System Of Parabolic Partial Differential Equations Research Articles

Adaptive Event-Triggered Control of Networked Fuzzy PDE Systems Under Hybrid Cyber Attacks

This study focuses on the security control problem of networked fuzzy partial differential equation (PDE) systems under hybrid cyber attacks. First, the nonlinear parabolic PDE system is exactly represented by a Takagi–Sugeno fuzzy PDE model. Second, with taking the insecure transmission network into consideration, a more practical stochastic hybrid cyber attacks model for PDE-based networked control systems is introduced. This model contains both denial-of-service attacks and deception attacks in a unified framework. Then, based on spatially pointwise nonuniform sampled-data measurements, an adaptive event-triggered control scheme with an adjustable threshold is newly designed to counter the hybrid cyber attacks’ impact while saving communication resources. Moreover, by constructing a novel Lyapunov–Krasovskii functional and employing some inequality techniques, stability conditions with less conservatism are derived. Finally, the developed method is applied to the Fisher equation to demonstrate its viability and superiority.

A reduced model for compressible viscous heat-conducting multicomponent flows

In the present paper we propose a reduced temperature non-equilibrium model for simulating multicomponent flows with inter-phase heat transfer, diffusion processes (including the viscosity and the heat conduction) and external energy sources. We derive three equivalent formulations for the proposed model. The first formulation consists of balance equations for partial densities, the mixture momentum, the mixture total energy, and phase volume fractions. The second formulation is symmetric and obtained by replacing the equations for the mixture total energy and volume fractions in the first formulation with balance equations for the phase total energy. Replacing one of the phase total energy equation of the second formulation with the mixture total energy equation gives the third formulation. All the three formulations assume velocity and pressure equilibrium across the material interface. These equivalent forms provide different physical perspectives and numerical conveniences. Temperature equilibration and continuity across the material interfaces are achieved with the instantaneous thermal relaxation. Temperature equilibrium is maintained during the heat conduction process. The proposed models are proved to respect the thermodynamical laws. For numerical solution, the model is split into a hyperbolic partial differential equation (PDE) system and parabolic PDE systems. The former is solved with the high-order Godunov finite volume method that ensures the pressure–velocity–temperature (PVT) equilibrium conduction. The parabolic PDEs are solved with both the implicit and the explicit locally iterative method (LIM) based on Chebyshev parameters. Numerical results are presented for several multicomponent flow problems with diffusion processes. Furthermore, we apply the proposed model to simulate the target ablation problem that is of significance to inertial confinement fusion. Comparisons with one-temperature models in literature demonstrate the ability to maintain the PVT property and superior convergence performance of the proposed model in solving multicomponent problems with diffusions.

Fault Isolation of a Class of Uncertain Nonlinear Parabolic PDE Systems

This paper proposes a novel fault isolation (FI) scheme for distributed parameter systems modeled by a class of parabolic partial differential equations (PDEs) with nonlinear uncertain dynamics. A key feature of the proposed FI scheme is its capability of dealing with the effects of system uncertainties for accurate FI. Specifically, an approximate ordinary differential equation (ODE) system is first derived to capture the dominant dynamics of the original PDE system. An adaptive dynamics identification approach using radial basis function neural network is then proposed based on this ODE system, to achieve locally-accurate identification of the uncertain system dynamics under faulty modes. A bank of FI estimators with associated adaptive thresholds are finally designed for real-time FI decision making. Rigorous analysis on the fault isolatability is provided. Simulation study on a representative transport-reaction process is conducted to demonstrate the effectiveness of the proposed approach.

Hybrid-Triggered H∞ Control for Parabolic PDE Systems Under Deception Attacks

A hybrid-triggered controller is designed in this article to analyze the stabilization of parabolic type partial differential equations (PDEs) with deception attacks and disturbances. The hybrid-triggered controller is designed by combining time-triggered and event-triggered controllers with the aid of a Bernoulli random variable. A nonlinear function is considered to describe the deception attack signal and the probability of occurrence of deception attack signals will be determined by a Bernoulli random variable. An <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance is utilized to attenuate the occurrence disturbances. We employ a Lyapunov-Krasovskii functional (LKF) to analyze the stabilization of the chosen PDE under the proposed controller and the stabilization conditions are obtained in terms of linear matrix inequalities (LMIs). Numerical examples are given finally to check the efficacy of the derived results.

Finite‐time local piecewise control for parabolic PDEs with ODE output feedback

This paper is devoted to the finite-time local piecewise control for parabolic partial differential equations (PDEs) by using dynamic output feedback control strategy, where the controller is designed as an ordinary differential equation (ODE). This makes the closed-loop system PDE-ODE coupled, which is employed to accurately describe the dynamics of the PDE system. According to the constructed PDE-ODE coupled model, a local piecewise dynamic feedback control law is first proposed. Sufficient conditions on finite-time stabilisation of the parabolic PDE-ODE coupled system by the suggested feedback controller are then developed in the sense of both complete spatial measurement and incomplete spatial measurement of the observed output of the PDE system, respectively. Finally, the issues regarding the finite-time stabilisation of the closed-loop system is converted into the feasibility of matrix inequalities, and some simulation studies are provided to verify the effectiveness of the proposed results.

Machine learning-based model predictive control of diffusion-reaction processes

In this work, we develop a machine-learning-based predictive control design for nonlinear parabolic partial differential equation (PDE) systems using process state measurement time-series data. First, the Karhunen-Loève expansion is used to derive dominant spatial empirical eigenfunctions of the nonlinear parabolic PDE system from the data. Then, these empirical eigenfunctions are used as basis functions within a Galerkin's model reduction framework to derive the temporal evolution of a small number of temporal modes capturing the dominant dynamics of the PDE system. Subsequently, feedforward neural networks (FNN) are used to approximate the reduced-order dominant dynamics of the parabolic PDE system from the data within a desired operating region. Lyapunov-based model predictive control (MPC) scheme using FNN models is developed to stabilize the nonlinear parabolic PDE system. Finally, a diffusion-reaction process example is used to demonstrate the effectiveness of the proposed machine-learning-based predictive control method.

Control Design for Parabolic PDE Systems via T–S Fuzzy Model

In this article, we investigate the parabolic partial differential equations (PDEs) systems with Neumann boundary conditions via the Takagi–Sugeno (T–S) fuzzy model. On the basis of the obtained T–S fuzzy PDE model, a novel fuzzy state controller which is associated with the boundary state of position and the mean value coefficient matrix derived through the mean value theorem of integral is designed to analyze the asymptotic stability of the parabolic PDE system. Without sampling the nonlinear parameter of the system, new stability conditions are deduced in the form of linear matrix inequalities (LMIs). Moreover, compared with the novel fuzzy state controller, more conservative conditions based on another fuzzy state controller are also provided. Finally, we explore the state-feedback controller into the Fisher equation as an application. Simulation results show that the proposed method is effective.

Finite-Time Fuzzy Bounded Control for Semilinear PDE Systems With Quantized Measurements and Markov Jump Actuator Failures.

This article presents a novel reliable fuzzy output feedback controller for a class of semilinear parabolic partial differential equation systems with Markov jump actuator failures. First, the control strategy's novelties include the following aspects: 1) the considered system is represented by using a fuzzy modeling approach, based on which a new asynchronous fuzzy observer is constructed via utilizing a series of discrete output signals that are induced by samplers and quantizers; 2) a novel Markov jump input model, which is more fit for real applications, is introduced to depict various stochastically occurring actuator faults; and 3) inspired by the above discussion, a reliable mode-dependent fuzzy piecewise control strategy, which only needs limited actuators, is developed. Then, some new conditions, which can ensure that the closed-loop system is finite-time bounded, are established. Furthermore, some slave matrices are introduced to relax the strict constraints caused by asynchronous membership functions. Finally, two simulation examples are provided to support the validity of the proposed method.

Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control

We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equation fit within this unified framework. Secondly, we introduce the ongoing project SCRIMP (Simulation and Control of Interactions in Multi-Physics) developed for the numerical simulation of infinite-dimensional pHs. SCRIMP notably relies on the FEniCS open-source computing platform for the finite element spatial discretization. Finally, we illustrate how to solve the considered model problems within this framework by carefully explaining the methodology. As additional support, companion interactive Jupyter notebooks are available.

Fuzzy reduced-order filtering for nonlinear parabolic PDE systems with limited communication

This paper investigates a fuzzy reduced-order filter design for a class of nonlinear partial differential equation (PDE) systems. First, a Takagi-Sugeno (T-S) fuzzy model is considered to reconstruct the nonlinear PDE system. Then, the employment of an event-triggered mechanism (ETM) can effectively avoid signal redundancy and improve network resource utilization. Furthermore, based on the advantages of the fuzzy model and ETM, several Lyapunov functions are designed and the proposed filter parameters are obtained by adopting linear matrix inequality methods to satisfy the asymptotic stability condition with H∞ performance. Finally, a simulation example is presented to demonstrate the practicality and effectiveness of the proposed filter design method.

Fault Detection of a Class of Nonlinear Uncertain Parabolic PDE Systems

This letter investigates the fault detection (FD) problem of a class of uncertain distributed parameter systems modeled by nonlinear parabolic partial differential equations (PDEs). A novel FD scheme is proposed with a neural network-based adaptive dynamics learning approach. Specifically, based on the Galerkin method, a finite dimensional ordinary differential equation (ODE) system is first derived to capture the dominant dynamics of the PDE system. An adaptive dynamics learning approach using radial basis function neural networks (RBF NN) is then developed to achieve locally accurate identification of the associated uncertain system dynamics. The learned knowledge can be obtained and stored in a bank of constant RBF NN models, which can be used to further construct a bank of FD estimators. A threshold is finally designed for real-time FD decision making. One important feature of the proposed FD scheme is that the spatio-temporal uncertain dominant dynamics of the parabolic PDE system can be locally accurately identified, such that occurring faults can be effectively distinguished from system uncertainties for accurate FD. Extensive simulation studies are conducted to verify the effectiveness and advantages of the proposed FD scheme.

Guaranteed cost design for controlling semilinear parabolic PDE systems with mobile collocated actuators and sensors

This paper investigates the guaranteed cost design problem for controlling a class of semilinear parabolic partial differential equation (PDE) systems using mobile collocated actuators and sensors. Initially, a mode indicator function is employed to indicate the different modes for all actuator/sensor pairs according to whether each actuator/sensor pair is static or mobile. Subsequently, a mode-dependent switching control scheme is proposed and the well-posedness of the closed-loop PDE system is also analysed. Then, based on Lyapunov direct method, an integrated design of switching controllers and mobile actuator/sensor guidance laws is developed in the form of linear matrix inequalities, such that the closed-loop PDE system is exponentially stable while providing an upper bound for the prescribed quadratic cost function. Moreover, a suboptimal guaranteed cost design problem is also addressed to make the cost bound as small as possible. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method.

Fuzzy Control Design of Nonlinear Time-Delay Parabolic PDE Systems Under Mobile Collocated Actuators and Sensors.

Via mobile sensing measurements, this study applies the Takagi-Sugeno (T-S) fuzzy model to deal with the mobile fuzzy control design problem for nonlinear time-delay parabolic partial differential equation (PDE) systems. Initially, we use a T-S fuzzy model to accurately represent the nonlinear time-delay parabolic PDE system. Subsequently, under the assumption that the actuators and sensors are collocated while the spatial domain is divided by several subdomains, a control scheme containing the fuzzy controllers and the guidance of mobile actuator/sensor pairs is proposed based on the obtained T-S fuzzy model, where the projection modification guidance to be designed can guarantee that each mobile actuator/sensor pair moves within the prescribed area. Then, using the Lyapunov direct method and integral inequalities, a membership-function-dependent design of fuzzy controllers plus mobile actuator/sensor guidance laws is developed to render the resulting closed-loop time-delay system exponentially stable. Moreover, the exponential decay rate can also be increased by the proposed mobile guidance laws. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method and the application of mobile actuator/sensor pairs contributes to accelerating the convergence speed of the closed-loop state.

Simulation of THz Oscillations in Semiconductor Devices Based on Balance Equations

Instabilities of electron plasma waves in high-mobility semiconductor devices have recently attracted a lot of attention as a possible candidate for closing the THz gap. Conventional moments-based transport models usually neglect time derivatives in the constitutive equations for vectorial quantities, resulting in parabolic systems of partial differential equations (PDE). To describe plasma waves however, such time derivatives need to be included, resulting in hyperbolic rather than parabolic systems of PDEs; thus the fundamental nature of these equation systems is changed completely. Additional nonlinear terms render the existing numerical stabilization methods for semiconductor simulation practically useless. On the other hand there are plenty of numerical methods for hyperbolic systems of PDEs in the form of conservation laws. Standard numerical schemes for conservation laws, however, are often either incapable of correctly handling the large source terms present in semiconductor devices due to built-in electric fields, or rely heavily on variable transformations which are specific to the equation system at hand (e.g. the shallow water equations), and can not be generalized easily to different equations. In this paper we develop a novel well-balanced numerical scheme for hyperbolic systems of PDEs with source terms and apply it to a simple yet non-linear electron transport model.

Exponential stabilization for 1-D linear Itô-type state-dependent stochastic parabolic PDE systems via static output feedback

This paper deals with the problem of exponential stabilization for 1-D linear stochastic parabolic partial differential equation (PDE) systems with state-multiplicative noise in the form of Itô type. A static output feedback (SOF) control scheme is proposed to stabilize the stochastic PDE system in a stochastic framework via locally collocated piecewise uniform actuators and sensors. The closed-loop well-posedness analysis is provided by virtue of the semi-group theory. Based on the infinite-dimensional infinitesimal operator and Lyapunov technique, the SOF controller design is developed to guarantee the exponential stability of the resulting closed-loop system in the mean square sense. Finally, simulation results on a stochastic heat equation are presented to verify the effectiveness of the proposed design method.

Estimator-Based $H_\infty$ Sampled-Data Fuzzy Control for Nonlinear Parabolic PDE Systems

This paper considers the estimator-based $H_{\infty }$ sampled-data fuzzy control (SDFC) problem of nonlinear parabolic partial differential equation (PDE) systems. First, a Takagi–Sugeno (T–S) fuzzy parabolic PDE model is proposed to represent the nonlinear PDE system. Second, with the aid of the T–S fuzzy PDE model, an estimator-based SDFC design ensuring the exponential stability of the closed-loop fuzzy PDE system with an $H_{\infty }$ performance is developed via a Lyapunov functional. The outcome of the estimator-based $H_{\infty }$ SDFC problem is formulated as a bilinear matrix inequality optimization problem, which is solved by an iterative algorithm on the basis of the linear matrix inequalities. Finally, for demonstrating the effectiveness of the proposed method, simulation results are provided to control the diffusion equation and the FitzHugh–Nagumo equation.

$H_{\infty }$ Sampled-Data Fuzzy Observer Design for Nonlinear Parabolic PDE Systems

This article considers the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> sampled-data fuzzy observer (SDFO) design problem for nonlinear parabolic partial differential equation (PDE) systems under spatially local averaged measurements (SLAMs). Initially, the nonlinear PDE system is accurately represented by the Takagi-Sugeno (T-S) fuzzy PDE model. Then, based on the T-S fuzzy PDE model, an SDFO under SLAMs is constructed for the state estimation. To attenuate the effect of the exogenous disturbance and the design disturbance, an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> SDFO design under SLAMs is developed in terms of linear matrix inequalities by utilizing Lyapunov functional and inequality techniques, which can guarantee the exponential stability and satisfy an H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance for the estimation error fuzzy PDE system. Finally, simulation results on the state estimation of the FitzHugh-Nagumo equation are given to support the presented H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> SDFO design method.

The inverse first passage time problem for killed Brownian motion

The classical inverse first passage time problem asks whether, for a Brownian motion $(B_{t})_{t\geq0}$ and a positive random variable $\xi$, there exists a barrier $b:\mathbb{R}_{+}\to\mathbb{R}$ such that $\mathbb{P}\{B_{s}>b(s),0\leq s\leq t\}=\mathbb{P}\{\xi>t\}$, for all $t\geq0$. We study a variant of the inverse first passage time problem for killed Brownian motion. We show that if $\lambda>0$ is a killing rate parameter and $𝟙_{(-\infty,0]}$ is the indicator of the set $(-\infty,0]$ then, under certain compatibility assumptions, there exists a unique continuous function $b:\mathbb{R}_{+}\to\mathbb{R}$ such that $\mathbb{E}[-\lambda\int_{0}^{t}𝟙_{(-\infty,0]}(B_{s}-b(s))\,ds]=\mathbb{P}\{\zeta>t\}$ holds for all $t\geq0$. This is a significant improvement of a result of the first two authors (Ann. Appl. Probab. 24 (2014) 1–33). The main difficulty arises because $𝟙_{(-\infty,0]}$ is discontinuous. We associate a semilinear parabolic partial differential equation (PDE) coupled with an integral constraint to this version of the inverse first passage time problem. We prove the existence and uniqueness of weak solutions to this constrained PDE system. In addition, we use the recent Feynman–Kac representation results of Glau (Finance Stoch. 20 (2016) 1021–1059) to prove that the weak solutions give the correct probabilistic interpretation.

Static output feedback stabilization for a linear parabolic PDE system with time-varying delay via mobile collocated actuator/sensor pairs

This paper addresses a static output feedback (SOF) stabilization problem for a linear parabolic partial differential equation (PDE) system with time-varying delay via mobile collocated actuator/sensor pairs. Initially, an SOF stabilization scheme via mobile actuator/sensor pairs is proposed for the delayed PDE system, where the spatial domain is divided into multiple subdomains according to the number of actuator/sensor pairs and the projection modification algorithm is employed to ensure each collocated actuator/sensor pair only can move within the respective subdomain. Subsequently, the well-posedness of the closed-loop delayed PDE system is analyzed using the operator semigroup theory. Then, by constructing an appropriate Lyapunov–Krasovskii functional candidate, a delay-dependent control-plus-guidance design is developed in the form of bilinear matrix inequalities (BMIs), such that the resulting closed-loop system is exponentially convergent and the mobile actuator/sensor guidance can improve the transient response of closed-loop state. Moreover, an iterative algorithm based on linear matrix inequalities is provided to solve the BMIs. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed method.

Design of Fuzzy State Observer and Mobile Sensor Guidance for Semilinear Parabolic PDE Systems.

This article investigates the issue of the fuzzy observer design for the semilinear parabolic partial differential equation (PDE) systems with mobile sensing measurements. Initially, we employ a Takagi-Sugeno (T-S) fuzzy PDE model to represent the semilinear parabolic PDE system accurately in a local region. Afterward, via the T-S fuzzy model and under the hypothesis that the spatial domain is divided by several subdomains in the light of the number of sensors, a state observation scheme which contains a fuzzy observer and the mobile sensor guidance is proposed. Then, by means of the Lyapunov direct method and integral inequalities, a design method of the fuzzy observer and mobile sensor guidance is provided to render the resulting state estimation error system exponentially stable, while the designed mobile sensor guidance can increase the exponential decay rate. Finally, numerical simulations are presented to show that the proposed fuzzy observer design approach is effective and the employment of mobile sensors contributes to improving the response speed of the state estimation error in comparison with the static ones.