Class Of Symmetric Hyperbolic Systems Research Articles

On the characteristic initial value problem for nonlinear symmetric hyperbolic systems, including Einstein equations

We consider a characteristic initial value problem for a class of symmetric hyperbolic systems with initial data given on two smooth null intersecting characteristic surfaces. We prove existence of solutions on a future neighborhood of the initial surfaces. The general result is applied to general semilinear wave equations, as well as the Einstein equations with or without sources, and conformal variations thereof.

THE MIXED PROBLEM FOR A CLASS OF NONLINEAR SYMMETRIC HYPERBOLIC SYSTEMS WITH DISCONTINUOUS DATA

This paper studies the nonlinear mixed problem for a class of symmetric hyperbolic systems with the boundary condition satisfying the dissipative condition about discontinuous data in higher dimension spaces, establishes the local existence theorem by using the method of a prior estimates, and obtains the structure of singularities of the solutions of such problems.

Boundary Control of a Class of Hyperbolic Systems

The paper studies a class of symmetric hyperbolic systems with boundary control. We prove the exact controllability of these systems. The exact controllability result is exploited to investigate the associated optimal control problem. For some special cases, we construct explicitly the optimal control law under the feedback form. Our construction illustrates the difficulty that we would meet for the boundary optimal control of hyperbolic systems. This difficulty does not appear if the control operator is bounded as has been observed in the existing literature.

Proportional and Integral Regulation of Irrigation canal Systems governed by the St Venant Equation

The purpose of our paper is to study control and regulation of irrigation canal systems using the St Venant equation. As the linearized models of these systems around equilibrium points are described by symmetric hyperbolic partial differential equations of two independent variables, we consider a class of symmetric hyperbolic systems with boundary observations. We present the exact observability and exponential stability results of Rauch and Taylor in the general case. These results are made explicit by providing estimates of necessary observation length for observability and estimates of exponential decay rate for stability in function of structure parameters. Then, wc apply these results to show that for an one-reach irrigation canal system governed by the St Venant equation, the linearized model around an equilibrium point is exponentially stable for the open loop system. To guarantee a robust regulation arid a robust stability of the dosed loop system wc propose proportional and integral output feedback controllers (PI-Controllers) which assure exponential stabilization of the linearized model and suppress (known or unknown) constant disturbances in the system. The work that we present here brings out a systematic synthesis method for designing boundary stabilizing PI-Controllers for irrigation systems based on linearized models of the St Venant equation

Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws

We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave.

Wave fronts for solutions of boundary-value problems for a class of symmetric hyperbolic systems

ly concave domains these results do not enable us to obtain precise theorems; the reason for this is that the symbol q) should not change at the moment of reflection of the bicharaeteristic from the boundary. In these domains results have been obtained by completely different methods [8-10]. Finally, in domains of arbitrary type results have been obtained by the energy method but with very essential appeal to results obtained for bicharacteristic ally concave domains [Ii, 12]. Analogous results were obtained independently and practically simultaneously by the author and were announced in [13]; however, the author used an improved energy method which made it possible to remove the condition that ~ be constant on reflection by replacing it by a condition that ~ not decrease. This, in turn, made it possible to obtain the results directly without appealing to results for bicharacteristic aIiy concave domains. It is true that here it is necessary to consider only boundary-value problems satisfying the Shapiro-Lopatins

Vector potentials for symmetric hyperbolic systems

0. Introduction. Potentials are widely used in the physics literature in the study of a great variety of wave-propagation problems, but it is not altogether clear how (or if) a potential for a particular problem might be discovered. In a recent paper [4] J. Schulenberger showed that the Lagrange identities and for Maxwell's equations and the equations of two-dimensional elasticity derive in a natural way from the spectral resolutions of the corresponding operators. More recently, J. Schulenberger and the author have shown [3] that there is a class of symmetric hyperbolic systems, describing most wave-propagation phenomena of classical physics, which is characterized by admitting of a special form, called there potentials of classical type. These are very useful in studying symmetries, degeneracies, and conserved quadratic forms, but they suffer the disadvantage of being nonlocal. It is the purpose of this paper to show that this same class of symmetric hyperbolic systems is also characterized by another type of potential decomposition which is very often in applications local and which even more strongly emphasizes the special nature of this class of equations. It is, furthermore, easier to compute these local than those of [3](cf. §3).

A class of symmetric hyperbolic systems with special properties

A class of symmetric hyperbolic systems with special properties

The solution of positive-symmetric hyperbolic systems by difference methods

where u and f are m-vector-valued functions, Ai and B are m Xrm matrix-valued functions, and x = (X1, X2, . , x.n) varies over all points of the real Euclidean n-space En which have non-negative coordinates. The system (1.1) is called positive-symmetric hyperbolic if the matrices Ai are symmetric and positive-definite. Such systems are included in the general class of symmetric hyperbolic systems introduced by Friedrichs [1]. Symmetric systems are of particular interest because they possess an energy inequality; that is, the square integral of the solution over a space-like hyperplane can be estimated in terms of square integrals of initial values of the solution and the nonhomogeneous term in the differential equation. We shall not consider the Cauchy problem for (1.1), but rather, we prescribe the initial values on the coordinate hyperplanes xi= 0, i.e.,