Stability in measure for uncertain delay differential equations based on new Lipschitz conditions

2021 ◽  
pp. 1-13
Author(s):  
Yin Gao ◽  
Lifen Jia
Keyword(s):  
Control System ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Delay Time ◽  
Delay Differential Equations ◽  
Special Class ◽  
Liu Process ◽  
The Stability ◽  
Delay Differential ◽  
Lipschitz Conditions

Uncertain delay differential equations (UDDEs) charactered by Liu process can be employed to model an uncertain control system with a delay time. The stability of its solution always be a significant matter. At present, the stability in measure for UDDEs has been proposed and investigated based on the strong Lipschitz condition. In reality, the strong Lipschitz condition is so strictly and hardly applied to judge the stability in measure for UDDEs. For the sake of solving the above issue, the stability in measure based on new Lipschitz condition as a larger scale of applications is verified in this paper. In other words, if it satisfies the strong Lipschitz condition, it must satisfy the new Lipschitz conditions. Conversely, it may not be established. An example is provided to show that it is stable in measure based on the new Lipschitz conditions, but it becomes invalid based on the strong Lipschitz condition. Moreover, a special class of UDDEs is verified to be stable in measure without any limited condition. Besides, some examples are investigated in this paper.

scholarly journals Numerical Solutions to Neutral Stochastic Delay Differential Equations with Poisson Jumps under Local Lipschitz Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Jianguo Tan ◽  
Hongli Wang ◽  
Yongfeng Guo ◽  
Zhiwen Zhu
Keyword(s):  
Differential Equations ◽  
Lipschitz Condition ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Stochastic Delay Differential Equations ◽  
Poisson Jumps ◽  
Stochastic Delay ◽  
Local Lipschitz Condition ◽  
The Stability ◽  
Delay Differential

Recently, Liu et al. (2011) studied the stability of a class of neutral stochastic delay differential equations with Poisson jumps (NSDDEwPJs) by fixed points theory. To the best of our knowledge to date, there are not any numerical methods that have been established for NSDDEwPJs yet. In this paper, we will develop the Euler-Maruyama method for NSDDEwPJs, and the main aim is to prove the convergence of the numerical method. It is proved that the proposed method is convergent with strong order 1/2 under the local Lipschitz condition. Finally, some numerical examples are simulated to verify the results obtained from theory.

Stability analysis of uncertain delay differential equations

2021 ◽  
pp. 1-11
Author(s):  
Jian Wang ◽  
Yuanguo Zhu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Liu Process ◽  
Uncertain Dynamical System ◽  
Analytic Expression ◽  
Functional Differential ◽  
The One ◽  
Delay Differential

Uncertain delay differential equation is a class of functional differential equations driven by Liu process. It is an important model to describe the evolution process of uncertain dynamical system. In this paper, on the one hand, the analytic expression of a class of linear uncertain delay differential equations are investigated. On the other hand, the new sufficient conditions for uncertain delay differential equations being stable in measure and in mean are presented by using retarded-type Gronwall inequality. Several examples show that our stability conditions are superior to the existing results.

scholarly journals Stability of delay differential equations in the sense of Ulam on unbounded intervals

2019 ◽  
Vol 9 (2) ◽  
pp. 125-131 ◽  
Author(s):  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Stability Problem ◽  
The Stability ◽  
Delay Differential ◽  
Unbounded Intervals

In this paper, we consider the stability problem of delay differential equations in the sense of Hyers-Ulam-Rassias. Recently this problem has been solved for bounded intervals, our result extends and improve the literature by obtaining stability in unbounded intervals. An illustrative example is also given to compare these results and visualize the improvement.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximate Model ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
Approximate System ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Least Squares Approach

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

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