- Email: [email protected]
Boundary stabilization of a fluid-rigid body interaction system⁎
3rd IFAC/IEEE CSS Workshop on Control of Systems Governed by 3rd IFAC/IEEE CSS Workshop on Control of Systems Governed by Partial 3rd IFAC/IEEE IFAC/IEEE CSS CSS Workshop Workshop on on Control Control of of Systems Systems Governed Governed by by 3rd Partial Available online at www.sciencedirect.com Differential Equation, and XI Workshop Control of Distributed 3rd IFAC/IEEE CSS Workshop on Control of Systems Governed by Partial Partial Differential Equation, and XI Workshop Control of Distributed Parameter Systems Partial Differential Equation, Differential Equation, and and XI XI Workshop Workshop Control Control of of Distributed Distributed Parameter Systems Oaxaca, Mexico, May 20-24, Differential Equation, and XI 2019 Workshop Control of Distributed Parameter Systems Parameter Systems Oaxaca, Mexico, May 20-24, 2019 Parameter Systems Oaxaca, May Oaxaca, Mexico, Mexico, May 20-24, 20-24, 2019 2019 Oaxaca, Mexico, May 20-24, 2019 IFAC PapersOnLine 52-2 (2019) 64–69
ScienceDirect
Boundary Boundary Boundary Boundary
stabilization stabilization of of a a fluid-rigid fluid-rigid stabilization of a interaction system stabilization of a fluid-rigid fluid-rigid interaction system interaction system interaction Mehdi Badra ∗∗ Tak´ esystem o Takahashi ∗∗ ∗∗
body body body body
Mehdi Badra ∗∗∗ Tak´ eo Takahashi ∗∗ ∗∗ Mehdi Badra ∗ Tak´ eo Takahashi ∗∗ ∗∗ ∗ Mehdi Badra Tak´ e o Takahashi Institut de Math´ e matiques de Toulouse, UMR5219; Universit´ ∗ ∗ Institut de Toulouse, UMR5219; Universit´ee de de Math´ e matiques de ∗ ∗ Institut Toulouse, CNRS; UPS, F-31062 Toulouse Cedex 9, France e de de Math´ e matiques de Toulouse, UMR5219; Universit´ ∗ ∗∗ Toulouse, CNRS; UPS, F-31062 Toulouse Cedex 9, France Institut de Math´ matiques de Toulouse, UMR5219; Universit´ e de Universit´ e de eLorraine, CNRS, Inria, IECL, F-54000 Nancy ∗∗ UPS, F-31062 Toulouse Cedex 9, France ∗∗ Toulouse, Universit´eCNRS; de Lorraine, CNRS, Inria, IECL, F-54000 Nancy ∗∗ Toulouse, CNRS; UPS, F-31062 Toulouse Cedex 9, France ∗∗ Universit´ e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy ∗∗ Universit´e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy Abstract: Let us consider a fluid-rigid body interaction system. We are interested in the Abstract: Let us consider fluid-rigid system. We are interested in the feedback stabilization of thisa system by body using interaction a finite-dimensional control localized on the Abstract: Let us consider a system fluid-rigid body interaction system. We are interested in the feedback stabilization of this by using a finite-dimensional control localized on the Abstract: Let us the consider a system fluid-rigid body interaction system. We are interested in and the interface between structure and the fluid. The fluid is assumed to be viscous feedback stabilization of this by using a finite-dimensional control localized on the interface stabilization between the ofstructure and theusing fluid.a The fluid is assumed to localized be viscous and feedback thisthe system finite-dimensional control on and the incompressible and to follow Navier-Stokes system and for body interface between the structure and by the fluid. The fluidwe isconsider assumed tothe berigid viscous incompressible and to follow the Navier-Stokes system and we consider for the rigid body the interface between the structure and the for fluid. The fluidweisconsider tothe berigid viscous and Newton laws. We follow a general method the stabilization ofassumed nonlinear parabolic systems incompressible and to follow the Navier-Stokes system and for body the Newton laws. We follow a general method for the stabilization of nonlinear parabolic systems incompressible and to follow the Navier-Stokes system consider for the body the combinedlaws. withWe a change variables to handle thestabilization factand thatwethe domain isrigid moving with Newton follow aofgeneral method for the of fluid nonlinear parabolic systems combined with a change variables to handle the fact that the fluid domain is moving with Newton laws. We follow aof general method for the stabilization of nonlinear parabolic systems time. We prove that for small initial velocities and if the initial position and the final position combined with a change of variables to handle the fact that the fluid domain is moving with time. We prove that for small initial velocities and iffact thethat initial position and theis final position combined with change of to handle the if theposition fluid moving are close, we canathat stabilize thevariables position and the velocity of the rigid bodydomain and the velocity ofwith the time. We we prove for small initial velocities and the initial andthe thevelocity final position are close, can stabilize the position and the velocity of the rigid body and of the time. We prove that for small initial velocities and if the initial position and the final position fluid. are close, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid. are close, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. fluid. Keywords: Controllability and Observability Analysis, Flow Control, Semigroup and Operator Keywords: Controllability and Analysis, Flow Control, Theory, Systems, Navier-Stokes Keywords: ControllabilityInteraction and Observability Observability Analysis, Flow system Control, Semigroup Semigroup and and Operator Operator Theory, Fluid-Structure Fluid-Structure Interaction Systems,Analysis, Navier-Stokes system Keywords: Controllability and Observability Flow Control, Semigroup and Operator Theory, Fluid-Structure Interaction Systems, Navier-Stokes system Theory, Interaction Systems, Navier-Stokes system equations of the fluid–structure system. We are interested in Fluid-Structure the stabilizability of a fluid-structure We We write write now now the the equations of the fluid–structure system. We are interested in the stabilizability of a fluid-structure interaction system composed by body and equations valid long S(t) We are interested in the stabilizability a fluid-structure We write now theare equations theas fluid–structure system. interaction systemin composed by a a rigid rigid of body and a a viscous viscous These These equations valid as as of long S(t) ⊂ ⊂ Ω. Ω. We are interested the stabilizability of a fluid-structure We write now theare equations of theas fluid–structure system. incompressible fluid. The control takes the form of a These equations interaction system composed by a rigid body and a viscous are valid as long as S(t) ⊂ Ω. The fluid is described by a velocity vector field v(t, incompressible fluid. The control takes the form of a interaction system composed by a rigid body and a viscous These equations are valid as long as S(t) ⊂ Ω. x) and and Dirichlet condition, localized on the takes interface between thea The fluid is described by a velocity vector field v(t, x) incompressible fluid.localized The control thebetween form ofthe a pressure function p(t, x) satisfying the incompressible The fluid is described by a velocity vector field v(t, x) and Dirichlet condition, on the interface incompressible The control takes thenull form a a pressure function p(t, x) satisfying the incompressible structure condition, and thefluid. fluid. The target state is the stateofthe for Dirichlet localized on the interface between The fluid is described by a velocity vector field v(t, x) and Navier-Stokes equations: structure and the fluid. The state is null state for pressure function p(t, x) satisfying the incompressible Dirichlet condition, localized on the interface between the equations: the velocity prescribed position for the rigid body. structure andand thea fluid. The target target state is the the null state for aNavier-Stokes a pressure function p(t, x) satisfying the incompressible the velocity and a prescribed position for the rigid body. d structure andand theafluid. The target statefor is the for Navier-Stokes d vv −equations: the velocity prescribed position thenull rigidstate body. div T(v, p) = 0 equations: Let us precisely our Let Ω be aa Navier-Stokes the and amore prescribed position for the rigid body. − div T(v, p) = 0 d v t Let velocity us describe describe more precisely our model. model. Let Ω be (3) in F(t), t > 0, 2,1 3 bounded smooth domain (say, class C of that Let us describe more precisely our model. Ω33 be a div T(v, p) = 0 (3) dd 2,1 ) bounded smooth more domain (say, of ofour class C 2,1 )Let of R R that d vtt − in in F(t), F(t), tt > > 0, 0, (3) div v − div T(v, p) = 0 2,1 3 Let us describe precisely model. Let Ω be a 2,1 3 contains both the domain fluid and the ofstructure. We consider div v = 0 in F(t), t > 0, bounded smooth (say, class C 2,1 ) of R3 thata dt (3) contains both the fluid and the structure. We consider div v = 0 bounded smooth (say, ofstructure. class CWeWe )assume of R thata with rigid body of shape S moving inside Ω. contains both the domain fluid and the consider rigid body of shape S moving inside Ω. We assume that div v = 0 with 2,1 contains both the fluid andCthe structure. consider S is smooth (say, class ), and compact v(t, (4) rigid of shape moving Ω. WeWe assume thata with 2,1inside S is aabody smooth (say, of ofS class C 2,1 ), connected connected and compact v(t, x) x) = = 0, 0, tt > > 0, 0, x x∈ ∈ ∂Ω, ∂Ω, (4) 3of shape 2,1 with rigid body S moving inside Ω. We assume that 2,1 subset of R with non empty interior. S is a smooth (say, of class C ), connected and compact v(t, x) = 0, t > 0, x ∈ ∂Ω, (4) 3 and 3 with non empty 2,1 subset of R interior. 3 (say, of class C S is a smooth ), connected and compact and v(t, x) = 0, t > 0, x ∈ ∂Ω, (4) 3 3 non empty interior. subset of R with and For R and for R v(t, x) = V (t) + r(t) × (x − h(t)) subset For all allofh hR3∈ ∈ with R3333 non and3empty for all allinterior. R ∈ ∈ SO(3) SO(3) (the (the special special and v(t, x) = V (t) + r(t) × (x − h(t)) orthogonal group R3 ), set For all h ∈ R of and forwe ∈ SO(3) (the special + v(t, x) = V (t) + r(t) × (x − h(t)) orthogonal group weall setR 3 of R3 3 ), + u(t, u(t, x), x), x x∈ ∈ ∂S(t). ∂S(t). (5) (5) For all h ∈ R and for all R ∈ SO(3) (the special 3 v(t, x) = V (t) + r(t) × (x − h(t)) = h + RS, F = Ω \ S . orthogonal S group of R ), we set h,R h,R h,R 3 + u(t, x), x ∈ (5) =ofh R + RS, Fh,R = Ω \ Sh,R . Here V (t)+r(t)×(x−h(t)) is the velocity of the∂S(t). rigid body h,R orthogonal S group ), we set h,R h,R h,R +our u(t, x), ofx the ∈ ∂S(t). (5) Sh,R h +the RS, Fh,R Ω \ Sh,R Here V (t)+r(t)×(x−h(t)) is the velocity rigid body h,R = in h,R =positions h,R .of the rigid We are interested admissible and u = u(t, x) is the control of system. To obtain the Here V (t)+r(t)×(x−h(t)) is the velocity of the rigid body We are interested in admissible positions the rigid and u = u(t, x) is the control of our system. To obtain the h(h, +the RS, Fh,R Ω\S .of h,R = in h,R body, inScouples R)admissible such that=Spositions ⊂ Ω. We assume We arei.e.interested the of the rigid Here is the velocity of the rigid body h,R equations of the positions we apply the laws and uV=(t)+r(t)×(x−h(t)) u(t, the control our system. To obtain h,R body, i.e. in R) such that ⊂ We assume equations of x) theis we of apply the Newton’s Newton’s laws the h,R We are interested in(h, the admissible the rigid that in that case, F is connected. In what follows, we body, i.e. in couples couples (h, R) such that S Spositions ⊂ Ω. Ω. of We assume and u = u(t, x) ispositions the control of our system. To obtain the h,R h,R h,R that in that case, F is connected. In what follows, we equations of the positions we apply the Newton’s laws h,R h,R body, i.e. in couples (h, R) such that Sofh,R ⊂ Ω. We assume also suppose that the center of mass S is located at the equations of the positions we apply the Newton’s laws h,R that in that case, F is connected. In what follows, we h,R also that the center of mass ofInS what is located at the T(v, (6) M that suppose insothat case, F is connected. follows, we origin that h is the center mass R). h,R also suppose that the center ofof mass ofof SS(h, is located at the =− − T(v, p)n p)n dΓ, dΓ, (6) MV V = origin so that h is the center of mass of S(h, R). also suppose is located T(v, p)n dΓ, (6) origin so thatthat h is the thecenter centerofofmass massofofSS(h, R). at the M V = − ∂S(t) ∂S(t) ∂S(t) If body a tt → (h(t), (6) M V = − ∂S(t) origin so that h isfollows the center of mass of S(h, R).R(t)), ∂S(t) T(v, p)n dΓ, If the the rigid rigid body follows a trajectory trajectory → (h(t), R(t)), we we = − ∂S(t) (x − h) × T(v, p)n dΓ. (J(t)r) (7) can define its angular velocity given through the formula If the rigid body follows a trajectory t → (h(t), R(t)), we (J(t)r) = − (x − h) × T(v, p)n dΓ. (7) can define angular velocity given through theR(t)), formula = − ∂S(t) (x − h) × T(v, p)n dΓ. velocity If the rigidits body follows a trajectory t → (h(t), we (J(t)r) (7) can define its angular given through the formula ∂S(t) R (1) ∂S(t) (t) = A(r(t))R(t), R (t) = A(r(t))R(t), (1) In above(J(t)r) =we − use (x − h) × T(v, p)n dΓ. (7) ∂S(t) can define its angular velocity given through the formula ∂S(t) settings, the notation where R (1) In above settings, we use the notation ∂S(t) (t) = A(r(t))R(t), where R (1) In above settings, we (t) d v use 0 = −rA(r(t))R(t), the notation def ∂v 3 r2 where def d vv use ∂v 3 −r rr22 (r ∈ R3 ). = (v · ∇)v, In above settings, we the+notation def r0 3 −r 0 A(r) = d ∂v 0 −r = where 3 def ∂t + (v · ∇)v, 21 d t def 3 ). 2 0 33 −r A(r) = r0333 −r (r ∈ R 1 3 1 d t ∂t = + · ∇)v, v def ∂vof v (v r021 r332 rr013 −r A(r) = −r (r ∈ R33 ). for the material derivative and −r 01 d t = ∂tof +v (v · ∇)v, 2 1 2 1 for the material derivative and r 0 −r A(r) = ). (r ∈ R d t def ∂tof v and Finally, the body is defined by −r322 r11of 0 1 rigid for the material derivative def = 2νD(v) − pI , T(v, p) Finally, the the linear linear velocity velocity of the rigid body is defined by def −r2 r1def 0 = 2νD(v) T(v, p) def def for the material derivative of v and− pI333 , Finally, the linear velocity of the rigid body is defined (2) by def V = h = 2νD(v) − pI33 , T(v, p) . def with def Finally, the linear velocity by V of = the h . rigid body is defined (2) def def with pI3 , T(v,def p) 1= 2νD(v) − V def = h . (2) with def 1 (∇u) + ttt (∇u) . D(u) = def To simplify, in what follows, we set V = h . (2) D(u) def = 12 (∇u) + tt (∇u) . with def To simplify, in what follows, we set D(u) def = 21 (∇u) + t (∇u) . To simplify, in def what follows, we setdef def def S(t) = S , F(t) = F . def def h(t),R(t) h(t),R(t) D(u) = 2 (∇u) + (∇u) . To simplify, in def what follows, we set S(t) = S , F(t) = F . We assume that the density def h(t),R(t) def h(t),R(t) def h(t),R(t) h(t),R(t) 2 ρ We assume that the density ρSSS of of the the rigid rigid body body is is constant constant h(t),R(t) h(t),R(t) = S , F(t) = F . This workS(t) def h(t),R(t) def h(t),R(t) so that was supported by ANR IFSMACS We assume that the density ρ of the rigid body is constant S This workS(t) S = Sh(t),R(t) F(t) = Fh(t),R(t) . so that was supported by ,ANR IFSMACS We assume that the density ρ of the rigid body is constant This S This work work was was supported supported by by ANR ANR IFSMACS IFSMACS so that This work was supported by ANR IFSMACS so that 2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 64 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 64 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 64 Copyright © 2019 2019 IFAC IFAC 64 10.1016/j.ifacol.2019.08.012 Copyright © 2019 IFAC 64
2019 IFAC CPDE-CDPS Oaxaca, Mexico, May 20-24, 2019
def
M = ρS |S|,
def
J(t) = ρS
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
S(t)
(x − h(t)) ⊗ (x − h(t)) dx.
u(t, x) =
Nσ
uj v j ,
(10)
j=1
One can check that J(t) = RJS R∗ where def JS = ρS y ⊗ y dy.
def
with Nσ ∈ N∗ and u = (uj )j∈{1,...,Nσ } satisfying u = Λu + G(v ◦ X, V, r, h, R),
S
t > 0,
2
u(0) = 0, (11)
for a suitable map G : L (Fh1 ,R1 ) × R → RNσ and for a matrix Λ of size Nσ × Nσ . In above setting, X : Fh1 ,R1 → F(t) is a diffeomorphism. Theorem 1. For all σ > 0, there exist Nσ ∈ N, c0 , C > 0, a map G as above such that if (v 0 , V 0 , r0 , h0 , R0 ) ∈ H1 (F(h0 , R0 )) × R9 × SO(3) satisfies
Finally, we assume the following initial conditions for the positions: h(0) = h0 , R(0) = R0 , V (0) = V 0 , r(0) = r0 , v(0, ·) = v 0 .
65
(8) (9)
In the system (1)–(7), the state is (v, p, V, r, h, R). In particular, the domains F(t) and S(t) are evolving and unknowns. The Navier-Stokes system is thus written in an open non-cylindrical subset of R4 that depends on the trajectories t → (h(t), R(t)).
12
div v 0 = 0 in Fh0 ,R0 , v = V + r0 × (x − h0 ) on ∂Sh0 ,R0 , v 0 = 0 on ∂Ω, 0
0
and v 0 H1 + |V 0 | + |r0 | + |h0 − h1 | + |R0 − R1 | c0 , then there exists a strong solution (v, p, V, r, h, R, u) of (1)– (7), (8)–(9), (10), (11) such that
We assume that the initial data (v 0 , V 0 , r0 , h0 , R0 ) is close to the target (0, 0, 0, h1 , R1 ) and we want to find a feedback control u in (5) such that the resulting solution (v(t), V (t), r(t), h(t), R(t)) goes to (0, 0, 0, h1 , R1 ) as t → +∞ with an exponential rate of decrease. In order to this, we follow the general strategy described in Badra and Takahashi (2011, 2014b) and in particular we prove the stabilizability of the linearized system. This is done by using the classical Fattorini-Hautus test on the unstable modes. As explained above, a difficulty here is that the fluid system is written in a moving domain F(t). This is standard in the study of fluid–structure system, and a natural method consists in using a change of variables (see for instance Grandmont and Maday (2000), Takahashi (2003), Boulakia and Osses (2008), Imanuvilov and Takahashi (2007), Boulakia and Guerrero (2013)).
v(t)H1 + |V (t)| + |r(t)| + |h(t) − h1 | + |R(t) − R1 | + |u(t)| Ce−σt v 0 H1 + |V 0 | + |r0 |
+ |h0 − h1 | + |R0 − R1 | .
1. CHANGE OF VARIABLES Up to a rotation and a translation on S, we can assume that Sh1 ,R1 = S. In that case, we also write
The stabilization of fluid-structure systems has been studied for some years: first Raymond considered in Raymond (2010) the boundary stabilization of fluid-beam equation around the null state. In Badra and Takahashi (2014a) the authors consider the case of a rigid body moving into a viscous incompressible fluid and obtain a local stabilization result around any stationary state by taking a control on the exterior boundary ∂Ω. The main difference between this work and the presented work corresponds to the localization of the control, here we act at the interface fluid-structure. This leads to some additional difficulties and due to our method, we can only stabilize here around a stationary state with null velocity for the fluid and for the structure. The control can also be a force applied on the rigid body: this is the strategy considered in Takahashi et al. (2015) (or in Cˆındea et al. (2015) in the 1D case).
def
F = Fh1 ,R1 We rewrite our system in a cylindrical domain by using a change of variables X(t, ·) : Ω → Ω, such that X(t, F) = F(t). We denote by Y (t, ·) the inverse of X(t, ·). Several constructions are possible to obtain such a change of variables. Here, we consider def
X(t, y) = y + η(y) [h(t) + (R(t) − I3 )y] where η is a smooth function such that 1 if dist(y, S) < ε, η(y) = 0 dist(y, S) 2ε,
Let us point out that here, we assume that the initial conditions are in H 1 for the fluid velocity (“strong solutions”). Consequently (see Badra (2012), Badra (2009b)), there are compatibility conditions at t = 0 with the feedback control u. Several techniques exist to overcome this problem: Raymond (2007), Badra (2009a), Badra and Takahashi (2011). Here, we use the approach of Badra and Takahashi (2011): the control u satisfies an evolution equation with another control feedback and this leads to stabilize a system coupling the state and the feedback control u. More precisely we assume that the control u in (4) can be written as
(12)
with ε < dist(S, ∂Ω)/2.
The map X is a C ∞ -diffeomorphism of Ω onto itself if (13) ηW 1,∞ (Ω) (|h(t)| + |R(t) − I3 |) < c,
for c small enough. It satisfies X(t, y) = y for y in a neighborhood of ∂Ω and that X(t, y) = R(t)y + h(t) in a neighborhood of S. In what follows, we assume that ∀t 0, |h(t)| + |R(t) − I3 | < C , with C small enough so that (13) holds.
65
2019 IFAC CPDE-CDPS 66 Oaxaca, Mexico, May 20-24, 2019
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
2. STABILIZATION OF THE LINEAR SYSTEM
With the change of variables introduced above, we set def
w(t, y) = Cof(∇X(t, y))∗ v(t, X(t, y)),
The present subsection is dedicated to the abstract formulation of the following linear nonhomogeneous problem: q(t, y) = p(t, X(t, y)), (33) ∂t w − ν∆w + ∇q = F in (0, +∞) × F, def def (t) = R∗ (t)V (t), ω(t) = R∗ (t)r(t) div w = 0 in (0, +∞) × F, (34) where Cof(M ) is the cofactor matrix of M , which satisfies w =+ω×y+u ˜ on (0, +∞) × ∂S, (35) in particular M (Cof(M ))∗ = (Cof(M ))∗ M = det(M )I3 . w = 0 on (0, +∞) × ∂Ω, (36) Following Inoue and Wakimoto (1977), Takahashi (2003) M = − T(w, q)n dΓ + F , t > 0, (37) and Boulakia et al. (2012), one can prove that (v, p, V, r, h, R) ∂S satisfies (1), (2) and (3)–(7) if and only if (w, q, , ω, h, R) JS ω = − y × T(w, q)n dΓ + ωF , t > 0, (38) satisfies the following system def
∂S
K∂t w − νLw + Mw + N(w) + Gq = 0 in (0, +∞) × F, div w = 0 in (0, +∞) × F, w =+ω×y+u ˜ on (0, +∞) × ∂S, w = 0 on (0, +∞) × ∂Ω, M (R) = −R T(w, q)n dΓ, t > 0, ∂S (RJS ω) = −R y × T(w, q)n dΓ, t > 0, ∂S
h = R, t > 0, R = RA(ω), t > 0.
(39) h = + hF , t > 0, (40) θ = ω + θF , t > 0, with the initial conditions h(0) = h0 , θ(0) = θ0 , (41) 0 0 0 (0) = , ω(0) = ω , w(0, y) = w (y) y ∈ F. (42) Here, F , F , ωF , hF , QF are nonhomogeneous terms that replace the nonlinearities in (22)–(32) so that (33)–(40) is a linear system.
(14) (15) (16) (17) (18)
We write the above system in the form
(19) (20) (21)
˜ + P F in [D(A∗ )] , P W = AP W + B u P W(0) = P W
The operators K, M, N, L and G can be expressed through X and Y (see Badra and Takahashi (2014a) for the precise expressions).
θ = B(θ)ω, where B : R → M3 (R) is a smooth map such that B(0) = I3 . Here M3 (R) denotes the space of 3-by-3 real matrices. 3
def
V =
∗
We set W = [w, , ω, h, θ] and our system can write as follows
h = + εh (W), θ = ω + εθ (W), 0
(0) = 0 ,
t > 0, t > 0, 0
h(0) = h , θ(0) = θ , ω(0) = ω 0 , w(0, y) = w0 (y)
where εw , ε , εω , εh , εθ are nonlinearities.
(44)
w · n = ( + ω × y) · n on ∂S,
∂S
(43)
(45) (Id −P )W = (Id −P )DF u, ∗ ∗ where W = [w, , ω, h, θ] , F = [F, F , ωF , hF , θF ] and where A, P , B and DF are linear operators defined below. Formulation of type (44)-(45) is due to Raymond (2006) for Stokes type systems. Let us define def H = [w, , ω, h, θ]∗ ∈ L2 (F) × (R3 )4 ;
In order to linearize this system, we use a local chart of SO(3) around I3 : there exist smooth maps Ri , i = 1, 2, 3 such that any matrix R ∈ SO(3) can write in a unique way as R = R1 (θ1 )R2 (θ2 )R3 (θ3 ). A classical way to do this is to use the Euler angles. After a standard calculation, one can write R = RA(ω) as
∂t w − ν∆w + ∇q = εw (W, q) in (0, +∞) × F, div w = 0 in (0, +∞) × F, w =+ω×y+u ˜ on (0, +∞) × ∂S, w = 0 on (0, +∞) × ∂Ω, M = − T(w, q)n dΓ + ε (W), t > 0, ∂S JS ω = − y × T(w, q)n dΓ + εω (W), t > 0,
0
w · n = 0 on ∂Ω, div w = 0 [w, , ω, h, θ]∗ ∈ H ;
in F ,
w ∈ H1 (F), w = + ω × y on ∂S,
w = 0 on ∂Ω .
We define the linear operator A : D(A) ⊂ H → H as follows: we set def def D(A) = {[w, , ω, h, θ]∗ ∈ V ; w ∈ H2 (F)}, A = P A, ∗ where for [w, , ω, h, θ] ∈ D(A), we set ν∆w −M −1 w 2νD(w)n dΓ ∂S def ω = A −JS−1 h y × 2νD(w)n dΓ ∂S θ ω
(22) (23) (24) (25) (26) (27) (28) (29) (30)
and where P is the orthogonal projection from L2 (F)×R12 onto H.
(31) y ∈ F, (32)
As in Badra and Takahashi (2014a), we can check the following proposition.
66
2019 IFAC CPDE-CDPS Oaxaca, Mexico, May 20-24, 2019
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
Proposition 2. The operator A defined above is densely defined with compact resolvent and it is the infinitesimal generator of an analytic semigroup on H.
def
Σα,θ = {λ ∈ C ; | arg(λ − α)| < θ} ⊂ ρ(A) sup λ(λ − A)−1 L(H) < ∞.
λ∈Σα,θ
Then, we can prove the following characterization of the adjoint of A. Proposition 3. The adjoint of the operator A is given by D(A∗ ) = D(A) and ν∆ϕ −M −1 ϕ 2νD(ϕ)n dΓ + M −1 a ξ ∂S ∗ A ζ = P −1 . −1 −JS a y × 2νD(ϕ)n dΓ + J b S ∂S b 0 0 (46)
Now we prove that for any σ > 0, there exists a feedback control Fσ (W) =
and
(Φj , W)L2 (F )×R12 vj , Φj ∈ L2 (F) × R12 ,
(52)
such that the solution W of (33)-(40) with u = Fσ (W) and with (F, F , ωF , hF , θF ) = (0, 0, 0, 0, 0) tends to zero as t → +∞ with an exponential rate of decrease σ > 0. For that, we are going to show the existence of families Φj and vj , j = 1, . . . , Nσ such that the underlying closedloop linear operator of (33)-(40) with (52) generates and analytic and exponentially stable semigroup of type lower than −σ (see (Bensoussan et al., 2007, II-1, Cor. 2.1)). It then permits to deduce results for the case of non zero (F, F , ωF , hF , θF ) that are used in the next subsection to construct fixed-point solutions of the nonlinear system (22)-(32). Proposition 6. For σ > 0, there exist Nσ ∈ N∗ , families 3 Φj ∈ D(A∗ ) and vj ∈ V 2 (∂S), j = 1, . . . , Nσ , such that def def the linear operator Aσ = A + BFσ with domain D(Aσ ) = {X ∈ H | AX + BFσ X ∈ H} is the infinitesimal generator of an analytic and exponentially stable semigroup on H of type lower than −σ.
w = + ω × y + u on ∂S, w = 0 on ∂Ω,
Nσ j=1
Next, we take λ0 > 0 large enough so that λ0 ∈ ρ(A). Then we introduce the Dirichlet operator DF : V0 (∂Ω) → L2 (F) × R12 defined as follows: for u ∈ V0 (∂Ω) we denote def by DF u = [wu u ωu hu θu ]∗ the unique solution of λ0 w − ν∆w + ∇q = 0 in F, div w = 0 in F,
∇π
−M −1 πn dΓ ∂S −1 = ⊥ πn · y dΓ −JS ∂S 0 0 −λ 0 ϕ + ν∆ϕ −λ0 ξ − M −1 2νD(ϕ)n dΓ + M −1 a ∂S (Id −P ) −1 −1 y × 2νD(ϕ)n dΓ + JS b −λ0 ζ − JS ∂S −λ0 a −λ0 b (51)
We recall (see (Bensoussan et al., 2007, p.112)) that A is the infinitesimal generator of an analytic semigroup on H if there exist α ∈ R and θ > π2 such that and if
67
T(w, q)n dΓ = 0, λ0 M + ∂S λ0 JS ω + y × T(w, q)n dΓ = 0, ∂S
λ0 h − = 0, λ0 θ − ω = 0.
We recall (see, for instance, (Bensoussan et al., 2007, p.92)) that a strongly continuous semigroup (T (t))t 0 on H is exponentially stable of type lower than −σ if for any α < σ there exists Cα > 0 such that
Using standard techniques (see Badra and Takahashi (2014a) ), we can prove the following result. Proposition 4. The mapping DF defined above satisfies the following boundedness property: 1 3 12 s s+ 12 . (47) (F) × R ) s ∈ − , DF ∈ L(V (∂S), H 2 2
∀t 0,
T (t)L(H) Cα e−αt .
Proof of Proposition 6. The proof of the above proposition relies on the Hautus-Fattorini stabilizability criterion, see (Badra and Takahashi, 2011, Theorem 1) or Badra and Takahashi (2014b). Since A has compact resolvent and generates an analytic semigroup on H, and since B is relatively bounded with respect to A, then the homogeneous linear system is stabilizable by finite dimensional feedback control for any rate of decrease if and only if the following criterion is satisfied for all λ ∈ C:
Next, we define the input operator B : V0 (∂S) → [D(A∗ )] , Bu = (λ0 − A)P DF u. (48) Proposition 5. The operator B defined by (48) satisfies: 1 −1+ 0 (λ0 − A) . (49) B ∈ L(V (∂S), H), ∈ 0, 4 Moreover, the adjoint of B is defined by ∗ (50) B ∗ [ϕ, ξ, ζ, a, b] = −T(ϕ, π)n|∂S , where
λY − A∗ Y = 0
and
B∗Y = 0
=⇒
Y = 0. (53)
From (46) and (50) we deduce that (53) is true if and only if all [ϕ, ξ, ζ, b, κ]∗ ∈ D(A∗ ) which satisfies 67
2019 IFAC CPDE-CDPS 68 Oaxaca, Mexico, May 20-24, 2019
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
λϕ − ν∆ϕ + ∇π = 0 in F, πdΓ = 0, ∂Ω div ϕ = 0 in F, ϕ = ξ + ζ × y on ∂S, ϕ = 0 on ∂Ω, T(ϕ, π)n dΓ = a, M λξ + ∂S JS λζ + y × T(ϕ, π)n dΓ = b, ∂S λa = 0, λb = 0. and T(ϕ, π)n = 0 on ∂S. must be identically zero.
Then by setting def
F(X, q) = [F (X, q), ε (X), εω (X), εh (X), εθ (X)]∗ the nonlinear system (14)-(21), (57) and (58) rewrites P X = AP X + BΨu + P F(X, q), P X(0) = P X0 , (59) u = g, u(0) = 0, (Id −P )X = (Id −P )DF Ψu.
(54)
Thus, by following the general framework of Badra and Takahashi (2011) for the construction of dynamical control (see also Badra and Takahashi (2014b)) we introduce the def “extended” space HE = H × RNσ , the following extended operator on HE : A BΨ def , AE = 0 0
(55)
(56)
The above implication can be proved as follows: if λ = 0, then, a = 0, b = 0 and thus from (56), ξ = ζ = 0. Thus (ϕ, π) satisfies a Stokes system with homogeneous Dirichlet boundary conditions and using a classical uniqueness result for Stokes type systems (see e.g. Fabre and Lebeau (1996)) we deduce that ϕ = 0 and π = 0 on F. If λ = 0, then we deduce from (55) and (56) that a = 0 and b = 0. Then we multiply the first equation of (54) by ϕ: div T(ϕ, π) · ϕ dx 0=− F = |D(ϕ)|2 dx − T(ϕ, π)n · ϕ dΓ F
def
D(AE ) = {(Y, u) ∈ HE | AY + BΨu ∈ H} ,
as well as the control operator BE ∈ L(RNσ , HE ): def 0 . BE g = g
Thus, since the stabilizability of (A + σ, B) by finite dimensional control generated by the family (vj ) implies the stabilizability of (AE + σ, BE ) (see (Badra and Takahashi, 2011, Theorem 8)), we can find a family (Φj )1 j Nσ ⊂ D(A∗ ), a matrix Λ of size Nσ × Nσ and a corresponding feedback operator FE,σ : HE → RNσ defined by
∂S
and thus Dϕ = 0 in F which implies that that ϕ = ϕ + ωϕ × y. Using the boundary conditions of ϕ this leads to
ϕ = ξ = 0 and ωϕ = ζ = 0.
Then the general framework of Badra and Takahashi (2011, 2014b) can be applied and for a given σ > 0, there exist families Φj ∈ D(A∗ ) and vj ∈ V0 (∂S), j = 1, . . . , Nσ , and a feedback control of the form (52) such that the conclusions of the proposition hold. Moreover, 3 each vj can be chosen in V 2 (∂S). This comes from the fact that the set of admissible families (vj ) is a nonempty open set of (V0 (∂S))Nσ (see (Badra and Takahashi, 2011, Theorem 5) or (Badra and Takahashi, 2014b, Theorem 6)). Indeed, if a family (˜ vj ) is admissible then all families in a neighborhood of (˜ vj ) in (V0 (∂S))Nσ are admissible. 3 Then the conclusion follows from the density of V 2 (∂S) in V0 (∂S).
GE,σ [w, , ω, h, θ, u]∗ = Λu +
4. FIXED POINT ARGUMENT In order to prove the result, we are going to use a fixed point argument. We set def L2σ (H) = F ∈ L2 (0, ∞; H) ; t → eσt F(t) ∈ L2 (0, ∞; H) . (60)
As explained in the introduction, in order to guarantee the initial compatibility condition w 0 = 0 + ω 0 × y + u ˜(0) on ∂S, we complete the system (14)-(21) with a dynamical equation for u ˜: Nσ def u ˜(t, x) = Ψu = uj v j , (57)
We consider the Banach space def
E = L2σ (H) and the following mapping defined on a closed ball of E:
j=1
def
u = g,
u(0) = 0.
(Φj , [w, , ω, h, θ, u])HE ej
where {ej | j = 1, . . . , Nσ } is the canonical basis of RNσ , def such that the linear operator AE,σ = AE + BE GE,σ with domain D(AE,σ ) = D(AE ) (since BE is bounded in HE ) is the infinitesimal generator of an analytic and exponentially stable semigroup on HE of type lower than −σ.
3. EXTENDED SYSTEM
Nσ j=1
with u = (uj )j∈{1,...,Nσ } satisfying
def
0 We set YE = [P X0 , 0]∗ , YE = [P X, u]∗ and FE (YE , q) = [P F(P X + (Id −P )DF Ψu, q), 0]∗ , so that system (59) can be written as 0 = AE YE + BE g + FE (YE , q), YE (0) = YE . YE
Ψ : BE (0, δ) → E, ˇ → P F(P X + (Id −P )DF Ψu, q), 0]∗ , F
(58) 68
2019 IFAC CPDE-CDPS Oaxaca, Mexico, May 20-24, 2019
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
dimensional systems. Systems & Control: Foundations & Applications. Birkh¨auser Boston Inc., Boston, MA, second edition. Boulakia, M. and Guerrero, S. (2013). Local null controllability of a fluid−solid interaction problem in dimension 3. J. Eur. Math. Soc., 15(3), 825–856. Boulakia, M. and Osses, A. (2008). Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM Control Optim. Calc. Var., 14(1), 1– 42. Boulakia, M., Schwindt, E.L., and Takahashi, T. (2012). Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid. Interfaces Free Bound., 14(3), 273–306. Cˆındea, N., Micu, S., Rovent¸a, I., and Tucsnak, M. (2015). Particle supported control of a fluid-particle system. J. Math. Pures Appl. (9), 104(2), 311–353. Fabre, C. and Lebeau, G. (1996). Prolongement unique des solutions de l’equation de Stokes. Comm. Partial Differential Equations, 21(3-4), 573–596. Grandmont, C. and Maday, Y. (2000). Existence for an unsteady fluid-structure interaction problem. M2AN Math. Model. Numer. Anal., 34(3), 609–636. Imanuvilov, O.Y. and Takahashi, T. (2007). Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. (9), 87(4), 408–437. Inoue, A. and Wakimoto, M. (1977). On existence of solutions of the Navier-Stokes equation in a time dependent domain. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24(2), 303–319. Raymond, J.P. (2007). Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 24(6), 921–951. Raymond, J.P. (2010). Feedback stabilization of a fluidstructure model. SIAM J. Control Optim., 48(8), 5398– 5443. Raymond, J.P. (2006). Feedback boundary stabilization of the two-dimensional Navier-Stokes equations. SIAM J. Control Optim., 45(3), 790–828. Takahashi, T. (2003). Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Adv. Differential Equations, 8(12), 1499–1532. Takahashi, T., Tucsnak, M., and Weiss, G. (2015). Stabilization of a fluid–rigid body system. Journal of Differential Equations, 259(11), 6459–6493.
def
where X = [w, , ω, h, θ]∗ and u is the solution ˇ P X = AP X + BΨu + F, 0 P X(0) = P X , u = g, u(0) = 0, (Id −P )X = (Id −P )DF Ψu.
69
(61)
ˇ is a fixed point of the mapping Ψ, then We remark that if F the corresponding solution (w, q, , ω, h, θ) of (33)–(40), (41) and (52) is a solution of (22)–(32). Consequently, we are reduced to show that Ψ admits a fixed point. We prove that for δ small enough, Ψ is well-defined from BE (0, δ) onto itself and that the restriction of Ψ on this closed ball is a contraction mapping. This is done by estimating F (X, q), ε (X), εω (X), εh (X) and εθ (X) and thus by working on the change of variables X and Y defined (12). Such technical results are done in Badra and Takahashi (2014a). With the smallness condition on the initial data, we deduce that the restriction of Ψ on BE (0, δ) is a contraction mapping. The classical Banach fixed point theorem allows us to conclude. 5. CONCLUSIONS AND FUTURE WORKS By acting on ∂S instead of ∂Ω we are lead to a different unique continuation property: (54), (55) and (56) imply that ϕ ≡ 0. We would arrive to a similar unique continuation property by acting on the force and the torque of the rigid body, i.e. by putting controls in (6) and in (7). This was done in Takahashi et al. (2015) with a “spring” force. Next, we would like to act only on a part Γc of ∂S. In that case, we have to replace (56) by (62) T(ϕ, π)n = 0 on Γc and we do not know if this implies ϕ ≡ 0 as before. REFERENCES Badra, M. (2009a). Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system. ESAIM Control Optim. Calc. Var., 15(4), 934– 968. Badra, M. (2009b). Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations. SIAM J. Control Optim., 48(3), 1797–1830. Badra, M. (2012). Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete Contin. Dyn. Syst., 32(4), 1169–1208. Badra, M. and Takahashi, T. (2011). Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamic controllers. Application to NavierStokes system. SIAM J. Control Optim., 49(2), 420–463. Badra, M. and Takahashi, T. (2014a). Feedback stabilization of a fluid-rigid body interaction system. Adv. Differential Equations, 19(11-12), 1137–1184. Badra, M. and Takahashi, T. (2014b). On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems. ESAIM Control Optim. Calc. Var., 20(3), 924–956. Bensoussan, A., Da Prato, G., Delfour, M.C., and Mitter, S.K. (2007). Representation and control of infinite 69
ScienceDirect
Boundary Boundary Boundary Boundary
stabilization stabilization of of a a fluid-rigid fluid-rigid stabilization of a interaction system stabilization of a fluid-rigid fluid-rigid interaction system interaction system interaction Mehdi Badra ∗∗ Tak´ esystem o Takahashi ∗∗ ∗∗
body body body body
Mehdi Badra ∗∗∗ Tak´ eo Takahashi ∗∗ ∗∗ Mehdi Badra ∗ Tak´ eo Takahashi ∗∗ ∗∗ ∗ Mehdi Badra Tak´ e o Takahashi Institut de Math´ e matiques de Toulouse, UMR5219; Universit´ ∗ ∗ Institut de Toulouse, UMR5219; Universit´ee de de Math´ e matiques de ∗ ∗ Institut Toulouse, CNRS; UPS, F-31062 Toulouse Cedex 9, France e de de Math´ e matiques de Toulouse, UMR5219; Universit´ ∗ ∗∗ Toulouse, CNRS; UPS, F-31062 Toulouse Cedex 9, France Institut de Math´ matiques de Toulouse, UMR5219; Universit´ e de Universit´ e de eLorraine, CNRS, Inria, IECL, F-54000 Nancy ∗∗ UPS, F-31062 Toulouse Cedex 9, France ∗∗ Toulouse, Universit´eCNRS; de Lorraine, CNRS, Inria, IECL, F-54000 Nancy ∗∗ Toulouse, CNRS; UPS, F-31062 Toulouse Cedex 9, France ∗∗ Universit´ e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy ∗∗ Universit´e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy Abstract: Let us consider a fluid-rigid body interaction system. We are interested in the Abstract: Let us consider fluid-rigid system. We are interested in the feedback stabilization of thisa system by body using interaction a finite-dimensional control localized on the Abstract: Let us consider a system fluid-rigid body interaction system. We are interested in the feedback stabilization of this by using a finite-dimensional control localized on the Abstract: Let us the consider a system fluid-rigid body interaction system. We are interested in and the interface between structure and the fluid. The fluid is assumed to be viscous feedback stabilization of this by using a finite-dimensional control localized on the interface stabilization between the ofstructure and theusing fluid.a The fluid is assumed to localized be viscous and feedback thisthe system finite-dimensional control on and the incompressible and to follow Navier-Stokes system and for body interface between the structure and by the fluid. The fluidwe isconsider assumed tothe berigid viscous incompressible and to follow the Navier-Stokes system and we consider for the rigid body the interface between the structure and the for fluid. The fluidweisconsider tothe berigid viscous and Newton laws. We follow a general method the stabilization ofassumed nonlinear parabolic systems incompressible and to follow the Navier-Stokes system and for body the Newton laws. We follow a general method for the stabilization of nonlinear parabolic systems incompressible and to follow the Navier-Stokes system consider for the body the combinedlaws. withWe a change variables to handle thestabilization factand thatwethe domain isrigid moving with Newton follow aofgeneral method for the of fluid nonlinear parabolic systems combined with a change variables to handle the fact that the fluid domain is moving with Newton laws. We follow aof general method for the stabilization of nonlinear parabolic systems time. We prove that for small initial velocities and if the initial position and the final position combined with a change of variables to handle the fact that the fluid domain is moving with time. We prove that for small initial velocities and iffact thethat initial position and theis final position combined with change of to handle the if theposition fluid moving are close, we canathat stabilize thevariables position and the velocity of the rigid bodydomain and the velocity ofwith the time. We we prove for small initial velocities and the initial andthe thevelocity final position are close, can stabilize the position and the velocity of the rigid body and of the time. We prove that for small initial velocities and if the initial position and the final position fluid. are close, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid. are close, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. fluid. Keywords: Controllability and Observability Analysis, Flow Control, Semigroup and Operator Keywords: Controllability and Analysis, Flow Control, Theory, Systems, Navier-Stokes Keywords: ControllabilityInteraction and Observability Observability Analysis, Flow system Control, Semigroup Semigroup and and Operator Operator Theory, Fluid-Structure Fluid-Structure Interaction Systems,Analysis, Navier-Stokes system Keywords: Controllability and Observability Flow Control, Semigroup and Operator Theory, Fluid-Structure Interaction Systems, Navier-Stokes system Theory, Interaction Systems, Navier-Stokes system equations of the fluid–structure system. We are interested in Fluid-Structure the stabilizability of a fluid-structure We We write write now now the the equations of the fluid–structure system. We are interested in the stabilizability of a fluid-structure interaction system composed by body and equations valid long S(t) We are interested in the stabilizability a fluid-structure We write now theare equations theas fluid–structure system. interaction systemin composed by a a rigid rigid of body and a a viscous viscous These These equations valid as as of long S(t) ⊂ ⊂ Ω. Ω. We are interested the stabilizability of a fluid-structure We write now theare equations of theas fluid–structure system. incompressible fluid. The control takes the form of a These equations interaction system composed by a rigid body and a viscous are valid as long as S(t) ⊂ Ω. The fluid is described by a velocity vector field v(t, incompressible fluid. The control takes the form of a interaction system composed by a rigid body and a viscous These equations are valid as long as S(t) ⊂ Ω. x) and and Dirichlet condition, localized on the takes interface between thea The fluid is described by a velocity vector field v(t, x) incompressible fluid.localized The control thebetween form ofthe a pressure function p(t, x) satisfying the incompressible The fluid is described by a velocity vector field v(t, x) and Dirichlet condition, on the interface incompressible The control takes thenull form a a pressure function p(t, x) satisfying the incompressible structure condition, and thefluid. fluid. The target state is the stateofthe for Dirichlet localized on the interface between The fluid is described by a velocity vector field v(t, x) and Navier-Stokes equations: structure and the fluid. The state is null state for pressure function p(t, x) satisfying the incompressible Dirichlet condition, localized on the interface between the equations: the velocity prescribed position for the rigid body. structure andand thea fluid. The target target state is the the null state for aNavier-Stokes a pressure function p(t, x) satisfying the incompressible the velocity and a prescribed position for the rigid body. d structure andand theafluid. The target statefor is the for Navier-Stokes d vv −equations: the velocity prescribed position thenull rigidstate body. div T(v, p) = 0 equations: Let us precisely our Let Ω be aa Navier-Stokes the and amore prescribed position for the rigid body. − div T(v, p) = 0 d v t Let velocity us describe describe more precisely our model. model. Let Ω be (3) in F(t), t > 0, 2,1 3 bounded smooth domain (say, class C of that Let us describe more precisely our model. Ω33 be a div T(v, p) = 0 (3) dd 2,1 ) bounded smooth more domain (say, of ofour class C 2,1 )Let of R R that d vtt − in in F(t), F(t), tt > > 0, 0, (3) div v − div T(v, p) = 0 2,1 3 Let us describe precisely model. Let Ω be a 2,1 3 contains both the domain fluid and the ofstructure. We consider div v = 0 in F(t), t > 0, bounded smooth (say, class C 2,1 ) of R3 thata dt (3) contains both the fluid and the structure. We consider div v = 0 bounded smooth (say, ofstructure. class CWeWe )assume of R thata with rigid body of shape S moving inside Ω. contains both the domain fluid and the consider rigid body of shape S moving inside Ω. We assume that div v = 0 with 2,1 contains both the fluid andCthe structure. consider S is smooth (say, class ), and compact v(t, (4) rigid of shape moving Ω. WeWe assume thata with 2,1inside S is aabody smooth (say, of ofS class C 2,1 ), connected connected and compact v(t, x) x) = = 0, 0, tt > > 0, 0, x x∈ ∈ ∂Ω, ∂Ω, (4) 3of shape 2,1 with rigid body S moving inside Ω. We assume that 2,1 subset of R with non empty interior. S is a smooth (say, of class C ), connected and compact v(t, x) = 0, t > 0, x ∈ ∂Ω, (4) 3 and 3 with non empty 2,1 subset of R interior. 3 (say, of class C S is a smooth ), connected and compact and v(t, x) = 0, t > 0, x ∈ ∂Ω, (4) 3 3 non empty interior. subset of R with and For R and for R v(t, x) = V (t) + r(t) × (x − h(t)) subset For all allofh hR3∈ ∈ with R3333 non and3empty for all allinterior. R ∈ ∈ SO(3) SO(3) (the (the special special and v(t, x) = V (t) + r(t) × (x − h(t)) orthogonal group R3 ), set For all h ∈ R of and forwe ∈ SO(3) (the special + v(t, x) = V (t) + r(t) × (x − h(t)) orthogonal group weall setR 3 of R3 3 ), + u(t, u(t, x), x), x x∈ ∈ ∂S(t). ∂S(t). (5) (5) For all h ∈ R and for all R ∈ SO(3) (the special 3 v(t, x) = V (t) + r(t) × (x − h(t)) = h + RS, F = Ω \ S . orthogonal S group of R ), we set h,R h,R h,R 3 + u(t, x), x ∈ (5) =ofh R + RS, Fh,R = Ω \ Sh,R . Here V (t)+r(t)×(x−h(t)) is the velocity of the∂S(t). rigid body h,R orthogonal S group ), we set h,R h,R h,R +our u(t, x), ofx the ∈ ∂S(t). (5) Sh,R h +the RS, Fh,R Ω \ Sh,R Here V (t)+r(t)×(x−h(t)) is the velocity rigid body h,R = in h,R =positions h,R .of the rigid We are interested admissible and u = u(t, x) is the control of system. To obtain the Here V (t)+r(t)×(x−h(t)) is the velocity of the rigid body We are interested in admissible positions the rigid and u = u(t, x) is the control of our system. To obtain the h(h, +the RS, Fh,R Ω\S .of h,R = in h,R body, inScouples R)admissible such that=Spositions ⊂ Ω. We assume We arei.e.interested the of the rigid Here is the velocity of the rigid body h,R equations of the positions we apply the laws and uV=(t)+r(t)×(x−h(t)) u(t, the control our system. To obtain h,R body, i.e. in R) such that ⊂ We assume equations of x) theis we of apply the Newton’s Newton’s laws the h,R We are interested in(h, the admissible the rigid that in that case, F is connected. In what follows, we body, i.e. in couples couples (h, R) such that S Spositions ⊂ Ω. Ω. of We assume and u = u(t, x) ispositions the control of our system. To obtain the h,R h,R h,R that in that case, F is connected. In what follows, we equations of the positions we apply the Newton’s laws h,R h,R body, i.e. in couples (h, R) such that Sofh,R ⊂ Ω. We assume also suppose that the center of mass S is located at the equations of the positions we apply the Newton’s laws h,R that in that case, F is connected. In what follows, we h,R also that the center of mass ofInS what is located at the T(v, (6) M that suppose insothat case, F is connected. follows, we origin that h is the center mass R). h,R also suppose that the center ofof mass ofof SS(h, is located at the =− − T(v, p)n p)n dΓ, dΓ, (6) MV V = origin so that h is the center of mass of S(h, R). also suppose is located T(v, p)n dΓ, (6) origin so thatthat h is the thecenter centerofofmass massofofSS(h, R). at the M V = − ∂S(t) ∂S(t) ∂S(t) If body a tt → (h(t), (6) M V = − ∂S(t) origin so that h isfollows the center of mass of S(h, R).R(t)), ∂S(t) T(v, p)n dΓ, If the the rigid rigid body follows a trajectory trajectory → (h(t), R(t)), we we = − ∂S(t) (x − h) × T(v, p)n dΓ. (J(t)r) (7) can define its angular velocity given through the formula If the rigid body follows a trajectory t → (h(t), R(t)), we (J(t)r) = − (x − h) × T(v, p)n dΓ. (7) can define angular velocity given through theR(t)), formula = − ∂S(t) (x − h) × T(v, p)n dΓ. velocity If the rigidits body follows a trajectory t → (h(t), we (J(t)r) (7) can define its angular given through the formula ∂S(t) R (1) ∂S(t) (t) = A(r(t))R(t), R (t) = A(r(t))R(t), (1) In above(J(t)r) =we − use (x − h) × T(v, p)n dΓ. (7) ∂S(t) can define its angular velocity given through the formula ∂S(t) settings, the notation where R (1) In above settings, we use the notation ∂S(t) (t) = A(r(t))R(t), where R (1) In above settings, we (t) d v use 0 = −rA(r(t))R(t), the notation def ∂v 3 r2 where def d vv use ∂v 3 −r rr22 (r ∈ R3 ). = (v · ∇)v, In above settings, we the+notation def r0 3 −r 0 A(r) = d ∂v 0 −r = where 3 def ∂t + (v · ∇)v, 21 d t def 3 ). 2 0 33 −r A(r) = r0333 −r (r ∈ R 1 3 1 d t ∂t = + · ∇)v, v def ∂vof v (v r021 r332 rr013 −r A(r) = −r (r ∈ R33 ). for the material derivative and −r 01 d t = ∂tof +v (v · ∇)v, 2 1 2 1 for the material derivative and r 0 −r A(r) = ). (r ∈ R d t def ∂tof v and Finally, the body is defined by −r322 r11of 0 1 rigid for the material derivative def = 2νD(v) − pI , T(v, p) Finally, the the linear linear velocity velocity of the rigid body is defined by def −r2 r1def 0 = 2νD(v) T(v, p) def def for the material derivative of v and− pI333 , Finally, the linear velocity of the rigid body is defined (2) by def V = h = 2νD(v) − pI33 , T(v, p) . def with def Finally, the linear velocity by V of = the h . rigid body is defined (2) def def with pI3 , T(v,def p) 1= 2νD(v) − V def = h . (2) with def 1 (∇u) + ttt (∇u) . D(u) = def To simplify, in what follows, we set V = h . (2) D(u) def = 12 (∇u) + tt (∇u) . with def To simplify, in what follows, we set D(u) def = 21 (∇u) + t (∇u) . To simplify, in def what follows, we setdef def def S(t) = S , F(t) = F . def def h(t),R(t) h(t),R(t) D(u) = 2 (∇u) + (∇u) . To simplify, in def what follows, we set S(t) = S , F(t) = F . We assume that the density def h(t),R(t) def h(t),R(t) def h(t),R(t) h(t),R(t) 2 ρ We assume that the density ρSSS of of the the rigid rigid body body is is constant constant h(t),R(t) h(t),R(t) = S , F(t) = F . This workS(t) def h(t),R(t) def h(t),R(t) so that was supported by ANR IFSMACS We assume that the density ρ of the rigid body is constant S This workS(t) S = Sh(t),R(t) F(t) = Fh(t),R(t) . so that was supported by ,ANR IFSMACS We assume that the density ρ of the rigid body is constant This S This work work was was supported supported by by ANR ANR IFSMACS IFSMACS so that This work was supported by ANR IFSMACS so that 2405-8963 © © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright 64 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 64 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 64 Copyright © 2019 2019 IFAC IFAC 64 10.1016/j.ifacol.2019.08.012 Copyright © 2019 IFAC 64
2019 IFAC CPDE-CDPS Oaxaca, Mexico, May 20-24, 2019
def
M = ρS |S|,
def
J(t) = ρS
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
S(t)
(x − h(t)) ⊗ (x − h(t)) dx.
u(t, x) =
Nσ
uj v j ,
(10)
j=1
One can check that J(t) = RJS R∗ where def JS = ρS y ⊗ y dy.
def
with Nσ ∈ N∗ and u = (uj )j∈{1,...,Nσ } satisfying u = Λu + G(v ◦ X, V, r, h, R),
S
t > 0,
2
u(0) = 0, (11)
for a suitable map G : L (Fh1 ,R1 ) × R → RNσ and for a matrix Λ of size Nσ × Nσ . In above setting, X : Fh1 ,R1 → F(t) is a diffeomorphism. Theorem 1. For all σ > 0, there exist Nσ ∈ N, c0 , C > 0, a map G as above such that if (v 0 , V 0 , r0 , h0 , R0 ) ∈ H1 (F(h0 , R0 )) × R9 × SO(3) satisfies
Finally, we assume the following initial conditions for the positions: h(0) = h0 , R(0) = R0 , V (0) = V 0 , r(0) = r0 , v(0, ·) = v 0 .
65
(8) (9)
In the system (1)–(7), the state is (v, p, V, r, h, R). In particular, the domains F(t) and S(t) are evolving and unknowns. The Navier-Stokes system is thus written in an open non-cylindrical subset of R4 that depends on the trajectories t → (h(t), R(t)).
12
div v 0 = 0 in Fh0 ,R0 , v = V + r0 × (x − h0 ) on ∂Sh0 ,R0 , v 0 = 0 on ∂Ω, 0
0
and v 0 H1 + |V 0 | + |r0 | + |h0 − h1 | + |R0 − R1 | c0 , then there exists a strong solution (v, p, V, r, h, R, u) of (1)– (7), (8)–(9), (10), (11) such that
We assume that the initial data (v 0 , V 0 , r0 , h0 , R0 ) is close to the target (0, 0, 0, h1 , R1 ) and we want to find a feedback control u in (5) such that the resulting solution (v(t), V (t), r(t), h(t), R(t)) goes to (0, 0, 0, h1 , R1 ) as t → +∞ with an exponential rate of decrease. In order to this, we follow the general strategy described in Badra and Takahashi (2011, 2014b) and in particular we prove the stabilizability of the linearized system. This is done by using the classical Fattorini-Hautus test on the unstable modes. As explained above, a difficulty here is that the fluid system is written in a moving domain F(t). This is standard in the study of fluid–structure system, and a natural method consists in using a change of variables (see for instance Grandmont and Maday (2000), Takahashi (2003), Boulakia and Osses (2008), Imanuvilov and Takahashi (2007), Boulakia and Guerrero (2013)).
v(t)H1 + |V (t)| + |r(t)| + |h(t) − h1 | + |R(t) − R1 | + |u(t)| Ce−σt v 0 H1 + |V 0 | + |r0 |
+ |h0 − h1 | + |R0 − R1 | .
1. CHANGE OF VARIABLES Up to a rotation and a translation on S, we can assume that Sh1 ,R1 = S. In that case, we also write
The stabilization of fluid-structure systems has been studied for some years: first Raymond considered in Raymond (2010) the boundary stabilization of fluid-beam equation around the null state. In Badra and Takahashi (2014a) the authors consider the case of a rigid body moving into a viscous incompressible fluid and obtain a local stabilization result around any stationary state by taking a control on the exterior boundary ∂Ω. The main difference between this work and the presented work corresponds to the localization of the control, here we act at the interface fluid-structure. This leads to some additional difficulties and due to our method, we can only stabilize here around a stationary state with null velocity for the fluid and for the structure. The control can also be a force applied on the rigid body: this is the strategy considered in Takahashi et al. (2015) (or in Cˆındea et al. (2015) in the 1D case).
def
F = Fh1 ,R1 We rewrite our system in a cylindrical domain by using a change of variables X(t, ·) : Ω → Ω, such that X(t, F) = F(t). We denote by Y (t, ·) the inverse of X(t, ·). Several constructions are possible to obtain such a change of variables. Here, we consider def
X(t, y) = y + η(y) [h(t) + (R(t) − I3 )y] where η is a smooth function such that 1 if dist(y, S) < ε, η(y) = 0 dist(y, S) 2ε,
Let us point out that here, we assume that the initial conditions are in H 1 for the fluid velocity (“strong solutions”). Consequently (see Badra (2012), Badra (2009b)), there are compatibility conditions at t = 0 with the feedback control u. Several techniques exist to overcome this problem: Raymond (2007), Badra (2009a), Badra and Takahashi (2011). Here, we use the approach of Badra and Takahashi (2011): the control u satisfies an evolution equation with another control feedback and this leads to stabilize a system coupling the state and the feedback control u. More precisely we assume that the control u in (4) can be written as
(12)
with ε < dist(S, ∂Ω)/2.
The map X is a C ∞ -diffeomorphism of Ω onto itself if (13) ηW 1,∞ (Ω) (|h(t)| + |R(t) − I3 |) < c,
for c small enough. It satisfies X(t, y) = y for y in a neighborhood of ∂Ω and that X(t, y) = R(t)y + h(t) in a neighborhood of S. In what follows, we assume that ∀t 0, |h(t)| + |R(t) − I3 | < C , with C small enough so that (13) holds.
65
2019 IFAC CPDE-CDPS 66 Oaxaca, Mexico, May 20-24, 2019
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
2. STABILIZATION OF THE LINEAR SYSTEM
With the change of variables introduced above, we set def
w(t, y) = Cof(∇X(t, y))∗ v(t, X(t, y)),
The present subsection is dedicated to the abstract formulation of the following linear nonhomogeneous problem: q(t, y) = p(t, X(t, y)), (33) ∂t w − ν∆w + ∇q = F in (0, +∞) × F, def def (t) = R∗ (t)V (t), ω(t) = R∗ (t)r(t) div w = 0 in (0, +∞) × F, (34) where Cof(M ) is the cofactor matrix of M , which satisfies w =+ω×y+u ˜ on (0, +∞) × ∂S, (35) in particular M (Cof(M ))∗ = (Cof(M ))∗ M = det(M )I3 . w = 0 on (0, +∞) × ∂Ω, (36) Following Inoue and Wakimoto (1977), Takahashi (2003) M = − T(w, q)n dΓ + F , t > 0, (37) and Boulakia et al. (2012), one can prove that (v, p, V, r, h, R) ∂S satisfies (1), (2) and (3)–(7) if and only if (w, q, , ω, h, R) JS ω = − y × T(w, q)n dΓ + ωF , t > 0, (38) satisfies the following system def
∂S
K∂t w − νLw + Mw + N(w) + Gq = 0 in (0, +∞) × F, div w = 0 in (0, +∞) × F, w =+ω×y+u ˜ on (0, +∞) × ∂S, w = 0 on (0, +∞) × ∂Ω, M (R) = −R T(w, q)n dΓ, t > 0, ∂S (RJS ω) = −R y × T(w, q)n dΓ, t > 0, ∂S
h = R, t > 0, R = RA(ω), t > 0.
(39) h = + hF , t > 0, (40) θ = ω + θF , t > 0, with the initial conditions h(0) = h0 , θ(0) = θ0 , (41) 0 0 0 (0) = , ω(0) = ω , w(0, y) = w (y) y ∈ F. (42) Here, F , F , ωF , hF , QF are nonhomogeneous terms that replace the nonlinearities in (22)–(32) so that (33)–(40) is a linear system.
(14) (15) (16) (17) (18)
We write the above system in the form
(19) (20) (21)
˜ + P F in [D(A∗ )] , P W = AP W + B u P W(0) = P W
The operators K, M, N, L and G can be expressed through X and Y (see Badra and Takahashi (2014a) for the precise expressions).
θ = B(θ)ω, where B : R → M3 (R) is a smooth map such that B(0) = I3 . Here M3 (R) denotes the space of 3-by-3 real matrices. 3
def
V =
∗
We set W = [w, , ω, h, θ] and our system can write as follows
h = + εh (W), θ = ω + εθ (W), 0
(0) = 0 ,
t > 0, t > 0, 0
h(0) = h , θ(0) = θ , ω(0) = ω 0 , w(0, y) = w0 (y)
where εw , ε , εω , εh , εθ are nonlinearities.
(44)
w · n = ( + ω × y) · n on ∂S,
∂S
(43)
(45) (Id −P )W = (Id −P )DF u, ∗ ∗ where W = [w, , ω, h, θ] , F = [F, F , ωF , hF , θF ] and where A, P , B and DF are linear operators defined below. Formulation of type (44)-(45) is due to Raymond (2006) for Stokes type systems. Let us define def H = [w, , ω, h, θ]∗ ∈ L2 (F) × (R3 )4 ;
In order to linearize this system, we use a local chart of SO(3) around I3 : there exist smooth maps Ri , i = 1, 2, 3 such that any matrix R ∈ SO(3) can write in a unique way as R = R1 (θ1 )R2 (θ2 )R3 (θ3 ). A classical way to do this is to use the Euler angles. After a standard calculation, one can write R = RA(ω) as
∂t w − ν∆w + ∇q = εw (W, q) in (0, +∞) × F, div w = 0 in (0, +∞) × F, w =+ω×y+u ˜ on (0, +∞) × ∂S, w = 0 on (0, +∞) × ∂Ω, M = − T(w, q)n dΓ + ε (W), t > 0, ∂S JS ω = − y × T(w, q)n dΓ + εω (W), t > 0,
0
w · n = 0 on ∂Ω, div w = 0 [w, , ω, h, θ]∗ ∈ H ;
in F ,
w ∈ H1 (F), w = + ω × y on ∂S,
w = 0 on ∂Ω .
We define the linear operator A : D(A) ⊂ H → H as follows: we set def def D(A) = {[w, , ω, h, θ]∗ ∈ V ; w ∈ H2 (F)}, A = P A, ∗ where for [w, , ω, h, θ] ∈ D(A), we set ν∆w −M −1 w 2νD(w)n dΓ ∂S def ω = A −JS−1 h y × 2νD(w)n dΓ ∂S θ ω
(22) (23) (24) (25) (26) (27) (28) (29) (30)
and where P is the orthogonal projection from L2 (F)×R12 onto H.
(31) y ∈ F, (32)
As in Badra and Takahashi (2014a), we can check the following proposition.
66
2019 IFAC CPDE-CDPS Oaxaca, Mexico, May 20-24, 2019
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
Proposition 2. The operator A defined above is densely defined with compact resolvent and it is the infinitesimal generator of an analytic semigroup on H.
def
Σα,θ = {λ ∈ C ; | arg(λ − α)| < θ} ⊂ ρ(A) sup λ(λ − A)−1 L(H) < ∞.
λ∈Σα,θ
Then, we can prove the following characterization of the adjoint of A. Proposition 3. The adjoint of the operator A is given by D(A∗ ) = D(A) and ν∆ϕ −M −1 ϕ 2νD(ϕ)n dΓ + M −1 a ξ ∂S ∗ A ζ = P −1 . −1 −JS a y × 2νD(ϕ)n dΓ + J b S ∂S b 0 0 (46)
Now we prove that for any σ > 0, there exists a feedback control Fσ (W) =
and
(Φj , W)L2 (F )×R12 vj , Φj ∈ L2 (F) × R12 ,
(52)
such that the solution W of (33)-(40) with u = Fσ (W) and with (F, F , ωF , hF , θF ) = (0, 0, 0, 0, 0) tends to zero as t → +∞ with an exponential rate of decrease σ > 0. For that, we are going to show the existence of families Φj and vj , j = 1, . . . , Nσ such that the underlying closedloop linear operator of (33)-(40) with (52) generates and analytic and exponentially stable semigroup of type lower than −σ (see (Bensoussan et al., 2007, II-1, Cor. 2.1)). It then permits to deduce results for the case of non zero (F, F , ωF , hF , θF ) that are used in the next subsection to construct fixed-point solutions of the nonlinear system (22)-(32). Proposition 6. For σ > 0, there exist Nσ ∈ N∗ , families 3 Φj ∈ D(A∗ ) and vj ∈ V 2 (∂S), j = 1, . . . , Nσ , such that def def the linear operator Aσ = A + BFσ with domain D(Aσ ) = {X ∈ H | AX + BFσ X ∈ H} is the infinitesimal generator of an analytic and exponentially stable semigroup on H of type lower than −σ.
w = + ω × y + u on ∂S, w = 0 on ∂Ω,
Nσ j=1
Next, we take λ0 > 0 large enough so that λ0 ∈ ρ(A). Then we introduce the Dirichlet operator DF : V0 (∂Ω) → L2 (F) × R12 defined as follows: for u ∈ V0 (∂Ω) we denote def by DF u = [wu u ωu hu θu ]∗ the unique solution of λ0 w − ν∆w + ∇q = 0 in F, div w = 0 in F,
∇π
−M −1 πn dΓ ∂S −1 = ⊥ πn · y dΓ −JS ∂S 0 0 −λ 0 ϕ + ν∆ϕ −λ0 ξ − M −1 2νD(ϕ)n dΓ + M −1 a ∂S (Id −P ) −1 −1 y × 2νD(ϕ)n dΓ + JS b −λ0 ζ − JS ∂S −λ0 a −λ0 b (51)
We recall (see (Bensoussan et al., 2007, p.112)) that A is the infinitesimal generator of an analytic semigroup on H if there exist α ∈ R and θ > π2 such that and if
67
T(w, q)n dΓ = 0, λ0 M + ∂S λ0 JS ω + y × T(w, q)n dΓ = 0, ∂S
λ0 h − = 0, λ0 θ − ω = 0.
We recall (see, for instance, (Bensoussan et al., 2007, p.92)) that a strongly continuous semigroup (T (t))t 0 on H is exponentially stable of type lower than −σ if for any α < σ there exists Cα > 0 such that
Using standard techniques (see Badra and Takahashi (2014a) ), we can prove the following result. Proposition 4. The mapping DF defined above satisfies the following boundedness property: 1 3 12 s s+ 12 . (47) (F) × R ) s ∈ − , DF ∈ L(V (∂S), H 2 2
∀t 0,
T (t)L(H) Cα e−αt .
Proof of Proposition 6. The proof of the above proposition relies on the Hautus-Fattorini stabilizability criterion, see (Badra and Takahashi, 2011, Theorem 1) or Badra and Takahashi (2014b). Since A has compact resolvent and generates an analytic semigroup on H, and since B is relatively bounded with respect to A, then the homogeneous linear system is stabilizable by finite dimensional feedback control for any rate of decrease if and only if the following criterion is satisfied for all λ ∈ C:
Next, we define the input operator B : V0 (∂S) → [D(A∗ )] , Bu = (λ0 − A)P DF u. (48) Proposition 5. The operator B defined by (48) satisfies: 1 −1+ 0 (λ0 − A) . (49) B ∈ L(V (∂S), H), ∈ 0, 4 Moreover, the adjoint of B is defined by ∗ (50) B ∗ [ϕ, ξ, ζ, a, b] = −T(ϕ, π)n|∂S , where
λY − A∗ Y = 0
and
B∗Y = 0
=⇒
Y = 0. (53)
From (46) and (50) we deduce that (53) is true if and only if all [ϕ, ξ, ζ, b, κ]∗ ∈ D(A∗ ) which satisfies 67
2019 IFAC CPDE-CDPS 68 Oaxaca, Mexico, May 20-24, 2019
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
λϕ − ν∆ϕ + ∇π = 0 in F, πdΓ = 0, ∂Ω div ϕ = 0 in F, ϕ = ξ + ζ × y on ∂S, ϕ = 0 on ∂Ω, T(ϕ, π)n dΓ = a, M λξ + ∂S JS λζ + y × T(ϕ, π)n dΓ = b, ∂S λa = 0, λb = 0. and T(ϕ, π)n = 0 on ∂S. must be identically zero.
Then by setting def
F(X, q) = [F (X, q), ε (X), εω (X), εh (X), εθ (X)]∗ the nonlinear system (14)-(21), (57) and (58) rewrites P X = AP X + BΨu + P F(X, q), P X(0) = P X0 , (59) u = g, u(0) = 0, (Id −P )X = (Id −P )DF Ψu.
(54)
Thus, by following the general framework of Badra and Takahashi (2011) for the construction of dynamical control (see also Badra and Takahashi (2014b)) we introduce the def “extended” space HE = H × RNσ , the following extended operator on HE : A BΨ def , AE = 0 0
(55)
(56)
The above implication can be proved as follows: if λ = 0, then, a = 0, b = 0 and thus from (56), ξ = ζ = 0. Thus (ϕ, π) satisfies a Stokes system with homogeneous Dirichlet boundary conditions and using a classical uniqueness result for Stokes type systems (see e.g. Fabre and Lebeau (1996)) we deduce that ϕ = 0 and π = 0 on F. If λ = 0, then we deduce from (55) and (56) that a = 0 and b = 0. Then we multiply the first equation of (54) by ϕ: div T(ϕ, π) · ϕ dx 0=− F = |D(ϕ)|2 dx − T(ϕ, π)n · ϕ dΓ F
def
D(AE ) = {(Y, u) ∈ HE | AY + BΨu ∈ H} ,
as well as the control operator BE ∈ L(RNσ , HE ): def 0 . BE g = g
Thus, since the stabilizability of (A + σ, B) by finite dimensional control generated by the family (vj ) implies the stabilizability of (AE + σ, BE ) (see (Badra and Takahashi, 2011, Theorem 8)), we can find a family (Φj )1 j Nσ ⊂ D(A∗ ), a matrix Λ of size Nσ × Nσ and a corresponding feedback operator FE,σ : HE → RNσ defined by
∂S
and thus Dϕ = 0 in F which implies that that ϕ = ϕ + ωϕ × y. Using the boundary conditions of ϕ this leads to
ϕ = ξ = 0 and ωϕ = ζ = 0.
Then the general framework of Badra and Takahashi (2011, 2014b) can be applied and for a given σ > 0, there exist families Φj ∈ D(A∗ ) and vj ∈ V0 (∂S), j = 1, . . . , Nσ , and a feedback control of the form (52) such that the conclusions of the proposition hold. Moreover, 3 each vj can be chosen in V 2 (∂S). This comes from the fact that the set of admissible families (vj ) is a nonempty open set of (V0 (∂S))Nσ (see (Badra and Takahashi, 2011, Theorem 5) or (Badra and Takahashi, 2014b, Theorem 6)). Indeed, if a family (˜ vj ) is admissible then all families in a neighborhood of (˜ vj ) in (V0 (∂S))Nσ are admissible. 3 Then the conclusion follows from the density of V 2 (∂S) in V0 (∂S).
GE,σ [w, , ω, h, θ, u]∗ = Λu +
4. FIXED POINT ARGUMENT In order to prove the result, we are going to use a fixed point argument. We set def L2σ (H) = F ∈ L2 (0, ∞; H) ; t → eσt F(t) ∈ L2 (0, ∞; H) . (60)
As explained in the introduction, in order to guarantee the initial compatibility condition w 0 = 0 + ω 0 × y + u ˜(0) on ∂S, we complete the system (14)-(21) with a dynamical equation for u ˜: Nσ def u ˜(t, x) = Ψu = uj v j , (57)
We consider the Banach space def
E = L2σ (H) and the following mapping defined on a closed ball of E:
j=1
def
u = g,
u(0) = 0.
(Φj , [w, , ω, h, θ, u])HE ej
where {ej | j = 1, . . . , Nσ } is the canonical basis of RNσ , def such that the linear operator AE,σ = AE + BE GE,σ with domain D(AE,σ ) = D(AE ) (since BE is bounded in HE ) is the infinitesimal generator of an analytic and exponentially stable semigroup on HE of type lower than −σ.
3. EXTENDED SYSTEM
Nσ j=1
with u = (uj )j∈{1,...,Nσ } satisfying
def
0 We set YE = [P X0 , 0]∗ , YE = [P X, u]∗ and FE (YE , q) = [P F(P X + (Id −P )DF Ψu, q), 0]∗ , so that system (59) can be written as 0 = AE YE + BE g + FE (YE , q), YE (0) = YE . YE
Ψ : BE (0, δ) → E, ˇ → P F(P X + (Id −P )DF Ψu, q), 0]∗ , F
(58) 68
2019 IFAC CPDE-CDPS Oaxaca, Mexico, May 20-24, 2019
Mehdi Badra et al. / IFAC PapersOnLine 52-2 (2019) 64–69
dimensional systems. Systems & Control: Foundations & Applications. Birkh¨auser Boston Inc., Boston, MA, second edition. Boulakia, M. and Guerrero, S. (2013). Local null controllability of a fluid−solid interaction problem in dimension 3. J. Eur. Math. Soc., 15(3), 825–856. Boulakia, M. and Osses, A. (2008). Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM Control Optim. Calc. Var., 14(1), 1– 42. Boulakia, M., Schwindt, E.L., and Takahashi, T. (2012). Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid. Interfaces Free Bound., 14(3), 273–306. Cˆındea, N., Micu, S., Rovent¸a, I., and Tucsnak, M. (2015). Particle supported control of a fluid-particle system. J. Math. Pures Appl. (9), 104(2), 311–353. Fabre, C. and Lebeau, G. (1996). Prolongement unique des solutions de l’equation de Stokes. Comm. Partial Differential Equations, 21(3-4), 573–596. Grandmont, C. and Maday, Y. (2000). Existence for an unsteady fluid-structure interaction problem. M2AN Math. Model. Numer. Anal., 34(3), 609–636. Imanuvilov, O.Y. and Takahashi, T. (2007). Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. (9), 87(4), 408–437. Inoue, A. and Wakimoto, M. (1977). On existence of solutions of the Navier-Stokes equation in a time dependent domain. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24(2), 303–319. Raymond, J.P. (2007). Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 24(6), 921–951. Raymond, J.P. (2010). Feedback stabilization of a fluidstructure model. SIAM J. Control Optim., 48(8), 5398– 5443. Raymond, J.P. (2006). Feedback boundary stabilization of the two-dimensional Navier-Stokes equations. SIAM J. Control Optim., 45(3), 790–828. Takahashi, T. (2003). Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Adv. Differential Equations, 8(12), 1499–1532. Takahashi, T., Tucsnak, M., and Weiss, G. (2015). Stabilization of a fluid–rigid body system. Journal of Differential Equations, 259(11), 6459–6493.
def
where X = [w, , ω, h, θ]∗ and u is the solution ˇ P X = AP X + BΨu + F, 0 P X(0) = P X , u = g, u(0) = 0, (Id −P )X = (Id −P )DF Ψu.
69
(61)
ˇ is a fixed point of the mapping Ψ, then We remark that if F the corresponding solution (w, q, , ω, h, θ) of (33)–(40), (41) and (52) is a solution of (22)–(32). Consequently, we are reduced to show that Ψ admits a fixed point. We prove that for δ small enough, Ψ is well-defined from BE (0, δ) onto itself and that the restriction of Ψ on this closed ball is a contraction mapping. This is done by estimating F (X, q), ε (X), εω (X), εh (X) and εθ (X) and thus by working on the change of variables X and Y defined (12). Such technical results are done in Badra and Takahashi (2014a). With the smallness condition on the initial data, we deduce that the restriction of Ψ on BE (0, δ) is a contraction mapping. The classical Banach fixed point theorem allows us to conclude. 5. CONCLUSIONS AND FUTURE WORKS By acting on ∂S instead of ∂Ω we are lead to a different unique continuation property: (54), (55) and (56) imply that ϕ ≡ 0. We would arrive to a similar unique continuation property by acting on the force and the torque of the rigid body, i.e. by putting controls in (6) and in (7). This was done in Takahashi et al. (2015) with a “spring” force. Next, we would like to act only on a part Γc of ∂S. In that case, we have to replace (56) by (62) T(ϕ, π)n = 0 on Γc and we do not know if this implies ϕ ≡ 0 as before. REFERENCES Badra, M. (2009a). Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system. ESAIM Control Optim. Calc. Var., 15(4), 934– 968. Badra, M. (2009b). Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations. SIAM J. Control Optim., 48(3), 1797–1830. Badra, M. (2012). Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete Contin. Dyn. Syst., 32(4), 1169–1208. Badra, M. and Takahashi, T. (2011). Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamic controllers. Application to NavierStokes system. SIAM J. Control Optim., 49(2), 420–463. Badra, M. and Takahashi, T. (2014a). Feedback stabilization of a fluid-rigid body interaction system. Adv. Differential Equations, 19(11-12), 1137–1184. Badra, M. and Takahashi, T. (2014b). On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems. ESAIM Control Optim. Calc. Var., 20(3), 924–956. Bensoussan, A., Da Prato, G., Delfour, M.C., and Mitter, S.K. (2007). Representation and control of infinite 69