# Assignability of dichotomy spectrum for discrete time-varying linear control systems

## TL;DR

This paper proves that for discrete time-varying linear control systems, uniform complete controllability guarantees the ability to assign the dichotomy spectrum of the closed-loop system arbitrarily, extending previous pole placement results.

## Contribution

It establishes the assignability of the dichotomy spectrum under controllability conditions, strengthening prior results on Lyapunov spectrum assignment.

## Key findings

- Uniform complete controllability implies arbitrary dichotomy spectrum assignability.
- Extends pole placement theorems to dichotomy spectrum for time-varying systems.
- Strengthens previous results on spectrum assignability in control systems.

## Abstract

In this paper, we show that for discrete time-varying linear control systems uniform complete controllability implies arbitrary assignability of dichotomy spectrum of closed-loop systems. This result significantly strengthens the result in A. Babiarz et. al. Pole placement theorem for discrete time-varying linear systems.SIAM J. Control Optim. 55 (2017), no. 2, 671-692 about arbitrary assignability of Lyapunov spectrum of discrete time-varying linear control systems.