Pseudo-Backstepping and Its Application to the Control of Korteweg--de Vries Equation from the Right Endpoint on a Finite Domain

Özsarı, Türker; Batal, Ahmet

doi:10.1137/18m1211933

Cited by 10 publications

(20 citation statements)

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“…This implies that (8) cannot have a smooth solution and the standard algorithm of backstepping method fails. This issue was previously observed in Korteweg de-Vries equation [11] and later treated in the case of uncritical domains in [13] and in the case of critical domains in [25,2]. The idea of the latter work was to drop one of the boundary conditions from the kernel pde model and take r sufficiently small.…”

Section: Figure 1 Backsteppingmentioning

confidence: 89%

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

Özsarı

Yılmaz

2022

EECT

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<p style='text-indent:20px;'>Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [<xref ref-type="bibr" rid="b3">3</xref>] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.</p>

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Section: Figure 1 Backsteppingmentioning

confidence: 89%

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

Özsarı

Yılmaz

2022

EECT

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“…The same issue also occurs in other third order equations such as the Korteweg-de Vries (KdV) equation [10]. In addition, it is not difficult to show that such a kernel model will not have a smooth solution [22]. This problem was first treated by [12] via extending the overdetermined kernel model from a triangular domain into a rectangular domain and using the exact (Neumann) boundary controllability property for the underlying dynamics.…”

Section: 21mentioning

confidence: 99%

“…The drawback was that it only applied to domains of uncritical lengths since it relied on the exact controllability, which only holds for such domains. Most recently, the first two authors introduced another approach in [22] which is based on using an imperfect kernel by disregarding one of the boundary conditions from the overdetermined kernel model. This approach eliminated the dependence on the type of domain, but the exponential decay rate could not be made as large as possible.…”

Section: 21mentioning

confidence: 99%

“…There are also some recent works on the boundary feedback stabilization of other evolution equations which involve higher order dispersion such as the Korteweg-de Vries (see e.g. [3], [10], [12], [22]) and Korteweg-de Vries Burgers (see e.g. [21], [2], [13], [18]) equations.…”

mentioning

confidence: 99%

“…This is because the power type structure of the nonlinear term is distorted (see (53)) and standard multipliers yield Chini's type ODE inequalities which involve more than one nonlinearities, see (99) and (110) below. The Lyapunov analysis given in the context of the KdV equation (see e.g., [10] and [22]) does not involve ODE inequalities which involve several nonlinearities. This is an intrinsic feature of the higher order Schrödinger equation and contrasts with the KdV equation.…”

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confidence: 99%

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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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Boundary Control of Korteweg-de Vries and Kuramoto-Sivashinsky PDEs

Cerpa

2019

Encyclopedia of Systems and Control

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Pseudo-Backstepping and Its Application to the Control of Korteweg--de Vries Equation from the Right Endpoint on a Finite Domain

Cited by 10 publications

References 20 publications

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

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

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

Boundary Control of Korteweg-de Vries and Kuramoto-Sivashinsky PDEs

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