## 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.

Original language | American English |
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Journal | Preprint |

State | Published - Jan 1 2010 |

## Keywords

- Fejer Polynomials
- Nonlinear Discrete Systems

## DC Disciplines

- Education
- Mathematics