Fejer Polynomials and Control of Nonlinear Discrete Systems

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 $f: \mathbb{R}\rightarrow\mathbb{R}$ of the form $$x(k+1) = f(x(k)) + u(k)$$ where $$u(k) = (a_1 - 1)f(x(k)) + a_2 f(x(k-T)) + \cdots + a_N f(x(k-(N-1)T))\;,$$ with $a_1 + \cdots + a_N = 1$. Following an approach of Morg\"ul, we associate to each periodic orbit of $f$, $N \in \mathbb{N}$, and $a_1,\ldots,a_N$ 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 $a_1,\ldots,a_N$ 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 Fej\'er kernels found in classical harmonic analysis.