Discrete Time Pontryagin Maximum Principle Under State-Action-Frequency Constraints

We establish a Pontryagin maximum principle for discrete-time optimal control problems under the following three types of constraints: first, constraints on the states pointwise in time, second, constraints on the control actions pointwise in time, and, third, 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 include several special cases fine-tuned to control-affine nonlinear and linear system models.