Piecewise Continuous Arguments Research Articles

Stability equivalence between regime‐switching jump diffusion delayed systems and corresponding systems with piecewise continuous arguments and application to discrete‐time feedback control

AbstractIn this paper, we mainly study the equivalence of exponential stability for regime‐switching jump diffusion delayed systems (RSJDDSs) and RSJDDSs with piecewise continuous arguments (RSJDDSs‐PCA). Our results show that if one of the RSJDDS and the RSJDDS‐PCA is th moment exponentially stable, then another system is also th moment exponentially stable when time delay and segment step size have a common upper bound, while both equations are almost surely exponentially stable, and we also provided a method to calculate this upper bound. In addition, as an application of the stability equivalence theorem, we design discrete‐time state and mode observations feedback control to stabilize unstable RSJDDSs and investigate that controllers of the drift, diffusion, and jump terms are all able to play a stabilizing effect on the controlled system.

Strong convergence of explicit numerical schemes for stochastic differential equations with piecewise continuous arguments

Strong convergence of explicit numerical schemes for stochastic differential equations with piecewise continuous arguments

Convergence and stability of modified partially truncated Euler-Maruyama method for stochastic differential equations with piecewise continuous arguments

ABSTRACT This paper constructs a modified partially truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift and diffusion coefficients grow superlinearly. We divide the coefficients of SDEPCAs into global Lipschitz continuous and superlinearly growing parts. Our method only truncates the superlinear terms of the coefficients to overcome the potential explosions caused by the nonlinearities of the coefficients. The strong convergence theory of this method is established and the 1/2 convergence rate is presented. Furthermore, an explicit scheme is developed to preserve the mean square exponential stability of the underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results.

Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments

Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments

Convergence and stability of an explicit method for nonlinear stochastic differential equations with piecewise continuous arguments

This paper constructs a new explicit method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift coefficients grow superlinearly and the diffusion coefficients have at most linear growth. We show that this method converges strongly with the convergence rate 1/2 to the exact solutions of SDEPCAs over the finite time interval and demonstrate it can inherit the mean square exponential stability of the underlying SDEPCAs. Several numerical experiments are carried out to support our findings.

CONVERGENCE AND STABILITY OF GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH PIECEWISE CONTINUOUS ARGUMENTS OF ADVANCED TYPE

This paper deals with the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments of advanced type. First of all, we obtain the expression of analytic solution by the method of separation variable, then the sufficient conditions for stability are obtained. Semidiscrete and fully discrete schemes are derived by Galerkin finite element method, and their convergence are both analyzed in L2-norm. Moreover, the stability of the two schemes are investigated. The semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are presented to illustrate the theoretical results.

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

Convergence and stability of Galerkin finite element method for hyperbolic partial differential equation with piecewise continuous arguments

AbstractIn this paper, the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments are investigated. Firstly, the variation formulation is derived by applying Green's formula and Galerkin finite element method to spatial direction of the original equation. Next, semidiscrete and fully discrete schemes are obtained and the convergence is analyzed in ‐norm rigorously. Moreover, the stability analysis shows that the semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are also obtained under which the analytic solution is asymptotically stable. Finally, some numerical experiments are provided to demonstrate our theoretical results.

Strong convergence of the tamed Euler method for nonlinear hybrid stochastic differential equations with piecewise continuous arguments

This paper develops the tamed Euler method (TEM) for approximating solutions of nonlinear hybrid stochastic differential equations with piecewise continuous arguments (SDEPCAs). As far as we know, there are only two published articles (Hou et al., 2009; Zhang, 2020) considered the Lp-convergence rate of numerical methods for hybrid systems, and the rates are no more than 1/p(p≥2) in both of these two studies. In this work, we use the technique proposed in Song et al. (2022) to develop the TEM, and obtain the following two main results: (i) The TEM converges strongly to the exact solution of hybrid SDEPCAs; (ii) The Lp-convergence rate (p≥2) can reach 1/2. Finally, we give two numerical examples to verify the theoretical results.

Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation

<p style='text-indent:20px;'>For the stochastic differential equation with piecewise continuous arguments, multiplicative noises and dissipative drift coefficients, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to numerically preserve the invariant measure, we apply the backward Euler method to the equation, and prove that the numerical solution at integer time is also a Markov chain and possesses a unique numerical invariant measure. By establishing several a priori estimations, we present the time-independent weak error analysis for the method via Malliavin calculus, which implies that the numerical invariant measure converges to the original one with weak order 1. Numerical experiments verify the theoretical analysis.</p>

Convergence and stability of spectral collocation method for hyperbolic partial differential equation with piecewise continuous arguments

Convergence and stability of spectral collocation method for hyperbolic partial differential equation with piecewise continuous arguments

Strong convergence rate of the stochastic theta method for nonlinear hybrid stochastic differential equations with piecewise continuous arguments

Strong convergence rate of the stochastic theta method for nonlinear hybrid stochastic differential equations with piecewise continuous arguments

A numerical scheme for solving nonlinear parabolic partial differential equations with piecewise constant arguments

‎This article deals with the nonlinear parabolic equation with piecewise continuous arguments (EPCA)‎. ‎This study‎, ‎therefore‎, ‎with the aid of the $theta$‎ ‎-methods,‎ ‎aims at presenting a numerical solution scheme for solving such types of equations which has applications in certain ecological studies‎. ‎Moreover‎, ‎the convergence and stability of our proposed numerical method are investigated‎. ‎Finally‎, ‎to support and confirm our theoretical results‎, ‎some numerical examples are also presented‎.

Convergence and stability of stochastic theta method for nonlinear stochastic differential equations with piecewise continuous arguments

This paper deals with the strong convergence and exponential stability of the stochastic theta (ST) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs) with non-Lipschitzian and non-linear coefficients and mainly includes the following three results: (i) under the local Lipschitz and the monotone conditions, the ST method with θ∈[1/2,1] is strongly convergent to SDEPCAs; (ii) the ST method with θ∈(1/2,1] preserves the exponential mean square stability of SDEPCAs under the monotone condition and some conditions on the step-size; (iii) without any restriction on the step-size, there exists θ∗∈(1/2,1] such that the ST method with θ∈(θ∗,1] is exponentially stable in mean square. Moreover, for sufficiently small step-size, the rate constant can be reproduced. Some numerical simulations are provided to illustrate the theoretical results.

Stability analysis for semi-linear stochastic differential equation with piecewise continuous arguments

In this paper, the exponential stability in mean square of the exponential Euler method for semi-linear stochastic differential equation with piecewise continuous arguments is obtained. Firstly, the exponential Euler scheme of the equation is given. Secondly, under Lipschitz condition, sufficient conditions of exponential stability to the exact solution are achieved in mean square. Furthermore, it is proved that exponential Euler method preserves exponential stability of the exact solution without any restriction on the step-size. Finally, an example is provided to illustrate our theories.

Convergence and Stability of the Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments

Convergence and Stability of the Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments

Stability equivalence among stochastic differential equations and stochastic differential equations with piecewise continuous arguments and corresponding Euler-Maruyama methods

In this paper, we consider the equivalence of the pth moment exponential stability for stochastic differential equations (SDEs), stochastic differential equations with piecewise continuous arguments (SDEPCAs) and the corresponding Euler-Maruyama methods EMSDEs and EMSDEPCAs. We show that if one of the SDEPCAs, SDEs, EMSDEs and EMSDEPCAs is pth moment exponentially stable, then any of them is pth moment exponentially stable for a sufficiently small step size h and τ under the global Lipschitz assumption on the drift and diffusion coefficients.

Compensated split-step balanced methods for nonlinear stiff SDEs with jump-diffusion and piecewise continuous arguments

This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments. By combining compensated split-step methods and balanced methods, a class of compensated split-step balanced (CSSB) methods are suggested for solving the equations. Based on the one-sided Lipschitz condition and local Lipschitz condition, a strong convergence criterion of CSSB methods is derived. It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions. Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods. Moreover, in order to show the computational advantage of CSSB methods, we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.

Convergence and stability of extended BBVMs for nonlinear delay-differential-algebraic equations with piecewise continuous arguments

Delay-differential-algebraic equations have been widely used to model some important phenomena in science and engineering. Since, in general, such equations do not admit a closed-form solution, it is necessary to solve them numerically by introducing suitable integrators. The present paper extends the class of block boundary value methods (BBVMs) to approximate the solutions of nonlinear delay-differential equations with algebraic constraint and piecewise continuous arguments. Under the classical Lipschitz conditions, convergence and stability criteria of the extended BBVMs are derived. Moreover, a couple of numerical examples are provided to illustrate computational effectiveness and accuracy of the methods.

The truncated Euler–Maruyama method for stochastic differential equations with piecewise continuous arguments driven by Lévy noise

ABSTRACT This paper aims to consider stochastic differential equations with piecewise continuous arguments (SDEPCAs) driven by Lévy noise where both drift and diffusion coefficients satisfy local Lipschitz condition plus Khasminskii-type condition and the jump coefficient grows linearly. We present the explicit truncated Euler–Maruyama method. We study its moment boundedness and its strong convergence. Moreover, the convergence rate is shown to be close to that of the classical Euler method under additional conditions.