scholarly journals Stability of Nonlinear Systems of Fractional Order Differential Equations

2010 ◽  
Vol 7 (4) ◽  
pp. 1458-1461
Author(s):  
Baghdad Science Journal
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Fractional Order ◽  
Finite Time ◽  
Time Interval ◽  
Sufficient Condition ◽  
Finite Time Interval ◽  
Infinite Time ◽  
Fractional Order Differential Equations ◽  
Infinite Time Interval

In this paper, a sufficient condition for stability of a system of nonlinear multi-fractional order differential equations on a finite time interval with an illustrative example, has been presented to demonstrate our result. Also, an idea to extend our result on such system on an infinite time interval is suggested.

On the Interrelation of Two Linear NonStationary Problems with Multiple Evaders

2015 ◽  
Vol 17 (04) ◽  
pp. 1550013 ◽  
Author(s):  
N. N. Petrov ◽  
K. A. Shchelchkov
Keyword(s):  
Finite Time ◽  
Time Interval ◽  
Finite Time Interval ◽  
Infinite Time ◽  
Pursuit Problem ◽  
Infinite Time Interval ◽  
Admissible Controls

A linear nonstationary pursuit problem in which a group of pursuers and a group of evaders are involved is considered under the condition that the group of pursuers includes participants whose admissible controls set coincides with that of the evaders and participants whose admissible controls sets belong to interior of admissible controls set of the evaders. The aim of the group of pursuers is to capture all the evaders. The aim of the group of evaders is to prevent the capture, that is, to allow at least one of the evaders to avoid the rendezvous. It is shown that, if in the game in which all the participants have equal capabilities at least one of the evaders avoids the rendezvous on an infinite time interval, then as a result of the addition of any number of pursuers with less capabilities, at least one of the evaders will avoid the rendezvous on any finite time interval.

scholarly journals Infinite time interval BSDEs and the convergence of g-martingales

2000 ◽  
Vol 69 (2) ◽  
pp. 187-211 ◽  
Author(s):  
Zengjing Chen ◽  
Bo Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Unique Solution ◽  
Backward Stochastic Differential Equations ◽  
Time Interval ◽  
Sufficient Condition ◽  
Infinite Time ◽  
Infinite Time Interval

AbstractIn this paper, we first give a sufficient condition on the coefficients of a class of infinite time interval backward stochastic differential equations (BSDEs) under which the infinite time interval BSDEs have a unique solution for any given square integrable terminal value, and then, using the infinite time interval BSDEs, we study the convergence of g-martingales introduced by Peng via a kind of BSDEs. Finally, we study the applications of g-expectations and g-martingales in both finance and economics.

scholarly journals New Methods of Finite-Time Synchronization for a Class of Fractional-Order Delayed Neural Networks

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Weiwei Zhang ◽  
Jinde Cao ◽  
Ahmed Alsaedi ◽  
Fuad E. Alsaadi
Keyword(s):  
Neural Networks ◽  
Fractional Order ◽  
Finite Time ◽  
Time Synchronization ◽  
Sufficient Conditions ◽  
Time Interval ◽  
Finite Time Interval ◽  
Delayed Neural Networks ◽  
New Methods ◽  
Theoretical Results

Finite-time synchronization for a class of fractional-order delayed neural networks with fractional order α, 0<α≤1/2 and 1/2<α<1, is investigated in this paper. Through the use of Hölder inequality, generalized Bernoulli inequality, and inequality skills, two sufficient conditions are considered to ensure synchronization of fractional-order delayed neural networks in a finite-time interval. Numerical example is given to verify the feasibility of the theoretical results.

scholarly journals Generalized Memory: Fractional Calculus Approach

2018 ◽  
Vol 2 (4) ◽  
pp. 23 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Power Law ◽  
Memory Function ◽  
Time Interval ◽  
Infinite Time ◽  
Suggested Approach ◽  
Infinite Time Interval ◽  
History Of ◽  
With Memory

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

A method for approximating the probability functions of a Markov chain

1988 ◽  
Vol 25 (4) ◽  
pp. 808-814 ◽  
Author(s):  
Keith N. Crank
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
Finite Number ◽  
Continuous Time ◽  
Finite Time ◽  
The State ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Finite Time Interval ◽  
System Of Differential Equations

This paper presents a method of approximating the state probabilities for a continuous-time Markov chain. This is done by constructing a right-shift process and then solving the Kolmogorov system of differential equations recursively. By solving a finite number of the differential equations, it is possible to obtain the state probabilities to any degree of accuracy over any finite time interval.

New Result on Finite-Time Stability of Fractional-Order Nonlinear Delayed Systems

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Liping Chen ◽  
Wei Pan ◽  
Ranchao Wu ◽  
Yigang He
Keyword(s):  
Fractional Order ◽  
Finite Time ◽  
Lyapunov Method ◽  
Time Interval ◽  
Fractional Order Systems ◽  
Finite Time Interval ◽  
Time Stability ◽  
Delayed Systems ◽  
Finite Time Stability ◽  
Delayed System

Since Lyapunov method has not been well developed for fractional-order systems, stability of fractional-order nonlinear delayed systems remains a formidable problem. In this letter, finite-time stability of a class of fractional-order nonlinear delayed systems with order between 0 and 1 is addressed. By using the technique of inequalities, a new and simple delay-independent sufficient condition guaranteeing stability of fractional-order nonlinear delayed system over the finite time interval is obtained. Numerical examples are presented to demonstrate the validity and feasibility of the obtained results.

Bounded solutions for general time interval BSDEs with quadratic growth coefficients and stochastic conditions

2018 ◽  
Vol 18 (05) ◽  
pp. 1850034
Author(s):  
Huan-Huan Luo ◽  
Sheng-Jun Fan
Keyword(s):  
Differential Equations ◽  
Finite Time ◽  
Quadratic Growth ◽  
Time Interval ◽  
Finite Time Interval ◽  
Bounded Solutions ◽  
One Dimensional ◽  
Stochastic Conditions ◽  
New Ideas ◽  
General Time

This paper deals with bounded solutions for general time interval one-dimensional backward stochastic differential equations (BSDEs for short) with quadratic growth coefficients and stochastic conditions. Several general results of existence, uniqueness, stability and comparison for the bounded solutions are put forward and established, which improve considerably some existing works, even though for the case of finite time interval. Some new ideas are also developed to establish these results.

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