Discrete time Pontryagin maximum principle for optimal control problems under state-action-frequency constraints

TL;DR

This paper extends the discrete-time Pontryagin maximum principle to include frequency spectrum constraints on control trajectories, providing new theoretical insights and conditions for optimality in constrained control problems.

Contribution

It introduces a novel Pontryagin maximum principle that incorporates frequency constraints, which are not covered by existing control theory results.

Findings

Established a Pontryagin maximum principle with frequency constraints.

Derived conditions for normal and abnormal optimal controls in linear systems.

Provided proofs and special cases for control-affine nonlinear and linear models.

Abstract

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.