On Stability with Respect to a Part of the Variables for Nonlinear Discrete-Time Systems with a Random Disturbances

2021 ◽  
Vol 22 (1) ◽  
pp. 12-18
Author(s):  
V. I. Vorotnikov ◽  
Yu. G. Martyshenko
Keyword(s):  
Lyapunov Function ◽  
Finite Difference ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
General Problem ◽  
Equilibrium Position ◽  
Lyapunov Functions ◽  
Stochastic Version ◽  
Random Disturbances ◽  
Zero Equilibrium

Nonlinear discrete (finite-difference) system of equations subject to the influence of a random disturbances of the "white" noise type, which is a difference analog of systems of stochastic differential equations in the Ito form, is considered. The increased interest in such systems is associated with the use of digital control systems, financial mathematics, as well as with the numerical solution of systems of stochastic differential equations. Stability problems are among the main problems of qualitative analysis and synthesis of the systems under consideration. In this case, we mainly study the general problem of stability of the zero equilibrium position, within the framework of which stability is analyzed with respect to all variables that determine the state of the system. To solve it, a discrete-stochastic version of the method of Lyapunov functions has been developed. The central point here is the introduction by N. N. Krasovskii, the concept of the averaged finite difference of a Lyapunov function, for the calculation of which it is sufficient to know only the right-hand sides of the system and the probabilistic characteristics of a random process. In this paper, for the class of systems under consideration, a statement of a more general problem of stability of the zero equilibrium position is given: not for all, but for a given part of the variables defining it. For the case of deterministic systems of ordinary differential equations, the formulation of this problem goes back to the classical works of A. M. Lyapunov and V. V. Rumyantsev. To solve the problem posed, a discrete-stochastic version of the method of Lyapunov functions is used with a corresponding specification of the requirements for Lyapunov functions. In order to expand the capabilities of the method used, along with the main Lyapunov function, an additional (vector, generally speaking) auxiliary function is considered for correcting the region in which the main Lyapunov function is constructed.

scholarly journals Stabilization of Stochastic Differential Equations Driven by G-Brownian Motion with Aperiodically Intermittent Control

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 988
Author(s):  
Pengju Duan
Keyword(s):  
Brownian Motion ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Mild Solution ◽  
Intermittent Control

The paper is devoted to studying the exponential stability of a mild solution of stochastic differential equations driven by G-Brownian motion with an aperiodically intermittent control. The aperiodically intermittent control is added into the drift coefficients, when intermittent intervals and coefficients satisfy suitable conditions; by use of the G-Lyapunov function, the p-th exponential stability is obtained. Finally, an example is given to illustrate the availability of the obtained results.

scholarly journals Exact Finite-Difference Schemes ford-Dimensional Linear Stochastic Systems with Constant Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Peng Jiang ◽  
Xiaofeng Ju ◽  
Dan Liu ◽  
Shaoqun Fan
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Symplectic Structure ◽  
Stochastic Systems ◽  
First Integral ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Linear Stochastic Systems ◽  
Constant Coefficients

The authors attempt to construct the exact finite-difference schemes for linear stochastic differential equations with constant coefficients. The explicit solutions to Itô and Stratonovich linear stochastic differential equations with constant coefficients are adopted with the view of providing exact finite-difference schemes to solve them. In particular, the authors utilize the exact finite-difference schemes of Stratonovich type linear stochastic differential equations to solve the Kubo oscillator that is widely used in physics. Further, the authors prove that the exact finite-difference schemes can preserve the symplectic structure and first integral of the Kubo oscillator. The authors also use numerical examples to prove the validity of the numerical methods proposed in this paper.

scholarly journals Euler-type approximation for systems of stochastic differential equations

1989 ◽  
Vol 2 (4) ◽  
pp. 239-249 ◽  
Author(s):  
J. Golec ◽  
G. Ladde
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Sufficient Conditions ◽  
Taylor Formula ◽  
Time Varying ◽  
Stochastic Version ◽  
Type Approximation ◽  
Mean Square Convergence ◽  
The Mean ◽  
Time Invariant

By developing a stochastic version of the Taylor formula, the mean-square convergence of the Euler-type approximation for the solution of systems of Itô-type stochastic differential equations is investigated. Sufficient conditions are given to obtain time-varying and time-invariant error estimates.

ON THE STABILITY OF DYNAMIC SYSTEMS WITH CERTAIN SWITCHINGS, WHICH CONSISTS OF LINEAR SUBSYSTEMS WITHOUT DELAY

2021 ◽  
Vol 3 ◽  
pp. 5-17
Author(s):  
Denis Khusainov ◽  
◽  
Alexey Bychkov ◽  
Andrey Sirenko ◽  
Jamshid Buranov ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Dynamic Systems ◽  
Lyapunov Functions ◽  
Lyapunov Method ◽  
Positive Definite Function ◽  
Definite Function ◽  
The Stability ◽  
Further Development ◽  
Zero Equilibrium

This work is devoted to the further development of the study of the stability of dynamic systems with switchings. There are many different classes of dynamical systems described by switched equations. The authors of the work divide systems with switches into two classes. Namely, on systems with definite and indefinite switchings. In this paper, the system with certain switching, namely a system composed of differential and difference sub-systems with the condition of decreasing Lyapunov function. One of the most versatile methods of studying the stability of the zero equilibrium state is the second Lyapunov method, or the method of Lyapunov functions. When using it, a positive definite function is selected that satisfies certain properties on the solutions of the system. If a system of differential equations is considered, then the condition of non-positiveness (negative definiteness) of the total derivative due to the system is imposed. If a difference system of equations is considered, then the first difference is considered by virtue of the system. For more general dynamical systems (in particular, for systems with switchings), the condition is imposed that the Lyapunov function does not increase (decrease) along the solutions of the system. Since the paper considers a system consisting of differential and difference subsystems, the condition of non-increase (decrease of the Lyapunov function) is used.For a specific type of subsystems (linear), the conditions for not increasing (decreasing) are specified. The basic idea of using the second Lyapunov method for systems of this type is to construct a sequence of Lyapunov functions, in which the level surfaces of the next Lyapunov function at the switching points are either «stitched» or «contain the level surface of the previous function».

scholarly journals Stability and Boundedness of Solutions to a Certain Second-Order Nonautonomous Stochastic Differential Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
A. T. Ademola ◽  
S. Moyo ◽  
B. S. Ogundare ◽  
M. O. Ogundiran ◽  
O. A. Adesina
Keyword(s):  
Differential Equation ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Relevant Literature ◽  
Second Order ◽  
Boundedness Of Solutions ◽  
Nonlinear Functions ◽  
Lyapunov’S Second Method

This paper focuses on stability and boundedness of certain nonlinear nonautonomous second-order stochastic differential equations. Lyapunov’s second method is employed by constructing a suitable complete Lyapunov function and is used to obtain criteria, on the nonlinear functions, that guarantee stability and boundedness of solutions. Our results are new; in fact, according to our observations from the relevant literature, this is the first attempt on stability and boundedness of solutions of second-order nonlinear nonautonomous stochastic differential equations. Finally, examples together with their numerical simulations are given to authenticate and affirm the correctness of the obtained results.

scholarly journals The Global Solutions and Moment Boundedness of Stochastic Multipantograph Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Maosheng Tian ◽  
Xuejing Meng ◽  
Jihong Chen ◽  
Xiaoqi Tang
Keyword(s):  
Lyapunov Function ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Polynomial Growth ◽  
Global Solutions ◽  
Growth Conditions ◽  
Unbounded Delay ◽  
Moment Boundedness ◽  
Methods And Techniques

We consider the existence of global solutions and their moment boundedness for stochastic multipantograph equations. By the idea of Lyapunov function, we impose some polynomial growth conditions on the coefficients of the equation which enables us to study the boundedness more applicably. Methods and techniques developed here have the potential to be applied in other unbounded delay stochastic differential equations.

scholarly journals Multivalued monotone stochastic differential equations with jumps

2017 ◽  
Vol 17 (03) ◽  
pp. 1750018 ◽  
Author(s):  
Lucian Maticiuc ◽  
Aurel Răşcanu ◽  
Leszek Słomiński
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Monotone Operator ◽  
General Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Stability Of Solutions ◽  
Yosida Approximation ◽  
Methods Of Approximation

We study multivalued stochastic differential equations (MSDEs) with maximal monotone operators driven by semimartingales with jumps. We discuss in detail some methods of approximation of solutions of MSDEs based on discretization of processes and Yosida approximation of the monotone operator. We also study the general problem of stability of solutions of MSDEs with respect to the convergence of driving semimartingales.

scholarly journals Phase portraits, Lyapunov functions, and projective geometry

2020 ◽  
Author(s):  
Lilija Naiwert ◽  
Karlheinz Spindler
Keyword(s):  
Dynamical System ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Phase Portrait ◽  
Projective Geometry ◽  
Lyapunov Functions ◽  
Phase Portraits ◽  
Classroom Experiences

AbstractWe discuss two problems which grew out of an introductory differential equations class but were solved only later, each after having been put into a different context. First, how do you find a rather complicated Lyapunov function with your bare hands, without using a fully developed theory (while reconstructing the steps leading up to such a theory)? Second, how can you obtain a global picture of the phase-portrait of a dynamical system (thereby invoking ideas from projective geometry)? Since classroom experiences played an important part in the making of this paper, didactical aspects will also be discussed.

scholarly journals Asymptotic Behavior of Positive Solutions of a Competitive System Subject to Environmental Noise

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Huili Xiang ◽  
Zhuang Fang ◽  
Zuxiong Li ◽  
Zhijun Liu
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Stochastic Differential Equations ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Environmental Noise ◽  
Competitive System ◽  
Stochastic Boundedness

A competitive system subject to environmental noise is established. By using the theory of stochastic differential equations and Lyapunov function, sufficient conditions for the existence, uniqueness, stochastic boundedness, and global attraction of the positive solution of the above system are established, respectively. An example together with its corresponding numerical simulations is presented to confirm our analytical results.

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