Almost sure convergence of galerkin approximations for a heat equation with a random initial condition

We analyze probabilistic convergences of random Galerkin approximations for a heat equation with a random initial condition.

Almost sure L²-convergence results for both continuous time and discrete time Galerkin approximations are obtained by the Borel-Cantelli's lemma. A criterion for determining the sample size is suggested.     

On almost sure convergence of adaptive algorithms

We present an extension of the Furstenberg-Kesten theorem on the convergence of random matrices. This extension is applied to the study of almost sure convergence of certain adaptive algorithms. In particular, we establish that the NLMS algorithm is almost surely convergent under extremely weak necessary and sufficient conditions. We also discuss the relationship of sufficient conditions that have appeared in the literature with our results.     

Almost sure and mean square exponential stability of numerical solutions for neutral stochastic functional differential equations

In this paper, we investigate the almost sure and mean square exponential stability of the Euler method and the backward Euler method for neutral stochastic functional differential equations (NSFDEs). Moreover, the almost sure and pth moment exponential stability of exact solutions for NSFDEs are considered. It is shown that the Euler method and the backward Euler method can reproduce the property of almost sure and mean square exponential stability of exact solutions to NSFDEs under suitable conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.     

On almost sure sample stability of nonlinear stochastic dynamicsystems

In this note, an extension of Khasminskii's theorem on almost sure stability of linear stochastic differential equations to a class of nonlinear stochastic differential equations is presented. The necessary and sufficient conditions for almost sure stability are proved. It is shown that in the second-order case, the stable region can be exactly determined by studying the singular boundaries of one-dimensional diffusion processes. The authors present a modified form of Feller's criteria for classification of singular boundaries. The new criteria are equivalent to and much simpler for applications than Feller's criteria. Two examples of nonlinear stochastic dynamic systems with stable regions illustrate the application procedures     

On the almost sure and mean-square exponential convergence of somestochastic observers

It is pointed out that linear observers used for estimating the state of the discrete-time stochastic-parameter systems are both almost surely and mean-square (MS) exponentially convergent under the same conditions guaranteeing mean-square convergence. In addition to the mean-square convergence properties of linear observers constructed for mean-square stable stochastic-parameter systems, they also possess an almost-sure exponential convergence property, and the rate of MS convergence is exponential. This rate depends on the parameters used in the design     

思维进化算法的转移概率分析及几乎处处收敛性证明

思维进化算法已有的收敛性分析均是在依概率收敛意义下考虑的,而几乎处处收敛强于依概率收敛。在详细分析思维进化算法趋同算子和异化算子转移概率的基础上,利用种群最大适应度值函数描述思维进化算法的演化过程,将最大适应度值函数的进化过程转化为下鞅数列,并根据数学期望的性质和最大适应度值函数的特点,利用下鞅收敛定理严格证明了思维进化算法的几乎处处收敛性。     

一种改进型克隆选择算法及其几乎处处强收敛性研究

基于克隆选择原理,引入混沌机制和小生境技术,提出一种改进型克隆选择算法(ICSA).该算法比传统的克隆选择算法具有更好的种群多样性和全局寻优能力.以随机过程理论为数学工具,分析了ICSA所形成抗体种群的平均适应度函数的鞅性质,并由此得出算法几乎处处强收敛性的结论.进而证明了,当状态空间有限时,该算法能在有限步内以概率1收敛到全局最优.仿真实验表明,该算法能有效地抑制早熟,具有更好的全局收敛性.     

Almost sure convergence of stochastic approximation algorithms with non-additive noise

Stochastic approximation algorithms with non-additive noise are discussed. In studying strong convergence of such algorithms, traditionally one assumes that the iterates return to a bounded or compact set infinitely often, or that the function under consideration grows with certain rate. The usual projection algorithms require that the bounded projection region is known beforehand. It is desirable to weaken these ‘boundedness’ conditions. By introducing randomly varying truncations, Chen and Zhu (1986) achieved this for stochastic approximation algorithms with additive noise. Here, we extend their result to a more general setting.     

Almost sure convergence of iterative learning control for stochastic systems

This paper proposes an iterative learning control (ILC) algorithm with the purpose of controling the output of a linear stochastic system presented in state space form to track a desired realizable trajectory. It is proved that the algorithm converges to the optimal one a.s. under the condition that the product input-output coupling matrices are full-column rank in addition to some assumptions on noises. No other knowledge about system matrices and covariance matrices is required.     

Almost sure convergence of extremum seeking algorithm using stochastic perturbation

This paper presents an extremum seeking (ES) algorithm where the perturbation signal is a martingale difference sequence (m.d.s.) with a vanishing variance. The measurement noise at the plant output is modeled by a superposition of deterministic component, and a non-stationary colored noise signal. The optimizing set point of the uncertain reference-to-output equilibrium map is estimated by a stochastic approximation (SA)-type algorithm. The algorithm has a vanishing gain sequence dependent on the set point estimates. By utilizing powerful tools of the martingale convergence theory it is proved that with probability one the set point estimates converge to the optimizing equilibrium point, in spite of the presence of a measurement noise. This result is derived without requiring boundedness or any prior condition on the set point estimates.     

Almost surely continuous solutions of a nonlinear stochastic integral equation

A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ) = h(t, x(t; )) + _s k(t, s; )f(s, x(s; ); )d(s) is studied wheret S, a measure space with certain properties, , the supporting set of a probability measure space (,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be an almost surely continuousm-dimensional vector-valued stochastic process onS which is bounded with probability one for eacht S and which satisfies the equation almost surely. Several theorems are proved which give conditions such that a unique random solution exists. AMS (MOS) subject classifications (1970): Primary; 60H20, 45G99. Secondary: 60G99.     

pth moment and almost sure exponential stability of impulsive neutral stochastic functional differential equations with Markovian switching

In this paper, the problems on the pth moment and the almost sure exponential stability for a class of impulsive neutral stochastic functional differential equations with Markovian switching are investigated. By using the Lyapunov function, the Razumikhin-type theorem and the stochastic analysis, some new conditions about the pth moment exponential stability are first obtained. Then, by using the Borel–Cantelli lemma, the almost sure exponential stability is also discussed. The results generalise and improve some results obtained in the existing literature. Finally, two examples are given to illustrate the obtained results.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching

In this paper, we consider neutral stochastic delay differential equations with Markovian switching. Our key aim is to establish LaSalle-type stability theorems for the underlying equations. The key techniques used in this paper are the method of Lyapunov functions and the convergence theorem of nonnegative semi-martingales. The key advantage of our new results lies in the fact that our results can be applied to more general non-autonomous equations.     

Infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems

This paper studies exponential convergence index assignment of stochastic control systems from the viewpoint of backward stochastic differential equation. Like deterministic control systems, it is shown that the exact controllability of an open-loop stochastic system is equivalent to the possibility of assigning an arbitrary exponential convergence index to the solution of the closed-loop stochastic system, formed by means of suitable linear feedback of the states. As an application, a sufficient and necessary condition for the existence and uniqueness of the solution of a class of infinite horizon forward-backward stochastic differential equations is provided.     

不确定变时滞随机系统的鲁棒均方指数稳定性

研究了不确定变时滞随机系统的鲁棒均方指数稳定性问题, 不确定性是范数有界的. 通过构造Lyapunov泛函, 得到了基于线性矩阵不等式的鲁棒均方指数稳定的充分条件. 最后给出实例加以验证所提出方法的有效性.     

Stabilization in probability and mean square of controlled stochastic dynamical system with state delay

Although stochastic dynamical systems have received a great deal of attention in terms of stabilization studies, so far there are few works on controlled stochastic dynamical systems with state delay. In this paper, a controlled stochastic dynamical system represented by a stochastic differential equation with state delay is considered. Condition under which the system is exponentially stable in mean square and in probability is examined.