Geometric invariance of determining and resonating centers: Odd- and any-number limitations of Pyragas control

Abstract

In the spirit of the well-known odd-number limitation, we study the failure of Pyragas control of periodic orbits and equilibria. Addressing the periodic orbits first, we derive a fundamental observation on the invariance of the geometric multiplicity of the trivial Floquet multiplier. This observation leads to a clear and unifying understanding of the odd-number limitation, both in the autonomous and the non-autonomous setting. Since the presence of the trivial Floquet multiplier governs the possibility of successful stabilization, we refer to this multiplier as the determining center. The geometric invariance of the determining center also leads to a necessary condition on the gain matrix for the control to be successful. In particular, we exclude scalar gains. The application of Pyragas control on equilibria does not only imply a geometric invariance of the determining center but surprisingly also on centers that resonate with the time delay. Consequently, we formulate odd-and any-number limitations both for real eigenvalues together with an arbitrary time delay as well as for complex conjugated eigenvalue pairs together with a resonating time delay. The very general nature of our results allows for various applications.

Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

“…Hence space shifts play an indispensable role for stabilization. Failure of stabilization with pure time delays has also been documented for different models; see [31,32,38]. • As a direct consequence, for the variational case (η, β) = (0, 0) we can selectively stabilize all the unstable vortex equilibria obtained in Lemma 2.3 and also those with the nodal class j = 1 in Lemma 2.4, independently of the number of arms m ∈ N. Our stabilization results, Theorem 3.1 and Theorem 3.2, are novel in the following four aspects.…”

“…Pyragas control is widely applied in experimental and numerical settings, because it renders the unstable target solutions visible while its implementation is model-independent and requires no expensive calculations; see [17,22,28,29,40]. Mathematical results on Pyragas control, however, are delicate and rely on explicit properties of the model; see [8,12,30,37,38]. In the setting of PDEs, feedback controls of Pyragas type have been exploited for solutions which are periodic in space or time; see [18,20,25].…”