Stabilization and uncertainty quantification for systems of hyperbolic balance laws

Gerster, Stephan; Herty, Michael (Thesis advisor); Frank, Martin (Thesis advisor); Göttlich, Simone (Thesis advisor)

Aachen (2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

We consider wellposedness and stabilization of one-dimensional systems of hyperbolic partial differential equations on networks. The p-system is of special interest. It includes shallow water equations, which model water flow in open channels, and it includes isothermal Euler equations, which model gas dynamics in pipelines. Since these supply systems are usually operated in a state of equilibrium, we present boundary controls that guarantee the stabilization of perturbations of steady states. Here, we assume linearized models and we use Lyapunov functions that yield estimates on the decay rates for deviations from the states of equilibrium. Sufficient conditions are presented that guarantee explicit decay rates for exponentially decaying Lyapunov functions. Furthermore, we present an upwind and downwind discretization that inherits continuous stability results. Larger perturbations are modelled as stochastic processes, whose evolutions are described by nonlinear hyperbolic systems. The dependency of the solution on the stochastic input is stated a priori by a series expansion. A stochastic Galerkin framework is used to obtain deterministic equations for the coefficients in truncated series expansions. We present conditions that guarantee the solvability of the equations for the coefficients. In particular, hyperbolicity properties are proven for the stochastic Galerkin formulations. Since solutions to hyperbolic equations admit discontinuities, they must be considered in a weak sense. Then, the solution is not necessarily unique. Non-physical solutions are singled out by an entropy-entropy flux pair, which we derive for the stochastic Galerkin formulation of shallow water equations. Stable finite volume methods introduce viscosities, which smear the discontinuous solutions. Therefore, a Roe flux, which is less dissipative, is deduced for the stochastic Galerkin formulation of isothermal Euler equations.