Some results on the stabilization and on the controllability of - - PowerPoint PPT Presentation

Some results on the stabilization and on the controllability of nonlinear wave equations

Jean-Michel Coron

http://www.ann.jussieu.fr/~coron/

CESAME, Louvain-la-Neuve, October 21, 2008

Outline

Stabilization: Dissipative boundary conditions for hyperbolic equations The equations Main result Comparaison with prior results Proof of the exponential stability Application to the control of open channels

Outline

Stabilization: Dissipative boundary conditions for hyperbolic equations The equations Main result Comparaison with prior results Proof of the exponential stability Application to the control of open channels Controllability of hyperbolic systems The control problem Controllability theorem

Outline

Stabilization: Dissipative boundary conditions for hyperbolic equations The equations Main result Comparaison with prior results Proof of the exponential stability Application to the control of open channels Controllability of hyperbolic systems The control problem Controllability theorem Controllability of the Korteweg-de Vries (KdV) equation Local controllability Global controllability

La Sambre

Commercial break

Commercial break

JMC, Control and nonlinearity, Mathematical Surveys and Monographs, 136, 2007, 427 pp.

yt + A(y)yx = 0, y ∈ Rn, x ∈ [0, 1], t ∈ [0, +∞), (1)

yt + A(y)yx = 0, y ∈ Rn, x ∈ [0, 1], t ∈ [0, +∞), (1)

• Assumptions on A

yt + A(y)yx = 0, y ∈ Rn, x ∈ [0, 1], t ∈ [0, +∞), (1)

• Assumptions on A

A(0) = diag (Λ1, Λ2, . . . , Λn),

yt + A(y)yx = 0, y ∈ Rn, x ∈ [0, 1], t ∈ [0, +∞), (1)

• Assumptions on A

A(0) = diag (Λ1, Λ2, . . . , Λn), Λi > 0, ∀i ∈ {1, . . . , m}, Λi < 0, ∀i ∈ {m + 1, . . . , n},

yt + A(y)yx = 0, y ∈ Rn, x ∈ [0, 1], t ∈ [0, +∞), (1)

• Assumptions on A

A(0) = diag (Λ1, Λ2, . . . , Λn), Λi > 0, ∀i ∈ {1, . . . , m}, Λi < 0, ∀i ∈ {m + 1, . . . , n}, Λi = Λj, ∀(i, j) ∈ {1, . . . , n}2 such that i = j.

• Boundary conditions on y:

y+(t, 0) y−(t, 1)

• = G

y+(t, 1) y−(t, 0)

• , t ∈ [0, +∞),

(2) where

• Boundary conditions on y:

y+(t, 0) y−(t, 1)

• = G

y+(t, 1) y−(t, 0)

• , t ∈ [0, +∞),

(2) where (i) y+ ∈ Rm and y− ∈ Rn−m are defined by y = y+ y−

• ,

• Boundary conditions on y:

y+(t, 0) y−(t, 1)

• = G

y+(t, 1) y−(t, 0)

• , t ∈ [0, +∞),

(2) where (i) y+ ∈ Rm and y− ∈ Rn−m are defined by y = y+ y−

• ,

(ii) the map G : Rn → Rn vanishes at 0.

Notations

For K ∈ Mn,m(R), K := max{|Kx|; x ∈ Rn, |x| = 1}.

Notations

For K ∈ Mn,m(R), K := max{|Kx|; x ∈ Rn, |x| = 1}. If n = m, ρ1(K) := Inf {∆K∆−1; ∆ ∈ Dn,+}, where Dn,+ denotes the set of n × n real diagonal matrices with strictly positive diagonal elements.

Theorem 1.1 (JMC-G. Bastin-B. d’Andr´ ea-Novel (2008)).

If ρ1(G ′(0)) < 1, then the equilibrium ¯ y ≡ 0 of the quasi-linear hyperbolic system yt + A(y)yx = 0, with the above boundary conditions, is exponentially stable for the Sobolev H2-norm.

Estimate on the exponential decay rate

For every ν ∈ (0, − min{|Λ1|, . . . , |Λn|} ln(ρ1(G ′(0)))), there exist ε > 0 and C > 0 such that, for every y0 ∈ H2((0, 1), Rn) satisfying |y0|H2((0,1),Rn) < ε (and the usual compatibility conditions) the classical solution y to the Cauchy problem yt + A(y)yx = 0, y(0, x) = y0(x) + boundary conditions is defined on [0, +∞) and satisfies |y(t, ·)|H2((0,1),Rn) Ce−νt|y0|H2((0,1),Rn), ∀t ∈ [0, +∞).

The Ta-tsien Li condition

R2(K) := Max {

n

• j=1

|Kij|; i ∈ {1, . . . , n}}, ρ2(K) := Inf {R2(∆K∆−1); ∆ ∈ Dn,+}.

Theorem 1.2 (Ta-tsien Li (1994)).

If ρ2(G ′(0)) < 1, then the equilibrium ¯ y ≡ 0 of the quasi-linear hyperbolic system yt + A(y)yx = 0, with the above boundary conditions, is exponentially stable for the C 1-norm.

The Ta-tsien Li approach

The Ta-tsien Li approach

The Ta-tsien Li proof relies mainly on the use of direct estimates of the solutions and their derivatives along the characteristic curves.

The Ta-tsien Li approach

The Ta-tsien Li proof relies mainly on the use of direct estimates of the solutions and their derivatives along the characteristic curves. Robustness to small perturbations: C. Prieur, J. Winkin and G. Bastin (2008); V. Dos Santos and C. Prieur (2008).

C 1/H2-exponential stability

• 1. Open problem : Does there exists K such that one has

exponential stability for the C 1-norm but not for the H2-norm?

C 1/H2-exponential stability

• 1. Open problem : Does there exists K such that one has

exponential stability for the C 1-norm but not for the H2-norm?

• 2. Open problem : Does there exists K such that one has

exponential stability for the H2-norm but not for the C 1-norm?

Comparison of ρ2 and ρ1

Comparison of ρ2 and ρ1

Proposition 1.3.

For every K ∈ Mn,n(R), ρ1(K) ρ2(K). (3)

Comparison of ρ2 and ρ1

Proposition 1.3.

For every K ∈ Mn,n(R), ρ1(K) ρ2(K). (3) Example where (3) is strict: for a > 0, let Ka := a a −a a

• ∈ M2,2(R).

Then ρ1(Ka) = √ 2a < 2a = ρ2(Ka).

Comparison of ρ2 and ρ1

Proposition 1.3.

For every K ∈ Mn,n(R), ρ1(K) ρ2(K). (3) Example where (3) is strict: for a > 0, let Ka := a a −a a

• ∈ M2,2(R).

Then ρ1(Ka) = √ 2a < 2a = ρ2(Ka). Open problem: Does ρ1(K) < 1 implies the exponential stability for the C 1-norm?

Comparison with stability conditions for linear hyperbolic systems

For simplicity we assume that Λi are all positive and consider we consider the special case of linear hyperbolic systems yt + Λyx = 0, y(t, 0) = Ky(t, 1), where Λ := diag (Λ1, . . . , Λn), with Λi > 0, ∀i ∈ {1, . . . , n}.

Theorem 1.4.

Exponential stability for the C 1-norm is equivalent to the exponential stability in the H2-norm.

A Necessary and sufficient condition for exponential stability

Notation: ri = 1 Λi , ∀i ∈ {1, . . . , n}.

Theorem 1.5.

¯ y ≡ is exponentially stable for the system ˙ y + Λyx = 0, y(t, 0) = Ky(t, 1) if and only if there exists δ > 0 such that

• det (Idn − (diag (e−r1z, . . . , e−rnz))K) = 0, z ∈ C
• ⇒ (ℜ(z) −δ).

An example

Let us choose λ1 := 1, λ2 := 2 (hence r1 = 1 and r2 = 1/2 and Ka := a a a a

• , a ∈ R.

Then ρ1(K) = 2|a|. Hence ρ1(Ka) < 1 is equivalent to a ∈ (−1/2, 1/2).

An example

Let us choose λ1 := 1, λ2 := 2 (hence r1 = 1 and r2 = 1/2 and Ka := a a a a

• , a ∈ R.

Then ρ1(K) = 2|a|. Hence ρ1(Ka) < 1 is equivalent to a ∈ (−1/2, 1/2). However exponential stability is equivalent to a ∈ (−1, 1/2).

Robustness issues

For a positive integer n, let Λ1 := 4n 4n + 1, Λ2 = 4n 2n + 1. Then y1 y2

• :=

sin

• 4nπ(t − (x/Λ1))
• sin
• 4nπ(t − (x/Λ2))
• is a solution of yt + Λyx, y(t, 0) = K−1/2y(t, 1) which does not

tends to 0 as t → +∞. Hence one does not have exponential

• stability. However limn→+∞ Λ1 = 1 and limn→+∞ Λ2 = 2.

Robustness issues

For a positive integer n, let Λ1 := 4n 4n + 1, Λ2 = 4n 2n + 1. Then y1 y2

• :=

sin

• 4nπ(t − (x/Λ1))
• sin
• 4nπ(t − (x/Λ2))
• is a solution of yt + Λyx, y(t, 0) = K−1/2y(t, 1) which does not

tends to 0 as t → +∞. Hence one does not have exponential

• stability. However limn→+∞ Λ1 = 1 and limn→+∞ Λ2 = 2.The

exponential stability is not robust with respect to Λ: small perturbations of Λ can destroy the exponential stability.

Robust exponential stability

Notations: ρ0(K) := max{ρ(diag (eιθ1, . . . , eιθn)K); (θ1, . . . , θn)tr ∈ Rn},

Theorem 1.6 (R. Silkowski (1993)).

If the (r1, . . . , rn) are rationally independent, the linear system yt + Λyx = 0, y(t, 0) = Ky(t, 1) is exponentially stable if and only if ρ0(K) < 1.

Robust exponential stability

Notations: ρ0(K) := max{ρ(diag (eιθ1, . . . , eιθn)K); (θ1, . . . , θn)tr ∈ Rn},

Theorem 1.6 (R. Silkowski (1993)).

If the (r1, . . . , rn) are rationally independent, the linear system yt + Λyx = 0, y(t, 0) = Ky(t, 1) is exponentially stable if and only if ρ0(K) < 1. Note that ρ0(K) depends continuously on K and that “(r1, . . . , rn) are rationally independent” is a generic condition. Therefore, if one wants to have a natural robustness property with respect to the ri’s, the condition for exponential stability is ρ0(K) < 1.

Robust exponential stability

Notations: ρ0(K) := max{ρ(diag (eιθ1, . . . , eιθn)K); (θ1, . . . , θn)tr ∈ Rn},

Theorem 1.6 (R. Silkowski (1993)).

If the (r1, . . . , rn) are rationally independent, the linear system yt + Λyx = 0, y(t, 0) = Ky(t, 1) is exponentially stable if and only if ρ0(K) < 1. Note that ρ0(K) depends continuously on K and that “(r1, . . . , rn) are rationally independent” is a generic condition. Therefore, if one wants to have a natural robustness property with respect to the ri’s, the condition for exponential stability is ρ0(K) < 1. This condition does not depend on the Λi’s!

Comparison of ρ0 and ρ1

Comparison of ρ0 and ρ1

Proposition 1.7 (JMC-G. Bastin-B. d’Andr´ ea-Novel (2008)).

For every n ∈ N and for every K ∈ Mn,n(R), ρ0(K) ρ1(K). For every n ∈ {1, 2, 3, 4, 5} and for every K ∈ Mn,n(R), ρ0(K) = ρ1(K). For every n ∈ N \ {1, 2, 3, 4, 5}, there exists K ∈ Mn,n(R) such that ρ0(K) < ρ1(K).

Comparison of ρ0 and ρ1

Proposition 1.7 (JMC-G. Bastin-B. d’Andr´ ea-Novel (2008)).

For every n ∈ N and for every K ∈ Mn,n(R), ρ0(K) ρ1(K). For every n ∈ {1, 2, 3, 4, 5} and for every K ∈ Mn,n(R), ρ0(K) = ρ1(K). For every n ∈ N \ {1, 2, 3, 4, 5}, there exists K ∈ Mn,n(R) such that ρ0(K) < ρ1(K). Open problem : Is ρ0(G ′(0)) < 1 a sufficient condition for exponential stability (for the H2-norm) in the nonlinear case ?

Proof of the exponential stability if A is constant and G is linear

Main tool: a Lyapunov approach. A(y) = Λ, G(y) = Ky. For simplicity, all the Λi’s are positive. Lyapunov function candidate: V (y) := 1 ytrQye−µxdx, Q is positive symmetric.

Proof of the exponential stability if A is constant and G is linear

Main tool: a Lyapunov approach. A(y) = Λ, G(y) = Ky. For simplicity, all the Λi’s are positive. Lyapunov function candidate: V (y) := 1 ytrQye−µxdx, Q is positive symmetric. ˙ V = − 1 (ytr

x ΛQy + ytrQyx)Λe−µxdx

= −µ 1 ytrΛQy e−µxdx − B, with B := [ytrΛQye−µx]x=1

x=0 = y(1)tr(ΛQe−µ − K trΛQK)y(1)

Let D ∈ Dn,+ be such that DKD−1 < 1 and let ξ := Dy(1). We take Q = D2Λ−1. Then B = e−µ|ξ|2 − |DKD−1ξ|2. Therefore it suffices to take µ > 0 small enough.

Remark 1.8.

Introduction of µ:

Let D ∈ Dn,+ be such that DKD−1 < 1 and let ξ := Dy(1). We take Q = D2Λ−1. Then B = e−µ|ξ|2 − |DKD−1ξ|2. Therefore it suffices to take µ > 0 small enough.

Remark 1.8.

Introduction of µ:

• JMC (1998) for the stabilization of the Euler equations.

Let D ∈ Dn,+ be such that DKD−1 < 1 and let ξ := Dy(1). We take Q = D2Λ−1. Then B = e−µ|ξ|2 − |DKD−1ξ|2. Therefore it suffices to take µ > 0 small enough.

Remark 1.8.

Introduction of µ:

• JMC (1998) for the stabilization of the Euler equations.
• Cheng-Zhong Xu and Gauthier Sallet (2002) for symmetric

linear hyperbolic systems.

New difficulties if A(y) depends on y

We try with the same V : ˙ V = − 1

• ytr

x A(y)trQy + ytrQA(y)yx

• e−µxdx

= −µ 1 ytrA(y)Qye−µxdx − B + N1 + N2 with N1 := 1 ytr(QA(y) − A(y)Q)yxe−µxdx, N2 := 1 ytr A′(y)yx trQye−µxdx

Solution for N1

Take Q depending on y such that A(y)Q(y) = Q(y)A(y), Q(0) = D2F(0)−1. (This is possible since the eigenvalues of F(0) are distinct.) Now ˙ V = −µ 1 ytrA(y)Q(y)ye−µxdx − B + N2 with N2 := 1 ytr A′(y)yxQ(y) + A(y)Q′(y)yx trye−µxdx

Solution for N1

Take Q depending on y such that A(y)Q(y) = Q(y)A(y), Q(0) = D2F(0)−1. (This is possible since the eigenvalues of F(0) are distinct.) Now ˙ V = −µ 1 ytrA(y)Q(y)ye−µxdx − B + N2 with N2 := 1 ytr A′(y)yxQ(y) + A(y)Q′(y)yx trye−µxdx What to do with N2?

Solution for N2

New Lyapunov function: V (y) = V1(y) + V2(y) + V3(y) with V1(y) = 1 ytrQ(y)y e−µxdx, V2(y) = 1 ytr

x R(y)yx e−µxdx,

V3(y) = 1 ytr

xxS(y)yxx e−µxdx,

where µ > 0, Q(y), R(y) and S(y) are symmetric positive definite matrices.

Choice of Q, R and S

• Commutations:

A(y)Q(y) − Q(y)A(y) = 0, A(y)R(y) − R(y)A(y) = 0, A(y)S(y) − S(y)A(y) = 0,

Choice of Q, R and S

• Commutations:

A(y)Q(y) − Q(y)A(y) = 0, A(y)R(y) − R(y)A(y) = 0, A(y)S(y) − S(y)A(y) = 0,

• Q(0) = D2A(0)−1, R(0) = D2A(0), S(0) = D2A(0)3.

Estimates on ˙ V

Lemma 1.9.

If µ > 0 is small enough, there exist positive real constants α, β, δ such that, for every y : [0, 1] → Rn such that |y|C 0([0,1]) + |yx|C 0([0,1]) δ, we have 1 β 1 (|y|2 + |yx|2 + |yxx|2)dx V (y) β 1 (|y|2 + |yx|2 + |yxx|2)dx, ˙ V ≤ −αV . ...

La Sambre (G. Bastin, L. Moens, ...)

x

L

H(t,x) V(t,x)

u0 uL

Z0

The Saint-Venant equations

The index i is for the i-th reach. Conservation of mass: Hit + (HiVi)x = 0,

The Saint-Venant equations

The index i is for the i-th reach. Conservation of mass: Hit + (HiVi)x = 0, Conservation of momentum: Vit +

• gHi + V 2

i

2

• x

= 0.

The Saint-Venant equations

The index i is for the i-th reach. Conservation of mass: Hit + (HiVi)x = 0, Conservation of momentum: Vit +

• gHi + V 2

i

2

• x

= 0. Flow rate: Qi = HiVi.

Barr´ e de Saint-Venant (Adh´ emar-Jean-Claude) 1797-1886 Th´ eorie du mouvement non perma- nent des eaux, avec applications aux crues des rivi` eres et ` a l’introduction des mar´ ees dans leur lit, C. R. Acad.

• Sci. Paris S´
• er. I Math., vol. 53

(1871), pp.147–154.

Barr´ e de Saint-Venant (Adh´ emar-Jean-Claude) 1797-1886 Th´ eorie du mouvement non perma- nent des eaux, avec applications aux crues des rivi` eres et ` a l’introduction des mar´ ees dans leur lit, C. R. Acad.

• Sci. Paris S´
• er. I Math., vol. 53

(1871), pp.147–154.

Boundary conditions

Underflow (sluice)

u u

Overflow (spillway)

u u

La Sambre: Hydraulic gates

Closed loop versus open loop

56

Work in progress: La Meuse

Complements

• Slope and friction: G. Bastin, JMC and B. d’Andr´

ea-Novel (2008).

• Robust H∞-control design techniques: X. Litrico and D.

Georges (2001), X. Litrico (2001).

• Spectral methods: X. Litrico and V. Fromion (2006).
• Observer for water level: G. Besan¸

con, J.F. Dulhoste and D. Georges (2001).

• Integral action: V. Dos-Santos, G. Bastin, JMC and B.

d’andr´ ea-Novel (2007).

• Book: “Automatique pour la gestion des ressources en eau”
• D. Georges and X. Litrico (2002).

Stabilization: Dissipative boundary conditions for hyperbolic equations The equations Main result Comparaison with prior results Proof of the exponential stability Application to the control of open channels Controllability of hyperbolic systems The control problem Controllability theorem Controllability of the Korteweg-de Vries (KdV) equation Local controllability Global controllability

The control problem

Let T > 0. Given y0 : [0, 1] → Rn and y1 : [0, 1] → Rn. Does there exist u+ : [0, T] → Rm, u− : [0, T] → Rm such that the solution of the Cauchy problem yt+A(y)yx = 0, y+(t, 0) = u+(t), y−(t, 0) = u−(t), y(0, x) = y0(x), satisfies y(T, x) = yT(x) ?

The control problem

Let T > 0. Given y0 : [0, 1] → Rn and y1 : [0, 1] → Rn. Does there exist u+ : [0, T] → Rm, u− : [0, T] → Rm such that the solution of the Cauchy problem yt+A(y)yx = 0, y+(t, 0) = u+(t), y−(t, 0) = u−(t), y(0, x) = y0(x), satisfies y(T, x) = yT(x) ? Local controllability: y0 and yT are small.

Remark 2.1.

The control is on both sides. The case where the control is only on

• ne side can also be considered.

Controllability theorem

Theorem 2.2 (Ta-tsien Li and Bopeng Rao (2003)).

(Local controllability for the C 1-norm) ⇔

• T > max
• 1

Λ1 , . . . , 1 Λm , 1 |Λm+1|, . . . , 1 |Λn|

Controllability theorem

Theorem 2.2 (Ta-tsien Li and Bopeng Rao (2003)).

(Local controllability for the C 1-norm) ⇔

• T > max
• 1

Λ1 , . . . , 1 Λm , 1 |Λm+1|, . . . , 1 |Λn|

• Remark 2.3.
• If T > max
• 1

Λ1 , . . . , 1 Λm , 1 |Λm+1|, . . . , 1 |Λn|

• , the linearized

control system at (¯ y, ¯ u) = (0, 0) is controllable.

• One of the main ingredient of the proof: consider

yt + A(y)yx = 0 as an evolution equation in x: A(y)−1yt + yx = 0.

• Generalization: A(t, x, y): Zhiqiang Wang (2007).
• Applications to channels: M. Gugat and G. Leugering (2008).
• For the control on one side only, see Ta-tsien Li and Bopeng

Rao (2002).

Stabilization: Dissipative boundary conditions for hyperbolic equations The equations Main result Comparaison with prior results Proof of the exponential stability Application to the control of open channels Controllability of hyperbolic systems The control problem Controllability theorem Controllability of the Korteweg-de Vries (KdV) equation Local controllability Global controllability

History of the KdV equation

• 1834 John Scott Russell
• 1872 Joseph Valentin de Boussinesq
• 1895 Diederik Korteweg and Gustav de Vries
• 1995 Experiment

The 1995 experiment

The KdV control system

yt + yx + yxxx + yyx = 0, t ∈ [0, T], x ∈ [0, L], y(t, 0) = y(t, L) = 0, yx(t, L) = u(t), t ∈ [0, T]. where, at time t ∈ [0, T], the control is u ∈ R and the state is y(t, ·) ∈ L2(0, L).

Controllability of the linearized control system

The linearized control system (around 0) is yt + yx + yxxx = 0, t ∈ [0, T], x ∈ [0, L], y(t, 0) = y(t, L) = 0, yx(t, L) = u(t), t ∈ [0, T]. where, at time t ∈ [0, T], the control is u ∈ R and the state is y(t, ·) ∈ L2(0, L).

Theorem 3.1 (L. Rosier (1997)).

For every T > 0, the linearized control system is controllable in time T if and only L ∈ N :=

• 2π
• k2 + kl + l2

3

• .

Proof of the “only if” part

L is in N if and only if there exists ϕ = 0 : [0, L] → C and λ ∈ C such that ϕx + ϕxxx = λϕ, ϕ(0) = ϕx(0) = ϕ(L) = ϕx(L) = 0. With such a ϕ, whatever is u(t), the solution y of the linearized control system satisfies d dt L yϕ = λ L yϕ ...

Connection with the Hautus criterion

Hautus criterion: in finite dimension ˙ y = Ay + Bu is controllable if and only if (A∗ϕ = λϕ and B∗ϕ = 0) ⇒ (ϕ = 0). In infinite dimension the Hautus criterion is still necessary, but not sufficient in general. However, in infinite dimension, the Hautus criterion is sufficient provided that one has enough compactness. Here: T. Kato (1983) smoothing effect (for x ∈ R; for x ∈ [0, L]:

• L. Rosier (1997)).

Application to the nonlinear system

Theorem 3.2 (L. Rosier (1997)).

For every T > 0, the KdV control system is locally controllable (around 0) in time T if L ∈ N.

Application to the nonlinear system

Theorem 3.2 (L. Rosier (1997)).

For every T > 0, the KdV control system is locally controllable (around 0) in time T if L ∈ N. Question: Does one have controllability if L ∈ N?

Controllability when L ∈ N

Theorem 3.3 (JMC and E. Cr´ epeau (2003)).

If L = 2π (which is in N: take k = l = 1), for every T > 0 the KdV control system is locally controllable (around 0) in time T.

Theorem 3.4 (E. Cerpa (2007), E. Cerpa and E. Cr´ epeau (2008)).

For every L ∈ N, there exists T > 0 such that the KdV control system is locally controllable (around 0) in time T.

Strategy of the proof : power series expansion.

Example with L = 2π. For every trajectory (y, u) of the linearized control system around 0 d dt 2π (1 − cos(x))y(t, x)dx = 0. This is is the only “obstacle” to the controllability of the linearized control system:

Proposition 3.5 (L. Rosier (1997)).

Let H := {y ∈ L2(0, L); L

0 (1 − cos(x))y(x)dx = 0}. For every

(y0, y1) ∈ H × H, there exists u ∈ L2(0, T) such that the solution to the Cauchy problem yt + yx + yxxx = 0, y(t, 0) = y(t, L) = 0, yx(t, L) = u(t), t ∈ [0, T], y(0, x) = y0(x), x ∈ [0, L], satisfies y(T, x) = y1(x), x ∈ [0, L].

We explain the method on the control system of finite dimension ˙ x = f (x, u), where the state is x ∈ Rn and the control is u ∈ Rm. We assume that (0, 0) ∈ Rn × Rm is an equilibrium of the control system (7), i.e f (0, 0) = 0. Let H := Span {AiBu; u ∈ Rm, i ∈ {0, . . . , n − 1}} with A := ∂f ∂x (0, 0), B := ∂f ∂u (0, 0). If H = Rn, the linearized control system around (0, 0) is controllable and therefore the nonlinear control system ˙ x = f (x, u) is small-time locally controllable at (0, 0) ∈ Rn × Rm.

Let us look at the case where the dimension of H is n − 1. Let us make a (formal) power series expansion of the control system ˙ x = f (x, u) in (x, u) around 0. We write x = y1 + y2 + . . . , u = v1 + v2 + . . . . The order 1 is given by (y1, v1); the order 2 is given by (y2, v2) and so on. The dynamics of these different orders are given by ˙ y1 = ∂f ∂x (0, 0)y1 + ∂f ∂u (0, 0)v1, ˙ y2 = ∂f ∂x (0, 0)y2 + ∂f ∂u (0, 0)v2 + 1 2 ∂2f ∂x2 (0, 0)(y1, y1) + ∂2f ∂x∂u (0, 0)(y1, v1) + 1 2 ∂2f ∂u2 (0, 0)(v1, v1), and so on.

Let e1 ∈ H⊥. Let T > 0. Let us assume that there are controls v1

±

and v2

±, both in L∞((0, T); Rm), such that, if y1 ± and y2 ± are

solutions of ˙ y1

± = ∂f

∂x (0, 0)y1

± + ∂f

∂u (0, 0)v1

±,

y1

±(0) = 0,

˙ y2

± = ∂f

∂x (0, 0)y2

± + ∂f

∂u (0, 0)v2

± + 1

2 ∂2f ∂x2 (0, 0)(y1

±, y1 ±)

+ ∂2f ∂x∂u (0, 0)(y1

±, u1 ±) + 1

2 ∂2f ∂u2 (0, 0)(u1

±, u1 ±),

y2

±(0) = 0,

then y1

±(T) = 0,

y2

±(T) = ±e1.

Let (ei)i∈{2,...n} be a basis of H. By the definition of H, there are (ui)i=2,...,n, all in L∞(0, T)m, such that, if (xi)i=2,...,n are the solutions of ˙ xi = ∂f ∂x (0, 0)xi + ∂f ∂u (0, 0)ui, xi(0) = 0, then, for every i ∈ {2, . . . , n}, xi(T) = ei. Now let b =

n

• i=1

biei be a point in Rn. Let v1 and v2, both in L∞((0, T); Rm), be defined by the following

• If b1 0, then v1 := v1

+ and v2 := v2 +.

• If b1 < 0, then v1 := v1

− and v2 := v2 −.

Then let u : (0, T) → Rm be defined by u(t) := |b1|1/2v1(t) + |b1|v2(t) +

n

• i=2

biui(t). Let x : [0, T] → Rn be the solution of ˙ x = f (x, u(t)), x(0) = 0. Then one has, as b → 0, x(T) = b + o(b). (4) Hence, using the Brouwer fixed-point theorem and standard estimates on ordinary differential equations, one gets the local controllability of ˙ x = f (x, u) (around (0, 0) ∈ Rn × Rm) in time T, that is, for every ε > 0, there exists η > 0 such that, for every (a, b) ∈ Rn × Rn with |a| < η and |b| < η, there exists a trajectory (x, u) : [0, T] → Rn × Rm of the control system ˙ x = f (x, u) such that x(0) = a, x(T) = b, |u(t)| ε, t ∈ (0, T).

Bad and good news for L = 2π

• Bad news: The order 2 is not sufficient. One needs to go to

the order 3

• Good news: the fact that the order is odd allows to get the

local controllability in arbitrary small time. The reason: If one can move in the direction ξ ∈ H⊥ one can move in the direction −ξ. Hence it suffices to argue by contradiction (assume that it is impossible to enter in H⊥ in small time...)

With more controls: Global controllability result

yt + yx + yxxx + yyx = u0(t), t ∈ [0, T], x ∈ [0, L], y(t, 0) = u1(t), y(t, L) = u2(t), yx(t, L) = u3(t), t ∈ [0, T]. where, at time t ∈ [0, T],

• the control is (u0(t), u1(t), u2(t), u3(t)) ∈ R4,
• the state is y(t, ·) ∈ L2(0, L).

Theorem 3.6 (M. Chapouly (2008)).

For every L > 0, and for every T > 0 the KdV control system (with four controls) is globally controllable in time T.

Heuristics of the proof

Return method (Navier-Stokes, JMC (1996))

T y t

Heuristics of the proof

Return method (Navier-Stokes, JMC (1996))

T y t

Heuristics of the proof

Return method (Navier-Stokes, JMC (1996))

T y t B1 B2

Heuristics of the proof

Return method (Navier-Stokes, JMC (1996))

T y t B1 B2 ¯ y(t)

Heuristics of the proof

Return method (Navier-Stokes, JMC (1996))

T y t B1 B2 ¯ y(t) B3 B4

Open problems

• Can one remove the control u0 (i.e. yt + yx + yxxx + yyx = 0)

and keep the global controllability result?

• With only one control (yx(t, L)) : Is there a minimal time for

the local controllability for some L ∈ N? (For a Schr¨

• dinger

control system: JMC (2006).)

• “Rapid” stabilization. (For the linearized control system with
• nly one control and L ∈ N: E. Cerpa and E. Cr´

epeau (2008).)