Controls insensitizing the observation of a quasi-geostrophic ocean model

SIAM J. CONTROL OPTIM.c 2005 Society for Industrial and Applied Mathematics Vol. 43, No. 5, pp. 1616–1639 CONTROLS INSENSITIZING THE OBSERVATION OF A QUASI-GEOSTROPHIC OCEAN MODEL∗ ENRIQUE FERN´ ANDEZ-CARA†, GALINA C. GARCIA‡,AND AXEL OSSES§ Abstract. We consider a linear quasi-geostrophic ocean model with partially known initial conditions. We search for controls that make the observation locally insensitive to the perturbations of the initial data. Their existence is equivalent to the null controllability property for an associated cascade Stokes-like system. Thanks to the presence of the Coriolis term, we are able to prove the existence of such controls. Our strategy is the following. First, we prove a unique continuation property for the adjoint of the state system that leads to approximate controllability; then, under certain assumptions, an observability inequality is established for the adjoint. The proof is inspired by the arguments leading to the unique continuation property. This inequality leads to the desired null controllability result. Key words. insensitizing controls, Carleman inequalities, unique continuation, null controllability, ocean model AMS subject classifications. 93B05, 35B37, 35B60, 35Q30 DOI. 10.1137/S0363012903433607 1. Introduction and main results. 1.1. Incomplete initial data ocean model. Let Ω be a nonempty open bounded and connected subset of R2, with boundary Γ of class C2and outwards unit normal vector ν=ν(x). Let ωbe a nonempty open subset of Ω, T>0, Q=Ω×(0,T), and Σ = Γ ×(0,T). In this paper, we will consider a linear quasi-geostrophic ocean model [1, 15, 16] described by the following equations: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ut−A∆u+γu +(f0+βx2)k∧u+1 ρ0 ∇p=T+h1ωin Q, div u=0 inQ, u= 0 on Σ, u(0) = u0+τu0in Ω, (1.1) where u(x, t) and p(x, t), respectively, denote the velocity and the pressure of the fluid at (x, t)=(x1,x 2,t)∈R2×R+. In this model, Arepresents the horizontal eddy viscosity coefficient, γis the bottom friction coefficient, ρ0is the fluid density, and (f0+βx2)k∧uis the Coriolis term, with k∧u=(−u2,u 1). In the right-hand side, 1ω denotes the characteristic function of ωand Tis a given source in L2(Q)2. The term ∗Received by the editors August 22, 2003; accepted for publication (in revised form) July 1, 2004; published electronically March 11, 2005. http://www.siam.org/journals/sicon/43-5/43360.html †Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain ([email protected]). This author’s work was partially supported by D.G.E.S. (Spain) grants BFM2000-1317 and BFM2003-06446. ‡Facultad de Ingenier´ıa, Universidad Cat´olica de la Sant´ısima Concepci´on, Casilla 297, Concepci´on, Chile ([email protected]). This author’s work was supported by FONDAP in Applied Mathematics, CONICYT Ph.D. grants, and CONICYT-INRIA cooperation agreements (Chile). §Departamento de Ingenier´ıa Matem´atica, Universidad de Chile, Casilla 170/3 Correo 3, Santiago, Chile, and Centro de Modelamiento Matem´atico, UMI 2807/Universidad de Chile-CNRS, Santiago, Chile ([email protected]hile.cl). This author’s work was partially supported by FONDAP in Applied Mathematics, FONDECYT-CONICYT 1030808-7030059, and ECOS-CONICYT C01E02 grants (Chile). 1616 Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLS INSENSITIZING AN OCEAN MODEL 1617 τu0, where τ∈R, represents a small unknown perturbation of the initial velocity field u0, and h=h(x, t) is a control function to be determined. Notice that the Coriolis force is represented by a zero order coupling term in the equations. It introduces a different behavior of the system depending on the direction in space. To simplify the presentation of the results, we will assume that A=1,γ=1, f0=1,β= 1, and ρ0=1. We introduce the following spaces, which are usual in the analysis of Stokes systems: H={v∈L2(Ω)2: div v= 0 in Ω,v·ν= 0 on Γ}, V={v∈H1 0(Ω)2: div v= 0 in Ω},W=H2(Ω)2∩V. Recall that W→V→H≡H→V→W, where the embeddings are dense and compact. For any given u0,τu0∈Hwith u00,Ω=1,anyT∈L2(Q)2, and any h∈ L2(ω×(0,T))2, the linear system (1.1) possesses a unique solution (u, p), with u∈ L2(0,T;V)∩H1(0,T;V) and p∈W−1,∞(0,T;L2(Ω)). (pis unique up to an additive distribution only depending on t.) This is easily proved by adapting the arguments of [17] to the presence of a skew-symmetric Coriolis term in the equations. Notice that if we had u0+τu0∈V, then the couple (u, p) would satisfy u∈L2(0,T;W)∩H1(0,T;H) and p∈L2(0,T;H1(Ω)). We will be concerned with the search of controls such that the velocity measurements over an observation set are either insensitive or almost insensitive to small variations of the initial conditions. To do this, we will use insensitizing control theory. 1.2. Insensitizing controls and controllability. Let Obe an open nonempty subset of Ω and let us introduce the following functional, defined on the family of solutions to (1.1): Φ(u)=1 2T 0O |u(x, t)|2dx dt.(1.2) The notion of insensitizing controls was introduced by Lions [13]. In the context of (1.1)–(1.2), it reads as follows. Definition 1.1. We say that the control h∈L2(ω×(0,T))2is Φinsensitizing if d dτ Φ(u)τ=0 =0 ∀u0∈Hwith u00,Ω=1.(1.3) On the other hand, we say that h∈L2(ω×(0,T))2is Φε-insensitizing if  d dτ Φ(u)τ=0≤ε∀u0∈Hwith u00,Ω=1.(1.4) Of course, in (1.3) and in (1.4) uis, together with p, the solution to (1.1). The Φ insensitizing (resp., Φ ε-insensitizing) controls hmust be interpreted as those leading to an observation Φ(u) that is locally independent (resp., almost independent) at the initial perturbation τu0. The existence of such controls is a pertinent Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1618 E. FERN´ ANDEZ-CARA, G. C. GARCIA, AND A. OSSES question, since it is realistic to assume that the true initial conditions for (1.1) are unknown. In fact, as noticed in [13], it would be more convenient to search for Ψ insensitizing (or Ψ ε-insensitizing) controls, where Ψ(u)=1 2T 0O |curl u(x, t)|2dx dt, but this is beyond the scope of this article and will be the subject of future work. It is easy to characterize the insensitivity (resp., ε-insensitivity) property in terms of exact null controllability (resp., approximate controllability) of a related cascade system. Indeed, let (¯u, ¯p) and (q,r) be the solutions of the following systems: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ¯ut−∆¯u+¯u+(1+x2)k∧¯u+∇¯p=T+h1ωin Q, div ¯u=0 inQ, ¯u= 0 on Σ, ¯u(0) = u0in Ω, (1.5) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −qt−∆q+q−(1 + x2)k∧q+∇π=¯u1Oin Q, div q=0 inQ, q= 0 on Σ, q(T)=0 inΩ. (1.6) Then the control his Φ insensitizing (resp., Φ ε-insensitizing ) if and only if q(0) = 0 (resp., q(0)0,Ω≤ε).(1.7) Indeed, in view of (1.2), condition (1.3) is equivalent to T 0O ¯u·uτdx dt =0 resp., (1.4) is equivalent to T 0O ¯u·uτdx dt≤ε, where ¯uis the solution of (1.5) and uτis the solution of (1.1) differentiated with respect to τ. Using the definition of (q,π) and integrating by parts, we obtain Ω q(0) ·u0dx =0 resp., Ω q(0) ·u0dx≤ε∀u0∈Hwith u00,Ω=1. This is equivalent to (1.7). See [18] for more detail. Notice that since ¯u∈L2(0,T;V), we also have q∈L2(0,T;W)∩H1(0,T;H) and π∈L2(0,T;H1(Ω)). We are thus in the presence of a null controllability problem (resp., an approximate controllability problem) for a cascade system, where the control his not acting directly in the system satisfied by q(the function we want to drive to zero after a time interval of length T) but indirectly, through ¯u1O. 1.3. Main results. There have been several recent results concerning the existence of insensitizing and ε-insensitizing controls for parabolic problems. Thus, in [2] the existence of ε-insensitizing controls for linear heat equations with partially known initial and boundary conditions was established. The same was also obtained for semilinear heat equations with globally Lipschitz-continuous nonlinearities. Since then, it has been proved in [18] that insensitizing controls exist for the same equations completed with zero initial data, under suitable assumptions Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLS INSENSITIZING AN OCEAN MODEL 1619 on the source term. In [3], the authors extended these results to other more general (slightly superlinear) nonlinearities. In this paper, we deal with the insensitizing and ε-insensitizing problems for the case of the Stokes-type equations (1.1). Our results were sketched in [7]. These are the first insensitivity results in the literature for equations of this type, as far as we know. We will assume that the following geometrical hypothesis is satisfied, as in the previous references: ω∩O=∅.(1.8) Our main results are the following. Theorem 1.2. Let T>0and assume that (1.8) is satisfied. Then, for each ε>0there exists a control h∈L2(ω×(0,T))2which is Φε-insensitizing. Theorem 1.3. Under the assumptions of Theorem 1.2, if we also have u0=0 and T 0Ω exp Mt−4T2dx dt < +∞(1.9) for an appropriate constant Mdepending on Ω,ω,O, and T, then there exists a control h∈L2(ω×(0,T))2which is Φinsensitizing. It was proved in [18] for the linear heat equation that, in general, we cannot expect the existence of insensitizing controls for nonvanishing initial data in L2(Ω) when Ω \ω=∅. The proof of this result is based on a counterexample for which the appropriate observability inequality fails when the initial data belong to L2(Ω). Similar arguments could be used for Stokes systems. In view of this, it is reasonable to impose in Theorem 1.3 that u0=0. This paper is organized as follows. In section 2, we prove Theorem 1.2, where we obtain a unique continuation result for an adjoint cascade system thanks to the presence of the Coriolis term. In section 3, we prove Theorem 1.3. In this section, we show that insensitizing controls do exist if an appropriate observability inequality holds. We deduce this observability inequality in section 3.2 by means of an appropriate global Carleman inequality for the same adjoint cascade system. The proof of this global Carleman inequality is given in section 3.1 and follows a chain of estimates based on the steps of the unique continuation proof. At the end of this section and to be self-contained, we give the proof of a standard global Carleman estimate for Stokes-like systems that is needed in section 3.1. Finally, in section 4, we summarize the key points of this article in some final remarks. 2. Proof of Theorem 1.2. We can assume without loss of generality that T=0 and u0= 0 in (1.5)–(1.6). It is well known that the existence of ε-insensitizing controls for (1.5)–(1.6) is equivalent to a unique continuation property of the associate adjoint system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ φt−∆φ+φ+(1+x2)k∧φ+∇θ=0 inQ, div φ=0 inQ, φ= 0 on Σ, φ(0) = φ0in Ω, (2.1) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −zt−∆z+z−(1 + x2)k∧z+∇r=φ1Oin Q, div z=0 inQ, z= 0 on Σ, z(T)=0 inΩ (2.2) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1620 E. FERN´ ANDEZ-CARA, G. C. GARCIA, AND A. OSSES for a given φ0∈H. This coupled system possesses a unique solution (φ, θ), (z,r), with at least φ, z ∈L2(0,T;V)∩H1(0,T;V) and θ,r ∈W−1,∞(0,T;L2(Ω)). (Again, θand rare unique up to a distribution depending only on t.) Using (1.5)–(1.6) and (2.1)–(2.2) the following duality identity is easily deduced: T 0ω h·zdxdt=Ω q(0) ·φ0dx ∀h∈L2(ω×(0,T))2. It is clear from this last identity that the set {q(0) : h∈L2(ω×(0,T))2}is dense in Hif the following unique continuation result holds. Lemma 2.1. Assume (1.8).Let(φ, θ),(z,r)be the solution to (2.1)–(2.2) with φ0∈H. Then, if z=0in ω×(0,T), we necessarily have z≡φ≡0and ∇r≡∇θ≡0 in Q. Proof. This is a direct consequence of a more general unique continuation result. To state this result precisely, let ω=ω∩O=∅and let us set C1(ω)={(x1,x 2)∈Ω:∃x0 1s. t. (x0 1,x 2)∈ω},Σ1(ω)=(Γ∩C1)×(0,T).(2.3) (C1(ω) is the horizontal component of ω.) We will prove that if φ=(φ1,φ 2)is together with θ,z, and ra solution of ⎧ ⎨ ⎩ φt−∆φ+φ+(1+x2)k∧φ+∇θ=0 inQ, div φ=0 inQ, φ1= 0 on Σ1, (2.4) −zt−∆z+z−(1 + x2)k∧z+∇r=φ1ωin Q, div z=0 inQ, and z=0inω×(0,T), then φ≡0. To prove this assertion, we divide the proof into two steps. Without loss of generality, we can assume that ωis connected; otherwise we would replace ωby one of its connected components. In a first step, we deduce from the fact that z=0inω×(0,T) that φ2= 0 and φ1is constant if they are restricted to ω×(0,T). Thus, since z=0inω×(0,T)we notice that curl φ=0inω×(0,T) by applying the curl operator in the equation of zin (2.4). Using this fact, if we now apply the curl operator to the first equation in (2.4), thanks to the presence of the Coriolis term we obtain that curl ((1 + x2)k∧φ)=φ2+ div φ=φ2=0 in ω×(0,T). Now, since div φ= 0 and curl φ=0inω×(0,T), we have ∇φ1=0in ω×(0,T). Therefore φ1is constant in ω×(0,T) and we certainly obtain φ= (Const.,0) in ω×(0,T). In a second step, let us introduce ψ=∂φ ∂x1 ,π=∂θ ∂x1 (2.5) and the coefficient matrix: a=1−(1 + x2) (1 + x2)1 ∈L∞ loc(Q).(2.6) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLS INSENSITIZING AN OCEAN MODEL 1621 Then we have ψt−∆ψ+aψ +∇π=0 inQ, div ψ=0 inQ, (2.7) (ψ,π)∈L2 loc(Q)2×D (Q)(2.8) with ψ=0inω×(0,T). Here, we use a sharp uniqueness property for the Stokes system (2.7) proved in [5] that says that, under the regularity determined by (2.6) and (2.8), one has ψ≡0inQ. Now, from (2.5) we obtain ∂φi/∂x1≡0 for i=1,2in Q. Since div φ=0inQwe also have ∇φ2=0inQand, from the fact that φ2=0in ω×(0,T), we deduce that φ2≡0inQ(recall that Ω is connected). On the other hand, since ∂φ1/∂x1=0inQand φ1= 0 on Σ1, we see that φ1=0 in C1×(0,T). Finally we have φ=(φ1,φ 2)=0inC1×(0,T), which is an open subset of Q. We can conclude that φ≡0inQusing again the uniqueness property of [5]. (We can also use here the weaker result proved in [4].) Remark 1. The method used in the second part of the proof of Lemma 2.1 leads to the following uniqueness property in any dimension n. Let (φ, θ) be the solution of ⎧ ⎨ ⎩ φt−∆φ+aφ+∇θ=0 inQ, div φ=0 inQ, φ= 0 on Σ1(ω), (2.9) where Q=Ω×(0,T), Ω is a nonempty open bounded connected subset of Rn,ωis an open nonempty subset of Ω, Σ1and C1are as defined in (2.3), and a∈L∞(Q). If ais a function independent of x1in Qand φis independent of x1in ω×(0,T), then φvanishes in Q. Indeed, let us introduce ψ=∂φ/∂x1,π=∂θ/∂x1, which satisfy a Stokes problem similar to (2.9), and this problem does not involve φexplicitly since ais independent of x1. Now, from the uniqueness property in [5], ψ≡0inQ. Consequently ∂φ/∂x1=0inQand φ= 0 on Σ1,sowehaveφ=0inC1×(0,T). Using the unique continuation property in [5] once again, we obtain that φ≡0inQ. Remark 2. The previous remark shows that the Coriolis term plays a crucial role only in the first part of the proof of Lemma 2.1. In fact, the presence of the Coriolis term allows us to prove that curl φ=0inω×(0,T) implies that the second component of φvanishes in ω×(0,T). This will also be important in the deduction of the Carleman inequality later. Remark 3. In the proof of the previous lemma, it is not possible to use the results of [4] concerning uniqueness properties of the Stokes system when one of the components of φvanishes in ω×(0,T). This is because the results in [4] require that the coefficient a, introduced in (2.6), satisfy a12 =0. 3. Proof of Theorem 1.3. The proof of the existence of insensitizing controls for (1.1), i.e., the exact null controllability for (1.5)–(1.6), relies on the following observability result for the cascade adjoint system (2.1)–(2.2). Proposition 3.1. Assume that ω∩O =∅. There exist positive constants Mand K, depending only on Ω,ω,O, and T, such that the inequality T 0Ω exp −Mt−4|z|2dx dt ≤KT 0ω |z|2dx dt(3.1) holds for every solution of (2.1)–(2.2) with φ0∈H. The proof of this result is based on a global Carleman inequality (see Theorem 3.3), as will be seen in section 3.2. This Carleman inequality will be proved in section 3.1. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1622 E. FERN´ ANDEZ-CARA, G. C. GARCIA, AND A. OSSES Let us now give the proof of Theorem 1.3 using Proposition 3.1. Thus, let us assume that (1.8) is satisfied, u0= 0, and (1.9) holds with Mbeing the constant furnished by Proposition 3.1. The approximate control hof minimal norm in L2(ω×(0,T))2corresponding to u0= 0, a source term Tsatisfying (1.9), and tolerance ε>0 can be obtained by minimizing in L2(Ω)2the following convex functional [6, 14]: Jε(φ0)=1 2T 0ω |z|2dx dt +T 0Ω T·zdxdt+εφ00,Ω.(3.2) Thus, the minimum of Jεis attained at some  φ0ε∈L2(Ω)2. We denote by ( φε, θε), (zε,rε) the corresponding solution to (2.1)–(2.2) with φ0= φ0ε; then the control function defined as hε=zε1ω (3.3) is such that the associated solution (¯uε,¯pε), (qε,π ε) to (1.5)–(1.6) with u0= 0 satisfies qε(0)0,Ω≤ε. It is not difficult to see that lim inf φ00,Ω→∞ Jε(φ0) φ00,Ω ≥ε. The proof of this inequality is classical; see [6]. It is implied by the unique continuation property for the cascade adjoint system that we presented above (see Lemma 2.1). Furthermore, the following optimality condition must be satisfied at  φ0ε: T 0ω |zε|2dx dt +T 0Ω T·zεdx dt +ε φ0ε0,Ω=0.(3.4) By replacing (3.3) in (3.4), introducing the weight eMt−4, and using (3.1) and Young’s inequality, we easily deduce that T 0ω |hε|2dx dt ≤K2T 0Ω exp(Mt−4)|T |2dx dt. Since {hε}is uniformly bounded in L2(ω×(0,T))2, then up to a subsequence, still denoted {hε},wehave hεh weakly in L2(ω×(0,T))2, ¯uε→¯ustrongly in L2(Q)2,and qε→qstrongly in L2(Q)2, as ε→0. Of course, we have denoted here by (¯uε,¯pε), (qε,π ε) and (¯u, ¯p), (q,π) the solutions to (1.5)–(1.6) associated with hεand h, respectively. Notice that qε(0)0,Ω≤εand consequently we have q(0) = 0. This ends the proof of Theorem 1.3. 3.1. A global Carleman estimate. The goal of this section is to present an estimate of the Carleman kind for the solutions to the adjoint cascade system (2.1)–(2.2). As mentioned above, this estimate will be crucial for the proof of Proposition 3.1. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLS INSENSITIZING AN OCEAN MODEL 1623 Let us first introduce an open ball B0such that B0⊂⊂ ω∩O and an auxiliary function η0∈C 2(Ω) satisfying η0(x)>0∀x∈Ω,η 0=0 on∂Ω,|∇η0(x)|>0∀x∈Ω\B0.(3.5) The existence of such a function is proved in [9]. Let us also introduce the weight functions α(x, t)=e2λη0∞−eλη0 t4(T−t)4,α(t) = min Ω α(x, t),α ∗(t) = max Ω α(x, t), ϕ(x, t)= eλη0 t4(T−t)4,ϕ(t) = max Ω ϕ(x, t),ϕ ∗(t) = min Ω ϕ(x, t). The following property of the functions α∗and αwill be needed later. Lemma 3.2. For any a>1there exists λa>0such that aα(t)>α ∗(t)∀λ>λ a,∀t∈(0,T). Proof. The proof is elementary. It suffices to notice that we have a(e2x−ex)> e2x−1ifa>1 and xis sufficiently large. The main result in this section is the following. Theorem 3.3. Assume that ω∩O=∅and let the functions α,ϕ,α, and ϕbe as above. For each γ∈(0,1), there exist constants s, λ, and  Cdepending on Ω,ω, O,T, and γsuch that one has T 0Ω e−2sα 1 sϕ(|zt|2+|∆z|2)+sλ2ϕ|∇z|2+s3λ4ϕ3|z|2dx dt +T 0Ω e−2sα 1 sϕ(|φt|2+|∆φ|2)+sλ2ϕ|∇φ|2+s3λ4ϕ3|φ|2dx dt ≤ CT 0ω e−(1+γ)sαs63λ32 ϕ67|z|2dx dt(3.6) for any s>sand λ> λand for every solution (φ, θ),(z,r)to (2.1)–(2.2) associated with initial data φ0∈H. The proof will be divided in several steps and will be given in the following subsections. First, we will apply a global Carleman estimate for the Stokes system to (2.1) and (2.2). This will lead to the estimate (3.10). Then, to deduce (3.6), we will have to estimate the integral in the right-hand side of (3.10) containing φin terms of z. To this end, we will follow the steps of the proof of Lemma 2.1 in reverse order. 3.1.1. Step 1: A first direct Carleman estimate. Let I(s, λ;v) stand for the quantity I(s, λ;v)=T 0Ω e−2sα 1 sϕ(|vt|2+|∆v|2)+sλ2ϕ|∇v|2+s3λ4ϕ3|v|2dx dt(3.7) for any positive sand λand any sufficiently regular function v=v(x, t). We then have the following. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1624 E. FERN´ ANDEZ-CARA, G. C. GARCIA, AND A. OSSES Lemma 3.4. For each γ1∈(0,1) there exist positive constants s1,λ1, and C1, depending on Ω,ω,O,T, and γ1, with the following properties: I(s, λ;z)≤C1T 0B0 e−(1+γ1)sαs7λ4ϕ15/2|z|2dx dt +T 0B0 e−2sα(sλϕ)2|φ|2dx dt(3.8) +T 0O e−2sα (sϕ)1/2|φ|2+1 s3ϕ7/2|φt|2dx dt and I(s, λ;φ)≤C1T 0B0 e−(1+γ1)sαs7λ4ϕ15/2|φ|2dx dt(3.9) for any s>s 1and λ>λ 1and for every solution of (2.1)–(2.2) with φ0∈H. The proof of Lemma 3.4 is similar to the proof of other recent global Carleman inequalities for the Stokes system. The main ideas are due to Imanuvilov [10, 11]; also see [8] for other related results. The proof is presented in the appendix. Let us fix γ, with 0 <γ<1. We are now going to deduce several estimates that hold for “sufficiently large sand λ.” By this we mean that they are satisfied for any s>¯sand any λ>¯ λ, where ¯sand ¯ λare (large) positive constants depending only on Ω, ω,O,T, and γ. In what follows, Cdenotes a generic constant, not necessarily the same at each occurrence, depending on Ω, ω,O,T, and (possibly) γ. Let γ1be given in (γ,1). In view of Lemma 3.4 applied to γ1,weget I(s, λ;z)+I(s, λ;φ)≤CT 0B0 e−(1+γ1)sαs7λ4ϕ15/2(|z|2+|φ|2)dx dt(3.10) for sand λlarge enough. Indeed, the last two integrals in (3.8) can be absorbed by the left-hand side of I(s, λ;φ), since Cs−3ϕ−7/2≤1 2(sϕ)−1and C(sϕ)1/2≤1 2s3ϕ3 for sufficiently large s. 3.1.2. Step 2: An estimate of φin terms of curl φ.To simplify the notation, let us set a= 7 and b=15/2. Then I(s, λ;z)+I(s, λ;φ)≤CT 0B0 e−(1+γ1)sαsaλ4ϕb(|z|2+|φ|2)dx dt.(3.11) We will denote by B1,B 2,... a sequence of balls centered at the same point as B0and satisfying B0⊂⊂ B1⊂⊂ · · · ⊂⊂ ω∩O. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLS INSENSITIZING AN OCEAN MODEL 1631 Let us first prove that there exist positive constants M,C1such that T/2 0Ω exp(−Mt−4)|z|2dx dt +T T/2Ω |φ|2dx dt ≤C1T 0ω |z|2dx dt.(3.31) For this estimate, let us first notice that for some constants Mand C, one has e−2sα(x,t)ϕ(x, t)3≥Ce−Mt−4∀(x, t)∈Ω×(0,T/2); this is easy to see, in view of the definitions of αand ϕ. Now, using (3.6), we get T/2 0Ω exp(−Mt−4)|z|2dx dt ≤1 s3λ4I(s, λ;z) ≤CK T 0ω e−(1+γ)sαs63 ϕ67|z|2dx dt, and since the weight e−(1+γ)sαϕ67 is bounded, we can estimate the first term in the left-hand side of (3.31). On the other hand, to obtain an estimate of φin terms of z, let us recall the inequality (3.6). Since e−2sαt−12(T−t)−12 is bounded from below far from t= 0 and t=T, in view of (3.30), we have T T/2Ω |φ|2dx dt ≤3T/4 T/4Ω |φ|2dx dt ≤C3T/4 T/4Ω e−2sαs3λ4ϕ3|φ|2dx dt ≤CT 0ω e−(1+γ)sαϕ67|z|2dx dt. As before, from the fact that e−(1+γ)sαϕ67 is bounded, we are able to estimate the second term in the left-hand side of (3.31). Finally, the desired observability inequality (3.1) is obtained using the energy estimate (3.29) and (3.31): T 0Ω exp(−Mt−4)|z|2dx dt ≤T/2 0Ω exp(−Mt−4)|z|2dx dt +T T/2Ω |z|2dx dt ≤CT/2 0Ω exp(−Mt−4)|z|2dx dt +T T/2Ω |φ|2dx dt  ≤CT 0ω |z|2dx dt. Appendix. Proof of Lemma 3.4. Let us recall that this proof is given for the sake of completeness, but it is essentially an adaptation to our framework of the arguments presented in [8] and [11]. Let us consider the system ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −zt−∆z+z−(1 + x2)k∧z+∇r=φ1Oin Q, div z=0 inQ, z= 0 on Σ, z(T)=0 inΩ, (3.32) where φ∈L2(0,T;W)∩H1(0,T;H). Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1632 E. FERN´ ANDEZ-CARA, G. C. GARCIA, AND A. OSSES Recall that B0is an open ball satisfying B0⊂⊂ ω∩O and the auxiliary function η0satisfies η0∈C 2(Ω), η0(x)>0∀x∈Ω,η 0=0 on∂Ω,|∇η0(x)|>0∀x∈Ω\B0. We will need an additional open ball B00 ⊂⊂ B0, such that we still have |∇η0(x)|>0∀x∈Ω\B00 . We will divide the proof of Lemma 3.4 into several steps. Step 1. Following [8], we apply some well-known Carleman estimates for the heat equation to (3.32). Thus, there exist constants s0,λ0, and C>0 depending on Ω, ω, and Tsuch that for every λ>λ 0and s>s 0, the following estimate holds: I(s, λ;z)≤CT 0B00 e−2sαs3λ4ϕ3|z|2dx dt +T 0Ω e−2sα |∇r|2+|(1 + x2)k∧z|2+|φ1O|2dx dt.(3.33) Recall that the definitions of I(s, λ;z) and the weights αand ϕare given in section 3.1. Of course, we can choose slarge enough to absorb the previous term |(1+x2)k∧z|2 with the left-hand side (3.33). We then have I(s, λ;z)≤CT 0B00 e−2sαs3λ4ϕ3|z|2dx dt +T 0Ω e−2sα|∇r|2dx dt +T 0O e−2sα|φ|2dx dt (3.34) for any λ>λ 0and any s>s 01 . Step 2. To estimate the pressure gradient ∇rin (3.34), we first apply the divergence operator to (3.32), i.e., we write ∆r(t) = div((1 + x2)k ∧z)(t)inΩ,t∈(0,T),(3.35) and then we use the following result by Imanuvilov and Puel [12], which is satisfied by weak solutions to second order elliptic equations. Lemma 3.5. Letussetβ(x)=eλη0(x)and let v∈H1(Ω) be a solution of ∆v= div hin Ω,(3.36) where h∈L2(Ω)2. Then there exist positive constants τ2,λ01, and Csuch that Ω e2τβ|∇v|2dx ≤CτΩ e2τββ|h|2dx +τ1/2e2τg2 1/2,∂Ω +τ2λ2B00 e2τββ2|v|2dx +B00 e2τβ|∇v|2dx (3.37) for any τ>τ 2and any λ>λ 01 , where g=v|∂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 CONTROLS INSENSITIZING AN OCEAN MODEL 1633 In particular, we have the following for r(t) and g(t)=r(t)|∂Ω: Ω e2τβ|∇r(t)|2dx ≤CτΩ e2τββ|(1 + x2)k∧z(t)|2dx +τ1/2e2τg(t)2 1/2,∂Ω+τ2λ2B00 e2τββ2|r(t)|2dx +B00 e2τβ|∇r(t)|2dx.(3.38) To estimate the last integral in (3.38), let us introduce an open set B01 such that B00 ⊂⊂ B01 ⊂⊂ B0and a function ξ01 ∈C 2 0(B01) such that 0≤ξ01 ≤1 and ξ01 =1inB00 . Integrating by parts, it follows from (3.35) that B00 e2τβ|∇r(t)|2dx ≤B01 e2τβξ01|∇r(t)|2dx =−B01 e2τβξ01 div((1 + x2)k ∧z)(t)r(t)dx −1 2B01 e2τβ∇ξ01 ·∇|r(t)|2dx −B01 ξ01∇e2τβ ·∇|r(t)|2dx. Integrating again by parts, applying Young’s inequality, and taking into account that |∆(e2τβξ01)|≤Cτ2λ2β2e2τβ for some positive constant C, after some straightforward computations we deduce that B00 e2τβ|∇r(t)|2dx ≤Cτ2λ2B01 e2τββ2|r(t)|2dx +B01 e2τβ|z(t)|2dx. Replacing this inequality in (3.38), we obtain the following for each t∈(0,T): Ω e2τβ|∇r(t)|2dx ≤CτΩ e2τββ|z(t)|2dx +τ1/2e2τg(t)2 1/2,∂Ω +τ2λ2B01 e2τββ2|r(t)|2dx. Now, let us put τ=s/(t4(T−t)4) and let us choose s>s 02 = max(s01 ,τ 2(T/2)8). Then τ>τ 2. Let us multiply by exp(−2sexp(2λη0∞)/(t4(T−t)4)) the previous inequality and let us integrate with respect to tin (0,T). This leads to the estimate T 0Ω e−2sα|∇r|2dx dt ≤CT 0Ω e−2sαsϕ|z|2dx dt +T 0 e−2sα∗(sϕ∗)1/2g(t)2 1/2,∂Ωdt +T 0ω1 e−2sα(sλϕ)2|r|2dx dt,(3.39) where α∗and ϕ∗were introduced in section 3.1. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1634 E. FERN´ ANDEZ-CARA, G. C. GARCIA, AND A. OSSES The first term in the right-hand side of (3.39) can be absorbed by the left-hand side I(s, λ;z) in (3.33) for slarge enough. Hence, we obtain I(s, λ;z)≤CT 0ω0 e−2sαs3λ4ϕ3|z|2dx dt +T 0 e−2sα∗(sϕ∗)1/2g(t)2 1/2,∂Ωdt +T 0ω1 e−2sα(sλϕ)2|r|2dx dt +T 0O e−2sα|φ|2dx dt (3.40) for any λ>λ 01 and any s>s 03 . Step 3. Step 3 estimates the norm of the trace of the pressure on the boundary. To this end, we introduce three new functions: χ(t)=e−sα∗(t)(sϕ∗(t))1/4,˜z=χ(t)z, ˜r=χ(t)r. From (3.32), we see that (˜z, ˜r) satisfies ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −˜zt−∆˜z+˜z+∇˜r=−χz+χ(1 + x2)k∧z+χφ1Oin Q, div ˜z=0 inQ, ˜z= 0 on Σ, ˜z(T)=0 inΩ. Using the continuity of the trace operator and standard a priori estimates for the pressure, we deduce that T 0 ˜r(t)2 1/2,∂Ωdt ≤T 0 ˜r(t)2 1,Ωdt ≤CT 0Ω e−2sα∗s5/2(ϕ∗)3|z|2dx dt +T 0O e−2sα∗(sϕ∗)1/2|φ|2dx dt. We have used here that |χ(t)|2≤Ce−2sα∗s5/2(ϕ∗(t))3for all t∈(0,T). We thus obtain a new estimate from (3.40): I(s, λ;z)≤CT 0B00 e−2sαs3λ4ϕ3|z|2dx dt +T 0B01 e−2sα(sλϕ)2|r|2dx dt +T 0O e−2sα(sϕ)1/2|φ|2dx dt (3.41) for any λ>λ 01 and any s>s 04 . Step 4. It remains to estimate the local term in the right-hand side of (3.41) containing |r|2in terms of zand φ. Assume that the pressure rhas been normalized in such a way that B01 r(t)dx =0 ∀t∈(0,T). Then there exists C>0 such that B01 |r(t)|2dx ≤CB01 |∇r(t)|2dx ∀t∈(0,T) Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLS INSENSITIZING AN OCEAN MODEL 1635 and also T 0B01 e−2sα(sλϕ)2|r|2dx dt ≤CT 0B01 e−2sα(sλϕ)2|∇r|2dx dt, where the functions α=α(t) and ϕ=ϕ(t) were introduced in section 3.1. From (3.32), we see that T 0B01 e−2sα(sλϕ)2|∇r|2dx dt ≤CT 0B01 e−2sα(sλϕ)2(|z|2+|φ|2)dx dt +T 0B01 e−2sα(sλϕ)2(|zt|2+|∆z|2)dx dt. Therefore, in view of (3.41), we obtain I(s, λ;z)≤CT 0B01 e−2sαs3λ4ϕ3|z|2+(sλϕ)2|φ|2dx dt +T 0B01 e−2sα(sλϕ)2(|zt|2+|∆z|2)dx dt +T 0O e−2sα(sϕ)1/2|φ|2dx dt.(3.42) Step 5. The rest of the proof deals with the estimates of the local integrals containing |∆z|2and |zt|2. First, we will be concerned with |∆z|2. Let us introduce a function ξ0∈C 4 0(B0) such that 0≤ξ0≤1 and ξ0=1inB01 . Let us set z(x, t)=e−sαϕξ 0∆z(T−t). We want to estimate the norm zL2(B01×(0,T ))2. Following the arguments in [8] (see Step 4), we can deduce that T 0B01 e−2sα(sλϕ)2|∆z|2dx dt =T 0B01 s2λ2|z|2dx dt ≤CT 0B0 e−2sαs4λ2ϕ9/2|z|2dx dt +T 0B0 e−2sα(sλϕ)2|φ|2dx dt. (3.43) Thus, from (3.42) we have I(s, λ;z) ≤CT 0B0 e−2sαs4λ4ϕ9/2|z|2dx dt +T 0B0 e−2sα(sλϕ)2|φ|2dx dt +T 0B01 e−2sα(sλϕ)2|zt|2dx dt +T 0O e−2sα(sϕ)1/2|φ|2dx dt.(3.44) Step 6. Now we want to estimate |zt|2. Due to the regularity properties of φ,we can use here a more straightforward argument than in [8], where the right-hand side belongs only to L2(Q)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 1636 E. FERN´ ANDEZ-CARA, G. C. GARCIA, AND A. OSSES First, notice that T 0B01 e−2sα(sλϕ)2|zt|2dx dt ≤δT 0B01 e−2sα 1 sϕ|zt|2dx dt +δT 0B01 e−2sα∗1 s3(ϕ∗)7/2|ztt|2dx dt +CδT 0B01 e−4sα∗+2sα∗s7λ4ϕ15/2|z|2dx dt. This is easily obtained by integrating by parts in time. We will later choose δ>0 small enough. We have the following auxiliary result. Lemma 3.6. Let (z,r)be the solution of (3.32). Then the following estimate holds: T 0Ω e−2sα∗1 s3(ϕ∗)7/2|ztt|2dx dt ≤CI(s, λ;z)+T 0O e−2sα∗1 sϕ∗|φ|2+1 s3(ϕ∗)7/2|φt|2dx dt.(3.45) Proof. Multiply (3.32) by e−2sα∗s−2(ϕ∗)−9/4ztt and integrate in Q. Noticing that (e−2sα∗(ϕ∗)−9/4)t≤Ce−2sα∗s(ϕ∗)−1, after some computations we deduce that T 0Ω e−2sα∗1 s2(ϕ∗)9/4|∇zt|2dx dt ≤CT 0Ω e−2sα∗1 sϕ∗(|z|2+|zt|2)dx dt +T 0Ω e−2sα∗s(ϕ∗)1/4|∇z|2dx dt +T 0O e−2sα∗1 sϕ∗|φ|2dx dt +1 2T 0Ω e−2sα∗1 s3(ϕ∗)7/2|ztt|2dx dt.(3.46) On the other hand, if we compute the time derivative of (3.32) and then we multiply the result by e−2sα∗s−3(ϕ∗)−7/2ztt , we find that T 0Ω e−2sα∗1 s3(ϕ∗)7/2|ztt|2dx dt ≤T 0Ω e−2sα∗1 s2(ϕ∗)9/4|∇zt|2dx dt(3.47) +CT 0Ω e−2sα∗1 sϕ∗|zt|2dx dt +T 0O e−2sα∗1 s3(ϕ∗)7/2|φt|2dx dt. From (3.46) and (3.47), we see that (3.45) holds. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLS INSENSITIZING AN OCEAN MODEL 1637 In view of this lemma, we have T 0B01 e−2sα(sλϕ)2|zt|2dx dt ≤CδI(s, λ;z)+T 0O e−2sα∗1 sϕ∗|φ|2+1 s3(ϕ∗)7/2|φt|2dx dt +CδT 0B01 e−4sα∗+2sα∗s7λ4ϕ15/2|z|2dx dt. If we assume that γ1<1, then (3−γ1)/2>1, and from Lemma 3.2 we deduce that (3 −γ1)α/2>α ∗for sufficiently large λ,say,λ>λ 02 . Consequently, −4α+2α∗< −(1 + γ1)αand T 0B01 e−2sα(sλϕ)2|zt|2dx dt ≤CδI(s, λ;z)+T 0O e−2sα∗1 sϕ∗|φ|2+1 s3(ϕ∗)7/2|φt|2dx dt +CδT 0B01 e−(1+γ1)sαs7λ4ϕ15/2|z|2dx dt for any λ>λ 02 and any s>s 04 . From (3.44) and this estimate, choosing δ>0 small enough, we find I(s, λ;z) ≤CT 0B0 e−(1+γ1)sαs7λ4ϕ15/2|z|2dx dt +T 0B0 e−2sα(sλϕ)2|φ|2dx dt +T 0O e−2sα (sϕ)1/2|φ|2+1 s3(ϕ∗)7/2|φt|2dx dt for all λ>λ 02 and s>s 04 . Obviously, this yields (3.8). The proof of (3.9) is very similar and in fact much simpler, since the left-hand side of (2.1) is zero. Thus, we have proved Lemma 3.4 for λ1=λ02 and s1=s04 (two parameters depending on Ω, ω,O, and T). 4. Some final remarks. The geometrical hypothesis ω∩O=∅is required to prove the existence of both ε-insensitizing and insensitizing controls. In the first case, this assumption is used to prove a unique continuation property (Lemma 2.1). In the case of insensitizing controls, it is used to prove an observability inequality. The problem is completely open when ω∩O=∅(see [18]). The existence of insensitizing controls is guaranteed by the null controllability property of a cascade system of quasi-geostrophic equations (1.5)–(1.6). In this case, the control acts indirectly on one variable through the other one. Of course, this controllability property is stronger than the null controllability of a single quasigeostrophic system. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1638 E. FERN´ ANDEZ-CARA, G. C. GARCIA, AND A. OSSES To prove the null controllability property of the cascade system, there are two main difficulties. First is the need for a unique continuation result for the adjoint system (2.2)–(2.1). The presence of the Coriolis term permits us to relate the second component of the velocity and its associated vorticity, and this is a key point in the proof of uniqueness (see Remark 2). The second problem is the need for an observability inequality for the adjoint. This inequality comes from an appropriate (global) Carleman estimate. The main idea is to estimate φin terms of curl φin a ball contained in ω∩Oin the right-hand side of (3.10). This is possible again due to the presence of the Coriolis term (in fact, our method does not work in the case of the usual Stokes equations). We rewrite the system using the stream function and the vorticity and we see that the Coriolis term leads to an expression of the horizontal derivative of the stream function in terms of the vorticity. In this way, we are able to avoid estimates of pressure terms, which are in general very hard to deduce (see the appendix in section 3). Moreover, the weight in the right-hand side of (3.10) is larger than the weight in the left-hand side. Accordingly, the terms in the right cannot be absorbed directly as in the case of the heat equation (see [18]) and this fact requires some additional work. Acknowledgments. The authors wish to thank J.-P. Puel of the Laboratoire de Math´ematiques Appliqu´ees, Universit´e Versailles/Saint Quentin-en-Yvelines (France) and S. Guerrero of the Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla (Spain), for very useful discussions. REFERENCES [1] R. Bermejo and P. G. Del Sastre,Numerical studies of the long-term dynamics of the 2D Navier-Stokes equations applied to ocean circulation, in XVII CEDYA: Congress on Differential Equations and Applications, Universidad de Salamanca, 2001, pp. 15–34. [2] O. Bodart and C. Fabre,Controls insensitizing the norm of the solution of a semilinear heat equation, J. Math. Anal. Appl., 195 (1995), pp. 658–683. [3] O. Bodart, M. Gonz´ alez-Burgos, and R. P´ erez-Garc´ ıa,Insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, C. R. Math. Acad. Sci. Paris, 335 (2002), pp. 677–682. [4] 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. [5] C. Fabre and G. Lebeau,R´egularit´e et unicit´e pour le probl`eme de Stokes, Comm. Partial Differential Equations, 27 (2002), pp. 437–475. [6] C. Fabre, J.-P. Puel, and E. Zuazua,Approximate controllability of the semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), pp. 31–61. [7] E. Fern´ andez-Cara, G. C. Garc´ ıa, and A. Osses,Insensitizing controls for a large-scale ocean circulation model, C. R. Math. Acad. Sci. Paris, 337 (2003), pp. 265–270. [8] E. Fern´ andez-Cara, S. Guerrero, O. Y. Imanuvilov, and J.-P. Puel,Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), pp. 1501–1542. [9] A. Fursikov and O. Y. Imanuvilov,Controllability of Evolution Equations, Lecture Notes Ser. 34, Research Institute of Mathematics, Seoul National University, Korea, 1996. [10] O. Y. Imanuvilov,On exact controllability of Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 3 (1998), pp. 97–131. [11] O. Y. Imanuvilov,Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), pp. 39–72. [12] O. Y. Imanuvilov and J.-P. Puel,Global Carleman estimates for weak solutions of elliptic non homogeneous Dirichlet problems, Int. Math. Res. Not., 16 (2003), pp. 883–913. [13] J. L. Lions,Quelques notions dans l’analyse et le contrˆole de syst` emes `a donn´ees incompl`etes, in Proceedings of the XIth Congress on Differential Equations and Applications/First Congress on Applied Mathematics, University of M´alaga, 1990, pp. 43–54. [14] J. L. Lions,Remarks on approximate controllability, J. Anal. Math., 59 (1992), pp. 103–116. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CONTROLS INSENSITIZING AN OCEAN MODEL 1639 [15] G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev,Adjoint Equations and Perturbation Algorithms in Nonlinear Problems, CRC Press, Boca Raton, FL, 1996. [16] P. G. Myers and A. J. Weaver,A diagnostic barotropic finite-element ocean circulation model, J. Atmos. Ocean Tech., 12 (1995), pp. 511–526. [17] R. Temam,Navier-Stokes Equations, 2nd ed., North-Holland, Amsterdam, 1984. [18] L. de Teresa,Insensitizing control for a semilinear heat equation, Comm. Partial Differential Equations, 25 (2000), pp. 39–72. Downloaded 05/20/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php