Optimal control problems constrained by hyperbolic conservation laws.
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Date
2021
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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.
Description
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.