MC 5501
Speaker
Benedikt Florian Oppeneiger, TU Ilmenau
Title
Spatial decay of perturbations in transport equations with optimal boundary control
Abstract
We consider linear-quadratic PDE-constrained optimal control problems (OCPs). We show that under certain domain-uniform detectability and stabilizability assumption the influence of perturbations decays exponentially in space for optimally controlled general evolution equations. This justifies the use of adaptive spatial meshes in distributed optimal control of e.g. hyperbolic equations, where certain areas of particular interest are more finely resolved than others.
The domain-uniform stabilizability and detectability will be discussed by means of a one-dimensional transport equation. Due to the finite propagation velocity of the transport equation, the control input cannot instantly influence the systems solution on the whole domain. This property will be used to derive a necessary and sufficient condition on the control domain such that the stabilizability and detectability condition is fulfilled. Finally we give some insights on how this approach may be extended to boundary control systems.
In future works these results could for example be used, to efficiently solve optimal control problems on networks of hyperbolic equations. Possible applications are hydraulic networks (transport equations) and adaptive buildings (Timoshenko beams).