Finite time differential equations

1985 ◽  
Author(s):  
V. Haimo
Keyword(s):  
Differential Equations ◽  
Finite Time ◽  
Time Differential

scholarly journals Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 297-313 ◽  
Author(s):  
Desmond J. Higham ◽  
Xuerong Mao ◽  
Andrew M. Stuart
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Finite Time ◽  
Mean Square ◽  
Convergence Conditions ◽  
Finite Time Convergence ◽  
Mean Square Stability ◽  
Exponential Mean Square Stability ◽  
Positive Results

AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

scholarly journals MORSE SPECTRUM FOR NONAUTONOMOUS DIFFERENTIAL EQUATIONS

2008 ◽  
Vol 08 (03) ◽  
pp. 351-363 ◽  
Author(s):  
FRITZ COLONIUS ◽  
PETER E. KLOEDEN ◽  
MARTIN RASMUSSEN
Keyword(s):  
Differential Equations ◽  
Finite Time ◽  
Accumulation Point ◽  
Morse Decomposition ◽  
Finite Time Lyapunov Exponents ◽  
The Past ◽  
Accumulation Points ◽  
Morse Spectrum ◽  
Nonautonomous Differential Equations ◽  
Linear Cocycle

The concept of a Morse decomposition consisting of nonautonomous sets is reviewed for linear cocycle mappings w.r.t. the past, future and all-time convergences. In each case, the set of accumulation points of the finite-time Lyapunov exponents corresponding to points in a nonautonomous set is shown to be an interval. For a finest Morse decomposition, the Morse spectrum is defined as the union of all of the above accumulation point intervals over the different nonautonomous sets in such a finest Morse decomposition. In addition, Morse spectrum is shown to be independent of which finest Morse decomposition is used, when more than one exists.

scholarly journals Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses

2019 ◽  
Vol 3 (2) ◽  
pp. 28 ◽  
Author(s):  
Snezhana Hristova ◽  
Krasimira Ivanova
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Finite Time ◽  
Lyapunov Functions ◽  
Random Variable ◽  
Time Intervals ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Dini Derivatives ◽  
Moment Exponential Stability

The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time between two consecutive moments of impulses is the Erlang distributed random variable. The study is based on Lyapunov functions. The fractional Dini derivatives are applied.

A Path-Following Feedback Control Law of a Five-Axle, Three-Steering Coupled-Vehicle System

2014 ◽  
Vol 26 (5) ◽  
pp. 551-565 ◽  
Author(s):  
Hiroaki Yamaguchi ◽  
◽  
Ryota Kameyama ◽  
Atsushi Kawakami ◽  
Keyword(s):  
Feedback Control ◽  
Differential Equations ◽  
Path Following ◽  
Control Law ◽  
Operating Environment ◽  
Linear Control Theory ◽  
Single Carrier ◽  
Vehicle System ◽  
Time Differential ◽  
Lyapunov’S Second Method

<div class=""abs_img""><img src=""[disp_template_path]/JRM/abst-image/00260005/03.jpg"" width=""300"" />Experimental coupled-vehicle system</div> This paper presents a new path-following feedback control law of a five-axle, three-steering coupledvehicle system which enables specifying the movements and rotations of its two carriers quantitatively, according to the operating environment. The kinematical equations of the coupled-vehicle system are first converted into time differential equations in a threechain, single-generator chained form. The time differential equations in the chained form are secondly converted into new differential equations with a new variable. The new control law enables the relative orientation between the two carriers to be constant in either a straight-bed carrier configuration or a V-bed carrier configuration, and simultaneously enables the orientations of these carriers functioning as a single carrier relative to the direction of the tangent of the path to be changed quantitatively, according to the locations of obstacles for avoiding collision with them. Asymptotic stability of the new control law is guaranteed by the linear control theory and the Lyapunov’s second method. Especially, the form of the new differential equations facilitates the design of the Lyapunov functions. The validity of the new control law is verified by an experimental five-axle, three-steering coupledvehicle system. </span>

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