Delayed feedback control method for calculating space-time periodic solutions of viscoelastic problems

We are interested in fast techniques for calculating a periodic solution to viscoelastic evolution problems with a space-time periodic condition of "rolling" type. Such a solution is usually computed as an asymptotic limit of the initial value problem with arbitrary initial data. We want to invent a control method, accelerating the convergence. The main idea is to modify our problem by introducing a feedback control term, based on a periodicity error, in order to accelerate the convergence to the desired periodic solution of the problem. First, an abstract evolution problem has been studied. From the analytic solution of the modified (controlled) problem, an efficient control has been found, optimizing the spectrum of the problem. The proposed control term can be mechanically interpreted, and its efficiency increases with the relaxation time. In order to confirm numerically the theoretical results, a finite element simulation has been carried out on a full 2D model for a steady rolling of a viscoelastic tyre with periodic sculpture. It has demonstrated that the controlled solution converges indeed faster than the non-controlled one, and that the efficiency of the method increases with the problem's relaxation time, that is when the memory of the underlying problem is large.