Smoothing algorithms for state–space models

Two-filter smoothing is a principled approach for performing optimal smoothing in non-linear non-Gaussian state–space models where the smoothing distributions are computed through the combination of ‘forward’ and ‘backward’ time filters. The ‘forward’ filter is the standard Bayesian filter but the ‘backward’ filter, generally referred to as the backward information filter, is not a probability measure on the space of the hidden Markov process. In cases where the backward information filter can be computed in closed form, this technical point is not important. However, for general state–space models where there is no closed form expression, this prohibits the use of flexible numerical techniques such as Sequential Monte Carlo (SMC) to approximate the two-filter smoothing formula. We propose here a generalised two-filter smoothing formula which only requires approximating probability distributions and applies to any state–space model, removing the need to make restrictive assumptions used in previous approaches to this problem. SMC algorithms are developed to implement this generalised recursion and we illustrate their performance on various problems.     

Large deviations for invariant measures of general stochastic reaction–diffusion systems

In this paper we prove a large deviations principle for the invariant measures of a class of reaction–diffusion systems in bounded domains of $\text{R}{}^{\mathrm{d}}\text{,}\text{d?1}$, perturbed by a noise of multiplicative type. We consider reaction terms which are not Lipschitz-continuous and diffusion coefficients in front of the noise which are not bounded and may be degenerate. To cite this article: S. Cerrai, M. Röckner, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Maximum principle for differential games of forward–backward stochastic systems with applications

This paper is concerned with a maximum principle for both zero-sum and nonzero-sum games. The most distinguishing feature, compared with the existing literature, is that the game systems are described by forward–backward stochastic differential equations. This kind of games is motivated by linear-quadratic differential game problems with generalized expectation. We give a necessary condition and a sufficient condition in the form of maximum principle for the foregoing games. Finally, an example of a nonzero-sum game is worked out to illustrate that the theories may find interesting applications in practice. In terms of the maximum principle, the explicit form of an equilibrium point is obtained.     

An Ito–Taylor weak 3.0 method for stochastic dynamics of nonlinear systems

A new framework for development of order 3.0 weak Taylor scheme towards stochastic modeling and dynamics of coupled nonlinear systems is presented. The proposed method is derived by including third order multiple stochastic integral terms of Ito–Taylor expansion and developing them for a wide class of stochastic nonlinear systems. For computing the system responses of linear and a wide class of nonlinear structural systems, the use of lower order integration schemes is sufficient. But for highly non-linear stochastically driven systems like base isolated hysteretic systems and degrading stochastic systems the evaluation of higher order terms is necessary. Additionally, the use of higher order integration schemes for stochastic dynamics of higher dimensional nonlinear systems remains a challenge due to the arising mathematical complexities with the increase in the number of DOFs (degrees-of-freedom) which really necessitates the development of the proposed algorithm. The proposed algorithm is verified using a representative class of coupled nonlinear system in presence and absence of nonlinear degradation and hysteretic oscillators. The efficiency of the proposed numerical scheme over classical integration schemes is demonstrated through a practical engineering problem. Finally, an automated extension of the proposed algorithm is presented by generalizing it for a system of N-DOFs.     

Optimal control for 3D stochastic Navier–Stokes equations

Loeb space methods are used to prove existence of an optimal control for general 3D stochastic Navier–Stokes equations with multiplicative noise. The possible non-uniqueness of the solutions mean that it is necessary to utilize the notion of a non-standard approximate solution developed in the paper by N.J. Cutland and Keisler H.J. 2004, Global attractors for 3-dimensional stochastic Navier–Stokes equations, Journal of Dynamics and Differential Equations, pp. 16205–16266, for the study of attractors.     

A Fokker–Planck control framework for multidimensional stochastic processes

An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker–Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang–Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka–Volterra model and a noised limit cycle model.     

A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems

A general formulation of the Fokker–Planck–Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHSs). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example.     

Impulsive control of multiple Lotka–Volterra systems

In this work, we consider the control problem of multiple Lotka–Volterra system. Our means to control the population dynamics is via impulses not only in a single species, but also in multiple species, that is, some members of these populations are added to or removed from the environment impulsively at the same time. We establish the strategies for preventing all the species from going extinct by stabilizing some special positive points, which may not be the equilibrium points of the system. We give several Lotka–Volterra systems to illustrate our results by drawing their time-series graphs.     

A multirate W-method for electrical networks in state–space formulation

Subunits of coupled technical systems typically behave on differing time scales, which are often separated by several orders of magnitude. An ordinary integration scheme is limited by the fastest changing component, whereas so-called multirate methods employ an inherent step size for each subsystem to exploit these settings. However, the realization of the coupling terms is crucial for any convergence. Thus the approach to return to one-step methods within the multirate concept is promising. This paper introduces the multirate W-method for ordinary differential equations and gives a theoretical discussion in the context of partitioned Rosenbrock–Wanner methods. Finally, the MATLAB implementation of an embedded scheme of order (3)2 is tested for a multirate version of Prothero–Robinson's equation and the inverter-chain-benchmark.     

Optimal variational principle for backward stochastic control systems associated with Lévy processes

The paper is concerned with optimal control of backward stochastic differential equation (BSDE) driven by Teugel’s martingales and an independent multi-dimensional Brownian motion,where Teugel’s martin- gales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see e.g.,Nualart and Schoutens’ paper in 2000).We derive the necessary and sufficient conditions for the existence of the op- timal control by means of convex variation methods and duality techniques.As an application,the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (or backward linear-quadratic problem,or BLQ problem for short) is discussed and characterized by a stochastic Hamilton system.     

Robust L ∞ reliable control for impulsive switched nonlinear systems with state delay

This paper investigates the problem of robust L _∞ reliable control for a class of uncertain impulsive switched nonlinear systems with time-delay in the presence of actuator failure. Based on the dwell time approach, we firstly obtain a sufficient condition of exponential stability for the impulsive switched nonlinear system with time-delay, and L _∞ performance for the considered system is also analyzed. Then, based on above results, a state feedback controller, which guarantees the exponential stability with L _∞ performance of the corresponding closed-loop system, is constructed. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method.     

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.     

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.     

Approximate the dynamical behavior for stochastic systems by Wong–Zakai approaching

Random invariant manifolds and foliations play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations. In a general way, these random objects are difficult to be visualized geometrically or computed numerically. The current work provides a perturbation approach to approximate these random invariant manifolds and foliations. After briefly discussing the existence of random invariant manifolds and foliations for a class of stochastic systems driven by additive noises, the corresponding Wong–Zakai type of convergence result in path-wise sense is established.