Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law

Perrollaz, Vincent

doi:10.1016/j.anihpc.2012.12.003

Cited by 17 publications

(20 citation statements)

References 23 publications

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“…By integration by parts and by using the boundary condition (24) and the parameters q 1 and q 2 given in Lemma 1 as well as the relations (30) we prove the required identity (29). 2…”

Section: B Nonlinear Systemmentioning

confidence: 73%

“…Now we use the idea presented in [9] to extend the Lyapunov functional from the linear system (in L 2 norm) to the nonlinear system (in H 2 norm). Therefore local asymptotic stability of the equilibrium state and the set-point output regulation will be proved for the nonlinear closed-loop system (24). To do that, we consider the Lyapunov functional candidate S : X → R + such that…”

Section: B Nonlinear Systemmentioning

confidence: 99%

“…In particular they have considered the stabilization problem of weak entropy solutions by boundary control and internal control around a constant equilibrium state for a scalar 1D conservation law with strictly convex flux. In [24] an internal state feedback control law has been designed to asymptotically stabilize the entropy solutions around a constant equilibrium in the topologies L 1 and L ∞ . A stabilizing nonlocal boundary control law (depending on time and the whole initial data) has been proposed in [3] to get asymptotic stability of the constant equilibrium in the L 2 topology.…”

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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“…By integration by parts and by using the boundary condition (24) and the parameters q 1 and q 2 given in Lemma 1 as well as the relations (30) we prove the required identity (29). 2…”

Section: B Nonlinear Systemmentioning

confidence: 73%

Section: B Nonlinear Systemmentioning

confidence: 99%

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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“…As we mentioned earlier, very few results exist on the stabilization of hyperbolic systems of conservation laws in the context of entropy solutions. In fact, even in the scalar case, we are only aware of the work [22], in which a suitable stationary feedback law is shown to stabilize the solutions exponentially. Regarding the stabilization of hyperbolic systems of conservation laws, we know only two results obtaining asymptotic stabilization in open loop.…”

Section: Previous Resultsmentioning

confidence: 99%

Dissipative boundary conditions for 2 × 2 hyperbolic systems of conservation laws for entropy solutions in BV

Coron

Ervedoza

Ghoshal

et al. 2017

Journal of Differential Equations

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In this article, we investigate the BV stability of 2×2 hyperbolic systems of conservation laws with strictly positive velocities under dissipative boundary conditions. More precisely, we derive sufficient conditions guaranteeing the exponential stability of the system under consideration for entropy solutions in BV. Our proof is based on a front tracking algorithm used to construct approximate piecewise constants solutions whose BV norms are controlled through a Lyapunov functional. This Lyapunov functional is inspired by the one proposed in J. Glimm's seminal work [16], modified with some suitable weights in the spirit of the previous works [10,9].

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“…The asymptotic stabilization of entropy solutions to scalar conservation laws is the topic of [10,33,34].…”

Section: Introductionmentioning

confidence: 99%

Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions

Donadello¹,

Perrollaz²

2019

Comptes Rendus. Mathématique

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Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law

Cited by 17 publications

References 23 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

Dissipative boundary conditions for 2 × 2 hyperbolic systems of conservation laws for entropy solutions in BV

Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions

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