impulsive differential equations at fixed moments is made more precise.     

Stability of linear impulsive differential equations

In the present paper the notion of stability of linear impulsive differential equations at fixed moments is made more precise.     

Further study of the linear regulator with disturbances--The case of vector disturbances satisfying a linear differential equation

In a previous paper [1], the conventional optimal linear regulator theory was extended to accommodate the case of external input disturbancesomega(t)which are not directly measurable but which can be assumed to satisfyd^{m+1}omega(t)/dt^{m+1} = 0, i.e., represented asmth-degree polynomials in timetwith unknown coefficients. In this way, the optimal controlleru^{0}(t)was obtained as the sum of: 1) a linear combination of the state variablesx_{i}, i = 1,2,...,n, plus 2) a linear combination of the first(m + 1)time integrals of certain other linear combinations of the state variables. In the present paper, the results obtained in [1] are generalized to accommodate the case of unmeasurable disturbancesomega(t)which are known only to satisfy a givenrhoth-degree linear differential equationD: d^{rho}omega(t)/dt^{rho} + beta_{rho}d^{rho-1}omega(t)/dt^{rho-1}+...+beta_{2}domega/dt + beta_{1}omega=0where the coefficientsbeta_{i}, i = 1,...,rho, are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulationx(t) approx 0in the face of any and every external disturbance functionomega(t)which satisfies the given differential equationD-even steady-state periodic or unstable functionsomega(t). An essentially different method of deriving this result, based on stabilization theory, is also described, In each cases the results are extended to the case of vector control and vector disturbance.     

Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations

The nonlinear delay differential equation with exponential and quadratic nonlinearities is considered. It is assumed that the equation is exposed to stochastic perturbations of the white noise type, which are directly proportional to the deviation of the system state from the equilibrium point. Sufficient conditions for stability in probability of the zero and positive equilibriums of the considered system under stochastic perturbations are obtained. The research results are illustrated by numerical simulations. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with an order of nonlinearity higher than one. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability of numerical procedures for a linear barotropic vorticity equation

The stability analysis of some commonly used numerical procedures for a finite difference linear barotropic vorticity equation is performed. Upper bounds of the time step to be used are given to maintain stable the numerical processes. In some special cases it is verified that these iterative procedures are block successive relaxation methods. The optimum process is an underrelaxation process. Numerical calculations support the analytical results.     

Partial differential equation modeling of malware propagation in social networks with mixed delays

With the wide applications of social networks, government and individuals increasingly emphasize information networks security. This paper is devoted to investigating a reaction–diffusion malware propagation model with mixed delays to describe the process of social networks. Applying matrix theory for characteristic values, we establish the local stability conditions of a positive equilibrium point. Based on the linear approximation method of nonlinear systems, the Hopf bifurcation at the positive equilibrium point is considered. Additionally, we identify some sensitive parameters in the process of malware propagation that are significant for control theory. Finally, numerical simulations are performed to illustrate the theoretical results.     

线性微分博弈系统的识别域判别

针对一般非线性系统识别域的判别定理难以被具体应用这一现象,利用非光滑分析理论,研究线性微分博弈系统的一个有界识别域问题,即如果一个闭集是识别域,则此集合的凸包也是识别域,从而得到一个多面体识别域。提出了该问题的一种有效算法,该算法简便易行,利用转换定理即可得到系统的胜利域。最后通过算例验证了所提出算法的可行性和有效性。     

Stability of linear stochastic differential delay systems

Sufficient conditions for stability with probability one are developed for a class of linear stochastic differential delay systems. The conditions take a form that is readily computable.     

Integration of second order linear differential equation with mixed boundary conditions

A fourth order finite difference scheme for obtaining an approximate solution of second order linear differential equation lacking the first derivative, with mixed boundary conditions, is presented. The convergence of the method is proved. A numerical illustration is included to demonstrate the practical usefulness of our method.     

Stability criteria for linear delay differential systems

In this paper, we investigate the stability properties of linear delay differential systems with constant coefficients. Several stability criteria are derived. Each of these results is expressed by a certain relation between the coefficient matrices A and B.     

Stability of linear stochastic delay differential equations with infinite Markovian switchings

This paper investigates the stability of linear stochastic delay differential equations with infinite Markovian switchings. Some novel exponential stability criteria are first established based on the generalized It formula and linear matrix inequalities. Then, a new sufficient condition is proposed for the equivalence of 4 stability definitions, namely, asymptotic mean square stability, stochastic stability, exponential mean square stability with conditioning, and exponential mean square stability. In particular, our results generalize and improve some of the previous results. Finally, two examples are given to illustrate the effectiveness of the proposed results.     

Stability of linear discrete switched systems with delays based on average dwell time method

This paper deals with the problem of exponential stability for a class of linear discrete switched systems with constant delays.The switched systems consist of stable and unstable subsystems.Based on the average dwell time method, some switching signals will be found to guarantee exponential stability of these systems.The explicit state decay estimation is also given in the form of the solutions of linear matrix inequalities(LMIs).An example relating to networked control systems(NCSs) illustrates the effect...