Parabolic Parabolic Partial Differential Equations Research Articles

Probabilistic approach to singular perturbations of viscosity solutions to nonlinear parabolic PDEs

In this paper, we prove a convergence theorem for singular perturbations problems for a class of fully nonlinear parabolic partial differential equations (PDEs) with ergodic structures. The limit function is represented as the viscosity solution to a fully nonlinear degenerate PDEs. Our approach is mainly based on G-stochastic analysis argument. As a byproduct, we also establish the averaging principle for stochastic differential equations driven by G-Brownian motion (G-SDEs) with two time-scales. The results extend Khasminskii’s averaging principle to nonlinear case.

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.

Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation

In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.

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.

Distributed Adaptive Consensus of Parabolic PDE Agents on Switching Graphs With Relative Output Information

This article mainly solves the consensus issue of parabolic partial differential equation (PDE) agents with switching topology by output feedback. A novel edge-based adaptive control protocol is designed to reach consensus under the condition that the switching graphs are always connected at any switching instants. Different from the existing adaptive protocol associated with partial differential dynamics, the proposed adaptive observer-type law relies on the relative output information rather than relative state information. A proper Lyapunov functional is constructed and some important lemmas are used, then a sufficient condition is obtained for the consensus of parabolic PDE agents on switching graphs. Besides, a corollary about the distributed adaptive consensus of parabolic PDE agents on fixed undirected communication networks is given. Finally, the theoretical results are demonstrated by two numerical simulations.

Approximations of Lyapunov functionals for ISS analysis of a class of higher dimensional nonlinear parabolic PDEs

This paper introduces a novel method, namely the approximations of Lyapunov functionals, for input-to-state stability (ISS) analysis of a class of higher dimensional nonlinear parabolic partial differential equations (PDEs) with variable coefficients. Specifically, for any q∈[1,+∞] and the considered nonlinear parabolic PDEs with different types of boundary disturbances in Llocq(R+;L1(∂Ω)) and initial data in L1(Ω), we show that ISS-like estimates in L1-norm (or weighted L1-norm) can be established by constructing approximations of (coercive and non-coercive) Lyapunov functionals. Some examples are provided to illustrate the application of the proposed method.

ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs

This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed in two ways: by using an ISS Lyapunov Functional for the sup norm and by exploiting well-known maximum principles. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficiency of the proposed methodology for the derivation of ISS estimates in the sup norm of the state.

Nonlocal Fully Nonlinear Parabolic Differential Equations Arising in Time-Inconsistent Problems

We prove the well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter $t$ on top of the temporal and spatial variables $(s,y)$ and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point $(t,s,y)$ but also at the diagonal line of the time domain $(s,s,y)$. Such equations arise from time-inconsistent problems in game theory or behavioural economics, where the observations and preferences are (reference-)time-dependent. To address the open problem of the well-posedness of the corresponding nonlocal PDEs (or the time-inconsistent problems), we first study the linearized version of the nonlocal PDEs with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. With fixed-point arguments, we obtain the well-posedness of nonlocal linear PDEs and establish a Schauder-type prior estimate for the solutions. Then, by the linearization method, we analogously establish the well-posedness under the fully nonlinear case. Moreover, we reveal that the solution of a nonlocal fully nonlinear parabolic PDE is an adapted solution to a flow of second-order forward-backward stochastic differential equations.

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.

ISS-like estimates for nonlinear parabolic PDEs with variable coefficients on higher dimensional domains

This paper presents a maximum principle-based approach in the establishment of ISS-like estimates for a class of nonlinear parabolic partial differential equations (PDEs) with variable coefficients and different types of nonlinear boundary conditions on higher dimensional domains. Comparing with the existing literature, the ISS-like estimates established in this paper are independent of the nonlinear terms of the PDEs. Technical development on ISS analysis of the considered systems is detailed, and an example of establishing ISS-like estimates for a nonlinear parabolic equation with, respectively, a nonlinear Robin boundary condition and a nonlinear Dirichlet boundary condition is provided to illustrate the application of the developed 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.

Construction of a class of forward performance processes in stochastic factor models, and an extension of Widder’s theorem

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes, as well as the corresponding optimal portfolios, with power-utility initial data and for stock–factor correlation matrices with eigenvalue equality (EVE) structure, which we introduce here. This is done by solving the associated nonlinear parabolic partial differential equations (PDEs) posed in the “wrong” time direction. Along the way, we establish on domains an explicit form of the generalised Widder theorem of Nadtochiy and Tehranchi (Math. Finance 27:438–470, 2015, Theorem 3.12) and rely for that on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the “right” time direction.

Solving nonlinear parabolic PDEs in several dimensions: Parallelized ESERK codes

There is a very large number of very important situations which can be modeled with nonlinear parabolic partial differential equations (PDEs) in several dimensions. In general, these PDEs can be solved by discretizing in the spatial variables and transforming them into huge systems of ordinary differential equations (ODEs), which are very stiff. Therefore, standard explicit methods require a large number of iterations to solve stiff problems. But implicit schemes are computationally very expensive when solving huge systems of nonlinear ODEs. Several families of Extrapolated Stabilized Explicit Runge-Kutta schemes (ESERK) with different order of accuracy (3 to 6) are derived and analyzed in this work. They are explicit methods, with stability regions extended, along the negative real semi-axis, quadratically with respect to the number of stages s, hence they can be considered to solve stiff problems much faster than traditional explicit schemes. Additionally, they allow the adaptation of the step length easily with a very small cost.Two new families of ESERK schemes (ESERK3 and ESERK6) are derived, and analyzed, in this work. Each family has more than 50 new schemes, with up to 84.000 stages in the case of ESERK6. For the first time, we also parallelized all these new variable step length and variable number of stages algorithms (ESERK3, ESERK4, ESERK5, and ESERK6). These parallelized strategies allow to decrease times significantly, as it is discussed and also shown numerically in two problems. Thus, the new codes provide very good results compared to other well-known ODE solvers. Finally, a new strategy is proposed to increase the efficiency of these schemes, and it is discussed the idea of combining ESERK families in one code, because typically, stiff problems have different zones and according to them and the requested tolerance the optimum order of convergence is different.

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.

Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs

In this work, numerical approximation of the second order non-autonomous semilinear parabolic partial differential equations (PDEs) is investigated using the classical finite element method. To the best of our knowledge, only the linear case is investigated in the literature. Using an approach based on evolution operator depending on two parameters, we obtain the error estimate of the semi-discrete scheme based on finite element method toward the mild solution of semilinear non-autonomous PDEs under polynomial growth and one-sided Lipschitz conditions of the nonlinear term. Our convergence rate is obtained with general non-smooth initial data, and is similar to that of the autonomous case. Such convergence result is very important in numerical analysis. For instance, it is one step forward for numerical approximation of non-autonomous stochastic partial differential equations with the finite element method.

Dynamic Compensator Design of Linear Parabolic MIMO PDEs in $N$-Dimensional Spatial Domain

This article employs the observer-based output feedback control technique to deal with dynamic compensator design for a linear $N$ -D parabolic partial differential equation (PDE) with multiple local piecewise control inputs and multiple noncollocated local piecewise observation outputs. These control inputs and observation outputs are provided by only few actuators and noncollocated sensors active over partial areas (or entire) of the spatial domain. An observer-based dynamic feedback compensator is constructed for exponential stabilization of the linear PDE. Poincare–Wirtinger inequality and its variant in $N$ -D spatial domain are presented for the closed-loop stability analysis. By the Lyapunov direct method and Poincare–Wirtinger inequality and its variant, sufficient conditions on the existence of such feedback compensator of the linear PDE are developed, and presented in term of standard linear matrix inequalities. Well-posedness is also analyzed for both open-loop PDE and resulting closed-loop coupled PDEs within the $C_0$ -semigroup framework. Finally, numerical simulation results are presented to support the proposed design method.

Extremum seeking for unknown scalar maps in cascade with a class of parabolic partial differential equations

SummaryWe present a generalization of the scalar gradient extremum seeking (ES) algorithm, which maximizes static maps in the presence of infinite‐dimensional dynamics described by parabolic partial differential equations (PDEs). The PDE dynamics contains reaction‐advection‐diffusion (RAD) like terms. Basically, the effects of the PDE dynamics in the additive dither signals are canceled out using the trajectory generation paradigm. Moreover, the inclusion of a boundary control for the PDE process stabilizes the closed‐loop feedback system. By properly demodulating the map output corresponding to the manner in which it is perturbed, the ES algorithm maximizes the output of the unknown map. In particular, our parabolic PDE compensator employs the same perturbation‐based (averaging‐based) estimate for the Hessian of the function to be maximized applied in the previous publications free of PDEs. We prove local stability of the algorithm, real‐time maximization of the map and convergence to a small neighborhood of the desired (unknown) extremum by means of backstepping transformation, Lyapunov functional and the theory of averaging in infinite dimensions. Finally, we present the generalization to the scalar Newton‐based ES algorithm, which maximizes the map's higher derivatives in the presence of RAD‐like dynamics. By modifying the demodulating signals, the ES algorithm maximizes the nth derivative only through measurements of the own map. The Newton‐based ES approach removes the dependence of the convergence rate on the unknown Hessian of the higher derivative, an effort to improve performance and remove limitations of standard gradient‐based ES. Numerical examples support the theoretical results.

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.