Yacine Chitour

Sébastien Fueyo

Guilherme Mazanti

Mario Sigalotti

Hautus-Yamamoto criteria for approximate and exact controllability of linear difference delay equations

Keywords: difference delay equations, approximate controllability, exact controllability, realization theory, Bézout's identity. 2020 Mathematics Subject Classification: 39A06, 93B05, 93C05

The paper deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time t is obtained as a linear combination of the control evaluated at time t and of the state evaluated at finitely many previous instants of time t ´Λ1 , . . . , t ´ΛN . Based on the realization theory developed by Y. Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in L q spaces, q P r1, `8q. We also provide a necessary condition for L 1 exact controllability, which can be seen as the closure of the L 1 approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by d maxtΛ 1 , . . . , Λ N u, where d is the dimension of the state space.

Introduction

The present paper deals with the approximate and exact controllability of linear difference delay equations of the form

xptq " N ÿ j"1 A j xpt ´Λj q `Buptq, t ě 0, (1)
where, given three positive integers d, m, and N , A 1 , . . . , A N are fixed d ˆd matrices with real entries, the state x and the control u belong to R d and R m respectively, and B is a fixed d ˆm matrix with real entries. Without loss of generality, the delays Λ 1 , . . . , Λ N are positive real numbers so that Λ 1 ă ¨¨¨ă Λ N . One of the major interests in the study of difference delay equations is that some 1D hyperbolic partial differential equations (PDEs) and, more precisely, systems of linear conservation laws can be put in the form [START_REF] Avellar | On the zeros of exponential polynomials[END_REF] through the method of characteristics [START_REF] Baratchart | Sufficient stability conditions for time-varying networks of telegrapher's equations or difference-delay equations[END_REF][START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF][START_REF] Chitour | Stability of non-autonomous difference equations with applications to transport and wave propagation on networks[END_REF][START_REF] Coron | Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems[END_REF], yielding a system under the form (1) with a specific structure for the matrices A 1 , . . . , A N (which are, in particular, all of rank 1). Equation (1) has mostly been analysed from a stability viewpoint, in the case where there is no open-loop control u (or, equivalently, when B " 0). Necessary and sufficient criteria have been obtained for the exponential stability of the origin of the system [START_REF] Avellar | On the zeros of exponential polynomials[END_REF][START_REF] Chitour | Stability of non-autonomous difference equations with applications to transport and wave propagation on networks[END_REF][START_REF] Hale | Introduction to functional-differential equations[END_REF][START_REF] Henry | Linear autonomous neutral functional differential equations[END_REF].

The main efforts to study the controllability properties of delay systems have been made on neutral differential delay systems

d dt ˜xptq ´N ÿ j"1 A j xpt ´Λj q ¸" N ÿ j"0 r A j xpt ´Λj q `Buptq, t ě 0, (2)
where r A 0 , . . . , r A N are d ˆd real matrices and Λ 0 " 0. Due to the infinite-dimensional nature of neutral functional differential equations and difference delay equations, several notions of controllability arise, such as approximate, exact, or relative controllability. Such notions of controllability can be particularized according to whether one requires or not that controllability occurs in a uniform time T (see Definitions 3.2 and 3.4).

The mostly investigated controllability notion is that of approximate controllability (with no finite upper bound on the controllability time), usually in a control space made of square integrable functions and a state space equal to the Sobolev space W 1,2 pr´Λ N , 0s, R d q. Usual tools in this context are semigroups properties and Laplace transforms, which lead to Hautustype conditions for controllability, i.e., rank conditions on certain matrix-valued holomorphic functions. When the system is retarded, i.e., A j " 0 for every j, Manitius [START_REF] Manitius | Function space controllability of linear retarded systems: a derivation from abstract operator conditions[END_REF] gave a necessary and sufficient condition for the approximate controllability of System [START_REF] Baratchart | Sufficient stability conditions for time-varying networks of telegrapher's equations or difference-delay equations[END_REF]. This result has been extended by Jacob et al. [START_REF] Jacobs | Criteria for function space controllability of linear neutral systems[END_REF] and O'Connor et al. [START_REF] O'connor | On the function space controllability of linear neutral systems[END_REF] to the neutral case in the single-delay case N " 1. Salamon [START_REF] Salamon | Control and observation of neutral systems[END_REF] gave dual formulations of exact controllability (which can be seen as observability inequalities) as well as some insights for the general case for small solutions. Finally, Yamamoto [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF], through the infinite realization theory developed in [START_REF] Yamamoto | Realization theory of infinite-dimensional linear systems[END_REF], expanded the result on approximate controllabiity to the general neutral case with an arbitrary number of delays.

Surprisingly, the controllability of the difference delay system (1) has been less investigated until recently. Due to the form of equation [START_REF] Avellar | On the zeros of exponential polynomials[END_REF], both the control space and the state space can be chosen of the same nature, such as, for instance, L q for some q P r1, `8s. The notion of relative controllability is the simplest one and it requires steering the system in time T ą 0 from any given initial condition to any given target xpT q in R d . That question is now completely understood: after studies handling special cases, Mazanti [START_REF] Mazanti | Relative controllability of linear difference equations[END_REF] gave a necessary and sufficient condition for the relative controllability for the general system (1).

Chitour et al. [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] proposed algebraic conditions for the exact and approximate controllability in L 2 pr´Λ N , 0s, R d q in uniform time T and gave a bound on the minimal time of controllability in the case where the delays are rationally dependent, situation which is solved by a standard state augmentation technique, reducing the matters at hand to the case of a single delay. They also addressed the first nontrivial case of two rationally independent delays when the system has dimension two and has a single scalar control, i.e., when N " d " 2 and m " 1. They provided necessary and sufficient conditions for both approximate and exact controllability in terms of Kalman-type criteria on the parameters of the system and asked whether it is possible to obtain such criteria in a more general situation. However, the techniques of proof used in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] become intractable when working in higher dimension or with more than two delays.

The approach adopted in this paper to address approximate and exact controllability of System (1) is based instead on Yamamoto's realization theory developed in [START_REF] Yamamoto | Realization Theory of infinite-dimensional linear systems[END_REF][START_REF] Yamamoto | Realization theory of infinite-dimensional linear systems[END_REF][START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF][START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF][START_REF] Yamamoto | Coprimeness in the ring of psedorational transfer functions[END_REF][START_REF] Yamamoto | Bézout identity over a ring of distributions-multivariable case[END_REF][START_REF] Yamamoto | Behavioral controllability and coprimeness for a class of infinite-dimensional systems[END_REF]. Yamamoto's theory, which extends older ideas given in [START_REF] Kalman | Realization of continuous-time linear dynamical systems: rigorous theory in the style of schwartz[END_REF][START_REF] Kamen | Module structure of infinite-dimensional systems with applications to controllability[END_REF], considers controllability issues for linear systems without a bound on the controllability time by adopting a distributional framework (i.e., inputs and outputs belong to spaces of distributions) in which the system can be characterized by a pair pP, Qq of matrix-valued distributions. In such a framework, controllability issues are given in terms of Bézout's characterizations. More precisely, exact controllability from the origin (in the space of distributions) is equivalent to the existence of two distributions R, S solving Bézout's identity Q ˚R `P ˚S " δ 0 I d [START_REF] Yamamoto | Behavioral controllability and coprimeness for a class of infinite-dimensional systems[END_REF], while L 2 approximate controllability from the origin is equivalent to proving an approximate Bézout identity, i.e., the existence of sequences of distributions pR n q nPN and pS n q nPN such that Q ˚Rn `P ˚Sn converges to δ 0 I d in the distributional sense as n tends to infinity [START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF]. In realization theory, the solvability of a Bézout identity (respectively, an approximate Bézout identity) is usually called left coprimeness (respectively, approximate left coprimeness), cf. [START_REF] Polderman | Introduction to mathematical systems theory[END_REF][START_REF] Sontag | Mathematical control theory[END_REF]. Note that one of the virtues of relating controllability issues and Bézout's identities is that, if the latter has been solved, then one has a solution for the motion planning problem, i.e., a right-inverse to the endpoint map, cf. Proposition 5.10.

Our results deal with approximate and exact criteria for controllability in functional state spaces L q pr´Λ N , 0s, R d q, for q P r1, `8q. First, we prove that the range of the endpoint map from the origin associated with System (1) saturates (i.e., does not increase) for T ě dΛ N , enabling one to provide an upper bound on the minimal time of controllability, namely, proving that if L q approximate (or exact) controllability holds true, then such a controllability must occur in time less than or equal to dΛ N (see Theorem 3.5). This is based on an explicit representation of the endpoint map given in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF]. It should be noticed that, for hyperbolic systems of conservation laws, sharp results on the (exact and null) controllability time have been obtained in the remarkable papers [START_REF] Coron | Optimal time for the controllability of linear hyperbolic systems in one-dimensional space[END_REF][START_REF] Coron | Null-controllability of linear hyperbolic systems in one dimensional space[END_REF]. We also mention the striking results obtained in [START_REF] Coron | On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients[END_REF] on optimal controllability time in the case of hyperbolic systems with analytic time-varying coefficients. We recall, however, that hyperbolic systems of conservation laws correspond to special classes of linear difference delay equations of the form [START_REF] Avellar | On the zeros of exponential polynomials[END_REF].

Furthermore, specifying Yamamoto's Hautus-type approximate controllability criterion to System (1), the second contribution of this paper consists in providing sufficient and necessary Hautus-type approximate controllability criteria for (1) in L q pr´Λ N , 0s, R d q for all q P r1, `8q (see Theorem 3.6 and Proposition 3.11). In particular, since our criteria are independent of q, we deduce that, if L q approximate controllability holds for some q P r1, `8q, then it holds for every such q.

The third contribution of this paper is a characterization of the L 1 exact controllability of System (1) in terms of a Bézout identity over a Radon measure algebra (see Theorem 5.13), showing in particular that establishing L 1 exact controllability of System (1) is equivalent to solving a corona problem (see [START_REF] Carleson | Interpolations by bounded analytic functions and the corona problem[END_REF][START_REF] Fuhrmann | On the corona theorem and its application to spectral problems in Hilbert space[END_REF] for more details on corona problems). It is worth noticing that the saturation of the range of the endpoint map is a crucial step also to derive such a characterization. A necessary Hautus-type criterion for the solvability of the Bézout identity is given in Proposition 5.17, allowing us to obtain a necessary condition for the exact controllability in L 1 pr´Λ N , 0s, R d q (see Theorem 3.7 and Proposition 3.12). The necessary condition for the L 1 exact controllability can be seen as the closure of the conditions obtained for the L q approximate controllability.

We conjecture that such a necessary condition is also sufficient for the exact controllability in L q pr´Λ N , 0s, R d q, for any q P r1, `8q. This amounts to solving a Bézout identity over a Radon measure algebra, which turns out to be a challenging open question. Note that the conjecture holds true for N " d " 2 and m " 1 as regards the L 2 exact controllability, cf. [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF].

In addition, we also provide in this paper a result, Theorem 3.10, yielding a partial positive answer to the conjecture, stating that our Hautus-type condition implies exact controllability between regular enough functions.

The exact controllability conditions we provide in Proposition 3.12 are reminiscent of other controllability criteria for abstract equations in Banach spaces, such as Condition [START_REF] O'connor | On the function space controllability of linear neutral systems[END_REF] in [START_REF] Miller | Controllability cost of conservative systems: resolvent condition and transmutation[END_REF], obtained through semigroup theory. However, such results are strongly related to the skew-adjointness of the involved infinitesimal generator operators and the unitary character of the corresponding semigroups, which is not the case for general difference delay systems of the form [START_REF] Avellar | On the zeros of exponential polynomials[END_REF]. We also remark that our approximate controllability criterion is similar to that given in [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF] for the strong stabilization of System (1), i.e., for the existence of matrices K 1 , . . . , K N so that the feedback uptq " ř N j"1 K j xpt ´Λj q stabilizes System (1) for all choices of delays Λ 1 , . . . , Λ N . Namely, combining our result with [15, Theorem 3.1], approximate controllability of (1) for every choice of delays Λ 1 , . . . , Λ N implies strong stabilization of System [START_REF] Avellar | On the zeros of exponential polynomials[END_REF].

We conclude the paper by illustrating the applicability of our results. We start by recovering those of [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF], at least for what concerns the characterization of approximate controllability. We then highlight the generality of our criteria by discussing in more details the case of systems with two delays and a single input in dimension three. In that case, we obtain a frequency-free approximate controllability characterization in the spirit of that of [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF].

The sequel of the paper is organized as follows. Section 2 introduces the notations used in this article, while our main results are stated in Section 3. In Section 4 we establish the saturation of the range of the endpoint map for times larger than dΛ N . Using Yamamoto's realization theory, our two main Hautus-Yamamoto criteria are proved in Section 5. Finally, we apply these criteria in Section 6 to deduce controllability criteria of Kalman type, recovering previous results in the literature and extending them to more general cases.

Notation

In this paper, we denote by N and N ˚the sets of nonnegative and positive integers, respectively. The set t1, . . . , N u is represented by 1, N for any N P N ˚. We use Z, R, C, R `, and R ´to denote the sets of relative integers, real numbers, complex numbers, nonnegative, and nonpositive reals respectively. For p P C, ℜppq and ℑppq represent the real and imaginary parts of p. For N P N ˚and n " pn 1 , . . . , n N q P N N , the length of the N -tuple n is denoted by |n| and is equal to n 1 `¨¨¨`n N . For Λ " pΛ 1 , . . . , Λ N q P R N , we write Λ ¨n :" n 1 Λ 1 `¨¨¨`n N Λ N . Given two positive integers i and j, M i,j pKq is the set of i ˆj matrices with coefficients in K " R or C. For A P M i,j pKq, we note A ˚its conjugate transpose matrix. We use } ¨} to denote a norm for every finite-dimensional space (over K) and ||| ¨||| the induced norm for linear maps.

Otherwise stated, elements x P K i are considered as column vectors. The identity matrix in M i,i pKq is denoted by I i . For M P M i,j pKq, rank M denotes the rank of M . Given a positive integer k, A P M i,j pKq, and B P M i,k pKq, the bracket rA, Bs denotes the juxtaposition of the two matrices, which hence belongs to M i,j`k pKq.

Let k P N ˚and q P r1, `8q. Given an interval I of R, L q pI, R k q represents the space of q-integrable functions on the interval I with values in R k endowed of the L q -norm on I denoted } ¨}I, q . The space of q-integrable functions on compact subsets of R (respectively,

R `) with values in R k is denoted L q loc `R, R k ˘(respectively, L q loc `R`, R k ˘).
The semi-norms }ϕ} r0,as,q :"

ˆż a 0 ||ϕptq|| q dt ˙1{q , ϕ P L q loc `R`, R k ˘, a ě 0,
induce a topology on L q loc `R`, R k ˘, which is then a Fréchet space. For a linear operator f we denote Ran f its range and Ran f the closure of the range. More generally, if F is a matrix-valued holomorphic function, we use F pCq and F pCq to denote its image and the the closure of its image, respectively.

We next introduce the distributional framework needed in the paper. A detailed presentation with precise definitions can be found, e.g., in [START_REF] Schwartz | Théorie des distributions[END_REF][START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF][START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]. We use DpRq to denote the space of C 8 functions defined on R with compact support, endowed with its canonical LF topology. We also use D 1 pRq to denote the space of continuous linear forms acting on DpRq, i.e., the space of all distributions on R, endowed with the strong dual topology of uniform convergence on bounded subsets of DpRq. For α P D 1 pRq and ψ P DpRq, xα, ψy denotes the duality product. The support of a distribution α P D 1 pRq, denoted supppαq, is the complement of the largest open set on which α is zero. The order of a distribution α P D 1 pRq is the smallest integer p such that, for every compact set K Ă R, there exists C K ą 0 such that |xα, ψy| ď C K sup xPK |ψ ppq pxq|, for all ψ P DpRq with compact support in K, where ψ ppq denotes the p-th derivative of ψ. We note δ x P D 1 pRq the Dirac distribution at x P R. Notice that δ x has order zero.

To deal with our controllability issues, we introduce the following subspaces of D 1 pRq. We use E 1 pR ´q and E 1 pR `q to denote the spaces of distributions having compact support in R ´and R `, respectively. We denote by M pR ´q and M pR `q the subspaces of E 1 pR ´q and E 1 pR `q consisting of distributions of order zero. The spaces M pR ´q and M pR `q can be also characterized as the sets of Radon measures with compact support contained in R ánd R `, thanks to the Riesz representation theorem (see, e.g., [START_REF] Rudin | Real and complex analysis[END_REF]Theorem 6.19]). Let D 1 `pRq be the space of distributions having support bounded on the left, which becomes an algebra when endowed with the convolution product ˚. We also consider M `pRq as the space of Radon measures with support bounded on the left, or equivalently the subspace of D 1 `pRq of distributions of order zero. With a slight abuse of notation, we also write D 1 pRq, E 1 pR ´q, M pR ´q, D 1 `pRq, and M `pRq to refer to sets of matrices whose entries belong to those respective spaces.

Given a Radon measure µ P M `pRq, we use p µppq to denote the two-sided Laplace transform of µ at frequency p P C, that is,

p µppq " ż `8 ´8 dµptqe ´pt , (3)
provided that the integral exists. In particular, the Laplace transform of a Dirac distribution δ x , x P R, is the holomorphic map

p δ x : p Þ Ñ e ´px , p P C. (4)
In the more general setting of a distribution α P D 1 pRq, the Laplace transform p αppq of α at p P C is defined by xα, e ´¨p y, when this quantity makes sense, i.e., when the linear form α on DpRq can be extended by continuity (in the standard topology of C 8 pRq) to the function t Þ Ñ e ´tp .

We now define the truncation to positive times of a distribution in D 1 `pRq, following the presentation of [START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF][START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]. Let DpR `q Ă DpRq be the space of infinitely differentiable functions on R with compact support contained in R `and D 1 pR `q be its topological dual space. For α P D 1 `pRq, we define its truncation πα P D 1 pR `q by xπα, ψy " xα, ψy, ψ P DpR `q.

(5)

We have that πα is a well-defined element of D 1 pR `q. Furthermore, π is continuous with respect to the strong dual topology of D 1 `pRq and D 1 pR `q. Note that π truncates distributions to positive times only and, in particular, πδ 0 " 0. For further properties concerning the operator π, see Lemma 5.9.

Description of the problem and statement of the controllability criteria

We start by defining solutions of System (1) considered in this paper. The following proposition can be easily obtained by a direct step-by-step construction of the solution, as given in [6, Proposition 3.2] (cf. also [START_REF] Mazanti | Relative controllability of linear difference equations[END_REF]Remark 2.3]). Unless otherwise stated, q denotes any real number belonging to the interval r1, `8q.

Proposition 3.1. Let T ą 0, u P L q pr0, T s, R m q, and x 0 P L q pr´Λ N , 0s, R d q. There exists a unique solution x P L q pr´Λ N , T s, R d q such that xpθq " x 0 pθq for all θ P r´Λ N , 0s and xp¨q satisfies Equation (1) for all t P r0, T s.

From now on, given T ą 0, u P L q pr0, T s, R m q, and x 0 P L q pr´Λ N , 0s, R d q, we write x P L q pr´Λ N , T s, R d q to denote the solution given by Proposition 3.1. For all t P r0, T s, we denote by x t P L q pr´Λ N , 0s, R d q the function defined by x t pθq :" xpt`θq, θ P r´Λ N , 0s. Using such a notation, we introduce in the next definition the standard notions of approximate and exact controllability in finite time T ą 0 that we are interested to study in this paper (see, for instance, [START_REF] Coron | of Mathematical Surveys and Monographs[END_REF]Chapter 2]). Definition 3.2. System (1) is said to be: 1) L q approximately controllable in time T ą 0 if for every x 0 , ϕ P L q pr´Λ N , 0s, R d q and ϵ ą 0, there exists u P L q pr0, T s, R m q such that }x T ´ϕ} r´Λ N ,0s,q ă ϵ;

2) L q exactly controllable in time T ą 0 if for every x 0 , ϕ P L q pr´Λ N , 0s, R d q, there exists u P L q pr0, T s, R m q such that x T " ϕ.

Remark 3.3. If System (1) is L q approximately (respectively, exactly) controllable in time T ą 0 then it is L q approximately (respectively, exactly) controllable in any time T 1 ě T .

We present below weaker notions of controllability, corresponding to those used by Yamamoto [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF], in which the controllability time is not fixed in advance, but may depend on the target to be reached, and the initial state is always assumed to be the origin. Definition 3.4. System (1) is said to be:

1) L q approximately controllable (from the origin) if for x 0 " 0, every ϕ P L q pr´Λ N , 0s, R d q, and ϵ ą 0, there exist T ϵ,ϕ ą 0 and u P L q pr0, T ϵ,ϕ s, R m q such that }x T ϵ,ϕ ´ϕ} r´Λ N ,0s,q ă ϵ;

2) L q exactly controllable (from the origin) if for x 0 " 0 and every ϕ P L q pr´Λ N , 0s, R d q, there exist T ϕ ą 0 and u P L q pr0, T ϕ s, R m q such that x T ϕ " ϕ.

Note that obvious implications hold true between the different notions provided in Definitions 3.2 and 3.4, i.e., approximate (respectively, exact) controllability in time T implies approximate (respectively, exact) controllability from the origin. One of the results of the present paper is that one actually has equivalence between such notions, as stated in the next theorem.

Theorem 3.5. Let q P r1, `8q. System (1) is L q approximately (respectively, exactly) controllable from the origin if and only if it is L q approximately (respectively, exactly) controllable in time T " dΛ N .

The proof of Theorem 3.5 is provided in Section 4.

Hautus-Yamamoto criteria

Before stating our two main theorems, which give necessary and sufficient (respectively, necessary) Hautus-Yamamoto criteria to ensure the approximate (respectively, exact) controllability in time dΛ N of System (1), let us introduce the matrix-valued holomorphic map

Hppq :" I d ´N ÿ j"1 e ´pΛ j A j , p P C. (6)
The holomorphic function H defined above arises when one considers solutions of (1) with u P L q pR `, R m q and an initial condition xpθq " 0 for θ P r´Λ N , 0s. Indeed, extending x and u by zero for all negative times, ( 1) is satisfied for every t P R and then, taking the two-sided Laplace transform, we obtain that

p xppq " Hppqp uppq, (7)
for p in some right-half plane of C, where Hppq " Hppq ´1B is the transfer function of System (1), i.e., the matrix describing the linear relation between the frequencies of the input and of the output. We notice that the Laplace transform of x exists for all frequencies in a suitable right-half plane of C because the solutions of (1) grow at most exponentially.

Our main results are given next.

Theorem 3.6. Let q P r1, `8q. System (1) is L q approximately controllable in time dΛ N if and only if the two following conditions hold true: i) rank rHppq, Bs " d for every p P C, ii) rankrA N , Bs " d.

Theorem 3.7. If System (1) is L 1 exactly controllable in time dΛ N then the two following conditions hold true: i) rank rM, Bs " d for every M P HpCq, ii) rankrA N , Bs " d.

Theorems 3.6 and 3.7 are proved in Sections 5.3 and 5.4, respectively. Theorem 3.6 is a complete characterization of L q approximate controllability, providing a necessary and sufficient condition which can be seen as the counterpart for difference equations of Hautus controllability criterion. In particular, the property of L q approximate controllability in time dΛ N does not depend on q P r1, `8q, in the sense that if it holds for some q P r1, `8q then it holds for all of them. On the contrary, Theorem 3.7 provides only a necessary condition for L q exact controllability in the case q " 1. We expect in fact that the above condition is also sufficient for any q and, in that direction, we propose the following conjecture.

Conjecture 3.8. Let q P r1, `8q. System (1) is L q exactly controllable in time dΛ N if and only if the two following conditions hold true: i) rank rM, Bs " d for every M P HpCq, ii) rankrA N , Bs " d. This is motivated firstly by the fact that this conjecture actually holds true for the L 2 exact controllability in the case N " d " 2 and m " 1, as it follows from the results of [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] (see Section 6 for details). In addition, we provide below a result giving ground to the above conjecture. For that purpose, we introduce the following compatibility condition for the existence of regular enough solutions of (1). Definition 3.9. Let k P N and x 0 P C k pr´Λ N , 0s, R d q. We say that x 0 is C k -admissible for System (1) if, for every integer ℓ with 0 ď ℓ ď k, we have

x pℓq 0 p0q " N ÿ j"1 A j x pℓq 0 p´Λ j q.
Theorem 3.10. Assume that Conditions i) and ii) of Conjecture 3.8 hold true. Then there exists a nonnegative integer k such that, for every q P r1, `8q and x 0 , ϕ P C k pr´Λ N , 0s, R d q C k -admissible for System (1), there exists u P L q pr´Λ N , 0s, R m q such that x dΛ N " ϕ almost everywhere in r´Λ N , 0s.

The proof of Theorem 3.10 is provided in Section 5.5.

Further controllability characterizations

The approximate and exact controllability of System (1) can be described in several equivalent ways using Propositions 3.11 and 3.12, respectively. (c) For every p P C, one has det pHppqH ˚ppq `BB ˚q ą 0. Proposition 3.12 below is the counterpart of Proposition 3.11 for the case of exact controllability.

Proposition 3.12. Under the assumption that rank rA N , Bs " d, the following three statements are equivalent:

(a) rank rM, Bs " d for every M P HpCq.

(b) There exists α ą 0 such that, for every p P C,

inf ␣ˇˇˇˇg T Hppq ˇˇˇˇ`ˇˇˇˇg T B ˇˇˇˇˇˇg P C d , }g T } " 1 ( ě α.
(c) There exists α ą 0 such that, for every p P C,

det pHppqH ˚ppq `BB ˚q ě α.
The proofs of Propositions 3.11 and 3.12 are given in Section 5.2.

As an immediate consequence of Theorems 3.6 and 3.7 and Propositions 3.11 and 3.12, we have the following corollary.

Corollary 3.13. Under the assumption that rank rA N , Bs " d, we have: i) Given any q P r1, `8q, System (1) is L q approximately controllable in time dΛ N if and only if one of the items (a)-(c) of Proposition 3.11 holds true.

ii) If System (1) is L 1 exactly controllable in time dΛ N then all items (a)-(c) of Proposition 3.12 hold true.

Thanks to Item (c) in Propositions 3.11 and 3.12, Item i) of Theorems 3.6 and 3.7 can be reformulated in an equivalent way in terms of the non-vanishing properties of the function p Þ Ñ det pHppqH ˚ppq `BB ˚q. Our next result shows that this function can only vanish in a bounded vertical strip of the complex plane. Proposition 3.14. There exist β 1 , β 2 , ρ P R such that det pHppqH ˚ppq `BB ˚q ě |det pHppqq| 2 ě ρ ą 0 [START_REF] Coron | of Mathematical Surveys and Monographs[END_REF] for all p P C such that ℜppq ă β 1 or ℜppq ą β 2 .

Proof. The left inequality in ( 8) is trivial. As for the second one, notice first that one can assume, with no loss of generality, that Hp¨q ı I d . Moreover, for ℜppq large enough, Hppq is arbitrarily close to the identity. This implies that there exists β 2 ą 0 such that |det pHppqq| ě 1{2 for ℜppq ą β 2 .

On the other hand, the holomorphic function h defined by hppq " det pHppqq can be written as hppq " hppq `hn e ´ppΛ¨nq , p P C, where h n is a nonzero real number, n is in N N and satisfies 0 ď |n| ď N , and hp¨q is a holomorphic function such that lim ℜppqÑ´8 hppqe ppΛ¨nq " 0. We then deduce that h n e ´ppΛ¨nq is the dominant term of hppq as ℜppq Ñ ´8, and it goes to 8 in modulus. This implies that for every ρ ą 0 there exists β 1 P R such that |hppq| 2 ě ρ for ℜppq ă β 1 .

Properties of the endpoint map and consequences for controllability

The goal of this section is threefold. Firstly, we give a variation-of-constants formula for System (1) allowing us to write the solution operator as the sum of a flow operator and an endpoint map. As a consequence, we can give characterizations of the controllability of System (1) in terms of the range of the endpoint map. Secondly, through a Cayley-Hamilton theorem for multivariate polynomials, we show that the range of the endpoint map is constant from the time dΛ N and finally we deduce that the approximate (exact respectively) controllability of System ( 1) is equivalent to the approximate controllability (exact respectively) in finite time dΛ N , thus providing a proof for Theorem 3.5.

Definitions and preliminary remarks

We recall in Proposition 4.3 the explicit representation formula for solutions of System (1), often called variation-of-constants formula (and sometimes flow formula), which can be found in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF]. Before the statement of the proposition, let us give the following definitions.

Definition 4.1. The family of matrices Ξ n P M d,d pRq, n P Z N , is defined inductively as

Ξ n " $ ' & ' % 0 if n P Z N zN N , I d if n " 0, ř N k"1 A k Ξ n´e k if n P N N and |n| ą 0,
where e k denotes the k-th canonical vector of Z N for k P 1, N , i.e., all the coordinates of e k are zero except for the k-th one, which is equal to one.

Definition 4.2. For T P r0, `8q, we introduce the following two operators:

1) The flow operator Υ q pT q : L q pr´Λ N , 0s, R d q Ñ L q pr´Λ N , 0s, R d q of System (1) defined by pΥ q pT qx 0 q psq " ÿ pn,jqPN N ˆ 1,N ´Λj ďT `s´Λ¨nă0 Ξ n´e j A j x 0 pT `s ´Λ ¨nq, for x 0 P L q pr´Λ N , 0s, R d q and s P r´Λ N , 0s. The operator Υ q pT q represents the solution operator of System (1) with B " 0 and initial state x 0 .

2) The endpoint operator E q pT q : L q pr0, T s, R m q Ñ L q pr´Λ N , 0s, R d q of System (1) defined by pE q pT quq ptq "

ÿ nPN N
Λ¨nďT `t Ξ n BupT `t ´Λ ¨nq, for u P L q pr0, T s, R m q and t P r´Λ N , 0s. Proposition 4.3 (Variation-of-constants formula). For T P r0, `8q, u P L q pr0, T s, R m q, x 0 P L q pr´Λ N , 0s, R d q, and t P r0, T s, we have

x t " Υ q ptqx 0 `Eq ptqu. (9)
Remark 4.4. Notice that Proposition 4.3 has been stated for q " 2 in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] only but it holds true as well for q P r1, `8q.

In Proposition 4.5 below we use the variation-of-constants formula to express approximate and exact controllability in terms of the image of the operator E q pT q, T ą 0. Proposition 4.5. Let q P r1, `8q and T ą 0. i) System (1) is L q approximately controllable in time T ą 0 if and only if Ran E q pT q is dense in L q pr´Λ N , 0s, R d q.

ii) System (1) is L q approximately controllable from the origin if and only if

ď T ě0
Ran E q pT q " L q pr´Λ N , 0s, R d q.

(10)

iii) System (1) is L q exactly controllable in time T ą 0 if and only if Ran E q pT q " L q pr´Λ N , 0s, R d q.

iv) System (1) is L q exactly controllable from the origin if and only if

ď T ě0
Ran E q pT q " L q pr´Λ N , 0s, R d q.

(11)

Saturation of the range of the endpoint map in time dΛ N

Cayley-Hamilton theorem is instrumental to study controllability properties of System (1) containing one delay (see [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF]Remark 3.5]). It turns out that the following generalization of Cayley-Hamilton theorem in the case of multivariate polynomials plays a similar role in our subsequent arguments.

Lemma 4.6. Let Ξ n , n P Z N , be the matrices introduced in Definition 4.1. There exist real coefficients α k , for k P N N with 0 ă |k| ď d, such that, for every n P N N with |n| ě d,

Ξ n " ´ÿ kPN N 0ă|k|ďd α k Ξ n´k . (12)
Proof. For t " pt 1 , . . . , t N q P R N , set

Aptq " t 1 A 1 `t2 A 2 `¨¨¨`t N A N .
One deduces, by Definition 4.1 and an immediate induction argument that, for every j P N and t P R N , it holds Aptq j "

ÿ nPN N |n|"j Ξ n t n , ( 13
)
where t n :" t n 1 1 t n 2 2 ¨¨¨t n N N . Using Neumann series, we deduce from Equation ( 13) that, for t small enough,

`Id ´Aptq ˘´1 " ÿ jPN Aptq j " ÿ nPN N Ξ n t n . ( 14
)
Notice that P ptq " det `Id ´Aptq ˘is a multivariate polynomial of degree d, that is,

P ptq " ÿ kPN N 0ď|k|ďd α k t k , (15)
for some real numbers α k defined for k P N N such that |k| ď d and α 0 " 1. Let Adj `Id ´Aptq be the adjugate matrix of I d ´Aptq. We have, on the one hand, that there exist

M k P M d,d pRq for k P N N and 0 ď |k| ď d ´1 such that Adj `Id ´Aptq ˘" ÿ kPN N 0ď|k|ďd´1 M k t k (16)
and, on the other hand, that Equations ( 14)-( 15) lead to Adj `Id ´Aptq ˘" P ptq `Id ´Aptq ˘´1 "

ÿ kPN N 0ď|k|ďd α k t k ÿ nPN N Ξ n t n " ÿ nPN N ÿ kPN N 0ď|k|ďd α k Ξ n t n`k . (17)
The substitution l " n `k in [START_REF] Jacobs | Criteria for function space controllability of linear neutral systems[END_REF] allows us to write

Adj `Id ´Aptq ˘" ÿ nPN N ÿ lPN N , l´nPN N 0ď|l´n|ďd α l´n Ξ n t l . (18)
We deduce from Equations ( 16)-( 18) that, for l P N N and |l| ě d,

ÿ nPN N , l´nPN N 0ď|l´n|ďd α l´n Ξ n " 0. ( 19
)
Setting n 1 " l ´n in [START_REF] Kamen | Module structure of infinite-dimensional systems with applications to controllability[END_REF], one obtains

ÿ n 1 PN N , l´n 1 PN N 0ď|n 1 |ďd α n 1 Ξ l´n 1 " 0. ( 20
)
Since Ξ l´n 1 " 0 for l ´n1 P Z N zN N (see Definition 4.1), we deduce from Equation ( 20) that

Ξ l " ´ÿ n 1 PN N 0ă|n 1 |ďd α n 1 Ξ l´n 1 , (21)
hence the conclusion.

We prove now that the range of the operator E q pT q is constant with respect to T for T ě dΛ N . Theorem 4.7. For all T P rdΛ N , `8q and q P r1, `8q, we have Ran E q pT q " Ran E q pdΛ N q.

(22)

Proof. Let q P r1, `8q. The proof is divided in two steps. We first prove that Equation ( 22) is satisfied for T P rdΛ N , dΛ N `δs and some δ ą 0. We then deduce from the flow formula (9) that Equation ( 22) is actually satisfied for all T P rdΛ N , `8q. Let T ą dΛ N and u P L q pr0, T s, R m q. We define

u 1 psq :" ups `T ´dΛ N q, s P r0, dΛ N s, (23)
and

u 2 psq :" ´ÿ 0ă|k|ďd, kPN N săΛ¨kďs`T ´dΛ N α k ups ´Λ ¨k `T ´dΛ N q, s P r0, dΛ N s, (24)
where the coefficients α k are defined as in Proposition 4.6. The sum in ( 24) is understood to be zero when the indices are taken in an empty set. For t P r´Λ N , 0s, we have pE q pT quq ptq "

ÿ nPN N Λ¨nďT `t Ξ n BupT `t ´Λ ¨nq " ÿ nPN N Λ¨nďdΛ N `t Ξ n BupT `t ´Λ ¨nq `ÿ nPN N dΛ N `tăΛ¨nďT `t Ξ n BupT `t ´Λ ¨nq. (25) Notice that ÿ nPN N Λ¨nďdΛ N `t Ξ n BupT `t ´Λ ¨nq " E q pdΛ N qu 1 ptq. (26)
Since dΛ N `t ă Λ ¨n for n P N N implies that |n| ě d, we deduce from Lemma 4.6 that

ÿ nPN N dΛ N `tăΛ¨nďT `t Ξ n BupT `t ´Λ ¨nq " ´ÿ nPN N dΛ N `tăΛ¨nďT `t ÿ kPN N 0ă|k|ďd α k Ξ n´k BupT `t ´Λ ¨nq. ( 27
)
The substitution n 1 " n ´k in Equation ( 27) yields

ÿ nPN N dΛ N `tăΛ¨nďT `t Ξ n BupT `t ´Λ ¨nq " ´ÿ kPN N 0ă|k|ďd ÿ n 1 PN N dΛ N `tăΛ¨pn 1 `kqďT `t α k Ξ n 1 BupT `t ´Λ ¨pn 1 `kqq. ( 28
) Let δ ą 0 be such that for all T P rdΛ N , dΛ N `δs, t P r´Λ N , 0s, k P N N with 0 ă |k| ď d, and

n 1 P N N , if dΛ N `t ă Λ ¨pn 1 `kq ď T `t then Λ ¨n1 ď dΛ N `t.
Letting T P rdΛ N , dΛ N `δs, we can thus rewrite Equation ( 28) as

ÿ nPN N dΛ N `tăΛ¨nďT `t Ξ n BupT `t ´Λ ¨nq " ´ÿ n 1 PN N Λ¨n 1 ďdΛ N `t Ξ n 1 B ÿ kPN N , 0ă|k|ďd dΛ N `tăΛ¨pn 1 `kqďT `t α k upT `t ´Λ ¨pn 1 `kqq " E q pdΛ N qu 2 ptq. (29)
Equations ( 25), [START_REF] Pólya | Problems and theorems in analysis. I. Classics in Mathematics[END_REF], and (29) prove that, for T P rdΛ N , dΛ N `δs and t P r´Λ N , 0s, E q pT quptq " E q pdΛ N qu 1 ptq `Eq pdΛ N qu 2 ptq.

From Equation (30), we deduce that, Ran E q pT q " Ran E q pdΛ N q, T P rdΛ N , dΛ N `δs.

Let us now extend Equation ( 31) to all T P rdΛ N , `8q. Let V " Ran E q pdΛ N q and x P V . Fix u P L q pr0, dΛ N s, R m q such that x " E q pdΛ N qu. For t P r0, δs, define ũ P L q pr0, dΛ N `ts, R m q by setting ũ| r0,dΛ N s " u and ũpsq " 0 for s P rdΛ N , dΛ N `ts. From the variation-of-constants formula (9), we have Υ q ptqx " E q pdΛ N `tqũ P Ran E q pdΛ N `tq.

Thanks to Equation [START_REF] Sontag | Mathematical control theory[END_REF] we have proved that Υ q ptqx P V, t P r0, δs, x P V.

Let y P Ran E q pT q for T P rdΛ N `δ, dΛ N `2δs and u P L q pr0, T s, R m q such that y " E q pT qu. Define z " E q pdΛ N `δqu| r0,dΛ N `δs P V . The variation-of-constants formula (Equation ( 9)) gives y " Υ q pT ´dΛ N ´δqz `Eq pT ´dΛ N ´δqǔ,

where ǔpαq " upα `dΛ N `δq for α P r0, T ´dΛ N ´δs. We deduce from [START_REF] Sontag | Mathematical control theory[END_REF] and ( 32) that y P Ran E q pdΛ N q, proving that Ran E q pdΛ N q " Ran E q pT q, T P rdΛ N `δ, dΛ N `2δs.

The iteration of the same process proves that Equation [START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF] actually holds for all T ě dΛ N .

Upper bound on the minimal control time

We next provide a proof of Theorem 3.5, which is a direct consequence of the saturation of the range of the endpoint map from time dΛ N .

Proof Theorem 3.5. By Theorem 4.7 and since T Þ Ñ Ran E q pT q is monotone nondecreasing for the inclusion, we have that ď T ě0

Ran E q pT q " Ran E q pdΛ N q.

The conclusion directly follows from Proposition 4.5.

One of the major questions concerning the approximate and exact controllability in finite time T is to determine the minimal time of controllability. Definition 4.8. We define T min, ap, q and T min, ex, q the minimal time of the approximate and exact controllability respectively as follows:

T min, ap, q :" inf T PR `tSystem (1) is L q approximately controllable in time T u, T min, ex, q :" inf T PR `tSystem (1) is L q exactly controllable in time T u, with the convention that inf H " `8.

Since an immediate inspection of System [START_REF] Avellar | On the zeros of exponential polynomials[END_REF] shows that it is never approximately or exactly controllable before the time Λ N , we can recast Theorem 3.5 in terms of minimal time of controllability as follows.

Corollary 4.9. Both times T min, ap, q and T min, ex, q belong to the set rΛ N , dΛ N s Y t`8u.

In the case N " d " 2 and m " 1, it is proved in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] that either T min, ap, 2 " `8 or T min, ap, 2 " T min, ex, 2 " 2Λ N . In the general case however, dΛ N is not always minimal as proved in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] by looking at the case of commensurable delays. It was noticed in the same reference that dΛ N is minimal when controllability holds for systems with a single input and commensurable delays. We conjecture that this result holds true also for the case of nonnecessarily commensurable delays. It would be also interesting to investigate the question of equality between T min, ap, q and T min, ex, q when both of them are finite.

Hautus-Yamamoto criteria for approximate and exact controllability

In this section, we stick to Yamamoto's notations used in [START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF][START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF], which have been introduced in Section 2.

Realization theory

When the initial condition is fixed at 0 (which is the case in Definition 3.4), System (1) describes a linear relation between the control u on r0, T s and the state x on rT ´ΛN , T s, described by the operator E q pT q introduced in Definition 4.2. The main idea in realization theory is to represent such a linear relation between an input (here the control u) and an output (here the state x), by writing the output as a convolution of the input with a certain kernel. For an introduction to the terminology of realization theory and input-output systems in finite dimension, we refer to the textbooks of Polderman and Willems [START_REF] Polderman | Introduction to mathematical systems theory[END_REF] or Sontag [START_REF] Sontag | Mathematical control theory[END_REF].

Yamamoto's approach to the realization theory for infinite-dimensional systems considers systems in which the input u is applied during a time interval of the form r´T, 0s, with T ą 0 arbitrary, and the output is a certain function t Þ Ñ yptq defined for positive times, in a suitable functional space. In order to relate System (1) with Yamamoto's realization theory, we rewrite it, after suitable time translations, as

The distribution Q is invertible over D 1 `pRq in convolution sense and the inverse distribution Q ´1 belongs to M `pRq. More precisely, the distribution

Q ´1 :" δ Λ N ˚˜`8 ÿ n"0 ˜N ÿ j"1 δ Λ j A j ¸ni
s easily seen to be the inverse of Q by a Neumann series argument. We take the convolution product of Equation (41) on the left by Q ´1 and we obtain ỹptq " `Q´1 ˚P ˚u˘p tq, t P R.

(

) 42
Applying the operator π in Equation (42), we have yp¨q " π pA ˚uq p¨q, where A :"

Q ´1 ˚P. ( 43
)
Notice also that the measure A belongs to M `pRq. As a consequence, System ( 35) is pseudorational in the sense of Yamamoto (see, e.g., [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]).

Remark 5.1. Equation ( 36) provides an expression for the relation between the input u and the output y of System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]. Other expressions for this same relation can be obtained by other means, such as by relying on the operator E q pT q as done in Section 4 or by expressing y as a Stieltjes integral of u with respect to the fundamental solution of the system, as done in the variation-of-constants formula in [START_REF] Hale | Introduction to functional-differential equations[END_REF]. While clearly equivalent, some representations can be more suitable than others for a given purpose. The representation through the operator E q pT q was useful in Section 4 to prove Theorem 3.5, and in the remaining part of the paper we will use [START_REF] Yamamoto | Coprimeness in the ring of psedorational transfer functions[END_REF] to obtain the controllability results from Theorems 3.6 and 3.7.

We now characterize the controllability notions from Definition 3.4 in terms of the above realization theory formalism. Proposition 5.2. System (1) is i) L q approximately controllable if and only if for every ϕ P X Q, q there exists a sequence of inputs pu n q nPN P pΩ q q N such that its associated sequence of outputs py n q nPN P `Lq loc `R`, R d ˘˘N through System (35) satisfies

y n ÝÑ nÑ`8 ϕ in L q loc `R`, R d ˘;
ii) L q exactly controllable if and only if for every ϕ P X Q, q there exists u P Ω such that its associated output through System (35) satisfies yp¨q " ϕp¨q.

Remark 5.3. Following Yamamoto (see, e.g., [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]), we will refer in the sequel to the characterization of approximate and exact controllability of System (1) given in Proposition 5.2 as approximate and exact controllability of System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]. Note also that, in [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF], approximate (respectively, exact) controllability of System (1) is also called quasi-reachability (respectively, reachability).

Preliminary properties and proofs of Propositions 3.11 and 3.12

Before making use of the realization theory to provide criteria for approximate and exact controllabilities in Sections 5.3 and 5.4, respectively, we provide in this section preliminary technical results relating properties of the (Laplace transform of the) distributions Q and P introduced in the framework of realization theory with H associated with System (1). In particular, we shall also obtain from those technical results a proof of Propositions 3.11 and 3.12. We start by considering the Laplace transforms of the distributions Q and P , which are given by (see formulas ( 3) and ( 4))

p Qppq " e pΛ N I d ´N ÿ j"1
e ppΛ N ´Λj q A j , p P ppq " p P " B, p P C.

In the following we use in an equivalent way B and p P depending on the context. Actually, when referring to Yamamoto's articles, it is more practical to use p P , while it is better to use B when we deal with Hautus-Yamamoto criteria for System [START_REF] Avellar | On the zeros of exponential polynomials[END_REF].

We first notice that Hp¨q and p Qp¨q satisfy the relation

Hppq " e ´pΛ N p Qppq, p P C. (44)
In Proposition 5.5 given below, we link the rank condition of the operator Hp¨q associated with System (1) with the rank condition of the operator p Qp¨q associated with System (35). As a preliminary step, let us prove the following technical lemma. Lemma 5.4. With the notations introduced above, the following properties hold true: i) There exists M P HpCq such that rank rM, Bs ă d if and only if there exist g P C d and pp n q nPN P C N with bounded real part such that }g} " 1, g T B " 0, and lim nÑ8 g T Hpp n q " 0.

ii) Under the assumption that rank rA N , Bs " d, there exists r Q P p QpCq such that rank " r Q, B ‰ ă d if and only if there exist g P C d and pp n q nPN P C N with bounded real part such that }g} " 1, g T B " 0, and lim nÑ8 g T p Qpp n q " 0.

iii) Under the assumption that rank rA N , Bs " d, Condition (b) in Proposition 3.12 is not satisfied if and only if there exist g P C d and pp n q nPN P C N with bounded real part such that }g} " 1, g T B " 0, and lim nÑ8 g T Hpp n q " 0.

Proof. We start by proving Item i). Let g P C d and pp n q nPN with bounded real part be such that }g} " 1, g T B " 0 and lim nÑ8 g T Hpp n q " 0. Since Hp¨q is uniformly bounded on any bounded vertical strip, it follows that, up to a subsequence, Hpp n q converges to some matrix M P M d,d pCq as n Ñ 8. We deduce that g T M " 0 and g T B " 0, proving that rank rM, Bs ă d. Conversely, assume that g P C d and M are such that }g} " 1, g T B " 0, g T M " 0, and M " lim nÑ8 Hpp n q for some sequence pp n q nPN P C N . The sequence pp n q nPN has bounded real part because of the following properties of H: Hppq is nonsingular for p out of a bounded vertical strip (see Proposition 3.14), Hppq converges to I d when the real part of p tends to `8, and Hppq diverges in norm when the real part of p tends to ´8 (cf. the proof of Proposition 3.14). This concludes the proof of the converse implication. The proof of Item ii) is similar to that of Item i), with p Q playing the role of H. The only difference is that, in the second part of the argument, one needs to use the assumption that rank rA N , Bs " d in order to ensure that the sequence pp n q nPN P C N has bounded real part, since p Qppq converges to ´AN (instead of I d ) when the real part of p tends to ´8 (instead of `8).

Let us prove Item iii). If condition (b) in Proposition 3.12 is not satisfied, there exist a sequence pp n q nPN P C N and a sequence of vectors pg n q nPN P `Cd ˘N such that }g T n } " 1 for all n P N, lim nÑ8 g T n B " 0, and lim nÑ8 g T n Hpp n q " 0. Proposition 3.14 and rank rA N , Bs " d imply that the real part of pp n q nPN is bounded, and thus pHpp n qq nPN is also bounded. Without loss of generality, lim nÑ8 g n " g for some vector g P C d and we deduce that }g} " 1, g T B " 0, and lim nÑ8 g T Hpp n q " 0, which proves one of the two implications. The proof of the converse is obvious.

Thanks to Lemma 5.4, we are able to prove the following proposition establishing a link between the rank properties of Hp¨q and p Qp¨q.

Proposition Proof. Item i) is a straightforward consequence of Equation (44). Let us now prove Item ii). Note that, thanks to Lemma 5.4, there exists M P HpCq such that rank rM, Bs ă d if and only if there exists g P C d and pp n q nPN P C N with bounded real part such that }g} " 1, g T B " 0, and lim nÑ8 g T Hpp n q " 0. By Equation (44), we can rewrite the last relation as lim nÑ8 g T e ´pnΛN p Qpp n q " 0, which is equivalent to lim nÑ8 g T p Qpp n q " 0, since the real part of pp n q nPN P C N is bounded.

A simple adaptation of Lemma 5.4, giving an analogue of Item iii) in terms of p Q, allows us to prove the following proposition. We conclude this section by providing the proofs of Propositions 3.11 and 3.12.

Proof of Propositions 3.11 and 3.12. We just prove Proposition 3.12 because the proof of Proposition 3.11 can be obtained following similar arguments, based on a simpler version of Lemma 5.4, adapted to the case of approximate controllability.

The equivalence between (a) and (b) is a consequence of Items i) and iii) of Lemma 5.4. Let us now prove that (c) implies (b). If (b) is not satisfied, it follows by Lemma 5.4 that there exist g P C d and a sequence pp n q nPN P C N with a bounded real part such that }g} " 1, g T B " 0, and lim nÑ8 g T Hpp n q " 0. Since Hp¨q is uniformly bounded on every bounded vertical strip of C, we deduce that condition (c) is not satisfied. Thus (c) implies (b). The converse implication is obvious.

Approximate controllability

The approximate controllability criterion for the approximate controllability of System ( 35) is an application of the paper [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] by Yamamoto.

Theorem 5.7. Let q P r1, `8q. System ( 35) is L q approximate controllable if and only if the following conditions hold true:

i) rank " p Qppq, B ı " d for every p P C, ii) rank rA N , Bs " d.
Proof. Denote by G q : Ω q ÝÑ X Q, q the map defined in Equation ( 43), which associates with every u P Ω q the element πpA ˚uq in X Q, q . We have that System ( 35) is L q approximately controllable if and only if G q pΩ q q is dense in X Q, q . Furthermore, recall that P " Bδ 0 and note that Q can be written as To conclude the proof, it remains to show that L q and L 2 approximate controllabilities are equivalent. This is achieved next by classical density arguments. Let q 1 , q 2 P r1, `8q, assume that System (35) is L q 1 approximate controllable, and let us prove that System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] is also L q 2 approximate controllable. Let y P X Q,q 2 and consider two cases: I) if q 2 ě q 1 , then consider a sequence pu n q nPN P pΩ q 1 q N such that G q 1 pu n q Ñ y. Since Ω q 2 is dense in Ω q 1 and G q 1 is continuous on Ω q 1 , we can find a sequence pũ n q nPN P pΩ q 2 q N such that G q 1 pũ n q Ñ y. Since the restriction of G q 1 to Ω q 2 is equal to G q 2 , we get that G q 2 pũ n q Ñ y.

Q " Q 0 `Q1 , with Q 0 " δ 0 A N and supp Q 1 Ă r´Λ N , ´ΛN `ΛN´1
II) if q 2 ď q 1 , up to identifying X Q,q 2 with L q 2 pr0, Λ N s, R d q, we can find a sequence py n q nPN P `XQ,q 1 ˘N such that y n Ñ y in X Q,q 2 . By the L q 1 approximate controllability and a diagonal argument, there exists a sequence pũ n q nPN P pΩ q 1 q N Ă pΩ q 2 q N such that G q 1 pũ n q Ñ y. As G q 1 pũ n q " G q 2 pũ n q for all n P N, we obtain that G q 2 pũ n q Ñ y.

This concludes the proof of the theorem.

We are finally in position to provide a proof of Theorem 3.6.

Proof of Theorem 3.6. Theorem 3.5 proves that L q approximate controllability is equivalent to the L q approximate controllability in time T " dΛ N . The conclusion follows by combining Theorem 5.7 and Item i) of Proposition 5.5.

Exact controllability

In order to take advantage of realization theory, one needs to expound and improve some of the exact controllability results obtained by Yamamoto.

Bézout's identity characterization of Radon exact controllability

As explained in the introduction, Yamamoto's realization approach tackles exact controllability in distributional sense. Thanks to the specific features of our difference delay system, we are able to replace the general distributional framework of Yamamoto by the more structured setting of distributions of order zero (or, equivalently, Radon measures). More precisely, we first define the Radon measure space X Q :" ! πΨ ˇˇΨ P pM pR `qq d and πpQ ˚πΨq " 0

) .

It is then straightforward to see that outputs of the input-output system defined in Equation (43) corresponding to inputs in pM pR ´qq m belong to X Q . We next extend the definition of exact controllability to Radon measures as follows.

Definition 5.8. System ( 35) is Radon exactly controllable if, for every πΨ P X Q with Ψ P pM pR `qq d , there exists u P pM pR ´qq m such that πpA ˚uq " πΨ.

In Proposition 5.10 below, we give a characterization of Radon exact controllability through a Bézout identity over the ring of Radon measures. We then discuss the link between such a Bézout identity and L q exact controllability in Corollary 5.12. Let us start by summarizing Lemmas 4.3, A.2, and A.3 from [START_REF] Yamamoto | Pseudo-rational input/output maps and their realizations: a fractional representation approach to infinite-dimensional systems[END_REF] in the following statement.

Lemma 5.9. The following assertions hold true: i) Let Ψ be an element in D 1 `pRq such that πpQ ˚πΨq " 0. Then there exists a sequence Ψ n P X Q,2 such that lim nÑ8 Ψ n " πΨ.

ii) For α P D 1 pRq, we have πpαq " 0 if and only if supppαq Ă p´8, 0s.

iii) πpα ˚πβq " πpα ˚βq for every α P E 1 pR ´q, β P D 1 `pRq.

Proposition 5.10. System (35) is Radon exactly controllable if and only if there exist two matrices R and S with entries in M pR ´q such that

Q ˚R `P ˚S " δ 0 I d . ( 45
)
Moreover, if (45) holds true, then, for every target output πΨ P X Q with Ψ P pM pR `qq d , the input ω P M `pRq given by ω " S ˚Q ˚Ψ (46)

steers the origin to the state πΨ along the system (35), i.e., πpA ˚ωq " πΨ.

(47)

Proof. We first prove that (45) is a necessary condition for Radon exact controllability of [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]. The entries of Q ´1 are in the space M `pRq and we have πpQ ´1q " Q ´1 so that the columns of Q ´1 are in X Q . Since System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] is Radon exactly controllable, we can take Ψ equal to each column of Q ´1 in Definition 5.8 and thus deduce the existence of a matrix S of size m ˆd and with entries in M pR ´q such that

πpQ ´1 ˚P ˚Sq " π `Q´1 ˘. (48)
We define R :"

Q ´1 ´Q´1 ˚P ˚S. (49)
From Equation (48), we deduce that πpRq " 0 and Item ii) of Lemma 5.9 proves that the support of R is included in R ´. Since the supports of Q ´1, P , and S are bounded on the left, the same is true for the support of R, which is, therefore, compact. Moreover the entries of R are distributions of order zero because the same is true for Q ´1, P , and S. Hence the entries of R belong to M pR ´q, and we finally obtain (45) by taking the convolution product of (49) on the left by Q.

We now prove that Condition (45) is sufficient for Radon exact controllability of [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]. Let R and S be two matrices with entries in the space M pR ´q satisfying Equation (45). Let Ψ P pM pR `qq d be such that πpQ ˚πΨq " 0,

so that πΨ P X Q . Set ω :" S ˚Q ˚Ψ. Since Q and S have entries in M pR ´q, we have that ω belongs to the space M `pRq. Item iii) of Lemma 5.9 implies that π pωq " π pS ˚π pQ ˚πΨqq " 0, and we deduce that ω is in pM pR ´qq m by Item ii) of Lemma 5.9. By definition of ω and Equation (45), we also have

Q ´1 ˚P ˚ω `R ˚Q ˚Ψ " Q ´1 ˚pP ˚S `Q ˚Rq ˚Q ˚Ψ " Ψ. (51)
We deduce from Equation (51), Condition iii) of Lemma 5.9, and (43) that πΨ " π `Q´1 ˚P ˚ω˘`π pR ˚Q ˚Ψq " π pA ˚ωq `π pR ˚π pQ ˚πΨqq " π pA ˚ωq , where the last equality follows from (50). We have shown that πΨ is the output corresponding to ω for the input-output map (43), proving the Radon exact controllability of [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF].

Remark 5.11. Proposition 5.10 can be seen as a Radon counterpart of the distributional Bézout identity characterization of exact distributional controllability provided in [START_REF] Yamamoto | Behavioral controllability and coprimeness for a class of infinite-dimensional systems[END_REF].

As a consequence of Proposition 5.10, we can now deduce that Radon exact controllability implies L q exact controllability. Corollary 5.12. If System (35) is Radon exactly controllable then it is L q exactly controllable for every q P r1, `8q.

Proof. Let R and S be two matrices with entries belonging to the space M pR ´q satisfying Equation (45). Let q P r1, `8q. Let y P X Q, q and denote by ỹ the extension of y on p´8, `8q by setting ỹ equal to zero on p´8, 0q. Set ω " S ˚Q ˚ỹ as in (46). Then ω is in Ω q and it follows from (47) that y " π pA ˚ωq, which proves L q exact controllability thanks to Proposition 5.2.

A challenging question is to investigate a converse to the previous corollary, i.e., whether L q exact controllability for some q P r1, `8q implies Radon exact controllability or not. We bring a positive answer to that question for q " 1 in the following section.

Bézout's identity characterization of L 1 exact controllability

We next provide a sufficient and necessary condition for System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] to be L 1 exactly controllable.

Theorem 5.13. System (35) is L 1 exactly controllable if and only if there exist two matrices R and S with entries in M pR ´q such that (45) holds true.

Proof. We first notice that, by Proposition 5.10 and Corollary 5.12, Condition (45) implies that System (35) is L 1 exactly controllable.

Let us now prove the converse implication. We proceed in four steps.

Step 1. Let us define the map r G : r

Ω 1 ÝÑ L 1 pr0, Λ N s, R d q by r Gpuqptq " πpQ ´1 ˚P ˚uqptq, t P r0, Λ N s, u P r Ω 1 ,
where r Ω 1 denotes the subspace of Ω 1 made of inputs with compact support in r´dΛ N , 0s, endowed with the norm }.} r´dΛ N ,0s,1 . Firstly, we can see that the map r G is a bounded operator because Q ´1 ˚P is a distribution with a finite number of Dirac distributions on each compact interval of R. We deduce that r G is a continuous linear map. Secondly, the saturation of the endpoint map (Theorem 4.7) allows us to state that System (35) is L 1 exactly controllable if and only if the map r G is surjective. We can now apply the open mapping theorem (see, e.g, [START_REF] Rudin | Functional Analysis. International series in pure and applied mathematics[END_REF]Theorem 4.13]) and deduce that there exists δ ą 0 such that

r GpU q Ą δV, (52)
where U and V are the open unit balls of r Ω 1 and L 1 pr0, Λ N s, R d q respectively.

Step 2. Since πpQ ˚πpQ ´1qq " πpδ 0 q " 0 and the inclusion X Q,2 Ă X Q,1 holds, Item i) of Lemma 5.9 implies that there exists a sequence ψ n " pψ n,1 , . . . , ψ n,d q P pX Q,1 q d , n P N, such that ψ n Ñ πpQ ´1q in the distributional sense as n Ñ 8. In other words, for i, j P t1, . . . , du, if we define the Radon measures pQ ´1 n q i,j ptq " ş t 0 pψ n q i,j pxqdx for t P R `and n P N, we get that pQ ´1 n q i,j weak-star converges to pπpQ ´1qq i,j in the sense of [START_REF] Maggi | Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory[END_REF]Paragraph 4.3]. By [START_REF] Maggi | Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory[END_REF]Remark 4.35] and the Banach-Steinhaus theorem, we obtain that the total variation sup nPN `Q´1

n ˘i,j pr0, Λ N sq ă 8,

where

`Q´1 n ˘i,j pr0, Λ N sq :" sup φPCpr0,Λ N sq ||φ|| Cpr0,Λ N sq ď1 ż Λ N 0 φptqd `Q´1
n ˘i,j ptq , with Cpr0, Λ N sq the space of the continuous functions defined on the interval r0, Λ N s with values in R endowed with its natural norm ||¨|| Cpr0,Λ N sq By the Riesz representation theorem, see [START_REF] Rudin | Real and complex analysis[END_REF]Theorem 6.19], we have that | pQ ´1 n q i,j |pr0, Λ N sq denotes the total variation of the measure on the interval r0, Λ N s of pQ ´1 n q i,j in the sense of [START_REF] Rudin | Real and complex analysis[END_REF]Chapter 6,Equation (3)]. In particular, we have that

`Q´1 n ˘i,j pr0, Λ N sq " ż Λ N 0 pψ n q i,j ptq dt,
so that each column of pψ n q nPN is uniformly bounded in L 1 pr0, Λ N s, R d q, that is, there exists C ą 0 such that }ψ n,j } L 1 pr0,Λ N s,R d q ď C, @j P 1, d , @n P N.

Let M 1 ą 0 be such that δM 1 ą C. We get from Equation (52) that

r GpM 1 U q Ą δM 1 V
so that, for all ψ n,j with j P 1, d and n P N, there exists S n,j P r Ω 1 such that r GpS n,j q " ψ n,j and }S n,j } r´dΛ N ,0s, 1 ď M 1 .

Step 3. We define S n " pS n,1 , . . . , S n,d q. By construction, S n P r Ω d 1 and

πpQ ´1 ˚P ˚Sn q Ñ πpQ ´1q, as n Ñ 8, (53)
in a distributional sense. Since the columns of S n , for n P N, are uniformly bounded for the norm in r Ω 1 , by the weak compactness of Radon measures (see for instance [START_REF] Maggi | Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory[END_REF]Theorem 4.33]), there exist a matrix S with entries in M pR ´q and a subsequence of pS n q nPN (still denoted pS n q nPN by abuse of notation) such that lim nÑ`8 S n " S in distributional sense. Since the convolution is continuous in distributional sense (see, for instance, [4, Theorem 7.4.9]), and π is continuous with respect to the strong dual topology, we deduce from Equation (53) that πpQ ´1 ˚P ˚Sq " π `Q´1 ˘.

(54)

Step 4. We define R :" Q ´1 ´Q´1 ˚P ˚S (55)

and we conclude the proof as in Proposition 5.10: by Equation (54) we have πpRq " 0, which, together with Condition ii) of Lemma 5.9, implies that the support of R is compact and contained in p´8, 0s. Moreover, the entries of R are distributions of order zero because the same is true for Q ´1, P , and S. Bézout's identity is then obtained by multiplying Equation (55) on the left by Q in convolution sense.

Remark 5.14.

It is an open question whether, for each q P r1, `8q, the Bézout identity is equivalent to the L q controllability. We can stress that the proof given for the case q " 1 does not straightforwardly extend to q ą 1. Indeed, in Step 2, the convergence ψ n Ñ πpQ ´1q in the distributional sense implies the boundedness of pψ n q nPN in L 1 pr0, Λ N s, R d q, but such a sequence may fail to be bounded in L q pr0, Λ N s, R d q for q ą 1.

An immediate consequence of the Bézout identity characterization of the Radon and L 1 exact controllability is the following corollary. The open question raised in Remark 5.14 is tantamount to know if, for difference delay systems of the type (1), the L q exact controllability for some q P r1, `8q is equivalent to the L q exact controllability for every q P r1, `8q. It has to be pointed out that such a property holds when dealing with the exponential stability of such systems, as noticed in [START_REF] Baratchart | Sufficient stability conditions for time-varying networks of telegrapher's equations or difference-delay equations[END_REF][START_REF] Chitour | Stability of non-autonomous difference equations with applications to transport and wave propagation on networks[END_REF]. Definition 5.16. We say that the Bézout identity (45) is solvable if there exist two matrices R and S with entries in M pR ´q that satisfy it.

We next show that the solvability of the Bézout identity is equivalent to a corona problem, the latter having led a tremendous literature on the subject for some alike problems. We can cite for example the celebrated paper [START_REF] Carleson | Interpolations by bounded analytic functions and the corona problem[END_REF] resolving the corona problem in one dimension for holomorphic bounded functions in the unit disk. However the present corona problem arising from the Bézout identity over a Radon measure algebra has not received a great attention and it is still an open question. To the best of our knowledge, the closest result dealing with the solvability of a Bézout identity is that provided in [START_REF] Yamamoto | Bézout identity over a ring of distributions-multivariable case[END_REF]Theorem 5.1] over the algebra of distributions, which gives a sufficient condition on pQ, P q for such a solvability. In turn, solving a Bézout identity over Radon measures is fundamentally different than over distributions because of the different topologies endowing these two spaces.

Our first result for Radon measures is the following necessary condition for the solvability of the Bézout identity. Proposition 5.17. A necessary condition for the Bézout identity (45) to be solvable is the following i) there exists α ą 0 such that, for every p P C, inf

!ˇˇˇˇˇˇg T p Qppq ˇˇˇˇ`ˇˇˇˇg T B ˇˇˇˇˇˇˇg P C d , }g T } " 1 ) ě α. (56)
Notice that a proof of Conjecture 5.18 would imply a positive answer to Conjecture 3.8 in the case q equal to one, yielding a sufficient and necessary criterion for the L 1 exact controllability.

Sufficient exact controllability criterion for C k functions

We close this section by providing a positive exact controllability result which relies on the sufficient condition for the resolution of the Bézout identity over the distributional algebra in dimension d given in [START_REF] Yamamoto | Bézout identity over a ring of distributions-multivariable case[END_REF], i.e., we provide an exact controllability criterion for steering the origin to regular solutions along System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF]. More precisely, let C k pR `, R d q (respectively, C k pR, R d q), with k integer, denote the space of k times continuously differentiable functions defined on R `(respectively, R).

We need the following definition of controllability to agree with the C k functions.

Definition 5.20. Let k be an integer and set

X Q k :" ␣ y P C k pR `, R d q | πpQ ˚yq " 0 ( .
System ( 35) is C k exactly controllable if, for every y P X Q k , there exists u P C 0 pR, R m q compactly supported in R ´such that πpA ˚uq " y.

Remark 5.21. Note that, contrarily to the notions of controllability that we introduced previously, C k exact controllability requires the control u to belong only to the less regular space of continuous functions, instead of requiring it to belong to a space similar to that of the state y.

The proposition below states a sufficient controllability criteria for the C k functions. Proposition 5.22. If Condition i) of Proposition 5.17 holds true then System (35) is C k exactly controllable for some integer k.

Proof. In this proof, we fix ||¨|| as the Euclidean norm. We proceed in two steps.

Step 1. We aim to apply [START_REF] Yamamoto | Bézout identity over a ring of distributions-multivariable case[END_REF]Theorem 5.1]. Note first that, thanks to the classical Pólya-Szegő bound provided in [26, Part Three, Problem 206.2], there exists a positive integer D such that the zeros of det ´p Qp¨q ¯have multiplicity at most D.

Assume that Condition i) of Proposition 5.17 holds true. Pick p P C such that det p Qppq " 0 and denote by ℓ the dimension of the left kernel of p Qppq. Let G p P M ℓ,d pCq be such that its rows form an orthonormal basis of the left kernel of p Qppq. Hence, G p p Qppq " 0 and ˇˇˇˇz T G p ˇˇˇˇ" ||z|| for every z P C ℓ .

We claim that the rows of the matrix G p p P P M ℓ,m pCq are linearly independent. Indeed, if this were not the case, there would exist z P C ℓ such that ||z|| " 1 and z T G p p P " 0. Thus we would have z T G p p Qppq " 0, z T G p p P " 0, and ˇˇˇˇz T G p ˇˇˇˇ" 1, in contradiction with Condition i) of Proposition 5.17.

We deduce that G p p P has right inverses and we use pG p p P q ´1 to denote its Moore-Penrose right inverse, i.e., pG p p P q ´1 " pG p p P q T pG p p P p P T G T p q ´1. Define ˇˇˇˇ" 1 for every n P N, and G pn p Qpp n q " 0, we obtain a contradiction with Condition i) in Proposition 5.17, yielding (61).

The assumptions (18) 1 and (17) in the statement of [START_REF] Yamamoto | Bézout identity over a ring of distributions-multivariable case[END_REF]Theorem 5.1] are satisfied thanks to Equations (60) and (61), respectively. We can then apply [37, Theorem 5.1], which ensures the existence of two distribution R and S with entries in E 1 pR ´q such that Q ˚R `P ˚S " δ 0 I d .

Step 2. Since S is a distribution with compact support, it has a finite order k. Let y P X Q k and denote by ỹ P C k pR, R d q an extension on R of y having a support bounded on the left. Set ω " S ˚Q ˚ỹ. Then ω belongs to C 0 pR, R m q and it has a compact support included in R ´. It follows from a similar argument given to obtain the equation (47) that y " π pA ˚ωq, which proves the C k exact controllability.

Proof of Theorem 3.10. Notice that y P X Q k if and only if r´Λ N , 0s Q t Þ Ñ ypt `ΛN q is C kadmissible for System (1) in the sense of Definition 3.9. Recall that, by Proposition 3.12, Conditions i) and ii) of Conjecture 3.8 imply that Condition i) of Proposition 5.17 holds true.

1 Equation ( 18) in [START_REF] Yamamoto | Bézout identity over a ring of distributions-multivariable case[END_REF] is the Laplace transform of Equation ( 8) of the same article, which is an equation in pE 1 pRqq dˆd { pQq, where pQq is the ideal generated over E 1 pR ´q by Q. Thus the equality in [START_REF] Kalman | Realization of continuous-time linear dynamical systems: rigorous theory in the style of schwartz[END_REF] has to be understood as an equality modulo the ideal generated by p Qppq, which is equivalent to requiring that z T ´p P ppq p Φppq ´Id ¯" 0 for every z T in the left null space of p Qppq. Therefore, Equation ( 18) in [START_REF] Yamamoto | Bézout identity over a ring of distributions-multivariable case[END_REF] is equivalent to our Equation (73).

Assume, for instance, that pA 1 , Bq is controllable and pA 2 , Bq is not. Hence, up to a linear change of variables, we can assume with no loss of generality that

A 1 " ˆa1 a 2 a 3 a 4 ˙, A 2 " ˆb1 0 b 2 b 3 ˙, B " ˆ0 1 ˙,
with a 2 ‰ 0 and b 1 ‰ 0. For g P C 2 with ||g|| " 1, one has either g T B ‰ 0 or g T " pα, 0q for some α P C with |α| " 1. Hence, in order to show Condition (b) of Proposition 3.11, it suffices to show that ˇˇˇˇg T Hppq ˇˇˇˇi s nonzero for every p P C and with g T " p1, 0q. For every p P C, one checks that the second coordinate of g T Hppq is equal to ´a2 e ´pL , which never vanishes, yielding the conclusion. The case where pA 1 , Bq is not controllable and pA 2 , Bq is can be handled similarly. III) pA 1 , Bq and pA 2 , Bq are both controllable. Let B K P R 2 be the unique vector such that detpB, B K q " 1 and B T B K " 0. Set

β " det prB, A 2 Bsq det prB, A 1 Bsq , α " det `rB, pA 2 ´βA 1 qB K s ˘.
Up to a linear change of coordinates, we can assume that

A 1 " ˆ0 1 a 1 a 2 ˙, A 2 " ˆα β b 1 b 2 ˙, B " ˆ0 1 ˙. (62)
The holomorphic map H is now given by Hppq " I In the third case above, the fact that there exists p P C so that 1 ´αe ´p " 0 and e ´pL `βe ´p " 0 can be equivalently written as

M Y ppq " ˆ1 0 ˙, where M " ˆ0 α 1 β ˙and Y ppq " ˆe´pL e ´p ˙.
The above computations are a particular case of a more general situation, described next. ii) If M is invertible, set M ´1Z " pa 1 , a 2 q T and define Our two main results allowed us to recover partially, in the case of matrices with real coefficients, the algebraic controllability result stated in [7, Theorem 4.1] when N " d " 2 and m " 1, which consider L 2 (approximate and exact) controllability concepts while in the present paper, we are instead dealing with L q approximate controllability and L 1 exact controllability. We have shown that three possibilities occur: I) neither approximate nor exact controllability hold true; II) L q approximate controllability holds true but we cannot say anything about exact controllability; III) there exists a countable subset S of the complex plane, completely characterized by A 1 , A 2 , B, and L, which is dense in a circle C, such that L q approximate controllability is equivalent to the fact that 0 R S and a necessary condition for L 1 exact controllability is 0 R C. Remark 6.2. For Case III), we can assume with no loss of generality (see [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF]Lemma 4.5]) that a 1 " a 2 " b 1 " b 2 " 0 in (62), so that Theorem 5.13 implies that System (1) is L 1 exactly controllable if and only if the Bézout identity (45) is solvable in M pR ´q, that is, if and only if there exists ps 1 , s 2 , r 1 , r 2 , r 3 , r 4 q P M pR ´q6 such that $ ' ' ' & ' ' ' % pδ ´1 ´αδ 0 q ˚r1 ´pδ L´1 `βδ 0 q ˚r2 " δ 0 , pδ ´1 ´αδ 0 q ˚r3 ´pδ L´1 `βδ 0 q ˚r4 " 0, δ ´1 ˚r2 `s1 " 0, δ ´1 ˚r4 `s2 " δ 0 .

S " ␣ a 1 ´|a 2 | L e iLpθ 2 `
(64)

The second and fourth equations of (64) are satisfied if one chooses r 3 " δ L´1 `βδ 0 , r 4 " δ ´1 ´αδ 0 , s 2 " δ 0 `αδ ´1 ´δ´2 .

In addition, as soon as the first equation of (64) is satisfied, the third one can be satisfied by setting s 1 " ´δ´1 ˚r2 , and we hence focus in the first equation of (64) in the sequel, i.e., pδ ´1 ´αδ 0 q ˚r1 ´pδ L´1 `βδ 0 q ˚r2 " δ 0 .

(65)

Let q 1 " δ ´1 ´αδ 0 and q 2 " ´δL´1 ´βδ 0 , and notice that their Laplace transforms are given by p q 1 ppq " e p ´α and p q 2 ppq " ´ep1´Lqp ´β for p P C. By definition of S, we have that 0 P S if and only if there exists p P C such that p q 1 ppq " p q 2 ppq " 0, and one can show that 0 P S if and only if inf pPC |p q 1 ppq| `|p q 2 ppq| " 0. Hence, the condition 0 R S is equivalent to the existence of a constant c ą 0 such that |p q 1 ppq| `|p q 2 ppq| ě c for every p P C. If Conjecture 5.18 held true, then it would imply that there exist r 1 and r 2 in M pR ´q such that their Laplace transforms satisfy p q 1 ppqp r 1 ppq `p q 2 ppqp r 2 ppq " 1, which is exactly the Laplace transform of (65). In particular, this shows that, if 0 R S and Conjecture 5.18 holds true, then System (1) in the case of the present example is L 1 exact controllable.

Note that the condition 0 R S is shown in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] to be a necessary and sufficient condition for the L 2 exact controllability of System (1) in Case III). Unfortunately, even for this simple example, it is not clear if the L 2 exact controllability is equivalent to the L 1 exact controllability.

6.2 Two delays, three space dimensions, and a single input: a geometric locus controllability result

In this section, we assume that N " 2, d " 3, and m " 1. In this case A 1 , A 2 P M 3,3 pRq and B P M 3,1 pRq. Up to a linear change of variables, we assume in the sequel that B " p0, 0, 1q T . We use r 0 to denote the dimension of the vector space spanned by the three vectors B, A 1 B, A 2 B.

We subdivide the discussion in three cases.

I) rankrA 2 , Bs ď 2 or r 0 " 1. Then, in both subcases, System (1) is not even approximately controllable. Indeed, in the first subcase, the rank condition ii) of Theorem 3.6 is violated. In the second subcase, the matrices A 1 and A 2 are of the form

A 1 " ˆr A 1 0 2,1 ˚a1 ˙, A 2 " ˆr A 2 0 2,1 ˚a2
˙.

Then the subsystem of System (1) made of the first two coordinates is uncoupled to the third coordinate and is uncontrolled. The overall system is then not controllable. II) rankrA 2 , Bs " 3 and r 0 " 2. We only treat the case where A 2 B is not colinear to B (the other case being entirely similar). Up to a linear change of variables (in a basis having A 2 B as second vector and B as third one), one can transform the matrices A 1 and A 2 into the form

A 1 " ¨r A 1 0 α ˚˚' , A 2 " ¨r A 2 0 1 ˚0'
, where the matrices r A 1 and r A 2 are 2 ˆ2.

The holomorphic map H associated with System (1) is equal to

Hppq " ¨I2 ´e´pL r A 1 ´e´p r A 2 0 ´pe ´pL α `e´p q ˚˚' . ( 66
)
Then, by Theorem 3.6 and Equation (66), System (1) is not L q approximately controllable in time 3Λ 2 if and only if there exists p P C such that rank rHppq, Bs ă 3, i.e., either (a) e ´pL α `e´p " 0 and detpI 2 ´e´pL r A 1 ´e´p r A 2 q " 0 or (b) e ´pL α `e´p ‰ 0 and the p1, 1q and p1, 2q coefficients of the 3 ˆ3 matrix Hppq are both equal to zero.

In the subcase (a), the second condition is equivalent to the fact that det `epL I 2 ´p r A 1 ά r A 2 q ˘" 0, i.e., e pL is one of the eigenvalues of r A 1 ´α r A 2 . Hence we are in the situation of Lemma 6.1 with M " ˆα 1 1 0 ˙and at most two vectors Z corresponding to the nonzero eigenvalues of r A 1 ´α r A 2 . It follows that there exist at most two countable sets of complex numbers S 1 , S 2 , completely characterized in terms of L and the coefficients of A 1 , A 2 , which are dense in two circles C 1 , C 2 , respectively, and such that System (1) is L q approximately controllable if and only if 0 does not belongs to the union of S 1 and S 2 , while System (1) is not L 1 exactly controllable if 0 belongs to the union of C 1 and C 2 . (Note that each S i can be equal to C i if the latter reduces to a point.) It is not difficult to see that the subcase (b) also boils down to a similar situation, but with at most one circle. III) rankrA 2 , Bs " 3 and r 0 " 3. Up to a linear change of variables (in a basis having A 1 B as first vector, A 2 B as second vector, and B as third one), one can transform the matrices A 1 and A 2 into the form

A 1 " ¨r A 1 1 0 ˚0' , A 2 " ¨r A 2 0 1 ˚0'
, where the matrices r A 1 and r A 2 are 2 ˆ2.

The holomorphic map H associated with System (1) is equal to

Hppq " ¨I2 ´e´pL r A 1 ´e´p r A 2 ´e´pL ´e´p ˚0 '.
We denote by v 1 and v 2 the columns of the matrix r Hppq " I 2 ´e´pL r A 1 ´e´p r A 2 and we set v 3 " pe ´pL e ´pq T . Note that the rank assumption on rA 2 , Bs implies that the first row of r A 2 is not equal to zero.

It is immediate to see that there exists p P C so that rank rHppq, Bs ă 3 if and only if both v 1 and v 2 are colinear to the nonzero vector v 3 , i.e., detpv 1 , v 3 q " 0 and detpv 2 , v 3 q " 0. These conditions can be rewritten as two scalar equations v T 3 Q 1 v 3 `e´p " 0 and v T 3 Q 2 v 3 `e´pL " 0, where Q 1 , Q 2 are symmetric matrices with real coefficients and at least one between Q 1 and Q 2 is not equal to zero because of the rank assumption on rA 2 , Bs. The two previous equations in the unknowns e ´pL and e ´p define two distinct conic sections, at least one of them being nontrivial. Therefore, they have therefore k distinct intersection points with 0 ď k ď 4. For each of them, one can completely characterize, in terms of L and the coefficient of A 1 , A 2 , a countable set of complex numbers S j , 1 ď j ď k, which is dense in a circle C j , such that System (1) is L q approximately controllable if and only if 0 does not belongs to the union of the S j , while System (1) is not L 1 exactly controllable if 0 does belongs to the union of the C j .

In conclusion, if N " 2, d " 3, and m " 1, Theorem 3.6 (respectively, Theorem 3.7) allows one to derive frequency-free necessary and sufficient (respectively, necessary) criteria for L q approximate (respectively, L 1 exact) controllability, and the results are qualitatively similar to those obtained if N " d " 2 and m " 1, with the difference in Case III) where now we may have up to four distinct countable sets S j , 1 ď j ď k, each of them dense in a circle C j .

Proposition 3 . 11 . 1 (

 3111 The following three statements are equivalent: (a) rank rHppq, Bs " d for every p P C. (b) For every p P C, one has inf ␣ˇˇˇˇg T Hppq ˇˇˇˇ`ˇˇˇˇg T B ˇˇˇˇˇˇg P C d , }g} "

Proposition 5 . 6 .

 56 Under the assumption that rank rA N , Bs " d, the following statements are equivalent: there exists α ą 0 such that, for every p P C,inf !ˇˇˇˇˇˇg T p Qppq ˇˇˇˇ`ˇˇˇˇg T B ˇˇˇˇˇˇˇg P C d , }g T } " 1 ) ě α.

Corollary 5 . 15 .

 515 System (35) is Radon exactly controllable if and only if it is L 1 exactly controllable.

5. 4 . 3 7 Proposition 5 .

 4375 Solvability of Bézout's identity over Radon measure spaces and proof of Theorem 3.10 and Theorem 5.13 reduced the exact controllability problem to the problem of existence of solutions of a suitable Bézout identity.

Lemma 6 . 1 .

 61 Let M be a 2 ˆ2 matrix with complex coefficients, Z P C 2 zt0u, L P p0, 1qzQ, and define the map Y : p Þ Ñ pe ´pL , e ´pq T . i) If M is not invertible, then the following assertions are equivalent: (a) Z P Ran M ;(b) Z P M Y pCq; (c) Z P M Y pCq.

  where λ min p¨q denotes the smallest eigenvalue of its argument. Let pz n q nPN denote a sequence of vectors in C ℓ such that, for every n P N, ˇˇˇˇz

				Φppq " ´Gp p P	¯´1	G p
	and notice that p Φppq satisfies
				G p p P p Φppq " G p .	(60)
	We now claim that there exists c ą 0 (independent of the zero p of det p Q) such that
				ˇˇˇˇˇˇp Φppq ˇˇˇˇˇˇď c.	(61)
	Indeed, arguing by contradiction yields a sequence pp n q nPN of zeros of det p Q such that
	ˇˇˇˇˇˇˇˇˇ´G pn p P	¯´1	ˇˇˇˇˇˇˇˇˇt ends to infinity as n Ñ `8. We have ˇˇˇˇˇˇˇˇˇ´G pn p P	¯´1	ˇˇˇˇˇˇˇˇˇ" 1{λ min pG pn p P p P T G T pn q,
				T n	ˇˇˇˇ"	1 and z n is an eigenvector of G pn p P p P T G T pn
	associated with its smallest eigenvalue. Then
				ˇˇˇˇz T n G pn p P ˇˇˇˇ2 " λ min pG pn p P p P T G T pn q ÝÑ nÑ`8	0,
	and, since	ˇˇˇˇz n G pn T

  1´L e ipθ`2kπqp1´Lq ˇˇk P Z ( . Since L is irrational, notice that S, is the circle in C of center β and radius |α| 1´L , denoted hereafter by C. By Theorem 3.7 and Equation (63), one can prove in the same way that System (1) is not L 1 exactly controllable in time 2Λ 2 if 0 P C.

	2	´e´pL A 1	´e´p A 2 "	´p ˚˚˙, ˆ1 ´αe ´p ´e´pL ´βe	p P C.	(63)

Let α " |α|e iθ with θ P t0, πu. Since β ‰ 0, we have that rankrA 2 , Bs " 2. Then, by Theorem 3.6 and Equation (63), we have that System (1) is not L q approximately controllable in time 2Λ 2 if and only if Dp P C s.t. rank rHppq, Bs ă 2 ðñ Dp P C s.t. 1 ´αe ´p " 0 and e ´pL `βe ´p " 0 ðñ 0 P S, where S " ␣ β `|α|

  2kπq ˇˇk P Z ( , where a 2 " |a 2 |e iθ 2 and θ 2 P R. Then the following assertions hold true: ii-i) Z P M Y pCq if and only if 0 P S; ii-ii) Z P M Y pCq if and only if 0 P C, the latter being the circle in C of center a 1 and radius |a 2 | L .

G.M. was partially supported by ANR PIA funding: ANR-20-IDEES-0002.

A j xpt ´Λj q `Buptq, for t ě inf supppuq, xptq " 0, for t ă inf supppuq, yptq " xpt ´ΛN q, for t P r0, `8q, [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] where the input u belongs to Ω q " tu P L q pR ´, R m q | supppuq is compactu, with supppuq denoting the support of u.

We aim at writing System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] as a convolution operator with a kernel in the space of Radon measures, i.e., we want to find A P M `pRq such that the input-output system [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] can be represented as yptq " pA ˚uqptq " ż `8 ´8 dApτ qupt ´τ q, t P r0, `8q. [START_REF] Yamamoto | Coprimeness in the ring of psedorational transfer functions[END_REF] Note that the convolution of a d ˆm matrix-valued Radon measure with a compactly supported function in L q pR, R m q belongs to L q pR, R d q. Recalling that π is the truncation operator on positive times defined in Equation ( 5), we can rewrite Equation [START_REF] Yamamoto | Coprimeness in the ring of psedorational transfer functions[END_REF] in a more convenient way as y " πpA ˚uq.

To achieve the goal of finding a Radon measure A satisfying Equation [START_REF] Yamamoto | Bézout identity over a ring of distributions-multivariable case[END_REF], we define the zero-order distributions

The matrix-valued distributions Q and P are in M pR ´q, the space of Radon measures with compact support included in R ´. These distributions are naturally associated with System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] for two major reasons. We define the state space of System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] in terms of the distribution Q as

The system thus has an input u belonging to Ω q and an output y in X Q, q . We remark that the set X Q, q can be easily identified with the space L q `r0, Λ N s , R d ˘. In fact, y P X Q, q if and only if the restriction y| r0,Λ N s is in L q pr0, Λ N s, R d q and y is the unique extension of y| r0,Λ N s on the interval r0, `8q satisfying the condition πpQ ˚yq " 0.

The distributions Q and P allow us to obtain the Radon measure A representing System [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] as a convolution operator. Denoting by ỹ the natural extension of the output y on R, i.e., ỹptq " xpt ´ΛN q for t P R, the first equation of [START_REF] Yamamoto | Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems[END_REF] implies that pQ ˚ỹq ptq " pP ˚uq ptq, t P R.

(41)

Proof. Let R and S be the two matrices with entries in M pR ´q satisfying Bezout's identity (45). By contradiction, we assume that Condition i) is not satisfied, so that there exist a sequence pp n q nPN P C N and a sequence pg n q nPN P `Cd ˘N such that }g T n } " 1 for all n P N, lim nÑ8 g T n B " 0, and lim nÑ8 g T n p

Qpp n q " 0. We claim that there exists α ą 0 such that ℜpp n q ď α for every n P N. Indeed, p

Qpp n q is equivalent to e pnΛ N I d when n tends to `8, and unboundedness from above of ℜpp n q would contradict the relation lim nÑ8 g T n p Qpp n q " 0. We next get, by a classical estimate of the Laplace transform of an element of M pR ´q, that there exists C ą 0 such that, for all p P C with ℜppq ď α, max

We then deduce from Equation (45) and the Laplace transform that

Equations ( 57) and ( 58) imply that it is impossible to have both lim nÑ8 g T n B " 0 and lim nÑ8 g T n p Qpp n q " 0. We reached a contradiction, so that Condition i) is a necessary condition for the solvability of Bézout's identity.

We can now undertake the proof of Theorem 3.7.

Proof of Theorem 3.7. Theorem 3.5 states that L 1 exact controllability is equivalent to the L 1 exact controllability in time T " dΛ N . By Theorem 3.6, Condition ii) of Theorem 3.7 is necessary for L 1 approximate controllability, so that it is also necessary for L 1 exact controllability. Combining Theorem 5.13 and Proposition 5.17, we deduce that (56) is a necessary condition for L 1 exact controllability, and the equivalence between (56) and Condition i) of Theorem 3.7 is a consequence of Item i) of Proposition 5.5 and Proposition 5.6.

The necessity of Condition i) of Proposition 5.17 is the easy part for the solvability of Bézout's identity over the Radon measure algebra. The harder part would be to prove that Condition i) is also a sufficient condition to have the solvability of Bézout's identity. A first step of such a proof would consist in using exactly the trick from [START_REF] Fuhrmann | On the corona theorem and its application to spectral problems in Hilbert space[END_REF], which reduces a corona matrix problem to the corresponding 1D version, which can be written as follows.

Conjecture 5.18. Let k P N and q j P M pR ´q for j " 1, . . . , k. Assume the existence of a c ą 0 such that

Then there exists p j P M pR ´q for j " 1, . . . , k satisfying the equation k ÿ j"1 p q j ppqp p j ppq " 1, p P C.

Remark 5.19. In fact, due to the structure of the difference delay systems that we consider, i.e., with a finite number of delays, it would be sufficient to prove Conjecture 5.18 for the measures q j , for all j " 1, . . . , k, belonging to the sub-algebra of Radon measures M pR ´q finitely generated by the Dirac measures pδ Λ i ´ΛN q i"1,...,N .

Hence, by Proposition 5.22, every C k -admissible function for System (1) is in the range of E q pT q for some T ą 0 possibly depending on y. By Theorem 4.7, moreover, the space of C k -admissible functions for System (1) is contained in Ran E q pdΛ N q. The conclusion then follows by Proposition 4.3 and the fact that if x 0 is C k -admissible for System (1) then Υ q ptqx 0 is C k -admissible for System (1) for every t ě 0.

Applications

In this section, we show in simple cases how to derive from Theorems 3.6 and 3.7 Kalman-type conditions for controllability in the single input case m " 1. By a Kalman-type condition, we refer to a frequency-free criterion for controllability. For instance, in the single-delay case N " 1, approximate and exact controllability (in any fixed time T ě dΛ 1 ) coincide and are equivalent to the classical Kalman rank condition stating that the rank of the controllability matrix rB, A 1 B, . . . , A d´1 1 Bs is equal to the state space dimension d. We first partially recover criteria given in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] in the case of two delays, two space dimensions, and a single input, and then extend such a study to the case of two delays, three space dimensions, and a single input.

Up to a time-rescaling, we can assume from now on that pΛ 1 , Λ 2 q " pL, 1q with L P p0, 1q. The case L P Q was completely addressed in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] and we assume for the rest of this section that L is irrational.

Two delays, two space dimensions, and a single input

In this section, we recover some results given in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF]Theorem 4.1], which concern the particular case where N " d " 2 and m " 1. The results in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] are stated for matrices with complex coefficients, hence, what we actually recover here are some results of [7, Theorem 4.1] restricted to the case of matrices with real coefficients. On the other hand, the results in [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF] are stated only for L 2 controllability, but here we consider L q controllability for any q P r1, `8q.

Fix then A 1 , A 2 P M 2,2 pRq and B P M 2,1 pRq. Given a matrix A P M 2,2 pRq, we say that the pair pA, Bq is controllable if it satisfies the Kalman rank condition. We have three cases. I) Ran A 2 Ă Ran B or both pairs pA 1 , Bq, pA 2 , Bq are not controllable. In both subcases, System (1) is not even L q approximately controllable. Indeed, in the first subcase, Condition ii) of Theorems 3.6 and 3.7 is not satisfied. In the second subcase, one can assume with no loss of generality that

with b 1 ‰ 0 (otherwise we are back to the first subcase). Then the first coordinate x 1 of the state x is not controllable since one has x 1 ptq " a 1 x 1 pt ´Lq `x1 pt ´1q.

II) Ran A 2 Ć Ran B and exactly one of the pairs pA 1 , Bq, pA 2 , Bq is controllable. Then System (1) is L q approximately controllable in time 2Λ 2 . Indeed, notice first that rankrA 2 , Bs " 2, and we are thus left to show that Condition i) of Theorem 3.6 hold true, which is equivalent to proving that Condition (b) of Proposition 3.11 holds true.