Analysis of stability of stochastic dynamic systems under Poisson perturbations. II. Stability of solutions in quadratic mean of systems of linear stochastic differential equations under Poisson perturbations

Necessary and sufficient conditions of asymptotic stability in quadratic mean are obtained for trivial solutions of systems of linear stochastic differential equations under Poisson perturbations. Model problems are analyzed. Part I of this article is published in No. 4 (2005). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 50–66, November–December 2005.     

Construction and Implementation of General Linear Methods for Ordinary Differential Equations: A Review

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge–Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge–Kutta stability is also briefly discussed     

Stability Analysis of Stochastic Dynamic Systems under Poisson Perturbations. I. General Analysis of Stability of Solutions of Stochastic Differential Equations under Poisson Perturbations

Using the second Lyapunov method, the exponential p-stability, mean square stability, p-stability, and stability with probability 1 are established for nonlinear stochastic differential equations under Poisson perturbations. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 66–78, July–August 2005.     

Study of a stochastic model of the“dangling spider” problem with an infinite previous history and poisson switchings

To study the stability of the stochastic“dangling spider” model, the second Lyapunov method is substantiated for stochastic functional differential equations with the entire previous history. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 79–105, July–August, 2000.     

Some Continuous Models of Inventory Control

The problem of optimal control of a manufacturing process is investigated. The dynamics of the process is described by a stochastic differential equation. Two methods of solution of this problem are considered. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 175–178, May–June 2005.     

Stability of diffusion stochastic functional differential equations with Markov parameters

The paper justifies the second Lyapunov method for diffusion stochastic functional differential equations with Markov parameters, which generalize stochastic diffusion equations without aftereffect. Analogs of Lyapunov stability theorems, which generalize the results for systems with finite aftereffect, are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 74–88, January–February 2008.     

A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. II

The investigation pursued in the previous article is continued. Using a general algorithm of calculating the optimum prediction for a random process, an optimum extrapolation estimate is found in explicit form for the decision of a nonlinear evolutionary differential equation in a Hilbert space with unbounded linear operators. If a differential equation contains a small nonlinearity, then such an estimate is developed as a series in powers of a small parameter. Part I of this article is published in No. 3 (2008). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 110–119, July–August 2008.     

Multiplicative stability criteria based on difference solutions of ordinary differential equations

For computer analysis of Lyapunov stability, multiplicative criteria are proposed that are based on difference approximations to solutions of the Cauchy problem. These criteria can be applied to ordinary differential equations in normal form and include the necessary and sufficient stability conditions. For a system of linear equations with constant coefficients, information on the characteristic polynomial of the coefficient matrix and its roots is not used. The stability analysis is combined with difference solution and simulation of error accumulation. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 127–142, January–February 2006.     

On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties–in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge–Kutta methods for nonlinear problems and for linear problems as well as implicit Runge–Kutta methods and multi step methods will be collected     

基于策略迭代的连续时间系统的随机线性二次最优控制

针对模型参数部分未知的随机线性连续时间系统, 通过策略迭代算法求解无限时间随机线性二次(LQ) 最优控制问题. 求解随机LQ最优控制问题等价于求随机代数Riccati 方程(SARE) 的解. 首先利用伊藤公式将随机微分方程转化为确定性方程, 通过策略迭代算法给出SARE 的解序列; 然后证明SARE 的解序列收敛到SARE 的解, 而且在迭代过程中系统是均方可镇定的; 最后通过仿真例子表明策略迭代算法的可行性.

     

Decomposition and stability of linear systems with multiplicative noise

This paper concerns a representation of solutions and the stability of linear systems with multiplicative white noise, which is described by a vector Ito stochastic differential equation. The solution can be represented as a finite product of exponential matrices if Lie algebra generated by system matrices is solvable. If Lie algebra is not solvable, it is shown by the decomposition principle of Lie algebra that the problem of solving an equation can be reduced to the problem of solving a set of equations, whose corresponding Lie algebra is simple. Noting the structure of the sample solution, we present a technique of obtaining asymptotic stability conditions of sample solutions w.p.1, in the pth-order moment and in the pth-mean moment. The necessary and/or sufficient conditions of stability in some stochastic sense are obtained under certain conditions.     

Optimal control models for interregional migration under social risks

An organizational two-level hierarchic system is used to examine control models for interregional migration under conditions of social risks. The Cauchy problem is used to solve the multidimensional transport equation (Problem A1) and a system of ordinary linear differential equations (Problem A2). For the models proposed, sufficient conditions for the existence of optimal control on the classes of functions designed are established, and stable numeral methods of searching for optimum solutions are developed. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 107–122, May–June 2006.     

Stability of dynamic systems with aftereffect with due regard for Markov perturbations

Lyapunov-Krasovskii functionals and infinitesimal operators are employed to analyze a system for asymptotic stochastic stability on the whole and asymptotic stability on the whole. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 134–146, November–December 2007.     

Stability in impulsive systems with markov perturbations in averaging scheme. 3. Weak convergence of solutions of impulsive systems

For a stochastic dynamic system with a small parameter, the uniform boundedness of the p-th moment of the solution (p > 1), the weak convergence of the solution of the system to the solution of Ito stochastic differential equation, and the weak convergence of normalized deviations are proved. The stability of linear systems with a small parameter and Markov perturbations is analyzed.     

Reducing an integer linear equation to an equivalent system

A method is proposed for the creation of a system of linear integer equations that is equivalent to a given linear integer equation. The method is based on the theorem stating that one linear integer equation can be transformed into two linear integer equations with the identical set of nonnegative integer solutions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 180–183, January–February 2006.     

On the Linear Stability of the Fifth-Order WENO Discretization

We study the linear stability of the fifth-order Weighted Essentially Non-Oscillatory spatial discretization (WENO5) combined with explicit time stepping applied to the one-dimensional advection equation. We show that it is not necessary for the stability domain of the time integrator to include a part of the imaginary axis. In particular, we show that the combination of WENO5 with either the forward Euler method or a two-stage, second-order Runge–Kutta method is linearly stable provided very small time step-sizes are taken. We also consider fifth-order multistep time discretizations whose stability domains do not include the imaginary axis. These are found to be linearly stable with moderate time steps when combined with WENO5. In particular, the fifth-order extrapolated BDF scheme gave superior results in practice to high-order Runge–Kutta methods whose stability domain includes the imaginary axis. Numerical tests are presented which confirm the analysis.     

Iterative procedure for solution of sylvester generalized matrix equation

An iteration procedure for solving Sylvester generalized matrix equation is proposed in this paper. The sufficient conditions of stability of the iteration procedure for solving this equation are obtained. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 183–186, May–June, 2000.     

Existence of the l-th moment of a solution to a stochastic functional-differential equation with the entire prehistory

The definition of a strong solution to a stochastic differential-functional equation with the entire prehistory is given, and basic inequalities required for obtaining existence and uniqueness theorems are proved. Global existence and uniqueness theorems are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 142–151, July–August 2008.     

Stability of bundles of dynamic systems and related problems

Solutions of a generalization of classical stability problems of linear systems are considered from a unified viewpoint of computational complexity. The stability problems are solved for bundles (generally speaking, continual) that are given by the degree of the characteristic polynomial and domain of admissible values of parameters. The stability criteria of the objects considered are proposed, which are based on the classical results obtained for linear systems. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 17–22, September–October, 1999.     

Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations

The nonlinear delay differential equation with exponential and quadratic nonlinearities is considered. It is assumed that the equation is exposed to stochastic perturbations of the white noise type, which are directly proportional to the deviation of the system state from the equilibrium point. Sufficient conditions for stability in probability of the zero and positive equilibriums of the considered system under stochastic perturbations are obtained. The research results are illustrated by numerical simulations. The proposed investigation procedure can be applied for arbitrary nonlinear stochastic delay differential equations with an order of nonlinearity higher than one. Copyright © 2016 John Wiley & Sons, Ltd.