Parabolic PDE Systems Research Articles

Two-level implicit high-order compact scheme in exponential form for 3D quasi-linear parabolic equations

We discuss a new high accuracy compact exponential scheme of order four in space and two in time to solve the three-dimensional quasi-linear parabolic partial differential equations. The derived half-step discretization based scheme is implicit in nature and demands only two levels for computation. The generalization of the proposed exponential scheme for the system of the quasi-linear parabolic PDEs is also represented. We generate unconditionally stable alternating direction implicit scheme for the linear parabolic equation in general form. The accuracy and the theoretical results of the proposed scheme are verified for high Reynolds number by several numerical problems like linear and non-linear convection-diffusion equation, coupled Burgers' equations, Navier-Stokes equations, quasi-linear parabolic equation, etc.

Dynamic Event-triggered Fuzzy Filtering for Semi-linear Parabolic PDE Systems: A Reduced-order Approach

Dynamic Event-triggered Fuzzy Filtering for Semi-linear Parabolic PDE Systems: A Reduced-order Approach

Locally risk minimizing pricing of Asian option in a semi-Markov modulated market

We consider a regime-switching model where the stock volatility dynamics is a semi-Markov process. Under this model assumption, we find the locally risk-minimizing price of some Asian options with European-style exercise. The price function is shown to satisfy a non-local degenerate system of parabolic PDEs in dimension two with a terminal condition. We show this by deriving the F-S decomposition of the discounted contingent claim. The related Cauchy problem involving the PDE is shown to be equivalent to an integral equation (IE). The existence and uniqueness of the classical solution to the PDE are determined by studying the IE and using semigroup theory. To be more precise, we first obtain the mild solution of the PDE and then we show that the mild solution has sufficient regularity. The locally risk-minimizing hedging for the option has also been identified in this work. Finally, the computational aspects of Asian option prices have been discussed by solving the equation numerically.

Hybrid-triggered-based fault-tolerant dissipative control for T-S fuzzy parabolic PDE systems with external disturbances and deception attacks

This paper addresses the fault-tolerant hybrid-triggered control problem for a class of nonlinear parabolic PDE systems with Neumann-type boundary conditions subject to actuator faults, external disturbances and randomly occurring deception attacks. Precisely, in order to provide an accurate representation of the considered nonlinear parabolic PDE system, a Takagi–Sugeno fuzzy PDE model is utilized. Further, in an effort to reduce the amount of communication transmissions, a more general hybrid-triggering method is implemented which includes both time- and event-triggered scheme. For the descriptions of the hybrid-triggered strategy and deception attacks, stochastic variables that follows Bernoulli distribution are presented. At the same time, an actuator fault model is considered in order to manage the impact of faults that occur in the actuators of the system under consideration. By the construction of suitable Lyapunov-Krasovskii functional, a set of adequate requirements is procured that ensure the closed-loop system is asymptotically stable and dissipative. In addition, the necessary controller gain matrices are obtained by making use of linear matrix inequalities. In the end, two numerical examples are exhibited to articulate the efficacy of the proposed control design technique.

Finite-Time Fuzzy Boundary Control for 2-D Spatial Nonlinear Parabolic PDE Systems

Seldom existing studies directly focus on the control issues of two-dimensional (2-D) spatial PDE systems, although they have strong application backgrounds in production and life. Therefore, this article investigates the finite-time control problem of an 2-D spatial nonlinear parabolic PDE system via T–S fuzzy boundary control scheme. First, the overall fuzzy system model is constructed using T–S fuzzy rules to approximate the original nonlinear system. Second, based on the planar distributed measurement, boundary collocated measurement, and linear measurement methods, three novel kinds of fuzzy boundary controllers are designed. Then, by employing the variable substitution and integral inequality techniques, three criteria that ensure the finite-time boundness of the considered system are obtained. Finally, simulations of main results are provided to verify the effectiveness and practicability of the proposed measurement and control schemes.

The Steady State Bifurcation for General Network-Organized Reaction-Diffusion Systems and Its Application in a Metapopulation Epidemic Model

Because of the high dimensionality of reaction-diffusion systems on networks, exploration for the bifurcation properties of such systems has been far less thorough than their corresponding parabolic PDE systems. With the overcoming of this long-standing difficulty, we consider the steady state bifurcation for the general network-organized reaction-diffusion system and derive the algorithm of calculating the perturbation incorporated restricted system associated with the simple zero singularity, through seeking out the parameter dependent center manifolds belonging to the unique zero eigenvalue. Rigorous theoretical analysis for the third-order truncated restricted system shows that the complex underlying networks can trigger some bifurcation types differing from the usual supercritical and subcritical pitchfork bifurcations and so enrich the dynamical behaviors of reaction-diffusion process in networks. To illustrate the application for our obtained theoretical results, we conduct a strict steady state bifurcation analysis for a metapopulation epidemic system defined on four typical connected undirected underlying networks, and find that there exist some new multistability phenomena and hysteresis effects in the irregular small-word and scale-free lattice network.

Computer-assisted modeling of Quorum sensing in bacterial population exposed to antibiotics

A mathematical model for bacterial growth and control by antibiotics treatment, including Quorum sensing as a special kind of communication, is introduced. We aim in setting up a flexible model structure allowing for fast simulations and overview about the general behavior. The deterministic approach can be used for in silico studies of bacterial cooperative behavior in the special case of Quorum sensing. Since antibiotic treatment is the basic and vital way to fight pathogenic bacteria, in the present study, we propose a modification of a reaction-diffusion model of communication processes in a bacterial population exposed to antibiotics. The dynamical biological system is formalized by a system of semilinear parabolic PDEs. The numerical solution of the 2D problem is based on a hybrid computing procedure, which includes a finite difference method combined with a Monte-Carlo simulation of population dynamics. Computational experiments are performed to describe space-time distributions of key chemical compounds characterizing Quorum sensing during the growth of a bacterial population and its decrease resulting from the predetermined strategy of antibiotic treatment.

Quantized state feedback-based $$\varvec{\mathcal {H}_\infty }$$ control for nonlinear parabolic PDE systems via finite-time interval

Quantized state feedback-based $$\varvec{\mathcal {H}_\infty }$$ control for nonlinear parabolic PDE systems via finite-time interval

Robust non-fragile boundary control for non-linear parabolic PDE systems with semi-Markov switching and input quantization

In this paper, the robust boundary control problem for non-linear PDE systems is discussed. Specifically, the PDE under investigation is of parabolic type with semi-Markov jumping signals subject to non-linearities and parameter uncertainties. The main goal of this paper is to devise a non-fragile boundary control law which assures the robust stabilization of the addressed system in spite of gain fluctuations and quantization in its design. In particular, to reduce the burden in the communication medium, the input signals can be quantized before transmitted to the actuator by using logarithmic quantizers. Moreover, by utilizing the Lyapunov stability theory and vector-valued Wirtinger’s inequality, a set of sufficient conditions for robust stabilization of the underlying system is established in the form of linear matrix inequalities (LMIs). Based on the obtained conditions, the existence of the desired controller is confirmed and an explicit form of the controller gains are developed. Eventually, the efficacious of the extracted theoretical results are manifested by virtue of two simulation examples.

Second order local minimal-time mean field games

The paper considers a forward–backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain $$\Omega $$ in minimal expected time. Agents are constrained by a bound on the drift depending on the density of other agents at their location. Existence for a finite time horizon T is proven via a fixed point argument, but the natural setting for this problem is in infinite time horizon. Estimates are needed to treat the limit $$T\rightarrow \infty $$ , and the asymptotic behavior of the solution obtained in this way is also studied. This passes through classical parabolic arguments and specific computations for MFGs. Both the Fokker–Planck equation on the density of agents and the Hamilton–Jacobi–Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach $$\partial \Omega $$ . The initial datum for the density is given, and the long-time limit of the value function is characterized as the solution of a stationary problem.

Fuzzy adaptive-event-triggered control for semi-linear parabolic PDE systems with stochastic actuator failures

This paper investigates the fault-tolerant control problem for fuzzy semi-linear parabolic PDE systems with stochastic actuator failures. First, a pointwise measurement-based adaptive-event-triggered control scheme is newly proposed for semi-linear PDE systems to reduce the waste of communication resources. Second, by introducing a more practical semi-Markov jump model describing the random occurring actuator failures, the closed-loop system is exactly represented by a T–S fuzzy PDE model with semi-Markovian switching parameters. Moreover, by constructing a suitable LKF including an improved looped-functional and using some advanced inequalities, sufficient stability conditions with less conservatism are established. Finally, two simulation examples including comparative studies are employed to demonstrate the superiority and effectiveness of the developed method.

A spatial stochastic epidemic model: law of large numbers and central limit theorem

We consider an individual-based SIR stochastic epidemic model in continuous space. The evolution of the epidemic involves the rates of infection and recovering of individuals. We assume that individuals move randomly on the two-dimensional torus according to independent Brownian motions. We define the sequences of empirical measures, which describe the evolution of the positions of the susceptible, infected and removed individuals. We prove the convergence in probability, as the size of the population tends to infinity, of those sequences of measures towards the solution of a system of parabolic PDEs. We show that appropriately centrered renormalized sequences of fluctuations around the above limit converge in law, as the size of the population tends to infinity, towards a Gaussian distribution valued process, solution of a system of linear PDEs with highly singular Gaussian driving processes. In the case where the individuals do not move we also define and study the law of large numbers and central limit theorem for the same sequence.

Explicit model predictive control for PDEs: The case of a heat equation

Explicit model predictive control design is carefully developed for discrete-time linear plants on Hilbert spaces, and we highlight the role of the so-called Slater condition in the reliable explicit solution of the MPC optimization. We then proceed to present an explicit MPC algorithm that accounts for the stabilization and input constraints satisfaction. We do structure preserving temporal discretization of the infinite-dimensional parabolic PDE system by application of the Cayley transformation. The salient feature of explicit MPC design is the realization of the region-free approach in explicit MPC design with identification of active constraint sets to realize optimal stabilization and constraints satisfaction. Finally, the resulting design is illustrated by the application to the PDE model given by an unstable heat equation with boundary actuation and Neumann boundary conditions. The example demonstrates simultaneous stabilization and input constraints satisfaction on the one hand, and on the ability to deal with a relatively high plant dimension and a long optimization horizon on the other hand.

Kobayashi–Warren–Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation

In this paper we study the Dirichlet problem for the Kobayashi–Warren–Carter system. This system of parabolic PDE’s models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the ω-limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the ω-limit set belong to the space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constant solutions.

Reduction of systems of two nonlinear parabolic type partial differential equations to solvable forms using differential invariants

Differential invariants for linear and nonlinear ordinary and partial differential equations have been derived using Lie infinitesimal method. These invariants help in reducing differential equations to their simplest possible solvable forms through invertible transformations of the dependent and independent variables. Here we derive differential invariants for a class of systems of two second order nonlinear parabolic type partial differential equations. Deduced invariants are shown to reveal solvable forms of these systems of PDEs that are much simpler than the considered general systems of nonlinear parabolic type PDEs.

Robust Cooperative Output Regulation for a Network of Parabolic PDE Systems

This article considers the robust cooperative output regulation for a network of parabolic PDE systems. The solution to this problem is obtained by extending the cooperative internal model principle from finite to infinite dimensions. For a time-invariant digraph describing the communication topology, a two-step backstepping approach is presented to systematically design cooperative state feedback regulators. This allows to solve both the leader–follower and the leaderless output synchronization problem in the presence of disturbances and model uncertainty for a finite-dimensional leader. Solvability conditions of the robust cooperative output regulation problem are presented in terms of the communication graph and the agent transfer behavior. The results of this article are demonstrated for a network consisting of four uncertain parabolic agents with and without a finite-dimensional leader in the presence of disturbances.

The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System

We research a control problem for an ecological model given by a reaction–diffusion system. The ecological model is given by a nonlinear parabolic PDE system of three equations modelling the interaction of three species by considering the standard Lotka-Volterra assumptions. The optimal control problem consists of the determination of a coefficient such that the population density of predator decreases. We reformulate the control problem as an optimal control problem by introducing an appropriate cost function. Then, we introduce and prove three types of results. A first contribution of the paper is the well-posedness framework of the mathematical model by considering that the interaction of the species is given by a general functional responses. Second, we study the differentiability properties of a cost function. The third result is the existence of optimal solutions, the existence of an adjoint state, and a characterization of the control function. The first result is proved by the application of semigroup theory and the second and third result are proved by the application of Dubovitskii and Milyutin formalism.

Diagnostic observer-based fault detection for nonlinear parabolic PDE systems via dual sampling approaches

This paper investigates the fault detection problem for nonlinear parabolic PDE systems. In contrast to the existing works, the designed fault detection observer utilizes less state information in both time domain and space domain, the details of which are illustrated as follows. First, based on Takagi–Sugeno fuzzy theory, a novel fuzzy state observer is developed via sampling in time and space approach. Then, by using Lyapunov direct method, some sufficient conditions that guarantee both the disturbance/control-inputs attenuation performance and the faulty sensitivity performance of the fault detection systems are established in the form of linear matrix inequality, which provides an effective approach to optimize the systems’ performance. Finally, two simulation examples are given to illustrate the effectiveness of the designed fault detection systems.

Spatial-L∞-Norm-Based Finite-Time Bounded Control for Semilinear Parabolic PDE Systems With Applications to Chemical-Reaction Processes.

This article investigates a spatial- L∞ -norm-based reliable bounded control problem for a class of nonlinear partial differential equation systems in a finite-time interval. The main novelties are reflected in the following aspects: 1) inspired by the sector-nonlinearity approach, the considered nonlinear system is reconstructed by a Takagi-Sugeno fuzzy model, which provides an effective method for control design. Besides, several actuator failures, such as stuck faulty, outage faulty, and bias faulty, are taken into account and modeled by a novel Markov process; 2) partial areas' states are sampled and transmitted based on a new distributed event-triggered communication strategy, which reduces the cost of the system design and saves the limited network resources to some extent; and 3) on the basis of the first two works, a new piecewise fuzzy controller, which requires fewer actuators compared with the distributed control method, is constructed. Then, some sufficient conditions to guarantee the finite-time boundedness (in the sense of spatial L∞ norm) and mixed L2-L∞/H∞ disturbance attenuation performance are established, and a new linear matrix inequality relax technique is introduced to deal with the strict constraint that is caused by the asynchronous phenomenon between plant and controller. Finally, two simulation studies are given to illustrate the effectiveness and advantages of the developed controller.

Steady states and their qualitative properties of several classes of Keller-Segel models

Chemotaxis is the directed movement of bacterial organisms towards or away from stimulating chemicals in their environment. This intrinsic ability is important for species to profit from the good, avoid the harm and to survive their environment. In order to study cellular chemotactic movement, Evelyn Fox Keller and Lee Aaron Segel proposed a class of parabolic PDE systems to describe the evolution of bacterial population density and chemical concentration. Though these systems have relatively simple structures, they admit rich spatial-temporal dynamics and can well be used to model cell aggregation, one of the most important phenomena in chemotaxis, hence become one of the most extensively studied topics in PDEs over the past few decades. Cell aggregation can be modeled by the $\\delta$-profile out of finite-time blowups in the parabolic systems, and also by the steady states with striking spatial concentrating profiles such as spikes, transition layers, etc. The former approach has been surveyed by a few articles, while a review of the latter is not available, and therefore it is the goal of this paper to survey the steady states and their qualitative properties of several classes of Keller-Segel models, with a focus on the spikes and transition layers. In particular, we shall describe the classical results and recent developments of this problem, as well as inspiring mathematics developed meanwhile such as the variation method, singular perturbation, bifurcation theory. We shall also introduce some open problems concerning stationary systems.