Numerical analysis for Hamilton-Jacobi equations on networks and indirect controllability/stability of a system of 1D wave equations.

Summary This thesis is composed of two parts in which we study on the one hand error estimates for numerical schemes associated with first order Hamilton-Jacobi equations. On the other hand, we study the stability and the exact indirect boundary controllability of coupled wave equations. First, using the Crandall-Lions technique, we establish an error estimate of a monotone finite difference numerical scheme for flux-limited boundary conditions, for a first order Hamilton-Jacobi equation. Then, we show that this numerical scheme can be generalized to general junction conditions. We then establish the convergence of the discretized solution to the viscosity solution of the continuous problem. Finally, we propose a new approach, à la Crandall-Lions, to improve the error estimates already obtained, for a class of well chosen Hamiltonians. This approach relies on the interpretation of the optimal control type of the considered Hamilton-Jacobi equation.In a second step, we study the stabilization and the exact indirect boundary controllability of a one-dimensional system of coupled wave equations. First, we consider the case of a coupling with velocity terms, and by a spectral method, we show that the system is exactly controllable with a single boundary control. The results depend on the arithmetic nature of the quotient of the propagation velocities and on the algebraic nature of the coupling term. Moreover, they are optimal. Next, we consider the case of zero-order coupling and establish an optimal polynomial rate of the energy decay. Finally, we show that the system is exactly controllable with a single boundary control.