Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network

Controllability of coupled systems is a complex issue depending on the coupling conditions and the equations themselves. Roughly speaking, the main challenge is controlling a system with less inputs than equations. In this paper this is successfully done for a system of Korteweg-de Vries equations posed on an oriented tree shaped network. The couplings and the controls appear only on boundary conditions


Introduction and main result
Partial differential equations (PDE) appear in many contexts to model different phenomena.
Most of time, these equations are coupled and their study gets much more difficult than when a single equation appear.The control of such systems is not the exception and thus it is very important to understand how we can get controllable systems by using the properties of the single equations and the couplings.
The most known PDE are parabolic and hyperbolic equations.Thus, it is very natural to find many works concerning the controllability of coupled systems involving them.If we restrict our attention to boundary controllability, which is the main issue of this paper, we can mention among a huge literature, [1,5,16] for parabolic equations and [3,13,19] for hyperbolic equations.
In this context, a network is a particular kind of coupled systems in which different PDE are posed on different domains (edges of the network) with coupled boundary conditions (acting on the nodes of the network).Depending on the topology of the structure edges-nodes we define star-shaped, tree-shaped or just general networks.For this particular kind of coupled systems we already find some boundary controllability results.In fact, we can mention [7] for parabolic systems and [4,14,15,18,21,25] for hyperbolic systems.
In this work, we are interested in the controllability of oriented networks for the Korteweg-de Vries (KdV) equation.In the literature there is already a good understanding of the control of the single KdV equation.When we deal with the KdV equation with homogeneous Dirichlet conditions and right Neumann condition on a bounded domain, the length L of the interval where the equation is set plays a role in the ability of controlling the solution of the equation ( [9,12,23]).Indeed, it is well-known that if L = 2π, there exists a stationary solution (y(x, t) = 1 − cos x) of the linearized system around 0 which has constant energy.More generally, defining the set of critical lengths one can recall that the linearized equation around 0 is exactly controllable with only one right Neumann control if and only if L / ∈ N (see [23]) and the local exact controllability result holds for the nonlinear KdV equation (using a fixed point argument) if L / ∈ N .Further results show that the nonlinear KdV equation is in fact locally exactly controllable for all critical lengths contrary to the linear KdV equation (see [8,10,12]).See also [9] and [24] for a complete bibliographical review.
As we have a rather complete understanding of the boundary controllability of this equation, we deal here with the KdV equation posed on a network.Recently, we find two papers dealing with the controllability of the KdV equation on a network.In both, the topology considered is a star-shaped network, having in this way one central node and several external nodes.These two papers giving a positive answer to the controllability of the nonlinear KdV equation on network are [2] with N + 1 boundary controls for N edges (the main topic of that work is the stabilization) and [11] with N boundary controls for N edges.
Generally speaking, the main differences between papers [2] and [11] and the present work are: the sense of the propagation of the water wave on the first edge; the transmission conditions at the central node; and the fact that we improve the previous results having one control less here.
More precisely, in this paper we consider a tree-shaped network R of (N + 1) edges e i (where N ∈ N * ), of lengths l i > 0, i ∈ {1, .., N + 1}, connected at one vertex that we assume to be 0 for all the edges.We assume that the first edge e 1 is parametrized on the interval I 1 := (−l 1 , 0) and the N other edges e i are parametrized on the interval I i := (0, l i ) (see Figure 1).
On each edge we pose a nonlinear Korteweg-de Vries (KdV) equation.On the first edge (i = 1) Figure 1: A tree-shaped network with 3 edges (N = 2) we put no control and on the other edges (i = 2, • • • , N + 1) we consider Neumann boundary controls.Thus, we can write the system where y i (x, t) is the amplitude of the water wave on the edge e i at position x ∈ I i at time t, ) are positive constants.The initial data y i0 are supposed to be L 2 functions of the space variable.
It is worth to mention that the transmission conditions at the central node 0 are inspired by the recent papers [20] and [6].It is not the only possible choice, and the main motivation is that they guarantee uniqueness of the regular solutions of the KdV equation linearized around 0 (see [6,26]).A characterization of boundary conditions that imply a well-posedness dynamics for the linear Airy-type evolution equation (u t = αu xxx + βu x , where α ∈ R * , β ∈ R) on star graphs of half-lines are given in [20].
Let us introduce some notations.First, for any function f : R → R we set In the sequel, we shall use the following notations : Then the inner products and the norms of the Hilbert spaces L 2 (R) and H 1 0 (R) are defined by The main goal of this paper is to study the controllability of the nonlinear KdV equation on the tree shaped network of N + 1 edges with N controls.The controllability problem can be stated as following.For any T > 0, l i > 0, y 0 ∈ L 2 (R) and y T ∈ L 2 (R), is it possible to find N Neumann boundary controls h i ∈ L 2 (0, T ) such that the solution y to (1.1) on the tree shaped network of N + 1 edges satisfies y(•, 0) = y 0 and y(•, T ) = y T ?
The main result of this paper gives a positive answer if the time of control is large enough and the lengths of the edges are small enough.
and assume that There exists a positive constant T min such that the system (1.1) is locally exactly controllable in any time T > T min .More precisely, there exists r > 0 sufficiently small such that for any states there exist N Neumann boundary controls h i ∈ L 2 (0, T ) such that the solution y to (1.1) on the tree shaped network of N + 1 edges satisfies y(•, 0) = y 0 and y(•, T ) = y T for T > T min .
Remark 1.A same type of result can be obtained for a general tree with N +1 external vertices, we get the controllability result with only N Neumann controls.For a sake of clarity on the notations, we choose to write our result for a simplified tree with only one internal vertex.

Remark 2.
An open problem is whether it is possible to reduce the number of controls at the external vertices and still having a control result.What is clear is that if we make zero one of our controls h i , then in that branch i the exact controllability does not hold anymore.It is known that the single KdV equation controlled from the left (here, through the couplings) is null controllable only [17].Regarding other similar systems, we can mention the well known result of controllability of the wave equation on a network where we can reduce the number of control if the ratio of the lengths is not rational (see for instance [14]).In our case the conclusion is not easy to obtain.
In order to prove Theorem 1 we prove first the exact controllability result of the KdV equation linearized around 0. Our proof is based on an observability inequality for the linear backward adjoint system obtained by a multiplier approach.We recall that the KdV equation linearized around 0 writes We then get the local exact controllability result of the nonlinear KdV equation applying a fixed point argument.The drawback of this method is that we do not obtain sharp conditions on the lengths l i and on the time of control T min .However, we get an explicit constant of observability.
The paper is organized as follows.Section 2 is devoted to the necessary preliminary step dealing with the well-posedness and regularity of the solutions of the linear and nonlinear KdV equation.
Section 3 will develop the proof of the local controllability result stated in Theorem 1 with a first step concerning the linearized KdV equation and a second step dealing with the original nonlinear system.

Well-posedness and regularity results
In this section, we follow [23] (see also [2,9,11]).We first study the homogeneous linear system (without control), then the linear KdV equation with regular initial data and controls, and by density and the multiplier method, with less regularity on the data.Secondly, we consider the case of the linear system with a source term in order to pass to the nonlinear KdV equation by a fixed point argument.

Study of the linear equation
We begin by proving the well-posedness of the linear KdV equation (1.4) with h i = 0 for any We consider the operator A defined by with domain , where Then we can rewrite the homogeneous linear KdV equation (1.4) with (2.5) It is not difficult to show that the adjoint of A, denoted by A * , is defined by with domain where the operators A and A * are dissipative.
Proof.We first prove that the operator we have with Cauchy-Schwarz inequality If we take α i and β i such that (2.6) holds, then Ay, y L 2 (R) ≤ 0, which means that the operator We now prove that the adjoint operator Then we have If we take α i and β i such that (2.6) holds, then A * z, z L 2 (R) ≤ 0, which means that the operator Consequently, A generates a strongly continuous semigroup of contractions S on L 2 (R) (see [22]).
We denote by {S(t), t ≥ 0} the semigroup of contractions associated with A. For any there exists a unique mild solution y = S(.)y0 ∈ C([0, T ], L 2 (R)) of (2.5).Moreover, if y 0 ∈ D(A), then the solution of (2.5) is classical and satisfies We now prove the well-posedness result for the linear equation (1.4) with regular initial data and controls.More precisely, we assume that the N boundary controls This choice is possible by taking, for instance, for all i ∈ {2, • • • , N + 1}, the functions Moreover, we can define the function .
We now define z = y − φ, which satisfies where g(x, t) . We deduce from classical results on semigroup theory (see [22]) and from the fact that A generates a strongly continuous semigroup of contractions on L 2 (R) that there exists a unique classical solution ) of (2.7).Consequently, there exists a unique solution (1.4).
We now study the same system but with less regularity on the data, using a density argument and the multiplier method.

KdV linear equation with a source term
In order to prove the well-posedness result for the nonlinear KdV equation (1.1), we use a well-posedness and regularity result for the linear KdV equation with a source term: ) and h i ∈ L 2 (0, T ) for any i ∈ {2, • • • , N + 1}.Then, there exists a unique solution y ∈ ).Moreover, there exists C > 0 such that the following estimate holds: (2.17) Proof.Using Proposition 3, it suffices to consider the case y 0 = 0 and h i = 0 for any i ∈ {2, • • • , N + 1}.Since A generates a strongly continuous semigroup of contractions on L 2 (R), , there exists a unique mild solution y ∈ C([0, T ], L 2 (R)) (see [22]) given by the Duhamel's formula and there exists C > 0 such that It remains to prove that y ∈ L 2 (0, T, H 1 0 (R)) and that To prove this we follow exactly the steps of the proof of Proposition 3 paying attention to the fact that the right hand side terms are not homogeneous anymore but involve the source f .

Well-posedness result of the nonlinear equation
We endow the space To prove the well-posedness result of the nonlinear system (1.1), we follow [12] (see also [9]).
The first step is to show that the nonlinear term yy x can be considered as a source term of the linear equation (2.16).
Proof.The proof can be found in [23] or [9].
where r > 0 is chosen small enough later.Given y ∈ B, we consider the map Φ : B → B defined by Φ(y) = ỹ where ỹ is the solution of Clearly y ∈ B is a solution of (1.1) if and only if y is a fixed point of the map Φ.From (2.17) and Proposition 5, we get Moreover, for the same reasons, we have We consider Φ restricted to the closed ball B(0, R) = {y ∈ B, y B ≤ R} with R > 0 to be chosen later.Then Φ(y we take R and r satisfying Consequently, we can apply the Banach fixed point theorem and the map Φ has a unique fixed point.We have then shown the following proposition. Proposition 6.Let T > 0, l i > 0 and assume that (2.6) holds.Then, there exist r > 0 and C > 0 such that for every y 0 ∈ L 2 (R) and there exists a unique y ∈ B solution of system (1.1) which satisfies

Controllability results
We first prove the exact controllability result of the linear system (1.4) by using a duality argument and the multiplier method in order to prove the observability inequality.Then, we obtain the local exact controllability result of the nonlinear system (1.1) by a fixed point theorem.

Linear system
Due to the linearity of the system (1.4), we can consider the case of a null initial data, i.e. by taking y 0 = 0 on R. It can be easily seen that the exact controllability of (1.4) is equivalent to the surjectivity of the operator where y = (y 1 , y 2 , • • • , y N +1 ) is the solution of (1.4) when controls (h 2 , • • • , h N +1 ) are chosen.
It is known that the surjectivity of this operator is equivalent to an observability inequality for the adjoint operator of Λ, which is given by ) is the solution of the backward adjoint system The first step is to prove an observability inequality for the backward adjoint system (3.18),stated below and obtained by a multiplier method.
Theorem 2. Let l i > 0 for any i ∈ {1, • • • , N + 1} satisfying (1.2) and assume that (1.3) holds.There exists a positive constant T min such that if T > T min , then we have the following observability inequality ) . By multiplying ϕ t + ϕ x + ϕ xxx = 0 by qϕ and integrating by parts on R × [s, T ], we get after some computations • Let us first choose q(x, t) = t and s = 0 in (3.20).Then we obtain and using the boundary condition of (3.18) at the internal node 0, we have By using Poincaré inequality and the estimation of the trace of the function, we have, As we can not estimate the trace of ϕ x (0, t), we need to use the strong hypothesis in (1.3), i.e.
Then, from (3.21) we get • Taking now q(x, t) = 1 and s = 0 in (3.20), we obtain Using again the boundary condition of (3.18) at the internal node 0, we have • Picking s = 0, q 1 (x, t) = x and q i (x, t) = αi N βi x in (3.20), we obtain Using again the boundary condition of (3.18) at the internal node 0, we have Then, from (3.24) we get, with Poincaré inequality and the fact that the operator A * is dissipative, x dxdt ≤ 2 max 1, where we used the notation Note that Γ > 0 under the condition which is weaker than the hypothesis (1.2).
In order to have the observability inequality (3.19) from (3.26), we have to impose that leads us to

.27)
This condition on time T makes also appear other condition on L: which is equivalent to that is exactly hypothesis (1.2).This finishes the proof of Theorem 2 where the existence of time T min is given by condition (3.27) and the observability constant is Remark 3. From previous proof we can deduce that if we have the condition Once the observability inequality is established as in Theorem 2, then the exact controllability result of the linear system (1.4) is obtained by duality and the Hilbert Uniqueness Method (HUM).Thus, the following is true.that impose the second condition 2C 5 R < 1.These conditions are satisfied for instance if we choose r and R such that , that ends the proof of Theorem 1.