# Discrete time optimal control with frequency constraints for non-smooth systems

## TL;DR

This paper develops a Pontryagin maximum principle for discrete-time optimal control problems involving pointwise and frequency constraints, as well as nonsmooth systems, enabling high-fidelity control synthesis in complex, constrained scenarios.

## Contribution

It introduces a novel maximum principle that handles combined pointwise, frequency, and nonsmooth constraints in discrete-time optimal control, expanding the scope of existing methods.

## Key findings

- Provides necessary conditions for optimality with multiple constraints.
- Applicable to a broad class of nonsmooth dynamical systems.
- Demonstrates effectiveness through standard problem illustrations.

## Abstract

We present a Pontryagin maximum principle for discrete time optimal control problems with (a) pointwise constraints on the control actions and the states, (b) frequency constraints on the control and the state trajectories, and (c) nonsmooth dynamical systems. Pointwise constraints on the states and the control actions represent desired and/or physical limitations on the states and the control values; such constraints are important and are widely present in the optimal control literature. Constraints of the type (b), while less standard in the literature, effectively serve the purpose of describing important spectral properties of inertial actuators and systems. The conjunction of constraints of the type (a) and (b) is a relatively new phenomenon in optimal control but are important for the synthesis control trajectories with a high degree of fidelity. The maximum principle established here provides first order necessary conditions for optimality that serve as a starting point for the synthesis of control trajectories corresponding to a large class of constrained motion planning problems that have high accuracy in a computationally tractable fashion. Moreover, the ability to handle a reasonably large class of nonsmooth dynamical systems that arise in practice ensures broad applicability our theory, and we include several illustrations of our results on standard problems.