First-order Hyperbolic Partial Differential Equations Research Articles

Neural operators for robust output regulation of hyperbolic PDEs

The recently introduced neural operator (NO) has been employed as a gain approximator in the backstepping stabilization control of first-order hyperbolic and parabolic partial differential equation (PDE) systems. Due to the global approximation ability of the DeepONet, the NO provides approximate spatial gain function with arbitrary accuracy. The closed-loop system stability can be ensured by the backstepping controller involving the approximate gain with sufficiently small error. In this paper, the NO theory is leveraged to solve the robust output regulation problem for a class of uncertain hyperbolic PDE systems under the design framework of backstepping-based regulator. The NO is trained offline on a dataset containing a sufficient number of system parameters and corresponding prior solutions of the kernel equation, so as to generate feedback gain for the robust regulator. Once the NO is trained, the kernel equation does not need to be solved ever again, for any new system parameters that do not exceed the range of the training set. Based on the internal model principle, the regulator is inherently robust to a degree of parameter uncertainty and error in approximate gain. Therefore, the tracking error can still converge to 0 if the extended regulator equations are solvable and the parameter uncertainty leads to an asymptotically stable origin. We provide a series of theory proofs and a numerical test under the approximate control and observation gains to demonstrate the robust regulation problem. It is shown that the NO is almost three orders of magnitude faster than PDE solver in generating kernel function, and the loss remains on the order of 10−4 in the test. This provides an opportunity to use the NO methodology for accelerated gain scheduling regulation for PDEs with time-varying system parameters.

A one-dimensional computational model for blood flow in an elastic blood vessel with a rigid catheter.

Strokes are one of the leading causes of death in the United States. Stroke treatment involves removal or dissolution of the obstruction (usually a clot) in the blocked artery by catheter insertion. A computer simulation to systematically plan such patient-specific treatments needs a network of about 105 blood vessels including collaterals. The existing computational fluid dynamic (CFD) solvers are not employed for stroke treatment planning as they are incapable of providing solutions for such big arterial trees in a reasonable amount of time. This work presents a novel one-dimensional mathematical formulation for blood flow modeling in an elastic blood vessel with a centrally placed rigid catheter. The governing equations are first-order hyperbolic partial differential equations, and the hypergeometric function needs to be computed to obtain the characteristic system of these hyperbolic equations. We employed the Discontinuous Galerkin method to solve the hyperbolic system and validated the implementation by comparing it against a well-established 3D CFD solver using idealized vessels and a realistic truncated arterial network. The results showed clinically insignificant differences in steady flow cases, with overall variations between 1D and 3D models remaining below 10%. Additionally, the solver accurately captured wave reflection phenomena at domain discontinuities in unsteady cases. A primary advantage of this model over 3D solvers is its ease in obtaining a discretized geometry of complex vasculatures with multiple arterial branches. Thus, the 1D computational model offers good accuracy and applicability in simulating complex vasculatures, demonstrating promising potential for investigating patient-specific endovascular interventions in strokes.

Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

APPLICATION OF THE FOURIER TRANSFORMATION FOR SOLVING A FIRST ORDER PARTIAL DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

This work is devoted to the solution of a linear first-order hyperbolic partial differential equation with constant coefficients. The specific objective is to prove the existence and uniqueness of the solution of the proposed PDE. The existence and uniqueness of the solution have been proved. To demonstrate the existence of the solution, the Fourier transformation was used. The variational formulation was used to prove the uniqueness of the solution. The combination of the Fourier transformation and the variational formulation yielded the expected results: the existence and uniqueness of the solution.

Robust stabilization of 2 × 2 first-order hyperbolic PDEs with uncertain input delay

A backstepping-based compensator design is developed for a system of 2 × 2 first-order linear hyperbolic partial differential equations (PDE) in the presence of an uncertain long input delay at boundary. We introduce a transport PDE to represent the delayed input, which leads to three coupled first-order hyperbolic PDEs. A novel backstepping transformation, composed of two Volterra transformations and an affine Volterra transformation, is introduced for the predictive control design. The resulting kernel equations from the affine Volterra transformation are two coupled first-order PDEs and each with two boundary conditions, which brings challenges to the well-posedness analysis. We solve the challenge by using the method of characteristics and the successive approximation. To analyze the sensitivity of the closed-loop system to uncertain input delay, we introduce a neutral system which captures the control effect resulted from the delay uncertainty. It is proved that the proposed control is robust to small delay variations. Numerical examples illustrate the performance of the proposed compensator.

Robustification of stabilizing controllers for ODE-PDE-ODE systems: A filtering approach

In this paper, we present a filtering technique that robustifies stabilizing controllers for systems composed of heterodirectional linear first-order hyperbolic Partial Differential Equations (PDEs) interconnected with Ordinary Differential Equations (ODEs) through their boundaries. The actuation is either available through one of the ODE or at the boundary of the PDE. The proposed framework covers a broad general class of interconnected systems. Assuming that a stabilizing controller is available, we derive simple sufficient conditions under which appropriate low-pass filters can be combined with the control law to robustify the closed-loop system. Our approach is based on a rewriting of the distributed dynamics as a delay-differential algebraic equation and an analysis in the Laplace domain. The proposed technique will simplify the design of stabilizing controllers for the class of systems under consideration (which can now be done only on a case-by-case basis due to the complexity and generality of the underlying interconnections) since it dissociates the stabilization problem from the robustness aspects. Indeed, it becomes possible to use convenient (but non-robust) techniques for the stabilization of such systems (as the cancellation of the boundary coupling terms or the inversion of the ODE dynamics), knowing that the resulting control law can be made robust (to delays and uncertainties) using the proposed filtering methodology.

Time-dependent finite-difference model for transient and steady-state analysis of thermoelectric bulk materials

A mathematical model of coupled thermoelectricity is presented to investigate the transient and steady-state behaviour of thermoelectric bulk material. Governing partial differential equations (PDEs) for the coupled thermal and electrical behaviour of the thermoelectric model are discretised using the explicit finite-difference method. Differencing schemes like Upwind and Lax–Wendroff methods are employed to obtain solutions for the first-order hyperbolic PDEs, whereas FTCS (Forward Time, Centred Space) scheme is employed to solve second-order parabolic PDEs. Courant-Friedrichs-Lewy and Von Neumann stability analyses are done to ensure the stability and convergence of the model. The model considers the temperature dependency of thermal conductivity, electrical conductivity, and Seebeck coefficient of the P/N materials separately. and accounts for the Seebeck, Peltier, and Joule-Thomson effects in thermoelectric materials. The new model is practically useful to predict the transient and steady-state behaviours of a thermoelectric device with multiple P-N elements. The results of the presented finite-difference model are proven to agree well with experimental values as well as 3D simulations with ANSYS®.

New Fifth-Kind Chebyshev Collocation Scheme for First-Order Hyperbolic and Second-Order Convection-Diffusion Partial Differential Equations

The fifth type of Chebyshev polynomials was used in tandem with the spectral tau method to achieve a semianalytical solution for the partial differential equation of the hyperbolic first order. For this purpose, the problem was diminished to the solution of a set of algebraic equations in unspecified expansion coefficients. The convergence and error analysis of the proposed expansion were studied in-depth. Numerical trials have confirmed the applicability and the accuracy.

Symmetry Groups of the Forward Master Equation for Stochastic Neutron Populations

In this work, we apply Lie Group Theory (LGT) to aid in solving and understanding equations arising from the Forward Master Equation formulation for the neutron number distribution in a zero-dimensional setting. In particular, we focus our LGT study on a first-order hyperbolic partial differential equation satisfied by the probability generating function. We show the connection between solutions to the symmetry determining equations with established analytical solutions given by Bell and by Prinja and Souto. We derive global transformations for isolated neutron fission chains as well as neutron sources and provide a physical interpretation of the results throughout.

Adaptive output tracking of distributed parameter systems

In this paper, we consider the unknown trajectory tracking problem for stable distributed parameter systems. The main assumptions are that trajectory signals are generated by an unknown finite-dimensional exosystem that is a marginally stable system and the tracking error is available for measurement. In order to achieve perfect error regulation, a frequency estimator scheme is proposed to estimate unknown exosystem parameters, and the control law that is designed based on geometric output regulation theory is revisited. The success of the proposed method is demonstrated on a parabolic heat equation and a first-order hyperbolic partial differential equation.

Boundary Observer Design for Fleets of Plug-in Electric Vehicles in Bidirectional Charging Stations

In this paper, we consider the problem of estimating the density profiles of fleets of Plug-in Electric Vehicles (PEVs) in charging stations. We put the focus on the case of bidirectional exchange of energy between the grid and PEVs i.e. both vehicle-to-grid (V2G) and grid-to-vehicle (G2V) modes are considered. In both modes, the energy exchange is assimilated to a transfer phenomenon that is modeled by using a first-order hyperbolic partial differential equation (PDE). We design a boundary observer that provides online estimates of charge/discharge density profiles along the spatial domain. The observer is designed based on the system PDE model. It only makes use of boundary measurements and involves measurement injection in the boundary equation. The exponential convergence of the observer is proved using a suitable Lyapunov functional. Simulation results are provided that highlight the satisfactory observer performances.

PDE-Based Deployment of Multiagents Measuring Relative Position to One Neighbor

We develop a PDE-based approach to multi-agent deployment where each agent measures its relative position to only one neighbor. First, we show that such systems can be modeled by a first-order hyperbolic partial differential equation (PDE) whose L2-stability implies the stability of the multi-agent system for a large enough number of agents. Then, we show that PDE modelling helps to construct a Lyapunov function for the multi-agent system using spatial discretisation. Then, we use the PDE model to estimate the leader input delay preserving the stability.

Robust state-feedback stabilization of an underactuated network of interconnected [formula omitted] hyperbolic PDE systems

We detail in this article the development of a delay-robust stabilizing state feedback control law for an underactuated network of two subsystems of heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. Only one of the two subsystems is actuated. The proposed approach is based on the backstepping methodology. A backstepping transform allows us to construct a first feedback to tackle in-domain couplings present in the actuated PDE subsystem. Then, we introduce a predictive tracking controller to stabilize the second PDE subsystem. The stabilization of this subsystem implies the stabilization of the whole network. Finally, the proposed control law is combined with a low-pass filter to become robust with respect to small delays in the control signal and uncertainties on the system parameters.

Compensation of spatially-varying state delay for a first-order hyperbolic PIDE using boundary control

This paper develops a spatially-varying delay-compensated control for a first-order hyperbolic partial integro differential equation (PIDE). By introducing a 2-dimensional (2-D) transport partial differential equation (PDE) with a spatially-dependent propagation speed, we transform the delay from the time domain to space domain and which results an equivalent cascade system composed of a PIDE and a transport PDE. The control design is based on the backstepping method with a novel transformation which contains the weighted integration of the shifted state of the transport PDE (actuator state) besides the weighted integration of the PIDE state and the weighted double integration of the actuator state. The kernel equations and the inverse kernel equations are proved to be well-posedness by the successive approximation method, respectively. The finite-time stability of the closed-loop system is established and finally, a simulation example is provided to illustrate the effectiveness of the proposed delay-compensated controller.

Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods

This paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efficient implementations of the discontinuous Galerkin (DG) method for solving a wide range of nonlinear partial differential equations (PDEs). Although the C++ interfaces of Dune-Fem-DG are highly flexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work, easier user interfaces based on Python and the unified form language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and first-order hyperbolic PDEs.

Simulation of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure-preserving semi-implicit scheme

We discuss the applicability of a unified hyperbolic model for continuum fluid and solid mechanics to modeling non-Newtonian flows and in particular to modeling the stress-driven solid-fluid transformations in flows of viscoplastic fluids, also called yield-stress fluids. In contrast to the conventional approaches relying on the non-linear viscosity concept of the Navier-Stokes theory and representation of the solid state as an infinitely rigid non-deformable solid, the solid state in our theory is deformable and the fluid state is considered rather as a “melted” solid via a certain procedure of relaxation of tangential stresses similar to Maxwell’s visco-elasticity theory. The model is formulated as a system of first-order hyperbolic partial differential equations with possibly stiff non-linear relaxation source terms. The computational strategy is based on a staggered semi-implicit scheme which can be applied in particular to low-Mach number flows as usually required for flows of non-Newtonian fluids. The applicability of the model and numerical scheme is demonstrated on a few standard benchmark test cases such as Couette, Hagen-Poiseuille, and lid-driven cavity flows. The numerical solution is compared with analytical or numerical solutions of the Navier-Stokes theory with the Herschel-Bulkley constitutive model for nonlinear viscosity.

Output feedback compensation to state and measurement delays for a first-order hyperbolic PIDE with recycle

This paper develops a novel observer-based output-feedback control law for a class of interconnected first-order hyperbolic partial integral differential equations (PIDEs) with in-domain, non-local coupling terms and measurement delay compensation. A transport PDE is introduced to account for in-domain recycle delay which results in an extended spatial domain with the PIDE and transport PDE being cascaded. Since standard backstepping approach cannot be applied, an affine Volterra integral transform is introduced which allows for the construction of a boundary controller for the PIDE and delay compensation. An observer backstepping transformation containing four kernels is realized, reconstructing the states throughout the delayed measurement. By the use of the backstepping method, a set of coupled kernel equations where one of the kernels has two boundary conditions is established. The existence and invertibility for the control and observer transformations are proved and the finite-time stability for the output system is established while results are illustrated by numerical simulations.

Boundary Control of Nonlinear ODE/Wave PDE Systems With a Spatially Varying Propagation Speed

We consider the boundary control of a nonlinear ordinary differential equation (ODE) actuated through a wave equation whose propagation speed is spatially varying. The ODE state is driven by the uncontrolled boundary of the wave equation. We design a nonlinear backstepping compensator to enable global asymptotic stability of the closed-loop system. We deduce the controller design and the stability proof by introducing a two-step backstepping transformation. The first transformation recasts the original system into a coupled 2×2 first-order hyperbolic system with spatially varying coefficients cascading into a nonlinear ODE. The second transformation is used in the design of a compensator for the resulting cascaded system. Our design offers a global stability result that is guaranteed assuming that the spatially varying propagation speed is continuously differentiable and positive. Moreover, for nonlinear systems, our result is the first contribution enabling actual compensation of actuator delays governed by a coupled first-order hyperbolic partial differential equation (PDE) induced by a wave PDE dynamics with a spatially varying propagation speed. The validity of the proposed controller is illustrated by the benchmark system controlled via a cable.

A third-order accurate in time method for boundary layer flow problems

The boundary layer flow problem arises in numerous industrial applications. As a result, it has received considerable attention over the last five decades which has involved developing numerical procedures to approximate the solution of time-dependent parabolic and first-order hyperbolic partial differential equations (PDEs). In this paper, we develop a method that guarantees third-order temporal accuracy. Stability conditions are derived using Von Neumann stability analysis that guarantee convergence of the proposed algorithm. In addition, we present a consistency analysis. The performance of the proposed algorithm is demonstrated for linear and nonlinear parabolic PDEs that extend the models for the Stokes first and second problems by incorporating the effect of heat transfer using viscous dissipation and thermal radiation. A comparison of performance in terms of convergence rate and estimation accuracy is shown.

Existence of periodic solutions in abstract semilinear equations and applications to biological models

In this paper, we study the existence of mild periodic solutions of abstract semilinear equations in a setting that includes several other types of equations such as delay differential equations, first-order hyperbolic partial differential equations, and reaction-diffusion equations. Under different assumptions on the linear operator and the nonhomogeneous function, sufficient conditions are derived to ensure the existence of mild periodic solutions in the abstract semilinear equations. When the semigroup generated by the linear operator is not compact, Banach fixed point theorem is used whereas when the semigroup generated by the linear operator is compact, Schauder fixed point theorem is employed. In applications, we apply the main results to establish the existence of periodic solutions in delayed red-blood cell models, age-structured models with periodic harvesting, and the diffusive logistic equation with periodic coefficients.