Parabolic Partial Differential Equation Model Research Articles

Spectral-based optimal output-feedback boundary control of a cracking catalytic reactor PDE model

This research paper focuses on the design of a boundary optimal controller with output feedback for a tubular catalytic cracking reactor. The process is represented by a set of non-linear parabolic partial differential equations (PDEs). The approach involves utilising the linearised infinite-dimensional version of the model, taking advantage of the system generator being a Riesz spectral operator. A stabilising compensator is designed by leveraging the eigenvalues and eigenvectors of the system generator. This paper also includes numerical simulations to demonstrate the performance of the developed algorithm.

Fuzzy Boundary Sampled-Data Control for Nonlinear Parabolic DPSs.

For a nonlinear parabolic distributed parameter system (DPS), a fuzzy boundary sampled-data (SD) control method is introduced in this article, where distributed SD measurement and boundary SD measurement are respected. Initially, this nonlinear parabolic DPS is represented precisely by a Takagi-Sugeno (T-S) fuzzy parabolic partial differential equation (PDE) model. Subsequently, under distributed SD measurement and boundary SD measurement, a fuzzy boundary SD control design is obtained via linear matrix inequalities (LMIs) on the basis of the T-S fuzzy parabolic PDE model to guarantee exponential stability for closed-loop parabolic DPS by using inequality techniques and a LF. Furthermore, respecting the property of membership functions, we present some LMI-based fuzzy boundary SD control design conditions. Finally, the effectiveness of the designed fuzzy boundary SD controller is demonstrated via two simulation examples.

Chebyshev collocation method for an interface parabolic partial differential equation model of transdermal medication delivery and transcutaneous absorption

An effective scheme based on the finite difference and Chebyshev collocation method is introduced to acquire approximate solutions of a mass transfer problem model for transdermal medication delivery. A priori error estimation is derived. The reliability of the proposed scheme is shown by comparing the drug concentration profiles with their counterparts acquired by other analytical and numerical methods.

Observer-Based Boundary Fuzzy Control Design of Nonlinear Parabolic PDE Systems Using Mobile Sensors

This paper studies the boundary fuzzy control problem for nonlinear parabolic partial differential equation (PDE) systems under spatially noncollocated mobile sensors. In a real setup, sensors and actuators can never be placed at the same location, and the noncollocated setting may be beneficial in some application scenarios. The control design is very difficult due to the noncollocated mobile observation, which can be solved by an observer-based technique. At first, a Takagi-Sugeno fuzzy PDE model is devoted to accurately representing the nonlinear parabolic PDE system. Next, we present a state estimation scheme including fuzzy Luenberger-type PDE state observer plus mobile sensor guidance. Then, an observer-based boundary fuzzy controller is posed to render the resulting closedloop system exponentially stable, and the exponential decay rate is increased by the designed mobile sensor guidance laws. At last, two examples verify the proposed method.

Fault-Tolerant Stochastic Sampled-Data Fuzzy Control for Nonlinear Delayed Parabolic PDE Systems

For nonlinear delayed parabolic partial differential equation (PDE) systems, this paper addresses fault-tolerant stochastic sampled-data (SD) fuzzy control under spatially point measurements (SPMs). Initially, a T–S fuzzy PDE model is given to accurately describe the nonlinear delayed parabolic PDE system. Secondly, in consideration of possible actuator failure, a fault-tolerant SD fuzzy controller with stochastic sampling under SPMs is designed for nonlinear delayed parabolic PDE system, where two sampling periods are considered whose occurrence probabilities are given constants and satisfy the Bernoulli distribution. Then, by constructing a novel time-dependent Lyapunov functional, sufficient conditions that guarantee the mean square exponential stability of closed-loop delayed PDE system are obtained based on linear matrix inequalities (LMIs). Lastly, three examples are given to illustrate the designed approach.

Single parameter adaptive neural network control for multi-agent deployment with prescribed tracking performance

This study addresses the problem of deploying multi-agent systems using single-parameter adaptive neural network control in time and space where the system is modeled by a parabolic partial differential equation (PDE). We investigate the agent model, simplify the pointwise dynamics using a PDE model, and consider the deployment problem when the number of agents is relatively large. In order for the deployed agents to follow the desired trajectory, we augment the agent dynamics with individual control inputs, accounting for the unknown interference faced by each agent during the deployment process. In the proposed approach, a radial basis function neural network structure is introduced to enhance the systems’ adaptivity under unknown interference. The unknown parameter is estimated via the single-parameter idea for reducing the computation of the entire process and increasing the calculation speed. Asymmetric performance constraints are imposed on the tracking error of the system to ensure that each agent is deployed in the required position. The results of numerical simulation prove the effectiveness of the method.

Quantized-Output-Based Exponential Stabilization of Fuzzy DPSs With Time-Varying Delay and Nonlinearities

The exponential stabilization of nonlinear distributed parameter systems (DPSs) with time-varying delay is studied in this paper. With the employment of the Takagi-Sugeno (T-S) fuzzy model, the nonlinear DPSs are described as a fuzzy parabolic partial differential equation (PDE). Then, with the consideration of the measurement output and control input information quantized by a kind of dynamic quantizers, fuzzy static and dynamic output feedback controllers are proposed. The time-delayed feedback controllers are also provided in the forms of static and dynamic. On the basis of the derived T-S fuzzy PDE model, Lyapunov-based control design strategies are proposed with respect to the effect of quantization. Moreover, by constructing appropriate Lyapunov functionals, sufficient conditions for designing the output feedback control gains and the quantizers' adjusting parameters which guarantee the exponential stability of the fuzzy closed-loop system are provided in terms of standard linear matrix inequalities (LMIs). Finally, we explore the output feedback controllers into the FitzHugh-Nagumo (FHN) equation and Fisher equation as applications. The efficiency of the design strategies is indicated through the simulation results.

Formula omitted] fuzzy intermittent boundary control for nonlinear parabolic distributed parameter systems

For nonlinear space-varying parabolic distributed parameter systems, this paper introduces an H∞ fuzzy intermittent boundary control, where the output measurements are only available at some specified boundary position (i.e., boundary measurements). Initially, a Takagi–Sugeno fuzzy parabolic partial differential equation is used to precisely describe the nonlinear space-varying parabolic distributed parameter system. Then, under boundary measurements, an H∞ fuzzy intermittent boundary control design based on the Takagi–Sugeno fuzzy parabolic partial differential equation model ensuring the exponential stability with an H∞ performance for closed-loop space-varying distributed parameter system is subsequently obtained via spatial linear matrix inequalities by employing inequality techniques and piecewise switching-time-dependent Lyapunov function. Furthermore, in order to solve the H∞ fuzzy intermittent boundary controller design of nonlinear space-varying parabolic distributed parameter systems under boundary measurements, we express spatial linear matrix inequalities as linear matrix inequalities and further present some linear matrix inequality based fuzzy intermittent boundary control design conditions respecting the property of membership functions. Finally, two simulation examples are offered to demonstrate the effectiveness of the proposed H∞ fuzzy intermittent boundary control method.

Boundary Fuzzy Output Tracking Control of Nonlinear Parabolic Infinite-Dimensional Dynamic Systems: Application to Cooling Process in Hot Strip Mills

This paper utilizes a combination of integral control, fuzzy control, and observer-based output feedback control to deal with the issue of nonlinear output tracking control (OTC) design for nonlinear infinite-dimensional dynamic systems. The system dynamics model is represented by a semi-linear parabolic partial differential equation (PDE) with boundary control and non-collocated boundary measurement. Initially, a Takagi-Sugeno (T-S) fuzzy parabolic PDE model is constructed to surmount the OTC design difficulty from the infinite-dimensional nonlinear system dynamics. Subsequently, a fuzzy-observer-based OTC law is proposed via the T-S fuzzy PDE model and the integral control approach. Here, the integral control ensures asymptotic output regulation, and the observer-based output feedback control is employed to conquer the stabilizing control design difficulty caused by the non-collocation between control actuation and measurement. It is shown via the Lyapunov technique with variants of vector-valued Poincaré-Wirtinger's inequality that the suggested fuzzy OTC law drives the measurement output to asymptotically track the desired reference signal and ensures the boundedness of the resulting closed-loop system, provided that a sufficient condition given in the form of linear matrix inequalities is fulfilled. Moreover, the proposed fuzzy-model-based OTC design is also revised for the exponential stabilization case. Finally, extensive simulation results for a numerical example and a cooling process in hot strip mills are provided to examine the effectiveness and merit of the proposed fuzzy OTC scheme.

Error Analysis for a Parabolic PDE Model Problem on a Coupled Moving Domain in a Fully Eulerian Framework

We introduce an unfitted finite element method with Lagrange-multipliers to study an Eulerian time stepping scheme for moving domain problems applied to a model problem where the domain motion is implicit to the problem. We consider a parabolic partial differential equation (PDE) in the bulk domain, and the domain motion is described by an ordinary differential equation (ODE), coupled to the bulk partial differential equation through the transfer of forces at the moving interface. The discretisation is based on an unfitted finite element discretisation on a time-independent mesh. The method-of-lines time discretisation is enabled by an implicit extension of the bulk solution through additional stabilisation, as introduced by Lehrenfeld & Olshanskii (ESAIM: M2AN, 53:585-614, 2019). The analysis of the coupled problem relies on the Lagrange-multiplier formulation, the fact that the Lagrange-multiplier solution is equal to the normal stress at the interface and that the motion of the interface is given through rigid body motion. This paper covers the complete stability analysis of the method and an error estimate in the energy norm, under an assumption on the discrete interface velocity. This includes the dynamic error in the domain motion resulting from the discretised ODE and the forces from the discretised PDE. To the best of our knowledge this is the first error analysis of this type of coupled moving domain problem in a fully Eulerian framework. Numerical examples illustrate the theoretical results.

Observer-Based Fuzzy Quantized Control for a Stochastic Third-Order Parabolic PDE System

The work addresses an observer-based fuzzy quantized control for stochastic third-order parabolic partial differential equations (PDEs) using discrete point measurements. For the first time, we contribute in introducing three types of quantizer—logarithmic quantizer, uniform quantizer, and hysteresis quantizer into the controller designs for the stochastic PDE system. The main advantage of the quantized control law lies in reducing the energy that the system spends. The stability analysis is nontrivial due to the complex stochastic PDE model. Different stability results are presented for each case. Sufficient LMI-based conditions are investigated to ensure the internal mean-square exponential stability and <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 of the perturbed actuated system. Consistent simulation results that support the proposed theoretical statements are provided.

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.

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.

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.

Nonlinear dynamics of dewetting thin films

Fluid films spreading on hydrophobic solid surfaces exhibit complicated dynamics that describe transitions leading the films to break up into droplets. For viscous fluids coating hydrophobic solids this process is called "dewetting". These dynamics can be represented by a lubrication model consisting of a fourth-order nonlinear degenerate parabolic partial differential equation (PDE) for the evolution of the film height. Analysis of the PDE model and its regimes of dynamics have yielded rich and interesting research bringing together a wide array of different mathematical approaches. The early stages of dewetting involve stability analysis and pattern formation from small perturbations and self-similar dynamics for finite-time rupture from larger amplitude perturbations. The intermediate dynamics describes further instabilities yielding topological transitions in the solutions producing sets of slowly-evolving near-equilibrium droplets. The long-time behavior can be reduced to a finite-dimensional dynamical system for the evolution of the droplets as interacting quasi-steady localized structures. This system yields <i>coarsening</i>, the successive re-arrangement and merging of smaller drops into fewer larger drops. To describe macro-scale applications, mean-field models can be constructed for the evolution of the number of droplets and the distribution of droplet sizes. We present an overview of the mathematical challenges and open questions that arise from the stages of dewetting and how they relate to issues in multi-scale modeling and singularity formation that could be applied to other problems in PDEs and materials science.

Sampled-Data Fuzzy Control With Guaranteed Cost for Nonlinear Parabolic PDE Systems via Static Output Feedback

This article introduces a sampled-data (SD) static output feedback fuzzy control (FC) with guaranteed cost for nonlinear parabolic partial differential equation (PDE) systems. First, a Takagi–Sugeno (T–S) fuzzy parabolic PDE model is employed to represent the nonlinear PDE system. Second, with the aid of the T–S fuzzy PDE model, a SD FC design with guaranteed cost under spatially averaged measurements is developed in the formulation of linear matrix inequalities by utilizing a time-dependent Lyapunov functional and inequality techniques, which can stabilize exponentially the PDE system while providing an optimized upper bound on the cost function. The membership functions of the proposed controller are determined by the measurement output and independent of the fuzzy PDE plant model. Finally, simulation results are presented to control the diffusion equation and the FitzHugh–Nagumo equation for demonstrating the effectiveness of the proposed method.

Sampled-data fuzzy control with space-varying gains for nonlinear time-delay parabolic PDE systems

This paper introduces a sampled-data fuzzy control (SDFC) with space-varying gains for nonlinear time-delay parabolic partial differential equation (PDE) systems. First, a Takagi-Sugeno (T-S) fuzzy time-delay parabolic PDE model is employed to represent the nonlinear time-delay PDE system. Second, with the aid of the T-S fuzzy time-delay PDE model, a SDFC design with space-varying gains is developed in the formulation of space-dependent linear matrix inequalities (LMIs) by constructing an appropriate Lyapunov functional, which can stabilize exponentially the time-delay PDE system. These stabilization conditions can be applied to either slowing-varying time delay or fast-varying one. Finally, simulation results on two examples are presented to demonstrate the effectiveness of the proposed method.

Fuzzy Stabilization Design for Semilinear Parabolic PDE Systems With Mobile Actuators and Sensors

This paper deals with a fuzzy stabilization design problem for a class of semilinear parabolic partial differential equation (PDE) systems using mobile actuators and sensors. Initially, a Takagi–Sugeno (T–S) fuzzy PDE model is employed to accurately represent the semilinear parabolic PDE system. Subsequently, based on the T–S fuzzy model, a stabilization scheme containing the fuzzy controllers and the guidance of mobile actuator/sensor pairs is proposed, where the spatial domain is decomposed into multiple subdomains according to the number of actuator/sensor pairs and each actuator/sensor pair is capable of moving within the respective subdomain. Then, by a Lyapunov direct technique, an integrated design of fuzzy controllers plus mobile actuator/sensor guidance laws is developed in the form of bilinear matrix inequalities (BMIs), such that the resulting closed-loop system is exponentially stable and the mobile actuator/sensor guidance can enhance the transient performance of the closed-loop system. Furthermore, an iterative algorithm based on linear matrix inequalities is proposed to solve the BMIs. Finally, two examples are given to illustrate the effectiveness of the proposed method.

Sampled-data fuzzy control for a class of nonlinear parabolic distributed parameter systems under spatially point measurements

In this paper, the exponential stabilization problem is addressed for a class of nonlinear parabolic partial differential equation (PDE) systems via sampled-data fuzzy control approach. 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, a novel time-dependent Lyapunov functional is used to design a sampled-data fuzzy controller under spatially point measurements such that the closed-loop fuzzy PDE system is exponentially stable with a given decay rate. The stabilization conditions are presented in terms of a set of linear matrix inequalities (LMIs). Finally, simulation results on the control of the diffusion equation and the FitzHugh–Nagumo (FHN) equation to illustrate the effectiveness of the proposed design method.