Exponential input-to-state stabilization of a class of diagonal boundary control systems with delay boundary control

Lhachemi, Hugo; Shorten, Robert; Prieur, Christophe

doi:10.1016/j.sysconle.2020.104651

Cited by 15 publications

(9 citation statements)

References 40 publications

(126 reference statements)

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“…Then [25] generalizes the result of [31] in the case where the main operator is a Riesz spectral operator with simple eigenvalues. The work in [27] extends the result of [31] to the case where the control contains some disturbances and where the delay can depend on time.…”

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confidence: 74%

Feedback stabilization of parabolic systems with input delay

Djebour¹,

Takahashi²,

Valein³

2022

MCRF

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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

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confidence: 74%

Feedback stabilization of parabolic systems with input delay

Djebour¹,

Takahashi²,

Valein³

2022

MCRF

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“…A first class of problems deals with the feedback stabilization of PDEs in the presence of a state-delay [12,[15][16][17][18]26,25,41]. In this paper, we are concerned with a second class of problem, namely: the feedback stabilization of PDEs in the presence of a delay in the control input [14,[21][22][23]28,27,24,[31][32][33][34][35]. One of the very first contributions in this field was reported in [21].…”

Section: Introductionmentioning

confidence: 99%

“…It was shown in [14] that this approach is not limited to reaction-diffusion systems but can also be applied to the boundary feedback stabilization of a linear Kuramoto-Sivashinsky equation under a constant input delay. This approach was generalized to the boundary stabilization of a class of diagonal infinitedimensional systems in [22,27] for constant input delays and then in [23,28] for fast time-varying input delays.…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by the early work [19] dealing with the robustness of constant-delay predictor feedback w.r.t. uncertain and time-varying input delays for finite-dimensional systems (see also [28] in the context of input-to-state stabilization), this analysis is carried out in this paper via a small gain argument. We show that the constant-delay predictor feedback achieves the exponential stabilization of the closedloop infinite-dimensional system provided that the deviations of the uncertain time-and spatially-varying delay around its nominal value are small enough.…”

Section: Introductionmentioning

confidence: 99%

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

2021

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This paper discusses the in-domain feedback stabilization of reaction-diffusion PDEs with Robin boundary conditions in the presence of an uncertain time-and spatially-varying delay in the distributed actuation. The proposed control design strategy consists of a constant-delay predictor feedback designed based on the known nominal value of the control input delay and is synthesized on a finite-dimensional truncated model capturing the unstable modes of the original infinite-dimensional system. By using a small-gain argument, we show that the resulting closed-loop system is exponentially stable provided that the variations of the delay around its nominal value are small enough. The proposed proof actually applies to any distributedparameter system associated with an unbounded operator that 1) generates a C0-semigroup on a weighted space of square integrable functions over a compact interval; and 2) is self-adjoint with compact resolvent.

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“…This approach could likely be generalized to other control problems, as the ones considered in [5] where an infinite-dimensional dynamics is decomposed into two parts: one unstable operator having a finite-dimensional representation, and one stable operator. See also [9,11] for control design methods exploiting this idea.…”

Section: Introductionmentioning

confidence: 99%

Regulation of a reaction-diffusion equation with bounded observation

Lhachemi¹,

Prieur²

2021

2021 Proceedings of the Conference on Control and Its Applications

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This paper solves a regulation problem for a reactiondiffusion equation. More precisely, for a given bounded observation, we design a boundary controller so that the setpoint output of the equation converges to a prescribed reference signal. The control law is finite-dimensional and is obtained by coupling a pole-placement control law with an observer. The proofs are based on a Lyapunov function and relies on the properties of Sturm-Liouville operators.

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Exponential input-to-state stabilization of a class of diagonal boundary control systems with delay boundary control

Cited by 15 publications

References 40 publications

Feedback stabilization of parabolic systems with input delay

Feedback stabilization of parabolic systems with input delay

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

Regulation of a reaction-diffusion equation with bounded observation

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