Turnpike Property for Two-Dimensional Navier–Stokes Equations

In this paper we study the turnpike phenomenon arising in the optimal distributed control tracking-type problem for the Navier–Stokes equations. We obtain a positive answer to this property in the case when the control is time-dependent function and also when it is independent of time. In both cases we prove an exponential turnpike property assuming that the stationary optimal state satisfies certain properties of smallness.