Semilinear Parabolic Partial Differential Equations Research Articles

Improved H∞ sampled‐data control for semilinear parabolic PDE systems

SummaryIn this paper, an H∞ sampled‐data control problem is addressed for semilinear parabolic partial differential equation (PDE) systems. By using a time‐dependent Lyapunov functional and vector Poincare's inequality, a sampled‐data controller under spatially averaged measurements is developed to stabilize exponentially the PDE system with an H∞ control performance. The stabilization condition is presented in terms of a set of linear matrix inequalities. Finally, simulation results on the control of the diffusion equation and the FitzHugh‐Nagumo equation are given to illustrate the effectiveness of the proposed design method.

Two-locus clines maintained by diffusion and recombination in a heterogeneous environment

We study existence and stability of stationary solutions of a system of semilinear parabolic partial differential equations that occurs in population genetics. It describes the evolution of gamete frequencies in a geographically structured population of migrating individuals in a bounded habitat. Fitness of individuals is determined additively by two recombining, diallelic genetic loci that are subject to spatially varying selection. Migration is modeled by diffusion. Of most interest are spatially non-constant stationary solutions, so-called clines. In a two-locus cline all four gametes are present in the population, i.e., it is an internal stationary solution. We provide conditions for existence and linear stability of a two-locus cline if recombination is either sufficiently weak or sufficiently strong relative to selection and diffusion. For strong recombination, we also prove uniqueness and global asymptotic stability. For arbitrary recombination, we determine the stability properties of the monomorphic equilibria, which represent fixation of a single gamete.

Stochastic exponential integrators for a finite element discretisation of SPDEs with additive noise

We consider the numerical approximation of the general second order semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space–time noise. Our goal is to build two numerical algorithms with strong convergence rates higher than that of the standard semi-implicit scheme. In contrast to the standard time stepping methods which use basic increments of the noise, we introduce two schemes based on the exponential integrators, designed for finite element, finite volume or finite difference space discretisations. We prove the convergence in the root mean square L2 norm for a general advection diffusion reaction equation and a family of new Lipschitz nonlinearities. We observe from both the analysis and numerics that the proposed schemes have better convergence properties than the current standard semi-implicit scheme.

Spatially Piecewise Fuzzy Control Design for Sampled-Data Exponential Stabilization of Semilinear Parabolic PDE Systems

This paper employs a Takagi–Sugeno (T-S) fuzzy partial differential equation (PDE) model to solve the problem of sampled-data exponential stabilization in the sense of spatial $L^\infty$ norm $\Vert \cdot \Vert _\infty$ for a class of nonlinear parabolic distributed parameter systems (DPSs), where only a few actuators and sensors are discretely distributed in space. Initially, a T-S fuzzy PDE model is assumed to be derived by the sector nonlinearity method to accurately describe complex spatiotemporal dynamics of the nonlinear DPSs. Subsequently, a static sampled-data fuzzy local state feedback controller is constructed based on the T-S fuzzy PDE model. By constructing an appropriate Lyapunov–Krasovskii functional candidate and employing vector-valued Wirtinger's inequalities, a variation of vector-valued Poincare–Wirtinger inequality in one-dimensional spatial domain, as well as a vector-valued Agmon's inequality, it is shown that the suggested sampled-data fuzzy controller exponentially stabilizes the nonlinear DPSs in the sense of $\Vert \cdot \Vert _\infty$ , if sufficient conditions presented in term of standard linear matrix inequalities (LMIs) are fulfilled. Moreover, an LMI relaxation technique is utilized to enhance exponential stabilization ability of the suggested sampled-data fuzzy controller. Finally, the satisfactory and better performance of the suggested sampled-data fuzzy controller are demonstrated by numerical simulation results of two examples.

Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations

This paper studies the input-to-state stability (ISS) properties for a class of semi-linear parabolic partial differential equations (PDEs) with respect to boundary disturbances. In order to avoid the appearance of time derivatives of the disturbances in ISS estimates, some technical inequalities are first developed, which allow directly dealing with the boundary conditions and establishing the ISS based on the method of Lyapunov functionals. The well-posedness analysis of the considered problem is carried out and the conditions for ISS are derived. Two examples are used to illustrate the application of the developed method.

A note on exponential Rosenbrock–Euler method for the finite element discretization of a semilinear parabolic partial differential equation

In this paper, we consider the numerical approximation of a general second order semilinear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element method for space discretization and the exponential Rosenbrock–Euler method for time discretization, we provide a convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part, for both smooth and nonsmooth initial solutions. This is in contrast to restrictive assumptions made in the literature, where the authors have considered only approximation in time so far in their convergence proofs. The main result reveals how the convergence orders in both space and time depend heavily on the regularity of the initial data. In particular, the method achieves optimal convergence order Oh2+Δt2 when the initial data belongs to the domain of the linear operator. Numerical simulations to sustain our theoretical result are provided.

Fuzzy Control With Guaranteed Cost for Nonlinear Coupled Parabolic PDE-ODE Systems via PDE Static Output Feedback and ODE State Feedback

This paper investigates the guaranteed cost fuzzy control (GCFC) problem for a class of nonlinear systems modeled by an $n$ -dimension ordinary differential equation (ODE) coupled with a semilinear scalar parabolic partial differential equation (PDE). A Takagi–Sugeno (T–S) fuzzy coupled parabolic PDE-ODE model is initially proposed to accurately represent the nonlinear coupled system. Then, on the basis of the T–S fuzzy coupled model, a GCFC design is developed in terms of linear matrix inequalities to exponentially stabilize the coupled system while providing an upper bound for a prescribed quadratic cost function. The proposed fuzzy control scheme consists of the ODE state feedback and the PDE static output feedback employing locally collocated piecewise uniform actuators and sensors. Moreover, a suboptimal GCFC problem is also addressed to minimize the cost bound. Finally, the developed method is applied to the cruise control and surface temperature cooling of a hypersonic rocket car.

Observer-based boundary control of semi-linear parabolic PDEs with non-collocated distributed event-triggered observation

This paper deals with the problem of boundary control for a class of semi-linear parabolic partial differential equations (PDEs) with non-collocated distributed event-triggered observation. A semi-linear Luenberger PDE observer with an output error based event-triggering condition is constructed by using the event-triggered observation to exponentially track the PDE state. By the estimated state, a feedback controller is proposed. It has been shown by the Lyapunov technique, and a variant of Poincaré–Wirtinger inequality that the resulting closed-loop coupled PDEs is exponentially stable if a sufficient condition presented in terms of standard linear matrix inequality (LMI) is satisfied. Moreover, a rigorous proof is provided for existence of a minimal dwell-time between two triggering times. Finally, numerical simulation results are given to show the effectiveness of the proposed design method.

On a L∞ functional derivative estimate relating to the Cauchy problem for scalar semi-linear parabolic partial differential equations with general continuous nonlinearity

In this paper, we consider a L∞ functional derivative estimate for the first spatial derivatives of bounded classical solutions u:RN×[0,T]→R to the Cauchy problem for scalar second order semi-linear parabolic partial differential equations with a continuous nonlinearity f:R→R and initial data u0:RN→R, of the form,maxi=1,…,N⁡(supx∈RN⁡|uxi(x,t)|)≤Ft(f,u0,u)∀t∈[0,T]. Here Ft:At→R is a functional as defined in §1 and x=(x1,x2,…,xn)∈RN. We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence (fn,0,u(n)), where for each n∈N, u(n):RN×[0,T]→R is a solution to the Cauchy problem with zero initial data and nonlinearity fn:R→R, and for which there exists α>0 such thatmaxi=1,…,N⁡(supx∈RN⁡|uxi(n)(x,T)|)≥α, whilstlimn→∞⁡(inft∈[0,T]⁡(maxi=1,…,N⁡(supx∈RN⁡|uxi(n)(x,t)|)−Ft(fn,0,u(n))))=0.

Dynamic Boundary Fuzzy Control Design of Semilinear Parabolic PDE Systems With Spatially Noncollocated Discrete Observation.

The problem of dynamic boundary fuzzy control design is investigated in this paper for nonlinear parabolic partial differential equation (PDE) systems with spatially noncollocated discrete observation. Two cases of noncollocated discrete observation in space (i.e., pointwise observation in space and local piecewise uniform observation in space) are considered, respectively. The spatially noncollocated discrete observation makes the control design very difficult. Such design difficulty can be surmounted by an observer-based feedback control technique. It is assumed that the semilinear PDEs are accurately represented by a Takagi-Sugeno fuzzy PDE model. On the basis of the obtained fuzzy PDE model, a fuzzy Luenberger-type PDE state observer with above two cases of noncollocated discrete observation in space is first proposed for exponential estimation of the PDE system state. An observer-based dynamic fuzzy controller is then constructed such that the resulting closed-loop system is exponentially stable. Sufficient conditions on existence of such fuzzy controller are developed by Lyapunov technique with variations of vector-valued Poincaré-Wirtinger inequality, and presented in terms of linear matrix inequalities. Finally, extensive numerical simulation results of two examples are provided to support the proposed design method.

Dissipative observers for coupled diffusion–convection–reaction systems

The dissipativity-based observer design approach is extended to a class of coupled systems of 1-D semi-linear parabolic partial differential equations (PDEs) of diffusion–convection–reaction type with in-domain point measurements. This class of systems covers important application examples like tubular or catalytic reactors. By combining a dissipativity (sector) condition for the nonlinearity with a modal measurement injection for the linear differential operator sufficient conditions for the exponential convergence of the observer are derived in the form of a linear matrix inequality (LMI). The performance of the proposed approach is illustrated for an exothermic tubular reactor model.

Exponential Pointwise Stabilization of Semilinear Parabolic Distributed Parameter Systems via the Takagi–Sugeno Fuzzy PDE Model

This paper deals with the problem of exponential stabilization for nonlinear parabolic distributed parameter systems using the Takagi–Sugeno (T–S) fuzzy partial differential equation (PDE) model, where a finite number of actuators are active only at some specified points of the spatial domain (these actuators are referred to as pointwise actuators). Three cases of state feedback are respectively considered in this study as follows: full state feedback, piecewise state feedback, and collocated pointwise state feedback. It is initially assumed that a T–S fuzzy PDE model obtained via the sector nonlinearity approach is employed to accurately represent the semilinear parabolic PDE system. Based on the obtained T–S fuzzy PDE model, Lyapunov-based design methodologies of fuzzy feedback control laws are subsequently derived for the above three state feedback cases by using the vector-valued Wirtinger's inequality to guarantee locally exponential pointwise stabilization of the semilinear PDE system, and presented in terms of standard linear matrix inequalities (LMIs). Moreover, the favorable property offered by sharing all the same premises in the T–S fuzzy PDE models and fuzzy controllers is not applicable for the case of collocated pointwise state feedback. A parameterized LMI is introduced for this case to enhance the stabilization ability of the fuzzy controller. Finally, the merit and effectiveness of the proposed design methods are demonstrated by numerical simulation results of two examples.

Monte-Carlo algorithms for a forward Feynman–Kac-type representation for semilinear nonconservative partial differential equations

Abstract The paper is devoted to the construction of a probabilistic particle algorithm. This is related to a nonlinear forward Feynman–Kac-type equation, which represents the solution of a nonconservative semilinear parabolic partial differential equation (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments.

On the time evolution of Bernstein processes associated with a class of parabolic equations

In this article dedicated to the memory of Igor D. Chueshov, I first summarize in a few words the joint results that we obtained over a period of six years regarding the long-time behavior of solutions to a class of semilinear stochastic parabolic partial differential equations. Then, as the beautiful interplay between partial differential equations and probability theory always was close to Igor's heart, I present some new results concerning the time evolution of certain Markovian Bernstein processes naturally associated with a class of deterministic linear parabolic partial differential equations. Particular instances of such processes are certain conditioned Ornstein-Uhlenbeck processes, generalizations of Bernstein bridges and Bernstein loops, whose laws may evolve in space in a non trivial way. Specifically, I examine in detail the time development of the probability of finding such processes within two-dimensional geometric shapes exhibiting spherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goal is to allow approximate computations with controlled error terms of the various probability distributions involved.

Backward stochastic differential equations with rank-based data

In this paper, we investigate Markovian backward stochastic differential equations (BSDEs) with the generator and the terminal value that depend on the solutions of stochastic differential equations with rank-based drift coefficients.We study regularity properties of the solutions of this kind of BSDEs and establish their connection with semi-linear backward parabolic partial differential equations in simplex with Neumann boundary condition. As an application, we study the European option pricing problem with capital size based stock prices.

Nonlinear stability in three-layer channel flows

The nonlinear stability of viscous, immiscible multilayer flows in plane channels driven both by a pressure gradient and gravity is studied. Three fluid phases are present with two interfaces. Weakly nonlinear models of coupled evolution equations for the interfacial positions are derived and studied for inertialess, stably stratified flows in channels at small inclination angles. Interfacial tension is demoted and high-wavenumber stabilisation enters due to density stratification through second-order dissipation terms rather than the fourth-order ones found for strong interfacial tension. An asymptotic analysis is carried out to demonstrate how these models arise. The governing equations are $2\times 2$ systems of second-order semi-linear parabolic partial differential equations (PDEs) that can exhibit inertialess instabilities due to interaction between the interfaces. Mathematically this takes place due to a transition of the nonlinear flux function from hyperbolic to elliptic behaviour. The concept of hyperbolic invariant regions, found in nonlinear parabolic systems, is used to analyse this inertialess mechanism and to derive a transition criterion to predict the large-time nonlinear state of the system. The criterion is shown to predict nonlinear stability or instability of flows that are stable initially, i.e. the initial nonlinear fluxes are hyperbolic. Stability requires the hyperbolicity to persist at large times, whereas instability sets in when ellipticity is encountered as the system evolves. In the former case the solution decays asymptotically to its uniform base state, while in the latter case nonlinear travelling waves can emerge that could not be predicted by a linear stability analysis. The nonlinear analysis predicts threshold initial disturbances above which instability emerges.

Stochastic De Giorgi iteration and regularity of stochastic partial differential equations

Under general conditions, we devise a stochastic version of De Giorgi iteration scheme for semilinear stochastic parabolic partial differential equation of the form \[\partial_{t}u=\operatorname{div}(A\nabla u)+f(t,x,u)+g_{i}(t,x,u)\dot{w}^{i}_{t}\] with progressively measurable diffusion coefficients. We use the scheme to show that the solution of the equation is almost surely Holder continuous in both space and time variables.

Guaranteed cost fuzzy state observer design for semilinear parabolic PDE systems under pointwise measurements

This paper studies the guaranteed cost fuzzy state observer (GCFSO) design via pointwise measurement sensors for a class of distributed parameter systems described by semilinear parabolic partial differential equations (PDEs). Initially, a Takagi–Sugeno (T–S) fuzzy model is employed to accurately represent the original PDE system in a local region. Then, based on the T–S fuzzy model, a fuzzy state observer is constructed for the state estimation. By augmenting the state estimation error system with the fuzzy PDE system and utilizing Lyapunov technique and Wirtinger’s inequality, a fuzzy state observer design with a guaranteed cost is developed in terms of linear matrix inequalities (LMIs). The resulting fuzzy observer can exponentially stabilize the augmented system while providing an upper bound on the cost function of state estimation error. Moreover, a suboptimal GCFSO design problem is also addressed to make the upper bound as small as possible. Finally, the numerical simulation results on two examples demonstrate the effectiveness of the proposed method.

$H_\infty $ Disturbance Attenuation for Nonlinear Coupled Parabolic PDE–ODE Systems via Fuzzy-Model-Based Control Approach

An ${H_\infty }$ fuzzy control design is presented for the disturbance attenuation of a class of coupled systems described by a set of nonlinear ordinary differential equations (ODEs) and a semi-linear parabolic partial differential equation (PDE). The fuzzy control scheme consists of an ODE state feedback fuzzy subcontroller for the ODE subsystem and a PDE static output feedback fuzzy subcontroller for the PDE subsystem by using piecewise uniform actuators and pointwise sensors. Initially, the original nonlinear system is accurately represented by employing a Takagi–Sugeno fuzzy coupled parabolic PDE–ODE model. Then, an ${H_\infty }$ fuzzy controller is developed to exponentially stabilize the fuzzy coupled system while satisfying a prescribed ${H_\infty }$ performance of disturbance attenuation, whose existence condition is given by linear matrix inequalities. Finally, simulation results on a hypersonic rocket car are given to show the effectiveness of the proposed design method.

RETRACTED: Multidimensional viscosity solution theory of semi-linear partial differential equations

This article has been retracted: please see Elsevier Policy on Article Withdrawal ( https://www.elsevier.com/locate/withdrawalpolicy ). This article was retracted at the request of the Editors, in agreement with the authors. This paper has been retracted as the main results have a high level of similarity with the work published by E. Pardoux et al., Probabilistic interpretation of a system of semi-linear parabolic partial differential equations, Annales de l'Institut Henri Poincare (B) Probability and Statistics 33 (4) (1997) 467–490, http://www.sciencedirect.com/science/article/pii/S024602039780101X?via%3Dihub .