Lyapunov functions and non-trivial stationary solutions of stochastic differential equations

2001 ◽  
Vol 16 (4) ◽  
pp. 303-317 ◽  
Author(s):  
Björn Schmalfuss
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lyapunov Functions ◽  
Stationary Solutions

scholarly journals Second-order stochastic differential equations: stability, dissipativity, periodicity. IV. - A survey

2021 ◽  
pp. 15-26
Author(s):  
Магомет Мишаустович Шумафов
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lyapunov Functions ◽  
Stationary Process ◽  
Stationary Solutions ◽  
Second Order ◽  
Dissipative Systems ◽  
Qualitative Properties ◽  
Fourth Part ◽  
Basic Facts

Настоящая статья является продолжением предыдущей статьи и представляет собой четвертую часть работы автора. В работе делается обзор результатов исследований качественных свойств решений стохастических дифференциальных уравнений и систем второго порядка. В первой части был дан краткий обзор результатов работ по стохастической устойчивости решений дифференциальных уравнений и систем второго порядка с использованием аппарата функций Ляпунова. Были приведены некоторые предварительные сведения из теории вероятностей и теории случайных процессов. Во второй части дана конструкция стохастических интегралов Ито и Стратоновича. В третьей части дано понятие стохастического дифференциала, приведена формула Ито дифференцирования сложной функции для стохастических дифференциалов, дано определение стохастического дифференциального уравнения в форме Ито и в форме Стратоновича, сформулирована теорема существования и единственности для решений стохастических дифференциальных уравнений. В настоящей, четвертой, части работы даются вкратце основные сведения из теории устойчивости стохастических дифференциальных уравнений Ито. Приводятся основные определения устойчивости в различных смыслах стохастических дифференциальных систем, формулируются основные общие теоремы об устойчивости в терминах существования функций Ляпунова, являющиеся стохастическими аналогами классических теорем Ляпунова об устойчивости. Дается понятие о стохастических диссипативных системах. Приводится теорема, дающая условия существования периодических и стационарных решений в терминах вспомогательных функций для дифференциальных уравнений со случайной периодической по времени правой частью, представляющей собой периодический или стационарный процесс. This paper is a continuation of the previous papers and presents the fourth part of the author’s work. The paper reviews results concerning qualitative properties of second-order stochastic differential equations and systems. In the first part we gave a short overview on stability of solutions of the second-order stochastic differential equations and systems by Lyapunov functions techniques and introduced some mathematical preliminaries from probability theory and stochastic processes. In the second part the construction of Ito’s and Stratonovich’s stochastic integrals is given. In the third part, analog of the chain rule for stochastic differentials (Ito’s formula) is presented. The stochastic differential equations in the sense of Ito and in the sense of Stratonovich are introduced. The existence and uniqueness theorem for solutions of stochastic differential equations is formulated. In the present fourth part of the work basic facts from the theory of stability of stochastic differential equations are briefly given. The basic definitions of stability in different senses of stochastic differential systems are presented, the basic general theorems on stability are formulated in terms of the existence of Lyapunov functions, which are stochastic analogs of the classical Lyapunov’s theorems on stability. The concept of stochastic dissipative systems is given. A theorem is formulated which gives conditions for existence of periodic and stationary solutions in terms of auxiliary functions for differential equations with a random periodic in time right-hand side, which is a periodic or stationary process.

STATIONARY SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH MEMORY AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 07 (05) ◽  
pp. 553-582 ◽  
Author(s):  
YURI BAKHTIN ◽  
JONATHAN C. MATTINGLY
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stationary Solution ◽  
Landau Equation ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equation ◽  
With Memory

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.

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.

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