Stabilizing high-dimensional dynamical systems using time -delayed feedback

In many scientific and engineering problems there is a need to stabilize some specific behavior of a system. In some of these systems, we know that periodic behavior is possible but unstable, and that stabilizing this behavior has practical benefits. Researchers have shown that when standard control methods are not applicable for stabilizing some periodic state, it may be possible to achieve stability using a relatively new method known as time-delayed feedback (TDF), in which the current state of the system is compared to states one or more periods in the past.;In this thesis, I show how to design an efficient TDF controller for stabilizing multi-dimensional unstable periodic states. I derive a novel formula for analyzing the effects of noise on a TDF-controlled discrete-time system, and I determine the level of noise that can be tolerated. I also show that TDF fails to stabilize a broad class of multi-dimensional plane-waves.;I show how to design, from almost any standard proportional controller (SPC), a general form of a TDF controller (GETDAS) that stabilizes unstable fixed points of discrete-time systems. Numerical studies in the thesis support the theory of noise amplification and suggest that the maximal level of noise the GETDAS controller can suppress is higher compared to SPC, over a wide range of parameters. I also suggest an idea for designing a GETDAS controller based on SPC for a continuous systems. As an example I test this method, using tools developed by the nonlinear dynamics community, on a damped driven nonlinear pendulum having an unstable periodic orbit.;In another chapter, I examine the possibility of using TDF to stabilize multi-dimensional plane-waves. Using linear stability analysis and Floquet theory, I show that it fails in the multi-dimensional case, though it has been successfully used in the one-dimensional case. This conclusion follows from symmetry considerations and therefore applies to a wide class of models with simple plane wave solutions.;Designing a TDF controller and performing the noise analysis as shown can be helpful in many real systems where stabilization of an unstable periodic state is required.