Аннотация:
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.
