TY - JOUR
T1 - Fejér Polynomials and Control of Nonlinear Discrete Systems
AU - Dmitrishin, Dmitriy
AU - Hagelstein, Paul
AU - Khamitova, Anna
AU - Korenovskyi, Anatolii
AU - Stokolos, Alexander M.
PY - 2010/1/1
Y1 - 2010/1/1
N2 - 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.
AB - 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.
KW - Fejer Polynomials
KW - Nonlinear Discrete Systems
UR - https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/668
M3 - Article
JO - Preprint
JF - Preprint
ER -