On the approximate and null controllability of the Navier-Stokes equations

SIAM REVIEW c 1999 Society for Industrial and Applied Mathematics Vol. 41, No. 2, pp. 269–277 On the Approximate and Null Controllability of the Navier–Stokes Equations∗ Enrique Fern´andez-Cara† Abstract. This paper presents some known results on the approximate and null controllability of the Navier–Stokes equations. All of them can be viewed as partial answers to a conjecture of J.-L. Lions. Key words. approximate controllability, null controllability, Navier–Stokes equations AMS subject classifications. 93C20, 76D05 PII. S0036144598334588 1. Introduction. The Formulation and the Meaning. Let Ω ⊂RNbe an open, bounded, and connected set with smooth boundary ∂Ω(N=2orN= 3); let O⊂Ω be a (small) nonempty open subset; and assume T>0. We will set Q=Ω×(0,T), Σ=∂Ω×(0,T), and U=L2(O×(0,T))N. For simplicity, the notation will be abridged such that we will give L2(Ω) instead of L2(Ω)N, etc. The usual norms in L2(Q) and L∞(Q) will be denoted, respectively, by k·k 2and k·k ∞. Let us consider the usual space of “test” functions V={φ∈C∞ 0(Ω)N;∇·φ= 0 in Ω }, and let us denote by H(respectively V) the closure of Vin L2(Ω) (respectively H1 0(Ω)). Hcoincides with the space of all L2-functions whose divergence vanishes in Ω and whose normal trace vanishes on ∂Ω. The usual norm and scalar product in Hwill be denoted, respectively, by |·|and (·,·). On the other hand, Vcoincides with the space of all H1 0-functions whose divergence vanishes. For each v∈Uand each y0∈H, we consider the corresponding Navier–Stokes problem (1)        ∂ty+(y·∇)y−∆y+∇p=1 Ov, ∇·y=0 in Q, y= 0 on Σ, y(0) = y0, where (for convenience) the constant density and viscosity coefficients have been taken equal to 1. Here, 1Ois the characteristic function of O. In [8], Lions conjectured that ∗Received by the editors February 25, 1998; accepted for publication (in revised form) June 8, 1998; published electronically April 23, 1999. This work was partially supported by D.G.I.C.Y.T. (Spain), Proyecto PB95–1242. A preliminary version has appeared in the Proceedings of the XV CEDYA - V Congreso de Matem´atica Aplicada, held in Vigo, Spain, in September 1997. http://www.siam.org/journals/sirev/41-2/33458.html †Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain ([email protected]). 269 Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 270 ENRIQUE FERN´ ANDEZ-CARA (1) is approximately controllable in Hfor all T>0. That is, he claimed that, for any given yd∈Hand ε>0, a control vcan be found such that (1) possesses at least one solution {y,p}satisfying y∈C0 w(0,T;H),|y(T)−yd|≤ε (and, also, that this has to be true for all O,T,y0,yd, and ε). The conjecture is open, with the exception of some particular situations (for instance, when N= 2 and Ois a neighborhood of ∂Ω; see section 3). The meaning of approximate controllability for (1) is the following. Provided it is true, once y0and ydare fixed, it is always possible to drive the system described by (1) from y0to a final state y(T) arbitrarily close to yd. Let us emphasize that, here, the control region is O×(0,T) and Ois arbitrarily small. We can formulate (and interpret) in a similar way a conjecture concerning boundary controllability, with the control being exerted on a portion of the boundary. It is also meaningful to ask whether or not (1) is null controllable for all T, that is, if, for each y0, there exists v∈Usuch that (1) possesses at least one solution {y, p} with y∈C0 w(0,T;H),y(T)=0. Again, the answer to this question is unknown. In recent years, several authors have given partial answers to these or related questions. Some of them are reviewed in the following sections. 2. A Fixed-Point Argument. In this context, the most significant contribution is the work of Fabre [3]. In principle, this seems to be a natural strategy. The goal is to reformulate the conjecture as a fixed-point equation. In order to make things meaningful, we have to be able to control linear systems of the Stokes kind, that is, (2)        ∂ty+∇·(ay)−∆y+∇p=1 Ov, ∇·y=0 in Q, y= 0 on Σ, y(0) = y0, where a=a(x, t) is prescribed. In order to have compactness, we have to be able to do this when the coefficient ais not regular. Unfortunately, the argument does not provide approximate controllability for (1), but only for an approximation of it. Let us be more precise. Assume a∈L∞(Q) and consider system (2). It is not difficult to see that this system is approximately controllable in Hif and only if the adjoint system (3) (−∂tφ−(a·∇)φ−∆φ+∇q=0,∇·φ=0 in Q, φ= 0 on Σ, has the unique continuation property in O×(0,T), that is, if and only if any couple {φ, q}verifying (3) and the equality φ=0 in O×(0,T) necessarily satisfies φ≡0. Recently, this property was established in [4]. Consequently, (2) is approximately controllable. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLLABILITY AND NAVIER–STOKES EQUATIONS 271 Once y0,yd, and εare fixed, there is a completely natural method for determining the “best” control that drives (2) to a final state y(T) satisfying |y(T)−yd|≤ε. This is inspired by the usual convex duality techniques, and the conclusions can be sketched as follows (see [7] for a complete description). The control function ˆvfor which the norm in L2(O×(0,T)) attains a minimum is the restriction to O×(0,T)ofthe function ˆ φ, where (4)          −∂tˆ φ−(a·∇)ˆ φ−∆ˆ φ+∇ˆq=0,∇·ˆ φ=0 in Q, ˆ φ= 0 on Σ, ˆ φ(T)=ˆ φ0. In (4), ˆ φ0is the unique function satisfying (5) J(ˆ φ0;a)≤J(φ0;a)∀φ0∈H, ˆ φ0∈H, with J(φ0;a)=1 2ZZO×(0,T ) |φ|2dx dt +ε|φ0|−(y0 d,φ 0); finally, y0 d=yd−Y(T), with Ybeing, together with P, the unique solution to        ∂tY+∇·(aY )−∆Y+∇P=0,∇·Y=0 in Q, Y= 0 on Σ, Y(0) = y0. Notice that a7→ ˆvis a well-defined continuous mapping. This is implied by the fact that φ07→ J(φ0;a)isstrictly convex and continuous and satisfies lim inf|φ0|→∞ J(φ0;a) (φ0)≥ε (a consequence, again, of the unique continuation property). Now, let Λ0:L2(Q)7→ L∞(Q) be a continuous mapping. For each z∈L2(Q), let Λ(z)=y, with ybeing the solution to (2) with a=Λ 0(z) and v=ˆv(the corresponding minimal norm control). Then Λ : L2(Q)7→ L2(Q)iscontinuous and compact.Ifwe could affirm that Λ maps a ball into itself, we would be able to deduce, by virtue of Schauder’s theorem, that it possesses a fixed point ˆy. Of course, we would have (6)        ∂tˆy+∇·(Λ0(ˆy)ˆy)−∆ˆy+∇ˆp=1 Oˆv, ∇·ˆy=0 in Q, ˆy= 0 on Σ, ˆy(0) = y0 and also |ˆy(T)−yd|≤ε. Consequently, our task is “reduced” to finding a mapping Λ0such that: 1. Λ maps a ball of L2(Q) into itself; 2. there exists a fixed point ˆyof Λ such that Λ0(ˆy)=ˆy. At present, the way that Λ0has to be constructed is not known. In fact, the only thing we know is a triviality: the first condition above is verified whenever Λ0takes Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 272 ENRIQUE FERN´ ANDEZ-CARA values in a ball of L∞(Q). This leads easily to controllability results for systems like (6). For instance, assume M>0: TM(s)=(Mif s>M, sif |s|≤M, −Mif s<−M. Then set ΛM(ξ)=(TM(ξ1),...,T M(ξN)). Let us also denote by ΛMthe corresponding Nemitskii operator, which is well defined and continuous in L2(Q) and takes values in the ball B(0; M)⊂L∞(Q). We then have the following system: (60)       ∂tˆy+∇·(ΛM(ˆy)ˆy)−∆ˆy+∇ˆp=1 Oˆv, ∇·ˆy=0 in Q, ˆy= 0 on Σ, ˆy(0) = y0. In accordance with the previous argument, once y0,yd, and εare fixed, there exists a v∈U and a solution {y,p}to (60) satisfying |y(T)−yd|≤ε. 3. The Analysis of a Galerkin Approximation. In this section, we will present controllability results for a finite-dimensional (Galerkin) approximation to the Navier– Stokes equations. The main contributors to results of this kind have been Lions and Zuazua (see [9]). Since we will be working in finite-dimensional spaces, approximate and exact controllability will be equivalent. Furthermore, the state equation will be reversible in time. This means that, for practical purposes, we only have to consider the case in which the initial state vanishes. Thus, let Ebe a finite-dimensional subspace of V. For each v∈U, we consider the following approximation to (1): (7)        yE:[0,T]7→ E, (∂tyE,e)+((yE·∇)yE,e)+(∇yE,∇e)=(1 Ov,e),ta.e. in [0,T], e∈E, yE(0)=0. Let EObe the linear space formed by the restriction to Oof the functions of E. We will see that, under the assumption (8) dim EO= dim E, the finite-dimensional system (7) is exactly controllable in Eat time T. That is, for each zE∈E, there exists v∈Usuch that the corresponding solution to (7) satisfies yE(T)=zE. We will also provide an estimate of the associated cost. As in the previous section, it is convenient to begin with a similar linear problem. Let us fix a∈L2(0,T;E) and consider the following system: (9)        yE:[0,T]7→ E, (∂tyE,e)+((a·∇)yE,e)+(∇yE,∇e)=(1 Ov,e),ta.e. in [0,T], e∈E, yE(0)=0. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLLABILITY AND NAVIER–STOKES EQUATIONS 273 This is exactly controllable at time Tif and only if the corresponding adjoint system (10)        φE:[0,T]7→ E, −(∂tφE,e)+(aφE,∇e)+(∇φE,∇e)=0,ta.e. in [0,T], e∈E, φE(T)=φ0 has the following property: (11) if φ0∈Eand (yE(T),φ 0) = 0 for all v∈U, then φE≡0. But (11) is implied by (8). Consequently, we have exact controllability for (9). Let us fix zE∈E. We can use arguments similar to those in the previous section to determine the minimal L2-norm control ˆvthat drives (9) to zE. Thus, let us set I(φ0;a)=1 2ZZO×(0,T ) |φE|2dx dt −(zE,φ 0)∀φ0∈E and let ˆ φ0be the unique function in Esatisfying (12) I(ˆ φ0;a)≤I(φ0;a)∀φ0∈E. Then ˆvis given by the restriction to O×(0,T) of the function ˆ φE, which is determined by ˆ φ0through (10). The corresponding cost is C(zE,a)=1 2ZZO×(0,T ) |ˆ φE|2dx dt. Note that this quantity can be bounded independently of a. Indeed, there exist constants C1(O,E) and C2(E) such that ZO |e|2≥C1|e|2and |e|2≥C2kek2∀e∈E. This is a consequence of (8) and the fact that Eis finite-dimensional. Hence, (13) C(zE,a)=−inf φ0∈EI(φ0;a) ≤− inf φ0∈EC1C2T 2(2T+C2)|φ0|2−|zE||φ0| =2T+C2 2C1C2T|zE|2. Let us now go back to the nonlinear problem (9). Again, let us fix zEin E.For each a∈L2(0,T;E), let us denote by ˆyEthe solution to (9) corresponding to the control function ˆv. Then a7→ ˆyEis a well defined, continuous, and compact mapping from L2(0,T;E) into itself. Furthermore, (13) shows that it maps the hole space L2(0,T;E) into a ball. Consequently, there exists at least one fixed point for this mapping. In other words, there exists ˆv∈U, with norm (14) kˆvk2≤2T+C2 C1C2T|zE|2, Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 274 ENRIQUE FERN´ ANDEZ-CARA such that the unique solution to        ˆyE:[0,T]7→ E, (∂tˆyE,e)+((ˆy·∇)ˆyE,e)+(∇ˆyE,∇e)=(1 Oˆv, e),ta.e. in [0,T], e∈E, ˆyE(0)=0 satisfies ˆyE(T)=zE. Since zEis arbitrary in E, this implies exact controllability. Remark 1. Of course, the bound (14) depends on E. It may be interesting to modify the previous argument by changing exact controllability to ε-approximate controllability at the finite-dimensional level. At that point, it might be reasonable to search for a bound of the cost depending on εbut not on E. 4. AVariantoftheReturnMethodandItsConsequences. The methods in this section were introduced by Coron (see [1] and the references therein). The main idea is to construct specific solutions {yα,p α}of the Navier–Stokes equations such that the linearized Euler equations at {yα,p α}are, in a certain sense, “almost” exactly controllable. Once y0and ydare fixed, this furnishes a first control v0. In a second step, after a correction of v0, we are able to drive the Navier–Stokes system to a final state that is close to yd. This method has several important limitations. First, we must have N= 2; on the other hand, the boundary conditions have to be of the Navier slip type. In practice, this is equivalent to prescribing the values on the boundary of the stream function and the vorticity function. Furthermore, we obtain approximate controllability only in W−1,∞(Ω), not in L2(Ω) (however, the same arguments lead to approximate controllability in W1,∞(K) for each compact set K⊂Ω). Let us assume that the state equation is (15)        ∂ty+(y·∇)y−∆y+∇p=1 Ov, ∇·y=0 in Q, y·n=0,∇×y= 0 on Σ, y(0) = y0in Ω (where Ω ⊂R2). Let us fix yd(for simplicity, regular enough). The argument is as follows: 1. For each α>0, it is possible to find vαand a couple, {yα,p α}, such that yαis a gradient outside O×(0,T), (16)        ∂tyα+(yα·∇)yα−∆yα+∇pα=1 Ovα,∇·yα=0 in Q, yα·n=0,∇×yα= 0 on Σ, yα(0) = yα(T)=0 in Ω and, furthermore, the linearized Euler system at yα, that is, (17) (∂tz+(yα·∇)z+(z·∇)yα+∇π=1 Ow, ∇·z=0 in Q, z·n= 0 on Σ, is α-controllable in the following sense: for any given z0and zdof class C∞, there exists a control w∈U such that (17) possesses at least one solution satisfying z(0) = z0in Ω Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLLABILITY AND NAVIER–STOKES EQUATIONS 275 and also z(T)=zdin {x∈Ω; dist(x, ∂Ω) ≥α}, and is bounded in C3(Q) independently of α. (Note that we are saying that, for every α, this property is satisfied for all z0and zd.) 2. Let us set v(x, t) = 0 in (15) for t∈[0,(1 −δ)T](δwill be determined below). This defines {y,p}without ambiguity in Ω ×[0,(1 −δ)T], and, in particular, we can speak of y((1 −δ)T). In [(1 −δ)T,T], we do the following: a. First, vαand {yα,p α}are rescaled. On the basis of the α-controllability of (17), we introduce a first control function v0: v0(x, t)≡1 δvαx, 1 δ(t−(1 −δ)T)+wx, 1 δ(t−(1 −δ)T). The associated state is {y0,p 0}≡1 δ{yα,p α}x, 1 δ(t−(1 −δ)T)+{z,π}x, 1 δ(t−(1 −δ)T). Here, wand {z,π}are perturbations corresponding to α(which will also be fixed below), the initial state y((1 −δ)T), and the desired state yd. In order to drive (15) to a final state close to yd, it is natural (at least formally) to look for a control close to v0for t∈[(1 −δ)T,T]. b. We introduce a second control function by modifying v0as needed. Thus, we solve the following problem: (18)        ∂ty+(y·∇)y−∆y+∇p=1 Ov0+(∇×y)(y−y0)⊥,∇·y=0 in Q, y·n=0,∇×y= 0 on Σ, y(0) = y0in Ω. In (18), we have introduced the following notation: y(x, t)=1 δyαx, 1 δ(t−(1 −δ)T),(a1,a 2)⊥=(−a2,a 1). Since ∇×yvanishes outside O×(0,T), it is clear that (18) can be written in the form (15), with v=v0+(∇×y)(y−y0)⊥. Now, the task is reduced to showing that, for every ε>0, there exist positive α and δsuch that (19) ky(T)−ydkW−1,∞≤ε. This can be achieved in the following way. Let us set R=y−y0,ω=∇×R. Then ∂tω+(R+y0)·∇ω−∆ω=−(R+z)·∇(∇×z)+∆(∇×z) in Ω ×((1 −δ)T,T). Furthermore, R(x, (1 −δ)T)≡0 and zand all its derivatives of order ≤3 are uniformly bounded. This leads first to a pointwise estimate of ω Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 276 ENRIQUE FERN´ ANDEZ-CARA and then to an estimate of R(·,T)inW−1,∞(Ω) when αand δare sufficiently small. Consequently, for any given ε>0, there exist α0>0 and η:(0,α 0)7→ R+such that, whenever 0 <α<α 0and 0 <δ<η(α), one has (19) (for further details, see [1]). It is only at this last step of the argument that the type of boundary condition becomes important. At present, it is not known how the method has to be modified in order to maintain its validity in the context of (1). Moreover, it is not clear whether it can be generalized to a similar three-dimensional situation (see, however, [2]). 5. Local Results Concerning Null Controllability. In this section, we will refer to the null controllability of (1). The intention is, in a second step, to prove exact controllability to any regular solution and, as a consequence, to prove approximate controllability. The most important contributions in this context are those of Fursikov and Imanuvilov (see [5], [6], and the references therein). Again, there are important limitations. In particular, only local results can be proved: starting from an initial state that is close to zero, we can control the system in such a way that the final state is exactly zero at t=T. For simplicity, we will present the argument when Ω is a bounded simply connected domain of R2, the boundary conditions are of the Navier slip type (as in (15)), and the control is the trace of the vorticity function ωon a portion γof the boundary. Remark 2. When N= 2 and several other additional conditions are satisfied, a combination of the results presented in this and the previous section leads to global null controllability, that is, with no restriction on the size of the initial data (see [2]). Let γ⊂∂Ω be a nonempty open set. With standard notation, the problem is to find a real-valued function h=h(x, t), defined and regular enough on γ×(0,T), such that the solution {ω,ψ,y}to (20)        ∂tω+∇·(ωy)−∆ω=0,−∆ψ=ω, y =∇×ψin Q, ω=h1γ,ψ= 0 on Σ, ω(0) = ∇×y0in Ω verifies ω(T) = 0. Observe that, for any given y0∈H2(Ω) ∩H1 0(Ω) ∩W1,∞(Ω) and h∈C1(γ×[0,T]), there exists exactly one solution {ω,ψ,y}to (20). Among other things, one has ω∈C0(0,T;H1(Ω)). We are going to introduce a sequence {ωn,ψn,yn,h n}that converges, when the initial state y0is “small,” towards a solution {ω,ψ,y,h}to (20). By definition, each ωnwill solve a null controllability problem for a linear system. In particular, this will give ω(T) = 0. The sequence {ωn,ψn,yn,h n}is defined as follows: a. First, we choose a constant R>0 and we take ω0≡0, h0≡0. b. Then, once n≥0 and ωnand hnare given, we set (21) ωn=TR(ωn),ψ n=(−∆)−1ωn,y n=∇×ψn; finally, ωn+1 is (together with hn+1) the solution to the null controllability problem (22)        ∂tωn+1 +∇·(ωn+1yn)−∆ωn+1 =0 in Q, ωn+1 =hn+11γon Σ, ωn+1(0) = ∇×y0,ω n+1(T)=0 in Ω furnished by the method of Fursikov and Imanuvilov (see [5]). Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLLABILITY AND NAVIER–STOKES EQUATIONS 277 Let M>0 be such that kωkL∞≤R⇒k∇×((−∆)−1ω)kL∞≤M. Then every ynsatisfies kynkL∞≤M. A first consequence is that the corresponding functions ωnare uniformly bounded in L∞(Q) and L2(0,T;H2(Ω)). More precisely, kωnkL2(0,T ;H2(Ω)) +kωnk∞≤C(ky0kH2+ky0kW1,∞), where Conly depends on Ω, γ,T, and M. From these estimates, it can be deduced in a second step that, under the assumption (23) ky0kH2+ky0kW1,∞≤ε, one has (24) kωn+1 −ωnkL2(0,T ;H2(Ω)) ≤η(ε)kωn−ωn−1kL2(0,T ;H2(Ω)) ∀n≥1, where limε→0+η(ε) = 0. Hence, the complete sequences ωn,ψn,yn, and hnconverge in appropriate spaces provided (23) is satisfied for a sufficiently small ε. Remark 3. There is another way to present the same argument. Let ΦR:L2(0,T;H2(Ω)) 7→ L2(0,T;H2(Ω)) be as follows: ΦR(z)=ωif and only if ωis (together with ψ,y, and h) the solution to (25)        ∂tω+∇·(ωy)−∆ω=0,−∆ψ=TR(z),y=∇×ψin Q, ω=h1γon Σ, ω(0) = ∇×y0,ω(T)=0 in Ω, obtained as in [5]. If ky0kH2+ky0kW1,∞is sufficiently small, then ΦRis a contraction. REFERENCES [1] J.-M. Coron,On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var., 1 (1996), pp. 35–75. [2] J.-M. Coron and A.V. Fursikov,Global exact controllability of the 2-D Navier–Stokes equations on a manifold without boundary, Russian J. Math. Physics, 4 (1996), pp. 1–19. [3] C. Fabre,Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems, ESAIM Control Optim. Calc. Var., 1 (1996), pp. 267–302. [4] C. Fabre and G. Lebeau,Prolongement unique des solutions des ´equations de Stokes, Comm. Partial Differential Equations, 21 (1996), pp. 573–596. [5] A.V. Fursikov and O. Yu. Imanuvilov,Local exact controllability of the Navier–Stokes equations, Res. Inst. Math., GARC Preprint Series, 95-92, 1996; Controllability of evolution equations, Lecture Notes 34, Res. Inst. Math., GARC, Seoul National University, Korea, 1996. [6] O.Yu. Imanuvilov,On exact controllability for the Navier–Stokes equations, ESAIM Control Optim. Calc. Var., 3 (1998), pp. 97–131. [7] R. Glowinski and J.-L. Lions,Exact and approximate controllability for distributed parameter systems, Acta Numerica, 1994, pp. 269–378. [8] J.-L. Lions,Are there connections between turbulence and controllability?, in Lecture Notes in Control and Inform. Sci. 144, A. Bensoussan and J-L. Lions, eds., Springer-Verlag, New York, 1990; Remarques sur la contrˆolabilit´e approch´ee, in Jornadas Hispano-Francesas sobre Control de Sistemas Distribuidos, M´alaga, Spain, 1990. [9] J.-L. Lions and E. Zuazua,Contrˆolabilit´e exacte des approximations de Galerkin des ´equations de Navier–Stokes, C.R. Acad. Sci. Paris Ser. I Math., 324 (1997), pp. 1015–1021; Exact boundary controllability of Galerkin’s approximations of Navier–Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., to appear. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php