Piecewise Constant Delay Research Articles

A Computational Block Method with Five Hybrid-Points for Differential Equations Containing a Piece-wise Constant Delay

AbstractThis paper presents an efficient numerical approach for first-order delay differential equations containing a piece-wise constant delay. The strategy is based on a five-point hybrid block method that has been developed for ordinary differential equations. We will use the interpolation technique for the evaluation of delay terms that are not defined at the grid points. The main characteristics are discussed, including zero stability, local truncation errors, convergence and stability region. The method is easy to implement, and the numerical experiments show the efficiency and accuracy of the proposed method, compared to other methods appeared in the literature.

The Heat Equation with Piecewise Constant Delay Perturbation

The Heat Equation with Piecewise Constant Delay Perturbation

Periodicity and stability analysis of impulsive neural network models with generalized piecewise constant delays

<p style='text-indent:20px;'>In this paper, the global exponential stability and periodicity are investigated for impulsive neural network models with Lipschitz continuous activation functions and generalized piecewise constant delay. The sufficient conditions for the existence and uniqueness of periodic solutions of the model are established by applying fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with generalized piecewise constant delay, some sufficient conditions for the global exponential stability of the model are obtained. The methods, which does not make use of Lyapunov functional, is simple and valid for the periodicity and stability analysis of impulsive neural network models with variable and/or deviating arguments. The results extend some previous results. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. This paper ends with a brief conclusion.</p>

New stability results for bidirectional associative memory neural networks model involving generalized piecewise constant delay

Bidirectional associative memories (BAMs) have been extensively applied in autoassociative and heteroassociative learning. However, the research on the implementation of BAM neural networks model with the effects of the constant delay is relatively few. The present work accumulates the global exponential stability criteria for the BAM neural networks model with deviation arguments. Here the effects of the constant delay of generalized type are provided, namely piecewise constant delay of generalized type (in short, DEGPCD). This article is principally concerned with the existence and global exponential stability of the BAM neural networks model with the DEGPCD system by using approach based on the construction of an equivalent integral equation. Applying the linearization method, Banach’s fixed point theorem, a DEGPCD integral inequality of Gronwall type and some inequality techniques, we establish a new sufficient condition to ensure the existence and global exponential stability of the equilibrium point of the BAM neural networks model with the DEGPCD system. The research indicates that the generalized piecewise constant delay has a vital effect on global exponential stability of the BAM neural networks model with the DEGPCD system. At the end of this work, the hypothesis has been established with two illustrative examples along with the simulations.

Global exponential periodicity and stability of neural network models with generalized piecewise constant delay

Abstract In this paper, the global exponential stability and periodicity are investigated for delayed neural network models with continuous coefficients and piecewise constant delay of generalized type. The sufficient condition for the existence and uniqueness of periodic solutions of the model is established by applying Banach’s fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with generalized piecewise constant delay, some sufficient conditions for the global exponential stability of the model are obtained. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. This paper ends with a brief conclusion.

Existence and global exponential stability of equilibrium for impulsive neural network models with generalized piecewise constant delay

In this paper, we investigate the models of the impulsive cellular neural network with generalized constant piecewise delay (IDEGPCD). To guarantee the existence, uniqueness and global exponential stability of the equilibrium state, several new adequate conditions are obtained, which extend the results of the previous literature. The method is based on utilizing Banach’s fixed point theorem and a new IDEGPCD’s Gronwall inequality. The criteria given are easy to check and when the impulsive effects do not affect, the results can be extracted from those of the non-impulsive systems. Typical numerical simulation examples are used to show the validity and effectiveness of the proposed results. We end the paper with a brief conclusion.

Minimal and maximal solutions to first-order differential equations with piecewise constant generalized delay

In this paper we employ the method of maximal and minimal solutions coupled with comparison principles and the monotone iterative technique to obtain results of existence and approximation of solutions for differential equations with piecewise constant delay of generalized type (DEPCAG).

General Lagrange-hybrid functions and numerical solution of differential equations containing piecewise constant delays with bibliometric analysis

Delay differential equations have been the topic of significant interest in scientific and engineering problems. In this manuscript, we deal with the numerical solution of delay differential equations containing piecewise constant delays. The present method is based on general Lagrange-hybrid functions (GLHFs) and collocation method. First, we define GLHFs without considering the nodes of Lagrange polynomials. The integration operational matrix and delay operational matrix of GLHFs are obtained, generally. Using the integration and delay operational matrices and collocation method, the proposed problem transforms into a system of algebraic equations. An estimation of the error is derived in the sense of the Sobolev norm. Several numerical examples are given to demonstrate the accuracy and validity of the proposed computational procedure, as the mathematical model of tumor growth in mice and HIV infection of CD4+ T-cells. Moreover, delay differential equations are studied through a bibliometric viewpoint.

Parameter identification of a class of nonlinear multidelay systems with piecewise constant delays

AbstractThis paper deals with the parameter estimation of an important class of nonlinear time‐delay systems with a piecewise constant delay function. The developed procedure is based on a generalization of the hybrid of block‐pulse functions and Taylor polynomials. The operational matrix of integration related to the generalized hybrid functions is constructed. Furthermore, an upper bound with respect to the L2‐norm for the generalized hybrid functions is acquired, and convergence of the generalized hybrid functions is demonstrated. The merits of the hybrid basis are implemented to convert the problem under examination into an algebraic system. The methodology of least squares is adopted for solving the acquired algebraic system. An illustrative example is included in order to validate the efficiency and accuracy of the designed method. It is shown that the proposed approach provides accurate estimates for the desired parameters. The current approach reduces the computation complexity of the identification process.

Qualitative Behavior of a Liénard-Type Differential Equation with Piecewise Constant Delays

In this paper, a Lienard-type differential equation with piecewise constant argument of generalized type is proposed. Existence and uniqueness of the solutions are investigated. Uniform asymptotic stability of the trivial solution is examined by the Lyapunov–Krasovskii method. The obtained theoretic results are exemplified and supported by a simulation.

Lyapunov Characterization of Uniform Exponential Stability for Nonlinear Infinite-Dimensional Systems

In this article, we deal with infinite-dimensional nonlinear forward complete dynamical systems which are subject to uncertainties. We first extend the well-known Datko lemma to the framework of the considered class of systems. Thanks to this generalization, we provide characterizations of the uniform (with respect to uncertainties) local, semi-global, and global exponential stability, through the existence of coercive and non-coercive Lyapunov functionals. The importance of the obtained results is underlined through some applications concerning 1) exponential stability of nonlinear retarded systems with piecewise constant delays, 2) exponential stability preservation under sampling for semilinear control switching systems, and 3) the link between input-to-state stability and exponential stability of semilinear switching systems.

Global dynamics of tick-borne diseases.

A tick-borne disease model is considered with nonlinear incidence rate and piecewise constant delay of generalized type. It is known that the tick-borne diseases have their peak during certain periods due to the life cycle of ticks. Only adult ticks can bite and transmit disease. Thus, we use a piecewise constant delay to model this phenomena. The global asymptotic stability of the disease-free and endemic equilibrium is shown by constructing suitable Lyapunov functions and Lyapunov-LaSalle technique. The theoretical findings are illustrated through numerical simulations.

Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models

A new efficient type of Chebyshev wavelet is used to find the optimal solutions of general linear, continuous-time, multi-delay systems with quadratic performance indices and also to obtain the responses of linear time-delay systems. According to the new definition of Chebyshev wavelets, the operational matrices of integration, product, delay and inverse time and the integration matrix are derived. Furthermore, new operational matrices as the piecewise delay operational matrix and the stretch operational matrix of the desired Chebyshev wavelets are introduced to analyze systems with, in turn, piecewise constant delays and stretched arguments or proportional delays. Two novel algorithms based on newly Chebyshev wavelet method are proposed for the optimal control and the analysis of delay models. Some examples are solved to establish that the accuracy and applicability of Chebyshev wavelet method in delay systems are increased.

A composite pseudospectral method for solving multi-delay optimal control problems involving piecewise constant delay functions

An adaptive numerical method for solving multi-delay optimal control problems with piecewise constant delay functions is introduced. The proposed method is based on composite pseudospectral method using the well-known Legendre–Gauss–Lobatto points. In this approach, the main problem converts to a mathematical optimization problem whose solution is much more easier than the original one. The necessary conditions of optimality associated to nonlinear piecewise constant delay systems are derived. The method is easy to implement and provides very accurate results.

Stable invariant manifolds for delay equations with piecewise constant argument

We construct smooth stable invariant manifolds for a class of delay equations with piecewise constant delay, for any sufficiently small perturbation of a nonuniform exponential dichotomy. We build on former work for perturbations of a uniform exponential dichotomy, also for delay equations with piecewise constant delay. These equations can be described as delay equations with an impulsive behavior of the derivative, such that at certain times the derivative changes abruptly.

Flip and N-S bifurcation behavior of a predator-prey model with piecewise constant arguments and time delay

In this paper, the stability and bifurcation behaviors of a predator-prey model with the piecewise constant arguments and time delay are investigated. Technical approach is fully based on Jury criterion and bifurcation theory. The interesting point is that the model will produce two different branches by limiting branch parameters of different intervals. Besides, image simulation is also given.

Impulsive observer design for linear systems with delayed outputs

This paper investigates the problem of the state reconstruction for LTI systems with time-delayed outputs. The proposed observer allows the reconstruction of the states in two steps. In the first step, an impulsive observer provides an estimation of the delayed states. In the second step, the actual states is estimated by the second observer using a predictive method. The finite time reconstruction is achieved for two different cases: constant and piece-wise constant time delays. The problem of the residual error in the case of piece-wise delay is solved. A numerical example is provided in order to show the efficiency of the proposed method.

Optimal control of linear multi-delay systems with piecewise constant delays

Optimal control of linear multi-delay systems with piecewise constant delays

Analysis of Linear Piecewise Constant Delay Systems Using a Hybrid Numerical Scheme

An efficient computational technique for solving linear delay differential equations with a piecewise constant delay function is presented. The new approach is based on a hybrid of block-pulse functions and Legendre polynomials. A key feature of the proposed framework is the excellent representation of smooth and especially piecewise smooth functions. The operational matrices of delay, derivative, and product corresponding to the mentioned hybrid functions are implemented to transform the original problem into a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the proposed numerical scheme.

Costate Computation by an Adaptive Pseudospectral Method for Solving Optimal Control Problems with Piecewise Constant Time Lag

In this paper, we consider nonlinear optimal control problems involving piecewise constant delay functions. A discretization of the Bolza problem by an adaptive Legendre pseudospectral method is considered. The method is based on composite pseudospectral method using the well-known Legendre---Gauss---Lobatto collocation points. In this approach, the main problem is converted into a mathematical optimization problem, whose solution is much more easier than that of the original one. It is shown that, under a simple set of conditions, the operations of discretization and dualization are commutative. The method is used to solve an optimal control problem with continuous variable time lag.