Stabilizing the absolutely or convectively unstable homogeneous solutions of reaction-convection-diffusion systems

We study the problem of stabilization of a homogeneous solution in a two-variable reaction-convection-diffusion one-dimensional system with oscillatory kinetics, in which moving or stationary patterns emerge in the absence of control. We propose to use a formal spatially weighted feedback control to suppress patterns in an absolutely or convectively unstable system and pinning control for a convectively unstable system. The latter approach is very effective and may require only one actuator to adjust feed conditions. In the former approach, the positive diagonal elements of the appropriate dynamics matrix are shifted to the left-hand part of the complex plane to ensure linear (asymptotic) stability of the system according to Gershgorin criterion. Moreover, we construct a controller that (with many actuators) will approach the global stability of the solution, according to Liapunov's direct method. We apply two alternative approaches to reveal the unstable modes: an approximate one that is based on linear stability analysis of an unbounded system, and an exact one that uses a traditional eigenstructure analysis of bounded systems. The number of required actuators increases dramatically with system size and with the distance from the bifurcation point. The methodology is developed for a system with learning cubic kinetics and is tested on a more realistic cross-flow reactor model.