Linear Impulsive Differential Equations Research Articles

Regularity Coefficients for Impulsive Differential Equations

Abstract For a linear impulsive differential equation, we introduce a Lyapunov regularity coefficient following as far as possible the non-impulsive case. We recall that a regularity coefficient is a quantity that characterizes the Lyapunov regularity of the dynamics. In particular, we obtain lower and upper bounds for the Lyapunov regularity coefficient and we show that its computation can always be reduced to that of the corresponding coefficient of an impulsive dynamics defined by upper triangular matrices. We also relate the Lyapunov regularity coefficient with the Grobman regularity coefficient. Finally, we combine all the former results to establish a criterion for tempered exponential behavior in terms of the Lyapunov exponents and of the Lyapunov regularity coefficient.

Asymptotic Behavior and Stability in Linear Impulsive Delay Differential Equations with Periodic Coefficients

We study first order linear impulsive delay differential equations with periodic coefficients and constant delays. This study presents some new results on the asymptotic behavior and stability. Thus, a proper real root was used for a representative characteristic equation. Applications to special cases, such as linear impulsive delay differential equations with constant coefficients, were also presented. In this study, we gave three different cases (stable, asymptotic stable and unstable) in one example. The findings suggest that an equation that is in a way that characteristic equation plays a crucial role in establishing the results in this study.

Unpredictable Solutions of Linear Impulsive Systems

We consider a new type of oscillations of discontinuous unpredictable solutions for linear impulsive nonhomogeneous systems. The models under investigation are with unpredictable perturbations. The definition of a piecewise continuous unpredictable function is provided. The moments of impulses constitute a newly determined unpredictable discrete set. Theoretical results on the existence, uniqueness, and stability of discontinuous unpredictable solutions for linear impulsive differential equations are provided. We benefit from the B-topology in the space of discontinuous functions on the purpose of proving the presence of unpredictable solutions. For constructive definitions of unpredictable components in examples, randomly determined unpredictable sequences are newly utilized. Namely, the construction of a discontinuous unpredictable function is based on an unpredictable sequence determined by a discrete random process, and the set of discontinuity moments is realized by the logistic map. Examples with numerical simulations are presented to illustrate the theoretical results.

Scalar linear impulsive Riemann-Liouville fractional differential equations with constant delay-explicit solutions and finite time stability

AbstractRiemann-Liouville fractional differential equations with a constant delay and impulses are studied in this article. The following two cases are considered: the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point of impulse. The initial conditions as well as impulsive conditions are defined in an appropriate way for both cases. The explicit solutions are obtained and applied to the study of finite time stability.

Combining fractional differential transform method and reproducing kernel Hilbert space method to solve fuzzy impulsive fractional differential equations

The aim of this paper is to use the combination of Reproducing kernel Hilbert space method (RKHSM) and fractional differential transform method (FDTM) to solve the linear and nonlinear fuzzy impulsive fractional differential equations. Finding the numerical solutions of this class of equations are a difficult topic to analyze. In this study, convergence analysis, estimations error and bounds errors are discussed in detail under some hypotheses which provide the theoretical basis of the proposed algorithm. Some numerical examples indicate that this method is an efficient one to solve the mentioned equations.

二阶线性脉冲中立型时滞微分方程的稳定性

二阶线性脉冲中立型时滞微分方程的稳定性

On the solutions of first-order linear impulsive fuzzy differential equations

This paper gives a representation of solutions to first-order linear impulsive fuzzy differential equations with constant coefficients. Using a generalized differentiability concept and an impulsive fundamental matrix we present ideas similar to those in linear fuzzy differential equations for impulsive fuzzy Cauchy problems. Finally, numerical examples are given to illustrate the theory.

Sufficient conditions for the stability of linear periodic impulsive differential equations

linear periodic impulsive differential equations are considered. The impulse effect instants are assumed to satisfy the average dwell-time condition (the ADT condition). The stability problem is reduced to studying the stability of an auxiliary abstract impulsive differential equation. This is a perturbed periodic impulsive differential equation, which considerably simplifies the construction of a Lyapunov function. Sufficient conditions for the asymptotic stability of abstract linear periodic impulsive differential equations are obtained. It is shown that the ADT conditions lead to less conservative dwell-time estimates guaranteeing asymptotic stability. Bibliography: 24 titles.

Ulam type stability of first-order linear impulsive fuzzy differential equations

In this paper we consider Ulam stability concepts for first order linear impulsive fuzzy differential equations, which generalize standard definitions for impulsive differential equations. Sufficient conditions are established to guarantee that our equations are Ulam stable. Numerically examples are also provided to illustrate our results.

Stability of linear impulsive neutral delay differential equations with constant coefficients

In this article, we have developed a new method regarding asymptotic behavior and stability for the solutions of first order linear impulsive neutral delay differential equations with constant coefficients and constant delays. The asymptotic behavior of the solutions and the stability of the zero solution were investigated using a real root suitable for the characteristic equation. Furthermore, we obtained similar results for impulsive delay differential equations. Finally, three examples are given for the stability of the zero solution of the impulsive neutral delay differential equation.

Philos-type oscillation criteria for second-order linear impulsive differential equation with damping

In this paper, the problem of oscillation for a second-order linear impulsive differential equation with damping is investigated. This equation can be more accurately used to model the states of many evolutionary processes, which are often subject to short-term perturbations and experience abrupt changes at certain moments of time. By employing a generalized Riccati transformation technique, we derive several oscillation criteria which are either new or improve several recent results in the literature. In addition, we provide an example to illustrate the effect of impulses on the oscillation of the equation.

Analytic and numerical stability of delay differential equations with variable impulses

A stability theory of analytic and numerical solutions to linear impulsive delay differential equations(IDDEs) is established. The stability results in existing literature are extended to IDDEs with variable impulses. A convergent numerical process is proposed to calculate numerical solutions to IDDEs with variable impulses. Convergence and stability of the numerical solutions are studied in the paper. Numerical experiments are given in the end to confirm the conclusion.

Property of oscillation of first order impulsive neutral differential equations with positive and negative coefficients

In this paper, necessary and sufficient conditions for oscillation are obtained, so that every solution of the linear impulsive neutral differential equation with variable delays and variable coefficients oscillates. Most of authors who study the oscillatory criteria of impulsive neutral differential equations, investigate the case of constant delays and variable coefficients. However the points of impulsive in this paper are more general. An illustrate example is given to demonstrate our claim and explain the results.

Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments

In this paper, we introduce a piecewise reproducing kernel method for impulsive delay differential equations with piecewise constant arguments. The method is an improved reproducing kernel method. Compared with the classical reproducing kernel method, the solutions obtained using the present method can give good approximations for a larger time interval. Some numerical examples are used to show the effectiveness and simplicity of the method.

Application of commutator calculus to the study of linear impulsive systems

In this paper, the formulas of commutator calculus are applied to the investigation of the stability of linear impulsive differential equations. It is assumed that the moments of impulse action satisfy the average dwell-time (ADT) condition. Sufficient conditions for the asymptotic stability of linear impulsive differential equations in a Banach space are obtained. In the Hilbert space, the stability of the original linear differential equation is reduced to the investigation of a linear differential equation with equidistant moments of impulse action and perturbed discrete dynamics. This reduction simplifies the application of Lyapunov’s direct method and the construction of Lyapunov functions. We give examples in the spaces R2 and X=C[0,l] to illustrate the effectiveness of results obtained. Finally, a sufficient generality of the obtained results on the dynamic properties of linear operators of the linear impulsive differential equation is established.

A Generalization of Bochner’s Theorem and Its Applications in the Study of Impulsive Differential Equations

In this paper, we reveal the deep relation between Stepanov and piecewise continuous almost periodic functions and apply it to the study of almost periodic impulsive differential equations. Under the quasi-uniform continuity condition, the equivalence of Stepanov and piecewise continuous almost periodic functions is firstly established, which provides both a generalization of Bochner’s theorem and a powerful tool to investigate piecewise continuous almost periodic functions. As applications, the module containment for piecewise continuous almost periodic solutions to linear impulsive differential equations is studied.

Dichotomous solutions of linear impulsive differential equations

The notion of Lp(h,k) solutions of linear impulsive differential equations in Banach spaces is introduced. Sufficient conditions for existence of such solutions are derived. Possible applications to linear control systems with impulses are considered. An illustrative example is given.

Linear multistep methods for impulsive delay differential equations

This paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations. Numerical experiments show that the Simpson’s Rule and two-step BDF method are of order p=0 when applied to impulsive delay differential equations. An improved linear multistep numerical process is proposed. Convergence and stability conditions of the numerical solutions are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

Modified Broström procedure in patients with generalized joint laxity

This paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations. Numerical experiments show that the Simpson’s Rule and two-step BDF method are of order p=0 when applied to impulsive delay differential equations. An improved linear multistep numerical process is proposed. Convergence and stability conditions of the numerical solutions are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

Continuous approximation of linear impulsive systems and a new form of robust stability

The time-scale tolerance for linear ordinary impulsive differential equations is introduced. How large the time-scale tolerance is directly reflects the degree to which the qualitative dynamics of the linear impulsive system can be affected by replacing the impulse effect with a continuous (as opposed to discontinuous, impulsive) perturbation, producing what is known as an impulse extension equation. Theoretical properties related to the existence of the time-scale tolerance are given for periodic systems, as are algorithms to compute them. Some methods are presented for general, aperiodic systems. Additionally, sufficient conditions for the convergence of solutions of impulse extension equations to the solutions of their associated impulsive differential equation are proven. Counterexamples are provided.