Stochastic Partial Differential Systems Research Articles

Dynamic event–triggered security control for stochastic partial differential equation systems via a unified pointwise measurement/control strategy

This article focuses on the security control for parabolic partial differential equation systems driven by fractional Brownian motion based on a unified pointwise measurement/control strategy. Initially, by employing the Takagi–Sugeno fuzzy rule, nonlinear stochastic systems are transformed into a series of linear subsystems. Then, based on the improved dynamic event-triggered mechanism and pointwise measurements method, a security controller is presented to resist deception attacks. In addition, the Lyapunov direct method and some novel inequalities are used to ensure the considered system is mean-square exponentially stable, and the gain of the point controller is obtained. Finally, an application example is provided to verify the effectiveness of the proposed method.

A New Method to Compute Transition Probabilities in Multi‐Stable Stochastic Dynamical Systems: Application to the Wind‐Driven Ocean Circulation

AbstractThe Kuroshio Current in the North Pacific displays path changes on an interannual‐to‐decadal time scale. In an idealized barotropic quasi‐geostrophic model of the double‐gyre wind‐driven circulation under stochastic wind‐stress forcing, such variability can occur due to transitions between different equilibrium states. The high‐dimensionality of the problem makes it challenging to determine the probability of these transitions under the influence of stochastic noise. Here we present a new method to estimate these transition probabilities, using a Dynamical Orthogonal (DO) field approach. In the DO approach, the solution of the stochastic partial differential equations system is decomposed using a Karhunen–Loève expansion and separate problems arise for the ensemble mean state and the so‐called time‐dependent DO modes. The original method is first reformulated in a matrix approach which has much broader application potential to various (geophysical) problems. Using this matrix‐DO approach, we are able to determine transition probabilities in the double‐gyre problem and to identify transition paths between the different states. This analysis also leads to the understanding which conditions are most favorable for transition.

Recent Developments in Event-Triggered Control of Nonlinear Systems: An Overview

Event-triggered control is a most popular paradigm for transferring feedback information in an economical “as needed” manner. The study of event-triggered control can be traced back to the 1960s. Following significant advances on the topic of control over networks and the topic of nonlinear control systems over the last two decades, event-triggered control has quickly emerged as a major theoretical subject in control theory. Applications of event-triggered control are wide-spread ranging from embedded control systems and industrial control processes to unmanned systems and cyber-physical transportation networks. In this paper, we first review developments in the synthesis of event-triggered sampling mechanisms. Various event triggering mechanisms, such as static event trigger, dynamic event trigger, time-regularized event trigger, and event trigger with positive threshold offsets, are systematically discussed. Then, we study how to design a stabilizing controller that is robust with respect to the sampling errors. Finally, we review some recent results in the directions of self-triggered control, event-triggered tracking control and cooperative control, and event-triggered control of stochastic systems and partial differential equation systems. Practical applications of event-triggered control are also discussed.

Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions

Abstract In this paper, we introduce a stochastic partial differential equation model for the spatial dynamic of tumor–immune interactions. We perform numerical simulations in order to investigate the propagation of traveling waves in model system under the influence of random space-time fluctuations. One of methods is to solve a stochastic partial differential equation system for tumor–immune cell densities. The second method is to solve a stochastic partial differential algebraic equation system in order to assess the wave behavior of the solution in comparison with the deterministic approach. Finally, we discuss the implications of the model results.

Stochastic time-optimal control for time-fractional Ginzburg–Landau equation with mixed fractional Brownian motion

A theoretical approach for solving time-fractional stochastic Ginzburg–Landau equation with mixed fractional Brownian motion in Hilbert space is elaborated. Initially, the stochastic partial differential system is reformulated in the Hilbert space by using the properties of fractional order space and fractional Laplacian. We establish the existence of mild solutions by employing Mittag–Leffler functions, stochastic analysis, and Krasnoselskii’s fixed point theorem. A sufficient condition for the existence of a Lagrange optimal control problem is established via Balder’s theorem. Further, the existence of stochastic time-optimal control and stochastic optimal time are analyzed for the proposed control system. An example is given to illustrate the developed theory. Finally, an application to the stochastic optimal control of hydropower plant model is provided. The optimal control is termed as the amount of release of water through the reservoir and it is controlled with a suitable performance index.

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modeled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection–extraction problem.

Reduced basis method for the adapted mesh and Monte Carlo methods applied to an elliptic stochastic problem

In this paper, we consider a stochastic elliptic partial differential system and we aim to approximate the solution using the Monte Carlo method based on the finite elements method. To speed up the resolution and reduce the CPU time of computation, we propose to couple the reduced basis method with the adapted mesh method based on an a posteriori error estimate. Balancing the discretization and the Monte Carlo errors is very important to avoid performing an excessive number of iterations. Numerical experiments show and confirm the efficiency of our proposed algorithm.

Optimal control of a class of semi‐linear stochastic evolution equations with applications

In this study, the authors derive a new type of maximum principle for both stochastic evolution and parabolic type stochastic partial differential systems. The systems only require the Hamiltonian functional to be concave in the state variable rather than in both state and control variables. They also show a connection between these two types of systems. Finally, examples are given to illustrate the authors' theoretical results.

Robust Fuzzy Filter Design for Nonlinear Parabolic Partial Differential Systems with Continuous Wiener Noise and Discontinuous Poisson Noise

This paper addresses a robust \(H_{\infty }\) fuzzy filter design problem for nonlinear stochastic partial differential systems (NSPDSs) with continuous random fluctuation, discontinuous Poisson jumping noise, random external disturbance and measurement noise in the spatiotemporal domain. For NSPDSs, the robust \(H_{\infty }\) filter design problem through a measurement output needs to solve a complex second-order Hamilton Jacobi integral inequality. In order to simplify the design procedure, a fuzzy stochastic partial differential system based on a fuzzy interpolation approach is proposed to approximate the NSPDS. Then, the robust \(H_{\infty }\) fuzzy filter design problem can be reformulated as a diffusion matrix inequality (DMI) problem. Since the DMI problem is difficult to be solved via traditional algebraic techniques, we utilize the divergence theorem and Poincare inequality to transform the DMIs to a set of linear matrix inequalities (LMIs) which could be easily solved with the help of MATLAB LMI Toolbox. Finally, a robust state estimation problem of an ecology system with intrinsic spatiotemporal continuous Wiener noise and discontinuous Poisson jump fluctuation is provided as an example to illustrate the design procedure and to confirm the \(H_{\infty }\) filtering performance of the proposed \(H_{\infty }\) fuzzy filter design method.

Analysis on the stability of numerical schemes for a class of stochastic partial differential systems

ABSTRACTThe high-order accurate compact finite difference scheme which belongs to the finite difference methods is constructed to solve the system of partial differential equations with random noise. The error analysis and stability analysis are given and then the numerical simulation is executed. The simulation results verify the theoretical analysis results and have the faster computation speed and higher accuracy.

System Entropy Measurement of Stochastic Partial Differential Systems

System entropy describes the dispersal of a system’s energy and is an indication of the disorder of a physical system. Several system entropy measurement methods have been developed for dynamic systems. However, most real physical systems are always modeled using stochastic partial differential dynamic equations in the spatio-temporal domain. No efficient method currently exists that can calculate the system entropy of stochastic partial differential systems (SPDSs) in consideration of the effects of intrinsic random fluctuation and compartment diffusion. In this study, a novel indirect measurement method is proposed for calculating of system entropy of SPDSs using a Hamilton–Jacobi integral inequality (HJII)-constrained optimization method. In other words, we solve a nonlinear HJII-constrained optimization problem for measuring the system entropy of nonlinear stochastic partial differential systems (NSPDSs). To simplify the system entropy measurement of NSPDSs, the global linearization technique and finite difference scheme were employed to approximate the nonlinear stochastic spatial state space system. This allows the nonlinear HJII-constrained optimization problem for the system entropy measurement to be transformed to an equivalent linear matrix inequalities (LMIs)-constrained optimization problem, which can be easily solved using the MATLAB LMI-toolbox (MATLAB R2014a, version 8.3). Finally, several examples are presented to illustrate the system entropy measurement of SPDSs.

Mean square H∞ synchronization of coupled stochastic partial differential systems

In this paper, the criterion and control are considered for the mean square H∞ synchronization of coupled stochastic partial differential systems (SPDSs). Based on the integral Lyapunov-like functional and by virtue of completing squares technique, a sufficient criterion is provided to guarantee the mean square H∞ synchronization. The effect of spatial domain on the mean square H∞ synchronization is also embodied in this criterion. When the coupled SPDSs cannot achieve the mean square H∞ synchronization, the adaptive controllers are adopted for the coupled SPDSs, and the effectiveness of the adaptive controllers is verified via a rigorous mathematical analysis. When the number of the nodes in the complex networks is large, pinning control is a natural choice. The adaptive pinning control strategy is also presented and a criterion is obtained which guarantees the mean square H∞ synchronization of coupled SPDSs. Numerical experiments are also given to illustrate the correctness of our results.

Approximate controllability of nonlinear stochastic partial differential systems with infinite delay

This paper aims to investigate approximate controllability of stochastic nonlinear partial differential systems with infinite delay. In the systems under study, nonlinearity and control variable exist both in drift and diffusion terms, and controllability problems are considered in the framework with a novel Banach space, which not only leads to some difficulties in deriving the properties of interest but also bring some opportunities to study the system in a more general framework. With the help of a new fundamental lemma established in this paper and some useful inequality techniques, some improved results for approximate controllability of stochastic partial differential systems are obtained by using the Banach contraction theorem without introducing any additional restraints on the terms of the system. An example of stochastic heat equation is also provided to illustrate our results.

Identification of partially known non‐linear stochastic spatio‐temporal dynamical systems by using a novel partially linear Kernel method

The identification of non-linear stochastic spatio-temporal dynamical systems given by stochastic partial differential equations is of great significance to engineering practice, since it can always provide useful insight into the mechanism and physical characteristics of the underlying dynamics. In this study, based on the difference method for stochastic partial differential equations, a novel state-space model named multi-input–multi-output extended partially linear model for stochastic spatio-temporal dynamical system is proposed. A new Reproducing Kernel Hilbert Space-based algorithm named extended partially linear least square ridge regression is thus particularly developed for the identification of the extended partially linear model. Compared with existing identification methods available for spatio-temporal dynamics, the advantages of the proposed identification method include that (i) it can make full use of the partially linear structural information of physical models, (ii) it can achieve more accurate estimation results for system non-linear dynamics and (iii) the resulting estimated model parameters have clear physical meaning or properties closely related to the underlying dynamical system. Moreover, the proposed extended partially linear model also provide a convenient state-space model for system analysis and design (e.g. controller or filter design) of the class of non-linear stochastic partial differential dynamical systems.

Robust control design for nonlinear stochastic partial differential systems with Poisson noise: Fuzzy implementation

This study addresses the robust H∞ control design problem for nonlinear stochastic partial differential systems (NSPDSs) with Poisson noise under the environment of random external disturbance in the spatio-temporal domain. For NSPDSs with Poisson noise, the robust H∞ control design needs to solve a complex Hamilton–Jacobi integral inequality (HJII) for robust control despite random external disturbance. In general, it is very difficult to solve the nonlinear partial differential HJII. In order to simplify the design procedure, a fuzzy stochastic partial differential system is proposed to approximate the NSPDS based on fuzzy interpolation approach. Then a fuzzy stochastic spatial state space model is developed to represent the fuzzy stochastic partial differential system via the semi-discretization finite difference scheme and the Kronecker product. Based on this model the robust H∞ control design is proposed to achieve the robust control of NSPDSs via solving linear matrix inequalities (LMIs) instead of an HJII. The proposed robust fuzzy H∞ controller has an efficient ability to attenuate the effect of spatio-temporal external disturbance on the controlled output of NSPDSs from the area energy point of view. Finally, a robust H∞ control of the nervous system is given to confirm the control performance of the proposed robust control design method for NSPDSs with Poisson noise.

ANALYSIS OF VARIANCE-BASED MIXED MULTISCALE FINITE ELEMENT METHOD AND APPLICATIONS IN STOCHASTIC TWO-PHASE FLOWS

The stochastic partial differential systems have been widely used to model physical processes, where the inputs involve large uncertainties. Flows in random and heterogeneous porous media is one of the cases where the random inputs (e.g., permeability) are often modeled as a stochastic field with high-dimensional random parameters. To treat the high dimensionality and heterogeneity efficiently, model reduction is employed in both stochastic space and physical space. An analysis of variance (ANOVA)-based mixed multiscale finite element method (MsFEM) is developed to decompose the high-dimensional stochastic problem into a set of lower-dimensional stochastic subproblems, which require much less computational complexity and significantly reduce the computational cost in stochastic space, and the mixed MsFEM can capture the heterogeneities on a coarse grid to greatly reduce the computational cost in the spatial domain. In addition, to enhance the efficiency of the traditional ANOVA method, an adaptive ANOVA method based on a new adaptive criterion is developed, where the most active dimensions can be selected to greatly reduce the computational cost before conducting ANOVA decomposition. This novel adaptive criterion is based on variance-decomposition method coupled with sparse-grid probabilistic collocation method or multilevel Monte Carlo method. The advantage of this adaptive criterion lies in its much lower computational overhead for identifying the active dimensions and interactions. A number of numerical examples in two-phase stochastic flows are presented and demonstrate the accuracy and performance of the adaptive ANOVA-based mixed MsFEM.

RobustH∞Control for Linear Stochastic Partial Differential Systems with Time Delay

This paper investigates the problems of robust stochastic mean square exponential stabilization and robustH∞for stochastic partial differential time delay systems. Sufficient conditions for the existence of state feedback controllers are proposed, which ensure mean square exponential stability of the resulting closed-loop system and reduce the effect of the disturbance input on the controlled output to a prescribed level ofH∞performance. A linear matrix inequality approach is employed to design the desired state feedback controllers. An illustrative example is provided to show the usefulness of the proposed technique.

Robust stabilization design of nonlinear stochastic partial differential systems: Fuzzy approach

In this paper, we address the robust stabilization design problem for nonlinear stochastic partial differential systems (NSPDSs) in the presence of random external disturbances and measurement noise in the spatio-temporal domain. We propose a fuzzy stochastic partial differential system to approximate the NSPDSs based on fuzzy interpolation approach. Next, we develop a fuzzy stochastic spatial state space model to represent the fuzzy stochastic partial differential system by employing a semi-discretization finite difference scheme. Based on this model, we propose a robust fuzzy estimator-based stabilization controller to stabilize the NSPDSs. Furthermore, robust stochastic H∞ stabilization design is suggested to attenuate the effects of random external disturbances and measurement noise in the spatio-temporal domain from the area energy point of view. The LMI technique is applied to solve the issue of control gains and estimator gains of the controller by using a systematic control design procedure. Finally, a simulation is presented as an example to illustrate the design procedure and to confirm the performance of the proposed robust fuzzy estimator-based stabilization design for the NSPDSs.

Robust Stabilization Design for Stochastic Partial Differential Systems Under Spatio-temporal Disturbances and Sensor Measurement Noises

We investigate an H∞ robust observer-based stabilization design problem for linear stochastic partial differential systems (LSPDSs) under spatio-temporal disturbances and sensor measurement noises. A general theoretical H∞ robust observer-based stabilization method is introduced at the beginning for LSPDSs under intrinsic fluctuation, external disturbance and sensor measurement noise in the spatio-temporal domain. A complex Hamilton Jacobi integral inequality (HJII) needs to be solved when designing the robust H∞ observer-based stabilization for LSPDSs. For simplifying the design procedure, a stochastic state space model is first developed via the semi-discretization finite difference scheme to represent the stochastic partial differential system at each grid node. Then the stochastic state space models at all grid nodes are merged together into a stochastic spatial state space model. Based on this stochastic spatial state space model, an implementable H∞ robust stabilization design is proposed for LSPDSs via an iterative linear matrix inequality (ILMI) method. The proposed robust H∞ stabilization design can efficiently attenuate the effect of spatio-temporal external disturbances and measurement noises upon LSPDSs from the area energy point of view. Finally, a robust H∞ stabilization example simulation is shown to illustrate the design procedure and to confirm the performance of the proposed robust stabilization design method.

Stochastic optimal control for backward stochastic partial differential systems

This paper studies optimal controls for a class of backward stochastic partial differential systems in the abstract evolution form. Under the assumption of a convex control domain, necessary and sufficient conditions for an admissible control to be optimal are derived in the form of stochastic maximum principles by means of a convex variation method and a duality technique. As an application, the optimal control for a linear backward stochastic evolution equation (BSEE) with quadratic cost criteria (called BSEELQ problem) is discussed, and the corresponding optimal control is characterized via the stochastic Hamilton system which is a linear full-coupled forward–backward stochastic evolution equation (FBSEE) and consists of the state equation, the adjoint equation and the dual presentation of the optimal control.