A non standard approach to a data assimilation problem and ...

A non standard approach to a data assimilation problem and Tychonov regularization revisited Jean-Pierre Puel

∗

April 2008

Abstract We consider evolution problems like diffusion convection equations, or linearized Navier Stokes system, or weak coupling of them, which we would like to “predict” on a time interval (T0 , T0 + T ) but for which the initial value of the state variable is unknown. On the other hand “measurements” of the solutions are known on a time interval (0, T0 ) and, for example, on a subdomain in space variable. The classical approach in variational data assimilation is to look for the initial value at time 0 and this is known to be an ill-posed problem which has to be regularized. Here we propose to look for the value of the state variable at time T0 (the end time of the “measurements”) and we prove on some basic examples that this is a well-posed problem. We give a result of exact reconstruction of the value at T0 which is based on global Carleman inequalities and we give an approximation algorithm which uses classical optimal control auxiliary problems. We also show why Tychonov regularization for variational data assimilation works in practical situations corresponding to realistic applications, using the same mathematical arguments.

1

Introduction

Data assimilation problems are of great importance for practical purposes, in particular for meteorological and climate prediction, ocean models and environment sciences. They lead to very heavy computations and in order to give an idea of this importance, for example in meteorological prediction, the computation time dedicated to data assimilation corresponds to more than one half of the total computation time. An example of these problems can be roughly but simply described ∗

J.-P. Puel: ([email protected]) Laboratoire de Math´ematiques de Versailles, Universit´e de Versailles Saint-Quentin, 45 avenue des Etats Unis, 78035 Versailles

1

as follows (cf also an interesting description in [1]). The phenomenon under consideration, for example meteorology or climate is modelled by a (very complex in general) system of evolution equations which can be written in the form (1.1)

∂Y + AY + N Y = F ∂t

where Y is the vector representing the state variables that we want to “predict”, A is a partial differential (elliptic) operator in space variables which can be assumed to be linear, N is a nonlinear operator which is in general of lower order and F is the (known) vector of exterior forces which act on the system. The goal is to compute a (good) approximation of Y during a period of time of length T (prediction). Of course many people have been working on these questions in practice and have developped various efficient methods of discretization (even rather complex) for system (1.1) but they are confronted by the following problem : they don’t know the “initial data” for Y at a given time before T0 in order to compute the solution of the prediction model from T0 on. On the other hand they know “measurements” of Y in some spatial regions during a time interval (0, T0 ). The data assimilation problem is then to determine an approximation of an initial data at a time before T0 from the known “measurements”. A great number of works, both theoretical and practical, as well as computations, have been devoted to these questions. For articles related to the problem under consideration here we can refer to [15], [16], [2] and the references therein. In the classical approach the problem is considered as follows. In order to fix ideas we shall say that T0 is to-day, 0 is yesterday and T0 + T is to-morrow. We know “measurements” of Y on a time interval (0, T0 ) (from yesterday to to-day) . We look for the initial value Y (0) (value of the state variable yesterday) in order to “compute” Y on the time interval (0, T0 + T ) (from yesterday to to-morrow) . This problem is known to be ill-posed and one of the methods generally used is what is called variational data assimilation (see [1], [15], [16], [2]). This method uses optimal control techniques for minimizing a suitable cost function (usually a least square method) on a linearized problem together with a regularization method (for example Tychonov regularization) and an optimality system making use of the adjoint state. In a first part we present a new approach which is based on controllability techniques and which is non classical in the following sense. We know the “measurements” of Y on (0, T0 ). What is really important for prediction puposes is to be able to “compute” Y on the time interval (T0 , T0 + T ) (from to-day until to-morrow). We will then look for an approximation of Y (T0 ) (and no more of Y (0)). The interest of this approach is that we will prove on significant examples of linear problems (unfortunately not on a realistic model of meteorology which would be, by far, too much complex) that this problem is well posed and we can give a good estimate on 2

its sensitivity to errors in the measurements. We will also give an exact method of reconstruction of Y (T0 ) and an approximate method which turns out to be simpler in practice, in particular for numerical computations, and for which we will prove convergence towards the exact solution. The underlying mathematical techniques are those used for exact controllability to trajectories and are based on global Carleman inequalities. A preliminary version of these results has been given in [21]. In the next section we give two basic examples of equations, diffusion convection equations and linearized Navier-Stokes equations, which can be rigorously treated. The case of weakly coupled systems, like linearized Boussinesq equations, can also be rigorously proved but we do not present this case here in order to clarify the presentation. It is then straightforward to derive a general principle which can be applied to a large class of evolution systems, formally or not, depending on the possibility of proving a global Carleman estimate for the system under consideration. In a second part, we study Tychonov regularization in a way which is different from the classical works on the subject, for example [10] or [4]. and we try to explain why it works in practical situations for variational data assimilation methods without abstract assumptions on the data. In fact we show that, due to Carleman estimates, the minimization problem (without regularization), which is ill posed in the variational data assimilation context (where the control variable is the initial condition Y (0)), is well posed in a class of trajectories of the problem which may not have an initial value. Then, assuming an hypothesis on the regularity of the minimizer which can be viewed as natural in realistic models (the minimizer is here a trajectory and not an initial value), we show that the solution of the problem with Tychonov regularization converges to this minimizer. Using an additional hypothesis which is similar to the one used for example in [10], [5] or [4] (but different), we give a convergence rate for the previous convergence.

2 2.1

Non standard approach on two basic examples Diffusion convection equations

Let Ω be a bounded open subset of IRN of class C 2 with boundary Γ and let us consider the following diffusion convection equation (2.2)

N N N X X X ∂y ∂ ∂y ∂y ∂ − (aij )+ bi + (cj y) = f in Ω × (0, T0 ), ∂t i,j=1 ∂xi ∂xj ∂x ∂x i j i=1 j=1

(2.3)

y = 0 on Γ × (0, T0 ).

3

Let us notice that we don’t impose any initial condition on y or that we don’t know the initial value y(0). We assume that, for example, (2.4) f ∈ L2 (0, T0 ; L2 (Ω)), aij ∈ W 1,∞ (Ω × (0, T0 )), bi , cj ∈ L∞ (Ω × (0, T0 )) and the coefficients aij satisfy an ellipticity condition (2.5)

∃γ > 0, ∀(x, t) ∈ Ω × [0, T0 ], ∀ξ ∈ IRN ,

N X

aij (x, t)ξi ξj ≥ γ|ξ|2 .

i,j=1

Let ω be a nonempty open subset of Ω and let χω denote the characteristic function of ω. We suppose that we know a “measurement” of the solution (2.6)

y.χω = h

with (2.7)

h ∈ L2 (0, T0 ; L2 (ω)).

Remark 2.1 It will be very important to make precise the class of functions in which y satisfying (2.2) and (2.3) is taken. This will be a key point in the sequel. Remark 2.2 Here, for simplicity, we suppose that we know a “measurement” of the solution y in the interior of Ω. Without many changes, one could consider the ∂y case where “measurements” of the flux ∂ν are known on a non-empty part Γ0 of A the boundary Γ, where ν = (ν1 , ..., νN ) is the outward unit normal on Γ and N X ∂y ∂y aij = νi . ∂νA i,j=1 ∂xj

Let us first recall shortly what is usually done in variational data assimilation (see for example [1], [15], [16]). For y0 ∈ L2 (Ω) (which will be the control variable) we consider the solution of (2.2), (2.3) satisfying in addition (2.8) y(0) = y0 in Ω. It is then well known that problem (2.2), (2.3), (2.8) has a unique solution y = y[y0 ] with y ∈ C([0, T0 ]; L2 (Ω)) ∩ L2 (0, T0 ; H01 (Ω)). In the sequel, we will always consider this regularity when speaking of solutions of (2.2), (2.3), (2.8). This will no longer be the case when we drop the initial condition (2.8). 4

Let us now consider the cost functional (2.9)

J(y0 ) =

Z T0 Z

1 2

0

|y[y0 ] − h|2 dxdt.

ω

(We could have taken a different functional expressing the error between the observation on y[y0 ] and the measurement h.) We would like to solve the minimization problem Find y¯0 ∈ L2 (Ω) such that (2.10)

J(¯ y0 ) =

min

y0 ∈L2 (Ω)

J(y0 ).

It is well known that this problem is ill-posed as, if we take a minimizing sequence y0n for J, we cannot obtain any estimate on this sequence (in any known functional space). Therefore it is usual to add a Tychonov regularization to our functional, namely we consider for α > 0 the new functional (2.11)

1 Jα (y0 ) = 2

Z T0 Z 0

|y[y0 ] − h|2 dxdt +

ω

α |y0 |2L2 (Ω) , 2

and we now want to solve the optimal control problem Find yα ∈ L2 (Ω) such that (2.12)

Jα (yα ) =

min

y0 ∈L2 (Ω)

Jα (y0 ).

It is then standard, using classical optimal control methods, (see [17]), to show that this problem has a unique solution. But of course the functional has been changed and it is not clear what the solution yα represents. The main question is then what happens when α → 0 and what is the sensitivity to perturbations in the measurements h? We will give a partial answer to these questions later on. We now come to a non standard approach to the data assimilation problem using controllability techniques. In order to state our result we need to introduce the adjoint (backward) problem where we allow a control on the region where the measurement is given. For ϕ0 ∈ L2 (Ω) and v ∈ L2 (0, T0 ; L2 (ω)) we denote by ϕ the solution of (2.13) −

N N N X X X ∂ϕ ∂ ∂ϕ ∂ ∂ϕ − (aji )− (bi ϕ) − cj = v.χω in Ω × (0, T0 ), ∂t i,j=1 ∂xj ∂xi ∂xi ∂xj i=1 j=1

(2.14) ϕ = 0 on Γ × (0, T0 ), (2.15) ϕ(T0 ) = ϕ0 . 5

We know that this problem has a unique solution and ϕ ∈ C([0, T0 ]; L2 (Ω)) ∩ L2 (0, T0 ; H01 (Ω)). It is by now well known (cf. for example [8]) that we have null controllability for this problem which means that for every ϕ0 ∈ L2 (Ω), there exists v ∈ L2 (0, T0 ; L2 (ω)) such that ϕ(0) = 0. We have to make things more precise for our purposes. This requires some technicalities which can be skipped by the reader in a first step. Let us call y 0 the solution of (2.2), (2.3), (2.8) with y0 = 0. Then if we call y the solution of (2.2), (2.3), (2.8) with general initial condition y0 ∈ L2 (Ω) we have y = y 0 + z where z is solution of (2.2), (2.3), (2.8) with f = 0. Let us call (2.16)

V 0 = {z, z solution of (2.2), (2.3), (2.8) with f = 0, y0 ∈ L2 (Ω)}

and (2.17)

V = y0 + V 0.

Then V 0 is a vector space and of course V is an affine space. We can now derive a global Carleman estimate for elements of V but this requires the introduction of some weights. Let ω0 be a non empty open set such that ω 0 ⊂ ω (for example ω0 can be a small open ball). Then we know from [8] (see also [20] for a detailed proof) that there exists ψ ∈ C 2 (Ω) such that ψ(x) > 0, ∀x ∈ Ω, ψ(x) = 0, ∀x ∈ Γ, |∇ψ(x)| 6= 0, ∀x ∈ Ω − ω0 . We now use the function ψ to build new weights. Let us define for λ > 0 and for an integer k ≥ 1 (here we only need k = 1 but for further extensions it happens that we sometimes need to take k > 1 and this does not make any change in the sequel)

(2.18)

ξk (x, t) =

(2.19)

ηk (x, t) =

eλ(m|ψ|L∞ (Ω) +ψ(x)) , tk (T0 − t)k e(

k+1 )λm|ψ|L∞ (Ω) k

− eλ(m|ψ|L∞ (Ω) +ψ(x)) , tk (T0 − t)k

where m > k. We can notice that ηk tends rapidly to +∞ when t → T0 or t → 0 but that ηk is uniformly bounded in Ω × [δ, T0 − δ] if 0 < δ < T20 . 6

We will also need in Chapter 3 the weights η˜k and ξ˜k defined by (2.20)

η˜k (t) = ηk (t) if t ∈ [0,

T0 T0 T0 ], η˜k (t) = ηk ( ) if t ∈ [ , T0 ], 2 2 2

T0 T0 T0 ξ˜k (t) = ξk (t) if t ∈ [0, ], ξ˜k (t) = ξk ( ) if t ∈ [ , T0 ]. 2 2 2 Notice that for every δ > 0, η˜k and ξ˜k are bounded on [δ, T0 ]. Our final weight will depend on a second positive parameter s and will be of the form e−sη1 (x,t) . We can see that, for fixed s, this function tends very rapidly to 0 when t → T0 or t → 0. Then, following [8] or [20] where the complete proofs can be found and the method of [14] when c = (cj ) 6= 0, we can state a global Carleman estimate for y ∈ V (2.21)

Proposition 2.3 There exist parameters s0 > 0 and λ0 > 0 and there exists a constant C > 0 depending only on Ω, ω0 , ψ, T0 , on γ defined in (2.5) and on the coefficients ai,j such that for every s > s0 , for every λ > λ0 and for every y ∈ V we have (2.22)

sλ2

Z T0 Z 0

ξ1 e−2sη1 |∇y|2 dxdt + s3 λ4

Z T0 Z

Ω

0

Z T0 Z

e−2sη1 |f |2 dxdt + s3 λ4

≤ C(

0

0

Ω

Ω

Z T0 Z ω

ξ13 e−2sη1 |y|2 dxdt

ξ13 e−2sη1 |y|2 dxdt).

This inequality will turn out to be fundamental. From now on we fix s > s0 and λ > λ0 . As a first consequence of this proposition we have the following result Proposition 2.4 The bilinear form defined by ∀z, z˜ ∈ V 0 , (z, z˜)V 0 =

Z T0 Z

z.˜ z dxdt 0

ω

is a scalar product on V 0 . Remark 2.5 We could have taken a weight ξ13 e−2sη1 in the expression of the above scalar product as we can notice that ∃M > 0, such that ξ13 e−2sη1 ≤ M. This would not lead to real improvements in the sequel and would make the presentation more complicated.

7

The proof of Proposition 2.4 only requires the following unique continuation property for z ∈ V 0 ∀z ∈ V 0 , z = 0 in ω × (0, T0 ) ⇒ z = 0 in Ω × (0, T0 ). This is an extension (in terms of regularity) of the well known result by Mizohata [19] and is for example a consequence of inequality (2.22). If z ∈ V 0 , this corresponds to the case f = 0, and (2.22) says that if z = 0 in ω × (0, T0 ) then the left hand side must be zero which ensures that z = 0 in the whole domain Ω × (0, T0 ). This shows Proposition 2.4. Definition 2.6 We denote by V 0 the (abstract) completion of V 0 with respect to the norm |.|V 0 associated with the above defined scalar product (., .)V 0 and by V the translated space (2.23) V = y0 + V 0. We then have immediately from the completion argument Corollary 2.7 The space V 0 is a Hilbert space for the scalar product (., .)V 0 and V is the associated complete metric space. For every y ∈ V inequality (2.22) still holds true. Remark 2.8 1) The spaces V 0 and V can appear to be very abstract spaces, but in fact, because of inequality (2.22) we know that V is contained in the weighted Sobolev space of L2 functions with the weight ξ13 e−2sη1 with gradients in L2 with the weight ξ1 e−2sη1 such that (2.2) and (2.3) (which now make perfect sense) hold true. Elements of V are therefore solutions of (2.2) and (2.3), also called trajectories of the problem. 2) As the weight degenerates near t = 0, functions of V may not have any value at t = 0 in any sense (we will refer to the value at t = 0 as the initial value as these functions are solutions of an evolution problem) and this is crucial to observe. 3) The weights ξ1 e−2sη1 and ξ13 e−2sη1 are uniformly bounded from below by a positive constant in Ω × [δ, T0 − δ] if δ > 0. Therefore, if y ∈ V , then ∀δ > 0, y ∈ L2 (δ, T ; H01 (Ω)). If now θ ∈ C ∞ [0, T0 ] with δ 0 ≤ θ ≤ 1, θ(t) = 0 if t ∈ [0, ], θ(t) = 1 if t ∈ [δ, T0 ], 2

8

considering the equation satisfied by z = θ.y, multiplying this equation by z and using classical energy estimates for the diffusion convection operator (see also the proof of Theorem 2.9 below), we see that y ∈ C([δ, T0 ]; L2 (Ω)) ∩ L2 (δ, T0 ; H01 (Ω)) and there exists a constant C(δ) > 0 such that, for every y ∈ V 0 (2.24)

|y|2C([δ,T0 ];L2 (Ω))

+

||y||2L2 (δ,T0 ;H 1 (Ω)) 0

≤

C(δ)|y|2V 0

Z T0 Z

= C(δ) 0

|y|2 dxdt,

ω

In particular, for y ∈ V , the value y(T0 ) makes perfect sense in L2 (Ω). We can now state a first result giving stability and exact reconstruction of y(T0 ) with respect to the “measurement” h = y/ω×(0,T0 ) . Theorem 2.9 If Ω is a bounded open subset of IRN of class C 2 , if the coefficients ai,j , bi , cj , and the functions f and h satisfy the previous hypotheses (2.4), (2.5) and (2.7), for any nonempty ω ⊂ Ω, for any T0 > 0 and for any ϕ0 ∈ L2 (Ω), there exists v = v(ϕ0 ) ∈ L2 (0, T0 ; L2 (ω)) such that the solution ϕ of (2.13), (2.14), (2.15) verifies (2.25) ϕ(0) = 0. Taking v(ϕ0 ) of minimal norm among admissible controls, the mapping ϕ0 → v(ϕ0 ) is continuous and (2.26)

∃C > 0, ∀ϕ0 ∈ L2 (Ω), |v(ϕ0 )|L2 (0,T0 ;L2 (ω)) ≤ C|ϕ0 |L2 (Ω) .

We then have, if y ∈ V , which means that h = y/ω×(0,T0 ) ∈ L2 (0, T0 ; L2 (ω)), (2.27) ∀ϕ0 ∈ L2 (Ω),

Z T0 Z

Z

y(T0 )ϕ0 dx = Ω

0

f ϕdxdt −

Z T0 Z

hv(ϕ0 )dxdt. 0

Ω

ω

Moreover, there exists a constant C > 0 depending only on Ω, ω, T0 and the coefficients aij , bi , cj such that (2.28)

|y(T0 )|2L2 (Ω) ≤ C(

Z T0 Z 0

|h|2 dxdt +

ω

Z T0 Z 0

|f |2 dxdt).

Ω

Remark 2.10 Inequality (2.28) is of course independent of the unknown value y(0) which may not exist in fact. It depends only on the right hand side f and the “measurement” h = y/ω×(0,T0 ) . It is a stability inequality. Equality (2.27) enables us to calculate the component of y(T0 ) on ϕ0 for any ϕ0 ∈ L2 (Ω) knowing the “measurement” h = y/ω×(0,T0 ) , the right hand side f and the 9

“control” v(ϕ0 ) which has to be computed. Taking successively for ϕ0 elements of a Hilbert basis of L2 (Ω) we can therefore reconstruct exactly y(T0 ). Of course, when dealing with numerical approximations, we would take ϕ0 in a finite dimensional basis to obtain approximations of y(T0 ). The method has been used and numerical computations have been performed for a large scale ocean model in [9] which give very promising results. Of course the use of reduced basis will be of major importance in using this method for numerical experiments. A first work in this direction has been done in [11] and it gives very positive results. The method allows us also to measure the sensitivity of the reconstruction of y(T0 ) with respect to perturbations in h as will be seen in the Corollary 2.11 below and this is a very important feature in practice. Of course, in real applications, the measuremnts are not provided in an open set but on a finite number of points, but this case seems to be impossible to treat mathematically. Proof of Theorem 2.9. We already know that Carleman estimate (2.22) holds true for every y ∈ V . Let us now take a cut-off function θ ∈ C ∞ [0, T0 ] such that 0 ≤ θ(t) ≤ 1, ∀t ∈ [0, T0 ], T0 θ(t) = 0, ∀t ∈ [0, ], 4 3T0 θ(t) = 1, ∀t ∈ [ , T0 ], 4 and define y˜(x, t) = θ(t).y(x, t). Then from (2.2), (2.3) we see that y˜ satisfies the following problem N N N X X ∂ y˜ ∂ ∂ y˜ ∂ y˜ X ∂ − (aij )+ bi + (cj y˜) = θf + y.θ0 in Ω × (0, T0 ), ∂t i,j=1 ∂xi ∂xj ∂x ∂x i j i=1 j=1

y˜ = 0 on Γ × (0, T0 ) y˜(x, 0) = 0 in Ω. Using now classical energy estimates (multiplying the equation by y˜) we obtain, as θ0 = 0 on [0, T40 ] ∪ [ 3T4 0 , T0 ] (C may denote different constants) |˜ y (T0 )|2L2 (Ω) ≤ C(

Z T0 Z 0

|f |2 dxdt +

Ω

10

Z 3 T0 Z 4 T0 4

Ω

|y|2 dxdt).

But, now y˜(T0 ) = y(T0 ) and from (2.22), due to the fact that on [ T40 , 3T4 0 ] the weight ξ13 e−2sη1 is bounded from below we have |y(T0 )|2L2 (Ω)

≤ C(

Z T0 Z 0

2

|f | dxdt +

Z T0 Z 0

Ω

|y|2 dxdt)

ω

which is exactly the stability inequality (2.28). But when f = 0 this can also be viewed as an observability inequality for (2.2) with f = 0, (2.3) and (2.8) which corresponds to the adjoint problem of the backward control problem (in the ϕ variable) (2.13), (2.14), (2.15). It is now well known (see [8] or [20]) for example) that this observability inequality implies existence of a control v = v(ϕ0 ) such that the solution ϕ of (2.13), (2.14), (2.15) verifies ϕ(0) = 0 in Ω. The same observability inequality gives the continuity of the mapping ϕ0 ∈ L2 (Ω) → v(ϕ0 ) ∈ L2 (0, T0 ; L2 (ω)) when v(ϕ0 ) is taken to be the control of minimal norm among admissible controls. This proves inequality (2.26). It remains to show equality (2.27). If y ∈ V then multiplying equation (2.2) by ϕ and taking into account that ϕ(0) = 0 we immediately obtain (2.27). Now if y ∈ V , taking a Cauchy sequence (y n ) with y n ∈ V and y n → y in V , it is immediate to see from (2.28) that (y n (T0 )) is a Cauchy sequence in L2 (Ω) and y n (T0 ) → y(T0 ) in L2 (Ω). Therefore we can pass to the limit in (2.27) which therefore remains true for y ∈ V . This finishes the proof of Theorem 2.9. The previous recovery method also gives an estimate of the sensitivity to error measurements. We have the following result. Corollary 2.11 We write y(T0 ) for the recovery obtained by the previous method from an exact measurement h and yˆ(T0 ) for the recovery obtained using a measureˆ Then there exists a constant C > 0, independent of h and h ˆ such that ment h. ˆ L2 (0,T ;L2 (ω)) . |y(T0 ) − yˆ(T0 )|L2 (Ω) ≤ C|h − h| 0

(2.29)

Proof of the corollary : first of all we recall that the problem giving v(ϕ0 ) does not depend on the measurement. Then from (2.27) it is immediate to obtain 2

∀ϕ0 ∈ L (Ω),

Z

(y(T0 ) − yˆ(T0 ))ϕ0 dx =

Ω

Z T0 Z 0

ˆ (h − h)v(ϕ 0 )dxdt.

ω

Therefore, taking the supremum over all ϕ0 ∈ L2 (Ω) with |ϕ0 |L2 (Ω) = 1, we obtain ˆ L2 (0,T ;L2 (ω)) |y(T0 ) − yˆ(T0 )|L2 (Ω) ≤ |h − h| 0

11

sup |ϕ0 |L2 (Ω) =1

|v(ϕ0 )|L2 (0,T0 ;L2 (ω))

Because of (2.26), we immediately obtain (2.29) and the proof is complete. The previous result gives an “exact” method for reconstructing y(T0 ) but it relies on the resolution of null controllability problems. Hereafter we give an approximation method which makes use of more classical optimal control problems (other approximations could also be developped) and we prove convergence of these approximations. Let us consider the following optimal control problem for fixed ϕ0 ∈ L2 (Ω). Let ϕ be solution of (2.13), (2.14), (2.15) and for β > 0 let us define (2.30)

Kβ (v) =

1 2β

Z

|ϕ(0)|2 dx +

Ω

1 2

Z T0 Z 0

|v|2 dxdt.

ω

We look for vβ ∈ L2 (0, T0 ; L2 (ω)) such that (2.31)

Kβ (vβ ) =

min

v∈L2 (0,T0 ;L2 (ω))

Kβ (v)

We obtain the following result. Theorem 2.12 1) For every β > 0 there exists a unique solution vβ ∈ L2 (0, T0 ; L2 (ω)) to problem (2.31) and vβ is characterized by the optimality system N N N X X X ∂ϕβ ∂ϕβ ∂ϕβ ∂ ∂ − − cj (aji )− (bi ϕβ ) − = vβ χω in Ω × (0, T0 ), ∂t ∂x ∂x ∂x ∂xj j i i i,j=1 i=1 j=1

ϕβ = 0 on Γ × (0, T0 ), ϕβ (T0 ) = ϕ0 , in Ω N N N X X ∂pβ X ∂pβ ∂pβ ∂ ∂ − (aij )+ bi + (cj pβ ) = 0 in Ω × (0, T0 ), ∂t ∂xi ∂xj ∂xi j=1 ∂xj i,j=1 i=1

pβ = 0 on Γ × (0, T0 ), 1 pβ (0) = ϕβ (0), in Ω β pβ + vβ = 0, in ω × (0, T0 ). 2) When β tends to zero, we have vβ → v¯ in L2 (0, T0 ; L2 (ω)), ϕβ → ϕ¯ in C([0, T0 ]; L2 (Ω)) ∩ L2 (0, T0 ; H01 (Ω)) where v¯ and ϕ¯ satisfy (2.32) −

N N N X X X ∂ ϕ¯ ∂ ∂ ϕ¯ ∂ ∂ ϕ¯ − (aji )− (bi ϕ) ¯ − cj = v¯χω in Ω × (0, T0 ), ∂t i,j=1 ∂xj ∂xi ∂xi ∂xj i=1 j=1

(2.33) ϕ¯ = 0 on Γ × (0, T0 ), (2.34) ϕ(T ¯ 0 ) = ϕ0 , in Ω 12

and (2.35)

ϕ(0) ¯ = 0 in Ω.

Moreover, v¯ = v(ϕ0 ) is the element of minimal norm in L2 (0, T0 ; L2 (ω)) such that (2.32), (2.33), (2.34) and (2.35) occur. Finally, when β → 0, we have Z T0 Z

(2.36) 0

Ω

f ϕβ dxdt −

Z T0 Z 0

ω

h.vβ dxdt →

Z

y(T0 )ϕ0 dx. Ω

Proof. For β > 0, (2.31) is a classical optimal control problem which is known to have a unique solution vβ (see for example [17]) and the characterization given in 1) is standard. We also know from Theorem 2.9 that there exists v such that (2.32), (2.33), (2.34) and (2.35) are satisfied. It is clear that the set of such elements v is a nonempty closed convex set in L2 (0, T0 ; L2 (ω)) so that there exists a unique v¯ ∈ L2 (0, T0 ; L2 (ω)) which minimizes the norm in this set, and we have v¯ = v(ϕ0 ) where v(ϕ0 ) is given by Theorem 2.9. The corresponding solution of (2.32), (2.33), (2.34) will be denoted by ϕ. ¯ Now v¯ is admissible in (2.31) and, as ϕ(0) ¯ = 0, we have for every β > 0 Kβ (vβ ) ≤ Kβ (¯ v) =

1 2

Z T0 Z 0

|¯ v |2 dxdt.

ω

This shows that when β → 0, we have Z T0 Z 0

1 β

ω

Z Ω

|vβ |2 dxdt ≤

|ϕβ (0)|2 dx ≤

Z T0 Z

Z

0 T0

0

|¯ v |2 dxdt,

ω

Z

|¯ v |2 dxdt.

ω

Consequently we can extract a subsequence, still denoted by (vβ ), such that vβ * v˜ in L2 (0, T0 ; L2 (ω)). From standard results on diffusion convection equations (on continuity of the solution with respect to the right hand side, the initial data ϕ0 being fixed) and compactness embeddings we then have ϕβ → ϕ˜ in C([0, T0 ]; L2 (Ω)), where ϕ˜ denotes the solution of (2.32), (2.33), (2.34) and (2.35) corresponding to v˜. Therefore ϕβ (0) → ϕ(0) ˜ in L2 (Ω) and we must have ϕ(0) ˜ = 0. 13

Because of the definition of v¯ we then have Z T0 Z 0

2

|¯ v | dxdt ≤

Z T0 Z 0

ω

|˜ v |2 dxdt.

ω

But on the other hand because of the weak convergence of vβ to v˜ we have Z T0 Z 0

|˜ v |2 dxdt ≤

Z T0 Z 0

ω

|¯ v |2 dxdt,

ω

so that the convergence of vβ to v˜ is strong and v˜ minimizes the norm among the elements v such that (2.32), (2.33), (2.34) and (2.35) are satisfied. By uniqueness of this minimum, we must have v˜ = v¯. Now, in view of the previous results, (2.36) is clear and the proof of Theorem 2.12 is complete. Without any further assumption, it seems impossible to obtain a rate of convergence in the previous approximation. Nevertheless, we will obtain a rate of convergence under a regularity assumption on the “adjoint state” associated to the null controllability problem solved in Theorem 2.9. We first have to introduce this adjoint state. We recall that Z Z (p, q) ∈ V 0 → (p, q)V 0 =

T0

pqdxdt 0

ω

is the scalar product on V 0 . On the other hand the mapping q→

Z

ϕ0 q(T0 )dx Ω

is a linear continuous form on V 0 . Therefore, from Riesz Theorem, there exists a unique p ∈ V 0 such that 0

∀q ∈ V ,

(2.37)

Z T0 Z

Z

pqdxdt = 0

ω

ϕ0 q(T0 )dx. Ω

We now use the results of Theorem 2.12. The adjoint state pβ corresponding to the approximate optimal control problem is an element of V 0 . Moreover, because pβ = −vβ and vβ converges strongly to v¯ in L2 (0, T0 ; L2 (ω)), we see that pβ converges strongly to p¯ in V 0 with p¯ = v¯ in ω × (0, T0 ). Now let us multiply the equation satisfied by ϕβ by q where q ∈ V 0 . Integrating by parts and using the properties of q we obtain Z T0 Z 0

ω

vβ qdxdt = −

Z Ω

Z

ϕβ (T0 )q(T0 )dx + 14

Ω

ϕβ (0)q(0)dx.

Therefore

Z T0 Z 0

ω

Z

pβ qdxdt =

ϕ0 q(T0 )dx −

Ω

Z Ω

ϕβ (0)q(0)dx.

We can now pass to the limit when β → 0 using the convergence of ϕβ (0) to 0 and obtain Z Z T0 Z 0 p¯qdxdt = ϕ0 q(T0 )dx. ∀q ∈ V , 0

V0

ω

Ω

0

As is dense in V this is also valid for every q ∈ V 0 , which says that p¯ is solution of (2.37). By uniqueness of this solution we have p¯ = p. Notice that because of the definiton of V 0 the function (uniquely defined) p¯ may not have any value at t = 0 a priori, but satisfies the following system (2.38)

N N N X X ∂ p¯ ∂ p¯ ∂ p¯ X ∂ ∂ (aij )+ bi + (cj p¯) = 0 in Ω × (0, T0 ), − ∂t i,j=1 ∂xi ∂xj ∂xi j=1 ∂xj i=1

(2.39)

p¯ = 0 on Γ × (0, T0 ),

(2.40)

p¯ + v¯ = 0 in ω × (0, T0 ).

We are now ready to give a result concerning the convergence rates of the approximation under a regularity assumption on the adjoint state p¯. Theorem 2.13 We use the notations of Theorem 2.12. Let us assume that the function p¯, solution of (2.37), satisfies p¯ ∈ C([0, T0 ]; L2 (Ω)).

(2.41) Then we have

(2.42) |ϕβ (0)|L2 (Ω) ≤ 2β|¯ p(0)|L2 (Ω) , 1

(2.43) |vβ − v¯|L2 (0,T0 ;L2 (ω)) ≤ 2β 2 |¯ p(0)|L2 (Ω) , (2.44) |

Z

y(T0 )ϕ0 dx −

Ω

Z T0 Z

Z T0 Z 0

Ω

f ϕβ dxdt +

0

ω

1

h.vβ dxdt| ≤ Cβ 2 |¯ p(0)|L2 (Ω) ,

where C is independent of β and of ϕ0 . Proof. We can write 1 1 T0 |vβ − v¯|2 dxdt + |ϕβ (0)|2 dx = 2 0 2β ω Ω Z Z Z Z Z Z T0 Z 1 T0 1 1 T0 2 2 2 |vβ | dxdt + |ϕβ (0)| dx + |¯ v | dxdt − vβ v¯dxdt = 2 0 2β Ω 2 0 ω ω 0 ω Z

Z

Z

15

T0 1 T0 vβ v¯dxdt ≤ |¯ v |2 dxdt − 2 0 0 ω ω Z Z T0 Z Z 1 T0 Kβ (¯ v) + vβ v¯dxdt = |¯ v |2 dxdt − 2 0 0 ω ω

Z

Z

Z

Z

Kβ (vβ ) +

Z T0 Z 0

|¯ v |2 dxdt −

0

ω

Z T0 Z

Z T0 Z ω

vβ v¯dxdt =

0

ω

(¯ v − vβ )¯ v dxdt.

The function φ = (ϕ¯ − ϕβ ) satisfies the following system. N N N X X X ∂φ ∂φ ∂φ ∂ ∂ − − (aji )− (bi φ) − cj = (¯ v − vβ )χω in Ω × (0, T0 ), ∂t i,j=1 ∂xj ∂xi ∂x ∂x i j i=1 j=1

φ = 0 on Γ × (0, T0 ), φ(T0 ) = 0, in Ω φ(0) = −ϕβ (0) in Ω. Multiplying this equation by p¯, using (2.40) and integrating by parts we obtain Z T0 Z 0

ω

(¯ v − vβ )¯ v dxdt =

Z Ω

ϕβ (0)¯ p(0)dx.

Therefore we have 1 2

Z T0 Z 0

1 |vβ − v¯| dxdt + 2β ω 2

Z Ω

|ϕβ (0)|2 dx ≤ |ϕβ (0)|L2 (Ω) |¯ p(0)|L2 (Ω) .

This gives immediately (2.42) and (2.43). The estimate on (ϕ¯ − ϕβ ) is of the same order as the one on (¯ v − vβ ) and this implies (2.44) which finishes the proof of Theorem 2.13. R

Remark 2.14 For each ϕ0 , in order to find an approximation of Ω y(T0 )ϕ0 dx, we have to solve a classical optimal control problem for the adjoint system. We have to notice that for different elements ϕ0 the optimal control problems to be solved are essentially the same and they only differ in the initial data in (2.32). This is particularly important for numerical approximation because all the linear systems corresponding to different elements ϕ0 have the same matrices.

2.2

Linearized Navier Stokes equations

We consider here in dimension N = 3 the Navier Stokes equations linearized around a velocity y¯ such that (2.45) y¯ ∈ L∞ (0, T0 ; W 1,∞ (Ω)),

∂ y¯ 6 ∈ L2 (0, T0 ; W 1,σ (Ω)), σ > , div (¯ y ) = 0, ∂t 5 16

namely

(2.47)

∂y − µ∆y + (¯ y .∇)y + (y.∇)¯ y + ∇p = f in Ω × (0, T0 ), ∂t div y = 0 in Ω × (0, T0 ),

(2.48)

y = 0 on Γ × (0, T0 ).

(2.46)

where f ∈ L2 (0, T0 ; (L2 (Ω))3 ) and µ > 0. Here again we do not impose any initial condition on y or we don’t know y(0). We suppose that we know a measurement of the solution on a subdomain (2.49)

y/ω×(0,T0 ) = h

where ω is a nonempty open set contained in Ω, and h ∈ L2 (0, T0 ; (L2 (ω))3 ). Remark 2.15 1) We could also consider the case where normal stresses (σ.ν)i = −pνi + µ

3 X

Dij (y)νj , i = 1, .., 3,

j=1 ∂y

∂yi with Dij (y) = 21 ( ∂x + ∂xji ) are known on Γ0 × (0, T0 ) where Γ0 is a nonempty j relatively open set of the boundary Γ. 2) We have taken here the case of measurements on y only but we could also have in addition local measurements on the pressure p. This would correspond to a simpler situation. On the other hand, the case of measurements only on the pressure turns out to be impossible to treat.

In the case of classical variational data assimilation we take the initial value as a control variable (2.50) y(0) = y0 in Ω where (2.51)

y0 ∈ H = {z ∈ (L2 (Ω))3 , div z = 0, z.ν = 0 on Γ}.

We know (cf. [18] or [22]) that for every y0 ∈ H, there exists a unique solution y = y[y0 ] of (2.46), (2.47), (2.48), (2.50) with y[y0 ] ∈ C([0, T0 ]; H) ∩ L2 (0, T0 ; (H01 (Ω))3 ). We then want to find y0 such that the error between the actual measure h and the value of the solution y = y[y0 ] of (2.46), (2.47), (2.48), (2.50) on the subdomain ω × (0, T0 ) achieves its minimum. If we define (2.52)

J(y0 ) =

1 2

Z T0 Z 0

|y[y0 ] − h|2 dxdt

ω

17

we consider the optimal control problem Find y¯0 ∈ H such that (2.53)

J(¯ y0 ) = min J(y0 ). y0 ∈H

Again this problem is ill-posed and we must add a Tychonov regularization term by considering for α > 0 Jα (y0 ) =

(2.54)

1 2

Z T0 Z 0

|y[y0 ] − h|2 dxdt +

ω

α |y0 |2H . 2

We now solve the regularized optimal control problem Find yα ∈ H such that (2.55)

Jα (yα ) = min Jα (y0 ). y0 ∈H

This problem is classical and has a unique solution yα ∈ H and the questions are again to give a meaning to this solution, to understand what happens when α → 0 and to estimate the sensitivity to errors in the measurements h. Following the same ideas as in the previous section we will present a non standard approach using controllability techniques. Let us define W = {y, y solution of (2.46), (2.47), (2.48), (2.50), y0 ∈ H}. If we call y 0 the element of W such that y0 = 0 we have W = y0 + W 0 where W 0 is a vector space. We want to give a Carleman estimate for elements of W. We use here the results of [12], [13] and [7] concerning Navier Stokes equations and the Carleman estimate in this case is much more difficult to obtain than for diffusion convection equations. We take the same principal weight ψ as in the previous section but this time we need to take functions ξ4 and η4 defined in (2.18) and (2.19). Moreover we must define ξb4 (t) = max ξ4 (x, t), x∈Ω

ηb4 (t) = min η4 (x, t), x∈Ω

(2.56)

η4∗ (t) b = θ(t)

η4 b sλe−sb ξ4 ,

= max η4 (x, t), x∈Ω ∗

15/4

η4 +sη4 b θ(t) = s15/4 e−2sb ξ4

We then have 18

,

Proposition 2.16 There exist s1 > 0 and λ1 > 0, and there exists a constant C depending on Ω, ω, ψ, µ, T0 and y¯ (in the spaces corresponding to hypothesis (2.45))such that for every s > s1 , for every λ > λ1 and for every y ∈ W we have 3 4

Z T0 Z

s λ

0

Ω

e−2sη4 ξ43 |y|2 dx dt

+ sλ

Z T0 Z 0

16 40

Ω

e−2sη4 ξ4 |∇y|2 dx dt

Ω

η4 +2sη ∗ b15/2 ξ4 |f |2 dx dt e−4sb

!

Z T0 Z

+s λ

0

Z T0 Z 0

≤ C s15/2 λ20

(2.57)

2

ω

η4 +6sη ∗ b16 ξ4 |y|2 dx dt e−8sb

From now on, we fix s > s1 and λ > λ1 and repeat the arguments of Proposition 2.4. Using (2.57) we can show the following unique continuation property : ∀z ∈ W 0 z = 0 on ω × (0, T0 ) ⇒ z = 0 in Ω × (0, T0 ). Therefore the bilinear form defined by ∀z, z˜ ∈ W 0 , (z, z˜)W 0 =

Z T0 Z

z.˜ z dxdt 0

ω

is a scalar product on W 0 and we can set the following definition. Definition 2.17 We denote by W 0 the (abstract) completion of W 0 with respect to the norm |.|W 0 associated with the above defined scalar product (., .)W 0 and by W the translated space (2.58) W = y0 + W 0. Then W 0 is a Hilbert space for the scalar product (., .)W 0 and for every y ∈ W , inequality (2.57) still holds true. Remark 2.18 Because of inequality (2.57) and classical energy estimates for the linearized Navier-Stokes operator, we can easily show that for every δ > 0, any element y of W satisfies y ∈ C([δ, T0 ]; H) ∩ L2 (δ, T0 ; (H01 (Ω))3 ). In particular, the value y(T0 ) makes perfect sense in H. But, a priori, the function y may have no value at t = 0 (initial value) in any sense. We now consider the backward adjoint controlled problem for a distributed control v ∈ L2 (0, T0 ; (L2 (ω))3 )

(2.60)

∂ϕ − µ∆ϕ − (¯ y .∇)ϕ + (∇¯ y )ϕ + ∇π = v.χω in Ω × (0, T0 ), ∂t div ϕ = 0 in Ω × (0, T0 ),

(2.61)

ϕ = 0 on Γ × (0, T0 ),

(2.62)

ϕ(T0 ) = ϕ0 ,

(2.59)

−

19

where ϕ0 ∈ H and H is defined in (2.51). Using the same arguments as in Theorem 2.9 we obtain the following result of stability and reconstruction of y(T0 ) Theorem 2.19 Under the previous hypotheses, for every ω ⊂ Ω, for every T0 > 0 and for every ϕ0 ∈ H, there exists v = v(ϕ0 ) ∈ L2 (0, T0 ; (L2 (ω))3 ) such that the solution ϕ of (2.59), (2.60), (2.61), (2.62) satisfies (2.63)

ϕ(0) = 0.

We will choose v(ϕ0 ) of minimal norm among the controls such that (2.63) is satisfied and then the mapping ϕ0 → v(ϕ0 ) is continuous which says that (2.64)

∃C > 0, ∀ϕ0 ∈ H, |v(ϕ0 )|L2 (0,T0 ;(L2 (ω))3 ) ≤ C|ϕ0 |H .

We then have, if y = h on ω × (0, T0 ), (2.65)

∀ϕ0 ∈ H, (y(T0 ), ϕ0 )H =

Z T0 Z 0

f.ϕdxdt −

Z T0 Z

h.v(ϕ0 )dxdt. ω

0

Ω

Moreover, there exists a constant C > 0 depending only on Ω, ω, T0 , µ and y¯ (in the spaces corresponding to hypothesis (2.45)) such that (2.66)

|y(T0 )|2H ≤ C(

Z T0 Z 0

|f |2 dxdt +

Ω

Z T0 Z 0

|h|2 dxdt).

ω

We also obtain in the same way as in Corollary 2.11 a result measuring the sensitivity of the recovered state with respect to errors in the measurements. Corollary 2.20 We write y(T0 ) for the recovery obtained by the previous method ˆ from a measurement h and yˆ(T0 ) for the recovery obtained using a measurement h. ˆ such that Then there exists a constant C > 0, independent of h and h (2.67)

ˆ L2 (0,T ;(L2 (ω))3 ) . |y(T0 ) − yˆ(T0 )|H ≤ C|h − h| 0

Here also we can consider an optimal control problem which will provide an approximation for (y(T0 ), ϕ0 )H . For v ∈ L2 (0, T0 ; (L2 (ω))3 ) let ϕ be solution of (2.59), (2.60), (2.61), (2.62) and for β > 0 let us define a cost function Kβ (v) by (2.68)

Kβ (v) =

1 1 |ϕ(0)|2H + 2β 2

Z T0 Z 0

|v|2 dxdt

ω

We look for vβ ∈ L2 (0, T0 ; (L2 (ω))3 ) such that (2.69)

Jβ (vβ ) =

min

v∈L2 (0,T0 ;(L2 (ω))3 )

Jβ (v).

For fixed β > 0 this last problem is a classical optimal control problem. We then obtain, following the same argument as for Theorem 2.12, 20

Theorem 2.21 1) For every β > 0, there exists a unique solution vβ to (2.69) and vβ is characterized by the following optimality system. ∂ϕβ − µ∆ϕβ − (¯ y .∇)ϕβ + (∇¯ y )ϕβ + ∇π = vβ .χω in Ω × (0, T0 ), ∂t div ϕβ = 0 in Ω × (0, T0 ), −

ϕβ = 0 on Γ × (0, T0 ), ϕβ (T0 ) = ϕ0 , ∂rβ − µ∆rβ + (¯ y .∇)rβ + (rβ .∇)¯ y + ∇ρ = 0 in Ω × (0, T0 ), ∂t div rβ = 0 in Ω × (0, T0 ), rβ = 0 on Γ × (0, T0 ), 1 rβ (0) = ϕβ (0), β rβ + vβ = 0, in ω × (0, T0 ). 2) When β tends to zero , vβ → v¯ in L2 (0, T0 ; (L2 (ω))3 ), ϕβ → ϕ¯ in C([0, T0 ]; H) where ϕ¯ and v¯ satisfy (2.59)-(2.62) and (2.63). Moreover v¯ is the element with minimal norm such that (2.59)-(2.62) and (2.63) are satisfied. In addition when β → 0 we have Z T0 Z

(2.70) 0

Ω

f.ϕβ dxdt −

Z T0 Z 0

ω

y.vβ dxdt → (y(T0 ), ϕ0 )H .

We can obtain an adjont state corresponding to the nul controllability problem solved in Theorem 2.19. The mapping q → (ϕ0 , q(T0 ))H is a continuous linear form on W 0 . Therefore, from Riesz Theorem, there exists a unique function r¯ ∈ W 0 such that (2.71) ∀q ∈ W 0 , (¯ r, q)W 0 = (ϕ0 , q(T0 ))H . It is easy to show, as in the case of diffusion convection equations, that rβ converges to r¯ in W 0 and that r¯ + v¯ = 0 on ω × (0, T0 ). We only know a priori that r¯ is element of W 0 so that it may not have any initial value at t = 0. But under an additional regularity assumption on r¯ we can obtain an estimate on the convergence rate of the previous approximate procedure. Theorem 2.22 We use the notations of Theorem 2.21. Let us assume that the function r¯, solution of (2.71), satisfies (2.72)

r¯ ∈ C([0, T0 ]; H). 21

Then we have (2.73)

|ϕβ (0)|H ≤ 2β|¯ r(0)|H ,

(2.74)

|vβ − v¯|L2 (0,T0 ;(L2 (ω))3 ) ≤ 2β 2 |¯ r(0)|H ,

(2.75)

1

|(y(T0 ), ϕ0 )H −

Z T0 Z

Z T0 Z 0

Ω

f.ϕβ dxdt +

0

ω

1

h.vβ dxdt| ≤ Cβ 2 |¯ r(0)|H ,

where C is independent of β and of ϕ0 . The proof uses exactly the same arguments as those used for Theorem 2.13.

3

New results on Tychonov regularization

We are going to give an existence result for a non classical optimal control problem and this will enable us to explain the reason for which in practical situations, with natural hypotheses, Tychonov regularization works in the sense that the corresponding solution converges when the regularization parameter α tends to zero. Let us for the moment go back to the case of diffusion convection equations (we keep the same notations) as the other cases can be treated exactly in the same way. We recall that a function y in V is a solution of (2.2) and (2.3). But y may not have a value at t = 0 (which will be refered to as an initial value) in any sense. Let us define for y ∈ V Z Z (3.76)

T0

˜ = J(y)

0

|y − h|2 dxdt.

ω

˜ Of course we notice that if y has an initial value y0 then J(y) = J(y0 ) but it is no longer the case in general. It is essential to understand that we consider the same functional value but defined on a different argument. Notice also that J˜ is perfectly R T0 R defined for y ∈ V as we know that for y ∈ V , we have 0 ω |y|2 dxdt < +∞. It is now immediate to obtain the following result. Theorem 3.1 There exists a unique element y˜ ∈ V solution of the following minimization problem

(3.77)

Find y˜ ∈ V, such that ˜ y ) = min J(y). ˜ J(˜ y∈V

Moreover if h, h0 ∈ L2 (0, T0 ; L2 (ω)) and the corresponding solutions are called y˜ and y˜0 then we have Z Z (3.78)

|˜ y − y˜0 |2V 0 ≤

T0

0

|h − h0 |2 dxdt,

ω

22

which implies that for every δ > 0, there exists a constant C(δ) > 0 such that |˜ y − y˜0 |2C([δ,T0 ];L2 (Ω)) + ||˜ y − y˜0 ||2L2 (δ,T0 ;H 1 (Ω)) ≤ C(δ) 0

Z T0 Z 0

|h − h0 |2 dxdt,

ω

Proof. The proof of existence and of the stability inequality is elementary as V is a Hilbert space for the norm Z Z T0

1

|y|2 dxdt) 2 .

y→(

0

ω

Uniqueness also follows immediately from the unique continuation property which is valid on V 0 . The last estimate comes from (2.24). Of course the previous result is immediate once we know all the technicalities which are included in the Carleman estimates. Remark 3.2 Therefore, a problem which was ill-posed if we minimize with respect to the initial value has become well-posed when minimizing with respect to the trajectory in V . We also have an estimate of the sensitivity to errors in the measurements h. This estimate is given in the V -norm but thanks to (2.22) and (2.24) it is also valid in the weighted Sobolev spaces which occur in Carleman estimates and in classical Sobolev spaces away from t = 0. Now let us make a regularity hypothesis on the minimizer y˜, namely that it has an initial value in the sense that (3.79)

y˜ ∈ C([0, T0 ]; L2 (Ω)), y˜(0) = y˜0 in Ω.

This regularity hypothesis is quite natural in practical (realistic) situations, even if this depends strongly on the (real) measurements which we are dealing with and which are not usually the true value of a solution on the subdomain ω × (0, T0 ). Then we have the following convergence theorem. Theorem 3.3 Let us suppose that (3.79) is true. Then we have ˜ y) J(˜ y0 ) = J(˜ and y˜0 is solution to the optimal control problem (2.10). We also have the following estimate Z T0 Z

(3.80) 0

ω

|˜ y − y α |2 dxdt ≤ α|˜ y0 |L2 (Ω) .

Moreover, when α → 0, the solution yα of the Tychonov regularized optimal control problem converges strongly in L2 (Ω) to y˜0 . 23

Proof. Let us call y α the solution of (2.2), (2.3) and (2.8) with y α (0) = yα . We have Jα (yα ) ≤ Jα (˜ y0 ) and ˜ y ) ≤ J(y ˜ α ). J(˜ Therefore 1 2

Z T0 Z 0

α 1 T0 α |y − h| dxdt + |yα |2L2 (Ω) ≤ |˜ y − h|2 dxdt + |˜ y0 |2L2 (Ω) 2 2 0 2 ω ω Z Z 1 T0 α |y α − h|2 dxdt + |˜ ≤ y0 |2L2 (Ω) . 2 0 2 ω α

Z

Z

2

As a consequence we have y0 |2L2 (Ω) . ∀α > 0, |yα |2L2 (Ω) ≤ |˜ After extraction of a subsequence (still denoted by yα ), we can suppose that yα * yˆ0 in L2 (Ω) weakly and if yˆ denotes the solution of (2.2), (2.3) and (2.8) with yˆ(0) = yˆ0 we have y α → yˆ in L2 (0, T0 ; L2 (Ω)) and also in various topologies. Now we have α |yα |2L2 (Ω) → 0, when α → 0 2 so that necessarily, when α → 0, 1 2

Z T0 Z 0

1 |y − h| dxdt → 2 ω α

2

Z T0 Z 0

|˜ y − h|2 dxdt.

ω

Therefore we must have 1 2

Z T0 Z 0

1 |ˆ y − h| dxdt = 2 ω 2

Z T0 Z 0

|˜ y − h|2 dxdt.

ω

From uniqueness in problem (3.77), we see that necessarily we have yˆ = y˜ in Ω × (0, T0 ),

24

so that yˆ0 = y˜0 . We now know that yα * y˜0 in L2 (Ω) weakly and |yα |2L2 (Ω) ≤ |˜ y0 |2L2 (Ω) . This implies strong convergence in L2 (Ω) of yα towards y˜0 . As y˜ is the minimizer of J˜ we see from the Euler-Lagrange equation associated to this minimization problem that ∀z ∈ V,

(3.81)

Z T0 Z 0

(˜ y − h).(˜ y − z)dxdt = 0.

ω

Now a simple calculation gives Z T0 Z 0

|˜ y − y α |2 dxdt + α|˜ y0 − yα |2L2 (Ω) =

ω

Z T0 Z 0

|y α − h|2 dxdt + α|yα |2L2 (Ω) +

ω

α|˜ y0 |2L2 (Ω) − 2 Z T0 Z

2 0

ω

Z T0 Z 0

ω

Z T0 Z 0

|˜ y − h|2 dxdt +

ω

(˜ y − h).(y α − h)dxdt − 2α(˜ y0 , yα )L2 (Ω) ≤

|˜ y − h|2 dxdt − 2

Z T0 Z 0

(˜ y − h).(y α − h)dxdt +

ω

y0 , yα )L2 (Ω) ≤ 2α|˜ y0 |2L2 (Ω) − 2α(˜ Z T0 Z

2 0

ω

(˜ y − h).(˜ y − y α )dxdt + 2α(˜ y0 , y˜0 − yα )L2 (Ω) .

But as y α ∈ V , because of (3.81) we obtain Z T0 Z

(3.82) 0

ω

y0 , y˜0 − yα )L2 (Ω) . |˜ y − y α |2 dxdt + α|˜ y0 − yα |2L2 (Ω) ≤ 2α(˜

This gives immediately (3.80) and the proof of Theorem 3.3 is now complete. Remark 3.4 1) Our situation is quite different from the one which is considered classically, for example in [10], [5] or [4]. These authors make an a priori assumption, which is usually written in an abstract form, which essentially suppose that there exists a solution to the minimization problem for the functional J. Here, we prove the existence and uniqueness of a minimizer for J˜ (without additional hypothesis) and, afterwards, we make a regularity assumption on this minimizer. 25

2) We obtain a rate of convergence in the V -distance but without additional hypotheses this does not give any rate of convergence for the initial values (˜ y0 − yα ). 3) In the same way, the last theorem does not say anything about the behavior of solutions yα with very small α, when we have perturbations in the measurements h. Even if the hypothesis (3.79) seems “natural” in practical situations it is really unnatural to assume that this initial value, that we assume to exist, would be continuous with respect to h. This is exactly what is missing in the information given by Carleman estimates. In order to give an estimate for the rate of convergence for the initial values in the Tychonov regularization method, we will follow a method similar to the one used in [10], [5] and [4] but with a different hypothesis. We willl make an assumption on y˜0 , which will be made precise below, but which roughly speaking says that y˜0 is on a controlled trajectory of the adjoint operator with control acting everywhere in the domain, except in the neighborhood of t = 0 where it can only act on ω. Theorem 3.5 Let us assume in addition to (3.79), that there exist q0 ∈ L2 (Ω), es˜η1 w ∈ L2 (0, T0 ; L2 (ω)) and g such that 3 g ∈ L2 (0, T0 ; L2 (Ω)) (this will be the case ξ˜12 2 2 if, for example, g ∈ L (0, T0 ; L (Ω)) and g = 0 in a neighborhood of t = 0) such that the solution q of the following equation (3.83)

−

N N N X X X ∂ ∂ ∂q ∂q ∂q − cj (aji )− (bi q) − = g + w.χω ∂t i,j=1 ∂xj ∂xi ∂xi ∂xj i=1 j=1

in Ω × (0, T0 ), (3.84)

q = 0 on Γ × (0, T0 ),

(3.85)

q(T0 ) = q0 , in Ω

satisfies (3.86)

q(0) = y˜0 .

Then there exists a constant C > 0 depending on g, w and q0 such that Z T0 Z

(3.87) 0

(3.88)

|˜ y − y α |2 dxdt ≤ Cα2

ω

|˜ y0 − yα |2L2 (Ω) ≤ Cα.

Remark 3.6 Due to the null controllability property for the adjoint operator with control acting only on ω, it can be shown that the hypothesis made in Theorem 3.5 is equivalent to the same hypothesis with g = 0 and q0 = 0. 26

Proof. From (3.82) we know that Z T0 Z 0

ω

|˜ y − y α |2 dxdt + α|˜ y0 − yα |2L2 (Ω) ≤ 2α(˜ y0 , y˜0 − yα )L2 (Ω) .

Let us multiply equation (3.83) by (˜ y − y α ) and integrate by parts. We obtain, using the equation satisfied by (˜ y − y α ), (˜ y0 , y˜0 − yα )L2 (Ω) = (q0 , y˜(T0 ) − y α (T0 ))L2 (Ω) + Z T0 Z 0

ω

0

g(˜ y − y α )dxdt +

Ω

w(˜ y − y α )dxdt ≤ |q0 |L2 (Ω) |˜ y (T0 ) − y α (T0 )|L2 (Ω) +

Z T0 Z 2s˜η1 e

(

0

Z T0 Z

Ω

ξ˜13

1 2

2

Z T0 Z

|g| dxdt) (

0

Z T0 Z

( 0

Ω

1 ξ˜13 e−2s˜η1 |˜ y − y α |2 dxdt) 2 +

Z T0 Z

1 2

2

1

|˜ y − y α |2 dxdt) 2 .

|w| dxdt) (

0

ω

ω

But from the Carleman estimate (2.22) and the energy estimate (2.24) applied to y˜ − y α we know that |˜ y (T0 ) − y α (T0 )|2L2 (Ω) +

Z T0 Z 0

Ω

ξ˜13 e−2s˜η1 |˜ y − y α |2 dxdt ≤ C

Z T0 Z 0

|˜ y − y α |2 dxdt.

ω

Therefore we obtain with a different constant C depending on g, w and q0 Z T0 Z 0

α 2

|˜ y − y | dxdt + α|˜ y0 −

ω

yα |2L2 (Ω)

Z T0 Z

≤ Cα(

0

1

|˜ y − y α |2 dxdt) 2

ω

and this gives immediately the result of Theorem 3.5. Of course we obtain completeley similar results for the case of linearized Navier Stokes equations without any additional difficulty. Acknowledgements The author wants to thank the referees for their useful comments and suggestions.

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