Class Of Quasilinear Hyperbolic Systems Research Articles

Boundary Feedback Control of 2×2 Quasilinear Hyperbolic Systems: Predictive Synthesis and Robustness Analysis

We present a predictive feedback control method for a class of quasilinear hyperbolic systems with one boundary control input. Assuming exact model knowledge, convergence to the origin, or tracking at the uncontrolled boundary, are achieved in finite time. A robustness certificate is provided, showing that at least under more restrictive assumptions on the system coefficients, the control method has inherent robustness properties with respect to small errors in the model, measurements and control input. Rigorous, although conservative conditions on the time derivative of the initial condition and on the design parameter controlling the convergence speed are given to ensure global existence of the solution for initial conditions with arbitrary infinity-norm.

High-Gain Observer Design for a Class of Quasi-Linear Integro-Differential Hyperbolic Systems—Application to an Epidemic Model

This article addresses the problem of high-gain observer design for a class of quasi-linear hyperbolic systems (with one characteristic velocity), possibly including nonlocal terms, making them systems of partial integro-differential equations. The design relies on distributed measurement of a part of the state vector. The observer is presented and discussed and the exponential stability in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C^1$</tex-math></inline-formula> spatial norm of the origin for the error system is fully established via Lyapunov-based analysis. Its use is illustrated via an application to an age-dependent susceptible, infected, recovered (SIR) epidemic model.

Diagnostics of a gradient catastrophe for a class of quasilinear hyperbolic systems

Abstract A two-stage analysis do detect the appearance of a gradient catastrophe of the solution is proposed for quasilinear systems of hyperbolic equations of special form. Applications of the first stage are considered for the following simple cases: scalar Burgers’ equation and a quasi-orthogonal system generalizing it. The entire two-stage analysis is applied to systems of equations describing one-dimensional electron oscillations in plasma, namely, plane oscillations in the relativistic and non-relativistic cases and also axially symmetric non-relativistic cylindrical oscillations.

Cauchy problem for a class of quasilinear hyperbolic systems is well posed on a family of Besov spaces

In this paper we consider the two dimensional Cauchy problem for the quasilinear systems{∂tu+a(u)∂xu=0∂tu+au(0,x)=u0(x), with u=(u1,…,uN) and a(u)=(ajk(u))j,k=1N real N×N matrix, with entries C∞, such that the eigenvalues of a(0) are real and distinct, that is, the system is hyperbolic at u=0. We show that a family of Besov spaces, containing the Hölder spaces, near u=0 is continuously preserved by the flow of the above Cauchy Problem.

Global Classical Solutions to the Mixed Initial-boundary Value Problem for a Class of Quasilinear Hyperbolic Systems of Balance Laws

Global Classical Solutions to the Mixed Initial-boundary Value Problem for a Class of Quasilinear Hyperbolic Systems of Balance Laws

The global existence and the life span of smooth solutions to a class of complex conservation laws

In this paper, we consider the Cauchy problem for a class of complex conservation laws, which is introduced by P.D. Lax and essentially a class of quasilinear hyperbolic systems in two space dimensions. For a kind of flux functions, we prove the global existence of smooth solutions to the Cauchy problem with small initial data. On the other hand, for different kinds of flux functions, we give some estimates on the lower bound of the life span of smooth solutions.

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws II

This work is a continuation of our previous work. In the present paper, we study the existence and uniqueness of global piecewise C1 solutions with shock waves to the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping in the presence of a boundary. It is shown that the generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws with linear damping with nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0} admits a unique global piecewise C1 solution u = u (t, x) containing only shock waves with small amplitude and this solution possesses a global structure similar to that of a self-similar solution u = U (x /t) of the corresponding homogeneous Riemann problem, if each characteristic field with positive velocity is genuinely nonlinear and the corresponding homogeneous Riemann problem has only shock waves but no rarefaction waves and contact discontinuities. This result is also applied to shock reflection for the flow equations of a model class of fluids with viscosity induced by fading memory. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The Cauchy problem for quasilinear SG-hyperbolic systems

We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) ∈ [0, T ] × ℝn and presenting a linear growth for |x | ∞. We prove well-posedness in the Schwartz space (ℝn). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws

It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C 1 solution u = u ( t , x ) containing only n shock waves with small amplitude on t ⩾ 0 and this solution possesses a global structure similar to that of the similarity solution u = U ( x / t ) of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data.

Global Exact Boundary Controllability of a Class of Quasilinear Hyperbolic Systems of Conservation Laws II

In this paper, by a new constructive method, the authors reprove the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate characteristics. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C1 functions. In particular, the authors give the optimal control time of the system. Finally, a new example belonging to this kind of system is also provided.

Cauchy Problem for Quasilinear Hyperbolic Systems with Higher Order Dissipative Terms

In this paper, the author studies the global existence, singularities and life span of smooth solutions of the Cauchy problem for a class of quasilinear hyperbolic systems with higher order dissipative terms and gives their applications to nonlinear wave equations with higher order dissipative terms.

Global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws

This paper deals with the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate characteristics. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C 1 functions. Some physical and mechanical applications are given.

Global classical solutions for a class of quasilinear hyperbolic systems

In this paper we investigate the mechanism of singularity formation to the Cauchy problem of quasilinear hyperbolic system.Moreover, we present some examples to show some significant problems.

Global Exact Controllability of a Class of Quasilinear Hyperbolic Systems

In this paper we consider a distributed parameter boundary control system described by a reducible quasilinear hyperbolic system, [formula]posed on the domain 0≤x≤1, with the nonlinear boundary conditions[formula]In the case that the system islinear degenerate, we show that it isgloballyexactly boundary controllable.

Exceptional Quasilinear Hyperbolic Systems of Conservation Laws

In this note we take into consideration of Temple's systems, a class of quasilinear hyperbolic systems of conservation laws which enjoy a remarkable property that the shock and rarefaction curves coincide. We give a criterion of how to verify a system to be of Temple's type, state and construct some new examples of such systems and show that this kind of exceptional systems has infinite convex entropies.

Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the C0-norm of the initial data is sufficiently small, while the C1-norm of the initial data can be large. The analysis is based on a priori estimates, which are obtained by a generalised Lax transformation.

Exact solutions to a class of quasilinear hyperbolic systems

A class of quasilinear hyperbolic systems satisfied by a lot of models which are of physical interest are considered. By means of introduction of the characteristic parameters, exact solutions to this kind of systems are obtained. Within the present theoretical framework the governing systems of plasma physics; magnetohydrodynamics, elasticity and perfect relativistic fluids are investigated.

A Semigroup Approach to W1,∞-Solutions to a Class of Quasilinear Hyperbolic Systems

In this paper we present a local existence and uniqueness theorem for a W 1,∞-solution to a quasilinear first order system based on semigroup theory. We apply abstract theorems on nonautonomous linear evolution equations to the linearized system and solve the quasilinear system by a fixed point argument.

A class of completely integrable hyperbolic systems

The author discusses the characterisation of a class of quasilinear hyperbolic systems which enjoy the property of being separable in two independent equations. A sort of duality relating these with the Lax 'linearly degenerate' systems (1957) is also pointed out.

Reduction to linear canonical forms and generation of conservation laws for a class of quasilinear hyperbolic systems

We consider a quasilinear hyperbolic homogeneous system of two first order equations involving two dependent and two independent variables. For the associated hodograph equations we investigate the reducibility to canonical forms allowing for an explicit integration. Such a kind of requirement, in problems of physical interest, provides a suitable method for characterizing possible material model laws. The theoretical approach shown herein can be relevant for studying the existence of conservation laws to non-homogeneous first order systems and also for describing the evolution of weak shock waves.