Elementary Interactions in Quasi-linear Hyperbolic Systems
نویسندگان
چکیده
where t>0, o o < x < o o , U(t, x)=(u(t, x), v(t, x)), and the function F(U)= (f(u, v), g(u, v)) is smooth. GLIMM [2] has proved existence of a weak solution provided that the variation of the initial data Uo(x) is small. GLIMM & LAX [3] then improved this result by requiring that the oscillation of Uo(x) be small. SMOLLER [7] has proved the existence of a weak solution provided that Uo(x ) is constant except for a single arbitrary jump (the Riemann problem). Finally, NISmDA [6] has proved existence for general data Uo but for the special case f= l / v , g=u. Here we will prove existence under the assumption that Uo(x) is constant except for two jumps; that is, Uo (x) is of the form
منابع مشابه
Boundary Control Design for Cascades of Hyperbolic 2× 2 PDE systems via Graph Theory
This article is concerned with the design of boundary control laws for stabilizing systems of 2 × 2 first order quasi-linear hyperbolic PDEs. A new graph representation of such systems represents the interactions between the characteristic curves and the boundary control laws, the invariant graph, is introduced. The structure of the invariant graph is used to design stabilizing control laws and...
متن کاملLifespan of Classical Solutions to Quasi-linear Hyperbolic Systems with Small BV Normal Initial Data
In this paper, we first give a lower bound of the lifespan and some estimates of classical solutions to the Cauchy problem for general quasi-linear hyperbolic systems, whose characteristic fields are not weakly linearly degenerate and the inhomogeneous terms satisfy Kong’s matching condition. After that, we investigate the lifespan of the classical solution to the Cauchy problem and give a shar...
متن کاملStability analysis and feedback control of T-S fuzzy hyperbolic delay model for a class of nonlinear systems with time-varying delay
In this paper, a new T-S fuzzy hyperbolic delay model for a class of nonlinear systems with time-varying delay, is presented to address the problems of stability analysis and feedback control. Fuzzy controller is designed based on the parallel distributed compensation (PDC), and with a new Lyapunov function, delay dependent asymptotic stability conditions of the closed-loop system are derived v...
متن کاملStability of Quasi-linear Hyperbolic Dissipative Systems
1. Introduction In this work we want to explore the relationship between certain eigenvalue condition for the symbols of first order partial differential operators describing evolution processes and the linear and nonlinear stability of their stationary solutions. Consider the initial value problem for the following general first order quasi-linear system of equations
متن کاملBoundary observers for linear and quasi-linear hyperbolic systems with application to flow control
In this paper we consider the problem of boundary observer design for one-dimensional first order linear and quasi-linear strict hyperbolic systems with n rightward convecting transport PDEs. By means of Lyapunov based techniques, we derive some sufficient conditions for exponential boundary observer design using only the information from the boundary control and the boundary conditions. We con...
متن کامل