#### Document Type

Preprint

#### Publication Date

2010

#### Publication Title

Preprint

#### Abstract

We consider optimization problems associated to a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T-cycles of a differentiable function f : R → R of the form x(k + 1) = f(x(k)) + u(k) where u(k) = (a_{1}−1)f(x(k))+a_{2}f(x(k−T))+· · ·+a_{N} f(x(k−(N −1)T)) , with a_{1} + · · · + a_{N} = 1. Following an approach of Morgul, we associate to each periodic orbit of f, N ∈ N, and a_{1}, . . . , a_{N} an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. We prove that, given any 1- or 2-cycle of f, there exist N and a_{1}, . . ., a_{N} whose associated polynomial is Schur stable, and we find the minimal N that guarantees this stabilization. The techniques of proof will take advantage of extremal properties of the Fejer kernels found in classical harmonic analysis.

#### Recommended Citation

Dmitrishin, Dmitriy, Paul Hagelstein, Anna Khamitova, Anatolii Korenovskyi, Alexander M. Stokolos.
2010.
"Fejér Polynomials and Control of Nonlinear Discrete Systems."
*Preprint*: 1-29.
source: https://sites.google.com/a/georgiasouthern.edu/astokolos/files/Stokolos_SIAM.pdf?attredirects=0

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/668

## Comments

This is an author-provided preprint.