Order Quasilinear Hyperbolic Systems Research Articles

Global exact boundary controllability for first order quasilinear hyperbolic systems

Based on the local exact boundary controllability for 1-D first order quasilinear hyperbolic systems, by an extension method the author gets the global exact boundary controllability for 1-D first order quasilinear hyperbolic systems of diagonal form with applications to Saint-Venant system with friction.

The Goursat problem for the Einstein–Yang–Mills–Higgs system in weighted Sobolev spaces

We establish original Moser estimates to clarify and complete previous works of Christodoulou and Müller zum Hagen concerning local existence and uniqueness results for the Goursat problem associated to second order quasilinear hyperbolic systems. As an application we locally solve, in some weighted Sobolev spaces, the Goursat problem for the Einstein–Yang–Mills–Higgs system using harmonic and Lorentz gauges.

Global existence of classical solutions to the Cauchy problem for a kind of quasilinear hyperbolic systems in divergence form

This paper deals with the global existence of classical solutions to a kind of second order quasilinear hyperbolic systems subject to a null condition, with the linear elastodynamic system as its principal part and the nonlinear terms depending on the product of u 2 and the derivatives of u .

Exact controllability for first order quasilinear hyperbolic systems with vertical characteristics

We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningful internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.

A note on the one-side exact boundary controllability for quasilinear hyperbolic systems

The known theory on the one-side exact boundary observabil- ity for first order quasilinear hyperbolic systems requires that the unknown variables are suitably coupled or satisfy the Group Property in boundary conditions on the non-observation side (see (1)-(2), (11)). In this paper we illustrate, with an inspiring example, that the one-side exact boundary ob- servability can be realized by means of a suitable coupling of the unknown variables in quasilinear hyperbolic system itself instead of in boundary con- ditions. Moreover, an implicit duality between the one-side exact boundary controllability and one-side exact boundary observability is also revealed in this situation.

The Generalized Riemann Problem for First Order Quasilinear Hyperbolic Systems of Conservation Laws II

In this paper, we are concerned with the existence and uniqueness of the local solution to the generalized Riemann problem for first order quasi-linear hyperbolic systems of conservation laws in the presence of the shock wave with large amplitude and the centered wave. Apart from some exceptions, we prove the problem admits a unique piecewise smooth solution u=u(t,x), and this solution has a structure similar to the similarity solution u=u(x/t) of the corresponding Riemann problem in the neighborhood of the origin, provided that the coefficients of the system and the initial conditions are sufficiently smooth. The application of our results in rich system is also given.

A Note on the One-Side Exact Boundary Observability for Quasilinear Hyperbolic Systems

Abstract The known theory on the one-side exact boundary observability for first order quasilinear hyperbolic systems requires that the unknown variables are suitably coupled or satisfy the Group Property in boundary conditions on the non-observation side (see [Tatsien, C. R. Math. Acad. Sci. 342: 937–942, 2006]–[Tatsien, ESAIM Control Optim. Calc. Var.], [Russell, SIAM Rev. 20: 639–739, 1978]). In this paper we illustrate, with an inspiring example, that the one-side exact boundary observability can be realized by means of a suitable coupling of the unknown variables in quasilinear hyperbolic system itself instead of in boundary conditions. Moreover, an implicit duality between the one-side exact boundary controllability and one-side exact boundary observability is also revealed in this situation.

Exact boundary observability for quasilinear hyperbolic systems

By means of a direct and constructive method based on the theory of semi-global C 1 solution, the local exact boundary observability is established for one-dimensional first order quasilinear hyperbolic systems with general nonlinear boundary conditions. An implicit duality between the exact boundary controllability and the exact boundary observability is then shown in the quasilinear case.

Global existence of weakly discontinuous solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems

In this paper, the mixed initial-boundary value problem for general first order quasilinear hyperbolic systems with nonlinear boundary conditions in the domain D = {(t, x) | t ≥ 0, x ≥ 0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.

Conditional symmetries and Riemann invariants for hyperbolic systems of PDEs

This paper contains an analysis of rank-k solutions in terms of Riemann invariants, obtained from interrelations between two concepts, that of the symmetry reduction method and of the generalized method of characteristics for first order quasilinear hyperbolic systems of PDEs in many dimensions. A variant of the conditional symmetry method for obtaining this type of solutions is proposed. A Lie module of vector fields, which are symmetries of an overdetermined system defined by the initial system of equations and certain first order differential constraints, is constructed. It is shown that this overdetermined system admits rank-k solutions expressible in terms of Riemann invariants. Finally, examples of applications of the proposed approach to the fluid dynamics equations in (k + 1) dimensions are discussed in detail. Several new soliton-like solutions (among them kinks, bumps and multiple wave solutions) have been obtained. These solutions remain bounded even when the Riemann invariants admit a gradient catastrophe. Some physical interpretation of these results is discussed.

Global exact boundary controllability for first order quasilinear hyperbolic systems of diagonal form

In this paper, using a constructive method we establish the global exact boundary controllability for a first order quasilinear hyperbolic system of diagonal form. As an application, we obtain the global exact controllability for the one-dimensional isentropic flow system by controlling the velocity or pressure on the boundary.

Exact Controllability for Nonautonomous First Order Quasilinear Hyperbolic Systems*

By means of the theory on the semi-global C 1 solution to the mixed initial-boundary value problem (IBVP) for first order quasilinear hyperbolic systems, we establish the exact controllability for general nonautonomous first order quasilinear hyperbolic systems with general nonlinear boundary conditions.

On the existence in Gevrey classes of local solutions to the Cauchy problem for nonlinear hyperbolic systems with Hölder continuous coefficients

In this paper we shall solve locally in time the solutions to the Cauchy problem for first order quasilinear hyperbolic systems of which coefficients of principal part and of lower order terms are μ- Holder and $\mu'$ - Holder continuous in time variable respectively and in Gevrey class of index s with respect to space variables under the assumption $1\le s <\min\{1 + \frac\mu \nu,1+\frac{1-\mu+\mu'}{\nu}\}, 0<\mu\le1$ , where ν denotes the maximal muliplicity of characteristics of systems. Keywords: Nonlinear hyperbolic systems, Cauchy problem, Gevrey classes

Observabilité exacte frontière pour des systèmes hyperboliques quasi-linéaires

By means of a direct and constructive method based on the theory of semi-global C 1 solution, the local exact boundary observability is established for one-dimensional first order quasilinear hyperbolic systems with general nonlinear boundary conditions. To cite this article: T. Li, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Numerical methods for nonconservative hyperbolic systems: a theoretical framework.

The goal of this paper is to provide a theoretical framework allowing one to extend some general concepts related to the numerical approximation of 1-d conservation laws to the more general case of first order quasi-linear hyperbolic systems. In particular this framework is intended to be useful for the design and analysis of well-balanced numerical schemes for solving balance laws or coupled systems of conservation laws. First, the concept of path-conservative numerical schemes is introduced, which is a generalization of the concept of conservative schemes for systems of conservation laws. Then, we introduce the general definition of approximate Riemann solvers and give the general expression of some well-known families of schemes based on these solvers: Godunov, Roe, and relaxation methods. Finally, the general form of a high order scheme based on a first order path-conservative scheme and a reconstruction operator is presented.

Exact boundary controllability for higher order quasilinear hyperbolic equations

By means of the existence and uniqueness of semi-global C 1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues, the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.

Exact Controllability for First Order Quasilinear Hyperbolic Systems with Zero Eigenvalues

For a class of mixed initial-boundary value problem for general quasilinear hyperbolic systems with zero eigenvalues, the authors establish the local exact controllability with boundary controls acting on one end or on two ends and internal controls acting on a part of equations in the system.

Contrôlabilité exacte frontière pour les équations des ondes quasi linéaires unidimensionnelles

By means of the existence and uniqueness of semi-global C 1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues, we present a unified method to establish the exact boundary controllability for 1-D quasilinear wave equations with boundary conditions of different types. To cite this article: T.T. Li, L.X. Yu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Exact Boundary Controllability for Quasi-Linear Hyperbolic Systems

Using a result on the existence and uniqueness of the semiglobal C1 solution to the mixed initial-boundary value problem for first order quasi-linear hyperbolic systems with general nonlinear boundary conditions, we establish the exact boundary controllability for quasi-linear hyperbolic systems if the C1 norm of initial and final states is small enough.

FORMATION OF SINGULARITIES FOR A KIND OF QUASILINEAR NON-STRICTLY HYPERBOLIC SYSTEM

The author gets a blow-up result of C1 solution to the Cauchy problem for a first order quasilinear non-strictly hyperbolic system in one space dimension.