Optimal control problems constrained by hyperbolic conservation laws.

Abstract

This thesis deals with the solutions of optimal control problems constrained by hyperbolic conservation laws. Such problems pose significant challenges for mathematical analysis and numerical simulations. Those challenges are mainly because of the discontinuities that occur in the solutions of non-linear systems of conservation laws and become more acute when dealing with the multidimensional case. The problem is formulated as the minimisation of a flow matching cost functional constrained by multi-dimensional hyperbolic conservation laws. The control variable is the initial condition of the partial differential equations. In our analysis of the problem, we review extensively the constraints equation and we consider successively the one-dimensional and the multi-dimensional cases. In all the cases, we derive the optimality conditions in the adjoint approach at the continuous level, which are then discretised to arrive at a numerical algorithm for the solution. In the derivation of the optimality conditions, we replace the non-linear conservation laws either by the relaxation equation or the Lattice Boltzmann equation. We illustrate our findings on examples related to the multi-dimensional Burger and the Euler equations.