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Fonctions de Lyapunov pour les EDP: analyse de la stabilit e et des perturbations Christophe Prieur Gipsa-lab, CNRS, Grenoble GT- Contr ole et Probl` emes inverses, F evrier 2011 1/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT

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SLIDE 1

Fonctions de Lyapunov pour les EDP: analyse de la stabilit´ e et des perturbations

Christophe Prieur Gipsa-lab, CNRS, Grenoble GT- Contrˆ

  • le et Probl`

emes inverses, F´ evrier 2011

1/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 2

Introduction

Level and flow control in an horizontal reach of an open channel Control = two overflow spillways:

!

! "

#"

$"%&' $!%&' ("%&'

#!

(!%&'

Q(x, t) H(x, t)

where H(x, t) is the water level and Q(x, t) the water flow rate in the reach.

2/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Shallow Water Equations

Model [Chow, 54] or [Graf, 98]: mass conservation ∂tH(x, t) + ∂x Q(x, t) B

  • = q(x)

momentum conservation ∂tQ(x, t) + ∂x Q2(x, t) BH(x, t) + gBH2(x, t) 2

  • = gBH(I − J) + kq Q

BH where g and B are constant values q the water supply/removal function I is the bottom slope J(Q, H) =

n2

MQ2

S(H)2R(H)4/3 is the slope’s friction

3/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Motivations

Problem Compute the positions u0 and uL of the spillways s.t. the control actions depend only on the (measured) H(0, t) and H(L, t) ∃ a solution of our model (PDE) state →t→+∞ equilibrium stability properties even in presence of perturbations I, J and q

4/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Outline

1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE

  • Asymp. Stability ⇒ Input-to State Stability

2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion

5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Outline

1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE

  • Asymp. Stability ⇒ Input-to State Stability

2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion

5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Outline

1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE

  • Asymp. Stability ⇒ Input-to State Stability

2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion

5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Outline

1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE

  • Asymp. Stability ⇒ Input-to State Stability

2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion

5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Outline

1.1 Stability analysis of hyperbolic non-homogeneous systems 1.2 Related works 1.3 Applications 2 Sensitivity with respect to large perturbations Notion of ISS Lyapunov functions for PDE

  • Asymp. Stability ⇒ Input-to State Stability

2.1 An ISS Lyapunov function for hyperbolic linear systems 2.2 An ISS Lyapunov function for semilinear parabolic systems Conclusion

5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Many works in the literature

For a survey, see [Malaterre, Rogers, and Schuurmans, 98]. Finite dimensional approach: H∞ control design is developed in [Litrico, and Georges, 99]. Infinite dimensional approach: Delay-based control [G. Besan¸ con, D. Georges, 09] Lyapunov methods [Dos Santos, Bastin, Coron, and d’Andr´ ea-Novel, 07], [V.T. Pham, G. Besan¸ con, D. Georges, 10] And also new dissipativity condition for quasi-linear hyperbolic systems [Coron, Bastin, and d’Andr´ ea-Novel, 08]. See below. LQ methods [Winkin, Dochain] Backstepping transformations [Smyshlyaev, Cerpa, Krstic, 10] among others

6/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Many works in the literature

For a survey, see [Malaterre, Rogers, and Schuurmans, 98]. Finite dimensional approach: H∞ control design is developed in [Litrico, and Georges, 99]. Infinite dimensional approach: Delay-based control [G. Besan¸ con, D. Georges, 09] Lyapunov methods [Dos Santos, Bastin, Coron, and d’Andr´ ea-Novel, 07], [V.T. Pham, G. Besan¸ con, D. Georges, 10] And also new dissipativity condition for quasi-linear hyperbolic systems [Coron, Bastin, and d’Andr´ ea-Novel, 08]. See below. LQ methods [Winkin, Dochain] Backstepping transformations [Smyshlyaev, Cerpa, Krstic, 10] among others

6/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Contribution

Here: Perturbations are taken into account

  • asymp. stability when perturbations are vanishing

bounded state with bounded perturbations Methods that are used Riemann invariants [Li Ta-tsien, 94] Lyapunov method

7/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Contribution

Here: Perturbations are taken into account

  • asymp. stability when perturbations are vanishing

bounded state with bounded perturbations Methods that are used Riemann invariants [Li Ta-tsien, 94] Lyapunov method

7/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Contribution

Here: Perturbations are taken into account

  • asymp. stability when perturbations are vanishing

bounded state with bounded perturbations Methods that are used Riemann invariants [Li Ta-tsien, 94] Lyapunov method

7/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Related issue: Leak localization

Instead of a control problem, we may also consider an observation problem. Leak detection for quasi-linear system Instead of controlling the state, we may regulate the error using a similar approach, let us cite In Australia: E. Weyer, I. Mareels In France: X. Litrico, N. Bedjaoui, G. Besan¸ con

8/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Related issue: Leak localization

Instead of a control problem, we may also consider an observation problem. Leak detection for quasi-linear system Instead of controlling the state, we may regulate the error using a similar approach, let us cite In Australia: E. Weyer, I. Mareels In France: X. Litrico, N. Bedjaoui, G. Besan¸ con

8/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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General context: non-homogeneous systems in R2

When fixing an equilibrium and using the Riemann invariant coordinates (as in [Li, 94]), we may rewrite the previous eq. as a non-homogeneous quasi-linear hyperbolic system : Let us consider ξ: [0, L] × [0, +∞) → R2 such that ∂tξ + Λ(ξ)∂xξ = h(ξ) (1) where Λ: ε0B → R2×2 is a C 1 function satisfying Λ = diag(λ1, λ2), and λ1(0) < 0 < λ2(0), and h : ε0B → R2 is C 1 s.t. h(0) = 0 . The boundary conditions are ξ1(L, t) ξ2(0, t)

  • = g

ξ1(0, t) ξ2(L, t)

  • ,

(2) where g: ε0B → R2 is C 1 s.t. g(0) = 0. In [de Halleux, CP, Coron, d’Andr´ ea-Novel, Bastin, 03] and [Li, 94]: h ≡ 0

9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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General context: non-homogeneous systems in R2

When fixing an equilibrium and using the Riemann invariant coordinates (as in [Li, 94]), we may rewrite the previous eq. as a non-homogeneous quasi-linear hyperbolic system : Let us consider ξ: [0, L] × [0, +∞) → R2 such that ∂tξ + Λ(ξ)∂xξ = h(ξ) (1) where Λ: ε0B → R2×2 is a C 1 function satisfying Λ = diag(λ1, λ2), and λ1(0) < 0 < λ2(0), and h : ε0B → R2 is C 1 s.t. h(0) = 0 . The boundary conditions are ξ1(L, t) ξ2(0, t)

  • = g

ξ1(0, t) ξ2(L, t)

  • ,

(2) where g: ε0B → R2 is C 1 s.t. g(0) = 0. In [de Halleux, CP, Coron, d’Andr´ ea-Novel, Bastin, 03] and [Li, 94]: h ≡ 0

9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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General context: non-homogeneous systems in R2

When fixing an equilibrium and using the Riemann invariant coordinates (as in [Li, 94]), we may rewrite the previous eq. as a non-homogeneous quasi-linear hyperbolic system : Let us consider ξ: [0, L] × [0, +∞) → R2 such that ∂tξ + Λ(ξ)∂xξ = h(ξ) (1) where Λ: ε0B → R2×2 is a C 1 function satisfying Λ = diag(λ1, λ2), and λ1(0) < 0 < λ2(0), and h : ε0B → R2 is C 1 s.t. h(0) = 0 . The boundary conditions are ξ1(L, t) ξ2(0, t)

  • = g

ξ1(0, t) ξ2(L, t)

  • ,

(2) where g: ε0B → R2 is C 1 s.t. g(0) = 0. In [de Halleux, CP, Coron, d’Andr´ ea-Novel, Bastin, 03] and [Li, 94]: h ≡ 0

9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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General context: non-homogeneous systems in R2

When fixing an equilibrium and using the Riemann invariant coordinates (as in [Li, 94]), we may rewrite the previous eq. as a non-homogeneous quasi-linear hyperbolic system : Let us consider ξ: [0, L] × [0, +∞) → R2 such that ∂tξ + Λ(ξ)∂xξ = h(ξ) (1) where Λ: ε0B → R2×2 is a C 1 function satisfying Λ = diag(λ1, λ2), and λ1(0) < 0 < λ2(0), and h : ε0B → R2 is C 1 s.t. h(0) = 0 . The boundary conditions are ξ1(L, t) ξ2(0, t)

  • = g

ξ1(0, t) ξ2(L, t)

  • ,

(2) where g: ε0B → R2 is C 1 s.t. g(0) = 0. In [de Halleux, CP, Coron, d’Andr´ ea-Novel, Bastin, 03] and [Li, 94]: h ≡ 0

9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Definition A function ξ# ∈ C 1(0, L; R2) satisfies the compatibility condition C if ξ0

1(L)

ξ0

2(0)

  • = g

ξ0

1(0)

ξ0

2(L)

  • ,

and λ1(ξ0(L))∂xξ0

1(L) − h(ξ0(L))

λ2(ξ0(0))∂xξ0

2(0) − h(ξ0(0))

  • = ∇g

ξ0

1(0)

ξ0

2(L)

λ1(ξ0(0))∂xξ0

1(0) − h(ξ0(0))

λ2(ξ0(L))∂xξ0

2(L) − h(ξ0(L))

  • .

We denote by BC(ε0) the set of C1-functions ξ#: [0.L] → B(ε0) satisfying the compatibility assumption C.

10/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Stability analysis

Theorem [CP, Winkin, Bastin, 08] If ρ(∇g(0)) < 1, then there exist ε > 0, and H > 0 such that, for all C1-functions h : B(ε) → R2 such that h(0) = 0 and |∇h(0)| ≤ H , (3) for all ξ0 ∈ BC(ε), there exists one and only one function ξ ∈ C 1([0, L] × [0, +∞) ; R2) satisfying (1), (2) and ξ(x, 0) = ξ0(x) , ∀x ∈ [0, L]. Moreover, there exist µ > 0 and C > 0 such that |ξ(., t)|C 1(0,L) ≤ Ce−µt|ξ0|C 1(0,L) , ∀t ≥ 0.

11/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Stability analysis

Theorem [CP, Winkin, Bastin, 08] If ρ(∇g(0)) < 1, then there exist ε > 0, and H > 0 such that, for all C1-functions h : B(ε) → R2 such that h(0) = 0 and |∇h(0)| ≤ H , (3) for all ξ0 ∈ BC(ε), there exists one and only one function ξ ∈ C 1([0, L] × [0, +∞) ; R2) satisfying (1), (2) and ξ(x, 0) = ξ0(x) , ∀x ∈ [0, L]. Moreover, there exist µ > 0 and C > 0 such that |ξ(., t)|C 1(0,L) ≤ Ce−µt|ξ0|C 1(0,L) , ∀t ≥ 0.

11/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Other damping condition

For all K ∈ R2×2, K = max{|Kx|, x ∈ R2, |x| = 1} ρ1(K) = inf{∆K∆−1, ∆ ∈ D2,+} Theorem : [Coron, Bastin, d’Andr´ ea-Novel, 08] If ρ1(∇g(0)) < 1 then the non-perturbed system ∂tξ + Λ(ξ)∂xξ = 0 is exponential stable for the H2-norm. Weaker condition than [Ta-tsien Li, 94]’s condition: ρ(∇g(0)) < 1

12/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Open question: Exponential stability for the C 1-norm ⇐ ⇒ Exp. stability for the H2-norm? Natural research line Can we use the Lyapunov of [Coron et al, 08] to estimate the sensitivity to perturbations in the sense of [CP et al, 08]? We will come back to this latter question in a few slides.

13/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Open question: Exponential stability for the C 1-norm ⇐ ⇒ Exp. stability for the H2-norm? Natural research line Can we use the Lyapunov of [Coron et al, 08] to estimate the sensitivity to perturbations in the sense of [CP et al, 08]? We will come back to this latter question in a few slides.

13/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Experimental and numerical validation

Two different applications in [Dos Santos, CP, 08]: Numerical validation simulating one reach on the Sambre river between Charleroi and Namur: B = 40m, L = 11239m, I = 7.92e−5

14/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Experimental validation

Here experiments on a small reach in ESISAR, Valence, France Some physical parameters B = 0.1m, L = 7m, I = 1.6e−4m We may compute the following output feedback law:

U0 = H0

¯ Q0 B ¯ H0 −2√gα0

“√H0−√ ¯ H0 ” µ0√ 2g(zup−H(0,t))

, UL = HL − hs − “

HL h ¯

QL B ¯ HL +2√gαL

“√HL−√ ¯ HL ”i”2 2gµ2

L

1/3 ,

such that the closed-loop system in locally exponentially stable for the C 1-norm.

15/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Experimental validation

Here experiments on a small reach in ESISAR, Valence, France Some physical parameters B = 0.1m, L = 7m, I = 1.6e−4m We may compute the following output feedback law:

U0 = H0

¯ Q0 B ¯ H0 −2√gα0

“√H0−√ ¯ H0 ” µ0√ 2g(zup−H(0,t))

, UL = HL − hs − “

HL h ¯

QL B ¯ HL +2√gαL

“√HL−√ ¯ HL ”i”2 2gµ2

L

1/3 ,

such that the closed-loop system in locally exponentially stable for the C 1-norm.

15/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Experimental results

(E1) k0 = −0.0853, kL = −0.463, (k0kL = 0.0395); (E2) k0 = −0.2134, kL = −1.1575, (k0kL = 0.247); (E3) k0 = −0.3414, kL = −1.852, (k0kL = 0.6322).

50 100 150 200 250 300 350 400 450 500 1.5 2 2.5 3 t (s) (dm3.s−1) Upstream water flow (E1) (E2) (E3) equilibrium 50 100 150 200 250 300 350 400 450 500 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 t (s) (dm) Downstream water level

Small offset in the asymptotic value. Indeed at the equilibrium, the perturbations are not vanishing This offset may be canceled, by adding an integrator

16/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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With an integral action

(EI1) without an integral action (EI1=E2) (EI2) with a small integral action (to cancel the offset) (EI3) with a larger integral action, but in presence of an overshoot

50 100 150 200 250 300 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 t(s) (dm3.s−1) Upstream water flow (EI1) (EI2) (EI3) equilibrium 50 100 150 200 250 300 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 t (s) (dm) Downstream water level equilibrium (EI1) (EI2) (EI3)

The stability with an integral action is still an open question The linear case is considered in [Dos Santos, Bastin, Coron, d’Andr´ ea-Novel, 07] The nonlinear case is considered in [Drici, Coron, preprint]

17/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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  • 2. Sensitivity to larger perturbations

What can be done when perturbations are not vanishing at the equilibrium? are only bounded? in L∞(0, T)? → 0 as t → ∞?

18/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 33
  • 2. Sensitivity to larger perturbations

What can be done when perturbations are not vanishing at the equilibrium? are only bounded? in L∞(0, T)? → 0 as t → ∞?

18/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 34
  • 2. Sensitivity to larger perturbations

What can be done when perturbations are not vanishing at the equilibrium? are only bounded? in L∞(0, T)? → 0 as t → ∞?

18/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 35
  • 2. Sensitivity to larger perturbations

What can be done when perturbations are not vanishing at the equilibrium? are only bounded? in L∞(0, T)? → 0 as t → ∞?

18/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 36

Sensitivity to large perturbations

Related question: Question: for an asymptotically stable system Do bounded perturbations result bounded states? Stronger notion than the Asymptotic Stability: Input-to-State Stability consider e.g. ˙ ξ = Aξ + Bw where ξ is the state, w is the perturbations and A, B are matrices. Then if ˙ ξ = Aξ is asymp. stable then w        bounded eventually small integrally small → 0        ⇒ ξ        bounded eventually small integrally small → 0        It mainly comes from inequality: |ξ(t)| ≤ etA |ξ0| + B ∞ esAds w∞

19/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 37

Sensitivity to large perturbations

Related question: Question: for an asymptotically stable system Do bounded perturbations result bounded states? Stronger notion than the Asymptotic Stability: Input-to-State Stability consider e.g. ˙ ξ = Aξ + Bw where ξ is the state, w is the perturbations and A, B are matrices. Then if ˙ ξ = Aξ is asymp. stable then w        bounded eventually small integrally small → 0        ⇒ ξ        bounded eventually small integrally small → 0        It mainly comes from inequality: |ξ(t)| ≤ etA |ξ0| + B ∞ esAds w∞

19/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 38

Sensitivity to large perturbations

Related question: Question: for an asymptotically stable system Do bounded perturbations result bounded states? Stronger notion than the Asymptotic Stability: Input-to-State Stability consider e.g. ˙ ξ = Aξ + Bw where ξ is the state, w is the perturbations and A, B are matrices. Then if ˙ ξ = Aξ is asymp. stable then w        bounded eventually small integrally small → 0        ⇒ ξ        bounded eventually small integrally small → 0        It mainly comes from inequality: |ξ(t)| ≤ etA |ξ0| + B ∞ esAds w∞

19/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 39

Sensitivity to large perturbations

False for nonlinear linear (even in finite dimension) GAS ⇒ ISS for nonlinear systems. Indeed [Sontag, survey on the ISS property, 06] ˙ ξ = −ξ + (ξ2 + 1)w Globally asymp. stable (with w ≡ 0) but not Input-to-State Stable indeed with w(t) = (2t + 2)−1/2, we have ξ(t) = (2t + 2)1/2 → ∞, as t → ∞ and even w ≡ 1 ⇒ ξ(t) → ∞, in finite time

20/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 40

Sensitivity to large perturbations

False for nonlinear linear (even in finite dimension) GAS ⇒ ISS for nonlinear systems. Indeed [Sontag, survey on the ISS property, 06] ˙ ξ = −ξ + (ξ2 + 1)w Globally asymp. stable (with w ≡ 0) but not Input-to-State Stable indeed with w(t) = (2t + 2)−1/2, we have ξ(t) = (2t + 2)1/2 → ∞, as t → ∞ and even w ≡ 1 ⇒ ξ(t) → ∞, in finite time

20/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 41

Sensitivity to large perturbations

False for nonlinear linear (even in finite dimension) GAS ⇒ ISS for nonlinear systems. Indeed [Sontag, survey on the ISS property, 06] ˙ ξ = −ξ + (ξ2 + 1)w Globally asymp. stable (with w ≡ 0) but not Input-to-State Stable indeed with w(t) = (2t + 2)−1/2, we have ξ(t) = (2t + 2)1/2 → ∞, as t → ∞ and even w ≡ 1 ⇒ ξ(t) → ∞, in finite time

20/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

slide-42
SLIDE 42

Sensitivity to large perturbations

False for nonlinear linear (even in finite dimension) GAS ⇒ ISS for nonlinear systems. Indeed [Sontag, survey on the ISS property, 06] ˙ ξ = −ξ + (ξ2 + 1)w Globally asymp. stable (with w ≡ 0) but not Input-to-State Stable indeed with w(t) = (2t + 2)−1/2, we have ξ(t) = (2t + 2)1/2 → ∞, as t → ∞ and even w ≡ 1 ⇒ ξ(t) → ∞, in finite time

20/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Sensitivity to large perturbations

For infinite dimensional linear systems:

  • Asymp. Stability ⇒ Input-to-State Stability

Ex: ˙ ξn = − 1 n + 1ξn + wn in ℓ2(N). This system is asymp. stable when wn ≡ 0 but which is not bounded even assuming (wn(t))n∈N small in ℓ2(N).

21/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 44

Let us consider a linear hyperbolic system: ∂tξ(x, t) + Λ∂xξ(x, t) = 0 , x ∈ [0, L], t ≥ 0 with the boundary condition ξ1(L, t) ξ2(0, t)

  • = G

ξ1(0, t) ξ2(L, t)

  • Assumption

The boundary condition is such that ρ1(G) < 1 Theorem: [Coron, Bastin, d’Andr´ ea-Novel, 08] Then ∃ a sym. pos. def. Q and µ > 0 such that, letting V (ξ) = L

0 ξ(x)⊤Qξ(x)e−µxdx we have 1 α

L

0 |ξ(x, t)|2dz ≤ V (ξ) ≤ α

L

0 |ξ(x, t)|2dz

˙ V ≤ −εV for α > 0 and ε > 0 sufficiently small. In other words, V is a strict Lyapunov function

22/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 45

Let us consider a linear hyperbolic system: ∂tξ(x, t) + Λ∂xξ(x, t) = 0 , x ∈ [0, L], t ≥ 0 with the boundary condition ξ1(L, t) ξ2(0, t)

  • = G

ξ1(0, t) ξ2(L, t)

  • Assumption

The boundary condition is such that ρ1(G) < 1 Theorem: [Coron, Bastin, d’Andr´ ea-Novel, 08] Then ∃ a sym. pos. def. Q and µ > 0 such that, letting V (ξ) = L

0 ξ(x)⊤Qξ(x)e−µxdx we have 1 α

L

0 |ξ(x, t)|2dz ≤ V (ξ) ≤ α

L

0 |ξ(x, t)|2dz

˙ V ≤ −εV for α > 0 and ε > 0 sufficiently small. In other words, V is a strict Lyapunov function

22/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 46

Let us consider a linear hyperbolic system: ∂tξ(x, t) + Λ∂xξ(x, t) = 0 , x ∈ [0, L], t ≥ 0 with the boundary condition ξ1(L, t) ξ2(0, t)

  • = G

ξ1(0, t) ξ2(L, t)

  • Assumption

The boundary condition is such that ρ1(G) < 1 Theorem: [Coron, Bastin, d’Andr´ ea-Novel, 08] Then ∃ a sym. pos. def. Q and µ > 0 such that, letting V (ξ) = L

0 ξ(x)⊤Qξ(x)e−µxdx we have 1 α

L

0 |ξ(x, t)|2dz ≤ V (ξ) ≤ α

L

0 |ξ(x, t)|2dz

˙ V ≤ −εV for α > 0 and ε > 0 sufficiently small. In other words, V is a strict Lyapunov function

22/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 47

∂tξ(x, t) + Λ∂xξ(x, t) = Fξ(z, t) + w(t) , x ∈ [0, L], t ≥ 0 where w is a perturbation, F is constant and known in R2×2. Same boundary conditions Assumption ∃ a pos. def. matrix Q such that QΛ − G ⊤QΛG ≤ 0 and F ⊤Q + FQ ≤ 0. The first part of this assumption is implied by the previous assumption Theorem : [Mazenc, CP, 10] Then ∃ µ > 0, ε > 0 and ν > 0 such that we have ˙ V ≤ −εV (ξ) + ν||w(t)||2

23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 48

∂tξ(x, t) + Λ∂xξ(x, t) = Fξ(z, t) + w(t) , x ∈ [0, L], t ≥ 0 where w is a perturbation, F is constant and known in R2×2. Same boundary conditions Assumption ∃ a pos. def. matrix Q such that QΛ − G ⊤QΛG ≤ 0 and F ⊤Q + FQ ≤ 0. The first part of this assumption is implied by the previous assumption Theorem : [Mazenc, CP, 10] Then ∃ µ > 0, ε > 0 and ν > 0 such that we have ˙ V ≤ −εV (ξ) + ν||w(t)||2

23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 49

∂tξ(x, t) + Λ∂xξ(x, t) = Fξ(z, t) + w(t) , x ∈ [0, L], t ≥ 0 where w is a perturbation, F is constant and known in R2×2. Same boundary conditions Assumption ∃ a pos. def. matrix Q such that QΛ − G ⊤QΛG ≤ 0 and F ⊤Q + FQ ≤ 0. The first part of this assumption is implied by the previous assumption Theorem : [Mazenc, CP, 10] Then ∃ µ > 0, ε > 0 and ν > 0 such that we have ˙ V ≤ −εV (ξ) + ν||w(t)||2

23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 50

∂tξ(x, t) + Λ∂xξ(x, t) = Fξ(z, t) + w(t) , x ∈ [0, L], t ≥ 0 where w is a perturbation, F is constant and known in R2×2. Same boundary conditions Assumption ∃ a pos. def. matrix Q such that QΛ − G ⊤QΛG ≤ 0 and F ⊤Q + FQ ≤ 0. The first part of this assumption is implied by the previous assumption Theorem : [Mazenc, CP, 10] Then ∃ µ > 0, ε > 0 and ν > 0 such that we have ˙ V ≤ −εV (ξ) + ν||w(t)||2

23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 51

∂tξ(x, t) + Λ∂xξ(x, t) = Fξ(z, t) + w(t) , x ∈ [0, L], t ≥ 0 where w is a perturbation, F is constant and known in R2×2. Same boundary conditions Assumption ∃ a pos. def. matrix Q such that QΛ − G ⊤QΛG ≤ 0 and F ⊤Q + FQ ≤ 0. The first part of this assumption is implied by the previous assumption Theorem : [Mazenc, CP, 10] Then ∃ µ > 0, ε > 0 and ν > 0 such that we have ˙ V ≤ −εV (ξ) + ν||w(t)||2

23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 52

∂tξ(x, t) + Λ∂xξ(x, t) = Fξ(z, t) + w(t) , x ∈ [0, L], t ≥ 0 where w is a perturbation, F is constant and known in R2×2. Same boundary conditions Assumption ∃ a pos. def. matrix Q such that QΛ − G ⊤QΛG ≤ 0 and F ⊤Q + FQ ≤ 0. The first part of this assumption is implied by the previous assumption Theorem : [Mazenc, CP, 10] Then ∃ µ > 0, ε > 0 and ν > 0 such that we have ˙ V ≤ −εV (ξ) + ν||w(t)||2

23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 53

ISS Lyapunov function for hyperbolic systems

˙ V ≤ −εV (ξ) + ν||w(t)||2 V is called an ISS Lyapunov function. Indeed this implies exponential stability when w ≡ 0 ξ(., t)L2(0,L) ≤ C1e−tεξ(., 0)L2(0,L) + C2 sup

[0,t]

|w(s)| in other words w bounded ⇒ ξ bounded similarly we may prove w → 0 ⇒ ξ → 0, as t → ∞

24/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 54

ISS Lyapunov function for hyperbolic systems

˙ V ≤ −εV (ξ) + ν||w(t)||2 V is called an ISS Lyapunov function. Indeed this implies exponential stability when w ≡ 0 ξ(., t)L2(0,L) ≤ C1e−tεξ(., 0)L2(0,L) + C2 sup

[0,t]

|w(s)| in other words w bounded ⇒ ξ bounded similarly we may prove w → 0 ⇒ ξ → 0, as t → ∞

24/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 55

ISS Lyapunov function for hyperbolic systems

˙ V ≤ −εV (ξ) + ν||w(t)||2 V is called an ISS Lyapunov function. Indeed this implies exponential stability when w ≡ 0 ξ(., t)L2(0,L) ≤ C1e−tεξ(., 0)L2(0,L) + C2 sup

[0,t]

|w(s)| in other words w bounded ⇒ ξ bounded similarly we may prove w → 0 ⇒ ξ → 0, as t → ∞

24/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 56

ISS Lyapunov function for hyperbolic systems

˙ V ≤ −εV (ξ) + ν||w(t)||2 V is called an ISS Lyapunov function. Indeed this implies exponential stability when w ≡ 0 ξ(., t)L2(0,L) ≤ C1e−tεξ(., 0)L2(0,L) + C2 sup

[0,t]

|w(s)| in other words w bounded ⇒ ξ bounded similarly we may prove w → 0 ⇒ ξ → 0, as t → ∞

24/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 57

It parallels what is known for semilinear parabolic systems. More precisely consider ∂tξ(z, t) = ∂xxξ(x, t) + f (ξ(x, t)) Assumption # 1 ∃ a sym. pos. def. Q such that, letting V(ξ) = 1

2ξ⊤Qξ

−W1(ξ) := ∂ξV(ξ)f (ξ) ≤ 0 either Dirichlet conditions or the Neumann conditions or ξ(0, t) = ξ(L, t) and ∂xξ(0, t) = ∂xξ(L, t) [Krstic, Smyshlyaev, 08] and [Coron, Tr´ elat, 04] for instance The function V (ξ) = L

0 V(ξ(x))dx is a weak Lyapunov function:

˙ V = − L ∂xξ(x, t)⊤Q∂xξ(x, t)dx − L W1(ξ(x, t))dx

25/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 58

Assumption # 2 ∃ ca > 0, cb > 0, a C 2 M : R2 → R≥0, M(0) = 0 and ∂ξM(0) = 0, and a C 0 W2 : Rn → R≥0 such that W1 + W2 is pos. def. and ∂ξM(ξ)f (ξ) ≤ −W2(ξ) , |∂ξξM(ξ)| ≤ ca , ∀ξ ∈ R2 , W1(ξ) + W2(ξ) ≥ cb|ξ|2 , ∀ξ ∈ R2 : |ξ| ≤ 1 Theorem [Mazenc, CP, 10] Then ∃ a def. pos. function k : R → R such that V (ξ) = L k(V(ξ(x)) + M(ξ(x)))dx is a strict Lyapunov function for ∂tξ(z, t) = ∂xxξ(x, t) + f (ξ(x, t)) ,

26/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Useful for ∂tξ(x, t) = ∂xxξ(x, t) + f (ξ(x, t)) + w(t) where w is an unknown continuous function. Assumption #3 ∃ a C 2 M : R2 → R≥0 such that M(0) = 0, −∂ξM(ξ)f (ξ) =: W2(ξ) ≥ 0, and ∃ ca > 0, cb > 0 and cc > 0 such that, for all ξ ∈ R2 |∂ξM(ξ)| ≤ ca|ξ| , |∂ξξM(ξ)| ≤ cb , cc|ξ|2 ≤ [W1(ξ) + W2(ξ)]

27/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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SLIDE 60

Useful for ∂tξ(x, t) = ∂xxξ(x, t) + f (ξ(x, t)) + w(t) where w is an unknown continuous function. Assumption #3 ∃ a C 2 M : R2 → R≥0 such that M(0) = 0, −∂ξM(ξ)f (ξ) =: W2(ξ) ≥ 0, and ∃ ca > 0, cb > 0 and cc > 0 such that, for all ξ ∈ R2 |∂ξM(ξ)| ≤ ca|ξ| , |∂ξξM(ξ)| ≤ cb , cc|ξ|2 ≤ [W1(ξ) + W2(ξ)]

27/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Useful for ∂tξ(x, t) = ∂xxξ(x, t) + f (ξ(x, t)) + w(t) where w is an unknown continuous function. Assumption #3 ∃ a C 2 M : R2 → R≥0 such that M(0) = 0, −∂ξM(ξ)f (ξ) =: W2(ξ) ≥ 0, and ∃ ca > 0, cb > 0 and cc > 0 such that, for all ξ ∈ R2 |∂ξM(ξ)| ≤ ca|ξ| , |∂ξξM(ξ)| ≤ cb , cc|ξ|2 ≤ [W1(ξ) + W2(ξ)]

27/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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ISS property for nonlinear parabolic equation

Theorem : [Mazenc, CP, 10] Assume that Assumptions #1 and #3 with the periodic boundary conditions ξ(L, t) = ξ(0, t) and ∂xξ(L, t) = ∂xξ(0, t) , ∀t ≥ 0 . Then, ∃ K > 0 such that

  • V (ξ) =

L [KV(ξ(x)) + M(ξ(x))]dx is an ISS Lyapunov function for ∂tξ(x, t) = ∂xxξ(x, t) + f (ξ(x, t)) + w(t)

28/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Numerical simulations for a semilinear parabolic equation

        

∂ξ1 ∂t (z, t)

=

∂2ξ1 ∂z2 (z, t) − ∂ξ1 ∂z (z, t)

+ξ2(z, t)[1 + ξ1(z, t)2] + w1(z, t)

∂ξ2 ∂t (z, t)

=

∂2ξ2 ∂z2 (z, t) − ξ1(z, t)[1 + ξ1(z, t)2]

−ξ2(z, t)[2 + ξ1(z, t)2] + w2(z, t) (4) Two heat equations with a convection term in the first. Assumptions # 1, # 2, and # 3 hold with V(Ξ) = 1 2[ξ2

1 + ξ2 2]

and M(Ξ) = ξ2

1 + ξ2 2 + ξ1ξ2 ,

Therefore, with the Dirichlet boundary conditions, the function ˜ V (φ) = 1153 L

  • φ1(z)2 + φ2(z)2

dz + L φ1(z)φ2(z)dz is an ISS Lyapunov function for the system (4).

29/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Numerical scheme so that the CFL condition for the stability holds. w1(z, t) = sin2(t), w2(z, t) = 0 ∀z ∈ [0, L], and ∀t ∈ [0, 5] w1(z, t) = w2(z, t) = 0 ∀z ∈ [0, L] and ∀t ∈ (5, 10].

30/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Component ξ2 of the solution for t in [0, 10]

31/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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1 2 3 4 5 6 7 8 9 10 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 t U

Time-evolution of the function ˜ V

32/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

We have considered two problems 1 Stability analysis of nonlinear hyperbolic system using a Lyapunov approach [Coron et al, 08] in presence of perturbations then the solutions do not converge to the equilibrium Applications of the stability result only! numerical simulations on real data experiments on a set-up

33/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

We have considered two problems 1 Stability analysis of nonlinear hyperbolic system using a Lyapunov approach [Coron et al, 08] in presence of perturbations then the solutions do not converge to the equilibrium Applications of the stability result only! numerical simulations on real data experiments on a set-up

33/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

We have considered two problems 1 Stability analysis of nonlinear hyperbolic system using a Lyapunov approach [Coron et al, 08] in presence of perturbations then the solutions do not converge to the equilibrium Applications of the stability result only! numerical simulations on real data experiments on a set-up

33/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

We have considered two problems 1 Stability analysis of nonlinear hyperbolic system using a Lyapunov approach [Coron et al, 08] in presence of perturbations then the solutions do not converge to the equilibrium Applications of the stability result only! numerical simulations on real data experiments on a set-up

33/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

2 Sensitivity of stable non-homogeneous linear system wrt perturbations perturbations are bounded ⇒ state is bounded Input-to-State Stability Study of stable semilinear parabolic systems wrt perturbations Study of stable linear hyperbolic systems wrt perturbations

34/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

2 Sensitivity of stable non-homogeneous linear system wrt perturbations perturbations are bounded ⇒ state is bounded Input-to-State Stability Study of stable semilinear parabolic systems wrt perturbations Study of stable linear hyperbolic systems wrt perturbations

34/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

2 Sensitivity of stable non-homogeneous linear system wrt perturbations perturbations are bounded ⇒ state is bounded Input-to-State Stability Study of stable semilinear parabolic systems wrt perturbations Study of stable linear hyperbolic systems wrt perturbations

34/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

2 Sensitivity of stable non-homogeneous linear system wrt perturbations perturbations are bounded ⇒ state is bounded Input-to-State Stability Study of stable semilinear parabolic systems wrt perturbations Study of stable linear hyperbolic systems wrt perturbations

34/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

Open questions ISS for nonlinear hyperbolic systems. We are working on that! and also Applications of ISS? Does it give the offset that we have seen on the experimental channel? Other PDE?

35/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

Open questions ISS for nonlinear hyperbolic systems. We are working on that! and also Applications of ISS? Does it give the offset that we have seen on the experimental channel? Other PDE?

35/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

Open questions ISS for nonlinear hyperbolic systems. We are working on that! and also Applications of ISS? Does it give the offset that we have seen on the experimental channel? Other PDE?

35/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011

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Conclusion and open questions

Open questions ISS for nonlinear hyperbolic systems. We are working on that! and also Applications of ISS? Does it give the offset that we have seen on the experimental channel? Other PDE?

35/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, f´ evrier 2011