Multilevel control by duality

TL;DR

This paper introduces a duality-based method for designing multilevel controls in linear systems, providing a complete characterization, optimal control construction, and efficient algorithms for ensuring controllability with piecewise constant controls.

Contribution

It offers a novel duality approach to characterize and compute multilevel controls, including optimal controls and controllability time, with practical numerical algorithms.

Findings

Complete characterization of multilevel controls via duality

Development of efficient algorithms for control computation

Determination of controllability time for multilevel controls

Abstract

We discuss the multilevel control problem for linear dynamical systems, consisting in designing a piece-wise constant control function taking values in a finite-dimensional set. In particular, we provide a complete characterization of multilevel controls through a duality approach, based on the minimization of a suitable cost functional. In this manner we build optimal multi-level controls and characterize the time needed for a given ensemble of levels to assure the controllability of the system. Moreover, this method leads to efficient numerical algorithms for computing multilevel controls.