Solving approximately a prediction problem for stochastic jump-diffusion systems

In this paper, a new approach to solving a prediction problem for nonlinear stochastic differential systems with a Poisson component is discussed. In this approach, the prediction problem is reduced to an analysis of stochastic jump-diffusion systems with terminating and branching paths. The prediction problem can be approximately solved by using numerical methods for stochastic differential equations and methods for modeling inhomogeneous Poisson flows.     

Stabilization of Passive Nonlinear Stochastic Differential Systems by Bounded Feedback

Abstract

The purpose of this paper is to give a systematic method for global asymptotic stabilization in probability of nonlinear control stochastic differential systems the unforced dynamics of which are Lyapunov stable in probability. The approach developed in this paper is based on the concept of passivity for nonaffine stochastic differential systems together with the theory of Lyapunov stability in probability for stochastic differential equations. In particular, we prove that, as in the case of affine in the control stochastic differential systems, a nonlinear stochastic differential system is asymptotically stabilizable in probability provided its unforced dynamics are Lyapunov stable in probability and some rank conditions involving the affine part of the system coefficients are satisfied. Furthermore, for such systems, we show how a stabilizing smooth state feedback law can be designed explicitly. As an application of our analysis, we construct a dynamic state feedback compensator for a class of nonaffine stochastic differential systems.     

On the practical global uniform asymptotic stability of stochastic differential equations

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations (SDEs) by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of SDEs based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.     

The resonance theory for stochastic layers in nonlinear dynamic systems

A theory of stochastic layers is developed for a better understanding of the resonant mechanism of stochastic layers in nonlinear Hamiltonian systems. A criterion based on an accurate whisker map and resonant conditions is developed for prediction of the onset of resonance in the stochastic layer. The onset of a specific primary resonance between the periodic forcing and periodic orbit of the integrable Hamiltonian system in the stochastic layer is predicted analytically. A forced, twin-well Duffing oscillator is investigated as a sample problem for prediction of a specified resonance in the stochastic layer. Verification of the analytical prediction is carried out through a symplectic numerical integration scheme. The analytical and numerical results are in good agreement.     

Global stability analysis for stochastic coupled reaction–diffusion systems on networks

Coupled systems on networks (CSNs) can be used to model many real systems, such as food webs, ecosystems, metabolic pathways, the Internet, World Wide Web, social networks, and global economic markets. This paper is devoted to investigation of the stability problem for some stochastic coupled reaction–diffusion systems on networks (SCRDSNs). A systematic method for constructing global Lyapunov function for these SCRDSNs is provided by using graph theory. The stochastic stability, asymptotically stochastic stability and globally asymptotically stochastic stability of the systems are investigated. The derived results are less conservative than the results recently presented in Luo and Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493]. In fact, the system discussed in Q. Luo and Y. Zhang [Q. Luo, Y. Zhang, Almost sure exponential stability of stochastic reaction diffusion systems. Non-linear Analysis: Theory, Methods & Applications 71(12) (2009) e487–e493] is a special case of ours. Moreover, our novel stability principles have a close relation to the topological property of the networks. Our new method which constructs a relation between the stability criteria of a CSN and some topology property of the network, can help analyzing the stability of the complex networks by using the Lyapunov functional method.     

Multi-Model Communication and Data Assimilation for Mitigating Model Error and Improving Forecasts

Models for weather and climate prediction are complex, and each model typi-cally has at least a small number of phenomena that are poorly represented, such as perhaps the Madden-Julian Oscillation (MJO for short) or El Ni\~{n}o-Southern Oscillation (ENSO for short) or sea ice. Furthermore, it is often a very challenging task to modify and improve a complex model without creating new deficiencies. On the other hand, it is sometimes possible to design a low-dimensional model for a particular phenomenon, such as the MJO or ENSO, with significant skill, although the model may not represent the dynamics of the full weather-climate system. Here a strategy is proposed to mitigate these model errors by taking advantage of each model''s strengths. The strategy involves inter-model data assimilation, during a forecast simulation, whereby models can exchange information in order to obtain more faithful representations of the full weather-climate system. As an initial investigation, the method is examined here using a simplified scenario of linear models, involving a system of stochastic partial differential equations (SPDEs for short) as an imperfect tropical climate model and stochastic differential equations (SDEs for short) as a low-dimensional model for the MJO. It is shown that the MJO prediction skill of the imperfect climate model can be enhanced to equal the predictive skill of the low-dimensional model. Such an approach could provide a route to improving global model forecasts in a minimally invasive way, with modifications to the prediction system but without modifying the complex global physical model itself.     

Integral sliding mode control for T–S fuzzy descriptor systems

The present study focuses on designing the integral sliding-mode control (ISMC) for generalized Takagi–Sugeno (T–S) fuzzy singular stochastic systems by involving the Markovian jump type of system parameters. Distinct to the existing works, the present paper concerns about the derivation of sufficient conditions that ensure the global stability of considered T–S fuzzy stochastic singular Markovian jump systems with matched/mismatched uncertainties under fuzzy-based ISMC. In this regard, an improved fuzzy integral sliding manifold function with mode-dependent derivative-term coefficient is proposed where the matched uncertainties become unnecessary and the mismatched uncertainties have been disintegrated during the sliding mode phase. Based on Lyapunov stability theory, a suitable Lyapunov functional candidate by involving the information about the membership functions is constructed with singular and P-type matrices, the stochastic admissibility of corresponding sliding mode dynamics are derived. To proven the effectiveness of the proposed method, a physical experimental problem on cart and pendulum is adapted and simulated via the derived theoretical results and the corresponding results are provided.     

Synchronization for the coupled stochastic strict-feedback nonlinear systems with delays under pinning control

In this paper, a class of new coupled stochastic strict-feedback nonlinear systems with delays (CSFND) on networks without strong connectedness (NWSC) is considered, and the issue pertaining to the synchronization of the systems is discussed by pinning control. Towards CSFND, the controllers are approached by combining the back-stepping method and the design of virtual controllers. A key novel design ingredient is that the global Lyapunov function is obtained based on each Lyapunov function of stochastic strict-feedback nonlinear systems with delays (SFND). Moreover, a sufficient criterion is presented to realize the exponential synchronization by employing the graph theory and Lyapunov method. As a subsequent result, we apply the obtained theoretical results to the second-order oscillator systems and robotic arm systems. Meanwhile, numerical simulations are provided to demonstrate the validity and feasibility of our theoretical results.     

On the application of the asymptotic method of global instability in aeroelasticity problems

The asymptotic method of global instability developed by A.G. Kulikovskii is an effective tool for determining the eigenfrequencies and stability boundary of one-dimensional or multidimensional systems of sufficiently large finite length. The effectiveness of the method was demonstrated on a number of one-dimensional problems; and since the mid-2000s, this method has been used in aeroelasticity problems, which are not strictly one-dimensional: such is only the elastic part of the problem, while the gas flow occupies an unbounded domain. In the present study, the eigenfrequencies and stability boundaries predicted by the method of global instability are compared with the results of direct calculation of the spectra of the corresponding problems. The size of systems is determined starting from which the method makes a quantitatively correct prediction for the stability boundary.     

Hausdorff dimension of a random invariant set

The notion of random attractor for a dissipative stochastic dynamical system has recently been introduced. It generalizes the concept of global attractor in the deterministic theory. It has been shown that many stochastic dynamical systems associated to a dissipative partial differential equation perturbed by noise do possess a random attractor. In this paper, we prove that, as in the case of the deterministic attractor, the Hausdorff dimension of the random attractor can be estimated by using global Lyapunov exponents. The result is obtained under very natural assumptions. As an application, we consider a stochastic reaction-diffusion equation and show that its random attractor has finite Hausdorff dimension.     

A Stochastic/Perturbation Global Optimization Algorithm for Distance Geometry Problems

We present a new global optimization approach for solving exactly or inexactly constrained distance geometry problems. Distance geometry problems are concerned with determining spatial structures from measurements of internal distances. They arise in the structural interpretation of nuclear magnetic resonance data and in the prediction of protein structure. These problems can be naturally formulated as global optimization problems which generally are large and difficult. The global optimization method that we present is related to our previous stochastic/perturbation global optimization methods for finding minimum energy configurations, but has several key differences that are important to its success. Our computational results show that the method readily solves a set of artificial problems introduced by Moré and Wu that have up to 343 atoms. On a set of considerably more difficult protein fragment problems introduced by Hendrickson, the method solves all the problems with up to 377 atoms exactly, and finds nearly exact solution for all the remaining problems which have up to 777 atoms. These preliminary results indicate that this approach has very good promise for helping to solve distance geometry problems.     

Bayesian prediction of crack growth based on a hierarchical diffusion model

A general Bayesian approach for stochastic versions of deterministic growth models is presented to provide predictions for crack propagation in an early stage of the growth process. To improve the prediction, the information of other crack growth processes is used in a hierarchical (mixed‐effects) model. Two stochastic versions of a deterministic growth model are compared. One is a nonlinear regression setup where the trajectory is assumed to be the solution of an ordinary differential equation with additive errors. The other is a diffusion model defined by a stochastic differential equation where increments have additive errors. While Bayesian prediction is known for hierarchical models based on nonlinear regression, we propose a new Bayesian prediction method for hierarchical diffusion models. Six growth models for each of the two approaches are compared with respect to their ability to predict the crack propagation in a large data example. Surprisingly, the stochastic differential equation approach has no advantage concerning the prediction compared with the nonlinear regression setup, although the diffusion model seems more appropriate for crack growth. Copyright © 2016 John Wiley & Sons, Ltd.     

Incremental stability analysis of stochastic hybrid systems

In this paper, we study the incremental stability of stochastic hybrid systems, based on the contraction theory, and derive sufficient conditions of global stability for such systems. As a special case, the conditions to ensure the second moment exponential stability which is also called exponential stability in the mean square of stochastic hybrid systems are obtained. The theoretical results in this paper extend previous works from deterministic or stochastic systems to general stochastic hybrid systems, which can be applied to qualitative and quantitative analysis of many physical and biological phenomena. An illustrative example is given to show the effectiveness of our results.     

Net demand prediction for power systems by a new neural network‐based forecasting engine

Integration of renewable generations, such as wind and photovoltaic, into electrical power systems is rapidly growing throughout the world. Stochastic and variable nature of these resources makes some operational challenges to power systems. The most effective way to tackle these challenges is short‐term prediction of their available powers. Despite various developed methods to forecast generation of renewable resources, still they have large errors, which may lead to under/over‐commitment of conventional generators in power systems. Prediction of net demand (ND), defined as electrical load minus renewable generations, can provide useful information for accurate scheduling of conventional generators. In this article, characteristics of the time series of electric load, renewable generations and ND are analyzed, and a new hybrid prediction strategy is presented for direct prediction of ND. The training mechanism of the proposed forecasting engine is composed of a new stochastic search method and Levenberg–Marquardt learning algorithm based on an iterative procedure and greedy search. The suggested prediction strategy is tested on different real‐world power systems and its obtained results are compared with the results of several other forecast methods and published literature figures. These comparisons confirm the validity of the developed forecasting strategy. © 2016 Wiley Periodicals, Inc. Complexity 21: 296–308, 2016     

一类随机系统的有限时间镇定

讨论随机系统的有限时间镇定问题.首先提出了随机系统有限时间稳定的概念;其次证明了随机系统有限时间稳定的Lyapunov定理;然后,讨论了一类随机系统的镇定问题.     

Prediction of output response probability of sound environment system using simplified model with stochastic regression and fuzzy inference

The traditional standard stochastic system models, such as the autoregressive (AR), moving average (MA) and autoregressive moving average (ARMA) models, usually assume the Gaussian property for the fluctuation distribution, and the well-known least squares method is applied on the basis of only the linear correlation data. In the actual sound environment system, the stochastic process exhibits various non-Gaussian distributions, and there exist potentially various nonlinear correlations in addition to the linear correlation between input and output time series. Consequently, the system input and output relationship in the actual phenomenon cannot be represented by a simple model. In this study, a prediction method of output response probability for sound environment systems is derived by introducing a correction method based on the stochastic regression and fuzzy inference for simplified standard system models. The proposed method is applied to the actual data in a sound environment system, and the practical usefulness is verified.     

Nonparametric multi-step prediction in nonlinear state space dynamic systems

Filtering and smoothing of stochastic state space dynamic systems have benefited from several generations of estimation approaches since the seminal works of Kalman in the sixties. A set of global analytical or numerical methods are now available, such as the well-known sequential Monte Carlo particle methods which offer some theoretical convergence results for both types of problems. However except in the case of linear Gaussian systems, objectives of the third kind i.e. prediction objectives, which aim at estimating k time steps ahead the anticipated probability density function of the system state variables, conditional on past and present system output observations, still raise theoretical and practical difficulties. The aim of this paper is to propose a nonparametric particle multi-step prediction method able to consistently estimate such anticipated conditional pdf of the state variables as well as their expectations.     

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two‐by‐two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean‐value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix–vector multiplications for the off‐diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen–Loève expansion and a good preconditioned for the mean‐value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

A stochastic extra-step quasi-Newton method for nonsmooth nonconvex optimization

In this paper, a novel stochastic extra-step quasi-Newton method is developed to solve a class of nonsmooth nonconvex composite optimization problems. We assume that the gradient of the smooth part of the objective function can only be approximated by stochastic oracles. The proposed method combines general stochastic higher order steps derived from an underlying proximal type fixed-point equation with additional stochastic proximal gradient steps to guarantee convergence. Based on suitable bounds on the step sizes, we establish global convergence to stationary points in expectation and an extension of the approach using variance reduction techniques is discussed. Motivated by large-scale and big data applications, we investigate a stochastic coordinate-type quasi-Newton scheme that allows to generate cheap and tractable stochastic higher order directions. Finally, numerical results on large-scale logistic regression and deep learning problems show that our proposed algorithm compares favorably with other state-of-the-art methods.

     

Razumikhin method to global exponential stability for coupled neutral stochastic delayed systems on networks

Global exponential stability for coupled neutral stochastic delayed systems on networks (CNSDSNs) is investigated in this paper. By means of combining the Razumikhin method with graph theory, some sufficient conditions that can be verified easily are derived to ensure the global exponential stability for CNSDSNs. Finally, a specific model of CNSDSNs is discussed, and numerical test manifests the effectiveness of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.