Boundary Geometric Control of a Nonlinear Diffusion System with Time‐Dependent Spatial Domain

A Stefan problem represents a distributed parameter system with a time‐dependent spatial domain. This paper addresses the boundary control of the position of the moving liquid–solid interface in the case of nonlinear Stefan problem with Neumann actuation. The main idea consists in deriving an equivalent linear model by means of Cole‐Hopf tangent transformation, i.e. under a certain physical assumption, the original nonlinear Stefan problem is converted to a linear one. Then, the geometric control law is deduced directly from that developed, by the authors of the present paper, for the linear Stefan problem. Based on the fact that the Cole‐Hopf transformation is bijective, it is shown that the developed control law yields a stable closed‐loop system. The performance of the controller is evaluated through numerical simulation in the case of stainless steel melting characterized by a temperature‐dependent thermal conductivity, which is nonlinear. The objective is to control the position of the liquid–solid interface by manipulating a heat flux at the boundary.     

Impulsive semilinear heat equation with delay in control and in state

In this paper, we prove the interior approximate controllability of the impulsive semilinear heat equation with delay in control and in state by proving first that the linear heat equation with delay in control is approximately controllable. After that, we add impulses and a nonlinear perturbation with delay in state, and using Rothe's fixed point theorem, we prove that the interior approximate controllability of the impulsive semilinear system. Finally, we present some open problems and a possible general framework to study the controllability of impulsive semilinear diffusion process in Hilbert spaces with delay in control and in state.     

Efficient numerical solution of Fisher's equation by using B-spline method

This paper seeks to develop an efficient B-spline scheme for solving Fisher's equation, which is a nonlinear reaction–diffusion equation describing the relation between the diffusion and nonlinear multiplication of a species. To find the solution, domain is partitioned into a uniform mesh and then cubic B-spline function is applied to Fisher's equation. The method yields stable and accurate solutions. The results obtained are acceptable and in good agreement with some earlier studies. An important advantage is that the method is capable of greatly reducing the size of computational work.     

Mixed Volume Element-Characteristic Fractional Step Difference Method for Contamination from Nuclear Waste Disposal

A nonlinear system with boundary-initial value conditions of convection–diffusion partial differential equations is presented to describe incompressible nuclear waste disposal contamination in porous media. The flow pressure is determined by an elliptic equation, the concentrations of brine and radionuclide are formulated by convection–diffusion equations, and the transport of temperature is defined by a heat equation. The pressure appears in convection–diffusion equations and heat equation in the form of Darcy velocity and controls the physical processes. The fluid pressure and velocity are solved by the conservative mixed volume element and the computation accuracy of Darcy velocity is improved one order. A combination method of the mixed volume element and the approximation of characteristics is applied to solve the brine and heat, where the diffusion is discretized by a mixed volume element method and the convection is treated by the method of characteristics. The characteristics can confirm strong computation stability at sharp fronts and it can avoid numerical dispersion and nonphysical oscillation. Larger time-steps along the characteristics are shown to result in smaller time-truncation errors than those resulting from standard methods. The mixed volume element method has the property of conservation on each element and it can obtain numerical solutions of the brine and adjoint vectors. The radionuclide is solved by a coupled method of characteristics and fractional step difference. The computational work is reduced greatly by decomposing a three-dimensional problem into three successive one-dimensional problems and using the algorithm of speedup. Using numerical analysis of priori estimates of differential equations, we demonstrate an optimal second order estimate in \(l^2\) norm. Numerical data are appropriate with the scheme and it is shown that the method is a powerful tool to solve the well-known problems in porous media.     

Boundary geometric control of a linear Stefan problem

This paper addresses the geometric control of the position of a liquid–solid interface in a melting process of a material known as Stefan problem. The system model is hybrid, i.e. the dynamical behavior of the liquid-phase temperature is modeled by a heat equation while the motion of the moving boundary is described by an ordinary differential equation. The control is applied at one boundary as a heat flux and the second moving boundary represents the liquid–solid interface whose position is the controlled variable. The control objective is to ensure a desired position of the liquid–solid interface. The control law is designed using the concept of characteristic index, from geometric control theory, directly issued from the hybrid model without any reduction of the partial differential equation. It is shown by use of Lyapunov stability test that the control law yields an exponentially stable closed-loop system. The performance of the developed control law is evaluated through simulation by considering zinc melting.     

Lattice Boltzmann simulation of some nonlinear convection–diffusion equations

In this work we proposed a lattice Boltzmann model for the nonlinear convection–diffusion equation (NCDE) with anisotropic diffusion. The constraints on the model for correctly recovering macroscopic equation are also carefully analyzed, which are ignored in some existing work. Detailed simulations of some 1D/2D NCDEs, including the nonlinear Schrödinger equation (NLSE), Buckley–Leverett equation with discontinuous initial data, NCDE with anisotropic diffusion, and generalized Zakharov system, are performed. The numerical results obtained by the proposed model agree well with the analytical solutions and/or the numerical solutions reported in previous studies. It is also found that, for complex-valued NLSE, the model using a complex distribution function is superior to that using two real distribution functions for the real and imaginary parts of the NLSE separately.     

Non-unified Compact Residual-Distribution Methods for Scalar Advection–Diffusion Problems

This paper solves the advection–diffusion equation by treating both advection and diffusion residuals in a separate (non-unified) manner. An alternative residual distribution (RD) method combined with the Galerkin method is proposed to solve the advection–diffusion problem. This Flux-Difference RD method maintains a compact-stencil and the whole process of solving advection–diffusion does not require additional equations to be solved. A general mathematical analysis reveals that the new RD method is linearity preserving on arbitrary grids for the steady-state advection–diffusion equation. The numerical results show that the flux difference RD method preserves second-order accuracy on various unstructured grids including highly randomized anisotropic grids on both the linear and nonlinear scalar advection–diffusion cases.     

Non-fragile guaranteed cost fuzzy control for nonlinear first-order hyperbolic partial differential equation systems

This paper concerns the non-fragile guaranteed cost control for nonlinear first-order hyperbolic partial differential equations (PDEs), and the case of hyperbolic PDE systems with parameter uncertainties is also addressed. A Takagi–Sugeno (T–S) fuzzy hyperbolic PDE model is presented to exactly represent the nonlinear hyperbolic PDE system. Then, the state-feedback non-fragile controller distributed in space is designed by the parallel distributed compensation (PDC) method, and some sufficient conditions are derived in terms of spatial differential linear matrix inequalities (SDLMIs) such that the T–S fuzzy hyperbolic PDE system is asymptotically stable and the cost function keeps an upper bound. Moreover, for the nonlinear hyperbolic PDE system with parameter uncertainties, using the above-design approach, the robust non-fragile guaranteed cost control scheme is obtained. Furthermore, the finite-difference method is employed to solve the SDLMIs. Finally, a nonlinear hyperbolic PDE system is presented to illustrate the effectiveness and advantage of the developed design methodology.     

Development of a sixth-order two-dimensional convection–diffusion scheme via Cole–Hopf transformation

In this paper, we develop a two-dimensional finite-difference scheme for solving the time-dependent convection–diffusion equation. The numerical method exploits Cole–Hopf equation to transform the nonlinear scalar transport equation into the linear heat conduction equation. Within the semi-discretization context, the time derivative term in the transformed parabolic equation is approximated by a second-order accurate time-stepping scheme, resulting in an inhomogeneous Helmholtz equation. We apply the alternating direction implicit scheme of Polezhaev to solve the Helmholtz equation. As the key to success in the present simulation, we develop a Helmholtz scheme with sixth-order spatial accuracy. As is standard practice, we validated the code against test problems which were amenable to exact solutions. Results show excellent agreement for the one-dimensional test problems and good agreement with the analytical solution for the two-dimensional problem.     

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.     

Linearized Crank–Nicolson method for solving the nonlinear fractional diffusion equation with multi-delay

This paper is concerned with numerical solution of the nonlinear fractional diffusion equation with multi-delay. The studied model plays a significant role in population ecology. A linearized Crank–Nicolson method for such problem is proposed by combing the Crank–Nicolson approximation in time with the fractional centred difference formula in space. Using the discrete energy method, the suggested scheme is proved to be uniquely solvable, stable and convergent with second-order accuracy in both space and time for sufficiently small space and time increments. Several numerical experiments for solving the delay fractional Hutchinson equation and two real problems in population dynamics are provided to verify our theoretical results.     

Stabilization of a spatially non‐causal reaction–diffusion equation by boundary control

Stabilization of a reaction–diffusion equation, in which the heat source depends on the temperature of the whole space, is considered by using boundary control. A new backstepping transformation is introduced, in which there are two kernels. Through a series of mathematical tricks, the exact solutions of kernels are obtained, and a control law is obtained specifically. The inverse transformation is derived, and stability of the closed loop system established. Simulation results show that the closed loop system is stable. Copyright © 2012 John Wiley & Sons, Ltd.     

A stable Gaussian radial basis function method for solving nonlinear unsteady convection–diffusion–reaction equations

We investigate a novel method for the numerical solution of two-dimensional time-dependent convection–diffusion–reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor–corrector schemes such as Adams–Bashforth and implicit–explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems.     

Simultaneous optimization of slab permutation scheduling and heat controlling for a reheating furnace

In this report, for a reheating furnace, which is employed in one of the processes for producing steel sheets from slabs, we propose a modelling method that simultaneously optimizes both the permutation scheduling of slabs and the heat controlling of the furnace. The proposed modelling scheme is based on a hybrid model composed of a nonlinear advection equation that expresses the behavior of the slab temperature and a discrete model for feeding slabs. The model predictive control problem of this model, which will be reduced to a mixed integer programming problem, is formulated by discretizing the advection equation in time and space by means of the method of characteristics and spatially piecewise-linearizing the nonlinear term. It is shown by numerical simulations that the proposed model predictive control method is very effective from the viewpoint of the control performance and the computational burden.     

Some nonlinear optimal control problems with closed‐form solutions

Optimal controllers guarantee many desirable properties including stability and robustness of the closed‐loop system. Unfortunately, the design of optimal controllers is generally very difficult because it requires solving an associated Hamilton–Jacobi–Bellman equation. In this paper we develop a new approach that allows the formulation of some nonlinear optimal control problems whose solution can be stated explicitly as a state‐feedback controller. The approach is based on using Young's inequality to derive explicit conditions by which the solution of the associated Hamilton–Jacobi–Bellman equation is simplified. This allows us to formulate large families of nonlinear optimal control problems with closed‐form solutions. We demonstrate this by developing optimal controllers for a Lotka–Volterra system. Copyright © 2001 John Wiley & Sons, Ltd.     

融合边缘和区域信息的水平集矢量图像分割

在充分利用矢量图像各通道区域和边缘信息的基础上,变分IAC（集成活动轮廓）模型引入了非线性热方程的符号距离函数约束项,使水平集不用耗时的重新初始化而始终保持符号距离函数的特性。对非线性热方程传导率的均衡化,使水平集的演化分割过程快速稳定。另外,算法改进了曲线2维梯度和散度算子传统离散化方式,使梯度和散度算子保持空间旋转不变性。实验结果表明,该方法是有效的,提高了分割的准确性和鲁棒性。     

Optimal learning control of oxygen saturation using a policy iteration algorithm and a proof-of-concept in an interconnecting three-tank system

In this work, “policy iteration algorithm” (PIA) is applied for controlling arterial oxygen saturation that does not require mathematical models of the plant. This technique is based on nonlinear optimal control to solve the Hamilton–Jacobi–Bellman equation. The controller is synthesized using a state feedback configuration based on an unidentified model of complex pathophysiology of pulmonary system in order to control gas exchange in ventilated patients, as under some circumstances (like emergency situations), there may not be a proper and individualized model for designing and tuning controllers available in time. The simulation results demonstrate the optimal control of oxygenation based on the proposed PIA by iteratively evaluating the Hamiltonian cost functions and synthesizing the control actions until achieving the converged optimal criteria. Furthermore, as a practical example, we examined the performance of this control strategy using an interconnecting three-tank system as a real nonlinear system.     

Modeling and vibration control for a flexible pendulum inverted system based on a PDE observer

This study addresses the problem of trajectory control of a flexible pendulum inverted system on the basis of the partial differential equation (PDE) and ordinary differential equation (ODE) dynamic model. One of the key contributions of this study is that a new model is proposed to simplify the complex system. In addition, this study proposed a nonlinear PDE observer to estimate distributed positions and velocities along flexible pendulum. Singular perturbation method is proposed to solve the coupling system of nonlinear PDE observer. The nonlinear PDE observer is divided into a fast subsystem and a slow subsystem by the use of the singular perturbation method. To stabilise this fast subsystem, a boundary controller is proposed at the free end of the beam. The sliding-mode control method is proposed to design controller for slow subsystems. The asymptotic stability of both the proposed nonlinear PDE observer and controller is validated by theoretical analysis. The results are illustrated by simulation.     

Nonlinear observer-based Lyapunov boundary control of distributed heat transfer mechanisms for membrane distillation plant

This paper presents a nonlinear observer-based Lyapunov control for a membrane distillation (MD) process. The control considers the inlet temperatures of the feed and the permeate solutions as inputs, transforming it to boundary control process, and seeks to maintain the temperature difference along the membrane boundaries around a sufficient level to promote water production. MD process is modeled with advection diffusion equation model in two dimensions, where the diffusion and convection heat transfer mechanisms are best described. Model analysis, effective order reduction and parameters physical interpretation, are provided. Moreover, a nonlinear observer has been designed to provide the control with estimates of the temperature evolution at each time instant. In addition, physical constraints are imposed on the control to have an acceptable range of feasible inputs, and consequently, better energy consumption. Numerical simulations for the complete process with real membrane parameter values are provided, in addition to detailed explanations for the role of the controller and the observer.     

一个基于各向异性的热传导方程导出的图像锐化算子

Ｗｉｔｋｉｎ等人的迟度空间滤波办法是对原图像一个逐渐平滑的过程,这个过程可以用传统的热传导方程来描述,假设“热量”由低温向高温外流动,导出一个各向异性的反向热传导方程。将它用于图像处理,通过选择合适的各向异性函数,使得在盛兴趣的边缘处“热量”由低温向高温处流动,即“灰度”从“