Homogenization and exact controllability for problems with imperfect interface

The first aim of this paper is to study, by means of the periodic unfolding method, the homogenization of elliptic problems with source terms converging in a space of functions less regular than the usual \begin{document}$ L^2 $\end{document} , in an \begin{document}$ \varepsilon $\end{document} -periodic two component composite with an imperfect transmission condition on the interface. Then we exploit this result to describe the asymptotic behaviour of the exact controls and the corresponding states of hyperbolic problems set in composites with the same structure and presenting the same condition on the interface. The exact controllability is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads us to the construction of the exact controls as solutions of suitable transposed problem.


(Communicated by Benedetto Piccoli)
Abstract. The first aim of this paper is to study, by means of the periodic unfolding method, the homogenization of elliptic problems with source terms converging in a space of functions less regular than the usual L 2 , in an ε-periodic two component composite with an imperfect transmission condition on the interface. Then we exploit this result to describe the asymptotic behaviour of the exact controls and the corresponding states of hyperbolic problems set in composites with the same structure and presenting the same condition on the interface. The exact controllability is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads us to the construction of the exact controls as solutions of suitable transposed problem.

Introduction.
Let Ω be a domain of R n , n ≥ 2, made up of a connected set Ω ε 1 and a disconnected one, Ω ε 2 , consisting of ε-periodic connected inclusions of size ε. Let Γ ε = ∂Ω ε 2 denote the interface separating the two sub-domains of Ω and suppose that ∂Ω ∩ Γ ε = ∅ (see Figure 1).
In the first part of the paper, we consider the stationary heat equation in the two component composite modelized by Ω, assuming that on the interface Γ ε the heat flux is proportional to the jump of the temperature field, by means of a function of order ε γ (see Section 3, problem (3.1)). The order of magnitude of the parameter γ, with respect to the period ε, determines the influence of the thermal resistance in the heat exchange between the two materials (see [5] for the physical justification of the model). As observed by H.C. Hummel in [41], it is natural to suppose γ ≤ 1, otherwise one cannot expect to have boundedness of the solutions.
This interface problem was studied in [28,49,50] in the case of fixed source term in L 2 by the classical method of oscillating test functions due to L. Tartar (see also [9], Section 8.5). The authors proved that, as long as the interfacial resistance increases, one gets, at the limit, a composite where the two components become more and more isolated. More precisely, asymptotically, the composite behaves as in presence of just one temperature field. However, the effective thermal conductivity of the homogenized material changes according to γ. Indeed, -for γ < −1, it is the one obtained in the case of a classical composite without barrier resistance; -for γ = −1, it also takes into account the contact barrier; -for −1 < γ < 1, it is the one obtained in the case of a perforated composite with no material occupying the inclusions; -for γ = 1, it is the same of the previous case, but an additional term depending on the interface resistance appears in the limit behaviour of the solution. This means that the heat exchange is not sufficient to spread out the interfacial contribution and the heat source inside the inclusions.
Later on, in [26], the above results were recovered and completed by specifying the convergences of the flux by means of the periodic unfolding method, introduced for the first time by D. Cioranescu, A. Damlamian and G. Griso in [6].
In [35], with the further assumption of symmetry of the coefficients' matrix, these results were extended, only for −1 < γ ≤ 1, to the case of source terms converging in a space of functions less regular than the usual L 2 , by using the classical method of oscillating test functions due to L. Tartar (see also [9], Section 8.5). Some difficulties arose when considering the remaining values of γ.
In this paper, our first aim is to overcome these difficulties by means of the periodic unfolding method and to conclude the asymptotic analysis started in [35] by considering the remaining cases γ < −1 and γ = −1.
More precisely, in Theorems 3.14 and 3.18 (see also Corollaries 3.15 and 3.19), we prove that also in this framework, at the limit one gets the same effective thermal conductivities of [26,49]. Nevertheless, due to the less regularity of the source terms, a relevant difference appears. Indeed, here the heat source in the limit problem depends on subsequences of the heat sources at ε-level (see Remarks 3.16 and 3.20). We remark that, if fixed right-hand members are considered, the homogenization results of this paper exacltly recover the ones of [26,49]. Moreover, we point out that the arguments used in this work can be easily adapted to the cases −1 < γ < 1 and γ = 1. In fact we improve the results of [35] since we don't require the coefficients' matrix to be symmetric anymore.
Physically speaking, the weak data may model two different wiry heat sources positioned in the two components of the material, for n = 2, or two heat sources that can be represented as n − 1-dimensional varieties, for n ≥ 3.
The above mentioned homogenization results with less regular source terms, interesting in itself, have as relevant application the study of the exact controllability of hyperbolic problems set in composites with the same structure and presenting the same jump condition on the interface, that cannot be performed at all using the results of [26,49].
For an evolution problem, given a time interval [0, T ], the exact controllability issue consists in asking if it is possible to act on the solutions, by means of a suitable control, in order to drive the system to a desired state at time T , for all initial data. When homogenization processes are involved, a further interesting question arises: provided the exact controllabilities of the ε-problems and of the homogenized one, do the exact controls and the corresponding states at ε-level converge to the ones of the homogenized problem? Having in mind this question, the second aim of this paper is to study the asymptotic behaviour of the exact controls and the corresponding states of the wave equation in a medium made up of two components with very different coefficients of propagation, giving rise to the jump condition on the interface depending on γ (see Section 4, problem (4.1)). Taking into account the homogenization results of Section 3, in Theorem 4.3 we give a positive answer to the above question, for γ ≤ −1. For the remaining cases of γ we refer the reader to [36].
The plan of the paper is the following one. In Section 2, we describe in details the two component domain Ω. In Section 3, at first, we recall the definitions and the properties of specific functional spaces, suitable for the solutions of these kinds of interface problems, introduced in [21,23,28,49]. Then, we remind the definitions and the main properties of two unfolding operators for the two component domain Ω, defined for the first time in [7,26]. Finally, we develop the homogenization of the stationary imperfect transmission problem with less regular source term, by means of the periodic unfolding method. Section 4, is devoted to the study of the exact controllability of the hyperbolic imperfect transmission problem. Here we use a constructive method, known as Hilbert Uniqueness Method, introduced for the first time by Lions in [44,45]. The idea is to build the exact controls as the solutions of transposed problems associated to suitable initial conditions obtained by calculating at zero time the solutions of related backward problems. These controls, obtained by HUM, are also energy minimizing controls. More precisely, in Theorem 4.3, we describe the asymptotic behavior of the ε-controllability problem. To this aim, at first, we recall the homogenization results of [21] for the wave equation in the same two component domain Ω (cf. Theorem 4.5). Then, having in mind the transposed problem at ε-level given by HUM method, we prove a homogenization result for the wave equation but with less regular initial data and zero right-hand member (cf. Theorem 4.7). This requires the asymptotic analysis of a stationary ε-problem, with right-hand member converging in a space of functions less regular than the usual L 2 , which is possible thanks to the results of Section 3. Finally we prove that the exact control of the problem at ε-level and the corresponding state, converge, as ε → 0, to the exact control and to the solution of the homogenized problem respectively.
The exact controllability of hyperbolic problems with oscillating coefficients in fixed domains is treated in [44] and, in the case of perforated domains, in [8,11]. In [14]÷ [18], [31]÷ [33] and [53], the authors study the optimal control and exact controllability problems in domains with highly oscillating boundary. We refer the reader to [38,39] for the optimal control of hyperbolic problems in composites with imperfect interface and to [42] for the optimal control of rigidity parameters of thin inclusions in composite materials. We quote [23]÷ [25] and [34] for the correctors and the approximate control for a class of parabolic equations with interfacial contact resistance. In [30], the approximate controllability of linear parabolic equations in perforated domains is considered. In [57,58], the author treats the approximate controllability of a parabolic problem with highly oscillating coefficients in a fixed domain. Null controllability results for semilinear heat equations in a fixed domain can be found in [40], while the exact internal controllability and exact boundary controllability for semilinear wave equations are considered in [43] and [56], respectively.
2. The ε-periodic two component domain. Let Y := n i=1 ]0, l i [, n ≥ 2, be the reference cell, where l i , for i = 1, . . . , n, are positive real numbers. Then, let Y 1 and Y 2 be two nonempty open and disjoint subsets of Y such that Moreover we suppose that Y 1 is connected and Γ := ∂Y 2 is Lipschitz continuous. For any k ∈ Z n , we denote by Y k i and Γ k the following translated sets where k l = (k 1 l 1 , . . . , k n l n ). Moreover, for any given ε, we set where ε is a sequence of positive real numbers converging to zero.
Let Ω be a connected open bounded subset of R n , we define We explicitly observe that, by construction, the set Ω is decomposed into two components Ω = Ω ε 1 ∪ Ω ε 2 with Ω ε 1 connected and Ω ε 2 a disconnected union of ε-periodic disjoint translated sets of εY 2 . In view of (2.1), the interface separating the two components, Γ ε , is such that ∂Ω ∩ Γ ε = ∅ (see Figure 1).
Throughout the paper we denote by • u: the zero extension to the whole Ω of a function u defined on Ω ε 1 or Ω ε 2 , • χ E : the characteristic function of any measurable set E ⊆ R n , Let us recall (see for istance [9]) that, as ε −→ 0, θ i being the proportion of the material occupying Ω ε i .
3. Homogenization of an elliptic imperfect transmission problem with weakly converging data. Our first goal is to describe, for γ ≤ −1, the asymptotic behavior, as ε → 0, of the following stationary problem where n iε is the unitary outward normal to Ω ε i , i=1,2. We suppose that is the set of the n × n Y −periodic matrix-valued functions with bounded coefficients such that, for any λ ∈ R n , |Aλ| ≤ β |λ| a.e. in Y.

(3.3)
We assume that h is a Y −periodic function in L ∞ (Γ) and Moreover, for any fixed ε, A ε , h ε are given by 3.1. The functional space H ε γ and its dual (H ε γ ) . In this subsection, we recall the definition and some useful properties of a class of functional spaces introduced for the first time in [49], and successively in [28], when studying the analogous stationary problem but with regular data (see also [19,23]). These spaces take into account the geometry of the domain where the material is confined as well as the boundary and interfacial conditions, hence they are suitable for the solutions of this particular kind of interface problems.
equipped with the norm see [12].
The condition on ∂Ω in the definition of V ε has to be understood in a density sense, since we don't require any regularity on ∂Ω. Namely, V ε is the closure, with respect to the H 1 (Ω ε 1 )-norm, of the set of the functions in C ∞ (Ω ε 1 ) with a compact support contained in Ω. This can be done in view of (2.1). Proposition 3.2 ( [23,26]). There exists a positive constant C 1 , independent of ε, such that If γ ≤ 1, then there exists another positive constant C 2 , independent of ε, such that 26]). Let u ε = (u 1ε , u 2ε ) be a bounded sequence of H ε γ . Then, if γ ≤ 1, there exists a positive constant C, independent of ε, such that (3.12) We denote by (H ε γ ) the dual of H ε γ . As proved in [23], for any fixed ε, the norms of (H ε γ ) and For sake of simplicity, throughout this paper, we denote by L 2 ε (Ω) := L 2 (Ω ε 1 ) × L 2 (Ω ε 2 ). The space L 2 ε (Ω) will be equipped with the usual product norm, that is, Since the homogenization results proved in this section will be applied to study the exact controllability of the wave equation in composites with the same structure, we need to recall some further properties of the space H ε γ . Remark 3.4. We point out that H ε γ is a separable and reflexive Hilbert space dense in L 2 ε (Ω). Furthermore, H ε γ ⊆ L 2 ε (Ω) with continuous imbedding. On the other hand, one has that L 2 ε (Ω) ⊆ H ε γ , where L 2 ε (Ω) is a separable Hilbert space. This means that the triple (H ε γ , L 2 ε (Ω), H ε γ ) is an evolution triple. We refer the reader to [21,22] for an in-depth analysis on this aspect.

3.2.
Periodic unfolding operators in two-component domains. In this subsection, we recall the definitions and the main properties of two unfolding operators. The first one, T ε 1 , concerning functions defined in Ω ε 1 , is exactly that introduced in [7] for perforated domains. The second one, T ε 2 , acts on functions defined in Ω ε 2 and was defined for the first time in [26]. These operators map functions defined on the oscillating domains Ω ε 1 , Ω ε 2 into functions defined on the fixed domains Ω × Y 1 and Ω × Y 2 , respectively. Consequently, there is no need to introduce extension operators to pass to the limit in the problem. Using the notations of Section 2, let us introduce the following sets (see Figure 2) In the sequel, for z ∈ R n , we use [z] Y to denote its integer part k l , such that z − [z] Y ∈ Y and set a.e. in R n .
Then, for a.e. x ∈ R n , one has Definition 3.5. [ [7,26]] For any Lebesgue-measurable function φ on Ω ε i , i = 1, 2, the periodic unfolding operator T ε i is defined by Remark 3.6. In order to simplify the presentation, in the sequel if Φ is a function defined in Ω, we simply denote Let us collect the following results which are proved in [7,10,26].
The following convergence result holds: We now give a result concerning the jump on the interface proved in [26].
with h ε given by (3.6) Let us finally recall a known result about the convergences of the unfolding operators, previously introduced, applied to bounded sequences in H ε γ . We restrict our attention to the case we are interested in, γ ≤ −1.
In order to describe the asymptotic behaviour, as ε tends to zero, of the solution u ε of problem (3.1), we suppose that there exists a positive constant C, independent of ε, such that . Then it is easily seen that the functionals are linear and continuous. Therefore (3.18) and (3.19) can be rewritten as . (3.21) Moreover, due to (3.17), one has up to a subsequence, still denoted ε.
In the sequel, for sake of simplicity and where no ambiguity arises, in view of (3.20) and (3.21) we will still denote by f 1ε and f 2ε the functionals f 1ε and f 2ε respectively.
Let us first recall an a priori estimate proved in [28,49] in the case of fixed datum in L 2 (Ω) and extended in [35] to the case of weakly converging ones.
We describe the homogenized problems for every γ ≤ −1 by treating separately the two cases γ < −1, γ = −1. In the case γ = −1, when passing to the limit in problem (3.16), we meet an additional difficulty to treat the integral over the interface. In order to overcome that, we use Theorem 3.10 ii). Now, let us consider an auxiliary problem related to problem (3.1), already introduced in [35], i.e.
Theorem 3.14. Let γ < −1 and u ε be the solution of problem (3.1). Then, under the assumptions (3.2)÷(3.6) and (3.17), there exist a subsequence, still denoted ε, u ∈ H 1 0 (Ω) and u ∈ L 2 Ω, where the pair (u, u) is the unique solution of the following problem where ρ and ρ are as in Lemma 3.13, hence the term Proof. Arguing as in the proof of Lemma 3.13, we get that there exist a subsequence, in Ω and u 2 ∈ L 2 (Ω, H 1 (Y 2 )) such that the convergences (3.35) 2,4 hold and Then, from (3.12) of Corollary 3.3, (3.35) 2,4 and Proposition 3.8 we obtain that, for and, since u is constant with respect to y, we deduce (3.35) 1 . In order to get the limit problem, let v ε = εωψ ε as in the proof of Lemma 3.13 and ϕ ∈ D (Ω). If we take v 1 = v 2 = ϕ + v ε as test functions in (3.16), in view of Remark 3.11, we get (3.38) can be rewritten as In view of the definitions of Λ ε i , i = 1, 2, and v ε , by Proposition 3.7 ii), via unfolding, we get that, for ε sufficiently small, (3.39) can be rewritten as where we also used Proposition 3.7 i) and vi).
In the following result we point out that the limit problem (3.36) is equivalent to an elliptic problem set in the fixed domain Ω whose homogenized matrix is the same obtained in [49] for γ < −1, i.e. that of the classical elliptic homogenization in the fixed domain Ω (see [2]).

(3.45)
In (3.45) the constant matrices A l γ = a l ij n×n , l = 1, 2, are defined by where the functions χ j , j = 1, ..., n, are the unique solutions of the cell problems and the function χ, for a.e. x ∈ Ω, is the unique solution of the following problem where ρ and ρ are the same functions as in Lemma 3.13. Moreover the limit function u is the unique solution of the problem
From (3.42) and (3.51), we have where ζ Γ is a function in L 2 (Ω). On the other hand, from Proposition 3.7 i) and vi) and convergences (3.37), we have Then, using Proposition 3.8, we deduce that

3.3.2.
Homogenization results by periodic unfolding method for γ = −1. As in the previous case, let us start by using the unfolding method to prove a preliminary convergence result for a subsequence of the solutions of problem (3.23).
For i = 1, 2, let us take (3.56) Following the same argument as in Lemma 3.13, we have that, for i = 1, 2, In view of the definitions of Λ ε i and v iε , i = 1, 2, by Proposition 3.7 ii), via unfolding, we get that, for ε sufficiently small, (3.56) can be rewritten as 58) where we also used Proposition 3.7 i), vi) and Lemma 3.9.
where (u, u 1 , u 2 ) is the unique solution of the following problem

|Y | Ω×Y1
A (y) (∇u + ∇ y u 1 ) (∇ϕ + ∇ y (ω 1 ψ 1 )) dx dy A (y) (∇u + ∇ y u 2 ) (∇ϕ + ∇ y (ω 2 ψ 2 )) dx dy Then, by density we get the limit problem (3.62). Let us finally show that (3.62) admits a unique solution (u, To this aim, let As proved in [27], this last application is a norm on B. Now, for any V = (v 1 , v 2 , v 3 ), W = (w 1 , w 2 , w 3 ) ∈ B, consider the bilinear form on B defined by It is easily seen that a is continuous and coercive, and F is linear and continuous on B. Hence, applying the Lax-Milgram theorem, we obtain that problem (3.62) has a unique solution.
As for the previous case, in the following result we point out that the limit problem (3.62) is equivalent to an elliptic problem set in the fixed domain Ω whose homogenized matrix is the same obtained in [49] for γ = −1.  6) and (3.17), there exist a subsequence, still denoted ε, and u ∈ H 1 0 (Ω) such that (3.66) In (3.66), the constant matrices A l γ = a l ij n×n , l = 1, 2, are defined by where the couples χ j 1 , χ j 2 , j = 1, ..., n, are the unique solutions of the cell problems, The couple ( χ 1 , χ 2 ), for a.e. x ∈ Ω, is the unique solution of the following problem where ρ and ρ i , i = 1, 2, are the same functions as in Lemma 3.17. Moreover, the limit function u is the unique solution of the problem in Ω, u = 0 on ∂Ω, where the homogenized matrix is defined by ). By standard arguments, as in the two scale method (see [9], ch. 9), this gives where χ j 1 , χ j 2 , j = 1, ..., n, are the solutions of the cell problems (3.68) and χ 1 , χ 2 satisfy (3.69).
Arguing as in the last part of the proof of Corollary 3.15, when proving (3.53), but taking into account that in this case u 1 and u 2 are given by (3.72), we get (3.66) 2,3 . Remark 3.20. As in the previous case, in problem (3.70) the right-hand side of the limit equation is not exactly the sum of the weak limits of f 1ε and f 2ε as in the case of more regular data, but it is a more complicated function depending on a subsequence of f iε , i = 1, 2 (see Lemma 3.17 and (3.69) of Corollary 3.19). 4. Exact controllability of an imperfect transmission problem. The second issue we deal with concerns the study of the exact controllability of a hyperbolic imperfect transmission problem posed in the domain Ω described in Section 2. More precisely, let ζ ε := (ζ 1ε , ζ 1ε ) ∈ L 2 0, T ; L 2 ε (Ω) be a control. For any fixed T > 0 and γ ≤ −1, let us consider the following problem in Ω ε 2 , where n iε is the unitary outward normal to Ω iε , i = 1, 2, and Moreover A ε and h ε are as in (3.2)÷(3.6) but, as usual when dealing with hyperbolic problems, in this section we require the additional symmetry assumption on A a ij = a ji , i, j = 1, ...n. For clearness sake, throughout the paper, we denote by u ε (ζ ε ) := (u 1ε (ζ ε ) , u 2ε (ζ ε )) the solution of problem (4.1) and where no ambiguity arises, we omit the explicit dependence on the control.
, there exists a control ζ ex ε := (ζ ex 1ε , ζ ex 2ε ) belonging to L 2 0, T ; L 2 ε (Ω) such that the corresponding solution u ε of problem (4.1) satisfies It is well known that for a linear system, driving it to any state is equivalent to driving it to the null state and this is known as null controllability. Hence, in the sequel we study the null controllability of the considered systems, namely we take U 0 ε , U 1 ε = (0, 0).
We will prove that the system (4.1) is null controllable. We use a constructive method known as the Hilbert Uniqueness Method introduced by Lions (see [44,45]). The idea is to build a control as the solution of a transposed problem associated to some suitable initial conditions. These initial conditions are obtained by calculating at zero time the solution of a backward problem. Let us underline that the control obtained by HUM is unique being the one minimizing the norm in L 2 (0, T ; L 2 ε (Ω)). In [21], the asymptotic behaviour, as ε → 0, of the solutions of problem (4.1) has already been studied. Whence, a natural question arises: provided the exact controllability of the homogenized problem, do the exact control and its corresponding solution converge, as ε goes to zero, to the exact control of the homogenized problem and to the corresponding solution, respectively?
We give a positive answer to this question by proving the following main result:

4.1.
Asymptotic behaviour of two types of evolution imperfect transmission problems. In this subsection, for reader's convenience, we start by recalling some properties of the solution of the evolution imperfect transmission problem already studied in [21]. Although these results hold for γ ≤ 1, we restrict our attention to the case we are interested in.
Thanks to Remark 3.4, by using an approach to evolutionary problems based on evolution triples, we assume as variational formulation of the formal problem (4.9) the following one in Ω ε 2 . (4.12) As observed in [21], an abstract Galerkin's method provides the existence and uniqueness result for the solution of problem (4.9) and also some a priori estimates for any ε > 0. Let us point out that, for any fixed ε, the solution of problem (4.9) has some further regularity properties (see [46], Chapter 3, Theorem 8.2). In fact, under the same hypotheses of Theorem 4.4, the unique solution z ε of problem (4.9) is such that . Now, let us recall the homogenization result for problem (4.9), proved in [21].
Remark 4.6. Let us observe that (see for instance [9]) A 0 γ is a symmetric constant matrix such that where α and β are defined in (3.3).
In order to prove Theorem 4.3, we need to study the homogenization of another evolution imperfect transmission problem with less regular initial data (see Subsection 4.2).
More precisely, for T > 0 and γ ≤ −1, let ϕ ε := (ϕ 1ε , ϕ 2ε ) be the solution of the following problem in Ω ε 2 , where n iε is the unitary outward normal to Ω ε i , i = 1, 2 and i) ϕ 0 (4.17) Since the initial data are in a weak space, in order to give an appropriate definition of weak solution of problem (4.16), one needs to apply the so called transposition method (see [46], Chapter 3, Section 9, Theorems 9.3 and 9.4) to obtain a unique with C positive constant independent of ε.
Assume that the initial data satisfy with C positive constant independent of ε.
The results of Theorem 4.5 can't be applied directly to problem (4.16), hypotheses (4.17) and (4.19) being too weak, but, thanks to the homogenization results of Section 3, we overcome the difficulty and prove the following new result.
By classical regularity results for hyperbolic equations we have . Hence, by (4.25) and (4.30) Therefore, the function ϕ 1 := σ 1 =φ θ 1 is the unique solution in the sense of transposition of system (4.21) and ϕ 2 = θ 2 ϕ 1 . Now the proof is complete.

4.2.
Proof of Theorem 4.3. The proof of the main result of this section developes into two steps. At first we prove the null controllability (or equivalently the exact controllability, see Remark 4.2) of problem (4.1), by using HUM (Hilbert Uniqueness Method), a constructive method introduced by Lions in [44,45]. As already observed, the idea is to build a control as the solution of a transposed problem associated to some suitable initial conditions. These initial conditions are obtained by calculating at zero time the solution of a backward problem. The crucial point is constructing an isomorphism between L 2 ε (Ω) × (H ε γ ) and its dual with constants independent of ε. This result was already proved in [36], Theorem 3.1, for the case −1 < γ ≤ 1. The proof for the case γ ≤ −1 is exactly the same, hence here, for the reader's convenience, we detail only the noteworthy points.
In the second step, having in mind the homogenization result of the previous subsection (see Theorem 4.5), we show that the exact control of the problem at ε-level, found in the first step, and the corresponding state, converge, as ε → 0, to the exact control and to the solution of the homogenized problem, respectively. To this aim, we need to apply the homogenization result stated in Theorem 4.7 to the transposed problem at ε-level.
Step1. Let us start by proving that there exists a control ζ ex ε ∈ L 2 0, T ; L 2 ε (Ω) driving the corresponding solution of problem (4.1) to the null state, i.e.
Theorem 4.3 is now completely proved.