Turnpike properties of optimal boundary control problems with random linear hyperbolic systems

In many applications, in systems that are governed by linear hyperbolic partial differential equations some of the problem parameters are uncertain. If information about the probability distribution of the parametric uncertainty, distribution is available, the uncertain state of the system can be described using an intrinsic formulation through a polynomial chaos expansion. This allows to obtain solutions for optimal boundary control problems with random parameters. We show that similar to the deterministic case, a turnpike result holds in the sense that for large time horizons the optimal states for dynamic optimal control problems on a substantial part of the time interval approaches the optimal states for the corresponding uncertain static optimal control problem. We show turnpike results both for the full uncertain system as well as for a generalized polynomial chaos approximation.