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A formal approach for control design aimed at stabilization of front and pulse patterns in parabolic quasilinear PDE systems using proportional weighted-average feedback regulators and inhomogeneous actuators is suggested and demonstrated. The method capitalizes on the structure of the Jacobian matrix of the system, presented by a finite Fourier series in eigenfunctions. The finite bandwidth of this matrix and the dissipative nature of the parabolic PDE allow for the construction of the finite feedback regulator by direct application of the Gershgorin stability theorem. Whereas this formalism was rigorously proven for polynomial source functions, we expect this approach to apply to other systems as well. The control obtained stabilizes the solutions in a wide range within the bistability domain. The results are compared with those of other control approaches.

This paper considers various strategies for controlling a stationary planar front solution, in a rectangular domain with a diffusion-reaction distributed system, by pinning the solution to one or few points and using actuators with the simplest possible spatial dependence. We review previous results obtained for one-dimensional diffusion-reaction (with or without convection) systems, for which we applied two approaches: an approximate model reduction to a form that follows the front position while approximating the front velocity, and linear stability analysis. We apply the same two approaches for the planar fronts. The approximate model reduction allows us to analyze qualitatively various control strategies and to predict the critical width below which the control mode of the one-dimensional system is sufficient. These results are corroborated by linear analysis of a truncated model with the spectral methods representation, using concepts of finite and infinite zeros of linear multidimensional systems.