Dissipative Boundary Conditions for One-Dimensional Quasi-linear Hyperbolic Systems: Lyapunov Stability for the $C^1$-Norm

Coron, Jean‐Michel; Bastin, Georges

doi:10.1137/14097080x

Cited by 71 publications

(77 citation statements)

References 40 publications

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“…The problem of controlling systems governed by hyperbolic partial differential equations (PDE) with both inputs and outputs on the boundary has attracted a considerable amount of studies [14], [5], [13]. Interested readers can find a nice literature review on the fields in the section 2 of [10] and in [2]. Many available results have established appropriate boundary conditions to ensure asymptotic or exponential stability of the equilibrium state as in [20], [15], [9], [10], [18] and [31] or Université [32] (most of them in C 1 topology but also in the H 2 topology, see [9]).…”

Section: Introductionmentioning

confidence: 99%

“…Interested readers can find a nice literature review on the fields in the section 2 of [10] and in [2]. Many available results have established appropriate boundary conditions to ensure asymptotic or exponential stability of the equilibrium state as in [20], [15], [9], [10], [18] and [31] or Université [32] (most of them in C 1 topology but also in the H 2 topology, see [9]). In these papers the boundary conditions are given as a function of the output.…”

Section: Introductionmentioning

confidence: 99%

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Design of Integral Controllers for Nonlinear Systems Governed by Scalar Hyperbolic Partial Differential Equations

Trinh

Andrieu

2017

IEEE Trans. Automat. Contr.

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The paper deals with the control and regulation by integral controllers for the nonlinear systems governed by scalar quasi-linear hyperbolic partial differential equations. Both the control input and the measured output are located on the boundary. The closed-loop stabilization of the linearized model with the designed integral controller is proved first by using the method of spectral analysis and then by the Lyapunov direct method. Based on the elaborated Lyapunov function we prove local exponential stability of the nonlinear closed-loop system with the same controller. The output regulation to the set-point with zero static error by the integral controller is shown upon the nonlinear system. Numerical simulations by the Preissmann scheme are carried out to validatethe robustness performance of the designed controller to face tounknown constant disturbances.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Design of Integral Controllers for Nonlinear Systems Governed by Scalar Hyperbolic Partial Differential Equations

Trinh

Andrieu

2017

IEEE Trans. Automat. Contr.

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“…Scalar conservation laws have been studied extensively in the last two decades; see for instance [1,10,19]. Control problems related to 1-D scalar conservation laws have been investigated in many works; see [1,2,7,8,11,17,20,24,25]. Traffic control problems related to one or two 1-D conservation laws have recently been studied in [18,27,28,29,30].…”

Section: Introductionmentioning

confidence: 99%

Feedback control of scalar conservation laws with application to density control in freeways by means of variable speed limits

Karafyllis

Papageorgiou

2019

Automatica

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The paper provides results for the stabilization of a spatially uniform equilibrium profile for a scalar conservation law that arises in the study of traffic dynamics under variable speed limit control. Two different control problems are studied: the problem with free speed limits at the inlet and the problem with no speed limits at the inlet. Explicit formulas are provided for respective feedback laws that guarantee stabilization of the desired equilibrium profile. For the first problem, global asymptotic stabilization is achieved; while for the second problem, regional exponential stabilization is achieved. Moreover, the solutions for the corresponding closed-loop systems are guaranteed to be classical solutions, i.e., there are no shocks. The obtained results are illustrated by means of a numerical example.

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“…In [9], they constructed a strict H 2 -Lyapunov for quasilinear hyperbolic systems with dissipative boundary conditions without source term. More recently in [7], Coron and Bastin study the Lyapunov stability of the C 1 -norm for quasilinear hyperbolic systems of the first order. They consider W 1 p -Lyapunov functions for p < ∞ and look at the limit for p → ∞.…”

Section: Introductionmentioning

confidence: 99%

Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function

Gugat¹,

Leugering²,

Wang³

2017

Mathematical Control &Amp; Related Fields

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For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H 2 -Lyapunov function and show that the boundary feedback constant can be chosen such that the H 2 -Lyapunov function and hence also the H 2 -norm of the difference between the non-stationary and the stationary state decays exponentially with time.

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Dissipative Boundary Conditions for One-Dimensional Quasi-linear Hyperbolic Systems: Lyapunov Stability for the $C^1$-Norm

Cited by 71 publications

References 40 publications

Design of Integral Controllers for Nonlinear Systems Governed by Scalar Hyperbolic Partial Differential Equations

Design of Integral Controllers for Nonlinear Systems Governed by Scalar Hyperbolic Partial Differential Equations

Feedback control of scalar conservation laws with application to density control in freeways by means of variable speed limits

Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function

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