Controlling spatiotemporal chaos in oscillatory reaction-diffusion systems by time-delay autosynchronization

Diffusion-induced turbulence in spatially extended oscillatory media near a supercritical Hopf bifurcation can be controlled by applying global time-delay autosynchronization. We consider the complex Ginzburg-Landau equation in the Benjamin-Feir unstable regime and analytically investigate the stability of uniform oscillations depending on the feedback parameters. We show that a noninvasive stabilization of uniform oscillations is not possible in this type of systems. The synchronization diagram in the plane spanned by the feedback parameters is derived. Numerical simulations confirm the analytical results and give additional information on the spatiotemporal dynamics of the system close to complete synchronization.Comment: 19 pages, 10 figures submitted to Physica

Finite-parameter feedback control for stabilizing the complex Ginzburg-Landau equation

In this paper, we prove the exponential stabilization of solutions for complex Ginzburg-Landau equations using finite-parameter feedback control algorithms, which employ finitely many volume elements, Fourier modes or nodal observables (controllers). We also propose a feedback control for steering solutions of the Ginzburg-Landau equation to a desired solution of the non-controlled system. In this latter problem, the feedback controller also involves the measurement of the solution to the non-controlled system.Comment: 20 page

Boundary Control of Coupled Reaction-Advection-Diffusion Systems with Spatially-Varying Coefficients

Recently, the problem of boundary stabilization for unstable linear constant-coefficient coupled reaction-diffusion systems was solved by means of the backstepping method. The extension of this result to systems with advection terms and spatially-varying coefficients is challenging due to complex boundary conditions that appear in the equations verified by the control kernels. In this paper we address this issue by showing that these equations are essentially equivalent to those verified by the control kernels for first-order hyperbolic coupled systems, which were recently found to be well-posed. The result therefore applies in this case, allowing us to prove H^1 stability for the closed-loop system. It also shows an interesting connection between backstepping kernels for coupled parabolic and hyperbolic problems.Comment: Submitted to IEEE Transactions on Automatic Contro

Al'brekht's Method in Infinite Dimensions

In 1961 E. G. Albrekht presented a method for the optimal stabilization of smooth, nonlinear, finite dimensional, continuous time control systems. This method has been extended to similar systems in discrete time and to some stochastic systems in continuous and discrete time. In this paper we extend Albrekht's method to the optimal stabilization of some smooth, nonlinear, infinite dimensional, continuous time control systems whose nonlinearities are described by Fredholm integral operators