Fejér polynomials and control of nonlinear discrete systems

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 ))+· · ·+aNf(x(k−(N−1)T )) , with a1+ · · · + aN= 1.Following an approach of Morg¨ul, we associate to each periodic orbit of f, N ∈ N, and a1, . . . , aNan 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, . . ., aN 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 Fej´er kernels found in classical harmonic analysis.