Pierre Lissy

THE FICTITIOUS CONTROL METHOD FOR THE INTERNAL CONTROLLABILITY OF UNDERACTUATED SYSTEMS OF PDES

Keywords: 2010 Mathematics Subject Classification. 93B05, 93B25, 34H05, 35E99 Coupled systems, Controllability, Fictitious control method, Algebraic solvability

. The course was part of the 13th International Young Researchers Workshop on Geometry, Mechanics and Control. The aim of the lectures was to explain how some ideas initially developed by Gromov in his famous book "Partial Differential Equations" (1986, Springer), in order to find non-holonomic solutions to underdetermined partial differential equations, can be used in the context of control theory of coupled systems.

1. Introduction 1.1. Aim of the lecture notes. Studying the controllability for linear or nonlinear coupled systems of partial differential equations has been an intensive subject of interest these last years. The main issue is to try to control many equations with less controls than equations. We hope to act indirectly on the equations that are not directly controlled thanks to the coupling terms.

A typical situation is when the control appears as a source term (possibly localized in space and time) in our partial differential equation. Such a control is called an internal control. In this particular case, one way to prove a controllability property can be described as follows.

(1) Firstly, control the system with a control acting on each equation. This is in general simpler than the original underactuated problem and may be performed by using classical tools. This first step has firstly been used in [START_REF] González-Burgos | Controllability results for some nonlinear coupled parabolic systems by one control force[END_REF] in the context of coupled systems of partial differential equations.

(2) Secondly, try to find a way to get rid of the control that should not appear in order to obtain the desired result. This can be done by using the notion of "algebraic solvability", which relies on a particular construction of a solution for a related underdetermined linear partial differential equation with a source term: the solution will be written as a linear combination of the source term and its derivatives. This property is strongly related to the inversion of differential operators as presented in [START_REF] Gromov | Partial differential relations[END_REF]Section 2.3.8].

The above strategy has been firstly introduced in [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF], in the context of stabilization of ordinary differential equations, and has been extended in [START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF] in the context of controllability of partial differential equations. The goal of these lecture notes is to explain the spirit of this "fictitious control + algebraic solvability" method (that we will shorten in "fictitious control method"), in order to highlight the main features, advantages, and drawbacks of this method. The lecture notes are as self-contained as possible, up to two points: in Section 2.1, the proof of the well-posedness of our abstract setting is skipped, and in Section 3.1, the proofs of two controllability results on the scalar Schrödinger equation are also skipped. We hope that it should not cause difficulties for understanding the main ideas. They are organized as follows. Firstly, in Section 1.2, we give a simple example of the algebraic solvability of a differential system, in order to explain the spirit of the method we will use constantly in these lecture notes. Then, in Section 1.3, we explain the link with the h-principle. A very short introduction to controllability problems is provided in Section 2.1. In section 2.2, we will explain the main ideas on a very simple system of ordinary differential equations, following closely the presentation of [8, Second proof of Theorem 1.18]. In Section 2.3, we make some comments on the method. Section 3 is devoted to proving a result of controllability for a system of linear partial differential equations of Schrödinger type, which is a simplified version of the results proved in [START_REF] Liard | A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups[END_REF]. We give some concluding remarks and perspectives in Section 4.

1.2. An introducing example. In order to explain the main ideas of the following lecture notes, let us consider the following question. Let f ∈ C ∞ 0 (R, R) (that we denote from now on C ∞ 0 (R)) and a 1 , a 2 , a 3 , b 1 , b 2 , b 3 some real numbers.

Question: is it possible to find two functions x 1 and x 2 , that are also in C ∞ 0 (R), verifying the equation

a 1 x 1 -a 2 x 1 + a 3 x 1 + b 1 x 2 -b 2 x 2 + b 3 x 2 = f ? (1.1)
From an analytic point of view, this equation is underdetermined (there are more unknowns (2, the functions x 1 and x 2 ) than equations (only 1)). We follow here closely the ideas developed in [START_REF] Gromov | Partial differential relations[END_REF]Section 2.3.8].

We rewrite (1.1) under the form L(x 1 , x 2 ) = f and

L = a 1 -a 2 ∂ t + a 3 ∂ tt , b 1 -b 2 ∂ t + b 3 ∂ tt : C ∞ (R) 2 → C ∞ (R).
Since we impose the solution (x 1 , x 2 ) to be compactly supported, the following remark is essential: the set C ∞ 0 (R) is stable by differentiation and linear combination. Moreover, for any ψ ∈ C ∞ 0 (R), the support of any derivative ψ (k) is included in the support of ψ. This suggests the following procedure: we aim to find a solution to (1.1) that can be written as a linear combination of the source term f and its derivatives. Such a solution will then be automatically with compact support. In other words, we would like to find a solution (x 1 , x 2 ) under the form (x 1 , x 2 ) = Mf , where M : 2 is a linear differential operator. In an equivalent way, we would like to find M such that the relation L • M = Id C ∞ (R) holds. In order to solve this equation, we need to introduce the following tool.

C ∞ (R) → C ∞ (R)
Definition 1.1. Let (p, q, r) ∈ (N * ) 3 . For i ∈ [|1, r|], we consider C i ∈ M p,q (R). We also call t C i ∈ M q,p (R) the transpose of C i .
We consider the following linear differential operator of order r with constant coefficients

P : ϕ = (ϕ 1 , . . . ϕ q ) ∈ C ∞ (R) q → r i=1 C r (ϕ (r) ) ∈ C ∞ (R) p .
We call the formal adjoint of P, that we denote by P * , the following operator:

P * : ψ = (ψ 1 , . . . ψ p ) ∈ C ∞ (R) p → r i=1 (-1) r t C r (ψ (r) ) ∈ C ∞ (R) q .
The name "formal adjoint" comes from the fact that the following property is verified: for any ϕ ∈ C ∞ 0 (R) q and any ψ ∈ C ∞ 0 (R) p , we have, using the definition of t C i on the second line and an integration by parts on the third line (there are no boundary terms since the functions we consider are compactly supported),

R Pϕ, ψ R p = r i=1 R C r (ϕ (r) ), ψ R p = r i=1 R ϕ (r) , t C r ψ R q = r i=1 R ϕ, ((-1) r t C r ψ) (r) R q = r i=1 R ϕ, (-1) r t C r (ψ (r) ) R q = R ϕ, P * ψ R q .
As for the usual adjoint of an operator, we can verify the involution property (P) * * = P and it is very easy to compute the formal adjoint of a composition P • Q (where Q is another linear differential operator with appropriate size) and obtain the formula

(P • Q) * = Q * • P * .
Coming back to (1.1), our problem can be reduced to find some operator

N : C ∞ (R) 2 → C ∞ (R) such that (1.2) N • L * = Id C ∞ (R) .
Using the involution property of the formal adjoint, one can then recover M by posing M = N * . Obviously, the identity (1.2) implies the following injectivity property: for any ψ ∈ C ∞ (R), 2 according to the previous rule:

L * ψ = 0 0 ⇒ ψ(= M * (L * ψ)) = 0. Now, we compute L * : C ∞ (R) → C ∞ (R)
L * = a 1 + a 2 ∂ t + a 3 ∂ tt b 1 + b 2 ∂ t + b 3 ∂ tt .
Let us solve L * ψ = 0 R 2 . This system is now overdetermined from an analytic point of view (we have 2 equations and 1 unknown ψ). However, from an algebraic point of view, if we forget that ψ and ψ are derivatives of ψ and we consider ψ, ψ , ψ as independent algebraic unknowns, the system becomes underdetermined because we have 2 equations and 3 unknowns. Now, we write down explicitely what is the system of equations

L * ψ = 0 R 2 : (1.3) a 1 ψ + a 2 ψ + a 3 ψ = 0, b 1 ψ + b 2 ψ + b 3 ψ = 0.
In order to make this system being well-posed in an algebraic point of view (i.e. we would like to have as many equations as independent algebraic unknowns), we differentiate these two equations. We obtain

(1.4) a 1 ψ + a 2 ψ + a 3 ψ = 0, b 1 ψ + b 2 ψ + b 3 ψ = 0.
If we are interested in the system formed by (1.3) et (1.4), we remark that if we see ψ, ψ , ψ and ψ as independent algebraic unknowns, we obtain a system with 4 equations and 4 unknowns, that we can write under the form C(ψ, ψ , ψ , ψ ) = 0 with

C =     a 1 a 2 a 3 0 b 1 b 2 b 3 0 0 a 1 a 2 a 3 0 b 1 b 2 b 3     .
Hence, under some generic algebraic relation between the coefficients of C (so that C is invertible, i.e. the determinant of C is nonzero), necessarily, we have ψ ≡ 0. Moreover, inverting the previous idea, the matrix C -1 can also be seen as a differential operator, in the sense that C -1 (y, y , y , y ) = P(y) for some linear differential operator P : We have the identity

C ∞ (R) → C ∞ (R)
C -1 C     ψ ψ ψ ψ     =     ψ ψ ψ ψ     .
If we consider the first line of this system of 4 equations, we obtain notably

ψ = c 11 (a 1 ψ + a 2 ψ + a 3 ψ ) + c 12 (b 1 ψ + b 2 ψ + b 3 ψ ) + c 13 (a 1 ψ + a 2 ψ + a 3 ψ ) + c 14 (b 1 ψ + b 2 ψ + b 3 ψ ).
Since we want to write it under the form N (L * ψ) = ψ, where L * ψ is given by

L * ψ = a 1 ψ + a 2 ψ + a 3 ψ b 1 ψ + b 1 ψ + b 3 ψ ,
we choose

N := c 11 Id + c 13 ∂ t , c 12 Id + c 14 ∂ t : C ∞ (R) 2 → C ∞ (R),
so that

N * = M = c 11 Id -c 13 ∂ t c 12 Id -c 14 ∂ t : C ∞ (R) → C ∞ (R) 2 .
Here, we say that we have solved algebraically the system L(x 1 , x 2 ) = f . The use of this notion of algebraic solvability and its link with the controllability properties of underdetermined systems will be made more precise later on.

1.3. Link with the h-principle. In this section, we aim to explain briefly in a more abstract way where the ideas that underly the approach of the previous example come from. Interested readers may consult [START_REF] Eliashberg | Introduction to the h-principle[END_REF]Part 2] and [START_REF] Gromov | Partial differential relations[END_REF]Section 2.3.8].

Consider some (linear or nonlinear, ordinary or partial) differential equation that is written as (1.5) Φ(ϕ, J k ϕ ) = 0, where ϕ = (ϕ 1 , . . . ϕ n ) is some smooth function (defined for instance on R d with d ∈ N * ) and J k ϕ represents all the derivatives up to order k ∈ N * . Note that it is a particular case of a closed partial differential relation, in the sense of [START_REF] Eliashberg | Introduction to the h-principle[END_REF]Sections 5.1 and 5.2]. Clearly, one necessary condition for (1.5) to have a solution is that the algebraic equation

(1.6) Φ(ϕ, Y ) = 0,
where Y = (y 1 , . . . y N k ), has a solution (N k is the number of derivatives of the components of ϕ up to order k). This leads to the following definitions.

Definition 1.2. A solution (ϕ, Y ) of (1.6) is called a formal (or non-holonomic) solution of (1.5). A solution ϕ to the original problem (1.5) is called a genuine (or holonomic) solution of (1.5).

Assume that (1.5) is an underdetermined system, i.e. there are more equations than "unknowns" (which are just the components of ϕ). In general, the algebraic version (1.6) will be overdetermined if we consider all the algebraic unknowns (ϕ, y 1 , . . . y N k ). To bypass this difficulty, as in the previous example, we differentiate (with respect to all the variables) as many times as needed the equations of (1.5). Since new derivatives of ϕ appear, we make some new unknowns and new equations appear.

In the case of a underdetermined linear partial differential equation with constant coefficients, it can be proved theoretically (see [START_REF] Gromov | Partial differential relations[END_REF]Section 2.3.8]) that we can differentiate enough times in order to have as many equations as "algebraic unknowns" (meaning as before that ϕ and all its derivatives are seen as independent algebraic unknowns). Then, under generic conditions on the coefficients of Φ, we can invert this well-posed algebraic system in order to solve (1.6). Hence, as in the previous example, we have found a genuine solution to (1.5) just by selecting an appropriate equation between the ones at our disposal. This leads to the following definition.

Definition 1.3. The system (1.5) satisfies the h-principle if any formal solution (ϕ, Y ) of (1.5) can be deformed into a genuine solution g of (1.5) in the space of formal solutions of (1.5), in the following sense: one can find an homotopy (i.e. a continuous deformation) that brings (ϕ, Y ) to g, in the class of non-holonomic solutions.

Note that all these definitions can be extended to the more general setting of partial differential relations.

2. Controllability of a coupled system of ordinary differential equations 2.1. Controllability. Consider some linear operator A (for instance, a matrix, or a partial differential operator like -∆, where ∆ denotes the Dirichlet-Laplace operator on some bounded domain of R d for some d ∈ N * ), posed on some real or complex Hilbert space (H, •, • H ) (which is called the state space). We consider a linear dynamical system of the form (2.7)

y (t) = Ay(t), t ∈ [0, T ], y(0) = y 0 ∈ H,
where for any t ∈ [0, T ], y(t) ∈ H. y is the state of the system: one has to imagine some physical, chemical or biological quantity (e.g. the temperature, the concentration of some substance), that evolves during the time. Under reasonable assumptions on A (see Remark 2.2), this system is wellposed: for any y 0 ∈ H, there exists a unique solution to (2.7) in the space C 0 ([0, T ], H). This solution verifies: for any T > 0, there exists a constant C(T ) > 0 such that for any y 0 ∈ H, we have

||y|| C 0 ([0,T ],H) C(T )||y 0 || H .
This is a free evolution: As soon as y 0 is fixed, we cannot choose the value of y(T ). We cannot act on the dynamical system. Hence, one natural question is to understand how we can introduce some way to influence the dynamics of (2.7).

A good model is the following. Let us consider some other Hilbert space (U, •, • U ) (which will be called the control space) and some B ∈ L c (U, H) (the space of linear continuous maps from U to H). B is called a linear bounded control operator. Definition 2.1. A linear control system is an equation of the form (2.8)

y (t) = Ay(t) + Bu(t), t ∈ [0, T ], y(0) = y 0 ∈ H, where u ∈ L 2 ((0, T ), U ). u is called the control.
Remind that the definition of L 2 ((0, T ), U ) is given by

L 2 ((0, T ), U ) := {u : (0, T ) → U measurable s.t. T 0 ||u(t)|| 2 U dt < ∞}.
(We could have introduced other functional spaces, but this choice is justified by the fact that it is more convenient to work in Hilbert spaces). We assume that we also have well-posedness, in the following sense: for fixed u ∈ L 2 ((0, T ), U ), there exists a unique solution y to (2.8) in the space C 0 ([0, T ], H), verifying moreover: for any T > 0, there exists a constant C(T ) > 0 such that for any y 0 ∈ H, we have

||y|| C 0 ([0,T ],H) C(T ) ||y 0 || H + ||u|| L 2 ((0,T ),U ) .
This abstract setting is very useful to cover a wide range of situations, but not all possible situations (e.g. nonlinear control systems).

Remark 2.2. Concerning the well-posedness of (2.7), a good framework is the semigroup theory: we assume that A is a close unbounded operator on H with dense domain, that verifies two additional properties:

• A is dissipative, i.e., for any x ∈ H, Re ( x, Ax ) 0, • A is maximal, i.e., for any λ > 0, Id -λA is surjective.

We refer for instance to [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Theorem 4.3,Page 14]).

Concerning the well-posedness of (2.8) under the above assumptions on A, we refer to [8, Section 2.3] or [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Chapter 4] for more explanations. Now, if we consider (2.8) without fixing u, i.e. meaning that the pair (y, u) is considered as an unknown, (2.8) becomes what we have already called an underdetermined system: if we imagine that a well-posed system like (2.7) requires to have "as many equations as unknowns", in a system like (2.8), we have "more unknowns" than equations (u also appears at least in one equation). Moreover, it is likely that y(T ) depends strongly on the choice of our control u. We have gained some "degree of freedom" to act on the equation.

One very natural question associated to (2.8) is the following: for any initial condition, can we find a control that put the system at rest at some time T > 0? This is exactly the purpose of the following definition.

Definition 2.3. The linear control system (2.8) is said to be null-controllable if, for any y 0 ∈ H, there exists some u ∈ L 2 ((0, T ), H) such that y(T ) = 0 in H.

This notion is particularly interesting: as soon as y(T ) = 0, if we switch off the control (i.e., we set u = 0 on [T, ∞)), up to a translation in time, we are back to the free system (2.7) with "initial condition" 0 ∈ H. Since 0 is an equilibrium of (2.7), the only solution is 0 and the solution stays at rest forever for any t T .

Question: what kind of condition we need on A and B to ensure that the null-controllability property holds?

In such a general setting, this question does not really make sense, so we will investigate some precise examples. To begin, let us consider the following trivial example.

Example 2.4. We consider H = R, A = a ∈ R, B = Id R , and the following very simple scalar linear ordinary differential equation with constant coefficients (2.9)

y (t) = ay(t) + u(t), t ∈ [0, T ], y(0) = y 0 ∈ R.
Let us fix some y 0 ∈ R and some final time T > 0. Can we find u such that the solution of (2.9) verifies y(T ) = 0?

The answer is yes, and it is very easy to construct "by hand" a control. Consider any function y : R + → R such that:

• y ∈ C ∞ (R + , R), • y(0) = y 0 , • The support (in the ambient space R + ) of y is included in [0, T ]. We then introduce u(t) = y (t) -ay(t). It is clear that: • u ∈ C ∞ (R + , R) (so notably u ∈ L 2 ((0, T ), R)),
• (2.9) 1 is verified by construction of u (here and in what follows, (S) i means that we consider the i-th line of system (S)),

• (2.9) 2 is also verified by construction of y,

• y(T ) = 0 since y is supported in [0, T ].
Of course, the above example is quite trivial, and does not really highlight the deepness of the problems we will have to face in more complex situations. Remark 2.5. In the above example, we have an infinite choice of functions for y. Hence, we do not have uniqueness of the control u and the trajectory y in the problem of null-controllability raised in Definition 2.1.

Systems of two linear

ODEs with one control. Now, let us consider the following system of two linear ordinary differential equations with constant coefficients (2.10)

     y 1 (t) = a 11 y 1 (t) + a 12 y 2 (t), t ∈ [0, T ], y 2 (t) = a 21 y 1 (t) + a 22 y 2 (t) + u(t), t ∈ [0, T ], y 1 (0) = y 0 1 ∈ R, y 2 (0) = y 0 2 ∈ R.
Let us put this system in the abstract setting of (2.8). Here,

H = R 2 , U = R. The operator A ∈ L(R 2 ) is the matrix of size 2 × 2 given by A = a 11 a 12 a 21 a 22 ,
and the control operator

B ∈ L(R, R 2 ) is the matrix of size 2 × 1 given by B = 0 1 .
We ask the same question: let us fix some initial condition (y 0 1 , y 0 2 ) ∈ R 2 and some final time T > 0. Can we find u such that the solution of (2.10) verifies y 1 (T ) = y 2 (T ) = 0? Of course, here, the method we used for the scalar equation fails: if we choose any (y 1 , y 2 ) regular enough verifying (y 1 (0), y 2 (0)) = (y 0 1 , y 0 2 ) and (y 1 (T ), y 2 (T )) = (0, 0), there is no reason for (2.10) 1 to be verified. Equation (2.10) is the prototype of what we will call an underactuated system:

• We have two unknowns, y 1 and y 2 , that are the states we would like to control. • But we have only one control u, on the last equation. In other worlds, we control only "directly" the quantity y 2 , and we hope that thanks to the coupling terms, we can "indirectly" control y 1 "through" y 2 (for this reason, this situation is called a problem of "indirect control"). This phenomenon can be observed in real-life models. For instance, in the model of tumour growth presented in [START_REF] Chakrabarty | Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method[END_REF], three quantities are involved: the drug concentration, the healthy and the non-healthy cells. The resulting system consists of three equations, that are coupled with some nonlinear reaction terms. The control is a drug bolus inside a small subdomain. It only influences directly the drug concentration, so that the structure of the coupling terms is crucial to understand how we can kill all the non-healthy cells.

Question: What are reasonable conditions we need to impose on the coupling coefficients a 11 , . . . , a 22 in order to ensure controllability?

One necessary condition is clearly the following.

Lemma 2.6. In order that (2.10) is null-controllable, it is necessary to have a 12 = 0.

Proof. If a 12 = 0, then y 1 verifies the equation

y 1 = a 11 y 1 .
The control u has no influence on the dynamics of y 1 . Hence, we can solve explicitly this very simple ordinary differential equation and find that y 1 (t) = y 0 1 e a11t . As soon as y 0 1 = 0, obviously y(T ) = 0, and we conclude that the system cannot be null-controllable in this case.

In fact, it is remarkable that this condition is also sufficient to ensure nullcontrollability.

Theorem 2.7. If a 12 = 0, then (2.10) is null-controllable.

Proof of Theorem 2.7

In order to prove this Theorem, we will use a fictitious control argument, in the spirit of [8, Second proof of Theorem 1.18]. The strategy can be briefly described as folllows:

(1) Firstly, we control our system of equations with one control on each equation, i.e. we solve the control problem

           y 1 (t) = a 11 y 1 (t) + a 12 y 2 (t) + u 1 (t), t ∈ [0, T ], y 2 (t) = a 21 y 1 (t) + a 22 y 2 (t) + u 2 (t), t ∈ [0, T ], y 1 (0) = y 0 1 ∈ R, y 2 (0) = y 0 2 ∈ R, y 1 (T ) = 0, y 2 (T ) = 0.
This can be done by a direct construction, which is very similar to Example 2.4. Moreover, it is easy to construct u 1 and u 2 in such a way that they are smooth and compactly supported in (0, T ) (their support will be included in [ T 3 , 2T 3 ]).

(2) Secondly, we solve the auxiliary control problem

           ỹ 1 (t) = a 11 ỹ1 (t) + a 12 ỹ2 (t) + u 1 (t), t ∈ [0, T ], ỹ 2 (t) = a 21 ỹ1 (t) + a 22 ỹ2 (t) + u 2 (t) + ũ(t), t ∈ [0, T ],
ỹ1 (0) = 0, ỹ2 (0) = 0, ỹ1 (T ) = 0, ỹ2 (T ) = 0.

( u 1 , u 2 ) is the same as the one introduced in the previous step and is seen as a source term. The unknowns are now (ỹ 1 , ỹ2 , ũ). We are typically in a situation that is similar to the introducing example explained in Section 1.2: the system is underdetermined, and we will find a solution that can be written as a linear combination of the source term ( u 1 , u 2 ) and its derivatives by applying the same procedure as in Section 1.2. (3) To conclude, we investigate the system that is verified by y 1 = y 1 -ỹ1 and y 2 = y 2 -ỹ2 , and we prove that it provides a solution to our desired control problem: there exists a control u such that (y 1 , y 2 ) verifies (2.10) together with y 1 (T ) = 0 and y 2 (T ) = 0. Moreover, during this procedure, the "fictitious control" ( u 1 , u 2 ) disappears, justifying the name of the method.

           y 1 (t) = a 11 y 1 (t) + a 12 y 2 (t) + u 1 (t), t ∈ [0, T ], y 2 (t) = a 21 y 1 (t) + a 22 y 2 (t) + u 2 (t), t ∈ [0, T ], y 1 (0) = y 0 1 ∈ R, y 2 (0) = y 0 2 ∈ R, y 1 (T ) = 0, y 2 (T ) = 0,
where we have now two controls u 1 and u 2 , each acting on one of the equations. This means that the control operator is now

B = 1 0 0 1 .
As in the case of one equation described in Example 2.4, it is very easy to exhibit a solution to (2.11). However, we will require some additional properties (for some reasons that will become clear later on). We introduce some function η : [0, T ] → R such that :

• η is of class C ∞ on [0, T ], • η = 1 on [0, T 3 ], • η = 0 on [ 2T 3 , T ].
Now, we consider the solution to the (free) equation (2.12)

       y F 1 (t) = a 11 y F 1 (t) + a 12 y F 2 (t), t ∈ [0, T ], y F 2 (t) = a 21 y F 1 (t) + a 22 y F 2 (t), t ∈ [0, T ], y 1 (0) = y 0 1 ∈ R, y 2 (0) = y 0 2 ∈ R. We introduce y 1 y 2 = ηy F 1 ηy F 2 and u 1 u 2 = y 1 -a 11 y 1 -a 12 y 2 y 2 -a 21 y 1 -a 22 y 2 .
( u 1 , u 2 ) is called the fictitious control (for reasons that will appear at the end of the next step). Then:

• ( y 1 , y 2 , u 1 , u 2 ) verifies (2.11) 1-2 by construction,
• We also have (2.11) 3 by definition of ( y 1 , y 2 ), since η(0) = 1,

• (2.11) 4 is also verified: η = 0 on [ 2T 3 , T ], so this is also the case for y 1 and y 2 ,

• y 1 , y 2 , u 1 , u 2 are C ∞ on [0, T ], since y F 1 , y F 2 are clearly in C ∞ ([0, T ], R) and η is also in C ∞ ([0, T ], R),
• The controls ( u 1 , u 2 ) are compactly supported in (0, T ). Indeed, on [0, T 3 ], by definition of ( u 1 , u 2 ) and since η = 1, using (2.12), we have

u 1 (t) u 2 (t) = y F 1 (t) -a 11 y F 1 (t) -a 12 y F 2 (t) y F 2 (t) -a 21 y F 1 (t) -a 22 y F 2 (t) = 0 0 ,
and on [ 2T 3 , T ], by definition of ( u 1 , u 2 ) and since η = 0, we also have ( u 1 , u 2 ) = (0, 0) on [ 2T 3 , T ].

2.2.2. Second step: algebraic part. Now, we would like to solve the following auxiliary control problem:

(2.13)

           ỹ 1 (t) = a 11 ỹ1 (t) + a 12 ỹ2 (t) + u 1 (t), t ∈ [0, T ], ỹ 2 (t) = a 21 ỹ1 (t) + a 22 ỹ2 (t) + u 2 (t) + ũ(t), t ∈ [0, T ],
ỹ1 (0) = 0, ỹ2 (0) = 0, ỹ1 (T ) = 0, ỹ2 (T ) = 0.

Note that ( u 1 , u 2 ) is the same as in (2.11), and has to be seen as a source term.

The unknowns are now (ỹ 1 , ỹ2 , ũ).

Remark that the system is now underdetermined : we have 2 equations and 3 unknowns. Hence, our intuition is that this system is likely to admit multiples solutions. We will select some particular solution, thanks to the algebraic solvability procedure that we explained in Section 1.2. This means that we will express (ỹ 1 , ỹ2 , ũ) as a linear combination of the source term ( u 1 , u 2 ) and its derivatives. For the moment, we are only interested in (2.13) 1-2 (the fact that (2.13) 3-4 are verified will be a consequence of our construction and will be explained at the end of the reasoning). We rewrite (2.13) 1-2 in an abstract way as

(2.14) L   ỹ1 ỹ2 ũ   = f, where L : C ∞ ([0, T ], R 3 ) → C ∞ ([0, T ], R 2
) is the ordinary differential operator given by

L   ỹ1 ỹ2 ũ   = ỹ 1 (t) -a 11 ỹ1 (t) -a 12 ỹ2 (t) ỹ 2 (t) -a 21 ỹ1 (t) -a 22 ỹ2 (t) -ũ(t) ,
and the source term f is given by

f = u 1 u 2 .
We can reformulate in an equivalent way our question as follows: find some right inverse to L, i.e. some linear differential operator M :

C ∞ ([0, T ], R 2 ) → C ∞ ([0, T ], R 3 ) such that (2.15) L • M = Id C ∞ ([0,T ],R 2 ) .
Indeed, the above equality applied to the source term gives L(M(f )) = f, so that if we define

M(f ) =   ỹ1 ỹ2 ũ   ,
we have found a solution to (2.14) by definition. Hence, our goal is now to solve (2.15). From (2.15), we get

(2.16) M * • L * = Id C ∞ ([0,T ],R 2 ) .
Here, * correspond to the formal adjoint with respect to the L 2 -norm introduced in Definition 1.1. According to this Definition, its expression is given here by

L * =   -∂ t -a 11 Id -a 21 Id -a 12 Id -∂ t -a 22 Id 0 Id   ,
i.e.

(2.17)

L * ϕ 1 ϕ 2 =   -ϕ 1 -a 11 ϕ 1 -a 21 ϕ 2 -ϕ 2 -a 12 ϕ 1 -a 22 ϕ 2 -ϕ 2   .
Here, Id means the identity on C ∞ ([0, T ], R). Now, let us go back to our goal: we want to find some linear differential operator M * such that (2.16) holds. It means that we want to differentiate and make linear combinations on the lines of (2.17) in order to recover ϕ 1 and ϕ 2 . In fact, in (2.17) 3 , ϕ 2 is already there. Hence, we just have to recover ϕ 1 . In order to reach this aim, we combine (2.17) 2 with (2.17) 3 in order to recover ϕ 1 , by applying a differential operator of order 1 to (2.17) 3 :

(2.18)

ϕ 1 = -(-ϕ 2 -a 12 ϕ 1 -a 22 ϕ 2 ) + (∂ t + a 22 )(-ϕ 2 ) a 12 ,
were we have used our hypothesis a 12 = 0. Taking into account this computation, it is natural to introduce M * as

M *   g 1 g 2 g 3   = -g2+g 3 +a22g3 a12 -g 3 .
By construction, using (2.17) and (2.18), we have

M * L * ϕ 1 ϕ 2 = ϕ 1 ϕ 2 .
From Definition 1.1 and the fact that * is an involution, it is easy to go back to M(= (M * ) * ). Indeed, M * can be written in matricial form as

M * = 0 -Id a12 ∂t+a22Id a12 0 0 -Id , so that M =    0 0 -Id a12 0 -∂t+a22Id a12 -Id    .
M is here a linear differential operator of order 1. Let us verify for the sake of security that such a M answers the initial problem (2.13) 1-2 . Indeed, from (2.14), we introduce

  ỹ1 ỹ2 ũ   = M u 1 u 2 , i.e.   ỹ1 ỹ2 ũ   =    0 -u1 a12 -u 2 + -u 1 +a22 u1 a12    .
Let us verify that (2.13) 1-2 holds. Indeed, we remark that:

• By definition of ỹ1 , we have

ỹ 1 = 0,
and by definition of ỹ1 and ỹ2 , we also have

a 12 ỹ2 + u 1 + a 11 ỹ1 = 0.
We deduce that (2.13) 1 is verified. • By definition of ỹ2 , we have

ỹ 2 = - u 1 a 12 ,
whereas by definition of ỹ1 and ỹ2 , we have

a 21 ỹ1 + a 22 ỹ2 + u 2 + ũ = - a 22 u 1 a 12 + u 2 -u 2 + -u 1 + a 22 u 1 a 12 = - u 1 a 12 .
We deduce that (2.13) 2 is verified. Remark that the source term f = ( u 1 , u 2 ) is compactly supported in (0, T ). Hence, this is also the case of (ỹ 1 , ỹ2 , ũ): by our construction, these functions only involve linear combinations and derivative (of order 1 here) of our source term f . This implies that the support of (ỹ 1 , ỹ2 , ũ) is included in the support of f . This is in fact a crucial point and the main interest of the method. Hence, the initial and final conditions (2.13) 3-4 are verified automatically. This concludes the algebraic part.

Conclusion.

We combine both the analytic and algebraic part. We introduce:

•

y 1 = y 1 -ỹ1 , • y 2 = y 2 -ỹ2 ,
• u = -ũ. Then, from (2.11) and (2.13), we have:

• y 1 (0) = y 1 (0) -ỹ1 (0) = y 0 1 -0 = y 0 1 , since ỹ1 is compactly supported in (0, T ) (so that it vanishes on a neighbourhood of 0). • y 2 (0) = y 2 (0) -ỹ2 (0) = y 0 2 -0 = y 0 2 , for the same reasons. • y 1 (T ) = y 1 (T ) -ỹ1 (T ) = 0 -0 = 0, since y 1 is controlled to 0 at time T and ỹ1 is compactly supported in (0, T ) (so that it vanishes on a neighbourhood of T ). • y 2 (T ) = y 2 (T ) -ỹ2 (T ) = 0 -0 = 0, for the same reasons.

• By linearity,

y 1 = y 1 -ỹ 1 = a 11 y 1 + a 12 y 2 + u 1 -a 11 ỹ1 -a 12 ỹ2 -u 1 = a 11 ( y 1 -ỹ1 ) + a 12 ( y 2 -ỹ2 ) = a 11 y 1 + a 12 y 2 and y 2 = y 2 -ỹ 2 = a 21 y 1 + a 22 y 2 + u 2 -a 21 ỹ1 -a 22 ỹ2 -u 2 - ũ = a 21 ( y 1 -ỹ1 ) + a 22 ( y 2 -ỹ2 ) - ũ = a 21 y 1 + a 22 y 2 + u.
Hence, (y 1 , y 2 , u) solves the initial control problem (2.10). We remark that the "fictitious control" ( u 1 , u 2 ) has disappeared. This justifies our terminology.

2.3.

The scope of the method. Let us enumerate some advantages and drawbacks of the method:

• We can deal both with constant and non-constant coefficients (in time and also in space, if we deal with PDEs). In fact, the non-constant case is in some sense richer, because one can get some help from the noncommutativity between the coefficients and the differential operators (which of course does not happen in the case of constant coefficients), that will add some extra coupling terms in our differentiation procedure. This has notably be strongly used in [START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF]. • It can be also used for the study of nonlinear systems, because it combines very well with the return method of Coron (see [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]). This method relies on a linearization procedure around some particular trajectories that we can choose in different ways (notably so that the algebraic solvability procedure described above works). This is exactly the spirit of the computations developed in [START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF]. • The method is purely local. It can be seen if we modify a little bit the example developed in Section 2.2. Assume that we change the constant coefficient a 12 to a time-varying coefficient a 12 (t) that we choose to be equal to 0 on [ T 3 , 2T 3 ]. Then, since we are working locally on [ T 3 , 2T 3 ] in the algebraic part, we do not "see" the coupling term a 12 on this time interval and then the algebraic solvability fails in this context. Of course, this example is quite artificial (instead of controlling on [ T 3 , 2T 3 ], one can choose to control on a time interval where a 12 is nonzero), but in more complex situations (notably in the context of partial differential equations, where the coupling region can be disjoint from the control region, notably in the space variable), it can become a real problem. Notably, we know situations where the algebraic solvability fails whereas the system is controllable (see e.g. [START_REF] Khodja | New phenomena for the null controllability of parabolic systems: minimal time and geometrical dependence[END_REF]).

• This method relies on a differentiation procedure, hence it consumes a lot of regularity: the control created by this procedure is less regular than it "should" be. To be more precise, let us consider the following example: assume that A is a differential operator and the initial condition of the free system (2.7) is very regular. Then, in general, the solution of (2.7) on [0, T ] × H will also be very regular, and will remain very regular if we add some source term f that is regular enough. However, when we use our notion of algebraic solvability, we will need to differentiate a system of the form "y = Ay + f ", so that the control obtained in the algebraic solvability procedure will be less regular than f and y. Hence, even if we want to control a system like (2.8) with controls in L 2 ((0, T ), U ) for instance, we will need to take initial data that are not in the natural energy space H but that will be much more regular (in the domain of some power of A).

For parabolic systems like systems of heat equations, this drawback can sometimes be avoided: we have a regularizing effect that ensure smoothness as soon as we are away from the initial condition and we have regular enough coefficients (see e.g. [17, Theorem 7, Page 367]). Hence, even if the initial condition is in H, after a short time on which we does not control, it becomes very regular and we can use our algebraic solvability procedure. However, for hyperbolic or dispersive systems (like the wave equation or the Schrödinger equation), the regularizing effect does not exist, and it means that even if we want to control a system with a low regularity control, we need to take very regular initial data (see the following Section 3 and notably system (3.21) and Theorem 3.6: the initial condition is very regular but the control is only in L 2 ). Hence, we lose sharpness in the results, in the sense that the of initial data that can be controlled is not the entire energy space H and has to be "artificially" reduced to make our argument work.

Let us mention that this method has been successfully used in different contexts in [START_REF] Alabau-Boussouira | Internal controllability of first order quasi-linear hyperbolic systems with a reduced number of controls[END_REF][START_REF] Coron | Control of three heat equations coupled with two cubic nonlinearities[END_REF][START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF][START_REF] Duprez | Indirect controllability of some linear parabolic systems of m equations with m-1 controls involving coupling terms of zero or first order[END_REF][START_REF] Duprez | Positive and negative results on the internal controllability of parabolic equations coupled by zero and first order terms[END_REF][START_REF] Duprez | Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control[END_REF][START_REF] Liard | A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups[END_REF][START_REF] Steeves | Controllability of coupled parabolic systems with multiple underactuations, parts 1 and 2[END_REF].

System of Schrödinger equations

Now, we move from ordinary differential equations to an application of the fictitious control method to systems of coupled partial differential equations. The forthcoming example is coming from [START_REF] Liard | A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups[END_REF].

3.1. Controllability of the scalar Schrödinger equation. For the sake of simplicity (in order to avoid problems with the boundary), we will consider the following setting. We consider (M, g) a smooth compact Riemannian manifold, without boundary. We denote by ∆ the Laplace-Beltrami operator on M . Since -∆ is a selfadjoint and positive operator with compact resolvent, we can introduce its eigenvalues {λ k } k∈N * , (note that each eigenvalue has finite multiplicity in this context), ordered such that 0 = λ 1 λ 2 . . ., each eigenvalue λ k being associated with a corresponding eigenfunction e k . We can construct the family {e k } k∈N * in such a way that {e k } k∈N * forms a Hilbert basis of L 2 (M ). Each function f ∈ L 2 (M ) can then be uniquely decomposed as

f = k∈N * a k e k ,
where (a k ) k∈N * ∈ l 2 (N * ). Moreover, Parseval's identity gives

||f || 2 L 2 (M ) = ∞ k=1 |a k | 2 .
Now, let us consider some s ∈ R + . We introduce the Sobolev space H s (M ) as

H s (M ) := {f = k∈N * a k e k ∈ L 2 (M ) s.t. ∞ k=1 1 + λ 2s k |a k | 2 < ∞}.
H s (M ) can alternatively be defined as the domain of the operator (-∆) s 2 . For more informations on Riemannian geometry, we refer to [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF].

Example 3.1. The simplest example of such a manifold is given by M = T d for some d ∈ N * , i.e. the d-dimensional torus. From the point of view of partial differential equations, this would correspond to the unit hypercube [-1, 1] d in R d endowed with the flat metric, with periodic boundary conditions. In T d seen as the unit hypercube in R d , the Laplace-Beltrami operator is just the usual Laplace operator given by

∆ = d i=1 ∂ 2 x 2 i . In this case, for s ∈ N * , H s (M ) is the space of periodic function f ∈ L 2 ((-1, 1) d ) verifying ∂ α1 x1 . . . ∂ α d x d f ∈ L 2 ((-1, 1) d ),
Figure 1. Illustration of GCC. We represent the 2dimensional torus, the black lines have to be thought as periodic boundary conditions. ω is represented in blue. One horizontal, vertical and oblique ray are drawn in red. On the left picture, GCC is not verified since neither the vertical ray nor the oblique ray never encounter the control region. Conversely, on the right picture, the control region verifies GCC.

for any multi-index (α 1 , α d ) ∈ N d such that α 1 + α 2 . . . + α d s.

We will need the following condition. This condition is strongly related to the controllability of the wave equation. It is notably proved in [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF] and [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] that it is a sufficient condition of controllability, respectively for a manifold without and with boundary (in this last case, some reasonable additional properties on the boundary are also needed). Conversely, this condition is also necessary for the controllability of the wave equation, see [START_REF] Ralston | Solutions of the wave equation with localized energy[END_REF] and [START_REF] Burq | Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes (necessary and sufficient condition for the exact controllability of the wave equation)[END_REF].

Let T > 0, ω an open subset of M verifying GCC, and u ∈ L 2 ((0, T ), M ). We will be interested in the following evolution equation, called the linear Schrödinger equation:

(3.19) i∂ t y + ∆y = 1 ω (x)u, y(0) = y 0 ∈ L 2 (M ).
If u = 0, this equation describes the evolution in time of the wave function associated to a free (i.e. without potential) quantic particle that moves into the manifold. Equation (3.19) is a linear control system in the sense of Definition 2.1. Moreover, as in Section 2.1, it can be proved that for any y 0 ∈ L 2 (M ) and any u ∈ L 2 ((0, T ), U ), there exists a unique solution y to (3.19) (see for instance [START_REF] Coron | Control and nonlinearity[END_REF]Section 2.3] or [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Chapter 4] for more explanations). As in the previous Section, our goal is to obtain the following null-controllability property. Definition 3.3. We say that (3. [START_REF] González-Burgos | Controllability results for some nonlinear coupled parabolic systems by one control force[END_REF]) is null-controllable if, for any y 0 ∈ L 2 (M ), there exists u ∈ L 2 ((0, T ) × ω) such that the corresponding solution y to (3.19) satisfies y(T ) = 0 L 2 (M ) .

Concerning controllability results, we have the following sufficient condition. Theorem 3.4. Assume that ω verifies GCC for some time T > 0. Then, (3.19) is null-controllable for any time T > 0.

This theorem was first proved in [START_REF] Lebeau | Contrôle de l'équation de Schrödinger (Control of the Schrödinger equations)[END_REF], in the case of a bounded domain of R n and boundary control (but the same strategy applies for an internal control). In a more elementary way, it can also be seen as a consequence of the abstract result given in [32, Theorem 6.7.5], taking into account the already mentioned result on the wave equation given in [START_REF] Rauch | Exponential decay of solutions to hyperbolic equations in bounded domains[END_REF].

Note that contrary to the wave equation, in the case of (3.19), GCC is in general far from being necessary, but it depends strongly and in a nontrivial way on the global geometry (i.e. on M ). For example, if M = T d , then any nonempty open subset ω can be chosen to obtain the null-controllability of (3.19) (see [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire (Internal exact controllability of the vibrations of a rectangular plate)[END_REF]).

We will also need the following refinement.

Proposition 3.5. Let n ∈ N * . For any open subset Ω of M , we define the following space:

(3.20) H n-1 0 ((0, T ), L 2 (Ω)) := {u ∈ H n-1 ((0, T ), L 2 (Ω))|u k t...t (0) = u k t...t (T ) = 0, ∀k ∈ [|0, n -2|]}.
Assume moreover that:

• y 0 ∈ H n-1 (M ) for some n ∈ N * , • 1 ω is replaced by some regularized version 1 ω ∈ C ∞ (M )
, which is supported on some ω containing ω with 1 ω = 1 on ω.

Then, there exists a control u such that

u ∈ H n-1 0 ((0, T ), L 2 (ω)) × n-1 k=0 C k ([0, T ], H 2n-2k-2 (ω))
and the solution of (3.19) verifies y(T ) = 0.

This proof of this result can be found in [START_REF] Ervedoza | A systematic method for building smooth controls for smooth data[END_REF] (see also [START_REF] Dehman | Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time[END_REF]).

A system version

. Let (n, m) ∈ (N * ) 2 , with m < n. Let A ∈ M n (R) and B ∈ M n,m (R). Let U = (U 1 , . . . U m ) ∈ L 2 ((0, T ) × Ω) m .
We consider the following system of coupled Schrödinger equations

(3.21) i∂ t Y + ∆Y = AY + BU 1 ω , Y (0) = Y 0 ∈ H n-1 (M ) n .
Here and in what follows, for Y = (Y 1 , . . . , Y n ), we will write

∂ t Y =   ∂ t Y 1 . . . ∂ t Y n   and ∆Y =   ∆Y 1 . . . ∆Y n   .
This is a system of n Schrödinger equations, that are coupled through the matrix A. Remind that 1 ω has been introduced in Proposition 3.5. Since m < n, we have less controls than equations, meaning that as in Section 2.2, we have an underactuated system. The typical situations are:

(1)

B   U 1 . . . U m   =         U 1 . . . U m 0 . . . 0         .
Some equations are directly controlled (the m first ones here) and some are not controlled at all: we hope that the coupling terms will help in order to control the equations. This situation is called indirect controllability, similarly to the case studied in Section 2.2.

(2)

B(U 1 ) =   U 1 . . . U 1   .
Here, all equations are controlled, with the same control on each equation. This situation (which is very different from the previous one) is called simultaneous controllability.

We have the following characterization. Theorem 3.6. We introduce the following matrix [A|B] ∈ M n,nm (R), called the Kalman matrix:

[A|B] = [B, AB, A 2 B, . . . A n-1 B]. (3.21) is null-controllable with U ∈ L 2 ((0, T ) × Ω) if and only if [A|B] is of full rank n.
This condition is exactly the celebrated Kalman rank condition for the controllability of linear ordinary differential equations (see [START_REF] Lasalle | The time optimal control problem[END_REF] and [START_REF] Kalman | Controllability of linear dynamical systems[END_REF]). Note that by the Cayley-Hamilton Theorem, it is useless to include the matrices A k B for k n + 1.

Remark 3.7.

• The regularity of the initial condition is very high (but it is necessary in our proof for reasons that will be clear later on). However, it is likely that this is a purely technical assumption. One should be able to obtain a result for initial condition in L 2 , but this is an open problem. • In the case where the Kalman matrix is not of full rank, we are also able to express the initial conditions that can be controlled (that are the ones in [A|B](H n-1 (M ))). This matrix is of rank 2 if and only if a 12 = 0. We find back the necessary and sufficient condition of controllability given in Section 2.2 for system (2.10).

Proof of Theorem 3.6 We only prove the "if" part (the "only if" part is easier, and is similar to the finite-dimensional case, see e.g. [8, Proof of Theorem 1.16]). The strategy is the same as in Section 2.2. Let us describe it briefly.

(1) Firstly, we control our system of equations with one control on each equation, i.e. we solve the control problem

     i∂ t Y + ∆ Y = A Y + U 1 ω , Y (0) = Y 0 , Y (T ) = 0,
where the fictitious control U = (u 1 , . . . u n ) is acting on each equation. This can be done by using a clever change of unknowns that enables to decouple the system and apply the scalar result of Theorem 3.6 on each equation. ( 2) Secondly, we solve the auxiliary control problem

i∂ t Ỹ + ∆ Ỹ = A Ỹ + B Ũ + U 1 ω , Ỹ (0) = Ỹ (T ) = 0.
The unknowns are ( Ỹ , Ũ ) and U is the control created in the previous analytic part, seen as a source term. We apply our algebraic solvability procedure, which will make naturally the Kalman matrix [A|B| appear. The Kalman rank condition will then be used to find a right inverse to [A|B|. [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] To conclude, we investigate the system that is verified by Y = Ỹ -Y , and we remark that there exists a control U ∈ L 2 ((0, T ), Ω) m such that Y verifies (3.21) together with Y (T ) = 0.

3.2.1. First step: analytic part. We look at the fictitious control problem

(3.22)      i∂ t Y + ∆ Y = A Y + U 1 ω , Y (0) = Y 0 , Y (T ) = 0,
where the fictitious control U = ( U 1 , . . . , U n ) is acting on each equation. We introduce the following change of unknowns: Z = e itA Y . Then, since Y verifies (3.22), we have

i∂ t Z = i e itA ∂ t Y + iAe itA Y = ie itA ∂ t Y -Ae itA Y = e itA -∆ Y + U 1 ω = -∆Z + W 1 ω ,
where we have used ∆e itA = e itA ∆ and we have introduced the new control W = e itA U . Moreover, we have Z(0) = Y 0 . Remark that Z verifies a new control problem where the coupling term A has disappeared, i.e. the system is uncoupled. Hence, we can apply Proposition 3.5 on each of the components of Z and we obtain that

W ∈ H n-1 0 ((0, T ), L 2 (ω)) × n-1 k=0 C k ([0, T ], H 2n-2k-2 )(ω).
Going back to the original variables ( Y , U ), we obtain that ( Y , U ) verifies (3.22) and moreover

(3.23) U ∈ H n-1 0 ((0, T ), L 2 (ω)) × n-1 k=0 C k ([0, T ], H 2n-2k-2 (ω)).
3.2.2. Second step: algebraic part. We look at the following auxiliary control problem:

(3.24)

i∂ t Ỹ + ∆ Ỹ = A Ỹ + B Ũ + U 1 ω , Ỹ (0) = Ỹ (T ) = 0.
The unknowns are ( Ỹ , Ũ ) and U is the control appearing in (3.22), seen as a source term. Remark that we have n + m unknowns and n equations, so that the system is underdetermined. We rewrite this system as

L( Ỹ , Ũ ) = U 1 ω , where L( Ỹ , Ũ ) = i∂ t Ỹ + ∆ Ỹ -A Ỹ -B Ũ .
We forget the regularity issues for the moment and we consider L as an operator acting on smooth functions. We want to find some differential operator

M : (C ∞ ((0, T ) × M )) n → (C ∞ ((0, T ) × M )) n+m such that (3.25) L • M = Id C ∞ ((0,T )×M ) n .
Applying this identity to our source term U 1 ω , we deduce that

L(M( U 1 ω )) = U 1 ω .
Hence, ( Ỹ , Ũ ) := M( U 1 ω ) will answer the question. Remark that here, we only need to work "locally" in space on ω. Using the same strategy as before, we pass to the formal adjoint and we rewrite the problem as

M * • L * = Id C ∞ ((0,T )×M ) n .
We remark that

L : (C ∞ ((0, T ) × M )) n+m → (C ∞ ((0, T ) × M )) n
can be written in a matricial form as

L = i∂ t + ∆ -A, -B .
Hence, using Definition 1.1 (that still apply for partial differential operators), we can compute its formal adjoint (using that ∆ is self-adjoint), which is given by

L * = -i∂ t + ∆ -A * -B * .
In other words, if ϕ = (ϕ 1 , . . . , ϕ n ) ∈ C ∞ ((0, T ), M ) n , we have

L * ϕ = -i∂ t ϕ + ∆ϕ -A * ϕ -B * ϕ .
Our goal is to make some linear combinations of the lines of L * ϕ and its derivatives, in order to recover ϕ. The trick is the following:

• the m last lines of L * ϕ are -B * ϕ. We multiply it by -1 to recover B * ϕ. • We would like to recover B * A * ϕ. We use the first n lines of L * ϕ, and we apply some appropriate differential operators to the m last lines. We have

-B * (-i∂ t ϕ + ∆ϕ -A * ϕ) + (i∂ t -∆)(-B * ϕ) = B * A * ϕ,
where we have used the commutativity relations B * (i∂ t ϕ) = i∂ t (B * ϕ) and B * (∆ϕ) = ∆(B * ϕ).

• Assume that we have recovered B * ϕ, B * A * ϕ, . . . , B * (A * ) k ϕ. Then, we can recover B * (A * ) k+1 ϕ. Indeed, it is easy to see that

B * (A * ) k+1 ϕ = -B * (A * ) k (-i∂ t ϕ + ∆ϕ -A * ϕ) + (i∂ t -∆)(-B * (A * ) k ϕ).
We can summarize the previous sequence of transformations in the following differential N : Moreover, ( Ỹ , Ũ ) := M( U 1 ω ) is a solution of (3.24) 1 , that is supported in space in ω (since it is made of linear combinations of the derivatives of U 1 ω ).

(3.26) N (L * ϕ) =     B * ϕ B * A * ϕ . . . B * (A * ) n-1 ϕ     , with (3.27) N         ψ 1 . . . ψ n ψ n+1 . . . ψ n+m         =                           -ψ n+1 . . . -ψ n+m   -B *   ψ 1 . . . ψ n   + (i∂ t -∆)   ψ n+1 . . . ψ n+m   . . . k-2 j=0 (-1) 1+j (i∂ t -∆) j B * (A * ) k-2-j   ψ 1 . . . ψ n   +(-1) k (i∂ t -∆) k-1   ψ n+1 . . . ψ n+m   . . . .                        
To finish, we have to verify that the initial and final conditions in time (3.24) 2 are also verified. Indeed, we have to look a little bit deeper at the structure of N * : for ψ 1 , . . . , ψ n , we apply differential operators up to order n -1 in time but, for ψ n+1 , . . . , ψ n+m , we apply differential operators up the order n -2 in time only. Hence, when we go back to the operator N , it can be seen that, for g ∈ C ∞ ((0, T ) × M ) nm , N (g) (which is of size n + m) is such that on the n first lines, we only have derivatives up to order n -2 in time of g. From this fact, one easily deduce that Ỹ (which is of size n) also involves derivatives of the source term U up to order n -2 in time only (this would not be the case for Ũ ). Hence, from the definition given in (3.20) and the conditions on the end points of U given in (3.23), it is clear that Y verifies (3.24) 2 . Hence, we have found a solution to our algebraic problem (3.24).

Conclusion. As in Section 2.2, we set

Y = Ỹ -Y , U = -Ũ .
Then, using (3.22) and (3.24), we conclude easily that (Y, U ) is a solution to the initial control problem (3.21) by similar arguments.

Conclusion

In these lecture notes, our goal was to present the fictitious method coupled with the algebraic solvability procedure, in order to derive results on underactuated systems of ordinary differential equations or partial differential equations. This method has the advantage of being quite systematic. We decouple the problems into two simple ones: one "analytic part" where we prove a controllability result with as many controls as equations, and one "algebraic part" where we take benefit of the structure of the coupling terms to perform the algebraic solvability procedure.

As highlighted before, this method has already been fruitful to obtain different controllability results.

• In [START_REF] Coron | Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components[END_REF], the local null-controllability of the Navier-Stokes system with a control only on one equation is obtained. The corresponding linearized system around the trajectory 0 (i.e. the Stokes system) is not controllable. Hence, we need to perform a different linearization procedure around particular families of trajectories. The first step is to establish a controllability result with controls on each equation and in a specific form. Then, we use a slightly different version of the previous algebraic solvability procedure in order to make the fictitious control disappear.

To finish, we go back to the nonlinear system by a standard inverse mapping theorem. A similar procedure is used in [START_REF] Coron | Control of three heat equations coupled with two cubic nonlinearities[END_REF] in order to study a system of three coupled semilinear heat equations, the coupling terms being in cascade form and cubic. In both cases, the choice of the family of linearized trajectories is crucial for the algebraic part. • In [START_REF] Duprez | Indirect controllability of some linear parabolic systems of m equations with m-1 controls involving coupling terms of zero or first order[END_REF] and [START_REF] Duprez | Positive and negative results on the internal controllability of parabolic equations coupled by zero and first order terms[END_REF], some coupled systems of two heat equations with zero and one order coupling terms is explored. In the case of constant coupling coefficients, it is possible to recover a necessary and sufficient condition. If we have non-constant coupling terms, the algebraic solvability procedure naturally gives a generic condition in order to ensure controllability. The case of constant coefficients was partially generalized in [START_REF] Steeves | Controllability of coupled parabolic systems with multiple underactuations, parts 1 and 2[END_REF] in the case of more than 2 equations and constant coupling coefficients.

• In [START_REF] Alabau-Boussouira | Internal controllability of first order quasi-linear hyperbolic systems with a reduced number of controls[END_REF], the authors study the case of quasi-linear hyperbolic systems, using a linearization procedure. Due to the difference between the regularity of the initial data and the control explained in Section 2.3 and highlighted in Section 3, going back to the quasi-linear system requires the use of a Nash-Moser type theorem.

• In [START_REF] Duprez | Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control[END_REF], a result of controllability to the non-zeros trajectories on the Fokker-Planck equation is derived. One more time, a linearization procedure is needed, since the control system is bilinear in the control and the state. Here we have a scalar equation, but the control (which in some sense acts on the gradient of the solution) has d components if we are working on R d . The main novelty is that the algebraic solvability is not performed on the control problem but on some dual problem, in order to reduce the number of controls needed. • Finally, in [START_REF] Liard | A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups[END_REF], we derive some Kalman-like conditions for abstract systems of group of operators with application to Schrödinger and wave systems, using ideas very similar to Section 3. It is likely that this method can also be applied successfully to many other problems. For instance, it would be interesting to investigate the following problems.

• Larger classes of semilinear parabolic systems of reaction-diffusion type, coming from real-life models. For instance, one could investigate the model studied in [START_REF] Balc'h | Local controllability of reaction-diffusion systems around nonnegative stationary states[END_REF] concerning reaction-diffusions systems arising in chemistry, where results on local controllability around particular classes of equilibriums. Is it possible to use the algebraic solvability method in order to get rid of the technical conditions present in this article? • Other models coming from fluid mechanics, for instance micropolar fluids as in [START_REF] Cornilleau | On the local exact controllability of micropolar fluids with few controls[END_REF], or the Boussinesq system as in [START_REF] Carreño | Local controllability of theN-dimensional Boussinesq system with N -1 scalar controls in an arbitrary control domain[END_REF], in order to reduce the number of controls. • Give an extension of the results of [START_REF] Liard | A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups[END_REF] to general systems of semigroups, with constant or time-dependant coupling terms, under the hypothesis that the "scalar" equation that is reproduced (for instance the Schrödinger equation in Section 3) is controllable. From a more theoretical point of view, the following issues to address would be very promising.

• Is it possible to modify the algebraic solvability procedure in order to handle the problem of the loss of derivatives that is in general artificial? One can think to try to both differentiate and integrate in the solvability procedure. However, integration does not preserve the support, so that it is not clear if such a procedure will give satisfying results. • In some sense, the algebraic solvability procedure gives "more" than one we need: the support of the resulting solution and control is included in the support of the fictitious control. However, in many situations, we have a little bit of latitude: it will be not a problem if the support of the resulting solution and control is "slightly larger" than the support of the fictitious control. Is it possible to modify the method in such a way? Both of these two perspectives are challenging and may require to develop new and different tools.

4 .C - 1 c 11 c 12 c 13 c 14 c 21 c 22 c 23 c 24 c 31 c 32 c 33 c 34 c 41 c 42 c 43 c 44

 4144 More precisely, let us write C -1 under the form

2. 2 . 1 .

 21 First step: analytic part. Let us introduce the following auxiliary control problem(2.11)

Definition 3 . 2 .

 32 Let ω be an open subset of M . We say that ω verifies the geometric control condition (shortened in GCC in what follows) if there exists T > 0 such that every geodesic starting from any point of M traveled at speed 1 enters ω before the time T .

  Here, n = 2 and m = 1 < 2. Then, the Kalman matrix [A|B] ∈ M 2 (R) is given by[A|B] = [B, AB] = 0 a 12 1 a 22 .

(

  we have written the generic term corresponding on the k-th line, for k ∈ [|2, n|]). Coming back to the definition of the Kalman matrix [A|B], we can rewrite (3.26) as N (L * ϕ) = [A|B] * ϕ. Hence, we have N • L * = [A|B] * . Passing one more time to the formal adjoint and using the fact that * is an involution, we have for the moment L • N * = [A|B]. N * involves only linear combinations of derivatives of the form (-i∂t -∆) j , for j ∈ [|1, n -1|]. Now, we introduce [A|B] -1 ∈ M n,nm (R) as any right inverse to [A|B| (such a right inverse exists, since [A|B] ∈ M nm,n (R) and [A|B] is of full rank). Hence, we have by linearity of all the operatorsL • N * [A|B] -1 = Id C ∞ ((0,T )×M ) n . Hence, (3.25) is verified with M = N * [A|B] -1 .Now, let us go back to the problem of the regularity. Taking into account that (3.23) holds, we see that M( U 1 ω ) ∈ L 2 ((0, T ) × M ) since M involves only linear combinations of derivatives of the form (-i∂t -∆) j , for j ∈ [|1, n -1|].

Acknowledgments

I would like to warmly thank the organizers of the 13th International Young Researchers Workshop on Geometry, Mechanics and Control, for having given me the opportunity to deliver these lectures, and also for the kind welcome at Coimbra and the very efficient organization of the Workshop. I also would like to thank the reviewers of these lecture notes for their numerous and relevant remarks and suggestions, that helped me to improve drastically the overall quality of the lecture notes.


Christine Keribin
email: christine.keribin@math.u-psud.fr

SOME ASYMPTOTIC PROPERTIES OF MODEL SELECTION CRITERIA IN THE LATENT BLOCK MODEL

Keywords: Latent block model, co-clustering, model selection, BIC, ICL

published or not. The documents may come

Some Asymptotic Properties of Model Selection Criteria in the Latent Block Model

Introduction

Clustering is an essential unsupervised tool to discover hidden structure from data by detecting groups of observations that are similar within a group and dissimilar from one group to another one. The challenge of modern data is to learn from observations x i ∈ R d with a large number n of units observed on a large number d of variables, and the question is not only to cluster the observations, but also to cluster simultaneously the observations and the variables, leading to a tremendous parsimonious data representation. This is called co-clustering and has many applications in many fields such as recommendation systems (to cluster simultaneously customers and goods), text mining (to co-cluster words and documents), genomics (to co-cluster genes and experimental conditions) for example. As for clustering, there are many ways to perform co-clustering, and we will focus here on the latent block model (LBM). We present the model and its asymptotical properties. In particular, we shall analyze the log-likelihood ratio under model order misspecifications, and derive a class of penalized log-likelihood criteria asymptotically consistent, results that are new for LBM. -row assignments (or labels) z i , i = 1, . . . , n, are independent from column assignments (or labels) w j , j = 1, . . . , d : p(z, w) = p(z)p(w); -row labels are independent, with a common multinomial distribution:

z i ∼ M (1, π = (π 1 , . . . , π g ));
in the same way, column labels are i.i.d. multinomial variables:

w j ∼ M (1, ρ = (ρ 1 , . . . , ρ m )).
-conditionally to row and column assignments (z 1 , . . . ,

z n ) × (w 1 , . . . , w d ),
the observed data X i j are independent, and their (conditional) distribution ϕ(., α) belongs to the same parametric family, which parameter α only depends on the given block:

X i j |{z ik w j = 1} ∼ ϕ(., α k )
where z ik is the indicator membership variable of whether row i belongs to row-group k and w j is the indicator variable of whether column j belongs to column-group .

Hence, the complete parameter set is θ = (π, ρ, α), with α = (α 11 , . . . , α gm ). With these assumptions, the likelihood of the complete data is

p(x, z, w; θ) = p(z; θ)p(w, θ)p(x|z, w; θ) = ∏ i,k π z ik k ∏ j, ρ w j ∏ i, j,k, ϕ (x i j ; α k ) z ik w j
The labels are usually unobserved, and the observed likelihood is obtained by marginalization over all the label configurations:

p(x; θ) = ∑ z∈Z,w∈W ∏ i,k π z ik k ∏ j, ρ w j ∏ i, j,k, ϕ (x i j ; α k ) z ik w j
LBM deals with matrix of homogeneous data, such as binary [START_REF] Govaert | Block clustering with bernoulli mixture models: Comparison of different approaches[END_REF], Gaussian [START_REF] Lomet | Sélection de modèles pour la classification de données continues[END_REF], categorical [START_REF] Keribin | Estimation and selection for the latent block model on categorical data[END_REF] or count [START_REF] Govaert | Latent block model for contingency table[END_REF] data. It involves a double missing data structure z for rows and w for columns, and the observed likelihood can not be factorized as a product of the mixing density as for simple mixture models. This implies that the likelihood is rapidly not tractable numerically even for few observations and few blocks, as the marginalization involves k n × d m terms. The estimation can however be performed either with numerical approximations (such as variational methods) or with Bayesian approaches (VBayes algorithm or Gibbs sampling).

Asymptotic properties

The double missing structure also leads to a very challenging and interesting study to state the asymptotic behavior of the maximum likelihood (MLE) and variational (VE) estimators. This question was first studied on the Stochastic Block Model (SBM) which is a LBM with the same statistical units in rows and columns, used to model graph adjacency matrices. In this case, there is only one set of latent variables z. These results were obtained when the true order (K × L) of the model is known. The question of the choice of K and L is crucial, and well-posed in the probability framework of LBM. Let K (resp. L ) be misspecifications of the number of row (resp. column) clusters. In this talk, we will study the D KK ,LL = log sup θ∈Θ K ,L p(x; θ) sup θ∈Θ K,L p(x; θ)

for K = K or L = L or both. Extending Wang et al., 2017 methodology for SBM, we deal with the LBM double asymptotic in row and column to provide an appropriate penalty term and define a class of selection criteria asymptotically consistent.

Figure 1 .

 1 Figure 1. n × d = 450 × 600 observations (left) and their reorganization according to the underlying structure in 4 × 5 blocks (right)

  [START_REF] Celisse | Consistency of maximum-likelihood and variational estimators in the stochastic block model[END_REF] first proved that under the true parameter value, the conditional distribution of the assignments of a binary SBM converges to a Dirac of the real assignments. Assuming the existence of an estimator of α converging at rate at least n -1 , they obtained the consistency of MLE and VE. Mariadassou & Matias, 2015 presented a unified framework for LBM and SBM for observations coming from an exponential family, but cannot get rid off the previous assumption to prove consistency. Using a different approach,[START_REF] Bickel | Asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels[END_REF] showed for binary SBM (i) the consistency and aymptotic normality of the MLE in the complete model where the labels are known (ii) these properties can be transferred to the MLE of the observed model. Recently, Brault et al., 2017 solved the consistency and the asymptotic normality of the MLE and VE for LBM observations coming from an exponential family.