Stabilization of Fronts in a Reaction−Diffusion System:  Application of the Gershgorin Theorem

Abstract

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.

Dissipative systems: from physics to robust nonlinear control

“…This problem is of relevance in the control of front and pulse pattern formation in distributed dissipative systems (see, for instance References [24,25]). In this context, subsystem (56) describes the desired periodic pattern which becomes unstable by the influence of the complementary subsystem.…”

Robustness of constant-delay predictor feedback for in-domain stabilization of reaction–diffusion PDEs with time- and spatially-varying input delays

“…Uncertain time-and spatially-varying delay occur due to network effects and the multiplicity of the devices used to generate the heating power density. A last example occurs in the context of the stabilization of fronts in a reaction-diffusion system with possible application to chemical reactors [40]. However, to the best of our knowledge, the design and/or robustness analysis of control strategies with respect to possibly spatially-varying delays is still an open problem.…”

Feedback control for parabolic distributed parameter systems with time-delay

“…Since the control development based on high-fidelity models has the potential to improve control performance, research have become increasingly active on developing efficient control using accurate PDE models in [6][7][8][9][10]. In some of these approaches, the methods based on spectral decomposition techniques have been appear in [11][12][13][14][15][16]. In [17][18][19][20], the infinite-dimensional DPS were controlled by representing DPS with their dominant finitedimensional modes such as eigenfunctions or singular functions.…”