Global exact controllability of ideal incompressible magnetohydrodynamic flows through a planar duct

This article is concerned with the global exact controllability for ideal incompressible magnetohydrodynamics in a rectangular domain where the controls are situated in both vertical walls. First, global exact controllability via boundary controls is established for a related Els\"asser type system by applying the return method, introduced in [Coron J.M., Math. Control Signals Systems, 5(3) (1992) 295--312]. Similar results are then inferred for the original magnetohydrodynamics system with the help of a special pressure-like corrector in the induction equation. Overall, the main difficulties stem from the nonlinear coupling between the fluid velocity and the magnetic field in combination with the aim of exactly controlling the system. In order to overcome some of the obstacles, we introduce ad-hoc constructions, such as suitable initial data extensions outside of the physical part of the domain and a certain weighted space.


Introduction
The property of global exact controllability by means of boundary controls has been proved for perfect fluids, which are described by the incompressible Euler equations, already more than two decades ago for various kinds of bounded domains under employment of the return method introduced by Coron in [7]. For instance in [6,8] Coron obtains controllability for two dimensional simply-and multi-connected domains while the situation of three dimensional multi-connected domains is treated in [16] by Glass. However, in the context of ideal incompressible magnetohydrodynamics, which contains the incompressible Euler system coupled in a nonlinear way with the induction equation for the evolution of the magnetic field, similar results on controllability are not available up to the present moment. The goal of this work is to make a first step in this direction for the very particular case of a rectangular domain.
Hereby, (0.3) is equivalent to (0.1) -(0.2) as long as ∇p + = ∇p − , which however would impose requirements on the controlled solutions along Γ 0 . Indeed, by taking the divergence in the equation for H and also by multiplying with n along Γ, one can obtain for the difference q : in Ω × (0, T ), ∂ n q = 0, on (Γ \ Γ 0 ) × (0, T ).
(0. 4) In this article we shall prove the small-time global exact boundary controllability for the Elsässer system (0.3) whereby the obtained solutions may have the property ∇p + = ∇p − . We are not able to verify for our controlled solutions that ∇p + = ∇p − holds, and for the sake of drawing conclusions regarding the original system (0.1) as well, our remedy is to allow for a pressure like corrector ∇q in the induction equation, see Theorem 1.2 below. We leave it as an open problem whether in our controllability result the harmonic contribution q actually satisfies ∇q ≡ 0 or not. One possible approach for showing ∇q ≡ 0, which is under ongoing investigation, would be to establish suitable well-posedness results for (0.1) with non-characteristic boundary conditions at Γ 0 .

Main results and outline of the work
For m ∈ N 0 := N ∪ {0}, n ∈ N, α ∈ (0, 1), the Hölder spaces of R n -valued functions f = (f 1 , . . . , f n ) ′ , in the case n = 1 written as f , on a bounded domain Above, | · | is the Euclidean norm in R n and if n = 1, the short form C m,α (O) might be used. Furthermore, L p , D ′ and C m stand for Lebesgue spaces with p ∈ [1, ∞], the spaces of distributions and m-times continuously differentiable functions, m ≥ 0, respectively. Regarding the domain Ω, the following space is occasionally employed: in Ω, f · n = 0 on Γ \ Γ 0 }. Moreover, concerning spatial derivatives, the following notations are mainly used: • ∂ β , with β ∈ (N 0 ) 2 , for spatial derivatives in general and ∂ m l (∂ l if m = 1), l = 1, 2, for the partial derivatives of order m ∈ N 0 in direction e l , where (e 1 , e 2 ) is the standard orthonormal basis of R 2 .
• ∇ and div denote the gradient and the divergence respectively.
• If ψ is a differentiable scalar function of two variables and v = (v 1 , v 2 ) ′ a differentiable vector field of two variables, then ∇ ⊥ ψ : The main results proved in this article are stated in Theorems 1.1 and 1.2 below.
Theorem 1.1. Let the integerm ≥ 3 and the control time T > 0 be fixed. Then, for all initial-and final data Regarding the original system of incompressible ideal magnetohydrodynamics (0.1), we conclude the following small-time global exact controllability result. Theorem 1.2. Let the integerm ≥ 3 and T > 0 be fixed. Then, for all initial-and final data with q(·, t) being for each t ∈ [0, T ] a harmonic function as in (0.4), to the system and fulfilling the terminal condition In order to avoid inconveniences caused by the corners of Ω, one could replace Ω for instance by a rounded rectangle. Then the controlled boundary would consist of the two lateral curves connecting the bottom and top side. For example, instead of Ω one could consider the system (0.1) directly in Ω 1 , as introduced below in Section 2, without changing the proofs.
Remark 1.4. Instead of constructing the controls acting on Γ 0 explicitly, (0.3) and (1.2) are treated as underdetermined problems. There are thus different possibilities for choosing boundary controls by means of suitably restricting solutions to Γ 0 . This approach has been discussed for instance in [6,[8][9][10]. In the present case, a natural example for such a choice of controls in the Elsässer system would be Since our results include the existence of solutions, knowing uniqueness renders the given approach already meaningful. For this purpose, one can use L 2 energy estimates for the difference Z ± = z ±,1 − z ±,2 of two solutions z ±,i , i = 1, 2 to (1.1), having the same data: Due to (1.4), the boundary terms which appear when deriving the above formula by means of integration by parts either vanish or have a good sign.
Remark 1.5. The assumptionm ≥ 3 in Theorems 1.1 and 1.2 is employed in (4.33). But also at other points of Section 4, for the sake of applying Banach's Fixed Point Theorem, the given data has to be of higher regularity, say C 2,α (Ω; R 2 ) compared to C 0 ([0, T ]; C 1,α (Ω; R 2 )) for a corresponding controlled solution, which can likewise be observed in related works such as [14,16].
Remark 1.6. The choice of a rectangular domain allows us to write (3.4), which is essential for using the weight ω k in Section 4 (for example in the estimate (4.22)). The crucial part in (4.22) is the possibility of making ω k (t) 1 0 ω k (s) −2 ds arbitrarily small by increasing k > 0. Hereby, the squared inverse of the weight inside of the integral is due to (3.4) and we could not obtain such a weight that also works for ω k (t) 1 0 ω k (s) −1 ds. 1.1. Related literature and outline of the proofs. Up to our knowledge, the question of exact controllability for incompressible ideal magnetohydrodynamics and the corresponding Elsässer systems has not been addressed in the mathematical literature before. This is in contrast to the situation for perfect fluids with H ≡ 0, where global exact controllability for the incompressible Euler system has been shown for instance in [8] by Coron,in [16] by Glass and with Boussinesq heat effects taken into account by Fernández-Cara et al. in [14].
In the case of magnetohydrodynamics involving viscous fluids, we can name for instance the work [22] by Tao concerning local exact controllability for compressible magnetohydrodynamics in a planar setting with controls on the whole boundary. Moreover, for incompressible fluids, Badra obtains in [2] local exact controllability to trajectories. Previous results in this direction have been achieved in [3] by Barbu et al. and by Havârneanu et al. in [17,18]. Moreover, Anh and Toi in [1] establish local exact controllability for a magneto-micropolar fluid model and Galan studies in [15] approximate controllability by means of interior controls in a three-dimensional torus.
Regarding basic well-posedness results for ideal incompressible magnetohydrodynamics in bounded domains, we refer to the study [20] by Schmidt and to further results in this direction which have been shown by Secchi in [21]. Moreover, in the article [5] by Bardos et al., long-time a priori estimates in Hölder norms for solutions in the full space case have been given.
In the absence of magnetic fields, when the Navier-Stokes or Euler equations are considered, many results on controllability have been obtained so far. Recent progress, open problems and further references may for instance be found in the articles [10,11] by Coron et al., which are concerned with global exact controllability for incompressible Navier-Stokes equations. With respect to compressible models we refer to [12,13] by Ervedoza et al. and to [19] by Molina. For proving Theorems 1.1 and 1.2 we follow in particular the approach of [8,14], but face several obstructions in the details when it comes to ideal magnetohydrodynamics: • The strong nonlinear coupling of the equations for u and H in (0.1) seems to prevent the strategy of [14], where the return method for the Euler equations is adapted to account for the additional coupling due to Boussinesq heat effects. In order to avoid loss of regularity in the fixed point iteration, we work with the system (0.3) instead of (0.1). • Employing the formulation (0.3) has the drawback that the idea of flushing the domain with zero vorticity, as previously used for instance in [8,14,16], is challenged by the coupling terms. • How to meet the requirements imposed by the induction equation at the controlled boundary parts, see (0.4), while allowing in-and outflow? This leads to the presence of ∇q in (1.2). In order to transship some of the mentioned hindrances, we work on the linearized level with suitably defined initial data in a nonphysical part of an extended domain and moreover introduce a certain weight that enables sufficient estimates, despite the presence of inhomogeneities and modified data in the linearized systems. Finding this weight heavily relies on the special geometry of the domain, which allows for a very simple return method trajectory that is compatible with the nature of the coupling terms.
This article is organized as follows. In Section 2, a return method trajectory is introduced and some auxiliaries are presented. In Section 3, Theorems 1.1 and 1.2 are established under the assumption that a certain local null controllability result holds. The main part of this article is then concerned with proving this local result in Section 4.

Preparations and notations
In this section, following the idea of [8], a return method trajectory for (0.1) is introduced and several auxiliary results will be recalled. For technical reasons, given a positive constant l > 0, the non-physical extensions Ω 1 , Ω 2 and Ω 3 are, as depicted in Figure 1 below, defined via being bounded open sets with smooth boundary and outward normal vectors denoted again by n. Moreover, let γ ∈ C ∞ ([0, 1]; [0, +∞)) be a non-negative smooth function such that Figure 1. A sketch of the rectangular domain Ω and its extensions, while the arrows indicate y(x, t).
Then, with χ ∈ C ∞ 0 (Ω 3 ) denoting a smooth cutoff function satisfying χ(x) = 1 for x ∈ Ω 2 , we define and in particular, the functions (y, H, p, q) which are given by the flow corresponding to y * which solves the ordinary differential equation (2.1) That (2.1) admits a unique solution is due to the global Lipschitz condition satisfied by y * . Notably, Y makes every point in Ω 2 leave this domain until t = 1 as long as M > 0 is sufficiently large, which is a result that in greater generality may be found in [8,16].
A detailed proof for the following result can be found in [14,Lemma 7].
The next lemma can be found in [4, Lemma 1], see also [14,, and will be employed mainly in form of Remark 2.5 below.
. Moreover, with a constant K = K(α, k) > 0 and d dt + denoting the right derivative in time, it holds for t ∈ (0, T ) that Remark 2.5. In the context of Lemma 2.4, for all t ∈ (0, T ) one has the following estimate: Hereby, it is enough to assume that v ∈ C 0 ([0, T ]; C m,α (Ω 3 )) due to regularization.

Proofs of the main results
In this section, Theorems 1.1 and 1.2 are concluded based on the assumption that a local controllability result, namely Proposition 3.2 as stated below, is true. The verification of Proposition 3.2 is postponed at this point and shall be carried out later out in Section 4.
By choosing T = 1 and z ± T = 0 one obtains, through applying the curl operator in (0.3), the control problem in Ω.
Denoting the components of z ± as z ± = (z ± 1 , z ± 2 ) ′ , the coupling terms g ± are given by Utilizing the assumption that both of z + and z − are divergence free, the functions g ± may in all of Ω × [0, 1] be represented in the following way: Since y = (y 1 , y 2 ) is constant in all of Ω with respect to the spatial variables, we can rewrite (3.3) as Remark 3.1. Writing the coupling terms in the form (3.4) will allow us in Section 4 to employ the weight ω k defined in (4.1). Instead of introducing (3.2) and then transforming via (3.3) to (3.4) it would be shorter, and sufficient for the proofs later on, to write directly However, (3.3) is valid for more general domains where the return method trajectory would be less simple and displays a structure that might be useful for the general case. Since, comparing with (3.5), the expression (3.4) is closer to (3.3), we choose to employ (3.4) from now on.
The following local null controllability result, established later in Section 4, constitutes the main step for proving Theorems 1.1 and 1.2.
By construction, there are p + , p − ∈ D ′ (Ω × (0, 1)) such that (z + , z − ) obtained in Proposition 3.2 together with (p + , p − ) satisfy (0.3) for the case of zero final data and the final time being fixed to T = 1.
Assuming first that Proposition 3.2 is true, we show now how to deduce Theorems 1.1 and 1.2 with the help of a scaling and gluing argument, as for instance in [8,14,16].
Proof of Theorems 1.1 and 1.2. We only verify Theorem 1.2, since the arguments for Theorem 1.1 are along the same line. We note that if (u, H, p, q) solve (1.2), then this is also true for (û,Ĥ,p,q) defined bŷ wheres > 0 is the constant introduced in Proposition 3.2. This leads, by applying Proposition 3.2 with T = 1 to solutions (u * , H * , p * , q * ) and (u * * , H * * , p * * , q * * ) to (1.2), obeying Here, the functions p * (·, 1), q * (·, 1), p * * (·, 1) and q * * (·, 1) might be nonzero constants at first, but we modify them to be zero as well. In order to utilize the reversibility in time, which was outlined in (3.6) above, we define , and observe that the functions

Proof of the local null controllability result
This section is devoted to proving Proposition 3.2. The first step is to define a certain map F on a suitable subset of a Banach space such that a fixed point of this map will be a solution to (3.1). The next task is then to assure that this map is a well defined self map and contractive. The idea for using this kind of fixed point argument is similar to [8,14,16], however most parts are carried out in a different way due to the peculiarities of ideal magnetohydrodynamics.

Definition of a fixed point space.
In what follows, λ ∈ C ∞ ([0, 1]; [0, 1]) is such that λ(t) = 1 for t ∈ [0, d] and λ(t) = 0 for t ∈ [2d, 1] with arbitrary but fixed d ∈ (0, 1/2). Let then the two spaces X + and X − be given by and define for k > 0 large, which will be fixed later, the weight function Next, we introduce a set X ν,k on which we shall construct a fixed point iteration below, Above, ν > 0 is the small constant from Lemma 2.3 and X ν,k is not empty as long as z + 0 m,α,Ω and z − 0 m,α,Ω are, depending on k, chosen to be small enough. Indeed, if this is the case, one has (y + λz + 0 , y + λz − 0 ) ∈ X ν,k . The definition of X ν,k above is motivated mainly by the following requirements: • In view of the return method trajectory (y, H ≡ 0, p, q ≡ 0) one should have for each (z + , z − ) ∈ X ν,k an estimate of the form • Elements of X ν,k should be near enough to y such that Lemma 2.3 can be applied.
• The weight should guarantee that for the map F defined below a fixed point can be found, but it has to be compatible with the previous requirement and satisfy for instance ω k (t) ≥ 1 for all t ∈ [0, 1]. In particular, ω k as defined above obeys 1 < ω k (t) < +∞ for each choice of t ∈ [0, 1], k > 0 and for every fixed t ∈ [0, 1] one has as k → +∞ that Moreover, for all t ∈ [0, 1] it holds that In order to make use of (4.2) later on, the representation (3.4) for g ± in Ω × [0, T ] is essential. • Elements of X ν,k should be uniformly bounded in C 0 ([0, 1]; Cm ,α (Ω; R 2 )) 2 by some constant depending only on fixed objects like ν and y. This is satisfied since y is fixed and for (z + , z − ) ∈ X ν,k one has

4.2.
Construction of a fixed point map. On X ν,k we define a map F by assigning to a given pair (z + , z − ) ∈ X ν,k the value F (z + , z − ) through the following steps.
In particular, this means that z ± = y * in Ω 3 \ Ω 2 and z ± = z ± in Ω. Moreover, for (x, s, t) ∈ Ω 3 × [0, 1] × [0, 1], the flow maps corresponding to z ± are denoted by Z ± (x, s, t) = (Z ± 1 (x, s, t), Z ± 2 (x, s, t)) ′ . With the help of the Cauchy-Lipschitz Theorem, they are defined through the ordinary differential equations (4.4) Step 2. Corresponding to g ± as represented in (3.4), and for the purpose of linearizing (3.1), we denote the following extended versions of the coupling terms by (4.5) By using the continuity of the extension operators from Lemma 2.2, G ± obey for all t ∈ [0, 1] the estimate The functions G ± are related to the representation for g ± in (3.4) only in Ω × [0, 1], where the vector fields z ± are divergence free and the spatial derivatives of y vanish. In Ω 3 \ Ω 2 × [0, 1] one observes that G ± = 0, since z ± = y * holds there.
Step 3. In this step suitable extensions j ± 0 for curl(z ± 0 ) are introduced in order to guarantee that a fixed point of the map F will vanish at the final time. LetÕ ⊆ Ω 3 be an open set with the following properties: The setÕ is independent of the above choice of (z + , z − ) ∈ X ν,k , hence it is independent of the fixed point iteration later on. (4) dist(Õ, Ω 2 ) ≥ c > 0 for arbitrarily small but fixed c > 0. The existence of a setÕ with the properties (1) -(4) is implied by the following points: • The constant ν > 0 that appears in the definition of X ν,k is chosen sufficiently small such that Lemma 2.3 guarantees that Y(Ω 2 , 0, 1) ∪ Z + (Ω 2 , 0, 1) ∪ Z − (Ω 2 , 0, 1) ∩ Ω 2 = ∅. • If every particle that resides in Ω 2 at time t = 0 is transported by Z ± to be at time t = 1 at a point of Ω 3 \ Ω 2 , then all particles which occupy Ω 1 at time t = 1 must originate, due to the continuity of Z ± , in some open set contained in Ω 3 \ Ω 2 . • By the previous considerations and the fact that dist(Ω 3 \ Ω 2 , Ω 1 ) > 0 is a fixed quantity, one can choosẽ O ⊆ Ω 3 \ Ω 2 independently of (z + , z − ) ∈ X ν,k .
Therefore, one may introduce independently of (z + , z − ) ∈ X ν,k a smooth cutoff functionχ ∈ C ∞ 0 (Ω 3 ) such that χ(x) = 0 when x ∈ Ω 2 andχ(x) = 1 for x ∈Õ. The extended data j ± 0 is then for each x ∈ Ω 3 defined by The proof for the next auxiliary lemma is postponed until the construction of F is complete.
Lemma 4.1. For σ ∈ [0, 1] and a constant C(ν, y * ) > 0, which is independent of the choice of (z + , z − ) ∈ X ν,k , the following estimate is valid: Together with Lemma 4.1, the extended data j ± 0 defined above in (4.7) is seen to satisfy the following properties: where the constant C > 0 depends on π 1 andχ, but not on the choice of (z + , z − ) ∈ X ν,k . • By the properties of π 1 from Lemma 2.2 and the cutoff functionχ, one has j ± 0 m−1,α,Ω2 ≤ C z ± 0 m,α,Ω . While the generic constants C > 0 or C(ν, y * ) > 0 can depend on several fixed objects such as the extension operators π i , i ∈ {1, 2} orm ≥ 3, we mostly choose to only point out the ν-and y * -dependence in order to indicate when the properties of X ν,k are used for obtaining some estimates.
It remains for this section to show Lemma 4.1.

(4.15)
It is left to move the second kind of term from (4.13) into a constant C(ν, y * ) > 0. Since X ν,k provides a uniform bound with respect to the higher norm · m,α,Ω , this can be done by obtaining for allβ ∈ N 2 0 , |β| ≤m, l ∈ {1, 2}, estimates of the form sup x∈Ω3 ∂βZ ∓ l (x, 0, σ) ≤ C(ν, y * ). (4.16) For this purpose, one can employ again the defining ordinary differential equations for Z ∓ , namely d dt We shall do this inductively: (1) Assume that |β| ≤ 1 in (4.16). By the Chain Rule, since |β| ≤ 1, (4.17) with initial conditions corresponds either to (4.4) or (4.14) and one has by integrating in time where C(ν, y * ) > 0 includes in particular a bound for all appearing factors of the form with some β * ∈ N 2 0 , |β * | ≤ 1, since the flows Z ∓ never leave Ω 3 and one can for all x ∈ Ω 3 estimatẽ (2) Now assume, that for all |β| <m a bound as in (4.16) holds. Then, by integrating (4.17) in time and from Faà di Bruno's Formula like in (4.13), one has sup x∈Ω3 |κ|=m,l=1,2 where In the above formula for V (x, s, κ, l), if |β| =m, thenπ i (β) = 1 for i = 1, 2. This fact comes from the requirement on the partitionsπ to satisfy 2 l=1 β βπ l (β) = κ. Therefore, all terms with exponent π l (β) > 1 can be integrated into a constant C(ν, y * ) by means of the induction assumption. Moreover, with the same explanation as in the first step of the induction argument, the terms of the form can likewise be estimated by C(ν, y * ). In conclusion, (4.18) simplifies to sup x∈Ω3 |κ|=m,l=1,2 and using Grönwall's inequality completes the estimate.

4.3.
Existence of a fixed point. The goal of this section is to show that F (X ν,k ) ⊆ X ν,k and that F is a contraction with respect to the topology of C 0 ([0, 1]; C 1,α (Ω; R 2 )) 2 , which allows then to conclude Proposition 3.2.
The main steps hereto are stated in Lemma 4.2 below.
There exists a possibly large k = k(ν, y * ) > 0 and a small number δ = δ(ν, k, y * ) > 0, depending on ν, k and y * , such that the following statements are true, provided that z + 0 m,α,Ω + z − 0 m,α,Ω < δ: (2) There exists a constant κ ∈ (0, 1) such that for all (z +,1 , z −,1 ), (z +,2 , z −,2 ) ∈ X ν,k one has with the notation The proof of Lemma 4.2 shall be given in Section 4.4 below. First, we conclude Proposition 3.2 under the assumption that Lemma 4.2 holds. Indeed, then F is a contraction and uniformly continuous on X ν,k with respect to · X . Therefore F can be uniquely extended to a continuous mapF : X ν,k → X ν,k , where X ν,k denotes the closure of X ν,k with respect to · X . Moreover,F is a self-map of X ν,k and contractive in the norm · X . Hence, with the help of Banach's Fixed Point Theorem,F admits a unique fixed point (z + , z − ) ∈ X ν,k , see also [14,Theorem 3]. By the construction of F and the definition of the set X ν,k , this fixed point is a solution to (3.1) with z + (x, 1) = z − (x, 1) = 0 for all x ∈ Ω. In order to verify z ± ∈ L ∞ ([0, T ]; Cm ,α (Ω; R 2 )), let (z + n , z − n ) n∈N ⊆ X ν,k be a sequence obeying Then, by the definition of the set X ν,k , there is a uniform bound R > 0 such that for all n ∈ N it holds max t∈[0,1] z + n m,α,Ω (t) + max which yields (z + , z − ) ∈ L ∞ ([0, T ]; Cm ,α (Ω; R 2 )) 2 .