Feedback stabilization of parabolic systems with input delay

Abstract

This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the N -dimensional linear reaction-convectiondiffusion equation with N ≥ 1 and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state.1. Introduction. Time delay phenomena appear in many applications, for instance in biology, mechanics, automatic control or engineering and are inevitable due to the time-lag between the measurements and their exploitation. For instance in control problems, one need to take into account the analysis time or the computation time. We aim at showing that, under quite general hypotheses, one can deduce the exponential stabilization with delay of a parabolic system from its exponential stabilization without delay. One of the first article devoted to the parabolic case is [23] with a backstepping method (see [13] for a similar method for the wave equation). We can also quote [12], [28], where the approach is to construct a feedback by a predictor approach. Several works have considered different extensions to this problem: the case of non constant delay (see, for instance, [9], [29]) or the case of multiple delay (see, for instance, [10]). Note that in the context of stability problems for partial differential equations with delay, some particular features can appear for hyperbolic systems: a small delay in the feedback mechanism can destabilize a system (see for instance [16,15]) and a delay term can also improve the

Continuous data assimilation and feedback control of fractional reaction-diffusion equations

“…Recently, Carlson et al [7,8] adopted the AOT method to dynamically learn the coefficients of nonlinear systems using partial observations. We also remark that just currently, Djebour et al [12] considered the feedback stabilization of parabolic systems with input delay, also see [34]. Stabilization by some feedback control is an active issue.…”

Event-triggered boundary control of an unstable reaction diffusion PDE with input delay

“…The stabilization of reaction-diffusion PDEs with an arbitrary level of instability and under arbitrarily long input delay is a challenging problem, first formulated and solved in [27] using the backstepping method for PDEs as the stabilization of a hyperbolic (transport) PDE which cascades into the reaction-diffusion PDE. Since then, control design for delay compensation (including known or unknown constant/timevarying delays) has evolved considerably and several results have been proposed for reaction-diffusion PDEs, see, e.g., [12,14,32,23,35,4,44] and the references therein.…”

Proportional Integral Regulation Control of a One-Dimensional Semilinear Wave Equation

“…In particular, one has w 1 = k∈Z w, f k e 1 k in H 1 -norm. Moreover, from (18) we have that Aw = k∈Z w, f k Ae k and thus…”

“…Hence, in the multi-dimensional case, a first challenging difficulty that must be addressed is to replace the Riesz basis study by more abstract projection operators. On this issue, it seems that the recent work [18] provides very interesting line of research, although the authors of that reference deal with a parabolic equation with a selfadjoint operator and then one has a spectral decomposition and spectral projectors. For a wave equation, the underlying operator is not selfadjoint and is not even normal, which creates deep difficulties for the spectral study.…”