The Fattorini Criterion for the Stabilizability of Parabolic Systems and its Application to MHD flow and fluid-rigid body interaction systems

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1 The Fattorini Criterion for the Stabilizability of Parabolic Systems and its Application to MHD flow and fluid-rigid body interaction systems Mehdi Badra Laboratoire LMA, université de Pau et des Pays de l Adour joint works with Takéo Takahashi (Nancy university) Kick-Off Meeting IFCAM, Nice, 21 novembre 2012

2 Outline Introduction Stabilizability of infinite dimensional systems Magnetohydrodynamic flow system Fluid-rigid body interaction system Conclusion

3 Outline Introduction Stabilizability of infinite dimensional systems Magnetohydrodynamic flow system Fluid-rigid body interaction system Conclusion

4 Infinite dimensional nonlinear system y = Ay + G(y, u) + Bu, y(0) = y 0 H A generates an analytic semigroup on H. A has compact resolvent. B : U [D(A )] is relatively bounded : (λ 0 A) γ B : U H is bounded for 0 γ < 1. G(0) = G (0) = 0.

5 Stabilization problem Problem. For σ > 0, find a finite dimensional feedback control : Goal. u(t) = F y(t) such that y(t) Ce σt y 0 1. To propose a methodology to contruct stabilizing finite dimensional feedback control : (v 1,..., v K ) U K, L : U H u(t) = F y(t) = K (y(t) Lv j )v j j=1 2. To give exemples of applications for a MHD flow system and a simple fluid-rigid body interaction system.

6 Stabilization of the Navier-Stokes equations Flow around a cylinder, Re = 200 :

7 Stabilization of the Navier-Stokes equations Flow around a cylinder, Re = 200 :

8 Stabilization of the Navier-Stokes equations Flow around a cylinder, Re = 200 :

9 Stabilization of the Navier-Stokes equations t v ν v + (v )v + r = f S in (0, ) Ω def = Q v = 0 in (0, ) Ω Stationary state V S : y def = v V S satisfies y t ν y + (V S )y + (y )V S + (y )y = p in Q, Dirichlet control u(t) on Γ Ω : (I P )y = (I P )Du (P y) = AP y + G(P y, u) + Bu Goal. to find u = F y stabilizing v around V S, i.e : lim y(t) = 0 t y = 0 in Q,

10 Stabilization of the Navier-Stokes equations Stabilizing control ([B., Takahashi 2011]) : s [ d 2 2, 1], y 0 H s ε = y(t) H s Ce σt y 0 H s Dirichlet (2D) : y(t) = Dynamical (3D) : y(t) = dū(t) dt K ( ) y(t) Lv j dγ Ω j=1 K u j (t)v j on Ω i=1 v j on Ω = D K ū(t) + (ḡ C 1 y(t))ḡ + (ḡ C 2 u(t))ḡ where ū = (u 1,..., u K ) References : [Barbu - Triggiani 2004], [Barbu - Lasieska - Triggiani 2006], [Raymond - Thevenet 2009].

11 Criterion for stabilizability N-S-E with Dirichlet control on Γ Ω is stabilizable iff the following problem : λε ν ε (V S )ε + t ( V S )ε + χ = 0 in Ω, ε = 0 in Ω, ε = 0 on Ω, with overdetermined conditions admits only the zero solution. χ n ν ε = 0 on Γ, n [Fabre, Lebeau 1996]

12 Criterion for stabilizability N-S-E with internal distributed control in ω Ω is stabilizable iff the following problem : λε ν ε (V S )ε + t ( V S )ε + χ = 0 in Ω, ε = 0 in Ω, ε = 0 on Ω, with overdetermined conditions admits only the zero solution. ε = 0 in ω, [Fabre, Lebeau 1996]

13 Criterion for stabilizability N-S-E with tangential Dirichlet control on Γ Ω is stabilizable iff the following problem : λε ν ε (V S )ε + t ( V S )ε + χ = 0 in Ω, ε = 0 in Ω, ε = 0 on Ω, with overdetermined conditions admits only the zero solution. ε = 0 on Γ, n Open Problem...

14 Criterion for stabilizability N-S-E with normal Dirichlet control on Γ Ω is stabilizable iff the following problem : λε ν ε (V S )ε + t ( V S )ε + χ = 0 in Ω, ε = 0 in Ω, ε = 0 on Ω, with overdetermined conditions admits only the zero solution. χ = 0 on Γ, Open Problem...

15 Outline Introduction Stabilizability of infinite dimensional systems Magnetohydrodynamic flow system Fluid-rigid body interaction system Conclusion

16 General methodology Step 1 : rewrite PDE system in the form y = Ay + G(y, u) + Bu, y(0) = y 0 H Step 2 : find F : U H such that system y = Ay + Bu, y(0) = y 0 H with satisfies u = F y y(t) H Ce σt y 0 H Step 3 : for y 0 small enough, obtain a stable solution to y = Ay + BF y + G(y, F y), y(0) = y }{{}}{{} 0 A F y N(y) with a fixed-point procedure.

17 General methodology Step 1 : rewrite PDE system in the form y = Ay + G(y, u) + Bu, y(0) = y 0 H Step 2 : find F : U H such that system y = Ay + Bu, y(0) = y 0 H with satisfies u = F y y(t) H Ce σt y 0 H Step 3 : for y 0 small enough, obtain a stable solution to y = Ay + BF y + G(y, F y), y(0) = y }{{}}{{} 0 A F y N(y) with a fixed-point procedure.

18 Step 2 : linear stabilization problem y = Ay + Bu [D(A )], y(0) = y 0 H (1) A : D(A) H H generates an analytic semigroup on H. A : D(A) H H has compact resolvent. B : U [D(A )]. Feedback stabilizability : For σ > 0 find u = F y and C > 0 such that y 0 H : y(t) H Ce σt y 0 H

19 Step 2 : linear stabilization problem y = Ay + Bu [D(A )], y(0) = y 0 H (1) A : D(A) H H generates an analytic semigroup on H. A : D(A) H H has compact resolvent. B : U [D(A )]. Feedback stabilizability : For σ > 0 find u = F y and C > 0 such that y 0 H : y(t) H Ce σt y 0 H Open-loop stabilizability is sufficient : For σ > 0 and y 0 find u L 2 (0, + ; U) and C > 0 s. t. y(t) H Ce σt.

20 Step 2 : open-loop stabilizability First strategy : prove that (1) is null controllable : for T > 0 find u L 2 (0, T ; U) such that y(t ) = 0. Or equivalently, prove the observability inequality : T 0 B e A (T τ) ε 2 dτ C T e A T ε H Use global Carleman inequalities...

21 Step 2 : open-loop stabilizability Second strategy : prove that the unstable part of (1) is null controllable. Spectrum σ(a) = {λ k k N } and σ > 0 such that Rλ N+1 < σ < Rλ N Rλ 2 Rλ 1 P N Projected systems : B N = P N B and B N = (I P N)B { y y = Ay + Bu N = A N y N + B N u, y N = A N y N + B N u, For T > 0 find u L 2 (0, T ; U) such that y N (T ) = 0 Approximate controllability is sufficient : B e A (T τ) ε = 0 τ [0, T ] = ε = 0

22 Step 2 : stabilizability criterion Finite dimensional projected system defined on H N = P N H : y N = A N y N + B N u (2) Approximate controllability criterion for the infinite dimensional system (1) ([Fattorini 1966]) [ ] λ C, A ε = λε and B ε = 0 = ε = 0. Controllability test for (2) ([Hautus 1969]) k = 1,... N, rank[a N λ k B N ] = dim(h N ) or equivalently, λ C, Rλ > σ, [ ] A ε = λε and B ε = 0 = ε = 0.

23 Step 2 : stabilizability criterion Finite dimensional projected system defined on H N = P N H : y N = A N y N + B N u (2) Approximate controllability criterion for the infinite dimensional system (1) ([Fattorini 1966]) [ ] λ C, A ε = λε and B ε = 0 = ε = 0. Controllability test for (2) ([Hautus 1969]) k = 1,... N, rank[a N λ k B N ] = dim(h N ) or equivalently, λ C, Rλ > σ, [ ] A ε = λε and B ε = 0 = ε = 0. Stabilizability criterion

24 Step 2 : finite dimensional control For v = (v 1,..., v K ) U K and ū = (u 1,..., u K ) L 2 (C K ) set : u(t) = and define V (v)ū = The linear system becomes : K u j (t)v j, j=1 K u j Bv j. j=1 y = Ay + V (v)ū [D(A )] (3)

25 Step 2 : finite dimensional control For k = 1,..., N introduce : Geometric multiplicity l k = dim Ker(A λ k ) Eigenvectors related to λ k Ker(A λ k ) = span C { ε i k, i = 1,..., l k } Controllability matrix W k (v) = (v 1 B ε 1 k )... (v j B ε 1 k )... (v K B ε 1 k )... (v 1 B ε i k )... (v j B ε i k )... (v K B ε i k )... (v 1 B ε l k k )... (v j B ε l k k )... (v K B ε l k k ) Theorem System (3) is stabilizable if and only if k = 1,..., N, rankw k (v) = l k

26 Step 2 : genericity of admissible families Necessary conditions : λ C, Rλ > σ, K max k} k=1,...,n (4) [ ] A ε = λε and B ε = 0 = ε = 0. (5) Those conditions are also sufficient : Theorem (B.,Takahashi 2011) The set of admissible families v = (v 1,..., v K ) U K is an open and dense subset of U K if and only if (4) and (5) are satisfied. For instance choose v as a basis of the set span C {B ε i k k = 1,..., N, i = 1,..., l k} ( or of span R {B Rε i k, B Iε i k k = 1,..., N, i = 1,..., l k})

27 Step 2 : application to linearized N-S-E For linearized N-S-E with Dirichlet control the criterion [ ] A ε = λε and B ε = 0 = ε = 0 is equivalent to the following unique continuation property : λε ν ε (V S )ε + t ( V S )ε + χ = 0 in Ω, ε = 0 in Ω, ε = 0 on Ω, χ n ν ε n = 0 on Ω, ε 0. Consequence : the linearized N-S-E are stabilizable ([Fabre, Lebeau 1996])

28 Step 2 : feedback stabilizability Goal : find a control of the form u = F y such that : y(t) Ce σt y 0 H Spectrum σ(a) = {λ k k N } and σ > 0 such that Rλ N+1 < σ < Rλ N Rλ 2 Rλ 1 P N def Projected systems : V N = P N V (v), V N def = (I P N )V (v), y = Ay + V (v)ū { y N = A N y N + V N ū, y N = A N y N + V N ū, If v = (v 1,..., v K ) U K is admissible then y N = A N y N + V N ū is controllable

29 Step 2 : finite dimensional quadratic cost problem { min I(y N, ū) with I(y N, ū) = 1 2 } y N = (A N + σ)y N + V N ū, y N (0) = P N y 0 0 y N (t) 2 dt ū(t) 2 dt. Because v is admissible then I(y N, ū) < + for some (y N, ū) and the above minimization problem is well-posed. Finite dimensional Riccati equation and closed-loop system : ΠA N + A NΠ ΠV N V NΠ + P N + 2σΠ = 0, Π = Π 0 y N = A N y N V N (V NΠ)y N, y N (t) H C P N y 0 e σt

30 Step 2 : infinite dimensional linear closed-loop system Expression of the finite dimensional feedback control : ū(t) = V NΠy N (t) u(t) = F y(t) = K (y(t) Lv j )v j, j=1 Complete stable controlled system : For the following set y = Ay + V (v)f y K ([B ΠP N ]y(t) v j )v j j=1 y(t) H Ce σt y 0 H def A F = A + V (v)f L def = [B ΠP N ]

31 Step 3 : nonlinear existence theorem Nonlinear closed-loop system : { y = A (NL) F y + N(y) y(0) = y 0 Functional framework : V s = D(( A F ) s 2 ) and V s = [ D(( A F ) s 2 ) ], V 0 = H Maximal regularity result : y = A F y + f L 2 (V s 1 ) y(0) = y 0 V s y L 2 (V s+1 ) H 1 (V s 1 ) ( and then y C(V s ) ) Fixed-point argument : { y z = A (NL) F y z + N(z), y z (0) = y 0

32 Conclusion : general abstract stabilization theorem With the following assumption on the nonlinear term N(ξ) V s 1 C ξ V s ξ V s+1, N(ξ) N(ζ) V s 1 C( ξ ζ V s ξ V s+1 + ζ V s ξ ζ V s+1) we have the following theorem : Theorem (B. Takahashi 2011) Assume that v is admissible. Then there exist ρ > 0 and µ > 0 such that, if y 0 V s obeys y 0 V s < µ, system (NL) admits a solution def = { y L 2 (V s+1 ) H 1 (V s 1 } ) y L 2 (V s+1 ) H 1 (V s 1 ) ρ y 0 V s B s δ unique within the class of functions in L loc (Vs ) L 2 loc (Vs+1 ). Moreover, there exists C > 0 such that y(t) V s C y 0 V se σt.

33 Outline Introduction Stabilizability of infinite dimensional systems Magnetohydrodynamic flow system Fluid-rigid body interaction system Conclusion

34 Stationary magnetohydrodynamic flow V S (x) r S (x) B S (x) Ω V S = V S (x) is the velocity of the fluid ; B S = B S (x) is the magnetic field ; r S = r S (x) is the pressure Steady state flow of a viscous, incompressible, electrically conducting fluid Example of 2D Hartmann Channel flow : V S = (V S 1 (x 2 ), 0), B S is constant, r S = αx 1 + β

35 Stationary magnetohydrodynamic flow Equations of stationary, incompressible magnetohydrodynamics in a bounded domain Ω R 3 V S + (V S )V S (curl B S ) B S + r S = f S in Ω, curl(curl B S ) curl(v S B S ) = 0 in Ω, div V S = div B S = 0 in Ω, V S = g S (curl B S V S B S ) n = e S τ B S n = b S n on Ω, on Ω, on Ω, V S def = velocity, r S def = pressure, B S def = magnetic field Reynolds number, interaction parameter and magnetic Reynolds number are set to 1 for simplicity

36 Instationary magnetohydrodynamic flow Instationary controlled flow (t > 0) v(t, x) r(t, x) ω u 1 (t, x) u 2 (t, x) B(t, x) Ω V = V (t, x) is the velocity of the fluid ; B = B(t, x) is the magnetic field ; u 1 = u 1 (t, x), u 2 = u 2 (t, x) are the controls

37 Stabilization problem v t v + (v )v (curl B) B + r = f S + 1 ω u 1 in Q, B t + curl(curl B) curl(v B) = 1 ω P ω u 2 in Q, div v = div B = 0 in Q, v = g S, B n = b S n on Σ, (curl B v B) n = e S τ on Σ, v(0) = v 0, B(0) = B 0 in Ω, def def 1 ω = characteristic function of ω Ω ; P ω = Leray projection operator related to ω. Goal. to find (u 1, u 2 ) stabilizing (v, B) around (V S, B S ) : lim v(t) V S + B(t) B S = 0 t Assumptions. Ω R 3 bounded, simply connected, Ω of class C 2,1 and (V S, B S ) (H 2 (Ω)) 3 (H 2 (Ω)) 3.

38 Theorem (B.,Takahashi 2012) Let s [ 1 2, 1]. For all σ > 0 there exist families (v1 j, v2 j ) and (ŵ j, ρ j ), j = 1,..., K, and µ > 0 such that if v 0 V S (H s (Ω)) 3 + B 0 B S (H s (Ω)) 3 µ, then the MHD system with feedback control ( u 1 (t) u 2 (t) ) = K ( v 1 j j=1 v 2 j ) admits a unique solution (v, B) in Ω ( (v(t) V S ) ŵ j + (B(t) B S ) ρ j ) dx, W loc ((H s+1 (Ω)) 3, (H s 1 (Ω)) 3 ) W loc ((H s+1 (Ω)) 3, (H s 1 (Ω; R)) 3 ). Moreover, there exists C > 0 such that : v(t) V S (H s (Ω)) 3+ B(t) BS (H s (Ω)) d Ce σt ( v 0 V S (H s (Ω)) 3 + B 0 B S (H s (Ω)) 3).

39 References Null controllability results. Non local control in the magnetic field equations : P 1 ω u 2 = 1 ω u 2 + q Barbu, Havârneanu, Popa, Sritharan (2003, 2005), Havârneanu, Popa, Sritharan (2006, 2007) Stabilizability results. 2D Hartmann Channel flow : Xu, Schuster, Vazquez, Krstic (2008), Schuster, Luo, Krstic (2008) 3D domain and non local control in M-F-E. : Lefter (2011) 2D domain and local control 1 ω P ω u 2 : Lefter (2012) (V S, B S W 2, (Ω))

40 Abstract system Define the state ( ) v V S y = B B S Define the control and control operator ( ) ( u 1 P 1ω u 1 ) u = u 2, Bu = 1 ω P ω u 2 Abstract system y = Ay + G(y) + Bu

41 Fattorini criterion for stabilizability of the linear system For λ C, Rλ > σ, prove : [λε = A ε, B ε = 0] = ε 0. Or equivalently, prove that the solution (w, ρ) of λw w (Dw)V S + S(B S )(curl ρ) + π = 0 in Ω, λρ ρ + (Dw)B S S(V S )(curl ρ) + κ = 0 in Ω, div w = div ρ = 0 in Ω, w = 0, ρ n = 0, (curl ρ) n = 0 on Ω, and w = 0, ρ = p in ω must be equal to zero in Ω.

42 Fattorini criterion for stabilizability of the linear system Set ζ def = curl ρ and apply curl to the equation of ρ to get λw w (Dw)w S + S(B S )ζ + π = 0 in Ω, λζ ζ + curl ( (Dw)B S S(V S )ζ ) = 0 in Ω, div z = 0 in Ω. and w = 0, ζ = 0 in ω The use Carleman ineq. for Laplace and Stokes eq. yields w = curl ρ = 0 and λρ = κ in Ω.

43 Fattorini criterion for stabilizability of the linear system w = curl ρ = 0 and λρ = κ in Ω. (6) 1. Case λ 0 : (6) with { div ρ = 0 in Ω ρ n = 0 on Ω = ρ = 0 in Ω 2. Case λ = 0 : the fact that Ω is simply connected guarantees that div ρ = 0, curl ρ = 0 in Ω and ρ n = 0 on Ω implies ρ = 0 in Ω. (If Ω is multiply connected then there are uncontrollable modes associated to the eigenvalue λ = 0)

44 Open question 1 Stabilizability if Ω is multiply connected? v(t, x) r(t, x) ω u 1 (t, x) u 2 (t, x) B(t, x) Ω

45 Open question 2 Stabilizability for controls with disjoint supports? v(t, x) r(t, x) u 2 (t, x) ω 2 ω 1 u 1 (t, x) B(t, x) Ω

46 Outline Introduction Stabilizability of infinite dimensional systems Magnetohydrodynamic flow system Fluid-rigid body interaction system Conclusion

47 Simplified 1d fluid particle system t = 0 : V S (x) V S (x) 1 H S 1

48 Simplified 1d fluid particle system t = 0 : V S (x) V S (x) 1 H S 1 t = 0 : v 0 (x) v 0 (x) 1 h 0 1

49 Simplified 1d fluid particle system V S (x) V S (x) t = 0 : 1 H S 1 v 0 (x) v 0 (x) t = 0 : 1 h 0 1 v(t, x) v(t, x) t > 0 : u(t) 1 h(t) H S 1

50 Stationary states t = 0 : V S (x) V S (x) 1 H S 1 For f S, r S, a S and b S given : Vyy S + V S Vy S = f S (y ( 1, 1) \ {H s }), V S ( 1) = a S, V S (H S ) = 0, V S (1) = b S, where [f](x) def = f(x + ) f(x ). 0 = [V S y ](H S ) + mr S. If f S = 0, r S = 0 then V S is of the form : V S (y) = 2c tan(c(y H S )), c 0 V S (y) = 2c tanh(c(y H S )) c 0

51 Simplified 1d fluid particle system v 0 (x) v 0 (x) t = 0 : 1 h 0 1 v(t, x) v(t, x) t > 0 : u(t) 1 h(t) H S 1 v = v(t, x) is the velocity of the fluid ; h = h(t) is the position of the particle ; u = u(t) is the control v t v xx + vv x = f S (t 0, x ( 1, 1) \ {h(t)}), v(t, h(t)) = ḣ(t), mḧ(t) = [v x](t, h(t)) + mr S (t 0), v(t, 1) = a S + u(t), v(t, 1) = b S, (t 0), h(0) = h 0, ḣ(0) = l 0, v(0, x) = v 0 (x) x ( 1, 1) \ {h 0 }.

52 Goal and assumptions Goal. For σ > 0 find u in feedback form such that v(t) V S L 2 ( 1,1) + ḣ(t) + h(t) HS Ce σt ( v 0 V S L 2 (( 1,1)) + h 1 + h 0 H S ). Assumptions. (A 1 ) f S W 2, (( 1, 1)) and V S W 1, (( 1, 1)) (A 2 ) One of the following two conditions holds 1. Vy S (H S ) < σ 2. If ϕ satisfies { V S y (H S )ϕ ϕ yy V S ϕ y = 0, y (H S, 1) ϕ(h S ) = ϕ(1) = 0 (7) then ϕ 0 on ( 1, 1).

53 Feedback Stabilization result Theorem (B.,Takahashi 2012) There exist µ > 0 and ( ϕ, ĝ, k) L 2 ( 1, 1) R 2, such that, if v 0 V S L 2 (( 1,1)) + h 1 + h 0 H S µ then there exists a unique solution v L 2 loc (R +, H 1 ( 1, 1)) C(R +, L 2 ( 1, 1)), h H 1 loc (R +), with 1 ( u(t) = v(t, X(h(t), y)) V S (y) ) ϕdy + mḣ(t)ĝ + h(t) k. 1 Moreover, there is a constant C > 0 such that v(t) V S L 2 ( 1,1) + ḣ(t) + h(t) Ce σt ( v 0 V S L 2 (( 1,1)) + h 1 + h 0 ).

54 References Well-posedness. Vázquez and Zuazua (2003, 2006) Controllability results. Doubova and Fernández-Cara (2005), V S 0, use of two controls, one at each boundary. Liu, Takahashi and Tucsnak (2012), V S 0, use of a single control.

55 Change of variables X(h(t), y) def = y + η(y)(h(t) H S ), with η C (( 1, 1)) satisfying η(h S ) = 1, η y (H S ) = 0, η(1) = η( 1) = 0. We set V (t, y) def = v(t, X(h(t), y)) and V = W + V S To simplify we assume for the following : H S = 0.

56 System after change of variables W t W yy + h ( η V S y ηf S) y + (V S W ) y ηlv S y = G (W, l, h) (t 0, y ( 1, 1) \ {0}), W (t, 0) = l(t) (t 0), l(t) = 1 m [W y](t, 0) (t 0), ḣ(t) = l(t) (t 0), W (t, 1) = u(t), W (t, 1) = 0 (t 0), h(0) = h 0, l(0) = l 0, W (0, y) = V 0 (y) V S (y) y ( 1, 1) \ {0}.

57 Abstract setting H def = L 2 (( 1, 1)) R 2 { W D(A) def = l H W (0) = l h W W yy h ( η Vy S A l def = h l ϕ B : R [D(A )], B g = ϕ y ( 1) k W H 2 (( 1, 1) \ {0}) H0 1 (( 1, 1)), ηf S) y (V S W ) y + ηlv S y 1 m [W y](0) W W W l = A l + G l + Bu, h h h }.

58 Fattorini criterion for stabilizability Assume and and λ C, Rλ σ, λϕ ϕ yy V S ϕ y = 0, y ( 1, 1) \ {0}, ϕ(0) = g, mλg = [ϕ y ](0) + k + λk = 1 1 ( η V S y ϕ( 1) = ϕ(1) = 0, 1 1 ηv S y ϕdy, + ηf S) y ϕdy, ϕ y ( 1) = 0. Do we have g = k = 0 and ϕ 0 in ( 1, 1)?

59 Fattorini criterion for stabilizability Standart uniqueness result yield ϕ 0 in ( 1, 0) and then g = ϕ(0) = 0 and λϕ ϕ yy V S ϕ y = 0, y (0, 1), ϕ(0) = ϕ(1) = 0, λϕ y (0) = 1 0 ( η Vy S ηf S) 1 y ϕdy 0 ληv S y ϕdy

60 Fattorini criterion for stabilizability Standart uniqueness result yield ϕ 0 in ( 1, 0) and then g = ϕ(0) = 0 and λϕ ϕ yy V S ϕ y = 0, y (0, 1), ϕ(0) = ϕ(1) = 0, λϕ y (0) = 1 0 ( η Vy S ηf S) 1 y ϕdy 0 ληv S y ϕdy Mutliplying by ηv S y and integrating by parts, we obtain 0 = (λ V S y (0))ϕ y (0). (8) Since assumption (A 2 ) exactly means that there is no eigenvalue λ with real part Rλ > σ which is equal to Vy S (0), then standart uniqueness result yield ϕ 0 in (0, 1).

61 Theorem There exist µ > 0 and ( ϕ, ĝ, k) L 2 ( 1, 1) R 2, such that, if then system W t W yy + h ( η V S y W 0 L 2 (( 1,1)) + l 0 + h 0 µ W (t, 0) = l(t) (t 0), m l(t) = [W y ](t, 0) (t 0), ḣ(t) = l(t) (t 0), ηf S) y + (V S W ) y ηlv S y = G(W, l, h) W (t, 1) = 1 1 W (t) ϕdy + mḣ(t)ĝ + h(t) k (t 0). W (0) = W 0, l(0) = l 0, h(0) = h 0. admits a unique solution W L 2 loc (R +, H 1 ( 1, 1)) C(R +, L 2 ( 1, 1)), h C 1 (R + ). Moreover, there exists C > 0 such that W (t) L 2 ( 1,1)+ l(t) + h(t) Ce σt ( W 0 L 2 (( 1,1)) + l 0 + h 0 ).

62 Open question Is it possible to find a non null solution to the following problem? { V S y (0)ϕ ϕ yy V S ϕ y = 0, y (0, 1), ϕ(0) = ϕ(1) = 0. ( )

63 Open question Is it possible to find a non null solution to the following problem? { V S y (0)ϕ ϕ yy V S ϕ y = 0, y (0, 1), ϕ(0) = ϕ(1) = 0. ( ) If we multiply by ϕ, we obtain V S y (0) 1 0 ϕ 2 dy ϕ y 2 dy V S y ϕ 2 dy = 0.

64 Open question Is it possible to find a non null solution to the following problem? { V S y (0)ϕ ϕ yy V S ϕ y = 0, y (0, 1), ϕ(0) = ϕ(1) = 0. ( ) If we multiply by ϕ, we obtain V S y (0) 1 0 ϕ 2 dy ϕ y 2 dy V S y ϕ 2 dy = 0. Using 2Vy S (y) = 2Vy S (0) + (V S (y)) 2 2F S, we deduce 3 2 V S y (0) 1 H S ϕ 2 dy ϕ y 2 dy F S ϕ 2 dy.

65 Open question Is it possible to find a non null solution to the following problem? { V S y (0)ϕ ϕ yy V S ϕ y = 0, y (0, 1), ϕ(0) = ϕ(1) = 0. ( ) If we multiply by ϕ, we obtain V S y (0) 1 0 ϕ 2 dy ϕ y 2 dy V S y ϕ 2 dy = 0. Using 2Vy S (y) = 2Vy S (0) + (V S (y)) 2 2F S, we deduce 3 2 V S y (0) 1 H S ϕ 2 dy + 1 Using Poincaré inequality, we deduce 0 ϕ y 2 dy 1 2 y (0) [F S ] + L 2π 2, F S (y) = 3 V S 1 0 y 0 F S ϕ 2 dy. f S (s)ds.

66 Conclusion We obtain the feedback stabilizability of a fluid-structure problem in 1d, with only one boundary control, provided σ is not too large or there does not exist a solution to ( ). We are working on the same problem in several dimensions.

67 Outline Introduction Stabilizability of infinite dimensional systems Magnetohydrodynamic flow system Fluid-rigid body interaction system Conclusion

68 Conclusion We present a general method to prove the feedback stabilizability of parabolic system and we apply it to : Coupled Navier-Stokes type systems : MHD but also Boussinesq equations, micropolar fluid system... Fluid structure system : 1D but also 2D and 3D fluid body interaction system (work in progress)

69 Research directions Stabilizability around a nonstationary solution : stabilize around zero the nonautonomous dynamical system y = A(t)y + Bu int. stab. of N-S-E : [Barbu, Rodrigues, Shirikyan 2011] Consider fluid with a nonhomogeneous density : add the transport equation ρ t + u ρ = 0 in Ω (0, + )