Fonctions de Lyapunov pour les EDP : Applications ` a deux syst` - - PowerPoint PPT Presentation

Fonctions de Lyapunov pour les EDP : Applications ` a deux syst` emes physiques

Christophe Prieur CNRS, Gipsa-lab, Grenoble GT EDP Valence, f´ evrier 2012

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Lyapunov functional based techniques are central in the study of partial differential equations (PDEs). See [Coron, Bastin, d’Andr´ ea-Novel, 2007], [Krstic, Smyshlyaev, 2008], [Cazenave, Haraux, 1998] ... Two motivations to use strict Lyapunov functions:

1

To demonstrate asymptotic stability by means of a weak Lyapunov functional, the LaSalle invariance principle may be used [Luo, Guo, Morgul, 1999]. It requires a precompactness property for the solutions (may be difficult to prove, or even not satisfied).

2

With strict Lyapunov functions in hands: robustness analysis

• f the stability with respect to uncertainties and sensitivity of

the solutions with respect to external disturbances.

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GT-EDP’12

Lyapunov functional based techniques are central in the study of partial differential equations (PDEs). See [Coron, Bastin, d’Andr´ ea-Novel, 2007], [Krstic, Smyshlyaev, 2008], [Cazenave, Haraux, 1998] ... Two motivations to use strict Lyapunov functions:

1

To demonstrate asymptotic stability by means of a weak Lyapunov functional, the LaSalle invariance principle may be used [Luo, Guo, Morgul, 1999]. It requires a precompactness property for the solutions (may be difficult to prove, or even not satisfied).

2

With strict Lyapunov functions in hands: robustness analysis

• f the stability with respect to uncertainties and sensitivity of

the solutions with respect to external disturbances.

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• C. Prieur

GT-EDP’12

Lyapunov functional based techniques are central in the study of partial differential equations (PDEs). See [Coron, Bastin, d’Andr´ ea-Novel, 2007], [Krstic, Smyshlyaev, 2008], [Cazenave, Haraux, 1998] ... Two motivations to use strict Lyapunov functions:

1

To demonstrate asymptotic stability by means of a weak Lyapunov functional, the LaSalle invariance principle may be used [Luo, Guo, Morgul, 1999]. It requires a precompactness property for the solutions (may be difficult to prove, or even not satisfied).

2

With strict Lyapunov functions in hands: robustness analysis

• f the stability with respect to uncertainties and sensitivity of

the solutions with respect to external disturbances.

3/44

• C. Prieur

GT-EDP’12

Lyapunov functional based techniques are central in the study of partial differential equations (PDEs). See [Coron, Bastin, d’Andr´ ea-Novel, 2007], [Krstic, Smyshlyaev, 2008], [Cazenave, Haraux, 1998] ... Two motivations to use strict Lyapunov functions:

1

To demonstrate asymptotic stability by means of a weak Lyapunov functional, the LaSalle invariance principle may be used [Luo, Guo, Morgul, 1999]. It requires a precompactness property for the solutions (may be difficult to prove, or even not satisfied).

2

With strict Lyapunov functions in hands: robustness analysis

• f the stability with respect to uncertainties and sensitivity of

the solutions with respect to external disturbances.

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• C. Prieur

GT-EDP’12

In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs. We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems.

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In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs. We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems.

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In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs. We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems.

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In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs. We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems.

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In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs. We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems.

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In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs. We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems.

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In this talk: new strict Lyapunov functionals for linear hyperbolic PDEs. We extend the strictification technique developed in [Malisoff, Mazenc, 2009] for ordinary differential equations and the Lyapunov design techniques of [Coron, Bastin, d’Andr´ ea-Novel, 2008] and [Diagne, Bastin, Coron, 2012] to families of PDEs with time-varying parameters. It parallels the semilinear parabolic case [Mazenc, CP, 11] It gives robustness properties of Input-to-State Stability (ISS) type See the surveys [Sontag, 2008] for nonlinear finite-dimensional systems and [Jayawardhana, Logemann, Ryan, 2011] for some classes of nonlinear infinite-dimensional systems.

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Outline

1 Motivations to design strict Lyapunov functions for non-homogeneous hyperbolic systems 2 Related works: Stability in presence of small perturbations 3 Sensitivity with respect to large perturbations using an ISS-Lyapunov function 4 Application to the Saint-Venant–Exner example 5 Related application: the Tokamak plasma Conclusion Sections 2 and 3 have been presented during the last CDC (sorry to Joseph and Yann).

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Outline

1 Motivations to design strict Lyapunov functions for non-homogeneous hyperbolic systems 2 Related works: Stability in presence of small perturbations 3 Sensitivity with respect to large perturbations using an ISS-Lyapunov function 4 Application to the Saint-Venant–Exner example 5 Related application: the Tokamak plasma Conclusion Sections 2 and 3 have been presented during the last CDC (sorry to Joseph and Yann).

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Outline

1 Motivations to design strict Lyapunov functions for non-homogeneous hyperbolic systems 2 Related works: Stability in presence of small perturbations 3 Sensitivity with respect to large perturbations using an ISS-Lyapunov function 4 Application to the Saint-Venant–Exner example 5 Related application: the Tokamak plasma Conclusion Sections 2 and 3 have been presented during the last CDC (sorry to Joseph and Yann).

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Outline

1 Motivations to design strict Lyapunov functions for non-homogeneous hyperbolic systems 2 Related works: Stability in presence of small perturbations 3 Sensitivity with respect to large perturbations using an ISS-Lyapunov function 4 Application to the Saint-Venant–Exner example 5 Related application: the Tokamak plasma Conclusion Sections 2 and 3 have been presented during the last CDC (sorry to Joseph and Yann).

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Outline

1 Motivations to design strict Lyapunov functions for non-homogeneous hyperbolic systems 2 Related works: Stability in presence of small perturbations 3 Sensitivity with respect to large perturbations using an ISS-Lyapunov function 4 Application to the Saint-Venant–Exner example 5 Related application: the Tokamak plasma Conclusion Sections 2 and 3 have been presented during the last CDC (sorry to Joseph and Yann).

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Outline

1 Motivations to design strict Lyapunov functions for non-homogeneous hyperbolic systems 2 Related works: Stability in presence of small perturbations 3 Sensitivity with respect to large perturbations using an ISS-Lyapunov function 4 Application to the Saint-Venant–Exner example 5 Related application: the Tokamak plasma Conclusion Sections 2 and 3 have been presented during the last CDC (sorry to Joseph and Yann).

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Hyperbolic systems in Rn

Consider the following non-homogeneous hyperbolic system, ∀z ∈ [0, 1], ∀t ≥ 0, ∂tX(z, t) + Λ(z, t)∂zX(z, t) = F(z, t)X(z, t) + δ(z, t) (1) where Λ = diag(λ1, . . . , λn), and λ1(z, t) < . . . < λm(z, t) < 0 < λm+1(z, t) < . . . < λn(z, t) δ stands for a disturbance, Λ, F and δ are of class C 1 and T-periodic with respect to t. Using the notation: X =

• X

− , X +

, the boundary conditions are X+(0, t) X−(1, t)

• = K

X+(1, t) X−(0, t)

• ,

(2) where K ∈ Rn×n.

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Many technics to Find sufficient conditions on K so that (1)-(2) is Locally Exponentially Stable in H2, or in C 1... This kind of models appear in many various applications such as the traffic flow control [Bressan, Han, 11], [Garavello, Piccoli, 06], [Gugat, Herty, Klar, Leugering, 06] the open-channel regulation [Bastin et al, 2009] see also the experimental channel here in Valence ...

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Many technics to Find sufficient conditions on K so that (1)-(2) is Locally Exponentially Stable in H2, or in C 1... This kind of models appear in many various applications such as the traffic flow control [Bressan, Han, 11], [Garavello, Piccoli, 06], [Gugat, Herty, Klar, Leugering, 06] the open-channel regulation [Bastin et al, 2009] see also the experimental channel here in Valence ...

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1 Motivations. Sensitivity to perturbations

As a first example, let us consider the following linear hyperbolic system: ∂tX + Λ∂zX = 0 , z ∈ [0, 1], t ≥ 0 X(0, t) = KX(1, t) (3) Notation: K = max{|Kx|, x ∈ Rn, |x| = 1} ρ1(K) = inf{∆K∆−1, ∆ ∈ Dn,+} ρ(K) = spectral radius of |K| [Coron et al, 08]: if ρ1(K) < 1 then the system (3) is Exp. Stable. This sufficient condition is weaker that the one of [Li, 94].

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Particular 2D system: ∂tX + Λ∂zX = 0 , z ∈ [0, 1], t ≥ 0 X(0, t) = KX(1, t) where K = − 1

2

1 1 1

• and Λ =

2 1

• .

The condition of [Coron et al, 08] (ρ1(|K|) < 1) is satisfied.Then this system is exponentially stable. Using a Lax-Friedrichs method, we may check the attractivity:

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Particular 2D system: ∂tX + Λ∂zX = 0 , z ∈ [0, 1], t ≥ 0 X(0, t) = KX(1, t) where K = − 1

2

1 1 1

• and Λ =

2 1

• .

The condition of [Coron et al, 08] (ρ1(|K|) < 1) is satisfied.Then this system is exponentially stable. Using a Lax-Friedrichs method, we may check the attractivity:

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Moreover let us consider the following finite-dimensional system: ∂tX = FX , t ≥ 0 No boundary condition (z is a parameter). where F = −4 −3 5 3

• (with eigenvalues having a negative real

part). It is Exp. Stable. With the same initial condition

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Now combining the two previous systems leads to ∂tX + Λ∂zX = FX , z ∈ [0, 1], t ≥ 0 X(0, t) = KX(1, t) which is unstable: Indeed, with the same initial condition

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For general non-homogeneous hyperbolic system

Let us consider the non-homogeneous and perturbed case: ∂tX + Λ(z, t)∂zX = F(z, t)X + δ(z, t) , z ∈ [0, 1], t ≥ 0 (4) X−(1, t) X+(0, t)

• = K

X−(0, t) X+(1, t)

• (5)

Thus When the homogeneous system (4)-(5) is stable then, with F ≡ 0, the non-homogeneous system (4)-(5) may be unstable.

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2 – Related works

In [Li, 94], and in [Coron et al, 08] the unperturbed case (F ≡ δ ≡ 0) is considered Following an analogous approach of [Li, 94], we may study the sensitivity for small source terms, i.e. when F is small and δ ≡ 0, see [CP, Winkin, Bastin, 08]. What happen for large source terms or with bounded perturbations? Question: for an asymptotically hyperbolic stable system Do bounded perturbations result bounded states?

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2 – Related works

In [Li, 94], and in [Coron et al, 08] the unperturbed case (F ≡ δ ≡ 0) is considered Following an analogous approach of [Li, 94], we may study the sensitivity for small source terms, i.e. when F is small and δ ≡ 0, see [CP, Winkin, Bastin, 08]. What happen for large source terms or with bounded perturbations? Question: for an asymptotically hyperbolic stable system Do bounded perturbations result bounded states?

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3 – Sensitivity to large source terms

Let us consider a linear, time-dependent hyperbolic system: ∂tX + Λ(z, t)∂zX = F(z, t)X + δ(z, t) , (6) up to a change of variables, we assume that Λ = diag(λ1, . . . , λn), and 0 < λ1(z, t) < . . . < λn(z, t) The boundary condition X(0, t) = KX(1, t) . (7) F is a source term. δ is an unknown perturbation Assumption 1 Λ, F and δ are are C 1 and T-periodic with respect to t. If Λ is constant and ρ1(K) < 1, then ∃ a diag. pos. def. matrix ∆ such that Sym(∆K∆−1) < Id. then ∃ a diag. pos. def. matrix Q := ∆2Λ−1, and ε > 0 such that Sym(QΛ − K QΛK) ≥ εId .

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3 – Sensitivity to large source terms

Let us consider a linear, time-dependent hyperbolic system: ∂tX + Λ(z, t)∂zX = F(z, t)X + δ(z, t) , (6) up to a change of variables, we assume that Λ = diag(λ1, . . . , λn), and 0 < λ1(z, t) < . . . < λn(z, t) The boundary condition X(0, t) = KX(1, t) . (7) F is a source term. δ is an unknown perturbation Assumption 1 Λ, F and δ are are C 1 and T-periodic with respect to t. If Λ is constant and ρ1(K) < 1, then ∃ a diag. pos. def. matrix ∆ such that Sym(∆K∆−1) < Id. then ∃ a diag. pos. def. matrix Q := ∆2Λ−1, and ε > 0 such that Sym(QΛ − K QΛK) ≥ εId .

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Assumption 2 ∃ a sym. pos. def. matrix Q, α ∈ (0, 1), a C 0 r : [0, ∞) → R, periodic of period T > 0 with a positive mean value, i.e. such that R = T max{r(m), 0} ||Q|| + min{r(m), 0} λmin(Q)

• dm > 0

such that, for all t ≥ 0 and for all z ∈ [0, 1], it holds Sym

• αQΛ(1, t) − K QΛ(0, t)K
• ≥ 0 ,

(8) Sym (QΛ(z, t)) ≥ r(t)Id , (9) Sym (Q∂zΛ(z, t) + 2QF(z, t)) ≤ 0 (10) Remark: If Λ is constant and ρ1(K) < 1, then ∃ a diag. pos. def. matrix Q, and ε > 0 such that Sym(QΛ − K QΛK) ≥ εId . and thus (8) and (9) of Assumption 2 hold.

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Assumption 2 ∃ a sym. pos. def. matrix Q, α ∈ (0, 1), a C 0 r : [0, ∞) → R, periodic of period T > 0 with a positive mean value, i.e. such that R = T max{r(m), 0} ||Q|| + min{r(m), 0} λmin(Q)

• dm > 0

such that, for all t ≥ 0 and for all z ∈ [0, 1], it holds Sym

• αQΛ(1, t) − K QΛ(0, t)K
• ≥ 0 ,

(8) Sym (QΛ(z, t)) ≥ r(t)Id , (9) Sym (Q∂zΛ(z, t) + 2QF(z, t)) ≤ 0 (10) Remark: If Λ is constant and ρ1(K) < 1, then ∃ a diag. pos. def. matrix Q, and ε > 0 such that Sym(QΛ − K QΛK) ≥ εId . and thus (8) and (9) of Assumption 2 hold.

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Under Assumption 2, let µ ∈ (0, ln(α)) and q(t) := µ

• max{r(t),0}

||Q||

+ min{r(t),0}

λmin(Q)

• − µR

2T .

Theorem : [CP, Mazenc, 11] Under Assumptions 1 and 2, letting V : L2(0, 1) × [0, +∞) → [0, +∞) defined, for all φ ∈ L2(0, 1) and t ≥ 0, by V (φ, t) := e

1 T

R t

t−T

R t

q(m)dmd

1 φ(z)Qφ(z)e−µzdz , we have, along the solutions of (6) and (7), for all t ≥ 0, ˙ V (X, t) ≤ −c1V (X, t) + c2δ(., t)2

L2(0,1)

c3X(., t)2

L2(0,1) ≤ V (X, t) ≤ c4X(., t)2 L2(0,1)

for suitable constant values ci > 0.

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Under Assumption 2, let µ ∈ (0, ln(α)) and q(t) := µ

• max{r(t),0}

||Q||

+ min{r(t),0}

λmin(Q)

• − µR

2T .

Theorem : [CP, Mazenc, 11] Under Assumptions 1 and 2, letting V : L2(0, 1) × [0, +∞) → [0, +∞) defined, for all φ ∈ L2(0, 1) and t ≥ 0, by V (φ, t) := e

1 T

R t

t−T

R t

q(m)dmd

1 φ(z)Qφ(z)e−µzdz , we have, along the solutions of (6) and (7), for all t ≥ 0, ˙ V (X, t) ≤ −c1V (X, t) + c2δ(., t)2

L2(0,1)

c3X(., t)2

L2(0,1) ≤ V (X, t) ≤ c4X(., t)2 L2(0,1)

for suitable constant values ci > 0.

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About the expression of the Lyapunov function

Time varying positive definite function V (φ, t) := e

1 T

R t

t−T

R t

q(m)dmd

1 φ(z)Qφ(z)e−µzdz , Introduction of µ: [Coron, 98] for the stabilization of the Euler equations. [Xu, Sallet, 02] for symmetric linear hyperbolic systems. Introduction of the time-varying term Quite usual for nonlinear finite-dimensional systems [Mazenc, Nesic, 07] among others but not so usual for PDEs?

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ISS Lyapunov function for hyperbolic systems

Input-to-State Stable Lyapunov function for hyperbolic systems ˙ V (X, t) ≤ −c1V (X, t) + c2δ(., t)2

L2(0,1)

c3X(., t)2

L2(0,1) ≤ V (X, t) ≤ c4X(., t)2 L2(0,1)

This implies exponential stability when δ ≡ 0 along the solutions of (6) and (7), for all t ≥ 0, X(., t)L2(0,1) ≤ C1e−tεX(., 0)L2(0,1)+C2 sup

s∈[0,t]

δ(., s)L2(0,1) in other words δ bounded ⇒ X bounded similarly we may prove δ → 0 ⇒ X → 0, as t → ∞

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ISS Lyapunov function for hyperbolic systems

Input-to-State Stable Lyapunov function for hyperbolic systems ˙ V (X, t) ≤ −c1V (X, t) + c2δ(., t)2

L2(0,1)

c3X(., t)2

L2(0,1) ≤ V (X, t) ≤ c4X(., t)2 L2(0,1)

This implies exponential stability when δ ≡ 0 along the solutions of (6) and (7), for all t ≥ 0, X(., t)L2(0,1) ≤ C1e−tεX(., 0)L2(0,1)+C2 sup

s∈[0,t]

δ(., s)L2(0,1) in other words δ bounded ⇒ X bounded similarly we may prove δ → 0 ⇒ X → 0, as t → ∞

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ISS Lyapunov function for hyperbolic systems

Input-to-State Stable Lyapunov function for hyperbolic systems ˙ V (X, t) ≤ −c1V (X, t) + c2δ(., t)2

L2(0,1)

c3X(., t)2

L2(0,1) ≤ V (X, t) ≤ c4X(., t)2 L2(0,1)

This implies exponential stability when δ ≡ 0 along the solutions of (6) and (7), for all t ≥ 0, X(., t)L2(0,1) ≤ C1e−tεX(., 0)L2(0,1)+C2 sup

s∈[0,t]

δ(., s)L2(0,1) in other words δ bounded ⇒ X bounded similarly we may prove δ → 0 ⇒ X → 0, as t → ∞

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ISS Lyapunov function for hyperbolic systems

Input-to-State Stable Lyapunov function for hyperbolic systems ˙ V (X, t) ≤ −c1V (X, t) + c2δ(., t)2

L2(0,1)

c3X(., t)2

L2(0,1) ≤ V (X, t) ≤ c4X(., t)2 L2(0,1)

This implies exponential stability when δ ≡ 0 along the solutions of (6) and (7), for all t ≥ 0, X(., t)L2(0,1) ≤ C1e−tεX(., 0)L2(0,1)+C2 sup

s∈[0,t]

δ(., s)L2(0,1) in other words δ bounded ⇒ X bounded similarly we may prove δ → 0 ⇒ X → 0, as t → ∞

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Sketch of the proof

∂tX + Λ(z, t)∂zX = F(z, t)X + δ(z, t) , (11) X(0, t) = KX(1, t) . (12) First Step: ˙ W ≤ 0??? Prove that the function W (φ) = 1 φ(z)Qφ(z)e−µzdz , is a weak Lyapunov function when δ is identically equal to zero We note first that, for all φ ∈ L2(0, 1), 1 β 1 |φ(z)|2 dz ≤ W (φ) ≤ β 1 |φ(z)|2 dz (13) with β = max

• Q,

eµ λmin(Q)

• .

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To do that, we compute the time-derivative of W along the solutions of (11) with (12): ˙ W = −RΛ(X(., t), t) + RF(X(., t), t) + Rδ(X(., t), t) , with RΛ(φ, t) = 2 1 φ(z)QΛ(z, t)∂zφ(z)e−µzdz , RF(φ, t) = 2 1 φ(z)QF(z, t)φ(z)e−µzdz , Rδ(φ, t) = 2 1 φ(z)Qδ(z, t)e−µzdz .

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Now, observe that RΛ(φ, t) = 1 ∂z(φ(z)QΛ(z, t)φ(z))e−µzdz − 1 φ(z)Q∂xΛ(x, t)φ(z)e−µzdz . Performing an integration by part on the first integral and using the boundary condition we get: ˙ W = −X(1, t)QΛ(1, t)X(1, t)e−µ + X(1, t)K QΛ(1, t)KX(1, t) +˜ RΛ(X, t) + RF(X, t) + Rδ(X, t) . with ˜ RΛ(X, t) = −µ 1 X(z)QΛ(z, t)X(z)e−µzdz + 1 X(z)Q∂zΛ(z, t)X(z)e−µzdz .

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By grouping the terms and using the notation N(t) = K QΛ(1, t)K , M(z, t) = µΛ(z, t) − ∂zΛ(z, t) − 2F(z, t) we obtain ˙ W = X(1, t) [N(t) − e−µQΛ(1, t)] X(1, t) − 1 X(z, t)QM(z, t)X(z, t)e−µzdz +2 1 X(z, t)Qδ(z, t)e−µzdz . With Assumption 2 and our choice for µ (sufficiently small), we get ˙ W ≤ −µr(t) 1 |X(z, t)|2e−µzdz + 2 1 X(z, t)Qδ(z, t)e−µzdz with r(t) ≥ 0, but the mean value of r is positive.

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It follows that, for all κ > 0, ˙ W ≤ − µ ||Q||r(t)W (X) + 2Qκ 1 |X(z, t)|2e−µzdz +Q 2κ 1 |δ(z, t)|2e−µzdz ≤ −qκ(t)W (X) + Q 2κ 1 |δ(z, t)|2dz , with qκ(t) =

µ Qr(t) − 2Qκ λmin(Q).

End of the first step W is not exactly a weak Lyapunov function when δ ≡ 0. But the mean value of r is positive and κ can be arbitrarily small Thus W is a weak Lyapunov function ”by mean”

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It follows that, for all κ > 0, ˙ W ≤ − µ ||Q||r(t)W (X) + 2Qκ 1 |X(z, t)|2e−µzdz +Q 2κ 1 |δ(z, t)|2e−µzdz ≤ −qκ(t)W (X) + Q 2κ 1 |δ(z, t)|2dz , with qκ(t) =

µ Qr(t) − 2Qκ λmin(Q).

End of the first step W is not exactly a weak Lyapunov function when δ ≡ 0. But the mean value of r is positive and κ can be arbitrarily small Thus W is a weak Lyapunov function ”by mean”

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It follows that, for all κ > 0, ˙ W ≤ − µ ||Q||r(t)W (X) + 2Qκ 1 |X(z, t)|2e−µzdz +Q 2κ 1 |δ(z, t)|2e−µzdz ≤ −qκ(t)W (X) + Q 2κ 1 |δ(z, t)|2dz , with qκ(t) =

µ Qr(t) − 2Qκ λmin(Q).

End of the first step W is not exactly a weak Lyapunov function when δ ≡ 0. But the mean value of r is positive and κ can be arbitrarily small Thus W is a weak Lyapunov function ”by mean”

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Second Step Use the positive mean value of r to modify W . Let us consider the time-varying candidate Lyapunov function V (t, φ) = esκ(t)W (φ) , with sκ(t) = 1

T

t

t−T

t

• qκ(m)dmd.

One get ˙ V ≤ −esκ(t)

• qκ(t)W (X) + Q

2κ 1 |δ(z, t)|2dz

• +esκ(t)
• qκ(t) − 1

T

t

t−T

qκ(m)dm

• W (X) .

Since r is periodic of period T, we have t

t−T

qκ(m)dm = µ QR − 2TQκ λmin(Q) , where R is the mean value of r. For a suitable choice of κ, we get the result.

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Second Step Use the positive mean value of r to modify W . Let us consider the time-varying candidate Lyapunov function V (t, φ) = esκ(t)W (φ) , with sκ(t) = 1

T

t

t−T

t

• qκ(m)dmd.

One get ˙ V ≤ −esκ(t)

• qκ(t)W (X) + Q

2κ 1 |δ(z, t)|2dz

• +esκ(t)
• qκ(t) − 1

T

t

t−T

qκ(m)dm

• W (X) .

Since r is periodic of period T, we have t

t−T

qκ(m)dm = µ QR − 2TQκ λmin(Q) , where R is the mean value of r. For a suitable choice of κ, we get the result.

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GT-EDP’12

Second Step Use the positive mean value of r to modify W . Let us consider the time-varying candidate Lyapunov function V (t, φ) = esκ(t)W (φ) , with sκ(t) = 1

T

t

t−T

t

• qκ(m)dmd.

One get ˙ V ≤ −esκ(t)

• qκ(t)W (X) + Q

2κ 1 |δ(z, t)|2dz

• +esκ(t)
• qκ(t) − 1

T

t

t−T

qκ(m)dm

• W (X) .

Since r is periodic of period T, we have t

t−T

qκ(m)dm = µ QR − 2TQκ λmin(Q) , where R is the mean value of r. For a suitable choice of κ, we get the result.

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Two applications

Applications of the design of ISS Lyapunov functions for Hyperbolic systems Parabolic systems

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4 – Application on a hydraulic problem

Saint-Venant–Exner equation, [Graf, 84], [Diagne, Bastin, Coron, 12]: ∂tH + V∂xH + H∂xV = δ1 , ∂tV + V∂xV + g∂xH + g∂xB = gSb − Cf V2

H + δ2 ,

∂tB + aV2∂xV = δ3 , (14) where H = H(x, t) is the water height at x in [0, L] V = V(x, t) is the water velocity B = B(x, t) is the bathymetry, i.e. the sediment layer g is the gravity constant Sb is the slope (which is assumed to be constant) Cf is the friction coefficient (also assumed to be constant) a is the effects of the porosity and of the viscosity δ(x, t) = (δ1(x, t), δ2(x, t), δ3(x, t)) is a disturbance, e.g. it can be a supply of water or an evaporation along the channel (see [Graf, 98]).

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4 – Application on a hydraulic problem

Saint-Venant–Exner equation, [Graf, 84], [Diagne, Bastin, Coron, 12]: ∂tH + V∂xH + H∂xV = δ1 , ∂tV + V∂xV + g∂xH + g∂xB = gSb − Cf V2

H + δ2 ,

∂tB + aV2∂xV = δ3 , (14) where H = H(x, t) is the water height at x in [0, L] V = V(x, t) is the water velocity B = B(x, t) is the bathymetry, i.e. the sediment layer g is the gravity constant Sb is the slope (which is assumed to be constant) Cf is the friction coefficient (also assumed to be constant) a is the effects of the porosity and of the viscosity δ(x, t) = (δ1(x, t), δ2(x, t), δ3(x, t)) is a disturbance, e.g. it can be a supply of water or an evaporation along the channel (see [Graf, 98]).

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4 – Application on a hydraulic problem

Saint-Venant–Exner equation, [Graf, 84], [Diagne, Bastin, Coron, 12]: ∂tH + V∂xH + H∂xV = δ1 , ∂tV + V∂xV + g∂xH + g∂xB = gSb − Cf V2

H + δ2 ,

∂tB + aV2∂xV = δ3 , (14) where H = H(x, t) is the water height at x in [0, L] V = V(x, t) is the water velocity B = B(x, t) is the bathymetry, i.e. the sediment layer g is the gravity constant Sb is the slope (which is assumed to be constant) Cf is the friction coefficient (also assumed to be constant) a is the effects of the porosity and of the viscosity δ(x, t) = (δ1(x, t), δ2(x, t), δ3(x, t)) is a disturbance, e.g. it can be a supply of water or an evaporation along the channel (see [Graf, 98]).

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4 – Application on a hydraulic problem

Saint-Venant–Exner equation, [Graf, 84], [Diagne, Bastin, Coron, 12]: ∂tH + V∂xH + H∂xV = δ1 , ∂tV + V∂xV + g∂xH + g∂xB = gSb − Cf V2

H + δ2 ,

∂tB + aV2∂xV = δ3 , (14) where H = H(x, t) is the water height at x in [0, L] V = V(x, t) is the water velocity B = B(x, t) is the bathymetry, i.e. the sediment layer g is the gravity constant Sb is the slope (which is assumed to be constant) Cf is the friction coefficient (also assumed to be constant) a is the effects of the porosity and of the viscosity δ(x, t) = (δ1(x, t), δ2(x, t), δ3(x, t)) is a disturbance, e.g. it can be a supply of water or an evaporation along the channel (see [Graf, 98]).

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Let us consider a steady-state H, V and B which is constant with respect to the x-variable. (It should satisfy gSbH = Cf V2.) The linearization of (14) is: ∂th + V∂xh + H∂zv = δ1 , ∂tv + V∂xv + g∂xh + g∂xb = Cf V2

H2 − 2Cf V H u + δ2 ,

∂tb + aV2∂xv = δ3 . In Riemann coordinates we get, for k ∈ {1, 2, 3}, ∂tyk + λk∂xyk +

3

• s=1

(2λs − 3V)θsys = δk , (15) where λk are some (distinct) constant values θk are some physical values.

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Let us consider a steady-state H, V and B which is constant with respect to the x-variable. (It should satisfy gSbH = Cf V2.) The linearization of (14) is: ∂th + V∂xh + H∂zv = δ1 , ∂tv + V∂xv + g∂xh + g∂xb = Cf V2

H2 − 2Cf V H u + δ2 ,

∂tb + aV2∂xv = δ3 . In Riemann coordinates we get, for k ∈ {1, 2, 3}, ∂tyk + λk∂xyk +

3

• s=1

(2λs − 3V)θsys = δk , (15) where λk are some (distinct) constant values θk are some physical values.

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This system is ∂ty + Λ∂xy = Fy + δ(x, t) , where y = (y1, y2, y3), Λ = diag(λ1, λ2, λ3), and, for all x ∈ [0, L], t ≥ 0, F =   α1 α2 α3 α1 α2 α3 α1 α2 α3   Λ and F are not simultaneously diagonalizable.

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Design of a stabilizing boundary control

Let us explain how out theorem can be applied to design a stabilizing boundary feedback control. Boundary conditions 1) Operation of the gate at outflow of the reach: H(L, t)V(L, t) = kg

• [H(L, t) − u1(t)]3

2) Value of the channel inflow rate H(0, t)V(0, t) = u2(t) 3) Physical constraint on the bathymetry B(0, t) = B Two boundary control laws u1 and u2

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By linearizing these boundary conditions, with suitable choice of the ui we get in Riemann coordinates: y1(L, t) = k12y2(L, t) + k13y3(L, t) y2(0, t) = k21y1(0, t) for tuning parameters k12, k13 and k21 in R. The last boundary condition is:

• i

[(λi − V)2 − gH]yi(0, t) = 0 How to compute k12, k13 and k21? How to compute an ISS Lyapunov function?

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To summarize we get: ∂ty + Λ∂xy = Fy + δ(x, t) y(0, t) = Ky(L, t) with K =   k12 k13 k21 ξ(k21)   , and ξ(k21) = −[(λ1 − V)2 − gH] + k21[(λ2 − V)2 − gH] (λ3 − V)2 − gH . Assumption 1 is ok.

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To summarize we get: ∂ty + Λ∂xy = Fy + δ(x, t) y(0, t) = Ky(L, t) with K =   k12 k13 k21 ξ(k21)   , and ξ(k21) = −[(λ1 − V)2 − gH] + k21[(λ2 − V)2 − gH] (λ3 − V)2 − gH . Assumption 1 is ok.

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Assumption 2 holds as soon as there exists a symmetric positive definite matrix Q such that Sym(QΛ − K QΛK) ≥ 0 , Sym(QF) ≤ 0 . (16) Note that, given K, computing Q in a cone is a convex problem Numerically tractable problem The equilibrium is chosen as in [Dos Santos, CP, 08]: H = 0.13 [m], V = 15 [ms−1], and B = 0 [m]. We use λ1 = −10, λ2 = 7.72 × 10−4, λ3 = 13. With K given by

k12 = 0 , k13 = 0 , k21 = −0.095 ,

we compute a solution of (16):

Q =   8.1 × 107 −2.7 × 103 −7.2 × 107

• 2.9 × 102

2.1 × 103

• 6.5 × 107

 

which ensures that Assumption 2 holds.

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Assumption 2 holds as soon as there exists a symmetric positive definite matrix Q such that Sym(QΛ − K QΛK) ≥ 0 , Sym(QF) ≤ 0 . (16) Note that, given K, computing Q in a cone is a convex problem Numerically tractable problem The equilibrium is chosen as in [Dos Santos, CP, 08]: H = 0.13 [m], V = 15 [ms−1], and B = 0 [m]. We use λ1 = −10, λ2 = 7.72 × 10−4, λ3 = 13. With K given by

k12 = 0 , k13 = 0 , k21 = −0.095 ,

we compute a solution of (16):

Q =   8.1 × 107 −2.7 × 103 −7.2 × 107

• 2.9 × 102

2.1 × 103

• 6.5 × 107

 

which ensures that Assumption 2 holds.

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Assumption 2 holds as soon as there exists a symmetric positive definite matrix Q such that Sym(QΛ − K QΛK) ≥ 0 , Sym(QF) ≤ 0 . (16) Note that, given K, computing Q in a cone is a convex problem Numerically tractable problem The equilibrium is chosen as in [Dos Santos, CP, 08]: H = 0.13 [m], V = 15 [ms−1], and B = 0 [m]. We use λ1 = −10, λ2 = 7.72 × 10−4, λ3 = 13. With K given by

k12 = 0 , k13 = 0 , k21 = −0.095 ,

we compute a solution of (16):

Q =   8.1 × 107 −2.7 × 103 −7.2 × 107

• 2.9 × 102

2.1 × 103

• 6.5 × 107

 

which ensures that Assumption 2 holds.

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Assumption 2 holds as soon as there exists a symmetric positive definite matrix Q such that Sym(QΛ − K QΛK) ≥ 0 , Sym(QF) ≤ 0 . (16) Note that, given K, computing Q in a cone is a convex problem Numerically tractable problem The equilibrium is chosen as in [Dos Santos, CP, 08]: H = 0.13 [m], V = 15 [ms−1], and B = 0 [m]. We use λ1 = −10, λ2 = 7.72 × 10−4, λ3 = 13. With K given by

k12 = 0 , k13 = 0 , k21 = −0.095 ,

we compute a solution of (16):

Q =   8.1 × 107 −2.7 × 103 −7.2 × 107

• 2.9 × 102

2.1 × 103

• 6.5 × 107

 

which ensures that Assumption 2 holds.

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Final remarks on this application

Thus selecting µ = 1.5 × 10−2, we compute the following ISS Lyapunov function, defined by, for all y in L2(0, L), V (y) = L y(x)Qy(x)e−µx dx for the Saint-Venant–Exner system. Note that the computed controller is a locally stabilizing boundary control. It depends only on the height at both ends of the channel and the bathymetry of the water. Does not depend on all the state. Output feedback law only. More details in [CP, Mazenc, 11]

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Final remarks on this application

Thus selecting µ = 1.5 × 10−2, we compute the following ISS Lyapunov function, defined by, for all y in L2(0, L), V (y) = L y(x)Qy(x)e−µx dx for the Saint-Venant–Exner system. Note that the computed controller is a locally stabilizing boundary control. It depends only on the height at both ends of the channel and the bathymetry of the water. Does not depend on all the state. Output feedback law only. More details in [CP, Mazenc, 11]

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5 – Control of the flux in a Tokamak plasma

Poloidal magnetic flux in a Tokamak plasma: ψ(R, Z) It is defined as the flux per radian of the magnetic field B(R, Z) through a disc centered on the toroidal axis at height Z, having a radius R and surface S Objective: Control of the safety factor profile, which is related to z := ∇ψ.

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5 – Control of the flux in a Tokamak plasma

Design of an ISS Lyapunov function for a parabolic PDE With [Blum, 1989], or [E. Witrant, et al, 2007], we have to consider ∂tz = ∂r η r ∂r [rz]

• + ∂r [ηu] , r ∈ (0, 1), t ≥ 0

(17) where r in the normalized position in the small disc. Tokamak = Torus but no dependence wrt the angle and to the height variable z is the inverse of the ”safety factor” that should be controlled η = η(r, t) is the diffusion u is the control from the ECCD1 antennas.

1ECCD=Electron Cyclotron Current Drive 35/44

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5 – Control of the flux in a Tokamak plasma

Design of an ISS Lyapunov function for a parabolic PDE With [Blum, 1989], or [E. Witrant, et al, 2007], we have to consider ∂tz = ∂r η r ∂r [rz]

• + ∂r [ηu] , r ∈ (0, 1), t ≥ 0

(17) where r in the normalized position in the small disc. Tokamak = Torus but no dependence wrt the angle and to the height variable z is the inverse of the ”safety factor” that should be controlled η = η(r, t) is the diffusion u is the control from the ECCD1 antennas.

1ECCD=Electron Cyclotron Current Drive 35/44

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The Dirichlet boundary conditions z(0, t) = z(1, t) = 0, ∀t ∈ [0, T) (18) and initial condition: z(r, 0) = z0(r), ∀r ∈ (0, 1) (19) Control Lyapunov function candidate: V (z) = 1 2 1 f (r)z2dr; f (r) > 0 ∀r ∈ [0, 1] with some function f : [0, 1] → (0, ∞) twice continuously differentiable.

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Theorem [Bribiesca, CP et al, 11] If there exist a C 1 f and α > 0 such that, ∀r ∈ [0, 1], ∀t ≥ 0, f (r)η + f (r)

• ∂rη − η

r

• + f (r)

∂rη r − η r2

• ≤ −αf (r),

then, along the solutions of (17), (18), (19), ˙ V ≤ −αV (z) + 1 f (r)∂r [ηu] zdr, ∀t ≥ 0 and thus with u = − γ

η

r

0 z(ρ, t)dρ, where γ ≥ 0 is a tuning

parameter, the system is globally exponentially stable.

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Numerical computation of the Lyapunov function

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 Radius (normalized) Weighting function f(r)

Figure: Function f for the computation of the Lyapunov function

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0.2 0.4 0.6 0.8 1 −0.1 0.1 0.2 0.3 0.4 0.5 Radius (normalized) Distributed state z [Tm] t=T0 t=Tf /4 t=Tf/2 t=3Tf/4 t=Tf

(a) Time-slices of the solution to the PDE.

0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Lyapunov Function V(z)

(b) Normalized evolution of the Lyapunov function.

Figure: Response of the nominal system without control action.

However, with control actions, the exponential convergence and the ISS gain may be improved. As proven by the Lyapunov computation, and checked on numerical simulations:

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Time Radius (normalized) 0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.1 0.1 0.2 0.3 0.4[Tm]

(a) Contour plot of the solution to the PDE.

0.2 0.4 0.6 0.8 1 −0.1 0.1 0.2 0.3 0.4 0.5 Radius (normalized) Distributed state z [Tm] t=T0 t=Tf /4 t=Tf/2 t=3Tf/4 t=Tf

(b) Time-slices of the solution to the PDE. (c) Evolution of the control u.

0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Lyapunov Function V(z)

(d) Normalized evolution of the Lyapunov function.

Figure: Response of the nominal system with control action

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Illustration of ISS property

Full-physics simulator to describe the evolution of η = η(r, t) Experimental data drawn from Tore Supra shot 35109 Actuator perturbation for t ∈ [8, 20] s control action for t ≥ 16 s (γ = 0.75).

(a) Solution of the PDE.

5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Lyapunov Function V(z)

(b) Normalized evolution of the Lyapunov function.

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Nonlinear constraint of the actuator

In the previous result, the control u = − γ

η

r

0 z(ρ, t)dρ was

• suggested. However a nonlinear expression should be considered

(see [Witrant et al, 2007]). Thus we prefer to consider, at each time step, a couple (P∗

lh, N∗ ) as follows:

(P∗

lh, N∗ ) = arg

min

(Plh,N)∈U

1 f (r)∂r

• ηu(Plh, N)
• zdr

subject to the constraints: 0 ≥ 1 f (r)

• ηu(P∗

lh, N∗ )

• r zdr ≥ −γV (z)

where U . = [Plh,min, Plh,max] × [N,min, N,max]. A simple gradient-descent algorithm on the discretized parameter space was implemented. See [Bribiesca, CP et al, 11] for more informations

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Nonlinear constraint of the actuator

In the previous result, the control u = − γ

η

r

0 z(ρ, t)dρ was

• suggested. However a nonlinear expression should be considered

(see [Witrant et al, 2007]). Thus we prefer to consider, at each time step, a couple (P∗

lh, N∗ ) as follows:

(P∗

lh, N∗ ) = arg

min

(Plh,N)∈U

1 f (r)∂r

• ηu(Plh, N)
• zdr

subject to the constraints: 0 ≥ 1 f (r)

• ηu(P∗

lh, N∗ )

• r zdr ≥ −γV (z)

where U . = [Plh,min, Plh,max] × [N,min, N,max]. A simple gradient-descent algorithm on the discretized parameter space was implemented. See [Bribiesca, CP et al, 11] for more informations

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Conclusion and open questions

We have dealt with the sensitivity of linear space-dependent time-varying hyperbolic systems wrt perturbations perturbations are bounded ⇒ state is bounded Input-to-State Stability It parallels what have been done for a class of semilinear parabolic PDEs Applications on an hydraulic problem and on the control of the flux in a Tokamak plasma

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Conclusion and open questions

We have dealt with the sensitivity of linear space-dependent time-varying hyperbolic systems wrt perturbations perturbations are bounded ⇒ state is bounded Input-to-State Stability It parallels what have been done for a class of semilinear parabolic PDEs Applications on an hydraulic problem and on the control of the flux in a Tokamak plasma

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Conclusion and open questions

We have dealt with the sensitivity of linear space-dependent time-varying hyperbolic systems wrt perturbations perturbations are bounded ⇒ state is bounded Input-to-State Stability It parallels what have been done for a class of semilinear parabolic PDEs Applications on an hydraulic problem and on the control of the flux in a Tokamak plasma

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Conclusion and open questions

Open questions ISS for nonlinear hyperbolic systems. See the Lyapunov function that is derived in the current work

• f Drici, Coron, Bastin.

Applications of ISS? Does it give the offset that we have seen on an experimental channel? Offset measured on experiments in [Dos Santos, CP, 08] may be interesting

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Conclusion and open questions

Open questions ISS for nonlinear hyperbolic systems. See the Lyapunov function that is derived in the current work

• f Drici, Coron, Bastin.

Applications of ISS? Does it give the offset that we have seen on an experimental channel? Offset measured on experiments in [Dos Santos, CP, 08] may be interesting

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