Fonctions de Lyapunov pour les EDP: analyse de la stabilit ...christophe.prieur/... · Fonctions de Lyapunov pour les EDP: analyse de la stabilit´e et des perturbations Christophe

Fonctions de Lyapunov pour les EDP:

analyse de la stabilite et des perturbations

Christophe PrieurGipsa-lab, CNRS, Grenoble

GT- Controle et Problemes inverses, Fevrier 2011

1/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011

Introduction

Level and flow control in an horizontal reach of an open

channel

Control = two overflow spillways:

x

0 L

uL

QL(t)

Q0(t)

HL(t)

u0

H0(t)

Q(x, t)

H(x, t)

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


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


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


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 6⇒ 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


Outline


1.2 Related works

1.3 Applications






Conclusion


Outline


1.2 Related works

1.3 Applications






Conclusion


Outline


1.2 Related works

1.3 Applications






Conclusion


Outline


1.2 Related works

1.3 Applications






Conclusion


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. Besancon, D. Georges, 09]

Lyapunov methods [Dos Santos, Bastin, Coron, andd’Andrea-Novel, 07], [V.T. Pham, G. Besancon, D. Georges,10]

And alsonew dissipativity condition for quasi-linear hyperbolic systems[Coron, Bastin, and d’Andrea-Novel, 08]. See below.

LQ methods [Winkin, Dochain]

Backstepping transformations [Smyshlyaev, Cerpa, Krstic, 10]

among others


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. Besancon, D. Georges, 09]

Lyapunov methods [Dos Santos, Bastin, Coron, andd’Andrea-Novel, 07], [V.T. Pham, G. Besancon, D. Georges,10]

And alsonew dissipativity condition for quasi-linear hyperbolic systems[Coron, Bastin, and d’Andrea-Novel, 08]. See below.

LQ methods [Winkin, Dochain]

Backstepping transformations [Smyshlyaev, Cerpa, Krstic, 10]

among others


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


Contribution






Lyapunov method


Contribution






Lyapunov method


Related issue: Leak localization

Instead of a control problem, we may also consider an observationproblem.

Leak detection for quasi-linear system

Instead of controlling the state, we may regulate the error using asimilar approach, let us cite

In Australia: E. Weyer, I. MareelsIn France: X. Litrico, N. Bedjaoui, G. Besancon


Related issue: Leak localization

Instead of a control problem, we may also consider an observationproblem.

Leak detection for quasi-linear system

Instead of controlling the state, we may regulate the error using asimilar approach, let us cite

In Australia: E. Weyer, I. MareelsIn France: X. Litrico, N. Bedjaoui, G. Besancon


General context: non-homogeneous systems in R2

When fixing an equilibrium and using the Riemann invariantcoordinates (as in [Li, 94]), we may rewrite the previous eq. asa non-homogeneous quasi-linear hyperbolic system :Let us consider ξ: [0,L] × [0,+∞) → R

2 such that

∂tξ + Λ(ξ)∂xξ = h(ξ) (1)

where Λ: ε0B → R2×2 is a C 1 function satisfying Λ = diag(λ1, λ2),

andλ1(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’Andrea-Novel, Bastin, 03]and [Li, 94]: h ≡ 0




2 such that

∂tξ + Λ(ξ)∂xξ = h(ξ) (1)


andλ1(0) < 0 < λ2(0),


are (ξ1(L, t)ξ2(0, t)

)= g

(ξ1(0, t)ξ2(L, t)

), (2)






2 such that

∂tξ + Λ(ξ)∂xξ = h(ξ) (1)


andλ1(0) < 0 < λ2(0),


are (ξ1(L, t)ξ2(0, t)

)= g

(ξ1(0, t)ξ2(L, t)

), (2)






2 such that

∂tξ + Λ(ξ)∂xξ = h(ξ) (1)


andλ1(0) < 0 < λ2(0),


are (ξ1(L, t)ξ2(0, t)

)= g

(ξ1(0, t)ξ2(L, t)

), (2)




Definition

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

ξ01(L)

ξ02(0)

)= g

(ξ01(0)

ξ02(L)

),

and(

λ1(ξ0(L))∂xξ0

1(L) − h(ξ0(L))λ2(ξ

0(0))∂xξ02(0) − h(ξ0(0))

)

= ∇g

(ξ01(0)

ξ02(L)

) (λ1(ξ

0(0))∂xξ01(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.


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) → R

2 such that h(0) = 0 and

|∇h(0)| ≤ H , (3)

for all ξ0 ∈ BC(ε),there exists one and only one functionξ ∈ 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)|C1(0,L) ≤ Ce−µt |ξ0|C1(0,L) ,∀t ≥ 0.


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) → R

2 such that h(0) = 0 and

|∇h(0)| ≤ H , (3)

for all ξ0 ∈ BC(ε),there exists one and only one functionξ ∈ 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)|C1(0,L) ≤ Ce−µt |ξ0|C1(0,L) ,∀t ≥ 0.


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’Andrea-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


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 thesensitivity to perturbations in the sense of [CP et al, 08]?

We will come back to this latter question in a few slides.


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 thesensitivity to perturbations in the sense of [CP et al, 08]?

We will come back to this latter question in a few slides.


Experimental and numerical validation

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


Experimental validation

Here experiments on a small reach in ESISAR, Valence, FranceSome physical parameters B = 0.1m, L = 7m, I = 1.6e−4m

We may compute the following output feedback law:

U0 = H0

Q0BH0

−2√

gα0

“

√

H0−

√H0

”

µ0

√2g(zup−H(0,t))

,

UL = HL − hs −[

“

HL

h

QLBHL

+2√

gαL

“

√

HL−

√HL

”i”2

2gµ2L

]1/3

,

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


Experimental validation

Here experiments on a small reach in ESISAR, Valence, FranceSome physical parameters B = 0.1m, L = 7m, I = 1.6e−4m

We may compute the following output feedback law:

U0 = H0

Q0BH0

−2√

gα0

“

√

H0−

√H0

”

µ0

√2g(zup−H(0,t))

,

UL = HL − hs −[

“

HL

h

QLBHL

+2√

gαL

“

√

HL−

√HL

”i”2

2gµ2L

]1/3

,

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


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 5001.5

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 5001.25

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 vanishingThis offset may be canceled, by adding an integrator


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 3001

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 3001.15

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 questionThe linear case is considered in[Dos Santos, Bastin, Coron, d’Andrea-Novel, 07]The nonlinear case is considered in[Drici, Coron, preprint]


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 → ∞?






→ 0 as t → ∞?






→ 0 as t → ∞?






→ 0 as t → ∞?


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 Stabilityconsider e.g. ξ = Aξ + Bw where ξ is the state, w is theperturbations and A, B are matrices.Then if ξ = Aξ is asymp. stable then

w

boundedeventually smallintegrally small

→ 0

⇒ ξ


→ 0

It mainly comes from inequality:

|ξ(t)| ≤ ‖etA‖ |ξ0| + ‖B‖∫ ∞

0‖esA‖ds ‖w‖∞



Related question:




w


→ 0

⇒ ξ


→ 0


|ξ(t)| ≤ ‖etA‖ |ξ0| + ‖B‖∫ ∞




Related question:




w


→ 0

⇒ ξ


→ 0


|ξ(t)| ≤ ‖etA‖ |ξ0| + ‖B‖∫ ∞




False for nonlinear linear (even in finite dimension)GAS 6⇒ 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 Stableindeed with w(t) = (2t + 2)−1/2, we haveξ(t) = (2t + 2)1/2 → ∞, as t → ∞and even w ≡ 1 ⇒ ξ(t) → ∞, in finite time




ξ = −ξ + (ξ2 + 1)w





ξ = −ξ + (ξ2 + 1)w





ξ = −ξ + (ξ2 + 1)w




For infinite dimensional linear systems:

Asymp. Stability 6⇒ Input-to-State Stability

Ex:

ξn = − 1

n + 1ξn + wn

in ℓ2(N). This system is asymp. stable when wn ≡ 0 but which isnot bounded even assuming (wn(t))n∈N small in ℓ2(N).


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’Andrea-Novel, 08]

Then ∃ a sym. pos. def. Q and µ > 0 such that, lettingV (ξ) =

∫ L

0 ξ(x)⊤Qξ(x)e−µxdx we have

1α

∫ 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 function22/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011


∂tξ(x , t) + Λ∂xξ(x , t) = 0 , x ∈ [0,L], t ≥ 0


(ξ1(L, t)ξ2(0, t)

)= G

(ξ1(0, t)ξ2(L, t)

)

Assumption




∫ L


1α

∫ L

0 |ξ(x , t)|2dz ≤ V (ξ) ≤ α∫ L

0 |ξ(x , t)|2dz

V ≤ −εV




∂tξ(x , t) + Λ∂xξ(x , t) = 0 , x ∈ [0,L], t ≥ 0


(ξ1(L, t)ξ2(0, t)

)= G

(ξ1(0, t)ξ2(L, t)

)

Assumption




∫ L


1α

∫ L

0 |ξ(x , t)|2dz ≤ V (ξ) ≤ α∫ L

0 |ξ(x , t)|2dz

V ≤ −εV



∂tξ(x , t) + Λ∂xξ(x , t) = F ξ(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 thatQΛ − G⊤QΛG ≤ 0 and F⊤Q + FQ ≤ 0.

The first part of this assumption is implied by the previousassumption

Theorem : [Mazenc, CP, 10]

Then ∃ µ > 0, ε > 0 and ν > 0 such that we have

V ≤ −εV (ξ) + ν||w(t)||2





Assumption





V ≤ −εV (ξ) + ν||w(t)||2





Assumption





V ≤ −εV (ξ) + ν||w(t)||2





Assumption





V ≤ −εV (ξ) + ν||w(t)||2





Assumption





V ≤ −εV (ξ) + ν||w(t)||2





Assumption





V ≤ −εV (ξ) + ν||w(t)||2


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−tε‖ξ(., 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 → ∞



V ≤ −εV (ξ) + ν||w(t)||2




[0,t]|w(s)|

in other words



w → 0 ⇒ ξ → 0, as t → ∞



V ≤ −εV (ξ) + ν||w(t)||2




[0,t]|w(s)|

in other words



w → 0 ⇒ ξ → 0, as t → ∞



V ≤ −εV (ξ) + ν||w(t)||2




[0,t]|w(s)|

in other words



w → 0 ⇒ ξ → 0, as t → ∞


It parallels what is known for semilinear parabolic systems. Moreprecisely consider

∂tξ(z , t) = ∂xxξ(x , t) + f (ξ(x , t))

Assumption # 1

∃ a sym. pos. def. Q such that, letting V(ξ) = 12ξ⊤Qξ

−W1(ξ) := ∂ξV(ξ)f (ξ) ≤ 0

either Dirichlet conditions or the Neumann conditions orξ(0, t) = ξ(L, t) and ∂xξ(0, t) = ∂xξ(L, t)

[Krstic, Smyshlyaev, 08] and [Coron, Trelat, 04] for instance

The function V (ξ) =∫ L

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

V = −∫ L

0∂xξ(x , t)⊤Q∂xξ(x , t)dx −

∫ L

0W1(ξ(x , t))dx


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

0k(V(ξ(x)) + M(ξ(x)))dx

is a strict Lyapunov function for

∂tξ(z , t) = ∂xxξ(x , t) + f (ξ(x , t)) ,


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 > 0such that, for all ξ ∈ R

2

|∂ξM(ξ)| ≤ ca|ξ| , |∂ξξM(ξ)| ≤ cb , cc |ξ|2 ≤ [W1(ξ) + W2(ξ)]


Useful for



Assumption #3

∃ a C 2 M : R2 → R≥0 such that M(0) = 0,


2



Useful for



Assumption #3

∃ a C 2 M : R2 → R≥0 such that M(0) = 0,


2



ISS property for nonlinear parabolic equation


Assume that Assumptions #1 and #3 with the periodic boundaryconditions

ξ(L, t) = ξ(0, t) and ∂xξ(L, t) = ∂xξ(0, t) , ∀t ≥ 0 .

Then, ∃ K > 0 such that

V (ξ) =

∫ L

0[KV(ξ(x)) + M(ξ(x))]dx

is an ISS Lyapunov function for



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 + ξ22]

andM(Ξ) = ξ2

1 + ξ22 + ξ1ξ2 ,

Therefore, with the Dirichlet boundary conditions, the function

V (φ) = 1153

∫ L

0

[φ1(z)2 + φ2(z)2

]dz +

∫ L

0φ1(z)φ2(z)dz

is an ISS Lyapunov function for the system (4).29/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011

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].


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


0 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


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 notconverge to the equilibrium

Applications of the stability result only!

numerical simulations on real data

experiments on a set-up






























2 Sensitivity of stable non-homogeneous linear system wrtperturbations

perturbations are bounded ⇒ state is bounded

Input-to-State Stability

Study of stable semilinear parabolic systems wrtperturbations

Study of stable linear hyperbolic systems wrt perturbations
























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 experimentalchannel?

Other PDE?



Open questions


and also


Other PDE?



Open questions


and also


Other PDE?



Open questions


and also


Other PDE?


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Fonctions de Lyapunov pour les EDP: analyse de la stabilit ...christophe.prieur/... · Fonctions de Lyapunov pour les EDP: analyse de la stabilit´e et des perturbations Christophe