Linear Parabolic Partial Differential Equations Research Articles

Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition - that we decide to call here local-time Kirchhoff's condition - induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

A posteriori error estimates of the weak Galerkin finite element methods for parabolic problems

In this paper, We propose the residual-based a posteriori error estimator of the weak Galerkin finite element method with the backward Euler time discretization for the linear parabolic partial differential equation. For the a posteriori error estimator, we introduce the Helmholtz decomposition technique to prove its reliability. We mainly study WG element (Pj(K),Pℓ(∂K),V(K,r)=RTj(K)). Numerical experiments based on the lowest order case, i.e. (P0(K),P0(∂K),RT0(K)), are provided to verify the theoretical research.

Sensor Fault Detection and Diagnosis of Linear Parabolic PDE Systems With Unknown Inputs

This paper investigates the sensor bias fault detection and diagnosis problem for linear parabolic partial differential equation (PDE) systems under the existence of unknown input signals. A variation of Wirtinger's inequality is used to design a Luenberger-type PDE observer and radial basis function (RBF) neural networks are applied to approximate the unknown inputs, guaranteeing the boundedness of the state estimation error. Taking the measurement error as the fault indication signal, a residual evaluation logic is developed to achieve fault detection. An adaptive diagnostic law is constructed to estimate the values of sensor bias faults once they are detected. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> -modification RBF neural networks are designed to compensate for the unknown inputs in the presence of sensor faults, ensuring that both state and sensor bias faults estimation errors converge to some adjustable sets. The effectiveness and applicability of the developed fault diagnosis strategy are demonstrated by simulation results on a heat diffusion process.

Model order reduction for optimality systems through empirical gramians

In the present article, optimal control problems for linear parabolic partial differential equations (PDEs) with time-dependent coefficient functions are considered. One of the common approach in literature is to derive the first-order sufficient optimality system and to apply a finite element (FE) discretization. This leads to a specific linear but high-dimensional time variant (LTV) dynamical system. To reduce the size of the LTV system, we apply a tailored reduced order modeling technique based on empirical gramians and derived directly from the first-order optimality system. For testing purpose, we focus on two specific examples: a multiobjective optimization and a closed-loop optimal control problem. Our proposed methodology results to be better performing than a standard proper orthogonal decomposition (POD) approach for the above mentioned examples.

Reinforcement learning for continuous-time mean-variance portfolio selection in a regime-switching market

We propose a reinforcement learning (RL) approach to solve the continuous-time mean-variance portfolio selection problem in a regime-switching market, where the market regime is unobservable. To encourage exploration for learning, we formulate an exploratory stochastic control problem with an entropy-regularized mean-variance objective. We obtain semi-analytical representations of the optimal value function and optimal policy, which involve unknown solutions to two linear parabolic partial differential equations (PDEs). We utilize these representations to parametrize the value function and policy for learning with the unknown solutions to the PDEs approximated based on polynomials. We develop an actor-critic RL algorithm to learn the optimal policy through interactions with the market environment. The algorithm carries out filtering to obtain the belief probability of the market regime and performs policy evaluation and policy gradient updates alternately. Empirical results demonstrate the advantages of our RL algorithm in relatively long-term investment problems over the classical control approach and an RL algorithm developed for the continuous-time mean-variance problem without considering regime switches.

Actuator Anomaly Detection in Linear Parabolic Distributed Parameter Cyber-Physical Systems

Distributed parameter cyber-physical systems (DPCPSs) are characterized by two features: 1) they exhibit spatio-temporal (distributed parameter) dynamics and 2) there exist cyber components in the form of communication channels between the physical plant and control system. DPCPSs are vulnerable to anomalies such as nonmalicious physical faults and malicious cyber-attacks. Although the safety and security aspects of CPSs modeled by ordinary differential equations (ODEs) have been extensively explored during the past decade, security of DPCPSs has not received its due attention despite its safety-critical nature. In this work, we explore anomaly diagnostics in DPCPSs from a system theoretic viewpoint. Specifically, we focus on DPCPSs modeled by linear parabolic partial differential equations (PDEs) subject to physical faults and cyber-attacks in the actuation channel. First, we explore the detectability of malicious anomalies (i.e., cyber-attacks) and derive conditions for stealthy attacks in the context of output-feedback-based monitoring. Such stealthy attacks essentially illustrate the fundamental limitations of feedback-based monitoring algorithms given perfect adversarial knowledge. Subsequently, focusing on the anomalies that are detectable, we develop a design framework for anomaly detection algorithms based on output injection observers. Such detection algorithm explicitly considers the conflicting objectives of robustness to uncertainties and sensitivity to anomalies in their design. Finally, theoretical analysis and simulation studies are performed to illustrate the effectiveness of the proposed approach.

Lyapunov-based stabilization mobile control design of linear parabolic PDE systems

This article considers the exponential mobile stabilization control of linear parabolic partial differential equation (PDE) systems. Firstly, a stabilization scheme based on static output feedback (SOF) control and mobile actuator/sensor guidance is proposed. Next, the operator theory is used to discuss the well-posedness of closed-loop PDE systems. Thus, the SOF control and mobile guidance design approach is simultaneously proposed to guarantee that the closed-loop PDE system is exponentially stable. In final, the effectiveness of the design method is verified by numerical simulation.

The Homotopy Perturbation Method for Solving Nonlocal Initial-Boundary Value Problems for Parabolic and Hyperbolic Partial Differential Equations

To obtain approximate-exact solutions to nonlocal initial-boundary value problems (IBVPs) of linear and nonlinear parabolic and hyperbolic partial differential equations (PDEs) subject to initial and nonlocal boundary conditions of integral type, the homotopy perturbation method (HPM) is utilized in this study. The HPM is used to solve the specified nonlocal IBVPs, which are then transformed into local Dirichlet IBVPs. Some examples demonstrate how accurate and efficient the HPM.&#x0D;

Approximating the Stationary Bellman Equation by Hierarchical Tensor Products

We treat infinite horizon optimal control problems by solving the associated stationary Hamilton-Jacobi-Bellman (HJB) equation numerically to compute the value function and an optimal feedback law. The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations (PDE), which means that the HJB is non linear and suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iteration algorithm, where the problem is reduced to a sequence of linear, hyperbolic PDEs. These equations remain the computational bottleneck due to their high dimensions. By the method of characteristics these linearized HJB equations can be reformulated via the Koopman operator in the spirit of dynamic programming. The resulting operator equations are solved using a minimal residual method. To overcome numerical infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors), and multi-polynomials, together with high dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidences are given.

Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems

We prove the local 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 behavioral economics, where the observations and preferences are (reference-)time-dependent. 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 solvability of nonlocal linear PDEs and establish a Schauder-type prior estimate for the solutions. Then, by the linearization method, we 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.

Nash Equilibria Strategies and Equivalent Single-Objective Optimization Problems. The Case of Linear Partial Differential Equations

In this paper we study the existence and uniqueness of Nash equilibria (solution to competition-wise problems, with several controls trying to reach possibly different goals) associated to linear partial differential equations and show that, in some cases, they are also the solution of suitable single-objective optimization problems (i.e. cooperative-wise problems, where all the controls cooperate to reach a common goal). We use cost functions associated with a particular linear parabolic partial differential equation and distributed controls, but the results are also valid for more general linear differential equations (including elliptic and hyperbolic cases) and controls (e.g. boundary controls, initial value controls,...).

Spatiotemporal adaptive state feedback control of a linear parabolic partial differential equation

AbstractThis article deals with the issue of asymptotic stabilization for a linear parabolic partial differential equation (PDE) with an unknown space‐varying reaction coefficient and multiple local piecewise uniform control. Clearly, the unknown reaction coefficient belongs to a function space. Hence, the fundamental difficulty for such issue lies in the lack of a conceptually simple but effective parameter identification technique in a function space. By the Lyapunov technique combined with a variant of Poincaré‐Wirtinger inequality, an update law is derived for estimate of the unknown reaction coefficient in a function space. Then a spatiotemporal adaptive state feedback control law is constructed such that the estimate of the unknown coefficient is bounded and the closed‐loop PDE is asymptotically stable in the sense of spatial norm if a sufficient condition given in terms of space‐time varying linear matrix inequalities (LMIs) is fulfilled for the estimated coefficient and the control gains. Both analytical and numerical approaches are proposed to construct a feasible solution to the space‐time varying LMI problem. With the aid of the semigroup theory, the well‐posedness and regularity of the closed‐loop PDE is also analyzed. Moreover, two extensions of the proposed adaptive control scheme are discussed: the PDE in ‐D space and the PDE with unknown diffusion and reaction coefficients. Finally, numerical simulation results are presented to support the proposed spatiotemporal adaptive control design.

Parabolic differential equations with bounded delay

We show the continuous dependence of solutions of linear nonautonomous second-order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-* topology of delay coefficients is required. The results are important in the applications of the theory of Lyapunov exponents to the investigation of PDEs with delay.

Closed-form solutions via the invariant approach for one-factor commodity models

We utilise the invariant method for scalar linear (1+1) parabolic partial differential equations so as to effect reductions via invertible transformations to Lie canonical forms of the first and third types. This is performed for the one-factor commodity models that have parameters which are constants or dependent on the price of stock or the time. The unknown parameters of the equations are determined. New closed-form solutions for the simple transformed equations are deduced by utilizing the equivalence maps and known solutions of the two Lie canonical forms.

Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics.

We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the "curse of dimensionality" related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to the original high-dimensional space using radial basis function interpolation and geometric harmonics. The proposed numerical data-driven scheme can also be applied as a reduced-order model procedure for the numerical solution/propagation of the (transient) dynamics of partial differential equations (PDEs). We assess the performance of the proposed scheme via three different families of problems: (a) the forecasting of synthetic time series generated by three simplistic linear and weakly nonlinear stochastic models resembling electroencephalography signals, (b) the prediction/propagation of the solution profiles of a linear parabolic PDE and the Brusselator model (a set of two nonlinear parabolic PDEs), and (c) the forecasting of a real-world data set containing daily time series of ten key foreign exchange rates spanning the time period 3 September 2001-29 October 2020.

A semi-analytical solution of Richards Equation for two-layered one-dimensional soil

A semi-analytical solution of the Richards Equation posed on a two-layered one-dimensional soil supplied with various boundary conditions is derived under a constraint that the constitutive relations are exponentially dependent on the pressure head. It allows for a transformation of the Richards Equation into a linear parabolic partial differential equation that governs a spatial–temporal function that represents the hydraulic conductivity. The procedure is proceeded with expressing this function as a linear combination of a set of eigenfunctions associated with a novel Sturm—Liouville problem that reflects the layer system and an auxiliary function that depends only on the spatial variable and the pressure head at the interface at the time of interest. All the relevant coefficients in the representation satisfy a nonlinear differential–algebraic system gathered from imposing the continuity of the pressure head and its flux at the interface. Two different approximations of the derivatives yield algebraic systems to be solved by the Newton method of iteration. Several pertinent numerical experiments demonstrating the approach are discussed and compared with the standard finite volume method.

Indirect Optimal Control of Advection-Diffusion Fields through Distributed Robotic Swarms

We consider the problem of optimally guiding a large-scale swarm of underwater vehicles that is tasked with the indirect control of an advection-diffusion environmental field. The microscopic vehicle dynamics are governed by a stochastic differential equation (SDE) with drift. The drift terms model the self-propelled velocity of the vehicle and the velocity field of the currents. In the mean-field setting, the macroscopic vehicle dynamics are governed by a Kolmogorov forward equation in the form of a linear parabolic advection-diffusion partial differential equation (PDE). The environmental field is governed by an advection-diffusion PDE in which the advection term is defined by the fluid velocity field. The vehicles are equipped with on-board actuators that enable the swarm to act as a distributed source in the environmental field, modulated by a scalar control parameter that determines the local source intensity. In this setting, we formulate an optimal control problem to compute the vehicle velocity and actuator intensity fields that drive the environmental field to a desired distribution within a specified amount of time. After proving an existence result for the solution of the optimal control problem, we discretize and solve the problem using the Finite Element Method (FEM). We show through numerical simulations the effectiveness of our control strategy in regulating the environmental field to zero or to a desired distribution in the presence of a double-gyre flow field.

Exponential Time Differencing Schemes for Fuel Depletion and Transport in Molten Salt Reactors: Theory and Implementation

A numerical framework for modeling depletion and mass transport in liquid-fueled molten salt reactions is presented based on exponential time differencing. The solution method involves using the finite volume method to transform the system of partial differential equations (PDEs) into a much larger system of ordinary differential equations. The key part of this method involves solving for the exponential of a matrix. We explore six different algorithms to compute the exponential in a series of progression problems that explore physical transport phenomena in molten salt reactors. This framework shows good results for solving linear parabolic PDEs with each of the six matrix exponential algorithms. For large problems, the series solvers such as Padé and Taylor have large run times, which can be mitigated by using the Krylov subspace.

Extending a new two-grid waveform relaxation on a spatial finite element discretization

In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the twodimensional heat equation.

On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.