Boundary feedback stabilization of Fisher’s equation

Abstract

The aim of this work is to design an explicit finite dimensional boundary feedback controller for locally exponentially stabilizing the equilibrium solutions to Fisher's equation in both L 2 (0, 1) and H 1 (0, 1). The feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation. This stabilizing procedure is applicable for any level of instability, which extends the result of [2] for nonlinear parabolic equations. The effectiveness of the approach is illustrated by a numerical simulation.

Boundary sampled-data feedback stabilization for parabolic equations

“…The above feedback is exactly the same as that given in [8] and [9], which was used to construct a continuous-time boundary feedback control.…”

“…al. applied this control to Fisher's equation in [9], and proved that it can locally stabilize this kind of semilinear parabolic equations. In last decade, a different approach to construct explicit boundary feedback controller to stabilize the 1-D linear and nonlinear parabolic equations, which is the so-called backstepping method, was developed.…”

“…In this work, we shall develop the technique introduced in [8] and [9] to construct a sampleddata feedback control for stabilizing the semilinear parabolic equations. Several novelties of this work should be stressed.…”

Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback

“…In the literature there are plenty of results concerning the stabilization of the deterministic Burgers equation, for example we refer to [15] and [13]. The last one provides a global stabilization result, with some consequences on the stabilizability of the stochastic version.…”

“…Indeed, the feedback law, we propose here, requires full state knowledge. However, taking into account the numerical results in [15], where a similar feedback is proposed for the stabilization of the Fischer equation, we may expect that only the knowledge on a part of the domain is enough. In [15] it is shown that measurements are needed only on (a, 1) instead of the whole interval (0, 1), where a ≤ 0.25.…”