Cone-valued lyapunov functions and the stability of stochastic differential equations

1996 ◽  
Vol 14 (1) ◽  
pp. 1-21
Author(s):  
Edet P. Akpan
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lyapunov Functions ◽  
The Stability

scholarly journals Stability of a Class of Hybrid Neutral Stochastic Differential Equations with Unbounded Delay

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Boliang Lu ◽  
Ruili Song
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Lyapunov Functions ◽  
Stability Criteria ◽  
Bounded Delay ◽  
Unbounded Delay ◽  
The Stability ◽  
Generalized Itô Formula ◽  
Stability Results

This paper studies the stability of hybrid neutral stochastic differential equations with unbounded delay. Some novel exponential stability criteria and boundedness conditions are established based on the generalized Itô formula and Lyapunov functions. The factor e-εδ(t) is used to overcome the difficulties caused by the unbounded delay δ(t) effectively. In particular, our results generalize and improve some previous stability results from bounded delay to unbounded delay conditions. Finally, an example is presented to demonstrate the effectiveness of the proposed results.

scholarly journals The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Rui Zhang ◽  
Yinjing Guo ◽  
Xiangrong Wang ◽  
Xueqing Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Stability Criteria ◽  
Two Measures ◽  
The Mean ◽  
The Stability

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.

scholarly journals Ultimate Boundedness of the Systems Governed by Stochastic Differential Equations

1972 ◽  
Vol 47 ◽  
pp. 111-144 ◽  
Author(s):  
Yoshio Miyahara
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stochastic Differential Equations ◽  
Functional Differential Equations ◽  
Ultimate Boundedness ◽  
Functional Differential ◽  
The Stability

The stability of the systems given by ordinary differential equations or functional-differential equations has been studied by many mathematicians. The most powerful tool in this field seems to be the Liapunov’s second method (see, for example [6]).

scholarly journals Almost sure exponential stability of the θ-Euler-Maruyama method for neutral stochastic differential equations with time-dependent delay when θ ∈ [0; 1 2]

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5629-5645 ◽  
Author(s):  
Maja Obradovic ◽  
Marija Milosevic
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Stability Result ◽  
Time Dependent ◽  
Almost Sure Exponential Stability ◽  
Linear Growth Condition ◽  
Highly Nonlinear ◽  
The Stability ◽  
Drift Coefficient

This paper represents a generalization of the stability result on the Euler-Maruyama solution, which is established in the paper M. Milosevic, Almost sure exponential stability of solutions to highly nonlinear neutral stochastics differential equations with time-dependent delay and Euler-Maruyama approximation, Math. Comput. Model. 57 (2013) 887 - 899. The main aim of this paper is to reveal the sufficient conditions for the global almost sure asymptotic exponential stability of the ?-Euler-Maruyama solution (? ? [0, 1/2 ]), for a class of neutral stochastic differential equations with time-dependent delay. The existence and uniqueness of solution of the approximate equation is proved by employing the one-sided Lipschitz condition with respect to the both present state and delayed arguments of the drift coefficient of the equation. The technique used in proving the stability result required the assumption ? ?(0, 1/2], while the method is defined by employing the parameter ? with respect to the both drift coefficient and neutral term. Bearing in mind the difference between the technique which will be applied in the present paper and that used in the cited paper, the Euler-Maruyama case (? = 0) is considered separately. In both cases, the linear growth condition on the drift coefficient is applied, among other conditions. An example is provided to support the main result of the paper.

scholarly journals Stability of Perturbed Set Differential Equations Involving Causal Operators in Regard to Their Unperturbed Ones considering Difference in Initial Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Coșkun Yakar ◽  
Hazm Talab
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Initial Conditions ◽  
Lyapunov Functionals ◽  
Stability Of Solutions ◽  
Causal Operators ◽  
The Difference ◽  
The Stability

We investigate the stability of solutions of perturbed set differential equations with causal operators in regard to their corresponding unperturbed ones considering the difference in initial conditions (time and position) by utilizing Lyapunov functions and Lyapunov functionals.

scholarly journals New Results on Impulsive Functional Differential Equations with Infinite Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jie Yang ◽  
Bing Xie
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Functional Differential Equations ◽  
Stability Behavior ◽  
Razumikhin Technique ◽  
Infinite Delays ◽  
Functional Differential ◽  
The Stability

We investigate the stability for a class of impulsive functional differential equations with infinite delays by using Lyapunov functions and Razumikhin-technique. Some new Razumikhin-type theorems on stability are obtained, which shows that impulses do contribute to the system’s stability behavior. An example is also given to illustrate the importance of our results.

scholarly journals Stability of the solutions of impulsive functional-differential equations by Lyapunov's direct method

2001 ◽  
Vol 43 (2) ◽  
pp. 269-278 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Direct Method ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Continuous Functions ◽  
Piecewise Continuous ◽  
Functional Differential ◽  
The Stability

AbstractWe consider the stability of the zero solution of a system of impulsive functional-differential equations. By means of piecewise continuous functions, which are generalizations of classical Lyapunov functions, and using a technique due to Razumikhin, sufficient conditions are found for stability, uniform stability and asymptotical stability of the zero solution of these equations. Applications to impulsive population dynamics are also discussed.

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