Feedback Control of Traveling Wave Solutions of the Complex Ginzburg Landau Equation

Through a linear stability analysis, we investigate the effectiveness of a noninvasive feedback control scheme aimed at stabilizing traveling wave solutions of the one-dimensional complex Ginzburg Landau equation (CGLE) in the Benjamin-Feir unstable regime. The feedback control is a generalization of the time-delay method of Pyragas, which was proposed by Lu, Yu and Harrison in the setting of nonlinear optics. It involves both spatial shifts, by the wavelength of the targeted traveling wave, and a time delay that coincides with the temporal period of the traveling wave. We derive a single necessary and sufficient stability criterion which determines whether a traveling wave is stable to all perturbation wavenumbers. This criterion has the benefit that it determines an optimal value for the time-delay feedback parameter. For various coefficients in the CGLE we use this algebraic stability criterion to numerically determine stable regions in the (K,rho) parameter plane, where rho is the feedback parameter associated with the spatial translation and K is the wavenumber of the traveling wave. We find that the combination of the two feedbacks greatly enlarges the parameter regime where stabilization is possible, and that the stability regions take the form of stability tongues in the (K,rho)--plane. We discuss possible resonance mechanisms that could account for the spacing with K of the stability tongues.Comment: 33 pages, 12 figure

Feedback control of flow alignment in sheared liquid crystals

Based on a continuum theory, we investigate the manipulation of the non-equilibrium behavior of a sheared liquid crystal via closed-loop feedback control. Our goal is to stabilize a specific dynamical state, that is, the stationary "flow-alignment", under conditions where the uncontrolled system displays oscillatory director dynamics with in-plane symmetry. To this end we employ time-delayed feedback control (TDFC), where the equation of motion for the ith component, q_i(t), of the order parameter tensor is supplemented by a control term involving the difference q_i(t)-q_i(t-\tau). In this diagonal scheme, \tau is the delay time. We demonstrate that the TDFC method successfully stabilizes flow alignment for suitable values of the control strength, K, and \tau; these values are determined by solving an exact eigenvalue equation. Moreover, our results show that only small values of K are needed when the system is sheared from an isotropic equilibrium state, contrary to the case where the equilibrium state is nematic

Tools for Stability of Switching Linear Systems: Gain Automata and Delay Compensation.

The topic of this paper is the analysis of stability for a class of switched linear systems, modeled by hybrid automata. In each location of the hybrid automaton the dynamics is assumed to be linear and asymptotically stable; the guards on the transitions are hyperplanes in the state space. For each location an estimate is made of the gain via a Lyapunov function for the dynamics in that location, given a pair of ingoing and outgoing transitions. It is shown how to obtain the best possible estimate by optimizing the Lyapunov function. The estimated gains are used in defining a so-called gain automaton that forms the basis of an algorithmic criterion for the stability of the hybrid automaton. The associated gain automaton provides a systematic tool to detect potential sources of instability as well as an indication on to how to stabilize the hybrid systems by requiring appropriate delays for specific transitions

Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation

We show that Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system. This extends results of Fiedler et al. [PRL 98, 114101 (2007)], who demonstrated that such feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis of this degenerate bifurcation problem reveals two qualitatively distinct cases when unfolded in a two-parameter plane. In each case, Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the phase angle satisfies a certain restriction.Comment: 35 pages, 19 figure

Distributed delays stabilize negative feedback loops

Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillation around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that the linear equation with distributed delays is asymptotically stable if the associated differential equation with a discrete delay of the same mean is asymptotically stable. Therefore, distributed delays stabilize negative feedback loops