SPARSE INITIAL DATA IDENTIFICATION FOR PARABOLIC PDE AND ITS FINITE ELEMENT APPROXIMATIONS

We address the problem of inverse source identification for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a finite number of pointwise locations. Moreover, we discuss the finite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.

1. Introduction. In this paper we address the issue of the backward resolution of parabolic equations formulated as a control problem, the control being the initial datum aiming to steer the solution to a given final value in a given time horizon.
We adopt the point of view of approximate controllability as in [13] where this problem was formulated and solved for initial data in L p with 1 < p ≤ ∞. In the present paper we consider the case p = 1 or, to be more precise, the one in which the initial data to be optimized are Borel measures. The range of the semigroup departing from this class of initial data is dense, since that is the case when p > 1 too.
Here we focus on the characterization of the initial measures of minimal norm ensuring that the final target is reached in an approximate sense. As we shall see, this can be done minimizing a suitable quadratic functional on the solutions of the adjoint system, following the arguments in [10].
Our analysis leads to sparsity results showing that the optimal initial measures have as support a set of zero Lebesgue measure. This also leads to effective inversion results within the class of initial data constituted by finite combinations of Dirac measures.
Our results apply in a broad class of parabolic equations, possibly semilinear with globally Lipschitz nonlinearities, as in [13]. But we shall focus on the classical heat equation with constant coefficients where the initial datum u, that plays the role of the control, is assumed to be a Borel measure, Ω ⊂ R n is an open connected bounded set and Γ is the boundary of Ω, that we will assume to be Lipschitz.
We view u as the control that we would like to choose such that the associated state y u at time T , y u (T ), is in the L 2 (Ω)-ballB ε (y d ), where y d represents the desired final state and > 0 the admissible distance to the target.
It is well known that for any > 0 it is possible to find u ∈ L 2 (Ω) such that y u (T ) ∈B ε (y d ). This is a consequence of the fact that the range of the semigroup generated by the heat equation is dense (see [13]), which is equivalent to the well known and classical backward uniqueness property for the heat equation, see [24]. In fact the same holds when the control u has its support in a subset ω of Ω of positive measure. The interested reader is referred to [14] where this issue is investigated in a more general frame in order to obtain sharp bounds on the cost of approximate control.
We are interested on analyzing the structure of the initial data u of minimal norm. In the L 2 -setting this can be done by considering the following minimization problem min yu(T )∈Bε(y d ) It can be checked that this problem has a unique solution that is given bȳ whereφ is the unique solution of the adjoint equation Above,ȳ denotes the state associated to the optimal initial datumū. In this way, the obtained initial datum, being the trace at t = 0 of a solution of the adjoint system in the time interval 0 ≤ t ≤ T , is smooth but non-zero at almost every point of Ω. This makes these initial data to be of little practical use in applications where one looks for initial data with small support.
This issue was recently addressed in [23], the goal being to develop efficient numerical algorithms to compute initial data constituted by a finite combination of Dirac measures. This was achieved by means of a fast Bregman iterative algorithm for 1 optimization. In that frame, of course, identifying the initial datum consists in determining a finite number of points for the support of the measure and the weight given to each of them.
In the present paper we develop the theory showing that sparse initial data exist and are characterized by means of a minimization principle over the class of solutions of the adjoint system. The proof requires two ingredients that are by now well known in the literature. On one hand, as mentioned above, the backward uniqueness for parabolic problems and, on the other, analyticity properties of solutions of the heat equation and, more precisely, the analyticity of solutions in Ω with respect to the space variable at the final time.
We will develop this program following closely the previous work [10], where the control was assumed to be a space-time dependent measure, acting as an exterior source. There it was shown that, by replacing the L 2 -norm of the control, by its measure-norm turns out to be an efficient way to obtain sparse optimal controls with support in small regions.
To be more precise, we shall consider the problem above but by replacing in (2) the L 2 -norm of the initial datum u, that plays the role of the control, by its total measure.
The main difficulty in the problem under consideration is the very strong irreversibility of the heat equation. The backward heat equation is ill-posed and, due to parabolic regularizing effects, the range of the generated semigroup at time t = T is only constituted by very smooth functions. Thus, of course, it is impossible to reach exactly any given target in L 2 (Ω). On the other hand, the range of the semigroup is dense and accordingly, the target can be approximated at any distance ε. This ensures the existence of solutions of the optimal control problem above. Note, however, that this existence result requires the regularizing effect ensuring that solutions of the heat equation departing from a measure belongs to L 2 at the final time. As we shall see, at the level of the adjoint equation this is reflected by the fact that the solutions of the adjoint system departing (at time t = T ) from an L 2 -datum, belong to the space of continuous functions at the initial time t = 0.
Using this adjoint methodology we shall prove the existence and uniqueness of the optimal measure.
The second issue we discuss is the sparsity of the obtained optimal initial measures. As we shall see, these initial measures, by duality, are supported on the set where the adjoint solution at time t = 0 reaches its L ∞ -norm that, by the space analyticity turns out to be a set of null Lebesgue measure.
As proved in [4], a measure in Ω can be efficiently approximated by a combination of Dirac measures. As a consequence, we deduce that the approximate reachability can be achieved by activating the initial datum in some finite number of pointwise locations at the time t = 0. These pointwise locations are all of them placed in a very small region of Ω. Even more, in the one dimensional case, we prove that the optimal measure is a finite combination of Dirac measures, which provides a rigorous justification of the existence of the objects that are numerically approximated in [23].
Moreover, we provide a systematic approach to the discretization of the sparse optimal control problem under consideration with finite elements. To this end we discretize the state equation with a discontinuous Galerkin method in time, which is here a variant of the implicit Euler scheme, and using (conforming) linear finite elements in space. The discretiztaion of the optimal control problem is done in the spirit of [4]. For the resulting finite dimensional optimization problem we provide a convergence result for discretization parameters tending to zero, see Theorem 4.10 for details.
Let us briefly comment on some other related papers. Sparse optimal control problems in measure spaces are analyzed in [11,26,7] for elliptic and in [5,21] for parabolic equations, where in both last papers also numerical analysis of spacetime discretizations is performed, see also [15,22] for related works on numerical analysis of pointwise control. Sparse optimal control problems with controls which are functions instead of measures are considered, e.g., in [6,9,19,30,32].
The plan of the paper is as follows. In the next section, we formulate the control problem in a precise way and, by means of backward uniqueness and standard regularizing effects for the heat equation (from L 2 into C 0 ) we infer that it has a unique solution. Later, in section §3, we consider the adjoint system and formulate the dual problem. In Section §4 we present a discrete version of the optimal control problem under consideration and prove the convergence as the discretization parameters tend to zero. Some final remarks are given in §5.

2.
The control problem: Main results. In this paper, we consider the following control problem where y u is the solution of the equation (1) associated to u, and M (Ω) = C 0 (Ω) * denotes the Banach space of real and regular Borel measures in Ω, C 0 (Ω) being the space of continuous functions inΩ vanishing on Γ. In this space, the norm is defined by with |u| being the total variation measure associated to u; see, for instance, [28,Chapter 6]. The function y d is fixed in L 2 (Ω) and ε > 0 is given. To avoid trivial situations, we assume that y d L 2 (Ω) > ε. The case y d L 2 (Ω) ≤ ε obviously leads toū = 0 as the unique solution of the problem (P).
This allows to introduce the spaces Φ = ϕ ∈ L 2 (0, T ; H 1 0 (Ω)) : Using the space Φ T we define a solution concept for the state equation (1).
Definition 2.1. We say that a function y ∈ L 1 (Q) is a solution of (1) if the following identity holds As a consequence we have the following: There exists a unique solution y of (1). Moreover, y belongs to the space L r (0, T ; W 1,p 0 (Ω)) for all p, r ∈ [1, 2) with (2/r) + (n/p) > n + 1, and the following estimate holds Furthermore, y satisfies Proof. For the proof of the first part of the Lemma, the reader is referred to [8,Theorem 2.2]. We only need to prove the estimate for y(T ) and (6). The identity (6) is obvious for regular data u ∈ L 2 (Ω). Then, it is enough to take a sequence {u k } ∞ k=1 ⊂ L 2 (Ω) such that u k * u in M (Ω), to write the equation for (u k , y k ) and to pass to the limit as k → ∞. In this limit, we only need to pay attention to the fact that ϕ(0) ∈ C 0 (Ω) due to the regularizing effect of the heat equation. To prove the estimate for y(T ) let us define ϕ ∈ L 2 (0, T ; in Ω Classical results on the gain of integrability and smoothing for the heat equation imply that Then, using (6) we get which implies (6).

Remark 1.
Choosing r = 1 in this lemma, we get y ∈ L 1 (0, T ; W 1,p (Ω)) for all 1 ≤ p < n n−1 . This implies y t ∈ L 1 (0, T ; W −1,p (Ω)) and therefore we conclude The next lemma makes again use of the smoothing property of the heat equations.
and y denote the corresponding states, solutions of (1), then the convergence y k (T ) → y(T ) holds strongly in L 2 (Ω).
Proof. We consider the linear mapping A t : M (Ω) → L 2 (Ω), which maps a control u ∈ M (Ω) to the solution y(t) of the (1) at time 0 < t ≤ T . By the previous lemma this mapping is linear and continuous. Moreover we consider an operator B t1,t2 : L 2 (Ω) → L 2 (Ω) mapping an initial condition z 1 ∈ L 2 (Ω) of the following equation to the solution For a given z 1 we consider a sequence {z 1,m } ⊂ H 1 0 (Ω) with z 1,m → z 1 in L 2 (Ω). Let z m be the solution of (7) with z 1,m instead of z 1 . By standard regularity we have that z m ∈ H 1 (t 1 , t 2 ; L 2 (Ω)) ∩ C([t 1 , t 2 ]; H 1 0 (Ω)). Then we test the weak formulation for z m with (t − t 1 ) ∂zm ∂t and obtain This results in The last term can be estimated as (8) resulting in . By the linearity of (7) we have for all m, m ∈ N and therefore {z m (t 2 )} is a Cauchy sequence in H 1 0 (Ω) converging strongly to z(t 2 ). This implies z(t 2 ) ∈ H 1 0 (Ω) and the well-known smoothing estimate of the heat equation Due to the compact embedding of H 1 0 (Ω) into L 2 (Ω) the operator B t1,t2 : L 2 (Ω) → L 2 (Ω) is compact. Moreover, there holds for any 0 < t 1 < T . Therefore, the operator A T : M (Ω) → L 2 (Ω) is also compact. This implies the statement of the lemma. Now we prove the existence and uniqueness of solution of (P). Let us observe that the uniqueness is proved despite that the cost functional J is not strictly convex.
Proof. First, from the approximate controllability properties of the heat equation, we know that the set of feasible controls is not empty; see [13].
The existence of a solution can be easily proved by taking a minimizing sequence and using the compactness the mapping of u ∈ M (Ω) → y u (T ) ∈ L 2 (Ω) established in Lemma 2.3.
Let us prove the uniqueness. To this end, we first observe that if u ∈ M (Ω), u = 0 and y u (T ) − y d L 2 (Ω) < ε, then we can take 0 < λ < 1 such that hence u is not a solution of (P). Therefore, to any solutionū of (P) corresponds an optimal stateȳ such thatȳ(T ) is on the boundary of the ballB ε (y d ).
Let us assume that u 1 and u 2 are two solutions of (P). We will prove that u 1 = u 2 . First we note that the identity y u1 (T ) = y u2 (T ) holds. Indeed, if y u1 (T ) = y u2 (T ), then we take u = (u 1 + u 2 )/2. Using the convexity of J, the strict convexity of the L 2 (Ω)-norm and the identity which is not possible as we proved above. Finally, we take u = u 1 − u 2 = 0. For this control we obtain y u (T ) = y u1 (T ) − y u2 (T ) = 0. Then, for every g ∈ L 2 (Ω) let ϕ g be the solution of the problem Then, since y u is a regular function in [ 1 k , T ] × Ω for every k ≥ 1, we have that Since the space S = {ϕ g ( 1 k ) : g ∈ L 2 (Ω)} is dense in L 2 (Ω) due to the approximate controllability properties of the heat equation, we conclude that y u ( 1 k ) = 0. By Remark 1, we have that y u ∈ C([0, T ], W −1,p (Ω)) for 1 ≤ p < n n−1 , and hence The next theorem characterizes the solutionū of (P).
whereȳ is the state associated toū. Then,ū is the solution of problem (P) if and only if there exist two Furthermore,φ andḡ are unique, and there exists a real numberλ > 0 such that Proof. Let us consider the linear mapping A ∈ L(M (Ω), L 2 (Ω)), defined by Au = y u (T ). The continuity of A follows from (5). We formulate (P) in an equivalent way. To this end we define the functional J : where IB ε(yd) denotes the indicator function of the ballB ε (y d ); which means that it vanishes inB ε (y d ) and takes the value +∞ outside. The problem (P) can be reformulated as the minimization of the convex functional J . Then,ū is a solution of (P) if and only if 0 ∈ ∂J (ū). Now, we apply the rules of the sub-differential calculus of convex functions; see, for instance, [12, Chapter 1, §5.3]. In particular, we can apply the chain rule because according to the proof of Theorem 2.4, there exists u 0 ∈ M (Ω) such that Au 0 ∈ B ε (y d ), which means that the Slater condition is fulfilled, consequently This implies that there existsḡ ∈ ∂IB ε(yd ) (ȳ(T )) such that −A * ḡ ∈ ∂J(ū). Relation (9) is precisely the definition ofḡ ∈ ∂IB ε(yd ) (ȳ(T )). Now, we takeφ as the solution of (10). Then, from (6) we deduce Combining this identity with the definition of −A * ḡ ∈ ∂J(ū) Taking u = 2ū andū/2, respectively, in the above inequality, we get (11). Therefore, the above inequality and (11) imply For any point x 0 ∈ Ω we select u = ±δ x0 in the above inequality, which shows that ±φ(x 0 , 0) ≤ 1. Since x 0 is arbitrary in Ω, we get that φ(0) C0(Ω) ≤ 1. This inequality along with (11) and the fact thatū = 0 imply (12). Finally, we prove the uniqueness ofḡ, the corresponding uniqueness forφ being an immediate consequence. From (9) it follows which implies the existence of some positive numberλ such thatḡ =λ(ȳ(T ) − y d ).
As a consequence of the previous theorem we get the desired sparsity of the optimal measure. In the sequel |A| will denote the Lebesgue measure of a measurable set A ⊂ Ω.
Corollary 1. Letū be the solution of (P) and consider the Jordan decomposition of the measureū:ū =ū + −ū − . Then, the following inclusions hold Furthermore, we have that |Ω + | = |Ω − | = 0. In addition, in dimension n = 1, there exist finitely many points {x j } m j=1 ⊂ Ω and real numbers {λ j } m j=1 such that Proof. Let us denote Ωū+ = supp(ū + ) and Ωū− = supp(ū − ). From (11) and (12) we deduce Hence, These identities imply (13) and (14). On the other hand, because of the properties of the heat equation, we know that the function x ∈ Ω −→φ(x, 0) ∈ R is analytic; see, for instance, [20]. Hence, the maximum and minimum values of this function are achieved in a set of points having zero Lebesgue measure, unless it is a constant function. This is not the case becauseφ(x, 0) = 0 for x ∈ Γ and φ C0(Ω) = 1.
3. The adjoint formulation. So far we have proved that the system (1) can be approximately controlled by using Borel measures with sparse support. To do this we have followed a direct approach, just looking for the minimum of problem (P). However, the analysis of the approximate controllability of the heat equation has traditionally followed a different approach. Indeed, the approximate controllability of (1) by using L 2 initial controls has been obtained by studying the adjoint optimization problem where ϕ g is the solution of the adjoint system This problem has a unique solution in L 2 (0, T ; H 1 0 (Ω)) ∩ C([0, T ]; L 2 (Ω)) for every element g ∈ L 2 (Ω). The functional J ε is well defined, continuous and strictly convex. The coercivity is more delicate to prove but, as pointed out in [13] it is a consequence of the backward uniqueness property ensuring that ϕ(x, 0) ≡ 0 implies that g ≡ 0.
The analysis in [13] covered also the case where the norm of ϕ g (x, 0) in L 2 (Ω) was replaced by the norm in any other space L q (Ω) with 1 ≤ q < ∞, thus leading to optimal initial data in any L p -setting with 1 < p ≤ ∞.
In the present paper however, we are interested in optimal data in the sense of measures. The functional above has to be then modified so to replace the L 2 (Ω)norm of ϕ g (x, 0) by the L ∞ -one. We shall do this following the arguments in [10].
The minimization problem in the adjoint system to be considered with that purpose is the following: Let us analyze this control problem. First of all, it is obvious that J ∞,ε is strictly convex and continuous. Moreover, it is coercive. Indeed, let {g k } ∞ k=1 ⊂ L 2 (Ω) such that g k L 2 (Ω) → +∞. We claim that To this end, we setg k = g k / g k L 2 (Ω) and, by taking a subsequence, we can assume thatg k g weakly in L 2 (Ω). Denote ϕ k = ϕ g k andφ k = ϕg k . Then,

4.1.
Motivation. In this section, for the sake of simplicity, we assume Ω ⊂ R n to be convex and n ≤ 3. The results of this section will probably hold without these restrictions. But such an extension would require further technical developments. We consider a family of triangulations {K h } h>0 ofΩ, defined in the standard way. To each element K ∈ K h we associate two parameters h K and K , where h K denotes the diameter of the set K and K is the diameter of the largest ball contained in K. The size of the mesh is defined by h = max K∈K h h K . We also assume the standard regularity assumptions on the triangulation: (i) -There exist two positive constants K and δ K such that h K K ≤ K and h h K ≤ δ K ∀K ∈ K h and ∀h > 0. (ii) -Define Ω h = ∪ K∈K h K, and let Ω h and Γ h denote its interior and its boundary, respectively. We assume that the vertices of K h placed on the boundary Γ h are points of Γ.
To each triangulation K h we associate the usual space of linear finite elements where {x j } N h j=1 are the interior nodes of K h and {e j } N h j=1 is the nodal basis formed by the continuous piecewise linear functions such that e j (x i ) = δ ij for every 1 ≤ i, j ≤ N h .
Following [4] we define the space of discrete controls by where δ xj denotes the Dirac measure centered at the point x j . For every σ we introduce the discrete state space The elements y σ ∈ Y σ can be represented in the form where χ k is the indicator function of I k and y k,h ∈ Y h . Moreover, by definition of Y h , we can write Thus U h and Y σ are finite dimensional spaces of dimension N h and N τ × N h , respectively, and bases are given by {δ xj } N h j=1 and {χ k e j } k,j .
Next we define the linear operators Λ h : M (Ω) → U h ⊂ M (Ω) and Π h : The operator Π h is the nodal interpolation operator for Y h . Concerning the operator Λ h we have the following result. (i) For every u ∈ M (Ω) and every y ∈ C 0 (Ω) and y h ∈ Y h we have (ii) For every u ∈ M (Ω) we have (iii) There exists a constant C > 0 such that for every u ∈ M (Ω) we have with 1/p + 1/p = 1.

Discrete state equation.
In this section we approximate the state equation. We recall that I k was defined as (t k−1 , t k ] and consequently y k,h = y σ (t k ) = y σ | I k , 1 ≤ k ≤ N τ . To approximate the state equation in time we use a dG(0) discontinuous Galerkin method, which can be formulated as an implicit Euler time stepping scheme. Given a control u ∈ M (Ω), for k = 1, . . . , N τ we set where (·, ·) denotes the scalar product in L 2 (Ω), a is the bilinear form associated to the operator −∆, i.e., a(y, z) = Ω ∇y∇z dx, and y 0h is the unique element of Y h satisfying The existence and uniqueness of the solution of (25) is obvious. Let us reformulate (25) in an equivalent form in terms of y σ ∈ Y σ with y σ | I k = y k,h . We define the bilinear form B σ : Then, it is immediate to check that y σ is the solution of (25) if and only if

EDUARDO CASAS, BORIS VEXLER AND ENRIQUE ZUAZUA
The following theorem establishes the convergence of these approximations.
Theorem 4.2. Let us assume that Γ is of class C 1,1 and let {u σ } σ ⊂ M (Ω) be a sequence such that u σ * u in M (Ω). If y is the state associated to u, solution of (1), and {y σ } σ denote the discrete states associated to {u σ } σ , then y σ y in L r (Q) for every r ∈ [1, 4 3 ) and y σ (T ) → y(T ) in L ∞ (Ω). First, we will prove the boundedness of {(y σ , y σ (T ))} σ in L r (Q) × L 2 (Ω). To do it we need some technical lemmas. We recall that given y h ∈ Y h , its discrete Laplacian ∆ h y h ∈ Y h is defined through the identity see, for instance, [31,Chapter 2]. The statement of the next lemma is similar to the discrete Gagliardo-Nirenberg inequality, see [18,Lemma 3.3].
The next lemma provides a discrete analog of the compact embedding of H 2 (Ω) into L ∞ (Ω).
Then, there exists a subsequence (denoted again by {y h }) with y h → y strongly in L ∞ (Ω).
Proof. As in the proof of the previous lemma we consider for each y h ∈ Y h the corresponding element y h ∈ H 2 (Ω) ∩ H 1 0 (Ω) as the solution of −∆y h = −∆ h y h in Ω, y h = 0 on Γ.
As noted above y h is the Ritz projection of y h . There holds Due to the compact embedding of H 2 (Ω) into L ∞ (Ω), there is a subsequence (denoted in the same way) with y h → y strongly in L ∞ (Ω). For this subsequence we obtain . For h → 0 within this subsequence we obtain y h → y strongly in L ∞ (Ω).
In the next lemma we provide a result on discrete smoothing, which is similar to [18], cf. also [29].
Lemma 4.5. Let g ∈ L 2 (Ω) and δ ∈ (0, 1 2 ] be given. For every σ we take ϕ σ ∈ Y σ as the unique element of Y σ satisfying Then, there exists C δ > 0 independent of g such that the following inequalities hold Proof. First, we observe that the inequalities for all 1 ≤ k ≤ N τ , can be established by repeating the arguments of [25,]. Now, it is enough to combine these inequalities with (30) to get (33). Lemma 4.6. Let y σ ∈ Y σ be the solution of (25) (or equivalently (28)), and let δ and C δ be as in Lemma 4.5. Then, the following inequalities hold and Proof. Obviously it is enough to prove (35) for k = N τ , hence t k = T . Indeed, for any other k we can replace the interval [0, T ] for [0, t k ] in the above estimates and proceed in the same way as we do for k = N τ . Let ϕ σ ∈ Y σ be as in Lemma 4.5 with g = y Nτ ,h . Taking z σ = y σ in (32), we get with (28) and (33)  Proof. Let δ ∈ (0, 1 2 ) arbitrary and take r ∈ [1, 4 3+2δ ). Using (35) we infer Since r( 3 4 + δ 2 ) < 1, then the above integral is finite and the boundedness of {y σ } σ in L r (Q) follows. Finally, for every r < 4 3 we can take δ > 0 sufficiently close to 0 such that r < 4 3+2δ . This concludes the proof.
Proof. First, we have to prove that the set of controls u ∈ M (Ω) for which y σ,u (T ) ∈ B ε (y d ) is non empty. From the approximate controllability property of the heat equation, we know that there exists a regular elements u 0 , for instance u 0 ∈ H 2 0 (Ω) ⊂ M (Ω) such that its associated state y u0 belongs to the open ball B ε (y d ). Due to the regularity of u 0 we have that y u0 (T ) − y σ,u0 (T ) L 2 (Ω) → 0; see [31,Chapter 9]. Hence, there exists σ 0 > 0 such that y σ,u0 ∈ B ε (y d ) for all |σ| ≤ σ 0 . Now, the proof of the existence of a solutionũ σ ∈ M (Ω) is like in Theorem 2.4.
We have the following result analogous to Corollary 1.
Corollary 2. Letū σ ∈ U h be the solution of (P σ ) for |σ| ≤ σ 0 . Then, we havē This corollary is an immediate consequence of (42), (43), and the definition of U h . We finish this section by proving the convergence of (P σ ).
whereū is the solution of (P) andȳ is its associated state.
Proof. Let u 0 ∈ M (Ω) be the control introduced in the proof of Theorem 4.8. Then, we know that u 0 is an admissible control for the problem (P σ ) for every |σ| ≤ σ 0 . Hence, J(ū σ ) ≤ J(u 0 ) for all |σ| ≤ σ 0 , and therefore {ū σ } |σ|≤σ0 is bounded in M (Ω). Let us take a subsequence, denoted in the same way, such thatū σ * u in M (Ω), and let y be the associated continuous state. We will prove that u is the solutionū of (P) and y =ȳ. From Theorem 4.2 we have that y σ y in L r (Q) ∀r ∈ [1, 4 3 ) andȳ σ (T ) → y(T ) in L ∞ (Ω).