  We establish a Pontryagin maximum principle for discrete time optimal control
problems under the following three types of constraints: a) constraints on the
states pointwise in time, b) constraints on the control actions pointwise in
time, and c) constraints on the frequency spectrum of the optimal control
trajectories. While the first two types of constraints are already included in
the existing versions of the Pontryagin maximum principle, it turns out that
the third type of constraints cannot be recast in any of the standard forms of
the existing results for the original control system. We provide two different
proofs of our Pontryagin maximum principle in this article, and include several
special cases fine-tuned to control-affine nonlinear and linear system models.
In particular, for minimization of quadratic cost functions and linear time
invariant control systems, we provide tight conditions under which the optimal
controls under frequency constraints are either normal or abnormal.
