On the controllability of the N-dimensional Navier–Stokes and Boussinesq systems with N − 1 scalar controls

C. R. Acad. Sci. Paris, Ser. I 340 (2005) 275–280 http://france.elsevier.com/direct/CRASS1/ Partial Differential Equations/Optimal Control On the controllability of the N-dimensional Navier–Stokes and Boussinesq systems with N−1 scalar controls Enrique Fernández-Caraa, Sergio Guerreroa, Oleg Yurievich Imanuvilovb, Jean-Pierre Puelc aDepartamento E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain bDepartment of Mathematics, Iowa State University, 400 Carver Hall, Ames, IA 50011-2064, USA cLaboratoire de mathématiques appliquées, université de Versailles – St Quentin, 45, avenue des Etats Unis, 78035 Versailles, France Received 4 November 2004; accepted 3 December 2004 Available online 18 January 2005 Presented by Pierre-Louis Lions Abstract In this Note we present several controllability results for nonlinear systems of the Navier–Stokes and Boussinesq kind. We discuss the existence of particular controls with a small number of degrees of freedom. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Sur la contrôlabilité des systèmes de Navier–Stokes et Boussinesq N-dimensionnels avec N−1 contrôles scalaires. Dans cette Note on présente quelques résultats de contrôlabilité pour des systèmes non linéaires du type Navier–Stokes et Boussinesq. On analyse l’existence de contrôles particuliers avec un nombre petit de degrés de liberté. Pour citer cet article:E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. Version française abrégée Cette Note contient quelques résultats de contrôlabilité pour des systèmes non linéaires du type Navier–Stokes et Boussinesq. On cherche des contrôles particuliers, avec un nombre petit de degrés de liberté. E-mail addresses: [email protected] (E. Fernández-Cara), [email protected] (S. Guerrero), [email protected] (O.Y. Imanuvilov), [email protected] (J.-P. Puel). 1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crma.2004.12.013 276 E. Fernández-Cara et al. / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 275–280 On se donne un domaine borné et régulier Ω⊂RN(N=2ouN=3), un ouvert non vide (et petit) O⊂Ωet un nombre T>0. On considère d’abord le système yt−y +(y ·∇)y +∇p=v1O,∇·y=0 dans Q=Ω×(0,T), y=0surΣ=∂Ω ×(0,T), y(0)=y0dans Ω,(1) où 1Oest la fonction caractéristique de O. Considérons l’espace de Banach E=H∩L4(Ω)N,où H={w∈L2(Ω)N;∇·w=0 dans Ω, w ·n=0sur∂Ω}.(2) On dira que (1) est localement exactement contrôlable aux trajectoires en temps Tsi, pour toute solution suffisamment régulière (¯y, ¯p) du système non contrôlé ¯yt−¯y+(¯y·∇)¯y+∇¯p=0,∇·¯y=0 dans Q, ¯y=0surΣ, il existe δ>0 tel que, si y0−¯y(0)E⩽δ, alors on peut trouver des contrôles v∈L2(O×(0,T)) Net des états associés (y, p) solution de (1) satisfaisant y(T) =¯y(T). On considère maintenant le système de Boussinesq      yt−y +(y ·∇)y +∇p=v1O+θe N,∇·y=0 dans Q, θt−θ +y·∇θ=h1Odans Q, y=0,θ=0surΣ, y(0)=y0,θ(0)=θ0dans Ω, (3) où les contrôles sont v∈L2(O×(0,T)) Net h∈L2(O×(0,T)).Ondiraque(3)estlocalement exactement contrôlable aux trajectoires en temps Tsi, pour toute solution suffisamment régulière (¯y, ¯p, ¯ θ) du système    ¯yt−¯y+(¯y·∇)¯y+∇¯p=¯ θeN,∇·¯y=0 dans Q, ¯ θt−¯ θ+¯y·∇¯ θ=0 dans Q, ¯y=0,¯ θ=0surΣ, il existe δ>0 tel que, si (y0,θ0)−(¯y(0), ¯ θ(0))E×L2⩽δ, alors on peut trouver des contrôles v∈L2(O× (0,T)) Net h∈L2(O×(0,T)) et des états associés (y,p,θ)satisfaisant y(T) =¯y(T) et θ(T)=¯ θ(T). Finalement, lorsque N=2, on considèrera les systèmes avec terme non linéaire tronqué yt−y +(y ·∇)TM(y) +∇p=v1O,∇·y=0 dans Q, y·n=0,∇×y=0surΣ, y(0)=y0dans Ω,(4) où M>0, TM(y) =(TM(y1), TM(y2)) et TM(s) =−Msi s⩽−M, ssi −M⩽s⩽M, Msi s⩾M. On dira que (4) est (globalement) contrôlable à zéro en temps Tsi, pour tout y0∈H,ilexistev∈L2(O× (0,T)) 2tel que l’état associé satisfait y(T ) =0. On fait les hypothèses suivantes sur O,¯yet ¯ θ: ∃x0∈∂Ω, ∃ε>0 tels que O∩∂Ω ⊃B(x0;ε) ∩∂Ω, (5) ¯y∈L∞(Q)N,¯yt∈L20,T;Lσ(Ω)Nσ>1siN=2 σ>6/5siN=3(6) et ¯ θ∈L∞(Q), ¯ θt∈L20,T;Lσ(Ω)σ>1siN=2 σ>6/5siN=3.(7) E. Fernández-Cara et al. / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 275–280 277 On a alors les résultats suivants : Théorème 0.1. Supposons que Osatisfait (5). Alors, pour tout T>0,(1) est localement exactement contrôlable en temps Taux trajectoires (y,p) qui satisfont (6) avec des contrôles v∈L2(O×(0,T)) Nqui possédent une composante nulle. Théorème 0.2. Supposons que Osatisfait (5) avec ni(x0)= 0, pour un indice i<N. Alors, pour tout T>0,(3) est localement exactement contrôlable en temps Taux trajectoires (¯y, ¯p, ¯ θ)qui satisfont (6),(7) avec des contrôles vet htels que vi≡vN≡0. En particulier, si N=2, on a la contrôlabilité exacte locale aux trajectoires avec des contrôles v≡0et h∈L2(O×(0,T)). Pour notre troisième résultat, on introduit l’espace W=∇×z=(∂2z, −∂1z);z∈L20,T;H1(Ω). Alorsona: Théorème 0.3. Soit N=2. Alors, pour tout T>0et tout M>0,lesystème(4)est contrôlable à zéro en temps Tavec des contrôles v1O,oùv∈W. Les démonstrations détaillées de ces résultats seront données dans un travail à paraître. 1. Introduction and main results Let Ω⊂RNbe a bounded and regular domain (N=2orN=3), let O⊂Ωbe a nonempty (small) open subset and let T>0 be given. In this Note, we present several controllability properties for some nonlinear systems of the Navier–Stokes and Boussinesq kind. We will first be concerned with the Navier–Stokes system yt−y +(y ·∇)y +∇p=v1O,∇·y=0inQ=Ω×(0,T), y=0onΣ=∂Ω ×(0,T), y(0)=y0in Ω,(8) where 1Ois the characteristic function of O. Let us introduce the Banach space E=H∩L4(Ω)N, where H=w∈L2(Ω)N;∇·w=0inΩ, w ·n=0on∂Ω.(9) It will be said that (8) is locally exactly controllable to the trajectories at time Tif, for each sufficiently regular solution (¯y, ¯p) to the uncontrolled system ¯yt−¯y+(¯y·∇)¯y+∇¯p=0,∇·¯y=0inQ, ¯y=0onΣ, there exists δ>0 such that, whenever y0−¯y(0)E⩽δ, we can find controls v∈L2(O×(0,T)) Nand associated states (y, p) satisfying y(T) =¯y(T). We will also consider the Boussinesq system      yt−y +(y ·∇)y +∇p=v1O+θe N,∇·y=0inQ, θt−θ +y·∇θ=h1Oin Q, y=0,θ=0onΣ, y(0)=y0,θ(0)=θ0in Ω, (10) 278 E. Fernández-Cara et al. / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 275–280 where the controls are now v∈L2(O×(0,T)) Nand h∈L2(O×(0,T)). It will be said that (10) is locally exactly controllable to the trajectories at time Tif, for any sufficiently regular solution (¯y, ¯p, ¯ θ) of the system    ¯yt−¯y+(¯y·∇)¯y+∇¯p=¯ θeN,∇·¯y=0inQ, ¯ θt−¯ θ+¯y·∇¯ θ=0inQ, ¯y=0,¯ θ=0onΣ, there exists δ>0 such that, if (y0,θ0)−(¯y(0), ¯ θ(0))E×L2⩽δ, then we can find controls v∈L2(O×(0,T)) N and h∈L2(O×(0,T)) and associated states (y,p,θ) satisfying y(T ) =¯y(T ) and θ(T)=¯ θ(T). Finally, for N=2 we will also consider the systems with truncated nonlinearity yt−y +(y ·∇)TM(y) +∇p=v1O,∇·y=0inQ, y·n=0,∇×y=0onΣ, y(0)=y0in Ω,(11) where the boundary conditions are of the Navier kind. Here, M>0, TM(y) =(TM(y1), TM(y2)) and TM(s) =−Mif s⩽−M, sif −M⩽s⩽M, Mif s⩾M. It will be said that (11) is (globally) null controllable at time Tif, for each y0∈H, there exists v∈L2(O× (0,T)) 2such that the associated state satisfies y(T ) =0. In the previous systems, v=v(x,t) and h=h(x, t) are control functions. The goal of this Note is to prove the existence of controls with a reduced number of degrees of freedom such that the previous controllability properties hold. Some hypotheses will be imposed on the control domain and the trajectories. More precisely, we will frequently assume that ∃x0∈∂Ω, ∃ε>0 such that O∩∂Ω ⊃B(x0;ε) ∩∂Ω, (12) (B(x0;ε) is the ball centered at x0of radius ε), ¯y∈L∞(Q)N,¯yt∈L20,T;Lσ(Ω)Nσ>1ifN=2 σ>6/5ifN=3(13) and ¯ θ∈L∞(Q), ¯ θt∈L20,T;Lσ(Ω)σ>1ifN=2 σ>6/5ifN=3.(14) Our first two results are the following: Theorem 1.1. Assume that Osatisfies (12). Then, for any T>0,(8) is locally exactly controllable at time Tto the trajectories (¯y, ¯p) satisfying (13) with controls v∈L2(O×(0,T)) Nsuch that vi≡0for some i. Theorem 1.2. Assume that Osatisfies (12) with nk(x0)= 0for some k<N. Then, for each T>0,(10) is locally exactly controllable at time Tto the trajectories (¯y, ¯p, ¯ θ) satisfying (13), (14) with controls vand hsuch that vk≡vN≡0. In particular, if N=2, we have local exact controllability to the trajectories with controls v≡0and h∈L2(O×(0,T)). For our last result, let us introduce the space W=∇×z=(∂2z, −∂1z);z∈L20,T;H1(Ω). We have the following: E. Fernández-Cara et al. / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 275–280 279 Theorem 1.3. Let N=2. Then, for any T>0and any M>0,(11)is null controllable at time Twith controls of the form v1O, where v∈W. In the following sections, we will indicate the main ideas used in the proofs of the previous results (for simplicity, we will only refer to Theorems 1.1 and 1.3). The detailed proofs will be given in a forthcoming paper. 2. The null controllability of some similar linear systems Following well known arguments, we will first deduce null controllability results for linearized versions of (8) and (11), namely: yt−y +(¯y·∇)y +(y ·∇)¯y+∇p=f+v1O,∇·y=0inQ, y=0onΣ, y(0)=y0in Ω (15) (where ¯ysatisfies (13) and f=f(x,t)satisfies appropriate decay assumptions near t=T) and yt−y +(y ·∇)¯y+∇p=v1O,∇·y=0inQ, y·n=0,∇×y=0onΣ, y(0)=y0in Ω (16) (where we assume that N=2 and ¯y∈L∞(Q)2). For the null controllability of (15) with vk≡0, the main tool is a global Carleman estimate for the solutions of the associated adjoint system −ϕt−ϕ −(Dϕ) ¯y+∇π=g, ∇·ϕ=0inQ, ϕ=0onΣ, ϕ(T ) =ϕ0in Ω,(17) where Dϕ =∇ϕ+t∇ϕand g∈L2(Q)N. Indeed, assume that (for instance) N=3 and n1(x0)= 0. Then the task is to prove that the solutions of (17) satisfy  Q ρ2 1|ϕ|2dxdt⩽C(Ω,O,T, ¯y) Q ρ2 2|g|2dxdt+ O×(0,T ) ρ2 3|ϕ2|2+|ϕ3|2dxdt(18) for some appropriate weights ρi=ρi(x, t) and some C(Ω,O,T, ¯y) > 0. This can be proved using first a global Carleman inequality established in [1]. At this point |ϕ1|2appears in the second term of the right hand side of (18). Using hypothesis (12) and the facts that ∇·ϕ=0inQand ϕ1=0onΣwe can get rid of this local term in |ϕ1|2. On the other hand, we can also deduce a null controllability result for (16) whenever ¯y∈L∞(Q)2. To this end, let us introduce the streamline-vorticity formulation of (16), namely    ωt−ω +∇×(∇×ψ)·∇¯y=∇×(v1O), ψ =ωin Q, ψ=0,ω=0onΣ, ω(0)=∇×y0in Ω (19) and the associated adjoint system    −ρt−ρ −∇×(¯y·∇×)∇θ=0,θ=ρin Q, θ=0,ρ=0onΣ, ρ(T)=ρ0in Ω.(20) Then the task amounts to prove the observability inequality  ∇θ(0)  2 L2⩽C O×(0,T ) |∇θ|2dxdt(21) for the solutions of (20). The proof of (21) relies on some global Carleman inequalities established in [3] and [4]. 280 E. Fernández-Cara et al. / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 275–280 3. The local exact controllability of the Navier–Stokes systems (8) and (11) Theorem 1.1 is proved by applying an inverse mapping theorem. In the framework of the Navier–Stokes equations, this strategy was introduced by O.Yu. Imanuvilov in [2] and has been used recently in [1], relaxing considerably the regularity requirements on the trajectories (¯y, ¯p). On the other hand, in order to prove Theorem 1.3 we can use a fixed point argument. This is possible because the unique assumption on ¯yneeded for the null controllability of (16) is ¯y∈L∞(Q)2. At this level, the fact that N=2 and the particular form of the boundary conditions in (11) are essential. Thus, we can introduce a set-valued mapping z→ Λ(z) where, for each z∈L2(Q)2,Λ(z) is the family of functions ywhich solve (together with some p) the linear system (16) with ¯y=TM(z) and satisfy y(T) =0 (and suitable estimates). It can be seen that an appropriate version of Kakutani’s fixed point theorem can be applied to Λ. Remark 1. Assume that N=2. The arguments in [1] implicitly show that, under hypotheses (13), we can find controls v1Owith v∈Wsuch that the associated solutions to (8) satisfy y(T) =¯y(T). Observe that the assumption (12) on the control domain is not necessary here. Remark 2. Assume that N=3. It is natural to ask whether a result similar to Theorem 1.1 holds with controls having two zero components. In general, the answer is no. In fact, it seems difficult to identify the open sets Ωand Osuch that one has null controllability for all T>0, even for the linear problems, with a reduced number of controls. This is unknown even for the classical Stokes equations for which, up to now, the unique known results concern approximate controllability; see [5]. Acknowledgements The two first authors have been partially supported by D.G.E.S. (Spain), Grant BFM2003-06446. The third author has been supported by NSF Grant DMS 0205148. References [1] E. Fernández-Cara, S. Guerrero, O.Yu. Imanuvilov, J.P. Puel, Local exact controllability of the Navier–Stokes system, J. Math. Pures Appl., in press. [2] O.Yu. Imanuvilov, Remarks on exact controllability for the Navier–Stokes equations, ESAIM Control Optim. Calc. Var. 6 (2001) 39–72. [3] O.Yu. Imanuvilov, J.P. Puel, Global Carleman estimates for weak elliptic non homogeneous Dirichlet problem, Int. Math. Res. Notices 16 (2003) 883–913. [4] O.Yu. Imanuvilov, M. Yamamoto, Carleman Estimate for a Parabolic Equation in a Sobolev Space of Negative Order and its Applications, Lecture Notes in Pure and Appl. Math., vol. 218, Dekker, New York, 2001. [5] J.-L. Lions, E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in: P. Marcellini, G. Talenti, E. Visentini (Eds.), Partial Differential Equations and Applications, in: Lecture Notes in Pure and Appl. Math., vol. 177, Dekker, New York, 1996, pp. 221–235.