On the Numerical Controllability of the Two-Dimensional Heat, Stokes and Navier–Stokes Equations

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ Esta es la versión aceptada del artículo publicado en: This is a accepted manuscript in: Journal of Scientific Computing (2017): 28 January 2024 DOI: Copyright: El acceso a la versión publicada del artículo puede requerir la suscripción de la revista. Access to the published version may require subscription. “This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s10915-016-0266-x” On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations Enrique FERN´ ANDEZ-CARA∗ ,Arnaud M ¨ UNCH† and Diego A. SOUZA‡ Abstract The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier-Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov-Imanuvilov formulation, see [A.V. Fursikov, O.Yu. Imanuvilov: Controllability of Evolutions Equations, Lectures Notes Series, Vol. 34, Seoul National University, 1996]; this approach has been followed recently for the onedimensional heat equation by the first two authors. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables x1,x2and tthat is second–order in time and fourth–order in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are relatively easy to handle. Finally, we exhibit some numerical experiments. Contents 1 Introduction. The controllability problems 2 2 A strategy for the computation of null controls for the heat equation 4 2.1 First mixed formulation with modified variables . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Second mixed formulation with modified variables . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Areformulationof(26) ..................................... 9 2.4 A numerical approximation based on Lagrangian finite elements . . . . . . . . . . . . . . . 10 2.5 The Arrow-Hurwicz algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.6 Anumericalexperiment ..................................... 11 3 A strategy for the computation of null controls for the Stokes equations 12 3.1 A first mixed formulation of (42) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 A second mixed formulation of (42) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 A third mixed reformulation of (54) with an additional multiplier . . . . . . . . . . . . . . 20 3.4 Another formulation related to (44) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5 A fifth (and final) mixed formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 ∗Dpto. EDAN, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. E-mail: [email protected]. Partially supported by grant MTM2013–41286–P (Spain). †Laboratoire de Math´ematiques, Universit´e Blaise Pascal (Clermont-Ferrand 2), UMR CNRS 6620, Campus des C´ezeaux, 63177 Aubi`ere, France. E-mail: [email protected]. ‡Dpto. EDAN, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. E-mail: [email protected]. Partially supported by grant MTM2013–41286–P (Spain). 1 3.6 A numerical approximation of (62) (without justification) . . . . . . . . . . . . . . . . . . 22 3.7 Anumericalexperiment ..................................... 23 4 An application : numerical local exact controllability to the trajectories of the NavierStokes equations 23 4.1 A fixed-point algorithm and a mixed formulation . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Numericalexperiments...................................... 28 5 Additional comments and conclusions 29 1 Introduction. The controllability problems Let Ω ⊂R2be a bounded domain whose boundary Γ := ∂Ω is regular enough. Let ω⊂Ω be a (possibly small) nonempty open subset and assume that T > 0. We will use the notation Qτ= Ω ×(0, τ), Στ= Γ ×(0, τ), qτ=ω×(0, τ) and n=n(x) will denote the outward unit normal to Ω at any point x∈Γ. Throughout this paper, Cwill denote a generic positive constant (usually depending on Ω, ωand T) and the bold letters and symbols will stand for vector-valued functions and spaces; for instance L2(Ω) is the Hilbert space of the functions u= (u1, u2) with u1, u2∈L2(Ω). This paper is concerned with the global null controllability of the heat equation      yt−∆y+G(x, t)y=v1ωin QT, y= 0 on ΣT, y(·,0) = y0in Ω (1) and the Stokes equations          yt−ν∆y+∇π=v1ωin QT, ∇ · y= 0 in QT, y=0on ΣT, y(·,0) = y0in Ω (2) and the local exact controllability to the trajectories of the Navier-Stokes equations          yt−ν∆y+ (y· ∇)y+∇π=v1ωin QT, ∇ · y= 0 in QT, y=0on ΣT, y(·,0) = y0in Ω. (3) Here, v=v(x, t) and v=v(x, t) stand for the controls (they are assumed to act on ωduring the time interval (0, T); the symbol 1ωstands for the characteristic function of ω). Moreover, in (1), we assume that G∈L∞(QT); in (2) and (3), ν > 0. Let us first consider the system (1). It is well known that, for any y0∈L2(Ω), T > 0 and v∈L2(qT), there exists exactly one solution yto (1), with y∈C0([0, T]; L2(Ω)) ∩L2(0, T;H1 0(Ω)). The null controllability problem for (1) at time Tis the following: For any y0∈L2(Ω) find a control v∈L2(qT)such that the associated solution to (1) satisfies y(x, T)=0 in Ω.(4) The following result is also well known; for a proof, see [13]: 2 Theorem 1. The heat equation (1) is null-controllable at any time T > 0. Let us now consider the systems (2) and (3). Let us recall the definitions of some usual spaces in the context of incompressible fluids: H:= ϕ∈L2(Ω) : ∇ · ϕ= 0 in Ω,ϕ·n= 0 on Γ, V:= ϕ∈H1 0(Ω) : ∇ · ϕ= 0 in Ω, U:= ψ∈H1(Ω) : ZΩ ψ(x)dx= 0. For any y0∈H, T > 0 and v∈L2(qT), there exists exactly one solution (y, π) to the Stokes equations (2) and (since we are in the 2Dcase), also one solution (y, π) to the Navier-Stokes equations (3). In both cases y∈C0([0, T]; H)∩L2(0, T;V), π ∈L2 loc(0, T ;U). In the context of the Stokes system (2), the null controllability problem at time Tis the following: For any y0∈Hfind a control v∈L2(qT)such that the associated solution to (2) satisfies y(x, T) = 0in Ω.(5) Again, the following result is well known; for a proof, see [13]: Theorem 2. The Stokes system (2) is null-controllable at any time T > 0. Let us recall the concept of exact controllability to the trajectories. The idea is that, even if we cannot reach every element of the state space exactly, we can try to reach (in finite time T) any state on any trajectory. Thus, let (y, π) be a solution to the uncontrolled Navier-Stokes equations:          yt−ν∆y+ (y· ∇)y+∇π=0in QT, ∇ · y= 0 in QT, y=0on ΣT, y(·,0) = y0in Ω. (6) We will search for controls v∈L2(qT) such that the associated solutions to (3) satisfy y(x, T) = y(x, T) in Ω.(7) The problem of exact controllability to the trajectories for (3) is the following: For any y0∈Hand any trajectory (y, π), find a control v∈L2(qT)such that the associated solution to (3) satisfies (7). The following result shows that this problem can be solved at least locally when yis bounded; for a proof, see [9, 18]: Theorem 3. The Navier-Stokes equations (3) are locally exact controllable to the trajectories (y, π)with y∈L∞(QT),y(·,0) ∈V.(8) In other words, for any T > 0and any solution to (6) satisfying (8), there exists ε > 0with the following property: if y0∈Vand ky0−y(·,0)kV≤ε, one can find controls v∈L2(qT)such that the associated solutions to (3) satisfy (7). 3 The aim of this paper is to present efficient strategies for the numerical solution of the previous controllability problems. These problems concern the computation of external heat sources or force fields that can be applied in a small part of the working domain and control the whole system at a prescribed positive time. There are lots of particular situations where this is highly desired; in particular, for theoretical and numerical information on control problems for fluid flows, see [14, 16]. However, the numerical resolution of control problems as those above is not easy. This is due to several reasons: •In the case of the “linear” problems (1)–(4) and (2)–(5), due to the regularizing effect of the PDEs, the equalities (4) and (5) can be satisfied only in a very small space and standard numerical approximations of the PDEs are unable to capture this. For instance, if we look for a minimal L2 norm null control for (1), we are led by duality to an unconstrained extremal problem in a huge space that cannot be approximated efficiently with usual finite dimensional spaces; see however [2, 1] for detailed comments on this issue. •On the other hand, in (3)–(7) we find the Navier-Stokes system and, obviously, this adds major difficulties. Note that, at present, it is unknown whether or not the exact controllability to the trajectories of (3) holds without smallness assumptions even when y≡0. Remark 1. In this paper, we will only deal with distributed controls. In fact, in the case of the Stokes and Navier-Stokes equations, it would be more appropriate from the viewpoint of applications to consider boundary controls acting on a part of Σ; this will be the subject of a forthcoming paper. Note however that, in general terms, a boundary control problem can be re-formulated in the form (2) or (3) by modifying slightly the domain, choosing ωappropriately (outside the original Ω) and then considering the restriction of the controlled state to the original Ω. 2 The paper is organized as follows. In Section 2, we deal with the numerical null controllability of the heat equation. Following ideas from [13], we reduce the task to the solution of a boundary-value problem that is fourth-order in space and second-order in time. We present a mixed approximate formulation where we avoid the use of C1 finite elements. In Sections 3 and 4, we present similar numerical strategies to solve numerically the controllability problems considered above for the Stokes and the Navier-Stokes equations. The methods are illustrated with several numerical experiments. Finally, Section 5 contains several additional comments. 2 A strategy for the computation of null controls for the heat equation In this Section, we will start from a formulation of the null controllability problem for (1) introduced and extensively used by Fursikov and Imanuvilov, see [13]. We will present some numerical methods essentially obtained by finite dimensional reduction. Note that this is not the unique efficient approach. The first contribution to the numerical solution of null controllability problems of this kind was due to Carthel, Glowinski and Lions in [4], using duality arguments. However, the resulting problems involve some dual spaces which are very difficult (if not impossible) to approximate numerically. In [20], in the context of approximate controllability, a relaxed observability inequality was given for general semi-discrete (in space) schemes, with the parameter εof the order of ∆x. The work [2] extends the results in [20] to the fully discrete situation and proves the convergence towards a semi-discrete control, as the time step ∆ttends to zero; let us also mention [8], where the authors prove that any controllable parabolic equation, be it discrete or continuous in space, is null-controllable after time discretization through the application of an appropriate filtering of the high frequencies. For a comparison of the results furnished by various methods, see the numerical experiments in [11, 12, 21]. 4 Let us fix the notation Ly := yt−∆y+G(x, t)y, L∗p:= −pt−∆p+G(x, t)p and let the weights ρ,χand ρibe given by ρ(x, t) := eχ(x)/(T−t), χ(x) := K1eK2−eχ0(x), ρi(x, t) := (T−t)3/2−iρ(x, t), i = 0,1,2,(9) where K1and K2are sufficiently large positive constants (depending on T) and χ0=χ0(x) is a regular bounded function that is positive in Ω, vanishes on Γ and satisfies |∇χ0|>0 in Ω \ω; for a justification of the existence of χ0, see [13]. The main idea relies on considering the extremal problem      Minimize J(y, v) = 1 2ZZQT ρ2|y|2dxdt +ZZqT ρ2 0|v|2dxdt Subject to (y, v)∈ H(y0, T). (10) Here, for any y0∈L2(Ω) and any T > 0, the linear manifold H(y0, T) is given by H(y0, T) := {(y, v) : v∈L2(qT),(y, v) satisfies (1) and (4)}. We have the following result: Theorem 4. For any y0∈L2(Ω) and any T > 0, there exists exactly one solution to (10). This result is a consequence of an appropriate Carleman inequality for the heat equation. More precisely, let us introduce the space P0:= {p∈C2(QT) : p= 0 on ΣT}.(11) Then, one has: Proposition 1. There exists C0, only depending on Ω,ωand T, such that the following holds for all p∈P0: ZZQTρ−2 2(|pt|2+|∆p|2) + ρ−2 1|∇p|2+ρ−2 0|p|2dxdt ≤C0ZZQT (ρ−2|L∗p|2+ρ−2 0|p|21ω)dxdt. (12) Let us introduce the bilinear form k(·,·), with k(p, p0) := ZZQTρ−2L∗p L∗p0+ 1ωρ−2 0p p0dxdt ∀p, p0∈P0.(13) In view of the unique continuation property of the heat equation, k(·,·) is a scalar product in P0. Indeed, if p∈P0,L∗p= 0 in QT,p= 0 on ΣTand p= 0 in qT, then we necessarily have p≡0. Let Pbe the completion of P0with respect to this scalar product. Then Pis a Hilbert space, the functions p∈Psatisfy ZZQT ρ−2|L∗p|2dxdt +ZZqT ρ−2 0|p|2dxdt < +∞(14) and, from Proposition 1 and a standard density argument, we also have (12) for all p∈P. Another consequence of Proposition 1 is that we can characterize the space Pas follows: P=p:p, pt, ∂jp, ∂jkp∈L2(0, T −δ;L2(Ω)) ∀δ > 0,(14) holds, p = 0 on Σ .(15) In particular, we see that any p∈Psatisfies p∈C0([0, T −δ]; H1 0(Ω)) for all δ > 0 and, moreover, kp(·,0)kH1 0≤C k(p, p)1/2∀p∈P. (16) The main ideas used in this paper to solve numerically (10) rely on the following result : 5 Theorem 5. Let the weights ρand ρ0be chosen as in Proposition 1. Let (y, v)be the unique solution to (10). Then one has y=ρ−2L∗p, v =−ρ−2 0pqT,(17) where pis the unique solution to the following variational equality in the Hilbert space P:    ZZQTρ−2L∗p L∗p0+ 1ωρ−2 0p p0dxdt =ZΩ y0(x)p0(x,0) dx ∀p0∈P;p∈P. (18) We can interpret (18) as the weak formulation of a boundary-value problem for a PDE that is fourthorder in xand second-order in t. Indeed, taking “test functions” p0∈Pfirst with p0∈C∞ 0(QT), then p0∈C2(Ω ×(0, T)) and finally p0∈C2(QT), we see easily that pmust necessarily satisfy :      L(ρ−2L∗p)+1ωρ−2 0p= 0 in QT, p= 0, ρ−2L∗p= 0 on ΣT, ρ−2L∗pt=0 =y0, ρ−2L∗pt=T= 0 in Ω. (19) By introducing the linear form `0, with h`0, pi:= ZΩ y0(x)p(x,0) dx∀p∈P, (20) we see from (16) that `0is continuous and (18) can be rewritten in the form k(p, p0) = h`0, p0i ∀p0∈P;p∈P. (21) Let Phdenote a finite dimensional subspace of P. A natural approximation of (21) is the following: k(ph, p0 h) = h`0, p0 hi ∀p0 h∈Ph;ph∈Ph.(22) Thus, to solve numerically the variational equality (21), it suffices to construct explicitly finite dimensional spaces Ph⊂P. Notice however that this is possible but needs some work. The reason is that, if p∈Ph, then ρ−1L∗p=ρ−1(−pt−∆p+G(x, t)p) and ρ−1 0p1ωmust belong to L2(QT). Consequently, ph must possess first-order time derivatives and up to second-order spatial derivatives in L2 loc(QT). Therefore, an approximation based on a standard triangulation of QTrequires spaces Phof functions that must be C0in (x, t) and C1in xand this can be complex and too expensive. Spaces of this kind are constructed for instance in [5]. For example, good behavior is observed for the so called reduced HTC, Bell or Bogner-Fox-Schmidt finite elements; the reader is referred to [11, 21] for numerical approximations of this kind in the framework of one spatial dimension. In spite of its complexity, the direct approximation of (22) has an advantage: it is possible to adapt the standard finite element theory to this framework and deduce strong convergence results for the numerical controls and states. 2.1 First mixed formulation with modified variables Let us introduce the new variable z:= L∗p(23) and let us set Z:= L2(ρ−1;QT). Then z∈Zand L∗p−z= 0 (an equality in Z). Notice that this identity can also be written in the form ZZQT (z−L∗p)ψ dxdt = 0 ∀ψ∈C∞ 0(QT); 6 Accordingly, we introduce the following reformulation of (21):              ZZQTρ−2z z0+ρ−2 0p p01ωdxdt +ZZQT (z0−L∗p0)λ dxdt =ZΩ y0(x)p0(x,0) dx, ZZQT (z−L∗p)λ0dxdt = 0, ∀(z0, p0, λ0)∈Z×P×Λ; ((z, p), λ)∈Z×P×Λ, (24) where, Λ := L2(ρ;QT). Notice that Z,Pand Λ are the appropriate spaces to keep all the terms in (24) meaningful. Let us introduce the bilinear forms α(·,·) and β(·,·), with α((z, p),(z0, p0)) := ZZQTρ−2z z0+ρ−2 0p p01ωdxdt ∀(z, p),(z0, p0)∈Z×P and β((z, p), λ) := ZZQT [L∗p−z]λ dxdt∀(z, p)∈Z×P, ∀λ∈Λ and the linear form `:X7→ R, with h`, (z, p)i:= ZΩ y0(x)p(x,0) dx∀(z, p)∈Z×P. Then, α(·,·), β(·,·) and `are well-defined and continuous and (24) reads:      α((z, q),(z0, p0)) + β((z0, p0), λ) = h`, (z0, p0)i, β((z, p), λ0)=0, ∀(z0, p0, λ0)∈Z×P×Λ; ((z, p), λ)∈Z×P×Λ. (25) This is a mixed formulation of the variational problem (10). In fact, the following result holds: Proposition 2. There exists exactly one solution to (25). Furthermore, (21) and (25) are equivalent problems in the following sense: 1. If ((z, p), λ)solves (25), then psolves (21). 2. Conversely, if psolves (21), there exists λ∈Λsuch that the triplet ((z, p), λ), with z:= L∗p solves (25). Proof. Let us introduce the space V:= {(z, p)∈Z×P:β((z, p), λ) = 0,∀λ∈Λ}. We will check that •α(·,·) is coercive in V. •β(·,·) satisfies the usual “inf-sup” condition with respect to Z×Pand Λ. This will be sufficient to guarantee the existence and uniqueness of a solution to (25); see for instance [3, 24]. The proofs of the previous assertions are straightforward. Indeed, we first notice that, for any (z, p)∈ V,z=L∗pand thus α((z, p),(z, p)) = ZZQTρ−2|z|2+ρ−2 0|p|21ωdxdt =1 2ZZQT ρ−2|z|2dxdt +1 2ZZQT ρ−2|L∗p|2dxdt +ZZQT ρ−2 0|p|21ωdxdt =1 2k(z, p)k2 Z×P+1 2ZZQT ρ−2 0|p|21ωdxdt ≥1 2k(z, p)k2 Z×P. 7 This proves that α(·,·) is coercive in V. On the other hand, for any λ∈Λ there exists (z0, p0)∈Xsuch that β((z0, p0), λ) = kλk2 Λand k(z0, p0)kZ×P≤CkλkΛ. Indeed, we can take for instance (z0, p0)=(−ρ2λ, 0). Consequently, sup (z,p)∈X β((z, p), λ) k(z, p)kZ×P ≥β((z0, p0), λ) k(z0, p0)kZ×P ≥1 CkλkΛ. Hence, β(·,·) certainly satisfies the “inf-sup” condition in Z×P×Λ. An advantage of (25) with respect to the previous formulation (21) is that the solution ((z, p), λ) furnishes directly the state-control couple that solves (10). Indeed, it suffices to take y=ρ−1z, v =−ρ−2 0p|qT. However, we still find spatial second-order derivatives in the integrals in (25) and, consequently, a finite element approximation of (25) still needs C1in space functions. 2.2 Second mixed formulation with modified variables Let us introduce the spaces ˜ P:= p:ZZQTρ−2 2|pt|2+ρ−2 1|∇p|2+ρ−2 0|p|2dxdt < +∞, pΣT= 0, ˜ Λ := λ:ZZQTρ2 2|λ|2+ρ2 1|∇λ|2dxdt < +∞, λΣT= 0, the bilinear forms ˜α(·,·) and ˜ β(·,·), with ˜α((z, p),(z0, p0)) := ZZQTρ−2z z0+ρ−2 0p p01ωdxdt ∀(z, p),(z0, p0)∈Zט P and ˜ β((z, p), λ) := ZZQT [(z+pt−G(x, t)p)λ− ∇p· ∇λ]dxdt ∀(z, p)∈Zט P, ∀λ∈˜ Λ and the linear form ˜ `, with h˜ `, (z, p)i:= ZΩ y0(x)p(x,0) dx∀(z, p)∈Zט P. Then ˜α(·,·) and ˜ β(·,·) are well-defined and continuous. The linear form ˜ `is also continuous on Zט P, since the functions in ˜ Psatisfy p∈C0([0, T −δ]; L2(Ω)) ∀δ > 0 and kp(·,0)kL2(Ω) ≤CZZQT (ρ2 2|pt|2+ρ2 1|∇p|2)dx dt1/2 ≤Ckpk˜ P. Let us consider the mixed formulation      ˜α((z, p),(z0, p0)) + ˜ β((z0, p0), λ) = h˜ `, (z0, p0)i, ˜ β((z, p), λ0)=0, ∀(z0, p0, λ0)∈Zט Pט Λ; (z, p, λ)∈Zט Pט Λ. (26) 8 and it is again assumed that the weights ρand ρ0satisfy (9). We have: Theorem 6. For any y0∈Hand T > 0, there exists exactly one solution to (31). Again, this result can be viewed as a consequence of a Carleman inequality. Thus, let us set Ly := yt−ν∆y,L∗p:= −pt−ν∆p and let us introduce the space Φ0={(p, σ) : pi, σ ∈C2(QT),∇ · p≡0, pi= 0 on ΣT,ZΩ σ(x, t)dx=0 ∀t}.(32) Then, one has the following (see [13, 18]): Proposition 3. The function χ0and the associated weights ρ,ρ0,ρ1and ρ2can be chosen such that there exists C, only depending on Ω,ωand T, with the following property: ZZQTρ−2 2(|pt|2+|∆p|2) + ρ−2 1|∇p|2+ρ−2 0|p|2+ρ−2|∇σ|2dxdt ≤CZZQT ρ−2|L∗p+∇σ|2dxdt +ZZqT ρ−2 0|p|2dxdt(33) for all (p, σ)∈Φ0. Let us introduce the bilinear form m((p, σ),(p0, σ0)) := ZZQTρ−2(L∗p+∇σ)·(L∗p0+∇σ0)+1ωρ−2 0p·p0dxdt. (34) In view of the unique continuation property of the Stokes system, m(·,·) is a scalar product in Φ0: if (p, σ)∈Φ0,L∗p+∇σ=0in QTand p=0in qT, then we have p≡0and σ≡0 (note that, in fact, under these circumstances, p(·, t) and σ(·, t) are analytic for all t). Let Φbe the completion of Φ0with respect to this scalar product. As before, Φis a Hilbert space, the functions (p, σ)∈Φsatisfy ZZQT ρ−2|L∗p+∇σ|2dxdt +ZZqT ρ−2 0|p|2dxdt < +∞(35) and, from Proposition 3 and a density argument, we also have (35) for any (p, σ)∈Φ. We also see from Proposition 3 that Φ={(p, σ) : pi, σ, ∂tpi, ∂xjpi, ∂xjσ, ∂xjxkpi∈L2(0, T −δ;L2(Ω)) ∀δ > 0, (35) holds,∇ · p≡0 in QT, pi= 0 on ΣT,ZΩ σ(x, t)dx= 0 ∀t}(36) and, in particular, any (p, σ)∈Φsatisfies p∈C0([0, T −δ]; V) for all δ > 0 and kp(·,0)kV≤Cm((p, σ),(p, σ))1/2∀(p, σ)∈Φ.(37) The following result is proved in [13]: Theorem 7. Let the weights ρand ρ0be chosen as in Proposition 3. Let (y,v)be the unique solution to (31). Then one has y=ρ−2(L∗p+∇σ),v=−ρ−2 0pω×(0,T ),(38) where (p, σ)is the solution to the following variational equality in Φ:    ZZQTρ−2(L∗p+∇σ)·(L∗p0+∇σ0) + ρ−2 0p·p01ωdxdt =ZΩ y0(x)·p0(x,0) dx ∀(p0, σ0)∈Φ; (p, σ)∈Φ. (39) 15 Once more, (39) can be viewed as the weak formulation of a (non-scalar) boundary-value problem for a PDE that is fourth-order in xand second-order in t. Indeed, arguing as in Section 2, we can easily deduce that (p, σ) satisfies, together with some π∈ D0(QT), the following:          L(ρ−2(L∗p+∇σ)) + ∇π+ 1ωρ−2 0p= 0 in QT, ∇ · (ρ−2(L∗p+∇σ)) = 0,∇ · p= 0 in QT, p=0, ρ−2(L∗p+∇σ) = 0on ΣT, ρ−2(L∗p+∇σ)t=0 =y0, ρ−2(L∗p+∇σ)t=T=0in Ω. (40) By setting h`0,(p, σ)i:= ZΩ y0(x)·p(x,0) dx,(41) it is found that (39) can be rewritten in the form m((p, σ),((p0, σ0)) = h`0,(p0, σ0)i ∀(p0, σ0)∈Φ; (p, σ)∈Φ.(42) Thus, if Φhdenotes a finite dimensional subspace of Φ, a natural approximation of (42) is the following: m((ph, σh),((p0 h, σ0 h)) = h`0,(p0 h, σ0 h)i ∀(p0 h, σ0 h)∈Φh; (ph, σh)∈Φh.(43) However, the couples (p, σ)∈Φsatisfy several properties that make it considerably difficult to construct explicitly finite dimensional spaces Φh⊂Φ. These are the following: •As in Section 2, since ρ−1(L∗p+∇σ) must belong to L2(QT) and ρ−1 0pqTmust belong to L2(qT), the pimust possess first-order time derivatives and up to second-order spatial derivatives in L2 loc(QT). As before, this means that, in practice, the functions in Φhmust be C0in (x, t) and C1in x. •We now have ∇ · p≡0. It is not simple at all to give explicit expressions of zero (or approximately zero) divergence functions associated to a triangulation of QTwith this regularity. The second inconvenient is classical in computational fluid dynamics when one considers incompressible fluids. As in many other works, it will be overcome by introducing additional “pressure-like” multipliers; see Section 3.2. On the other hand, the first difficulty will be circumvented as in Section 2, by introducing new variables and associated multipliers and eliminating all the second-order derivatives in the formulation. In the following Sections, we will present several mixed problems connected to (42). More precisely, in Sections 3.1 and 3.4, we consider mixed formulations where the constraint ∇·p≡0 is preserved. Accordingly, we only introduce one additional variable zand one multiplier, related to the identity z=L∗p+∇σ. Contrarily, Sections 3.2, 3.3 and 3.5 deal with other different formulations where the zero-divergence condition is not imposed and, therefore, another multiplier appears. 3.1 A first mixed formulation of (42) Arguing as in the case of the heat equation and introducing the variable z=L∗p+∇σ, we see that (42) is equivalent to:      a((z,p, σ),(z0,p0, σ0)) + b((z0,p0, σ0), λ) = h`,(z0,p0, σ0)i, b((z,p, σ), λ0)=0, ∀((z0,p0, σ0), λ0)∈Z×Φ×Λ; ((z,p, σ), λ)∈Z×Φ×Λ, (44) where Z=L2(ρ−1;QT),Λ=L2(ρ;QT) 16 and the bilinear forms a(·,·) and b(·,·) are given by a((z,p, σ),(z0,p0, σ0)) := ZZQTρ−2z·z0+ρ−2 0p·p01ωdxdt and b((z,p, σ), λ) := ZZQT [z−(L∗p+∇σ)] ·λ dxdt (45) and the linear form `is given by h`,(z,p, σ)i:= ZΩ y0(x)·p(x,0) dx. 3.2 A second mixed formulation of (42) As we have said, numerical difficulties are found when we try to introduce finite element approximations (C1in space, C0in time) of the space Φ, where the (p, σ) satisfy ∇ · p≡0. Accordingly, before approximating, we will reformulate (39) as a new mixed system involving a multiplier associated to this constraint. Let us introduce ˜ Φ0={(p, σ) : pi, σ ∈C2(QT), pi= 0 on ΣT,ZΩ σ(x, t)dx= 0 ∀t}.(46) We have the following Carleman estimates for the couples in ˜ Φ0: Proposition 4. There exist weights ρ,ρ0and ρ∗and a constant Conly depending on Ω,ωand T, with the following property: ZZQT ρ−2 0|p|2dxdt +kp(·,0)k2 V0≤CZZQT ρ−2 ∗|L∗p+∇σ|2dxdt +ZZqT ρ−2 0|p|2dxdt +ZZQT |∇ · p|2dxdt(47) for all (p, σ)∈˜ Φ0. Proof. The proof follows easily by splitting (p, σ)∈˜ Φ0in the form (p, σ)=(˜ p,˜σ)+(ˆ p,ˆσ) where (ˆ p,ˆσ) solves the linear problem          L∗ˆ p+∇ˆσ=fin QT, ∇ · ˆ p= 0 in QT, ˆ p=0on ΣT, ˆ p(·, T) = p(·, T) in Ω (48) with f:= L∗p+∇σand (˜ p,˜σ) solves the linear problem          L∗˜ p+∇˜σ=0in QT, ∇ · ˜ p=∇ · pin QT, ˜ p=0on ΣT, ˜ p(·, T) = 0in Ω. (49) 17 In view of the Carleman estimates (33) for (ˆ p,ˆσ), we have ZZQT ρ−2 0|ˆ p|2dxdt +kp(·,0)k2 V≤CZZQT ρ−2|f|2dxdt +ZZqT ρ−2 0|ˆ p|2dxdt. On the other hand, (˜ p,˜σ) solves (49) in the sense of transposition, that is, h˜ p,ψiL2(QT),L2(QT)+h˜ p(·,0),u0iV0,V=−ZZQT (∇ · p)h dxdt, for all (ψ,u0)∈L2(QT)×V, where (u, h) is the unique strong solution to          Lu +∇h=ψin QT, ∇ · u=0in QT, u=0on ΣT, u(·,0) = u0in Ω. Consequently, we can argue as in [23] and deduce that k˜ pk2 L2(QT)+k˜ p(·,0)k2 V0≤Ck∇ · pk2 L2(QT). Now, putting together the estimates for (ˆ p,ˆσ) and (˜ p,˜σ), we are easily led easily to (47). Let us introduce the bilinear form ˜ m((p, σ),(p0, σ0)) := m((p, σ),(p0, σ0)) + ZZQT (∇ · p)(∇ · p0)dxdt. Again, in view of the unique continuation property of the Stokes system, ˜ m(·,·) is a scalar product in ˜ Φ0. Let ˜ Φbe the completion of ˜ Φ0with respect to this scalar product. As before, ˜ Φis a Hilbert space, the functions (p, σ)∈˜ Φsatisfy ZZQT ρ−2|L∗p+∇σ|2dxdt +ZZqT ρ−2 0|p|2dxdt +ZZQT |∇ · p|2dxdt < +∞(50) and, from Proposition 4 and a density argument, we also have (47) for all (p, σ)∈˜ Φ. On the other hand, any (p, σ)∈˜ Φsatisfies ρ−1 0p∈L2(QT),p(·,0) ∈V0 and kp(·,0)k2 V0≤C˜ m((p, σ),(p, σ)) ∀(p, σ)∈˜ Φ.(51) By setting h˜ `,(p, σ)i:= hp(·,0),y0iV0,V(52) thanks to (51), we have that ˜ `is continuous on ˜ Φ. Let us introduce the space ˜ M={µ∈L2(QT) : ZΩ µ(x, t)dx= 0 a.e. } and the following reformulation of (42):              ˜ m((p, σ),(p0, σ0)) + ZZQT (∇ · p0)µ dxdt =h˜ `,(p, σ)i, ZZQT (∇ · p)µ0dxdt = 0, ∀(p0, σ0, µ0)∈˜ Φט M; (p, σ, µ)∈˜ Φט M. (53) 18 Once more, notice that the definitions of ˜ Φand ˜ Mare the appropriate to keep all the terms in (53) meaningful. Let the bilinear forms ˜ a(·,·) and ˜ b(·,·) be given by ˜ a((p, σ),(p0, σ0)) := m((p, σ),(p0, σ0)) and ˜ b(p, σ, µ) := ZZQT (∇ · p)µ dxdt. Then, ˜ a(·,·) and ˜ b(·,·) are well-defined and continuous and (53) reads:      ˜ a((p, σ),(p0, σ0)) + ˜ b(p0, σ0, µ) = h˜ `,(p0, σ0)i, ˜ b((p, σ), µ0)=0, ∀(p0, σ0, µ0)∈˜ Φט M; (p, σ, µ)∈˜ Φט M. (54) One has the following: Proposition 5. There exists exactly one solution to (54). Furthermore, (42) and (54) are equivalent problems in the following sense: 1. If (p, σ, µ)solves (54), then (p, σ)solves (42). 2. If (p, σ)solves (42), there exists µ∈˜ Msuch that (p, σ, µ)solves (54). Proof. Let us set ˜ Ψ:= {(p, σ)∈˜ Φ:˜ b(p, σ, µ) = 0 ∀µ∈˜ M}. We will check that •˜ a(·,·) is coercive in ˜ Ψ. •˜ b(·,·) satisfies the usual “inf-sup” condition in ˜ Φט M. The proofs of these assertions are straightforward. Indeed, we have ˜ Ψ=Φ(the completion of Φ0 with respect to m(·,·), see (34)). Thus, ˜ a((p, σ),(p, σ)) = m((p, σ),(p, σ)) = ˜ m((p, σ),(p, σ)) ∀(p, σ)∈˜ Ψ and this proves that ˜ a(·,·) is coercive in ˜ Ψ. On the other hand, the “inf-sup” condition is a consequence of the fact that, for any µ∈˜ M, there exists (p, σ)∈˜ Φsuch that ˜ b(p, σ, µ) = kµk2 ˜ Mand k(p, σ)k˜ Φ≤Ckµk˜ M.(55) This can be seen as follows: for any fixed µ∈˜ M, let (p, σ) be the solution to L∗p+∇σ=0in QT,∇ · p=µin QT,p=0on ΣT,p(·, T) = 0in Ω; then pbelongs to L2(QT) and kpkL2(QT)≤CkµkL2(QT)(56) (see [23]). Therefore, it is lear that (55) also holds. 19 3.3 A third mixed reformulation of (54) with an additional multiplier As in Section 3.1, introducing the variable z:= L∗p+∇σ, we observe that (54) is equivalent to:      ˆ a((z,p, σ),(z0,p0, σ0)) + ˆ b((z0,p0, σ0),(λ, µ)) = hˆ `,(z0,p0, σ0)i, ˆ b((z,p, σ),(λ0, µ0)) = 0, ∀((z0,p0, σ0),(λ0, µ0)) ∈Zט Φ×Λט M; ((z,p, σ),(λ, µ)) ∈Zט Φ×Λט M, (57) where the bilinear forms ˆ a(·,·) and ˆ b(·,·) are given by ˆ a((z,p, σ),(z0,p0, σ0)) := ZZQTρ−2z·z0+ρ−2 0p·p01ωdxdt and ˆ b((z,p, σ),(λ, µ)) := ZZQT [z−(L∗p+∇σ)] ·λdxdt +ZZQT (∇ · p)µ dxdt (58) and the linear form ˆ `is given by hˆ `,(z,p, σ)i:= ZΩ y0(x)·p(x,0) dx. Now, the following holds: Proposition 6. There exists exactly one solution to (57). Furthermore, (54) and (57) are equivalent problems in the following sense: 1. If ((z,p, σ),(λ, µ)) solves (57), then (p, σ, µ)solve (54). 2. If (p, σ, µ)solves (54), there exists λ∈Λsuch that ((z,p, σ),(λ, µ)), with z:= L∗p+∇σ, solves (57). Proof. Let us introduce the space ˆ Ψ={(z,p, σ)∈Zט Φ:ˆ b((z,p, σ),(λ, µ)) = 0 ∀(λ, µ)∈Λט M} and, as before, let us check that •ˆ a(·,·) is coercive in ˆ Ψ. •ˆ b(·,·) satisfies the usual “inf-sup” condition in (Zט Φ)×(Λט M). Again, the proofs of these assertions are easy. Indeed, for any (z,p, σ)∈ˆ Ψ,z=L∗p+∇σand ∇ · p= 0 and, therefore, ˆ a((z,p, σ),(z,p, σ)) = ZZQTρ−2|z|2+ρ−2 0|p|21ωdxdt =1 2k(z,p, σ)k2 Zט Φ+1 2ZZQT ρ−2 0|p|21ωdxdt ≥1 2k(z,p, σ)k2 Zט Φ, 20 whence ˆ a(·,·) is coercive in ˆ Ψ. On the other hand, the “inf-sup” condition is a consequence of the fact that, for any (λ, µ)∈Λט M, there exists (z,p, σ)∈Zט Φsuch that ˆ b((z,p, σ),(λ, µ)) = k(λ, µ)k2 Λט Mand k(z,p, σ)kZט Φ≤Ck(λ, µ)kΛט M.(59) This time, the argument is as follows: for any fixed (λ, µ)∈ˆ Y, let (p, σ) be the solution to L∗p+∇σ=0in QT,∇ · p=µin QT,p= 0 on ΣT,p(·, T ) = 0in Ω; then pbelongs to L2(QT) and kpkL2(QT)≤CkµkL2(QT).(60) Taking z=ρ2λ, one arrives easily at (59). 3.4 Another formulation related to (44) Let us introduce the spaces Φ∗:= {(p, σ) : ZZQTρ−2 2|pt|2+ρ−2 1|∇p|2+ρ−2 0|p|2+ρ−2|∇σ|2dxdt < +∞, ∇ · p≡0,p=0on ΣT}, Λ∗:= {λ:ZZQTρ2 2|λ|2+ρ2 1|∇λ|2dxdt < +∞,λ=0on ΣT}, the bilinear forms a∗(·,·) and b∗(·,·), with a∗((z,p, σ),(z0,p0, σ0)) := ZZQTρ−2z·z0+ρ−2 0p·p01ωdxdt and b∗((z,p, σ),λ) := ZZQT {[z+pt− ∇σ]·λ−ν∇p· ∇λ}dxdt and the linear form `∗, with h`∗,(z,p, σ)i:= ZΩ y0(x)·p(x,0) dx. The bilinear form b∗(·,·) appears when we integrate by parts the second–order terms in b(·,·), see (45). Accordingly, at least formally, we can reformulate (44) as follows:      a∗((z,p, σ),(z0,p0, σ0)) + b∗((z0,p0, σ0),λ0) = h`∗,(z0,p0, σ0)i, b∗((z,p, σ),λ)=0, ∀((z0,p0, σ0),λ0)∈Z×Φ∗×Λ∗; ((z,p, σ),λ)∈Z×Φ∗×Λ∗. (61) This can be viewed as a new mixed formulation of (42). However, that these two problems are equivalent in the sense of Propositions 5 and 6 is, at present, an open question. 3.5 A fifth (and final) mixed formulation Finally, let us introduce the space Φ:= {(p, σ) : ZZQT ρ−2 2|pt|2+ρ−2 1|∇p|2+ρ−2 0|p|2+|∇ · p|2+ρ−2|∇σ|2dxdt < +∞, p=0on ΣT}, 21 the bilinear forms a(·,·) and b(·,·), with a((z,p, σ),(z0,p0, σ0)) := ZZQTρ−2z·z0+ρ−2 0p·p01ωdxdt and b((z,p, σ),(λ, µ)) := ZZQT {[z+pt− ∇σ]·λ−ν∇p· ∇λ}dxdt +ZZQT (∇ · p)µ dxdt and the linear form `, with h`,(z,p, σ)i:= ZΩ y0(x)·p(x,0) dx. In accordance with (61), it can be accepted that, at least formally, (57) possesses the following reformulation:      a((z,p, σ),(z0,p0, σ0)) + b((z0,p0, σ0),(λ, µ)) = h`,(z0,p0, σ0)i, b((z,p, σ),(λ0, µ0)) = 0, ∀((z0,p0, σ0),(λ0, µ0)) ∈Z×Φ×Λ×M; ((z,p, σ),(λ, µ)) ∈Z×Φ×Λ×M. (62) Remark 3. The previous mixed formulations possess several relevant properties: •By constructing finite dimensional subspaces of Φ, we are led to standard mixed approximations of (44). But this is not a simple task: recall that, in order to have (p, σ)∈Φ, we need (among other things) L∗p+∇σ∈L2(ρ−1;QT) and ∇ · p= 0. •Contrarily, it is relatively easy to construct numerically efficient finite dimensional subspaces of ˜ Φ, for instance, based on the Bell triangle or the Bogner-Fox-Schmidt rectangle. Consequently, we can get finite element approximations of (54) for which, furthermore, a convergence analysis can be performed. •The same can be said for (57). In this case, the fact that the variable zappears explicitly is useful for a direct computation of an approximation of the state. •The mixed formulations (61) and (62) share an advantageous characteristic: they can be approximated in a rather standard way by C0finite elements since, after integration by parts, no secondorder spatial derivative appears. Unfortunately, up to our knowledge, it is unknown whether or not they are well posed. More precisely, the proof of the “inf-sup” condition is open and, moreover, the well-posedness of their associated discrete versions is not clear. 3.6 A numerical approximation of (62) (without justification) Let us conserve the notation in Section 2.4. For any couple of integers m, n ≥1, we will set Zh(m, n) := {zh∈C0(Qκ,T ) : zh|K∈(Pm,x⊗Pn,t)(K)∀K∈ Qh}, Vh(m, n) := {ph∈Zh(m, n) : ph=0on ΣT}, Mh(m, n) := {σh∈C0(Qκ,T ) : σh|K∈(Pm,x⊗Pn,t)(K)∀K∈ Qh}. Then, Zh(m, n) and Vh(m, n) are finite dimensional subspaces of the Hilbert space H1(Qκ,T ). Moreover, Vh(m, n)×Mh(m, n)⊂Φ,Vh(m, n)⊂Λand Mh(m, n)⊂˜ M. Therefore, for any m,n,m0,n0, m00,n00,m000,n000,m0000,n0000 ≥1, we can define Xh:= Zh(m, n)×Vh(m0, n0)×Mh(m00, n00) and Yh:= Vh(m000, n000)×Mh(m0000, n0000), that are finite dimensional subspaces of Z×Φand Λ×M, respectively. 22 The following mixed approximation of (62) makes sense:      a((zh,(ph, σh)),(z0 h,(p0 h, σ0 h))) + b((z0 h,(p0 h, σ0 h)),(λh, µh)) = h`,(z0 h,(p0 h, σ0 h))i, b((zh,(ph, σh)),(λ0 h, µ0 h)) = 0, ∀((z0 h,(p0 h, σ0 h)),(λ0 h, µ0 h)) ∈Xh×Yh; ((zh,(ph, σh)),(λh, µh)) ∈Xh×Yh. (63) 3.7 A numerical experiment This Section deals with some numerical results. We have solved (63) with the following data: Ω = (0,1) ×(0,1), ω= (0.2,0.6) ×(0.2,0.6), T= 1, K1= 1, K2= 2, χ0=χ(0.5,0.5) 0(as in Section 2); ν= 1, y0(x)≡(M, 0), with M= 1000. Again, the computations have been performed with the software Freefem++, using P2-Lagrange approximations in (x, t) for all the variables. Th domain and the mesh are depicted in Fig. 1. The linear system in (63) has been solved with the Arrow-Hurwicz algorithm, where we have taken r= 0.01 and s= 0.1. The convergence of this algorithm is illustrated in Table 2, where the first and the second relative errors are given by k(z(k+1) h,p(k+1) h, σ(k+1) h)−(z(k) h,p(k) h, σ(k) h)kL2(QT) k(z(k+1) h,p(k+1) h, σ(k+1) h)kL2(QT) and k(λ(k+1) h, µ(k+1) h)−(λ(k) h, µ(k) h)kL2(QT) k(λ(k+1) h, µ(k+1) h)kL2(QT) . The computed control and state are displayed in Fig. 5–8. Iterate Rel. error 1 Rel. error 2 1 0.659686 0.202439 10 0.063864 0.106203 20 0.016147 0.076083 30 0.008874 0.046212 40 0.000464 0.001397 50 0.000206 0.000762 Table 2: The behavior of the Arrow-Hurwicz algorithm for (63). 4 An application : numerical local exact controllability to the trajectories of the Navier-Stokes equations In this Section, we will present a numerical method for the computation of a solution to the local exact controllability problem to the trajectories of (3). This controllability property was proved in [9] under suitable regularity assumptions on the trajectories. More precisely, we have to assume that the trajectory satisfies y∈L2(0, T;D(A)) ∩C0([0, T]; V)∩L∞(QT),yt∈L20, T ;H,(64) where D(A) := H2(Ω) ∩Vis the domain of the usual Stokes operator A; see also [18] for a previous result. 23 Figure 5: ω= (0.2,0.6); y0(x) = (1000,0). Cuts of v1,h and 1010v2,h at x1= 0.28 (Left) and x1= 0.52 (Right). 24 Figure 12: Taylor-Green test – The domain and the mesh. Number of vertices: 3146. Number of elements (tetrahedra): 15900. Total number of variables: 22022. Figure 13: Taylor-Green test – First component of the initial datum (Left) and second component of the initial datum (Right). 31 Figure 14: Taylor-Green test – The initial data: first component (Left) and second component (Right). Figure 15: Taylor-Green test – The first component of the state (Left) and the second component of the state at T= 0.6 ((Right). 32 Figure 16: Taylor-Green test – The first component of the state (Left) and the second component of the state at T= 0.9 ((Right). 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 TIMES SPATIAL NORMS UNCONTROLLED STATE CONTROLLED STATE 0.8 1 1.2 1.4 1.6 1.8 2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 TIMES SPATIAL NORMS UNCONTROLLED STATE CONTROLLED STATE Figure 17: Poiseuille test – The L2-norms of the deviations y−yPand z−yPcorresponding to the computed controlled and uncontrolled states (Left) and a detail ((Right). Here, T= 2. 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