Finite-time internal stabilization of a linear 1-D transport equation

Zhang, Christophe

doi:10.1016/j.sysconle.2019.104529

Cited by 25 publications

(18 citation statements)

References 36 publications

(44 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are mainly two ways to prove the existence of the transformation, either by direct methods [18,19] or, more commonly, by proving the existence of a Riesz basis. For the latter, we again distinguish two cases: either the Riesz basis is deduced directly by an isomorphism applied on an eigenbasis [17,50,51] or the existence of a Riesz basis follows by controllability assumptions and sufficient growth of the eigenvalues of the spatial operator allowing in particular to prove that the family is quadratically close to the eigenfunctions [16,20,21,27] (see Section 2.2 and Section 4 for a definition).…”

Section: Related Results: the Heat Equation And The Backstepping Methodsmentioning

confidence: 99%

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

Gagnon¹,

Hayat²,

Xiang³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with Fredholm transformations on Laplace operators in a sharp functional setting, which is the main objective of this work. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.

show abstract

Section: Related Results: the Heat Equation And The Backstepping Methodsmentioning

confidence: 99%

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

Gagnon¹,

Hayat²,

Xiang³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…However the equation above is actually purely formal, and the "right way" to formulate it is the weak form in (310), which is not surprising, as, according to Corollary 5.1, T I ν is not defined and thus (311) has no real mathematical meaning. This is already the case for transport equations ( [73,74]).…”

Section: Finding a Suitable Candidate: The Operator Equalitymentioning

confidence: 70%

Stabilization of the linearized water tank system

Coron,

Hayat,

Xiang

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

In this article we study the so-called water tank system. In this system, the behavior of water contained in a 1-D tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimension. Here we will focus on the linearized systems around non-uniform steady states, corresponding to a constant acceleration of the tank. We prove that these systems are controllable in Sobolev spaces, using the moments method and perturbative spectral estimates. Then, for steady states corresponding to small enough accelerations, we design an explicit Proportional Integral feedback law (obtained thanks to a well-chosen dynamic extension of the system) that stabilizes these systems exponentially with arbitrarily large decay rate. Our design relies on feedback equivalence/backstepping.

show abstract

“…Also, several results (e.g. [16]) were obtained using a backstepping approach, a very powerful method based on a Volterra transformation, developed mainly for PDE in [24], and generalized recently with a Fredholm transformation for hyperbolic systems [14,34,35]. One may look at [22] for a more detailed survey about this method and its use for the Saint-Venant equations.…”

Section: Introductionmentioning

confidence: 99%

PI controllers for the general Saint-Venant equations

Hayat¹

2021

Preprint

View full text Add to dashboard Cite

We study the exponential stability in the H 2 norm of the nonlinear Saint-Venant (or shallow water) equations with arbitrary friction and slope using a single Proportional-Integral (PI) control at one end of the channel. Using a good but simple Lyapunov function we find a simple and explicit condition on the gain of the PI control to ensure the exponential stability of any steady-states. This condition is independent of the slope, the friction coefficient, the length of the river, the inflow disturbance and, more surprisingly, can be made independent of the steadystate considered. When the inflow disturbance is time-dependent and no steady-state exist, we still have the Input-to-State stability of the system, and we show that changing slightly the PI control enables to recover the exponential stability of slowly varying trajectories.

show abstract

Finite-time internal stabilization of a linear 1-D transport equation

Abstract: We consider a 1-D linear transport equation on the interval (0, L), with an internal scalar control. We prove that if the system is controllable in a periodic Sobolev space of order greater than 1, then the system can be stabilized in finite time, and we give an explicit feedback law.

Cited by 25 publications

References 36 publications

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

Fredholm transformation on Laplacian and rapid stabilization for the heat equation

Stabilization of the linearized water tank system

PI controllers for the general Saint-Venant equations

Contact Info

Product

Resources

About