26novembre2015Soutenancedethèse MichelDuprez Contrôlabilitédequelquessystèmesgouvernéspardesequationsparaboliques.

Contrôlabilité de quelques systèmes gouvernés par des equations paraboliques.

Michel Duprez

LMB, université de Franche-Comté

26 novembre 2015

Soutenance de thèse

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

Setting

Let T > 0, N ∈ N ^∗ , Ω be a bounded domain in R ^N with a C ² -class boundary ∂Ω and ω be a nonempty open subset of Ω.

We consider here the following system of n linear parabolic equations with m controls







∂ _t y = ∆y + G · ∇ y + Ay + B 1 ^ω u in Q T ,

y = 0 on Σ _T ,

y (0) = y 0 in Ω,

(S)

where Q T := Ω × (0, T ), Σ _T := ∂Ω × (0, T ), A := (a ij ) _ij , G := (g ij ) _ij and B := (b ik ) _ik with a ij , b ik ∈ L ^∞ (Q T ) and g ij ∈ L ^∞ (Q T ; R ^N ) for all 1 6 i, j 6 n and 1 6 k 6 m.

We denote by

P : R ^p × R ^n−p → R ^p ,

(y 1 , y 2 ) 7→ y 1 .

Definitions

We will say that system (S) is

null controllable on (0, T ) if for all y 0 ∈ L ² (Ω) ⁿ , there exists a u ∈ L ² (Q T ) ^m such that

y ( · , T ; y 0 , u) = 0 in Ω.

approximately controllable on (0, T ) if for all ε > 0 and y ₀ , y _T ∈ L ² (Ω) ⁿ there exists a control u ∈ L ² (Q _T ) ^m such that

k y ( · , T ; y 0 , u) − y T ( · ) k L

(Ω)

6 ε.

partially null controllable on (0, T ) if for all y ₀ ∈ L ² (Ω) ⁿ , there exists a u ∈ L ² (Q _T ) ^m such that

Py ( · , T ; y 0 , u) = 0 in Ω.

partially approximately controllable on (0, T ) if for all ε > 0 and y ₀ , y _T ∈ L ² (Ω) ⁿ there exists a control u ∈ L ² (Q _T ) ^m such that

k Py ( · , T ; y ₀ , u) − Py _T ( · ) k L

(Ω)

6 ε.

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

System of differential equations

Assume that A ∈ L ( R ⁿ ) and B ∈ L ( R ^m , R ⁿ ) and consider the system ∂ _t y = Ay + Bu in (0, T ),

y (0) = y ₀ ∈ R ⁿ . (SDE) P : R ^p × R ^n−p → R ^p , (y ₁ , y ₂ ) 7→ y ₁ . T HEOREM (see [1])

System (SDE) is controllable on (0, T ) if and only if rank(B | AB | ... | A ⁿ⁻¹ B) = n.

T HEOREM

System (SDE) is partially controllable on (0, T ) if and only if rank(PB | PAB | ... | PA ⁿ⁻¹ B) = p.

[1] R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in mathematical system theory.

McGraw-Hill Book Co., New-York, 1969.

Constant matrices

Assume that A ∈ L ( R ⁿ ) and B ∈ L ( R ^m , R ⁿ ) and consider the system







∂ _t y = ∆y + Ay + B 1 ^ω u in Q _T ,

y = 0 on Σ _T ,

y (0) = y ₀ in Ω.

(S)

T HEOREM (F. Ammar Khodja, A. Benabdallah, C. Dupaix and M.

González-Burgos, 2009)

System (S) is null controllable on (0, T ) if and only if rank(B | AB | ... | A ⁿ⁻¹ B) = n.

T HEOREM (F. Ammar Khodja, F. Chouly, M.D., 2015)

System (S) is partially null controllable on (0, T ) if and only if

rank(PB | PAB | ... | PA ⁿ⁻¹ B) = p.

Time dependent matrices

Let us suppose that A ∈ C ⁿ⁻¹ ([0, T ]; L ( R ⁿ )) and B ∈ C ⁿ ([0, T ]; L ( R ^m ; R ⁿ )) and define

B 0 (t ) := B(t),

B i (t ) := A(t)B i−1 (t) − ∂ _t B i−1 (t), 1 6 i 6 n − 1.

We can define for all t ∈ [0, T ], [A | B](t ) := (B ₀ (t) | B ₁ (t ) | ... | B n−1 (t )).

T HEOREM (F. Ammar Khodja, A. Benabdallah, C. Dupaix and M.

González-Burgos, 2009)

If there exist t ₀ ∈ [0, T ] such that

rank[A | B](t 0 ) = n, then System (S) is null controllable on (0,T).

T HEOREM (F. Ammar Khodja, F. Chouly, M. D., 2015) If

rank P[A | B](T ) = p,

then system (S) is partially null controllable on (0,T).

Space dependent matrices:

Example of non partial null controllability

Consider the control problem



 



 



∂ _t y 1 = ∆y 1 + αy 2 + 1 ^ω u in Q T ,

∂ _t y 2 = ∆y 2 in Q T .

y = 0 on Σ _T ,

y (0) = y ₀ , y ₁ (T ) = 0 in Ω.

T HEOREM (F. Ammar Khodja, F. Chouly, M.D., 2015) Assume that Ω := (0, 2π) and ω ⊂ (π, 2π). Let us consider α ∈ L ^∞ (0, 2π) defined by

α(x ) :=

∞

X

j=1

1

j ² cos(15jx) for all x ∈ (0, 2π).

Then system (S 2×2 ) is not partially null controllable.

Space dependent matrices:

Example of partial null controllability

Consider the control problem



 



 



∂ _t y ₁ = ∆y ₁ + αy ₂ + 1 ^ω u in Q _T ,

∂ _t y ₂ = ∆y ₂ in Q _T ,

y = 0 on Σ _T ,

y (0) = y ₀ , y ₁ (T ) = 0 in Ω.

(S 2×2 )

T HEOREM (F. Ammar Khodja, F. Chouly, M.D., 2015)

Let be Ω := (0, π) and α = P α _p cos(px) ∈ L ^∞ (Ω) satisfying

| α _p | 6 C 1 e ^−C

^p for all p ∈ N , where C 1 > 0 and C 2 > π are two constants.

Then system (S _2×2 ) is partially null controllable.

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

Numerical illustration

Let us consider the fonctionnal HUM J ε (u) = 1

2 Z

ω×(0,T )

u ² dxdt + 1 2ε

Z

Ω

y 1 (T ; y 0 , u) ² dx.

T HEOREM (Following the ideas of [1])

1

System (S 2x2 ) is partially approx. controllable from y 0 iff y ₁ (T ; y ₀ , u ε ) −→

ε→0 0.

2

System (S 2x2 ) is partially null controllable from y 0 iff M _y ²

_,T := 2 sup

ε>0

inf

L

(Q

) J ε

< ∞ . In this case k y ₁ (T ; y ₀ , u ε ) k ^Ω 6 M _y

_,T √

ε.

[1] F. Boyer, On the penalised HUM approach and its applications to the numerical

approximation of null-controls for parabolic problems, ESAIM Proc. 41 (2013).

10 ⁻³ 10 ⁻² 10 ⁻⁵

10 ⁻⁴ 10 ⁻³ 10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹

h

inf _L

δt

(0,T;U

) F _ε ^h,δt (slope =-0.103) k u _ε ^h,δt k L

(0,T ;U

) (slope =-6.34e-2) k y _ε ^h,δt (T ) k ^E

(slope =2.74)

Figure : Case α = 1.

10 ⁻³ 10 ⁻² 10 ⁻⁴

10 ⁻³ 10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹

h

inf _L

δt

(0,T ;U

) F _ε ^h,δt (slope =-3.80) k u ^h,δt _ε k L

(0,T ;U

) (slope =-1.70) k y _ε ^h,δt (T ) k E

(slope =8.34e-2)

Figure : Case α =

∞

P

j=1 1

j²

cos(15jx).

t 0.0

0.5 1.0

1.5 2.0

x 0

1 2

3 4 5 6 7

y1

−4

−3

−2

−1 0 1 2

Figure : Evolution of y

in system (S

) in the case α = 1.

t 0.0

0.5 1.0

1.5 2.0

x 0

1 2

3

4 5 6 7 y1

−0.20

−0.15

−0.10

−0.05 0.00 0.05 0.10 0.15 0.20

Figure : Evolution of y

in system (S

) in the case α =

∞

P

j=1 1

j²

cos(15jx).

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

Setting

Let T > 0, Ω be a bounded domain in R ^N (N ∈ N ^∗ ) of class C ² and ω be an nonempty open subset of Ω. Consider the system



 

 

 

 

∂ _t y ₁ = div(d ₁ ∇ y ₁ ) + g ₁₁ · ∇ y ₁ + g ₁₂ · ∇ y ₂ + a ₁₁ y ₁ + a ₁₂ y ₂ + 1 ω u in Q _T ,

∂ _t y 2 = div(d 2 ∇ y 2 ) + g 21 · ∇ y 1 + g 22 · ∇ y 2 + a 21 y 1 + a 22 y 2 in Q T ,

y = 0 on Σ _T ,

y( · , 0) = y ⁰ in Ω,

(S full ) where y ⁰ ∈ L ² (Ω; R ² ), u ∈ L ² (Q T ), g ij ∈ L ^∞ (Q T ; R ^N ), a ij ∈ L ^∞ (Q T ) for all i, j ∈ { 1, 2 } , Q _T := Ω × (0, T ), Σ _T := (0, T ) × ∂Ω and

( d _l ^ij ∈ W _∞ ¹ (Q _T ), d _l ^ij = d _l ^ji in Q _T ,

N

X

i,j =1

d _l ^ij ξ _i ξ _j > d ₀ | ξ | ² in Q _T , ∀ ξ ∈ R ^N .

Cascade condition

T HEOREM (see [1] and [2]) Let us suppose that

g ₂₁ ≡ 0 in q _T := ω × (0, T ) and

(a ₂₁ > a ₀ in q _T or a ₂₁ < − a ₀ in q _T ), for a constant a ₀ > 0 and q _T := ω × (0, T ).

Then System (S _full ) is null controllable at time T .

[1] F. Ammar Khodja, A. Benabdallah and C. Dupaix, Null-controllability of some reaction-diffusion systems with one control force, J. Math. Anal. Appl., 2006.

[2] M. González-Burgos and L. de Teresa, Controllability results for cascade systems of

m coupled parabolic PDEs by one control force. Port. Math., 2010.

Condition on the dimension

T HEOREM (S. Guerrero, 2007)

Let N = 1, d ₁ , d ₂ , a ₁₁ , g ₁₁ , a ₂₂ , g ₂₂ be constant and suppose that g ₂₁ · ∇ + a ₂₁ = P ₁ (θ · ) in Ω × (0, T ),

where θ ∈ C ² (Ω) with | θ | > C in ω ₀ ⊂ ω and P 1 := m 0 · ∇ + m 1 , for m 0 , m 1 ∈ R. Moreover, assume that

m 0 6 = 0.

Then System (S _full ) is null controllable at time T .

Boundary condition

T HEOREM (A. Benabdallah, M. Cristofol, P. Gaitan, L. De Teresa, 2014)

Assume that a _ij ∈ C ⁴ (Q _T ), g _ij ∈ C ³ (Q _T ) ^N , d _i ∈ C ³ (Q _T ) ^N

for all i, j ∈ { 1, 2 } and

∃ γ 6 = ∅ an open subset of ∂ω ∩ ∂Ω,

∃ x 0 ∈ γ s.t. g 21 (t, x 0 ) · ν (x 0 ) 6 = 0 for all t ∈ [0, T ], where ν is the exterior normal unit vector to ∂Ω.

Then System (S _full ) is null controllable at time T .

Necessary and sufficient condition

T HEOREM 1 (M. D., P. Lissy, 2015 )

Let us assume that d i , g ij and a ij are constant in space and time for all i, j ∈ { 1, 2 } .

Then System (S _full ) is null controllable at time T if and only if g 21 6 = 0 or a 21 6 = 0.

1

The term g 21 can be different from zero in the control domain

2

Theorem 1 is true for all N ∈ N ^∗

3

We have no geometric restriction

4

Concerning the controllability of a system with m equations controlled by m − 1 forces, the last condition becomes:

∃ i 0 ∈ { 1, ..., m − 1 } s.t. g mi

6 = 0 or a mi

6 = 0.

Additional result

T HEOREM 2 (M. D., P. Lissy, 2015 ) Let us assume that Ω := (0, L) with L > 0.

Then System (S full ) is null controllable at time T if for an open subset (a, b) × O ⊆ q _T one of the following conditions is verified:

(i) d _i , g _ij , a _ij ∈ C ¹ ((a, b), C ² ( O )) for i, j = 1, 2 and g 21 = 0 and a 21 6 = 0 in (a, b) × O ,

1

a

∈ L ^∞ ( O ) in (a, b).

(ii) d i , g ij , a ij ∈ C ³ ((a, b), C ⁷ ( O )) for i = 1, 2 and

| det(H(t , x )) | > C for every (t , x ) ∈ (a, b) × O , where

H:=















−a21+∂x g21 g21 0 0 0 0

−∂x a21+∂xx g21 −a21+2∂x g21 0 g21 0 0

−∂t a21+∂tx g21 ∂t g21 −a21+∂x g21 0 g21 0

−∂xx a21+∂xxx g21 −2∂x a21+3∂xx g21 0 −a21+3∂x g21 0 g21

−a22+∂x g22 g22−∂x d2 −1 −d2 0 0

−∂x a22+∂xx g22 −a22+2∂x g22−∂xx d2 0 g22−2∂x d2 −1 −d2













 .

Remarks

Even though the last condition seems complicated, it can be simplified in some cases :

1

System (S full ) is null controllable at time T if there exists an open subset (a, b) × O ⊆ q _T such that







g ₂₁ ≡ κ ∈ R ^∗ in (a, b) × O , a ₂₁ ≡ 0 in (a, b) × O ,

∂ _x a ₂₂ 6 = ∂ _xx g ₂₂ in (a, b) × O .

2

If the coefficients depend only on the time variable, the condition becomes :

∃ (a, b) ⊂ (0, T ) s.t.:

g ₂₁ (t)∂ _t a ₂₁ (t ) 6 = a ₂₁ (t )∂ _t g ₂₁ (t ) in (a, b).

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

The notion of the algebraic resolvability can be found in:

[1] M. Gromov, Partial differential relations, Springer-Verlag , 1986.

And was used for the first time in the control theory of pde’s in:

[2] J.-M. Coron & P. Lissy, Local null controllability of the three-dimensional

Navier-Stokes system with a distributed control having two vanishing components,

Invent. Math. 198 (2014), 833–880.

Let f ∈ C c ^∞ ([0, 1]). Consider the problem ( Find (w 1 , w 2 ) ∈ C c ^∞ ([0, 1]; R ² ) s. t. :

a ₁ w ₁ − a ₂ ∂ _x w ₁ + a ₃ ∂ _xx w ₁ + b ₁ w ₂ − b ₂ ∂ _x w ₂ + b ₃ ∂ _xx w ₂ = f , (1) with a 1 , a 2 , a 3 , b 1 , b 2 , b 3 ∈ R .

Problem (1) can be rewritten as follows : L

w 1

w 2

= f , where the operator L is given by

L := (a ₁ − a ₂ ∂ _x + a ₃ ∂ _xx , b ₁ − b ₂ ∂ _x + b ₃ ∂ _xx ) .

If there exists a differential operator M := P M

i=0 m i ∂ _x ⁱ with m 0 , ..., m M ∈ R, M ∈ N such that

L ◦ M = Id ,

then (w ₁ , w ₂ ) := M f is a solution to problem (1). The last equality is formally equivalent to

M ^∗ ◦ L ^∗ = Id , with

L ^∗ (ϕ) =

a ₁ + a ₂ ∂ _x + a ₃ ∂ _xx b ₁ + b ₂ ∂ _x + b ₃ ∂ _xx

.

Consider the operator defined for all ϕ ∈ C c ^∞ ([0, 1]) by







 L ^∗ 1

L ^∗ 2

∂ _x L ^∗ 1

∂ _x L ^∗ 2







 ϕ :=









a ₁ + a ₂ ∂ _x + a ₃ ∂ _xx b ₁ + b ₂ ∂ _x + b ₃ ∂ _xx a ₁ ∂ _x + a ₂ ∂ _xx + a ₃ ∂ _xxx b ₁ ∂ _x + b ₂ ∂ _xx + b ₃ ∂ _xxx







 ϕ = C







 ϕ

∂ _x ϕ

∂ _xx ϕ

∂ _xxx ϕ







 ,

where

C :=









a 1 a 2 a 3 0 b 1 b 2 b 3 0 0 a 1 a 2 a 3

0 b 1 b 2 b 3









.

Let us suppose that C is invertible, denote by

C ⁻¹ :=









e c 11 e c 12 e c 13 e c 14

e c 21 e c 22 e c 23 e c 24

e c 31 e c 32 e c 33 e c 34

e c 41 e c 42 e c 43 e c 44







 .

Then we have

C ⁻¹







 L ^∗ 1

L ^∗ 2

∂ _x L ^∗ 1

∂ _x L ^∗ 2







 ϕ =







 ϕ

∂ _x ϕ

∂ _xx ϕ

∂ _xxx ϕ







 ,

where the first line is given by

e c ₁₁ L ^∗ 1 ϕ + c e ₁₂ L ^∗ 2 ϕ + e c ₁₃ ∂ _x L ^∗ 1 ϕ + e c ₁₄ ∂ _x L ^∗ 2 ϕ = ϕ.

Thus the problem is solved if we define M ^∗ for all (ψ ₁ , ψ ₂ ) ∈ C ^∞ c ([0, 1]; R ² ) by

M ^∗ ψ ₁

ψ ₂

:= e c 11 ψ ₁ + e c 12 ψ ₂ + c e 13 ∂ _x ψ ₁ + e c 14 ∂ _x ψ ₂ .

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

Let us recall system (S _full ) and Theorem 1. Consider the system of two parabolic equations controlled by one force



 

 

∂ _t y = div(D ∇ y ) + G · ∇ y + Ay + e 1 1 ^ω u in Q T ,

y = 0 on Σ _T ,

y ( · , 0) = y ⁰ in Ω,

(S _full )

where y ⁰ ∈ L ² (Ω; R ² ), D := (d ₁ , d ₂ ) ∈ L ( R ^2N , R ^2N ), G := (g _ij ) ∈ L ( R ^2N , R ² ), A := (a _ij ) ∈ L ( R ² ).

T HEOREM 1 (M. D., P. Lissy, 2015 )

Let us assume that d _i , g _i,j and a _i,j are constant in space and time for all i, j ∈ { 1, 2 } .

Then System (S _full ) is null controllable at time T if and only if

g 2,1 6 = 0 or a 2,1 6 = 0.

Strategy: fictitious control method

Analytic problem

Find (z, v ) with Supp(v ) ⊂ ω × (ε, T − ε) s.t. :







∂ _t z = div(D ∇ z ) + G · ∇ z + Az + N ( 1 ω v ) in Q _T ,

z = 0 on Σ _T ,

z (0, · ) = y ⁰ , z(T , · ) = 0 in Ω, where the operator N is well chosen.

Difficulties: • the control is in the range of the operator N ,

• v has to be regular enough.

Algebraic problem:

Find ( b z, b v ) with Supp( b z , v b ) ⊂ ω × (ε, T − ε) s.t. :

∂ _t b z = div(D ∇b z) + G · ∇b z + A z b + B b v + N f in Q _T , Conclusion:

The couple (y , u) := (z − z b , −b v ) will be a solution to system (S _full )

and will satisfy y (T ) ≡ 0 in Ω.

Resolution of the algebraic problem

The algebraic problem can be rewritten as:

For f := 1 ^ω v , find ( b z , v b ) with Supp( b z, b v ) ⊂ ω × (ε, T − ε) such that L ( b z, b v ) = N f ,

where

L ( b z , v b ) := ∂ _t b z − div(D ∇b z) − G · ∇b z − A b z − B b v .

This problem is solved if we can find a differential operator M such that

L ◦ M = N .

The last equality is equivalent to

M ^∗ ◦ L ^∗ = N ^∗ , where L ^∗ is given for all ϕ ∈ C c ^∞ (Q _T ) ² by

L ^∗ ϕ :=



 L ^∗ 1 ϕ L ^∗ 2 ϕ L ^∗ 3 ϕ



 =







− ∂ _t ϕ ₁ − div(d ₁ ∇ ϕ ₁ ) + P 2

j=1 { g _j1 · ∇ ϕ _j − a _j1 ϕ _j }

− ∂ _t ϕ ₂ − div(d ₁ ∇ ϕ ₂ ) + P 2

j=1 { g _j2 · ∇ ϕ _j − a _j2 ϕ _j } ϕ ₁





 .

We remark that

(g 21 · ∇ − a 21 ) L ^∗ 1 ϕ

L ^∗ 1 ϕ + (∂ _t + div(d 1 ∇ ) − g 11 · ∇ + a 11 ) L ^∗ 3 ϕ

= N ^∗ ϕ, where

N ^∗ := g ₂₁ · ∇ + a ₂₁ . Thus the algebraic problem is solved by taking

M ^∗



 ψ ₁ ψ ₂ ψ ₃



 :=

(g ₂₁ · ∇ − a ₂₁ )ψ ₃

ψ ₁ + (∂ _t + div(d 1 ∇ ) − g 11 · ∇ + a 11 )ψ ₃

.

Resolution of the analytic problem: sketch of the proof

The system







∂ _t z = div(D ∇ z ) + G · ∇ z + Az + N ( 1 ^ω v ) in Q T ,

z = 0 on Σ _T ,

z (0, · ) = y ⁰ , z(T , · ) = 0 in Ω,

is null controllable if for all ψ ⁰ ∈ L ² (Ω; R ² ) the solution of the adjoint system







− ∂ _t ψ = div(D ^∗ ∇ ψ) − G ^∗ · ∇ ψ + A ^∗ ψ in Q T ,

ψ = 0 on Σ _T ,

ψ(T , · ) = ψ ⁰ in Ω

satisfies the following inequality of observability Z

Ω | ψ(0, x ) | ² dx 6 C _obs Z T

0

Z

ω

ρ ₀ |N ^∗ ψ(t , x ) | ² dxdt ,

where ρ ₀ can be chosen to be exponentially decreasing at times t = 0

and t = T .

This inequality is obtained by applying the differential operator

∇∇N ^∗ = ∇∇ ( − a ₂₁ + g ₂₁ · ∇ )

to the adjoint system. More precisely we study the solution φ _ij := ∂ _i ∂ _j N ^∗ ψ of the system







− ∂ _t φ _ij = D∆φ _ij − G ^∗ · ∇ φ _ij + A ^∗ φ _ij in Q _T ,

∂φ

∂n = ^∂(∇∇N _∂n

^ψ) on Σ _T ,

φ(T , · ) = ∇∇N ^∗ ψ ⁰ in Ω.

Remark : We need that ∇∇N ^∗ commutes with the other operators of

the system, what is possible with some constant coefficients.

Following the ideas of Barbu developed in [1], the control is chosen as the solution of the problem







minimize J k (v ) := 1 2 Z

Q

ρ ⁻¹ ₀ | v | ² dxdt + k 2

Z

Ω | z (T ) | ² dx, v ∈ L ² (Q T , ρ ^−1/2 ₀ ) ² .

We obtain the regularity of the control with the help of the following Carleman inequality

3

P

j =1

x

Q

ρ _j |∇ ^j ψ | ² dxdt 6 C T

Z T 0

Z

ω

ρ ₀ |N ^∗ ψ(t , x ) | ² dxdt ,

where ρ _j are some appropriate weight functions.

[1] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim.

42 (2000), 73–89.

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

Consider the following system



 

 

 

 

∂ _t z 1 = ∂ _xx z 1 + 1 ω u in (0, π) × (0, T ),

∂ _t z 2 = ∂ _xx z 2 + p(x )∂ _x z 1 + q(x )z 1 in (0, π) × (0, T ), z(0, · ) = z(π, · ) = 0 on (0, T ), z( · , 0) = z ⁰ in (0, π),

(S simpl. )

where z ⁰ ∈ L ² ((0, π); R ² ), u ∈ L ² ((0, π) × (0, T )), p ∈ W _∞ ¹ (0, π), q ∈ L ^∞ (0, π) and ω := (a, b) ⊂ (0, π). Consider the two following quantities



 



 



I a,k (p, q) :=

Z a 0

q − ¹ 2 ∂ _x p ϕ ² _k ,

I k (p, q) :=

Z π 0

q − ¹ 2 ∂ _x p ϕ ² _k ,

where for all k ∈ N ^∗ and x ∈ (0, π), we denote by ϕ _k (x ) :=

q 2

π sin(kx ) the eigenvector of the Laplacian operator, with

Dirichlet boundary condition.

When the supports of the control and the coupling terms are disjoint in System (S simpl . ), following the ideas F. Ammar Khodja et al, we obtain a minimal time of null controllability:

T HEOREM (M. D., 2015)

Let p ∈ W _∞ ¹ (0, π), q ∈ L ^∞ (0, π). Suppose that | I _k | + | I _a,k | 6 = 0 for all k ∈ N and

(Supp(p) ∪ Supp(q)) ∩ ω = ∅ . Let T 0 (p, q) be given by

T ₀ (p, q) := lim sup

k→∞

min( − log | I _k (p, q) | , − log | I _a,k (p, q) | )

k ² .

One has

1

If T > T 0 (p, q), then System (S simpl. ) is null controllable at time T .

2

If T < T ₀ (p, q), then System (S _simpl . ) is not null controllable at

time T .

Recall the studied system



 

 

 

 

∂ _t z ₁ = ∂ _xx z ₁ + 1 ω u in (0, π) × (0, T ),

∂ _t z ₂ = ∂ _xx z ₂ + p(x)∂ _x z ₁ + q (x )z ₁ in (0, π) × (0, T ), z (0, · ) = z (π, · ) = 0 on (0, T ), z ( · , 0) = z ⁰ in (0, π),

where z ⁰ ∈ L ² ((0, π); R ² ), u ∈ L ² ((0, π) × (0, T )), p ∈ W _∞ ¹ (0, π), q ∈ L ^∞ (0, π) and ω := (a, b) ⊂ (0, π).

T HEOREM (M. D., 2015)

Suppose that p ∈ W _∞ ¹ (0, π) ∩ W _∞ ² (ω), q ∈ L ^∞ (0, π) ∩ W _∞ ¹ (ω) and (Supp(p) ∪ Supp(q)) ∩ ω 6 = ∅ .

Then the system is null controllable at time T .

1 Partial controllability of parabolic systems Results

Numerical simulations

2 Indirect controllability of parabolic systems coupled with a first order term

Full system

Local inversion of a differential operator Proof in the constant case

Simplified system

3 Conclusion

Works in progress with P. Lissy

1

A general condition for the null controllability by m − 1 forces of m parabolic equations coupled by zero and first order terms:



 

 

 

 

∂ _t y 1 = div(d 1 ∇ y 1 ) + g 11 · ∇ y 1 + g 12 · ∇ y 2 + a 11 y 1 + a 12 y 2 + 1 ^ω u in Q T ,

∂ _t y ₂ = div(d ₂ ∇ y ₂ ) + g ₂₁ · ∇ y ₁ + g ₂₂ · ∇ y ₂ + a ₂₁ y ₁ + a ₂₂ y ₂ in Q _T ,

y = 0 on Σ _T ,

y ( · , 0) = y ⁰ in Ω.

(S _full ) System (S _full ) is null controllable at time T if

g ₂₁ 6 = 0 in q _T and a ₂₂ belong to an appropriate space.

Using a non linear variable change it can be proved that the

second condition is artificial.

Works in progress with M. Gonzalez-Burgos and D.

Souza

2

The null controllability for a parabolic systems with non-diagonalizable diffusion matrix:

Consider the parabolic systems







y t − Dy xx + Ay = B 1 ^ω u in (0, π) × (0, T ), y ( · , 0) = 0 on (0, T ),

y (0, · ) = y ⁰ in (0, π),

where y ⁰ ∈ L ² (0, π; R ⁿ ) is the initial data, u ∈ L ² (Q _T ) is the control and D, A ∈ L ( R ⁿ ) and B ∈ L ( R , R ⁿ ) are given, for d > 0, by

D =













d 1 0 · · · 0 0 . .. ... ... ...

.. . . .. ... ... 0 .. . . .. ... 1 0 · · · · 0 d











 , A =









0 0 · · · 0 .. . .. . .. . 0 .. . .. . 1 0 · · · 0









and B =





 0 .. . 0 1





 .

Comments and open problems

1

Partial null controllability of a parabolic system:

Consider the system



 



 



∂ _t y ₁ = ∆y ₁ + αy ₂ in Q _T ,

∂ _t y ₂ = ∆y ₂ + 1 ^ω u in Q _T .

y = 0 on Σ _T ,

y (0) = y ₀ in Ω.

When Supp(α) ∩ ω = ∅ , there exists a minimal time T ₀ for the null controllability.

If T < T ₀ , is it possible, for all y ₀ ∈ L ² (Ω; R ² ), to find a control

u ∈ L ² (Q _T ) such that y ₁ (T ) = 0 in Ω?

Comments and open problems

2

Partial controllability of semilinear systems:

Let Ω ⊂ R ³ , T > 0, Q T := Ω × (0, T ) and Σ _T := ∂Ω × (0, T ).

Consider the system



 

 

 



 

 

 



∂ _t y ₁ = d ₁ ∆y ₁ + a ₁ (1 − y ₁ /k ₁ )y ₁ − (α _1,2 y ₂ + κ _1,3 y ₃ )y ₁ in Q _T ,

∂ _t y ₂ = d ₂ ∆y ₂ + a ₂ (1 − y ₂ /k ₂ )y ₂ − (α _2,1 y ₁ + κ _2,3 y ₃ )y ₂ in Q _T ,

∂ _t y 3 = d 3 ∆y 3 − a 3 y 3 + u in Q T ,

∂ _n y i := ∇ y i · ~ n = 0 ∀ 1 6 i 6 3 on Σ _T ,

y (x , 0) = y 0 in Ω,

where

y

is the density of tumor cells, y

is the density of normal cells, y

is the drug concentration,

u is the rate at which the drug is being injected (control), d

, a

, k

, α

, κ

are known constants.

[1] S. Chakrabarty & F. Hanson, Distributed parameters deterministic model for

treatment of brain tumours using Galerkin finite element method, 2009.

Comments and open problems

3

Null controllability of a general parabolic system:

Consider the system of n equations controlled by m forces







∂ _t y = ∆y + G · ∇ y + Ay + B 1 ^ω u in Q T ,

y = 0 on Σ _T ,

y ( · , 0) = y ⁰ in Ω,

where y ⁰ ∈ L ² (Ω; R ⁿ ), G ∈ L ^∞ (Q _T ; L ( R ^nN , R ⁿ )),

A ∈ L ^∞ (Q _T ; L ( R ⁿ )), B ∈ L ^∞ (Q _T ; L ( R ^m , R ⁿ )) and u ∈ L ² (Q _T ; R ^m ).

Comments and open problems

4

Numerical scheme using the algebraic resolvability:

Is it possible to deduce a numerical approximation of the solution to the system with one control force



 

 

 

 

∂ _t y 1 = div(d 1 ∇ y 1 ) + a 11 y 1 + a 12 y 2 + 1 ^ω u in Q T ,

∂ _t y ₂ = div(d ₂ ∇ y ₂ ) + a ₂₁ y ₁ + a ₂₂ y ₂ in Q _T ,

y = 0 on Σ _T ,

y ( · , 0) = y ⁰ in Ω,

using the study of the system with two control forces



 

 

 

 

∂ _t z 1 = div(d 1 ∇ z 1 ) + a 11 z 1 + a 12 z 2 + 1 ω v 1 in Q T ,

∂ _t z ₂ = div(d ₂ ∇ z ₂ ) + a ₂₁ z ₁ + a ₂₂ z ₂ + 1 ^ω v ₂ in Q _T ,

z = 0 on Σ _T ,

z( · , 0) = z ⁰ in Ω

and the fictitious control method?

Merci de votre attention !