ISBN 978-3-8439-4943-9

978-3-8439-4943-9, Reihe Mathematik

Paloma Schäfer Aguilar
Numerical approximation of optimal control problems for hyperbolic conservation laws

271 Seiten, Dissertation Technische Universität Darmstadt (2021), Softcover, B5

Zusammenfassung / Abstract

Hyperbolic conservation laws are often used to model physical processes such as gas transport in networks or traffic flow. The crucial issue of nonlinear hyperbolic conservation laws is that even for smooth input data solutions develop moving discontinuities after finite time, so-called shocks. For this reason, the numerical approximation of hyperbolic conservation laws is quite involved, since several challenges in the analytical study as well as in the numerical analysis of the solutions appear.

The first part of this thesis provides a rigorous sensitivity and adjoint calculus for conservation laws with source terms. Due to the shock formations, the control-to-state mapping is at best differentiable in the weak topology of measures. Pfaff and S. Ulbrich developed an adjoint-based gradient formulation for tracking type functionals by using a suitable adjoint state, which is a solution to the associated adjoint equation. In the case of shocks, the correct definition of appropriate solutions to the adjoint and the associated sensitivity equation requires care. We analyze solutions to these equations and introduce convenient characterizations, which are suitable to show that the limit of discrete adjoints and discrete sensitivities are in fact the desired solution of the adjoint and sensitivity equation, respectively.

In the second part of this thesis, we derive consistent numerical discretization schemes for optimal boundary control problems of hyperbolic conservation laws with source terms. We use monotone difference schemes for the state equation and derive the associated sensitivity and adjoint scheme. We derive convergence results for these classes of schemes, wherefore we use suitable numerical flux functions to guarantee that the limit functions of the adjoint and sensitivity schemes attain the corresponding boundary data in the correct sense.