Linear Hyperbolic Systems Research Articles

On the decay of one-dimensional locally and partially dissipated hyperbolic systems

We study the time-asymptotic behavior of linear hyperbolic systems subject to partial dissipation that is localized in suitable subsets of the domain. Specifically, we recover the classical decay rates of partially dissipative systems that satisfy the stability condition (SK), with a time-delay that depends only on the velocity of each component and the size of the undamped region. To quantify this delay, we assume that the undamped region is a bounded space interval and that the system, without space-restriction on the dissipation, satisfies the stability condition (SK). The former assumption ensures that the time spent by the characteristics of the system in the undamped region is finite, and the latter ensures that the solutions decay whenever the damping is active. Our approach consists of reformulating the system into $n$ coupled transport equations and showing that the time-decay estimates are delayed by the sum of the times that each characteristic spends in the undamped region.

Composite Learning Based Adaptive Control of Linear 2×2 Hyperbolic PDE Systems.

This article considers the adaptive stability control of a class of 2×2 linear hyperbolic PDE systems. The PDE model is subject to constant but in-domain and boundary unknown parameters. A novel adaptive controller is developed by leveraging the swapping design technique and composite parameter learning law. With swapping design, several linear and static combinations, including carefully designed filters, unknown parameters, and error terms, are constructed to express the system states. From the static combinations, a composite learning based forgetting-factor least squares law is introduced to guarantee exponential parameter convergence without the persistent excitation (PE). Although inaccurate parameter estimation in the adaptive backstepping control results in asymptotic stability of the system, accurate parameter estimation ensures the exponential convergence of closed-loop system and concomitantly improves the transient performance. Finally, a comparative numerical simulation is performed to validate the effectiveness and advantage of the developed adaptive control scheme.

Symmetric form of the magnetohydrodynamics equations

It is well known that many linear hyperbolic systems of mathematical physics are symmetric hyperbolic systems. In the case of nonlinear equations, the question of whether they can be written in symmetric form is far from simple.

Pulse and Point Optimal Control of Linear Hyperbolic Integro-Differential Systems with Mixed Boundary Conditions

Pulse and Point Optimal Control of Linear Hyperbolic Integro-Differential Systems with Mixed Boundary Conditions

Regional Controllability of the Position and Fractional Speed of Linear Hyperbolic Systems

Regional Controllability of the Position and Fractional Speed of Linear Hyperbolic Systems

Observers for hyperbolic systems with multiple delays in the nonlocal boundary condition and its application to secure image encryption

In this article, we propose a step-by-step observer design for a first-order linear hyperbolic system with multiple delays in the nonlocal boundary condition. In the proposed approach inspired by the backstepping methodology, the observer gains are obtained following seven successive steps. Next, we present the application of this observer design in the context of image encryption. We consider a system where a local nonlinear term causing a chaotic vibration is added to the nonlocal boundary condition with time lags. A synchronization system for this chaotic system can be constructed using the same observer design method. The chaotic states generated by both systems can be used to encrypt and decrypt image data. The time lags, and weight functions included in the nonlocal boundary condition play the role of common encryption keys. By considering a system that includes arbitrarily many delays, this approach allows for a significant increase in the number of common encryption keys compared to conventional methods, resulting in greatly improved communication security.

Quantum algorithms for nonlinear partial differential equations

Computations of nonlinear partial differential equations (PDEs) are some of the most important problems in science and engineering, but come with tremendous costs. Quantum algorithms have the potential to speed up these computations, but current methods are bound to weak or polynomial nonlinearity, short time of validity, or display no quantum advantage otherwise. We construct new quantum algorithms–based on level set methods – for Hamilton-Jacobi and scalar hyperbolic PDEs that generate algorithms with advantages with respect to critical numerical parameters, even for computing the physical observables. These algorithms hold for arbitrary nonlinearity for the equations under study and are valid globally in time. In addition, we also introduce a new quantum encoding, called level set encoding, that allows the efficient computation of physical observables. These PDEs are important for many applications like geometric optics, semi-classical limit of the Schrödinger equations, and high-frequency limit of symmetric linear hyperbolic systems (for example the elastic wave and Maxwell equations). It can display up to exponential advantage –depending on the initial conditions and domain of the relevant observables– in both the dimension of the PDE and the error in computing its observables with respect to classical finite difference methods for these PDEs. Compared to classical finite difference methods, our algorithm also has an advantage with respect to the number of initial data, which has potential important applications in Monte Carlo simulations and uncertainty quantification. For more general PDEs, while an advantage with respect to a very large number of initial data is possible, this is mostly unrealistic for problems of practical interest, and new methods are required.

Classical Elastodynamics as a Linear Symmetric Hyperbolic System in Terms of $({\mathbf{u}}_{\mathbf{x}}, {\mathbf{u}}_{t})$

Classical Elastodynamics as a Linear Symmetric Hyperbolic System in Terms of $({\mathbf{u}}_{\mathbf{x}}, {\mathbf{u}}_{t})$

Uniform accuracy of implicit-explicit Runge-Kutta (IMEX-RK) schemes for hyperbolic systems with relaxation

Implicit-explicit Runge-Kutta (IMEX-RK) schemes are popular methods to treat multiscale equations that contain a stiff part and a non-stiff part, where the stiff part is characterized by a small parameter ε \varepsilon . In this work, we prove rigorously the uniform stability and uniform accuracy of a class of IMEX-RK schemes for a linear hyperbolic system with stiff relaxation. The result we obtain is optimal in the sense that it holds regardless of the value of ε \varepsilon and the order of accuracy is the same as the design order of the original scheme, i.e., there is no order reduction.

A Conservative Eulerian–Lagrangian Runge–Kutta Discontinuous Galerkin Method for Linear Hyperbolic System with Large Time Stepping

A Conservative Eulerian–Lagrangian Runge–Kutta Discontinuous Galerkin Method for Linear Hyperbolic System with Large Time Stepping

The dichotomy property in stabilizability of 2 × 2 linear hyperbolic systems

This paper is devoted to discuss the stabilizability of a class of 2 × 2 non-homogeneous hyperbolic systems. Motivated by the example in the Section 5.6 of Bastin and Coron’s book in 2016, we analyze the influence of the interval length L on stabilizability of the system. By spectral analysis, we prove that either the system is stabilizable for all L > 0 or it possesses the dichotomy property: there exists a critical length Lc > 0 such that the system is stabilizable for L ∈ (0, Lc) but unstabilizable for L ∈ [Lc,+∞). In addition, for L ∈ [Lc, +∞), we obtain that the system can reach equilibrium state in finite time by backstepping control combined with observer. Finally, we also provide some numerical simulations to confirm our developed analytical criteria.

On the bounded smooth solutions to exponentially stable linear nonautonomous hyperbolic systems

On the bounded smooth solutions to exponentially stable linear nonautonomous hyperbolic systems

Boundary feedback control for hyperbolic systems

We are interested in the feedback stabilization of general linear multi-dimensional first order hyperbolic systems in ℝd. Using a Lyapunov function with a suited weight function depending on the system under consideration we show stabilization in L2 for the studied system using a suitable feedback control. Therefore the controllability of the studied system is related to the feasibility of an associated linear matrix inequality.We show the applicability discussing the barotropic Euler equations.

Exact internal controllability and exact internal synchronization for a kind of first order hyperbolic system

In this paper we investigate the exact controllability and exact synchronization in L2 space for a 1-D first order linear hyperbolic system with internal controls located on some part of the domain. Based on the exact boundary controllability theory, we first use the constructive method to establish the exact controllability by internal controls for a time reversible system with inhomogeneous boundary conditions. The method is then adapted to prove the exact internal null controllability for a general system with homogeneous boundary conditions. These results can be then used to establish the exact internal synchronization for the system, and related subjects are further studied.

On the solvability and the dimension of the solution set of abstract linear boundary value problems and applications to ODE'S

A solvability theory for linear abstract boundary value problems, that is of linear operator equations subject to an additional (“boundary”) condition given by a linear operator, is developed using a new approach based on normal solvability of the induced map or a pseudoinverse of the boundary map when restricted to the null space of the homogeneous problem. Uniqueness and dimension of their solution sets is also established. Applications to BVP's for finite-dimensional as well as infinite-dimensional systems of ordinary differential equations defined on the half line or the whole line are given. Systems having exponential dichotomy, or satisfying Riccati type inequalities are studied, as well as systems with asymptotically hyperbolic linear part and systems between pairs of admissible Banach spaces.

Lyapunov Functions for Linear Hyperbolic Systems

We develop new methods to construct a Lyapunov function for 1-D linear hyperbolic equations with variable coefficients. The main focus is on the nonstrictly hyperbolic case for which we give an example demonstrating that existing approaches cannot provide sufficient conditions for the asymptotic stability, but our approach does. Sufficient conditions for exponential <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}$</tex-math></inline-formula> -stability for a connected <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2 \times 2$</tex-math></inline-formula> system of linear 1-D hyperbolic systems are obtained. By means of examples we compare the capabilities of our approach with the existing ones.

Adaptive Fuzzy Boundary Observer Design for Uncertain Linear Coupled Hyperbolic Partial Differential Equation Systems

Joint uncertainties and state estimation of a class of linear coupled hyperbolic partial differential equation systems in the presence of unstructured and structured uncertainties are studied in this paper. For unstructured uncertainties which are completely unknown, by employing Takagi-Sugeno fuzzy logic system to approximate the unstructured uncertainties, a novel adaptive fuzzy boundary observer is developed to estimate both unknown system states as well as unknown weights in the fuzzy logic system, and the estimation errors are ultimately bounded. Therein, in the design of the proposed observer, a set of swapping filters and infinite dimensional backstepping technique are combined. On the other hand, for structured uncertainties that can be described in a concrete parameterized form, the proposed method can easily achieve the exact estimation of weights and states to their true values. The rigorous proof is provided to show that the ultimately bounded estimation errors for the case of unstructured uncertainties and the exponential convergent estimation errors for the case of structured uncertainties can be realized. Finally, three illustrative simulations are carried out to show the feasibility and effectiveness of the developed methods in this paper.

Boundary and Contact Conditions of Higher Order of Accuracy for Grid-Characteristic Schemes in Acoustic Problems

Seismic wave propagation through geological media is described by linear hyperbolic systems of equations. They correspond to acoustic, isotropic, and anisotropic linear elastic porous fluid-saturated models. They can be solved numerically by applying grid-characteristic schemes, which take into account propagation of solution discontinuities along characteristics. An important property of schemes used in practice is their high order of accuracy, due to which signal wavefronts can be clearly resolved. Previously, much attention was given to this property at interior points of the computational domain. In this paper, we study the order of a scheme up to the boundary of the domain inclusive. An approach is proposed whereby arbitrary linear boundary and contact conditions can be set up to high accuracy. The presentation is given for the system of one-dimensional acoustic equations with constant coefficients.

Stability properties of multidimensional symmetric hyperbolic systems with damping, differential constraints and delay

Multidimensional linear hyperbolic systems with constraints and delay are considered. The existence and uniqueness of solutions for rough data are established using Friedrichs method. With additional regularity and compatibility on the initial data and initial history, the stability of such systems are discussed. Under suitable assumptions on the coefficient matrices, we establish standard or regularity-loss type decay estimates. For data that are integrable, better decay rates are provided. The results are applied to the wave, Timoshenko, and linearized Euler–Maxwell systems with delay.

Hybrid UWB-Inertial TDoA-Based Target Tracking With Concentrated Anchors

In this paper, hybrid radio/inertial mobile target tracking for accurate and smooth path estimation is considered. The proposed tracking approach builds upon an Ultra WideBand (UWB)-based positioning algorithm, based on the Linear Hyperbolic Positioning System (LinHPS), with Time Difference of Arrival (TDoA) processing and anchors concentrated on a single hotspot at the center of the environment where the target moves. First, we design an Adaptive Radio-based Extended Kalman Filter (AREKF), which does not require a priori statistical knowledge of the noise in the target movement model and estimates the measurement noise covariance, at each sampling time, according to a proper LookUp Table (LUT). In order to improve the performance of AREKF, we incorporate inertial data collected from the target and propose three “hybrid” radio/inertial algorithms, denoted as Hybrid Inertial Measurement Unit (IMU)-aided Radio-based EKF (HIREKF), Hybrid Noisy Control EKF (HNCEKF), and Hybrid Control EKF (HCEKF). Our results on experimentally acquired paths show that the proposed algorithms achieve an average instantaneous position estimation error on the order of a few centimeters. Moreover, the minimum target path length estimation error, obtained with HCEKF, is on the order of 6% and 1% for two paths with lengths equal to approximately 17 m and 46 m, respectively.