Feedback control of flow alignment in sheared liquid crystals

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

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.

Feedback control of flow vorticity at low Reynolds numbers

Our aim is to explore strategies of feedback control to design and stabilize novel dynamic flow patterns in model systems of complex fluids. To introduce the control strategies, we investigate the simple Newtonian fluid at low Reynolds number in a circular geometry. Then, the fluid vorticity satisfies a diffusion equation. We determine the mean vorticity in the sensing area and use two control strategies to feed it back into the system by controlling the angular velocity of the circular boundary. Hysteretic feedback control generates self-regulated stable oscillations in time, the frequency of which can be adjusted over several orders of magnitude by tuning the relevant feedback parameters. Time-delayed feedback control initiates unstable vorticity modes for sufficiently large feedback strength. For increasing delay time, we first observe oscillations with beats and then regular trains of narrow pulses. Close to the transition line between the resting fluid and the unstable modes, these patterns are relatively stable over long times.

Delay-induced transport in a rocking ratchet under feedback control

Based on the Fokker-Planck equation we investigate the transport of an overdamped colloidal particle in a static, asymmetric periodic potential supplemented by a time-dependent, delayed feedback force, F(fc). For a given time t, F(fc) depends on the status of the system at a previous time t-τ(D), with τ(D) being a delay time, specifically on the delayed mean particle displacement (relative to some "switching position"). For nonzero delay times F(fc)(t) develops nearly regular oscillations, generating a net current in the system. Depending on the switching position, this current is nearly as large or even larger than that in a conventional open-loop rocking ratchet. We also investigate thermodynamic properties of the delayed nonequilibrium system and we suggest an underlying Langevin equation which reproduces the Fokker-Planck results.