  We consider averages convergence as the time-horizon goes to infinity of
optimal solutions of time-dependent optimal control problems to optimal
solutions of the corresponding stationary optimal control problems. Control
problems play a key role in engineering, economics and sciences. To be more
precise, in climate sciences, often times, relevant problems are formulated in
long time scales, so that, the problem of possible asymptotic behaviors when
the time-horizon goes to infinity becomes natural. Assuming that the controlled
dynamics under consideration are stabilizable towards a stationary solution,
the following natural question arises: Do time averages of optimal controls and
trajectories converge to the stationary optimal controls and states as the
time-horizon goes to infinity? This question is very closely related to the
so-called turnpike property that shows that, often times, the optimal
trajectory joining two points that are far apart, consists in, departing from
the point of origin, rapidly getting close to the steady-state (the turnpike)
to stay there most of the time, to quit it only very close to the final
destination and time. In the present paper we deal with heat equations with
non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the
fractional Laplace operator $(-\Delta)^s$ ($0<s<1$). We prove the turnpike
property for the nonlocal Robin optimal control problem and the exponential
turnpike property for both Dirichlet and nonlocal Robin optimal control
problems.
