joint work with Stefan Ulbrich
We study the consistent numerical approximation of optimal boundary control problems for scalar hyperbolic conservation laws. We consider conservative finite difference schemes with corresponding sensitivity and adjoint schemes. To analyze the convergence of the adjoint schemes, we propose two convenient characterizations of reversible solutions for the adjoint equation in the case of boundary control, which are suitable to show that the limit of discrete adjoints is in fact the correct reversible solution.
The talk is dedicated to the analysis of the Receding-Horizon Control (RHC) method for a class of infinite dimensional stabilisation problems. An error estimate, bringing out the effect of the prediction horizon, the sampling time, and the terminal cost used for the truncated problems will be provided. Reference: Karl Kunisch and Laurent Pfeiffer. The effect of the Terminal Penalty in Receding Horizon Control for a Class of Stabilization Problems. ArXiv preprint, 2018.
The first order system of a regularized version of the control problem is reformulated as an optimality condition in terms of only the state variable, which renders an elliptic boundary value problem of second order in time, and of fourth order in space. We suggest a $C^0$ interior penalty method on the space-time domain with tailored terms that enforce suitable regularity in space direction. A combined numerical analysis leads to a solution algorithm that intertwines mesh refinement with an update strategy for the regularization parameter.