BOUNDARY CONTROLLABILITY OF A SYSTEM MODELLING A PARTIALLY IMMERSED OBSTACLE

. In this paper, we address the problem of boundary controllability for the one-dimensional nonlinear shallow water system, describing the free surface ﬂow of water as well as the ﬂow under a ﬁxed gate structure. The system of diﬀerential equations considered can be interpreted as a simpliﬁed model of a particular type of wave energy device converter called oscillating water column. The physical requirements naturally lead to the problem of exact controllability in a prescribed region. In particular, we use the concept of nodal proﬁle controllability in which at a given point (the node) time-dependent proﬁles for the states are required to be reachable by boundary controls. By rewriting the system into a hyperbolic system with nonlocal boundary conditions, we at ﬁrst establish the semi-global classical solutions of the system, then get the local controllability and nodal proﬁle using a constructive method. In addition, based on this constructive process, we provide an algorithmic concept to calculate the required boundary control function for generating a solution for solving these control problem. The dates will be set by the publisher.


Context
Free surface interactions with fixed or floating structures have been intensively studied by the mathematical community in the last years with respect to modelling, well-posedness, numerical simulations, etc. Recently, Lannes proposed in [7] a new formulation of water-waves problem in order to take into account the presence of a floating body. More precisely, in his work Lannes implemented a method for the full water wave equations and for reduced asymptotic models, such as the Boussinesq and the nonlinear shallow-water equations, where the pressure exerted by the fluid on the partially immersed structure appears as a Lagrange multiplier associated with the Keywords and phrases: Nonlinear shallow water equations, exact controllability, nodal profile controllability, boundary control, semi-global piecewise C 1 solution.
constraint that under the floating structure, the surface of the fluid coincides with the bottom of the structure. In the case of the nonlinear shallow water equations, the resulting fluid-structure model with vertical lateral walls has been studied in [7] and in the more general case of non vertical walls in [6]. An extension to a system modelling a floating structure on a viscous shallow water regime has been recently studied by Maity et al. in [14], and by Matignon, Tucsnak and Vergara-Hermosilla in [15] and [16].
Bresch, Lannes and Métivier in [3] treat the derivation and mathematical analysis of a fluidstructure interaction problem with a configuration where the motion of the fluid is governed by the Boussinesq system, which is a dispersive perturbation of the hyperbolic nonlinear shallow water equations, and in the presence of a fixed partially immersed obstacle. They showed that the fluid-structure interaction problem can be reduced to a transmission problem.
In a similar way and motivated for mathematical modeling and simulations of a specific type of wave energy converting device, the so-called oscillating water column (OWC) device, Bocchi, He and Vergara-Hermosilla discuss in [1] about the fluid-structure interaction of the partially submerged fixed wall structure of the OWC device, considering the nonlinear shallow water equations to describe the fluid movement, and obtain explicit transmission conditions for the system and respective reduced transmission problems. Recently, Bocchi, He and Vergara-Hermosilla propose in [2] a new and general approach on the mathematical modelling of the OWC, where include the presence of the time-dependent air pressure in the device and prove a local well-posedness result in a Sobolev setting. Hence, by considering their results, in this work we deep on a particular kind of boundary controllability on the transmission problems studied in [1] on an equivalent physical configuration. More precisely, we deal with the exact boundary controllability of nodal profile on a system modelling a structure partially immersed in a fluid governed by the nonlinear shallow water equations [7], considering a discontinuity in the height of the fluid bottom and the transmission conditions developed in [1]. This physical situation is presented in a graphical sketch in Figure 1.
This kind of boundary controllability was motivated by practical applications on gas networks and introduced recently in the literature by Gugat, Herty and Schleper, in [5]. Their new approach was almost immediately generalized by Li to general 1-D first order quasilinear hyperbolic systems with general nonlinear boundary conditions in [9].
As is well known, the usual exact boundary controllability that asks the solution to the system under certain boundary controls to satisfy a given final state at a suitably large time t = T , however, the exact boundary controllability of nodal profile, requires that the value of solution satisfies the given profiles on one or more nodes for t ≥ T by using boundary controls. This approach can be established by means of a constructive method with modular structure, by using the following three ingredients: existence and uniqueness of semi-global classical solution to the mixed initial-boundary value problem, exchanging the role of the the space variable x and time variable t, and the uniqueness of classical solution to the one-sided mixed initial-boundary value problem. This approach was synthesized in a systematic way in the book of Li et al. [11], where their also deal with the local exact boundary controllability of nodal profile on a tree-like network with general topology. For more details, see [4], [17] and [18].

General settings
Mathematically speaking, in this work we consider an incompressible, irrotational, inviscid and homogeneous fluid in a shallow water regime. The motion of the fluid is governed by the 1D nonlinear shallow water equations [7], which are given by where ζ(t, x) is free surface elevation, h(t, x) is the fluid height given by h b + ζ with h b denoting the height of the bottom of the fluid, q(t, x) is the horizontal discharge defined by with u(t, x, z) denoting the horizontal component of the fluid velocity vector field. In the following, we consider the interval (−l, l 1 ) as spacial domain, and we assume that the height of the bottom h b is given by where h 1 and h 0 are real constants such that h 1 < h 0 . Furthermore, we assume that For our analysis, we divide the domain of the problem (−l, l 1 ) into two parts: the interior domain I = (l 0 − r, l 0 + r), and its complement E = (−l, l 1 )\I, called exterior domain, and which is the union of three intervals Let where ζ w is the parameterization of the bottom of the solid. Therefore, the fluid heights are defined by whereh i (i = 0, 1, 2) are the fluid height at rest withh 1 =h 2 =h 0 − s, and s is the height of the step. h w =h 1 + ζ w .
where ∂ t q w (t) = − hw ρ ∂ x P w with the fluid density ρ and an unknown surface pressure P w . Thus, the coupled system of nonlinear shallow water equations for x ∈ E i can be written as with the boundary conditions given as x = l 1 : q 2 = 0, (10) where f (t) denotes a prescribed boundary function or a boundary control to be determined.
In addition, we consider the transmission conditions developed in [1], wich read as follows: x = l 0 ± r : where α = 2r hw . The initial data of the system is given at rest, ζ w , x ∈ I. and q(0, x) = 0.

Outline of the paper
In Section 2 we present the main results about exact controllability of final data given on the whole or a part of fluid domain at a given finite time T > 0 (see in §2.1), and exact controllability of nodal profile for a given demand at one end x = l 1 (See in §2.2). In §3.1 we prove some results relative to the well-posedness result of semi-global C 1 solutions, and in §3.2 we prove the main theorems of the paper by considering a constructive method with modular structure. Finally, in §2.3, we proposed an algorithm that synthesise the ideas of the proof of Theorem 2.4, which could be utilised to implement numerically our results on exact controllability of nodal profile.

Main results
Now we come to our results about control problems for the model given in (8). In order to ensure the well-posedness of quasilinear system (8) with (9)-(12), we consider the exact controllability problem in the neighbourhood of the equilibrium state. In the following, Theorem 2.1 and Theorem 2.4 show the local exact controllability and of nodal profile near the rest state given as (13), respectively. The corresponding results near other equilibrium state can be obtained in a similar way, we explain with some words in Remark 2.2.

Exact controllability with one boundary control.
The aim of exact controllability for system (8) is looking for a boundary control f (t) acting at x = −l though the boundary condition (9), such that the prescribed final data (ζ iT (x), q iT (x)) is attained in the entire water regime at a given time T : where the U i is generated by the control f as a solution of system (8)- (12) with the initial condition: In order to ensure the existence and uniqueness of piece-wise C 1 semi-global solution for this problem, here we give some assumptions on the initial data and final data in advance: For , q i0 (x) satisfy the C 1 compatibility at the points (t, x) = (0, −l), (0, l 1 ) and the piecewise C 1 compatibility at the adjoint points (t, x) = (0, 0), (0, l 0 − r), (0, l 0 + r); Accordingly, , q iT (x) satisfy the C 1 compatibility at the points (t, x) = (T, −l), (T, l 1 ) and the piecewise C 1 compatibility at the adjoint points (t, x) = (T, 0), (T, l 0 − r), (T, l 0 + r).
The following result states the existence of such a control, which can be constructed explicitly in Section 3.
In the above results, we consider only the steady-state at rest, but we can easily establish the corresponding local exact controllability in the neighborhood of a stationary subsonic continuously differentiable state.
Remark 2.2. The exact controllability result given in Theorem 2.1 still holds for initial states in a C 1 -neighborhood of a stationary subsonic continuously differentiable state (q is , ζ is ), d dx Remark 2.3. The result given in Theorem 2.1 illustrates that after a finite time, the control given at one end x = −l can effect the state functions ζ and q in the entire space horizon x ∈ E.
In fact, this result can be generalized to a special case where, the final condition (14) is replaced by a demand given on a part of the horizon: where ζ 2T (x), q 2T (x) are regarded as given final state functions in the 'indoor water regime' E 2 . Now, the infimum controllability time (22) should be modified by a smaller lower bound: A limit case, with a given demand only at the end node x = l 1 , suggests a controllability problem of nodal profile, which will be shown in details in the following subsection.

2.2.
Exact controllability of nodal profile: a given demand at the end x = l 1 .
Stimulated by some practical applications, Gugat et al. [5] and Li [9] proposed in 2010 another kind of exact boundary controllability, called the nodal profile control. This kind of controllability does not ask to exactly attain any given final state at a suitable time t = T by means of boundary controls, instead it asks the state to exactly fit any given profile on a node after a suitable time t = T by means of boundary controls.
For equation (8), we consider a given demand in fluid height and horizontal discharge at the end x = l 1 by : It's worth to mention that the given nodal profile function U B := (h 2 + ζ B (t), q B (t)) should be compatible with the boundary condition (10) in the time interval [0, T ] at the node x = l 1 , which implies that the demand q B must be set as 0. While, the other one, ζ B (t), can be chosen as any given C 1 function of time after a finite time T . In the following, we give a positive answer to the exact controllability of nodal profile: and let T be an arbitrarily given number satisfying T > T . Then for any given initial data (ζ i0 , q i0 ) satisfying assumptions (S1)-(S2), and for any given demand of the surface elevation ζ B (t) with small norm ζ B C 1 [T,T ] , there exists a boundary control f (t) ∈ C 1 [0, T ], such that the mixed initial-boundary value problem for equation (8) with initial condition (15), boundary conditions (9)-(10) and transmission conditions (11)-(12) admits a unique piecewise C 1 solution x ∈ E i }, which exactly satisfies the given nodal profile condition at the end x = l 1 : Remark 2.5. In the above results, we consider only the steady state at rest, but we can easily replace it with any other steady state that is compatible with the boundary and transmission conditions (10)- (12) and establish local exact controllability of nodal profile near a given subsonic continuously differentiable stationary state (q is (x), ζ is (x)), x ∈ E i , i = 0, 1, 2: There exists a C 1 -neighborhood of the stationary state such that for all initial data in this neighborhood that satisfies the C 1 -compatibility conditions and for any given smooth subsonic desired fluid height h 2 + ζ B (t, l 1 ) and discharge profile q B (t, l 1 ) at the end x = l 1 that is in a sufficiently small C 1 -neighborhood of the boundary data corresponding to the stationary state we can construct a continuously differentiable control f = f (t) such that the demand is fulfilled exactly for all t ∈ [T, T ]. Moreover, this control generates a continuously differentiable system state in the entire domain.

Existence and uniqueness of semi-global C 1 solution
We reduce the system in the exterior domain E to a compact form by introducing the couple U i = (ζ i , q i ) T , i = 0, 1, 2: which has two distinct eigenvalues: and the corresponding left eigenvectors can be taken as Introduce the Riemann invariants to the nonlinear shallow water equations Then, (28) can be equivalently rewritten in Thus, The 1D nonlinear shallow water equations in E can be rewritten as the following diagonal form: Obviously, for each i = 0, 1, 2, L i (t, x) is the Riemann invariant corresponding to the negative eigenvalue λ − i , while R i (t, x) is the Riemann invariant corresponding to the positive eigenvalue λ + i . λ − i (resp. λ + i ) is the entering (resp. departing) characteristic on the right-side boundary, while λ − i (resp. λ + i ) is the departing (resp. entering) characteristic on the left-side boundary. In order to guarantee the well-posedness of the mixed initial-boundary value problem on this coupled network, the boundary conditions on each boundary must satisfy (see [13]): (1) the number of the boundary conditions must be equal to that of the entering characteristics; (2) the boundary conditions can be written in the form that the Riemann invariants corresponding to the entering characteristics can be explicitly expressed by all other Riemann invariants (corresponding to the departing characteristics).
The boundary conditions (9) and (10) can be rewritten in Riemann invariants as and which imply that and where R 0 and L 2 are the Riemann invariants corresponding to entering characteristic on x = −l and x = l 1 , respectively, and At x = 0, the Riemann invariants corresponding to entering characteristic are R 0 and L 1 . Let By a direct calculation and noting (29)-(30), we have Similarly, Thus, then by implicit function theorem, in a neighborhood of (R 0 , L 1 ) = (0, 0), the transmission conditions (11) can be equivalently rewritten as Similarly, at x = l 0 ± r, the Riemann invariants corresponding to entering characteristic are R 1 and L 2 . Using the initial data and integral from 0 to t, the second transmission condition in (12) can be replaced by a nonlocal condition as follows Then, by the first transmission condition in (12), we rewrite (12) into Now, we introduce two boundary functions G 1 and G 2 : In fact, by substituting (30) into (44), G 1 and G 2 can be also regarded as functions with respect to Riemann invariants R 1 , L 1 , R 2 , L 2 . For simplicity, we denote R 1 (t, l 0 − r), L 1 (t, l 0 − r) and R 2 (t, l 0 + r), L 2 (t, l 0 + r) by R 1 , L 1 and R 2 , L 2 , respectively, in the following expression: in which G is the function determined by then by implicit function theorem, (43) can be solved as Indeed, by calculation, we obtain Therefore, which implies (47). Using a method similar to [13], we get the existence and uniqueness of piecewise local C 1 solution to the mixed initial-boundary value problem (8) and (9)- (12). And then, by a method similar to [8], [10], and the generalized result for a kind of non-local boundary condition as Lemma 4.1 in [12], we can obtain the corresponding results on existence and uniqueness of piecewise semi-global C 1 solution.
Step 2: Solve the following leftward problem from x = l 1 to x = l 0 + r to get the solution whereζ(t) is a C 1 function satisfyinḡ and ζ 2T (t) is given by the final data (14). The problem (53) is well-posed (by exchanging the role of t and x in hyperbolic system) and admits a solution U 2 = (ζ 2 , q 2 )(t, x) on the domain R 2 := {(t, x)|0 ≤ t ≤ T, x ∈ E 2 }. Then, we get the value of trace ζ 2 (t, l 0 + r) and q 2 (t, l 0 + r) at x = l 0 + r.
At first, by the continuity condition, we have and Substituting the value of q 1 (t, l 0 − r), q 2 (t, l 0 + r), d dt q w (t) and ζ 2 (t, l 0 + r) into the second formula of (12), it becomes an equation for ζ 1 (t, l 0 − r) that provided with the hyperbolicity near the rest state. Thus, we can solve (57) near the rest state and uniquely determine ζ 1 (t, l 0 − r) by Implicit Theorem. We denote the values of ζ 1 (t, l 0 − r) and q 1 (t, l 0 − r) obtained in above asζ 1 (t) andq 1 (t). Hence, we can verify that (1)ζ 1 (t) andq 1 (t) are both C 1 functions; (2) U 1 = (ζ 1 (t),q 1 (t)) is compatible with the value of U 1f (t, l 0 − r) and U 1b (t, l 0 − r) on the corresponding intervals.
Similarly, we solve the leftward problem from x = 0 to x = −l to get the solution U 0 = U 0 (t, x) in the domain {0 ≤ t ≤ T, x ∈ E 0 }: Then we take the boundary control f (t) as the trace ζ 0 (t, −l). We can verify that the solutions U i = U i (t, x), i = 0, 1, 2 constructed in the Step 2, 4, 5 satisfy all requirements of the Lemma.

Proof of Theorem 2.4.
Proof. The prescribed boundary data ζ B (t) and a fixed control function f (t) in (8) would generate an overdetermined initial-boundary value problem. To prove Theorem (2.4), we will find a piecewise C 1 solution (ζ i (t, x), q i (t, x)) for system (8) on the domain R(T ), which satisfies simultaneously the initial condition (15), the interface conditions (11)-(12), the null flux boundary condition (10), and the given nodal profile (23). Substituting this solution into the boundary condition (9), we obtain the desired boundary control f (t).
We construct a solution to the control problem using the following steps explained in more detail below, where T 1 , T 2 and T 3 are defined as in (52).
Step 3: We first solve the leftward problem (53) in the domain R 2 to get the solution U 2 = (ζ 2 , q 2 )(t, x) and then do the same construction procedure shown as the Step 3, Step 4 and Step 5 in the proof of Lemma 3.3 to get solution U i = (ζ i , q i )(t, x), i = 1, 0 in the domain R 1 and R 0 , respectively.
Step 5: Using the uniqueness theorem, we verify that: at time t = 0, U i (0, x) ≡ U if (0, x), x ∈ E i . Thus, the solution U i (i = 0, 1, 2) is the piecewise C 1 solution to solve the overdetermined initial-boundary value problem

Numerical algorithm
In this Subsection we present an algorithm (see in Algorithm 1) that could be utilised to implement numerically our results on the exact controllability of nodal profile of the system modelling the partially immersed structure considered. More precisely, we synthesize the ideas of the proof of Theorem 2.4 in a consecutive scheme, which gives rise to the following algorithm in order to obtain numerical approximations of the control f (t) associated with our problem. The implementation, however, is beyond the scope of this article and is deferred to a forthcoming publication.