Fejér Polynomials and Control of Nonlinear Discrete Systems

Dmitriy Dmitrishin, Odessa National Polytechnic University
Paul Hagelstein, Baylor University
Anna Khamitova, Georgia Southern University
Anatolii Korenovskyi, Odessa National Polytechnic University
Alexander M. Stokolos, Georgia Southern University

This is an author-provided 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) = (a₁−1)f(x(k))+a₂f(x(k−T))+· · ·+a_N f(x(k−(N −1)T)) , with a₁ + · · · + a_N = 1. Following an approach of Morgul, we associate to each periodic orbit of f, N ∈ N, and a₁, . . . , 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 a₁, . . ., 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.