Contrôlabilité de quelques systèmes gouvernés par des equations paraboliques.

Transcription

1 Contrôlabilité de quelques systèmes gouvernés par des equations paraboliques. Michel Duprez LMB, université de Franche-Comté 26 novembre 2015 Soutenance de thèse M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 0 / 41

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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 1 / 41

3 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 + B1 ω u in Q T, y = 0 on Σ T, y(0) = y 0 in Ω, 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 i, j n and 1 k m. (S) We denote by P : R p R n p R p, (y 1, y 2 ) y 1. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 2 / 41

4 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 y(, T ; y 0, u) y T ( ) L2 (Ω) n 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 Py(, T ; y 0, u) Py T ( ) L2 (Ω) p ε. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 3 / 41

5 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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 3 / 41

6 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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 3 / 41

7 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 ) y 1. THEOREM (see [1]) System (SDE) is controllable on (0, T ) if and only if rank(b AB... A n 1 B) = n. THEOREM 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, M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 4 / 41

8 Constant matrices Assume that A L(R n ) and B L(R m, R n ) and consider the system t y = y + Ay + B1 ω u in Q T, y = 0 on Σ T, (S) y(0) = y 0 in Ω. THEOREM (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. THEOREM (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. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 5 / 41

9 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 { B0 (t) := B(t), B i (t) := A(t)B i 1 (t) t B i 1 (t), 1 i n 1. We can define for all t [0, T ], [A B](t) := (B 0 (t) B 1 (t)... B n 1 (t)). THEOREM (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). THEOREM (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). M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 6 / 41

10 Space dependent matrices: Example of non partial null controllability Consider the control problem t y 1 = y 1 + αy 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. THEOREM (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) := j=1 1 cos(15jx) for all x (0, 2π). j2 Then system (S 2 2 ) is not partially null controllable. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 7 / 41

11 Space dependent matrices: Example of partial null controllability Consider the control problem t y 1 = y 1 + αy 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 ) THEOREM (F. Ammar Khodja, F. Chouly, M.D., 2015) Let be Ω := (0, π) and α = α p cos(px) L (Ω) satisfying α p C 1 e C 2p for all p N, where C 1 > 0 and C 2 > π are two constants. Then system (S 2 2 ) is partially null controllable. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 8 / 41

12 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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 8 / 41

13 Numerical illustration Let us consider the fonctionnal HUM J ε (u) = 1 u 2 dxdt ω (0,T ) 2ε Ω y 1 (T ; y 0, u) 2 dx. THEOREM (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 System (S 2x2 ) is partially null controllable from y 0 iff ( M 2 y 0,T := 2 sup ε>0 In this case y 1 (T ; y 0, u ε ) Ω M y0,t ε. ) inf J ε <. L 2 (Q T ) [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). M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 9 / 41

14 inf L 2 δt (0,T ;U h ) F ε h,δt (slope =-0.103) (slope =-6.34e-2) u h,δt ε L 2 δt (0,T ;U h ) y h,δt ε (T ) Eh (slope =2.74) h Figure : Case α = 1. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 10 / 41

15 inf L 2 δt (0,T ;U h ) F ε h,δt (slope =-3.80) uε h,δt L 2 δt (0,T ;U h ) (slope =-1.70) yε h,δt (T ) Eh (slope =8.34e-2) h Figure : Case α = j=1 1 j 2 cos(15jx). M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 11 / 41

16 2 1 y t 2 x Figure : Evolution of y1 in system (S2x2 ) in the case α = 1. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 12 / 41

17 y t 2 x Figure : Evolution of y1 in system (S2x2 ) in the case α = P j=1 M. Duprez Controllability of some systems governed by parabolic equations 1 j2 cos(15jx). 26/11/15 13 / 41

18 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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 13 / 41

19 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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 13 / 41

20 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 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 ij d ij l l W (Q 1 T ), = d ji l in Q T, N i,j=1 d ij l ξ i ξ j d 0 ξ 2 in Q T, ξ R N. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 14 / 41

21 Cascade condition THEOREM (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., [2] M. González-Burgos and L. de Teresa, Controllability results for cascade systems of m coupled parabolic PDEs by one control force. Port. Math., M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 15 / 41

22 Condition on the dimension THEOREM (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 0. Then System (S full ) is null controllable at time T. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 16 / 41

23 Boundary condition THEOREM (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 ) N2 for all i, j {1, 2} and { γ an open subset of ω Ω, x 0 γ s.t. g 21 (t, x 0 ) ν(x 0 ) 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. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 17 / 41

24 Necessary and sufficient condition THEOREM 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 0 or a 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 mi0 0 or a mi0 0. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 18 / 41

25 Additional result THEOREM 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 { g21 = 0 and a 21 0 in (a, b) O, 1 a 21 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 := a 21 + g 21 g x a 21 + xx g 21 a x g 21 0 g t a 21 + tx g 21 t g 21 a 21 + x g 21 0 g 21 0 xx a 21 + xxx g 21 2 x a xx g 21 0 a x g 21 0 g 21 a 22 + g 22 g 22 x d 2 1 d x a 22 + xx g 22 a x g 22 xx d 2 0 g 22 2 x d 2 1 d 2. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 19 / 41

26 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 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) a 21 (t) t g 21 (t) in (a, b). M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 20 / 41

27 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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 20 / 41

28 The notion of the algebraic resolvability can be found in: [1] M. Gromov, Partial differential relations, Springer-Verlag, 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), M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 21 / 41

29 Let f C c ([0, 1]). Consider the problem { Find (w1, 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 : ( ) w1 L = f, w 2 where the operator L is given by L := (a 1 a 2 x + a 3 xx, b 1 b 2 x + b 3 xx ). M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 22 / 41

30 If there exists a differential operator M := M i=0 m i i x with m 0,..., m M R, M N such that L M = Id, then (w 1, w 2 ) := Mf is a solution to problem (1). The last equality is formally equivalent to M L = Id, with ( L a1 + a (ϕ) = 2 x + a 3 xx b 1 + b 2 x + b 3 xx ). M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 23 / 41

31 Consider the operator defined for all ϕ Cc ([0, 1]) by L 1 a 1 + a 2 x + a 3 xx L 2 x L ϕ := b 1 + b 2 x + b 3 xx 1 a 1 x + a 2 xx + a 3 xxx ϕ = C x L 2 b 1 x + b 2 xx + b 3 xxx ϕ x ϕ xx ϕ xxx ϕ, where C := a 1 a 2 a 3 0 b 1 b 2 b a 1 a 2 a 3 0 b 1 b 2 b 3. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 24 / 41

32 Let us suppose that C is invertible, denote by c 11 c 12 c 13 c 14 C 1 := c 21 c 22 c 23 c 24 c 31 c 32 c 33 c 34. c 41 c 42 c 43 c 44 Then we have C 1 L 1 L 2 x L 1 x L 2 where the first line is given by ϕ = ϕ x ϕ xx ϕ xxx ϕ, c 11 L 1ϕ + c 12 L 2ϕ + c 13 x L 1ϕ + c 14 x L 2ϕ = ϕ. Thus the problem is solved if we define M for all (ψ 1, ψ 2 ) Cc ([0, 1]; R 2 ) by ( ) M ψ1 := c ψ 11 ψ 1 + c 12 ψ 2 + c 13 x ψ 1 + c 14 x ψ 2. 2 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 25 / 41

33 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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 25 / 41

34 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, (S full ) y(, 0) = y 0 in Ω, 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 ). THEOREM 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 0 or a 2,1 0. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 26 / 41

35 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 (ẑ, v) with Supp(ẑ, v) ω (ε, T ε) s.t. : t ẑ = div(d ẑ) + G ẑ + Aẑ + B v + N f in Q T, Conclusion: The couple (y, u) := (z ẑ, v) will be a solution to system (S full ) and will satisfy y(t ) 0 in Ω. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 27 / 41

36 Resolution of the algebraic problem The algebraic problem can be rewritten as: For f := 1 ω v, find (ẑ, v) with Supp(ẑ, v) ω (ε, T ε) such that where L(ẑ, v) = N f, L(ẑ, v) := t ẑ div(d ẑ) G ẑ Aẑ B v. This problem is solved if we can find a differential operator M such that L M = N. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 28 / 41

37 The last equality is equivalent to M L = N, where L is given for all ϕ Cc (Q T ) 2 by L L 1 ϕ ϕ := L 2 ϕ = L 3 ϕ t ϕ 1 div(d 1 ϕ 1 ) + 2 j=1 {g j1 ϕ j a j1 ϕ j } t ϕ 2 div(d 1 ϕ 2 ) + 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 ψ 1 ( ) M ψ 2 (g := 21 a 21 )ψ 3. ψ 1 + ( t + div(d 1 ) g 11 + a 11 )ψ 3 ψ 3 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 29 / 41.

38 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 T ψ(0, x) 2 dx C obs ρ 0 N ψ(t, x) 2 dxdt, Ω 0 where ρ 0 can be chosen to be exponentially decreasing at times t = 0 and t = T. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 29 / 41 ω

39 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 ψ) n 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. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 30 / 41

40 Following the ideas of Barbu developed in [1], the control is chosen as the solution of the problem minimize J k (v) := 1 ρ v 2 dxdt + k z(t ) 2 dx, Q T 2 Ω 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 ρ j j ψ 2 dxdt C T j=1 Q T T where ρ j are some appropriate weight functions. 0 ω ρ 0 N ψ(t, x) 2 dxdt, [1] V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim. 42 (2000), M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 31 / 41

41 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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 31 / 41

42 Consider the following system t z 1 = xx z 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 (0, 1 π), q L (0, π) and ω := (a, b) (0, π). Consider the two following quantities I a,k (p, q) := I k (p, q) := a 0 π 0 ( q 1 2 xp ) ϕ 2 k, ( q 1 2 xp ) ϕ 2 k, where for all k N and x (0, π), we denote by 2 ϕ k (x) := π sin(kx) the eigenvector of the Laplacian operator, with Dirichlet boundary condition. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 32 / 41

43 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: THEOREM (M. D., 2015) Let p W 1 (0, π), q L (0, π). Suppose that I k + I a,k 0 for all k N and (Supp(p) Supp(q)) ω =. Let T 0 (p, q) be given by One has T 0 (p, q) := lim sup k min( log I k (p, q), log I a,k (p, q) ) k 2. 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. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 33 / 41

44 Recall the studied system t z 1 = xx z 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). THEOREM (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. Then the system is null controllable at time T. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 34 / 41

45 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 M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 34 / 41

46 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 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 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. M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 35 / 41

47 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 = B1 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 = , A = d and B = M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 36 / 41

48 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 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? M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 37 / 41

49 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 i 3 on Σ T, y(x, 0) = y 0 in Ω, where y1 is the density of tumor cells, y 2 is the density of normal cells, y 3 is the drug concentration, u is the rate at which the drug is being injected (control), d i, a i, k i, α i,j, κ i,j are known constants. [1] S. Chakrabarty & F. Hanson, Distributed parameters deterministic model for treatment of brain tumours using Galerkin finite element method, M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 38 / 41

50 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 + B1 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 ). M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 39 / 41

51 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 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 ω v 1 in Q T, t z 2 = div(d 2 z 2 ) + a 21 z 1 + a 22 z ω v 2 in Q T, z = 0 on Σ T, z(, 0) = z 0 and the fictitious control method? in Ω M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 40 / 41

52 Merci de votre attention! M. Duprez Controllability of some systems governed by parabolic equations 26/11/15 41 / 41