The cost of large eddy simulation (LES) in the near-wall region of attached turbulent boundary layers scales as the square of the friction Reynolds number, thus limiting LES to moderate Reynolds numbers. Wall stress boundary conditions are frequently used to alleviate this resolution requirement, but commonly used models are shown to perform poorly at high Reynolds numbers even in turbulent channel flow. Techniques from optimal control theory are used to find wall stresses that yield much better results in turbulent channel flow at high Reynolds numbers than existing models even on extremely coarse grids. In this approach, a suboptimal control strategy is used in which the objective is to force the outer LES towards a desired solution by using the wall stress boundary conditions as control. The suboptimal wall stresses are not necessarily physical, rather they are whatever is necessary to overcome the numerical and modeling errors present in the near-wall region to yield the correct mean velocity profile. Furthermore, the suboptimal control strategy generates reference data for comparing and deriving new wall models. Using linear stochastic estimation it is shown that the dynamically relevant part of the suboptimal wall stresses can be predicted from the local velocity field. A wall model derived from linear stochastic estimation yields good mean flow predictions in LES of turbulent channel flow on a 32(3) uniform and for friction velocity Reynolds numbers from 640 to 20 000.