Pinning stationary planar fronts in diffusion-convection-reaction systems

Abstract

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.

Robust Control of Stationary Planar Fronts in Reaction-Diffusion Systems

“…(ρ = 1 when i = j = 1, ρ = 2 when i, j > 1 and ρ = √ 2 when i = 1, j > 1 or j = 1,i > 1, see [4] for derivation ). Substituting (6) into (5),(4) with set points y * d = y s (L/2, r d ) and integrating with a weight eigenfunctions φ e (z, r) results in the spectral representation of closed-loop linearized system (5),(4)…”

“…15, 11) may be presented in the usual vector-matrix form as the linear infinite-dimensional dynamical system with η -dimensional input v and output w vectors The obvious way of designing of a finite-dimensional control (18) for an infinite-dimensional system (16)(17) is to use a truncated (finite-dimensional) approximation of the PDEs. To truncate the system we capitalize on the dissipative nature of the parabolic PDEs: the truncation order N is estimated by calculating the leading eigenvalues of the dynamics matrix (16) (see [4] for details). As follows from [24], for a sufficiently large truncated order N, with actuator functions ψ d (z, r) that coincide with eigenfunctions φ e (z, r), the finite-dimensional control guarantees the stability of infinite-dimensional system.…”

“…Let us apply the above method for stabilization of planar front all R's and two complex eigenvalues with real parts that becomes positive for R ≥ 5.6( see Fig. 4a in [4]). At first we try to apply control (20) with one space-independent actuator and the sensor at (z * = 10, r 1 = R/2).…”

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

“…where ␦ ij = 1 when i = j and ␦ ij = 0 otherwise, and i ͑z͒ ͓ j a ͑z͔͒ are the eigenfunctions (adjoint eigenfunctions) of the linear operator (22) subject to corresponding boundary conditions (Eqs. (22a)-(22c), while j are the related eigenvalues.…”

“…where j = i 2 + 0.25V 2 , i ͑z͒, and j a ͑z͒ are eigenvalues, eigenfunctions, and adjoint eigenfunctions of linear operator (3) with flux boundary conditions, and j Ͼ 0, i =1,2,... satisfy the transcendental equation (see [22] for concrete formulas), q ji = ͗u͑ i ͒ j a ͘, where u͑ i ͒ satisfies Eq. (2b) with x = i and u͑0͒ = 0.…”