Small time global null controllability for a viscous Burgers' equation despite the presence of a boundary layer

Marbach, Frédéric

doi:10.1016/j.matpur.2013.11.013

Cited by 28 publications

(27 citation statements)

References 32 publications

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“…It would also be interesting to consider the case of nonlinear equations like the Burger's equation y ε t − εy ε xx + y ε y ε x = 0 introduced to model turbulence. The asymptotic analysis of this equation posed for x ∈ R is mentioned in [13, Section 4.3.1] (we also refer to [14]).…”

Section: Remarksmentioning

confidence: 99%

Asymptotic analysis of an advection‐diffusion equation involving interacting boundary and internal layers

Amirat

Münch

2020

Math Methods in App Sciences

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As ε goes to zero, the unique solution of the scalar advection-diffusion equation y ε t −εy ε xx +M y ε x = 0, (x, t) ∈ (0, 1) × (0, T ) submitted to Dirichlet boundary conditions exhibits a boundary layer of size O(ε) and an internal layer of size O( √ ε). If the time T is large enough, these thin layers where the solution y ε displays rapid variations intersect and interact each other. Using the method of matched asymptotic expansions, we show how we can construct an explicit approximation P ε of the solution y ε satisfying y ε − P ε L ∞ (0,T ;L 2 (0,1)) = O(ε 3/2 ) and y ε − P ε L 2 (0,T ;H 1 (0,1)) = O(ε), for all ε small enough.

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Section: Remarksmentioning

confidence: 99%

Asymptotic analysis of an advection‐diffusion equation involving interacting boundary and internal layers

Amirat

Münch

2020

Math Methods in App Sciences

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“…However, since there is only one scalar control on the boundary, the global controllability in small time is a real challenge and might be false, as it is the case for the Burgers equations [34]. If there are more controls there are several global controllability results in small time for the nonlinear KdV equations [11] and for the Burgers equations [12,26,28,29,32,48,49]. Note that, using the backstepping approach, [26] allows to recover the global controllability result of [12] obtained by means of the return method.…”

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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“…This is probably the reason for which there exists in the literature only very few asymptotic analysis for controllability problem, a fortiori for singular partial differential equations. We mention [18], [1] following [6]. We also mention the book [13] and the review [8] in the close context of optimal control problems.…”

Section: Introductionmentioning

confidence: 99%

Singular asymptotic expansion of the exact control for the perturbed wave equation

Castro

Münch

2021

ASY

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The Petrowsky type equation y ε tt + εy ε xxxx − y ε xx = 0, ε > 0 encountered in linear beams theory is null controllable through Neumann boundary controls. Due to the boundary layer of size of order √ ε occurring at the extremities, these boundary controls get singular as ε goes to 0. Using the matched asymptotic method, we describe the boundary layer of the solution y ε then derive a rigorous second order asymptotic expansion of the control of minimal L 2 −norm, with respect to the parameter ε. In particular, we recover that the leading term of the expansion is a null Dirichlet control for the limit hyperbolic wave equation, in agreement with earlier results due to J-.L. Lions in the eighties. Numerical experiments support the analysis.

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Small time global null controllability for a viscous Burgers' equation despite the presence of a boundary layer

Cited by 28 publications

References 32 publications

Asymptotic analysis of an advection‐diffusion equation involving interacting boundary and internal layers

Asymptotic analysis of an advection‐diffusion equation involving interacting boundary and internal layers

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

Singular asymptotic expansion of the exact control for the perturbed wave equation

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