Exact Boundary Controllability And Observability For Quasilinear Hyperbolic Systems On A Tree-Like Network

In this Ph.D thesis, based on the theory of the semi-global C~1 solution to the mixed initial-boundary value problem for first order quasilinear hyperbolic systems, by means of a constructive method, we deal with the exact boundary controllability and the exact boundary observability for quasilinear hyperbolic systems on a tree-like network with general topology.Two physical models are considered in this thesis, one is the unsteady flows in a tree-like network of open canals (Saint-Venant systems), and another one is a tree-like network of vibrating strings (quasilinear wave equations). For the first one, we get the exact boundary controllability and the exact boundary observability in subcritical and supercritical situations, respectively, and we show some duality properties between controllability and observability. For the second one, we get similar results for quasilinear wave equations with various boundary conditions, in particular, the previous results on the exact boundary controllability for linear wave equations with Dirichlet boundary conditions on a tree-like network are extended to the quasilinear case with various boundary conditions.The arrangement of the thesis is as follows:First of all, in Chapter 1, a brief introduction is given for the background and present situation on the study and for the results in this thesis.In Chapter 2 and Chapter 3, the exact boundary controllability of unsteady flows in a tree-like network of open canals is obtained for the subcritical and supercritical situations, respectively.In Chapter 4, by transforming a second order quasilinear wave equation to a first order quasilinear system, the exact boundary controllability on a tree-like network of vibrating strings is realized.In Chapter 5 and Chapter 6, the exact boundary observability of unsteady flows in a tree-like network of open canals is given for the subcritical and supercritical situations, respectively. Then, some duality properties between controllability and observability are obtained by comparing with the results mentioned in Chapter 2 and Chapter 3.Corresponding to Chapter 4, in Chapter 7, the exact boundary observability on a tree-like network of vibrating strings is constructed.At last, Chapter 8, deals with some discussions about the interface conditions. We show that it is possible to further reduce the number of controls under certain additional hypotheses, and it gives the possibility to improve the result on the controllability for some physical models.