Null controllability of the Grushin equation : non-rectangular control region M. Duprez

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Null controllability of the Grushin equation : non-rectangular control region

M. Duprez

¹

, A. Koenig

²

,

1Laboratoire Jacques-Louis Lions, Paris

2Laboratoire Jean-Alexandre Dieudonné, Nice

Séminaire d’analyse

Institut de Mathématiques de Bordeaux 18/10/18

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Outline

1 Controllability of the heat equation

2 Controllability of the Grushin equation

3 Comments and open problems

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Notions of controllability

Problematic

We searchu, calledcontrol, such that the solutionfto system

∂tf=A(f) +B(u) f(0) =f⁰

satisfies :

ε

f⁰

f¹ f(T)

• fnear a given target at timeT

∀ε >0, f⁰, f¹,∃u s.t. d(f(T), f¹)6ε.

â Approximate controllability

• freach a target at timeT

∀f⁰, f¹, ∃us.t.f(T) =f¹. â Exact controllability

• freach a target at timeT

∀f⁰, f¹, ∃us.t.f(T) = 0.

â Null controllability

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Internal null controllability of the heat equation

LetΩbe an open bounded set ofRⁿ, ωan non-empty open set ofΩ andT >0.

Consider the heat equation ( _∂

tf−∆f=1ωu, f|∂Ω= 0,

f(0) =f⁰.

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ω Ω

Theorem (Internal null controllability of the heat equation)

The heat equation(1)isnull controllableat timeT,i.e.for each initial condition f⁰∈L²(Ω), there exists a controlu∈L²([0, T]×ω)such that solutionfto (1) satisfiesf(T) = 0.

→Fattorini-Russell 71’ (dim 1) : moment method

→Fursikov-Imanuvilov / Lebeau-Robbiano 95’ : Carleman inequality / spectral inequality

Remark

• No minimal timeof controllability

• No geometrical conditiononω

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Boundary null controllability of the heat equation

LetΩbe an open bounded set ofRⁿ,γan non-empty open set of∂ΩandT >0.

( _∂

tf−∆f= 0, f|∂Ω=1γu, f(0) =f⁰.

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Corollary (Boundary null controllability of the heat equation)

The heat equation(2)isnull controllableat timeT,i.e.for each initial condition f⁰∈L²(Ω), there exists a controlu∈L²([0, T]×γ)such that solutionfto (2) satisfiesf(T) = 0.

Sketch of proof: the fictitious control method

• ExtensionΩ˜ofΩthroughγ

• ω⊂Ω˜

• f˜: = solution of the int. contr. problem onΩ∪Ω˜

• u:= ˜f|γ

Ω γ

Remark

• Equivalence between internal and boundarynull controllability

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Boundary null controllability of the heat equation

( _∂

tf−∆f= 0, f|∂Ω=1γu, f(0) =f⁰.

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• ω⊂Ω˜

• u:= ˜f|γ

Ω˜ Ω

γ

Remark

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Boundary null controllability of the heat equation

( _∂

tf−∆f= 0, f|∂Ω=1γu, f(0) =f⁰.

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• ω⊂Ω˜

• u:= ˜f|γ

ω Ω˜ Ω

γ

Remark

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Boundary null controllability of the heat equation

( _∂

tf−∆f= 0, f|∂Ω=1γu, f(0) =f⁰.

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• ω⊂Ω˜

• u:= ˜f|γ

ω Ω˜ Ω

γ

Remark

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Boundary null controllability of the heat equation

( _∂

tf−∆f= 0, f|∂Ω=1γu, f(0) =f⁰.

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• ω⊂Ω˜

• u:= ˜f|γ

ω Ω˜ Ω

γ

Remark

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Boundary null controllability of the heat equation

( _∂

tf−∆f= 0, f|∂Ω=1γu, f(0) =f⁰.

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• ω⊂Ω˜

• u:= ˜f|γ

ω Ω˜ Ω

γ

Remark

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Duality

Denote byf(t;f⁰,u)the solution to







∂tf−∆f=1ωu, f|∂Ω= 0, f(0) =f⁰.

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LetQT:= Ω×(0, T)and FT : L²(Ω) → L²(Ω),

f⁰ 7→ f(T;f⁰,0)

GT : L²(QT) → L²(Ω), u 7→ f(T; 0,u) One has :

Null controllability of (1) ⇔ ∀f⁰,∃u:FT(f⁰) +GT(u) = 0

⇔ Im(FT)⊂Im(GT)

⇔ kFT^∗ϕ^Tk6CkG^∗Tϕ^Tk,∀ϕ^T Approximate controllability of (1) ⇔ Im(GT) =L²(Ω)

⇔ G^∗_Tϕ^T = 0⇒ϕ^T= 0 ,∀ϕ^T

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Duality







∂tf−∆f=1ωu, f|∂Ω= 0,

f(0) =f⁰.

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





−∂tϕ= ∆ϕ inQT, ϕ= 0 onΣT, ϕ(T) =ϕ^T inΩ.

Proposition (see Coron’s book 2007 or Tucsnak-Weiss’s book 2009)

1 System(1)is null controllableon(0, T), if and only if there existsCobs>0 such that for allϕ^T ∈L²(Ω)

kϕ(0)kL²(Ω)6Cobs

Z T 0

kϕkL²(ω)dt

withϕthe solution to the dual system.

2 System(1)isapprox. controllableat timeT, if and only if for allϕ^T∈L²(Ω) the solution to the dual system satisfies

ϕ= 0inω×(0, T) ⇒ϕ= 0inQT.

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Approximate controllability of the heat equation







∂tf−∆f=1ωu inQT, f|∂Ω= 0 onΣT, f(0) =f⁰ inΩ.

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





−∂tϕ= ∆ϕ inQT, ϕ= 0 onΣT, ϕ(T) =ϕ^T inΩ.

Since (1) is null controllable,

kϕ(0)k_L2(Ω)6Cobs

Z T 0

kϕk_L2(ω)dt

Hence

ϕ= 0inω×(0, T) ⇒ ϕ(0) = 0inΩ

⇒ ϕ= 0inΩ×(0, T)

Corollary (Approximate controllability of the heat equation) The heat equation(1)isapprox. controllableat timeT, i.e. for each

f⁰, f¹∈L²(Ω), there exists a controlu∈L²([0, T]×ω)such that solutionfto (1)satisfieskf(T)−f¹k6ε.

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The heat equation

Conclusion

• Non exact controllability

âRegularizing effect of the heat equation

• Null controllability

âEquivalence of int. and bound. null contr. thanks to fictitious control method

• Approximate controllability

âCome from the null controllability thanks to the duality

Question

How a modification of−∆can influx these notions of controllability ?

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The Grushin equation

Consider ( _(∂

t−∂x²−x²∂y²)f(t, x, y) =1ωu t∈[0, T],(x, y)∈Ω := (−1,1)×(0,1), f(t, x, y) = 0 t∈[0, T],(x, y)∈∂Ω,

f(0, x, y) =f⁰(x, y) (x, y)∈Ω,

x y

a a)

x y

a b)

x y

c)

a) minimal time of internal null controllabilityT^∗=a²/2 âBeauchard-Miller-Morancey 2015

b) minimal time of internal null controllabilityT^∗> a²/2 âBeauchard-Cannarsa-Guglielmi 2014

minimal time of boundary null controllabilityT^∗=a²/2 âBeauchard-Dardé-Ervedoza 2018

c) No internal null controllability âKoenig 2017

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Positive result

−a

x y

ω₀ γ

Theorem (D.-Koenig)

Assume that there existε >0andγ∈C⁰([0,1],Ω)withγ(0)∈(−1,1)× {0}and γ(1)∈(−1,1)× {π}such that

ω0:={z∈Ω,distance(z,Range(γ))< ε} ⊂ω, then the Grushin equation isnull-controllablein any timeT>a²/2, where a:= sup

(x,y)∈Ω\ω₀

{|x|:∃x0∈(−1,1),|x|<|x0|,sgn(x) = sgn(x0),(x0, y)∈ω0}.

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Positive result : sketch of proof

Proof :

−a a

x y

ω0

γ ωleft

ωright

Consider that controlled solutions [Beauchard Darde Ervedoza 2018]

( _(∂

t−∂_x²−x²∂_y²)fleft=1ω_leftuleft on[0, T]×Ω

(∂t−∂x²−x²∂y²)fright=1ω_righturight on[0, T]×Ω

fleft(0) =fright(0) =f0, fleft(T) =fright(T) = 0 onΩ There exists a functionθ∈C^∞( ¯Ω)such that

θ= 0onωleft\ω0, θ= 1onωright\ω0,supp(∇θ)⊂ω0. We remark thatf:=θfleft+ (1−θ)fright,solves

(∂t−∂_x²−x²∂_y²)f= 1ω₀u on[0, T]×Ω, f(0) =f0, f(T) = 0 onΩ,

whereu:=θ1ω_leftuleft+ (1−θ)1ω_righturight+ (fright−fleft)(∂²x+x²∂y²)θ

+2∂x(fright−fleft)∂xθ+ 2x²∂y(fright−fleft)∂yθ.

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Positive result : sketch of proof

Proof :

−a a

x y

ω0

γ ωleft

ωright

( _(∂

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Positive result : sketch of proof

Proof :

−a a

x y

ω0

γ ωleft

ωright

( _(∂

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Negative result

ω

y0

−a a

x y

Theorem (D.-Koenig)

If for somey0∈(0, π)anda >0one has

{(x, y0),−a < x < a} ∩ω=∅,

then the Grushin equation isnot null controllablein timeT<a²/2.

Remark :Since the Grushin equation is not null-controllable whenωis the complement of a horizontal strip [Koenig 2017], our Assumption is quasi optimal.

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Observability

Proposition (see Coron’s book 2007 or Tucsnak-Weiss’s book 2009)

The Grushin equation isnull controllabilityif, and only if, there existsC >0such that for allf0inL²(Ω), the solutionfto

( _(∂

t−∂_x²−x²∂_y²)f(t, x, y) = 0 t∈[0, T],(x, y)∈Ω, f(t, x, y) = 0 t∈[0, T],(x, y)∈∂Ω, f(0, x, y) =f0(x, y) (x, y)∈Ω,

satisfies

Z

Ω

|f(T, x, y)|²dxdy≤C Z

[0,T]×ω

|f(t, x, y)|²dtdxdy.

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Negative result : sketch of proof

Assume thatΩ =R×T. Forn >0,

e^−nx²^/2is the first eigenfunction of −∂²x+ (nx)² and with associated eigenvaluen.

Then

e^−nx²^/2e^inyis an eigenfunction of −∂_x²−x²∂_y² and with eigenvaluen.

For the functions

f(t, x, y) =X

ane^{−nt+iny−nx}²^/2, the observability inequality reads :

X _2π n

^1/2

|an|²e^−2nT ≤C Z

[0,T]×ω

Xane^{−nt+iny−nx}²^/2

2

dtdxdy

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Negative result : sketch of proof

Assume thatΩ =R×T. Forn >0,

e^−nx²^/2is the first eigenfunction of −∂²x+ (nx)² and with associated eigenvaluen.

Then

e^−nx²^/2e^inyis an eigenfunction of −∂_x²−x²∂_y² and with eigenvaluen.

For the functions

f(t, x, y) =X

ane^{−nt+iny−nx}²^/2, the observability inequality reads :

X _2π n

^1/2

|an|²e^−2nT ≤C Z

[0,T]×ω

Xane^{−nt+iny−nx}²^/2

2

dtdxdy

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Negative result : sketch of proof

Consider the change of variablezx(t, y) := e^−t+iy−x²^/2 and denoteDx:={e^−x²^/2−t+iy,0< t < T,(x, y)∈ω}:

Z

[0,T]×ω

Xane^{iny−nt−nx}²^/2

2

dtdxdy= Z 1

−1

Z

D_x

X

n>0

anzⁿ⁻¹

2

dλ(z) dx.

ω x

|x|> a y |x|< a

−a a

Dx

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Negative result : sketch of proof

Thus Z

D(0,e^−T)

X

n>0

anzⁿ⁻¹

2

dλ(z)≤C

X

n>0

anzⁿ⁻¹ L^∞(K)

.

e^−a²^/2 K=S1

x=−1Dx

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Negative result : sketch of proof

Z

D(0,e^−T)

X

n>0

anzⁿ⁻¹

2

dλ(z)≤C

X

n>0

anzⁿ⁻¹

2 L^∞(K)

.

Letz0∈D(0, e^−T)\K and

f:z∈C\z0[1,+∞)7→(z−z0)⁻¹.

K

D(0, e^−T)

z0

Proposition (Runge’s theorem)

LetKbe a connected and simply connected open subset ofC, and letfbe a holomorphic function onK. There exists a sequence(pk)of polynomials that converges uniformly on every compact subsets ofKtof.

Then, the familypkis a counter example to the inequality on entire polynomials.

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Negative result : sketch of proof

Assume now thatΩ = (−1,1)×(0,1).

Forn >0, let us note

vnthe first eigenfunction of −∂²_x+ (nx)²

with Dirichlet boundary condition on(−1,1), and with associated eigenvalueλn. Then

vn(x) sin(ny)is an eigenfunction of −∂x²−x²∂²y

with Dirichlet boundary condition on∂Ω, and with eigenvalueλn. For the functions

f(t, x, y) =X

anvn(x)e^−λⁿ^tsin(ny), the observability inequality reads :

X|an|²|vn|²_L2e^−2λⁿ^T≤C Z

[0,T]×ω

Xanvn(x)e^−λⁿ^tsin(ny)

2

dtdxdy Let us estimate the right hand-side of this inequality

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Negative result : sketch of proof

•Notingω˜=ω∪ {(x,−y),(x, y)∈ω}, this implies Z

[0,T]×ω

Xanvn(x)e^−λⁿ^tsin(ny)

2

dtdxdy

≤C Z

[0,T]×˜ω

Xanvn(x)e^iny−λⁿ^t

2

dtdxdy.

˜ ω

x y

−a⁰ a⁰

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Negative result : sketch of proof

•Consider the change of variable







λn=n+ρn

vn(x) = e^{−(1−ε)nx}²^/2wn(x) zx(t, y) = e−t+iy−(1−ε)x²/2

Then Z

[0,T]×˜ω

Xanvn(x)e^iny−λⁿ^t

2

dtdxdy

= Z

[0,T]×˜ω

X

n>0

anwn(x)e^−ρⁿ^tzx(t, y)ⁿ

2

dtdxdy.

Remark (lemma)

The eigenfunctionvnof−∂x²+ (nx)²for the eigenvalue is closed to the eigenfunction of same operator onRfor the eigenvaluen,i.e.

vnclosed to _n 2π

^1/4

e^−nx²^/2 and λnclosed ton.

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Negative result : sketch of proof

•DenotingDx={e^−(1−ε)x²^/2e^−t+iy,0< t < T,(x, y)∈ω}, we get˜ Z

[0,T]×˜ω

X

n>0

anwn(x)e^−ρⁿ^tzx(t, y)ⁿ

2

dtdxdy

= Z 1

−1

Z

D_x

X

n>0

anwn(x)e^−ρⁿ^tzⁿ⁻¹

2

dλ(z) dx.

ω˜ x

|x|> a⁰ y |x|< a⁰

−a⁰ a⁰

Dx

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Negative result : sketch of proof

•Sinceρn,wnare closed to0,1, Z 1

−1

X

n>N

anwn(x)e^−ρⁿ^tzⁿ⁻¹

2 L^∞(D_x)

dx≤C Z 1

−1

X

n>N

anzⁿ⁻¹ L^∞(U)

.

U

e^−(1−)a⁰²^/2 K=S1

x=−1Dx

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Negative result : sketch of proof

Thus Z

D(0,e^−T)

X

n>0

anzⁿ⁻¹

2

dλ(z)≤C

X

n>N

anzⁿ⁻¹ L^∞(U)

.

U

e^−(1−)a⁰²^/2 K=S1

x=−1Dx

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Negative result : sketch of proof

Then Z

D(0,e^−T)

X

n>N

anzⁿ⁻¹

2

dλ(z)≤C

X

n>N

anzⁿ⁻¹

2 L^∞(U)

.

Letz0∈D(0, e^−T)\U and

f:z∈C\z0[1,+∞)z7→(z−z0)⁻¹.

U

D(0, e^−T)

z0

Proposition (Runge’s theorem)

LetUbe a connected and simply connected open subset ofC, and letfbe a holomorphic function onU. There exists a sequence(˜pk)of polynomials that converges uniformly on every compact subsets ofUtof.

Then, the familypk(z) :=z^N+1p˜k(z)is a counter example to the inequality on entire polynomials.

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Minimal time

ω0 ω

y0

−a a

x y

Corollary (T0=a²/2)

Assume that there existε >0andγ∈C⁰([0,1],Ω)withγ(0)∈(−1,1)× {0}and γ(1)∈(−1,1)× {π}such thatω0:={z∈Ω,distance(z,Range(γ))< ε} ⊂ω.

Assume that somey0∈(0, π),{(x, y0),−a < x < a}is disjoint fromωwhere a:= sup

(x,y)∈Ω\ω₀

{|x|:∃x0∈(−1,1),|x|<|x0|,sgn(x) = sgn(x0),(x0, y)∈ω0}.

One has

• IfT>a²/2, then the Grushin equation isnull controllablein timeT.

• IfT<a²/2, then the Grushin equation isnot null controllablein timeT.

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Open problems

ω

−a

ω ω

−a We can’t go from the bottom boundary

to the top boundary while staying in- sideω.

T^∗>a²/2

A pinched domain.

T^∗>a²/2

ω

−a

−b

A cave.

T^∗∈[a²/2, b²/2]

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The Grushin equation

Conclusion

• Non exact controllability

âRegularizing effect of the heat equation

• Approximate controllability âBeauchard-Cannarsa-Guglielmi 2014

• Minimal time null controllability

âDetailed description of the influence of the control region

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Comments

Other example : Parabolic coupled systems Consider







∂tf1−∆f1+q(x)f2= 0,

∂tf2−∆f2=1ωu, f|∂Ω= 0, f(0) =f⁰.

(1) System (1) is approx. contr. if and only if

Ik(q) = Z π

0

q(x)φk(x)²dx6= 0, ∀k >0. (2) whereφk:=p

2/πsin(kx).

If (2) is satisfied and Supp(q)∩ω=∅, then the minimal time of null controllability of system (1) is given by

T^∗= lim sup−log|Ik(q)|

k² . Ammar Khodja-Benabdallah-Gonzalez-Burgos-De Teresa, 2015 Question

Is there ageneral theoryto determine the minimal time of null controllability for parabolic systems ?

∂tf+Af=Bu

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Comments

Other example : Parabolic coupled systems Consider







∂tf1−∆f1+q(x)f2= 0,

∂tf2−∆f2=1ωu, f|∂Ω= 0, f(0) =f⁰.

(1) System (1) is approx. contr. if and only if

Ik(q) = Z π

0

q(x)φk(x)²dx6= 0, ∀k >0. (2) whereφk:=p

2/πsin(kx).

If (2) is satisfied and Supp(q)∩ω=∅, then the minimal time of null controllability of system (1) is given by

T^∗= lim sup−log|Ik(q)|

k² . Ammar Khodja-Benabdallah-Gonzalez-Burgos-De Teresa, 2015 Question

Is there ageneral theoryto determine the minimal time of null controllability for parabolic systems ?

∂tf+Af=Bu

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Thanks for your attention !

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