Output feedback stabilization of the Korteweg–de Vries equation

Marx, Swann; Cerpa, Eduardo

doi:10.1016/j.automatica.2017.07.057

Cited by 33 publications

(31 citation statements)

References 22 publications

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“…Based on the linear result and the Kato smoothing effect, we may expect the local controllability by standard perturbation. Actually, as it is shown in [5,9,21,50] that the backstepping method can be directly used to treat the nonlinear case. But as the main purpose of this paper is to extend the new method found by Coron and Nguyen to more general models (as we stated in the Introduction), we do not consider nonlinear cases here.…”

Section: Further Commentsmentioning

confidence: 99%

Null Controllability of a Linearized Korteweg--de Vries Equation by Backstepping Approach

Xiang¹

2019

SIAM J. Control Optim.

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We prove the null controllability of a linearized Korteweg-de Vries equation with a Dirichlet control on the left boundary. Instead of considering classical methods, i.e. Carleman estimates, moment method etc., we use a backstepping approach which is a method usually used to handle stabilization problems. type controls are less preferred than piecewise continuous controls (or even L p type conditions), especially for stabilization problems. In [9], Coron and Cerpa proved rapid stabilization of the system (1.1) by using the backstepping method. But since they used some stationary feedback laws, this boundary condition problem is avoided. Recently, by using the (piecewise) backstepping approach, Coron and Nguyen proved the null controllability and semi-global finite time stabilization for a class of heat equations (see [23]). They showed how the use of the maximum principle leads to the well-posedness of the closed-loop system. Their method turns out to be a potential way to solve the local (or even semi-global) finite time stabilization problem for systems which can be rapidly stabilized by means of backstepping methods. At the same time, this method provides a visible way to get null controllability directly instead of using observability inequalities and the duality between controllability and observability.Initially the backstepping is a method to design stabilizing feedback laws in a recursive manner for systems having a triangular structure. See, for example, [15, Section 12.5]. It was first introduced to deal with finite-dimensional control systems. But it can also be used for control systems modeled by means of partial differential equations (PDEs) as shown first in [18]. For linear partial differential equations, a major innovation is due to Krstic and his collaborators. They observed that, when applied to the classical discretization of these systems, the backstepping leads, at the PDEs level (as the mesh size tends to 0), to the transformation of the initial system into a new target system which can be easily stabilized. This transformation is accomplished by means of a Volterra transform of the second kind. An excellent introduction to this method is presented in [39]. Krstic's innovation has been shown to be very useful for many PDEs control systems as, in particular, heat equations [45,4,23], wave equations [60], hyperbolic systems [25,27,36,20] [2, Chapter 7], Korteweg de Vries equations [9,21], and Kuramoto-Sivashinsky equations [44,22]. It was observed later on that for some PDEs more general transforms than Volterra transforms of the second kind have to be considered: see [19,21,22]). Recently, the backstepping method has been adapted to coupled systems, for example the Boussinesq system of KdV-KdV type [5]. For the case of finite dimensional control system and Krstic's backstepping, see [16].Krstic's backstepping requires solving a kernel equation. In the case of the heat equation, the kernel equation is a wave equation; however, in this paper the kernel equation turns out to be a third-order equ...

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Section: Further Commentsmentioning

confidence: 99%

Null Controllability of a Linearized Korteweg--de Vries Equation by Backstepping Approach

Xiang¹

2019

SIAM J. Control Optim.

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“…T . This is nothing but the problem given in (16) which has a unique solution by Lemma 5. This defines an operator Γ :Q T →Q T given by Γ(w * ) =w.…”

Section: Wellposednessmentioning

confidence: 99%

“…In this section, we describe the steps to obtain the numerical solution of the plant-observer-error system given in (1), (6), and (7). We follow a different approach compared to for instance [16]. Our idea is based on first solving the models (7) and (32) with homogeneous boundary conditions and then obtaining the solutions of nonhomogeneous boundary value problems (1) and (6) by using the invertibility of the backstepping transformation given in Lemma 3.…”

Section: Algorithmmentioning

confidence: 99%

Output feedback stabilization of the linearized Korteweg–de Vries equation with right endpoint controllers

Batal

Özsarı

2019

Automatica

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In this paper, we prove the output feedback stabilization for the linearized Korteweg-de Vries (KdV) equation posed on a finite domain in the case the full state of the system cannot be measured. We assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution. This allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable. Stabilization of the original system is proved in the L 2 -sense, while the convergence of the observer system to the original plant is also proved in higher order Sobolev norms. The standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models. In order to deal with this difficulty we use the method of [18] which is based on using modified target systems involving extra trace terms. In addition, we show that the number of controllers and boundary measurements can be reduced to one, with the cost of a slightly lower exponential rate of decay. We provide numerical simulations illustrating the efficacy of our controllers. of the following problem (so called "target system"):in Ω × R + , w(0, t) = w(L, t) = w x (L, t) = 0 in R + , w(x, 0) = w 0 (x) ≡ u 0 − x 0 k(x, y)u 0 (y)dy in Ω.

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“…Finally, we also introduce a numerical approach for the backstepping problem which is different than numerical approaches of other authors who treated the KdV equation. For instance, in [20], the authors directly solve the original model with the boundary feedback whereas in our paper we solve the target systems that have homogeneous boundary values first and then use the bounded invertibility properties of the backstepping transformation to find the solution of the original model. In this way, we can use a finite element method which suits best for homogeneous boundary value problems and not susceptible to numerical errors which might happen due to inhomogeneous and rough boundary interactions.…”

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confidence: 99%

Stabilization of higher order Schrödinger equations on a finite interval: Part I

Batal¹,

Özsarı²,

Yılmaz³

2021

EECT

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We study the backstepping stabilization of higher order linear and nonlinear Schrödinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a prescribed rate of decay. The construction of the backstepping kernel is based on a challenging successive approximation analysis. This contrasts with the case of second order pdes. Second, we consider the case where the full state of the system cannot be measured at all times but some partial information such as measurements of a boundary trace are available. For this problem, we simultaneously construct an observer and the associated backstepping controller which is capable of stabilizing the original plant. Wellposedness and regularity results are provided for all pde models. Although the linear part of the model is similar to the KdV equation, the power type nonlinearity brings additional difficulties. We give two examples of boundary conditions and partial measurements. We also present numerical algorithms and simulations verifying our theoretical results to the fullest extent. Our numerical approach is novel in the sense that we solve the target systems first and obtain the solution to the feedback system by using the bounded invertibility of the backstepping transformation.

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Output feedback stabilization of the Korteweg–de Vries equation

Cited by 33 publications

References 22 publications

Null Controllability of a Linearized Korteweg--de Vries Equation by Backstepping Approach

Null Controllability of a Linearized Korteweg--de Vries Equation by Backstepping Approach

Output feedback stabilization of the linearized Korteweg–de Vries equation with right endpoint controllers

Stabilization of higher order Schrödinger equations on a finite interval: Part I

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