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

In many applications,  in systems that are governed by a 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 approach the optimal states for the  corresponding uncertain static optimal control problem. We show integral turnpike results both for the full uncertain system as well as for a generalized polynomial chaos approximation.