Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow

Vázquez, Rafael; Trélat, Emmanuel; Coron, Jean‐Michel

doi:10.3934/dcdsb.2008.10.925

Cited by 43 publications

(30 citation statements)

References 45 publications

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“…For n = 0, the result follows from (A. 33). Assume that the bound is correct for all i in ∆F n (x, ξ).…”

Section: )mentioning

confidence: 92%

“…By the assumptions on the coefficients and applying Theorem A.2, the direct and inverse transformations (3.23) and (3.38) have kernels that are C 2 (T ) functions. Differentiating twice with respect to x in these transformations, it can be shown that the H 2 norm of γ is equivalent to the H 2 norm of z (see for instance [33]). Thus, if we show H 2 local stability of the origin for (5.15)-(5.18), the same holds for z.…”

Section: Preliminary Definitionsmentioning

confidence: 99%

See 1 more Smart Citation

Local Exponential $H^2$ Stabilization of a $2\times2$ Quasilinear Hyperbolic System Using Backstepping

Coron¹,

Vázquez²,

Krstić³

et al. 2013

SIAM J. Control Optim.

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309

322

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Abstract. In this work, we consider the problem of boundary stabilization for a quasilinear 2 × 2 system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves H 2 exponential stability of the closedloop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type 4 × 4 system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them. This problem has been considered in the past for 2 × 2 systems [11] and even n × n systems [22], using the explicit evolution of the Riemann invariants along the characteristics. More recently, an approach using control Lyapunov functions has been developed for 2 × 2 systems [2] and n × n systems [3]. These results use only static output feedback (the output being the value of the state on the boundaries). However, they do not deal with the same class of systems considered in this work (which includes an extra term in the equations); with this term, it has been shown in [1] that there are examples (even for linear 2 × 2 system) for which there are no control Lyapunov functions of the "diagonal" form

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“…For n = 0, the result follows from (A. 33). Assume that the bound is correct for all i in ∆F n (x, ξ).…”

Section: )mentioning

confidence: 92%

Section: Preliminary Definitionsmentioning

confidence: 99%

Local Exponential $H^2$ Stabilization of a $2\times2$ Quasilinear Hyperbolic System Using Backstepping

Coron¹,

Vázquez²,

Krstić³

et al. 2013

SIAM J. Control Optim.

Self Cite

309

322

View full text Add to dashboard Cite

show abstract

“…For the determination of the solution k(z, s, t) of (24) using either formal integration and successive approximation or a suitable numerical scheme the reader is referred to, e.g., Meurer and Kugi (2009a); Jadachowski et al (2012). With Assumption 10 it can be shown that k(z, s, t) is a strong solution to (24) Vazquez et al, 2008;Meurer and Kugi, 2009a).…”

Section: Stabilisation Of Tracking Error Dynamicsmentioning

confidence: 99%

PDE-based multi-agent formation control using flatness and backstepping: Analysis, design and robot experiments

Freudenthaler

Meurer

2020

Automatica

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A PDE-based control concept is developed to deploy a multi-agent system into desired formation profiles. The dynamic model is based on a coupled linear, time-variant parabolic distributed parameter system. By means of a particular coupling structure parameter information can be distributed within the agent continuum. Flatness-based motion planning and feedforward control are combined with a backstepping-based boundary controller to stabilise the distributed parameter system of the tracking error. The tracking controller utilises the required state information from a Luenberger-type state observer. By means of an exogenous system the relocation of formation profiles is achieved. The transfer of the control strategy to a finite-dimensional discrete multi-agent system is obtained by a suitable finite difference discretization of the continuum model, which in addition imposes a leader-follower communication topology. The results are evaluated both in simulation studies and in experiments for a swarm of mobile robots realizing the transition between different stable and unstable formation profiles.

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“…The boundary stabilization of Navier-Stokes equations, with tangential controllers or normal controllers was studied in two dimensions, for example by Barbu [5,4], Munteanu [13], Coron [17], Krstic [18], [1], [2], Raymond [15]. In most of these papers, either there are sufficiently many boundary controls so there are no missing directions, or stabilizability is proved but with no specific decay rate (except in [15]).…”

Section: ∂U ∂Tmentioning

confidence: 99%

Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control

Chowdhury¹,

Mitra²,

Renardy³

2018

Evolution Equations &Amp; Control Theory

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In this paper, we consider the Stokes equations in a two-dimensional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Müntz-Szász Theorem.2010 Mathematics Subject Classification. Primary: 93C20, 93B05; Secondary: 76D05.

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Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow

Cited by 43 publications

References 45 publications

Local Exponential $H^2$ Stabilization of a $2\times2$ Quasilinear Hyperbolic System Using Backstepping

Local Exponential $H^2$ Stabilization of a $2\times2$ Quasilinear Hyperbolic System Using Backstepping

PDE-based multi-agent formation control using flatness and backstepping: Analysis, design and robot experiments

Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control

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