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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.
2/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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, fevrier 2011
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, fevrier 2011
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
5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
5/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
6/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
6/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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, fevrier 2011
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, fevrier 2011
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, fevrier 2011
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
8/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
8/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
9/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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.
10/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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.
11/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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.
11/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
12/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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.
13/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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.
13/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
14/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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.
15/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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.
15/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
16/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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]
17/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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, fevrier 2011
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, fevrier 2011
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, fevrier 2011
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, fevrier 2011
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
boundedeventually smallintegrally small
→ 0
It mainly comes from inequality:
|ξ(t)| ≤ ‖etA‖ |ξ0| + ‖B‖∫ ∞
0‖esA‖ds ‖w‖∞
19/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
boundedeventually smallintegrally small
→ 0
It mainly comes from inequality:
|ξ(t)| ≤ ‖etA‖ |ξ0| + ‖B‖∫ ∞
0‖esA‖ds ‖w‖∞
19/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
boundedeventually smallintegrally small
→ 0
It mainly comes from inequality:
|ξ(t)| ≤ ‖etA‖ |ξ0| + ‖B‖∫ ∞
0‖esA‖ds ‖w‖∞
19/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Sensitivity to large perturbations
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
20/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Sensitivity to large perturbations
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
20/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Sensitivity to large perturbations
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
20/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Sensitivity to large perturbations
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
20/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Sensitivity to large perturbations
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).
21/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
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
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) = 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
23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
∂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
23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
∂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
23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
∂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
23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
∂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
23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
∂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
23/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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 → ∞
24/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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 → ∞
24/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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 → ∞
24/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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 → ∞
24/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
25/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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)) ,
26/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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)]
27/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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)]
27/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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)]
27/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
ISS property for nonlinear parabolic equation
Theorem : [Mazenc, CP, 10]
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
∂tξ(x , t) = ∂xxξ(x , t) + f (ξ(x , t)) + w(t)
28/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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].
30/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Component ξ2 of the solution for t in [0, 10]
31/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
32/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
33/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
33/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
33/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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
33/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Conclusion and open questions
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
34/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Conclusion and open questions
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
34/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Conclusion and open questions
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
34/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
Conclusion and open questions
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
34/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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 experimentalchannel?
Other PDE?
35/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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 experimentalchannel?
Other PDE?
35/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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 experimentalchannel?
Other PDE?
35/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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 experimentalchannel?
Other PDE?
35/35 Christophe Prieur Gipsa-lab, CNRS, Grenoble GT EDP, fevrier 2011
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