Stabilization of stochastic systems under Markovian switching

The problem of state feedback stabilization of discrete-time stochastic processes under Markovian switching is considered. The jump Markovian switching is modeled by a discrete-time Markov chain, and the noise or stochastic environmental disturbance is modeled by a sequence of identically independently normally distributed random variables. Necessary and sufficient conditions based on linear matrix inequalities (LMI’s) for stochastic stability is obtained. The proposed control law for this stochastic stabilization result depends on the mode of the system as well as the environmental disturbances. The robustness results of such stability concepts against all admissible uncertainties are also investigated. An example is given to demonstrate the obtained results.     

Robust Process Control by Dynamic Stochastic Programming

Unnecessarily conservative behavior of standard process control techniques can be avoided by stochastic programming models when the distribution of random disturbances is known. In an earlier study we have investigated such an approach for tank level constraints of a distillation process. Here we address techniques that have accelerated the numerical solution of the large and expensive stochastic programs by a factor of six, and then present a refined optimization model for the same application. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Noise-induced Transitions for an SIV Epidemic Model with Medical-resource Constraints

We consider a stochastically forced epidemic model with medical-resource constraints. In the deterministic case, the model can exhibit two type bistability phenomena, i.e., bistability between an endemic equilibrium or an interior limit cycle and the disease-free equilibrium, which means that whether the disease can persist in the population is sensitive to the initial values of the model. In the stochastic case, the phenomena of noise-induced state transitions between two stochastic attractors occur. Namely, under the random disturbances, the stochastic trajectory near the endemic equilibrium or the interior limit cycle will approach to the disease-free equilibrium. Besides, based on the stochastic sensitivity function method, we analyze the dispersion of random states in stochastic attractors and construct the confidence domains (confidence ellipse or confidence band) to estimate the threshold value of the intensity for noise caused transition from the endemic to disease eradication.     

Nonnormal deterministic equivalents and a transformation in stochastic mathematical programming

This paper considers random variables of the continuous type in a stochastic programming problem and presents (1) a general approach to the development of deterministic equivalents of constraints to be satisfied within certain probability limits, and (2) a deterministic transformation of a stochastic programming problem with random variables in the objective function. Deterministic equivalents are developed for constraints containing uniform random variables, but the approach used can be applied to other types of continuous random variables, as well. When the random variables appear in the objective function, a deterministic transformation of the stochastic programming problem is obtained to yield a closed-form solution without resort to a Monte Carlo computer simulation. Extension of this approach to stochastic problems with discrete random variables and integer decision variables is discussed briefly. A numerical example is presented.     

一类随机时延网络控制系统的神经网络自适应容错控制

针对一类随机时延网络控制系统,提出一种基于RBF神经网络自适应动态补偿的容错控制策略.该方法通过在线估计时延将系统建模为随机切换系统,并在模型参考自适应方法的基础上设计RBF神经网络动态补偿容错控制器,利用Lyapunov稳定性理论给出神经网络补偿器的在线权值学习算法,以保证网络控制系统在故障情况下的跟踪性能和状态一致最终有界稳定.最后通过仿真验证了该方法的有效性.     

Distribution functions in stochastic programs with recourse: A parametric analysis

This paper studies the behavior of the optimum value of a two-stage stochastic program with recourse (random right-hand sides) as the mean and covariance matrices defining the random variables in the program are perturbed. Several results for convex programs are developed and are used to study the effect such perturbations have on the regularity properties of the stochastic programs. Cost associated with incorrectly specifying the mean and covariance matrices are discussed and estimated. A stochastic programming model in which the random variable is dependent on the first-stage decision is presented.     

Optimal Guaranteed Cost Control of Stochastic Discrete-Time Systems with States and Input Dependent Noise Under Markovian Switching

A problem of robust guaranteed cost control of stochastic discrete-time systems with parametric uncertainties under Markovian switching is considered. The control is simultaneously applied to both the random and the deterministic components of the system. The noise (the random) term depends on both the states and the control input. The jump Markovian switching is modeled by a discrete-time Markov chain and the noise or stochastic environmental disturbance is modeled by a sequence of identically independently normally distributed random variables. Using linear matrix inequalities (LMIs) approach, the robust quadratic stochastic stability is obtained. The proposed control law for this quadratic stochastic stabilization result depended on the mode of the system. This control law is developed such that the closed-loop system with a cost function has an upper bound under all admissible parameter uncertainties. The upper bound for the cost function is obtained as a minimization problem. Two numerical examples are given to demonstrate the potential of the proposed techniques and obtained results.     

A developed solution of the stochastic Milne problem using probabilistic transformations

This paper considers the solution of the Milne problem of radiative transfer with isotropic scattering in a continuous stochastic medium. Properties of the medium are assumed to be continuous random functions of the spatial dimensions. The available solutions - in literature - for this stochastic integro-differential equation (SIDE) are represented only by the ensemble average of the radiant energy density. In this paper, a developed algorithm, based on the implementation of the random variable transformation technique together with an integral transformation to the stochastic properties, is introduced. A complete stochastic solution represented by the probability-density function (p.d.f) of the radiant energy density is obtained. Using the closed form of the p.d.f, the nth moment of the stochastic solution is evaluated. In realization of this work, Exponential and Gaussian statistics for the medium properties are assumed. Results are physically acceptable and found to be compatible with those in the literature.     

A stochastic dynamic model of train-track-bridge coupled system based on probability density evolution method

An innovative stochastic dynamic model of a 3D train-track-bridge coupled system (TTBS) with refined wheel/rail interaction is established for a high-speed railway based on the random theory of probability density evolution method (PDEM). The multi-coupling effect of excitations can be simultaneously input into the new model, e.g. random track irregularity, random vehicle loads, stochastic system parameters, et al. Moreover, a new approach, named “Number theoretic method of multi-target probability functions” (NTM-mp), is developed to obtain the discrete point sets of multidimensional random parameters in hypercube space, aims to solve the point design of system uncertainty. The stochastic harmonic function (SHF) is applied to generate representative random track irregularity samples. The results of TTBS got by PDEM are verified with several typical case studies for its efficiency and reliability, which are the deterministic results in the representative publication, the Monte Carlo method (MCM) results, and the field testing results on the high-speed railway. At last, a typical case study of TTBS on a high-speed railway is presented for numerical analysis. Discussions and significant conclusions on the random dynamic responses are presented.     

On dynamic programming for sequential decision problems under a general form of uncertainty

We study the applicability of the method of Dynamic Programming (DP) for the solution of a general class of sequential decision problems under uncertainty, that may more commonly be referred to as discrete-time control problems under uncertainty. The uncertainty is due to the fact that the evolution of the state of the controlled system is affected by disturbances that are only known to belong to random sets, whose distributions are given a-priori. This includes as special cases the well known stochastic control problem and the robust min-max problem.     

Dominance stochastic models in data envelopment analysis

In this paper stochastic models in data envelopment analysis (DEA) are developed by taking into account the possibility of random variations in input-output data, and dominance structures on the DEA envelopment side are used to incorporate the modelbuilder's preferences and to discriminate efficiencies among decision making units (DMUs). The efficiency measure for a DMU is defined via joint dominantly probabilistic comparisons of inputs and outputs with other DMUs and can be characterized by solving a chance constrained programming problem. Deterministic equivalents are obtained for multivariate symmetric random errors and for a single random factor in the production relationships. The goal programming technique is utilized in deriving linear deterministic equivalents and their dual forms. The relationship between the general stochastic DEA models and the conventional DEA models is also discussed.     

Nonparametric adaptive control of discounted stochastic systems with compact state space

We are concerned with discrete-time stochastic control models for which the random disturbances are independent with a common unknown distribution. When the state space is compact, we prove that mild continuity conditions are sufficient to obtain adaptive policies which are asymptotically optimal with respect to the discounted reward criterion.This research was supported in part by the Consejo Nacional de Ciencia y Tecnología (CONACYT), Mexico City, Mexico, under Grant No. PCEXCNA-050156.     

Stochastic simulations based on mixed finite elements for solute transport in groundwater

An accurate mixed finite element method to solve both flow and transport is developed for stochastic simulations of transport in saturated aquifers characterized by random log-hydraulic conductivity fields. The main advantage of the mixed finite element is that it is local mass conservative. Unlike in stochastic finite element methods, this approach yields concentration fields and concentration moments for samples of the random field. In this way, it will be possible to analyze the behavior of different ensemble average observables of the transport process as well as the behavior of their fluctuations. Results of the stochastic simulations described here can be used to assess the reliability for real cases of the ensemble average quantities provided by stochastic modeling of transport in groundwater. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of Quasilinear Stochastic Problems in Abstract Colombeau Algebras

The main object of study is the stochastic Cauchy problem for a quasilinear equation with random disturbances in the form of a Hilbert-valued white noise process and with an operator generating an integrated semigroup in the space L²(R). We use the Colombeau theory of multiplication of distributions to introduce an abstract stochastic factor algebra and construct an approximate solution of the problem in this algebra.     

The dynamic multilevel assignment problem as a stochastic extremal process

This paper considers the multilevel assignment problem (i.e. the assignment problem where the supply alternatives are ranked in hierarchical levels) under the assumption that the utility components for each pairwise matching are stochastic. A dynamic version of the multilevel stochastic assignment model is developed, where both demand and supply evaluate alternatives according to a stochastic extremal process, i.e. a process where the maximum of a sequence of random variables is taken into account. The probability distributions of the random variables which describe the joint dynamic behaviour of demand and supply are found. It is also shown that the assignment probabilities assume the structure of a nested-logit model.     

Propagation of Disturbances in a Random Medium

In this paper we are concerned with the determination of thestochastic rays along which light disturbances propagate ina random inhomogeneous, time-dependent and isotropic medium.The analysis is formulated in the framework of stochastic optimalcontrol. We state an appropriate stochastic Fermat's Principlewhich upon invoking the principle of optimality in dynamic programmingleads to a parabolic functional differential equation for thetraversal time of a wavefront, the randomness entering as anadditive white-noise in the displacements of the field. Whenthe randomness constitutes small perturbations to the mean velocityfield, the problem becomes one of singular perturbation of theHamilton-Jacobi equation by a small second order term. Approximateexpressions are presented for the traversal time and mean stochasticpath for a time-independent mean-velocity of propagation. Asa specific example we consider these expressions for propagationsin a stratified medium.     

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.