Boundary Effects on the Controllability of Coupled KdV Systems

TL;DR

This paper investigates the boundary controllability of a coupled nonlinear KdV system modeling gravity waves, using spectral analysis and the contraction mapping theorem to establish local controllability results.

Contribution

It introduces a novel approach combining spectral analysis and entire function theory to analyze controllability of coupled KdV systems with boundary controls.

Findings

Controllability is achieved for the linearized system using duality and hidden regularity.

Spectral problem solved via Paley-Wiener method and entire function analysis.

Local controllability of the nonlinear system is established.

Abstract

We study the exact boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations on a bounded interval. The model describes the interactions of two weakly nonlinear gravity waves in a stratified fluid. Due to the nature of the system, six boundary conditions are required. However, to study the controllability property, we consider a different combination of the control inputs, with a maximum of four. Firstly, the results are obtained for the linearized system through a classical duality approach and some hidden regularity properties of the boundary terms. This approach reduces the controllability problem to the study of a spectral problem, which is solved by using the Paley-Wiener method introduced by Rosier. Then, the issue is to establish when a certain quotient of entire functions still turns out to be an entire function. It can be viewed as a problem of factoring an entire function that, depending on the control configuration, leads to the study of a transcendental equation. Finally, by using the contraction mapping theorem, we derive the local controllability for the full system.

Equations

$$\{\begin{array}{l}{u}_t+u{u}_x+u_{xxx}+a{v}_{xxx}+a_{1}v{v}_x+a_2(uv{)}_x=0,\\ c{v}_t+r{v}_x+v{v}_x+ab{u}_{xxx}+v_{xxx}+a_{2}bu{u}_x+a_{1}b(uv{)}_x=0,\\ u(0,x)=u_0(x),v(0,x)=v_0(x).\end{array}$$

$$\{\begin{array}{l}u(t,0)=h_0(t),u_x(t,L)=h_1(t),u_{xx}(t,L)=h_2(t),\\ v(t,0)=g_0(t),v_x(t,L)=g_1(t),v_{xx}(t,L)=g_2(t).\end{array}$$

$$u(T,\cdot )=u_T\text{and}v(T,.)=v_T.$$

$$\{\begin{array}{l}u(t,0)=0,u(t,L)=h_1(t),u_x(t,L)=h_2(t),\\ v(t,0)=0,v(t,L)=g_1(t),v_x(t,L)=g_2(t).\end{array}$$

$$\{\begin{array}{l}{u}_{xx}(t,0)=h_0,u_x(t,L)=h_1(t),u_{xx}(t,L)=h_2(t),\\ v_{xx}(t,0)=g_0,v_x(t,L)=g_1(t),v_{xx}(t,L)=g_2(t).\end{array}$$

$$z{e}^z=\alpha .$$

$$W(z)e^{W(z)}=\alpha .$$

$$W_k(z)=\log (z)–z–\log (z)+2k\pi i,k\in \mathbb{Z},$$

$$b,c>0\text{and}0<1-a^{2}b.$$

$$\Vert (u_0,v_0){\Vert }_{\mathcal{X}}+\Vert (u_T,v_T){\Vert }_{\mathcal{X}}\le \delta ,$$

$$(u,v)\in C\left([0,T];(L^2(0,L){)}^2\right)\cap L^2\left(0,T,(H^1(0,L){)}^2\right)$$

$$u(T,x)=u_T(x)\text{mbox}andv(T,x)=v_T(x).$$

$$0<C_1(1-a^{2}b)<c,$$

$$\mathcal{H}:=H^{\frac{1}{3}}(0,T)\times L^2(0,T)\times H^{-\frac{1}{3}}(0,T),$$

$$\mathcal{X}:=(L^2(0,L){)}^2,$$

$$⟨(u,v),(\phi ,\psi )⟩:=\frac{b}{c}{\int }_0^{L}u\phi dx+{\int }_0^{L}v\psi dx.$$

$$\{\begin{array}{l}{w}_t+w_{xxx}+w_x=f,\\ w(t,0)=k_1(t),w_x(t,L)=k_2(t),w_{xx}(t,L)=k_3(t),\\ w(0,x)=w_0.\end{array}$$

$$\{\begin{array}{l}{w}_t+w_{xxx}=f,\\ w(t,0)=k_1(t),w_x(t,L)=k_2(t),w_{xx}(t,L)=k_3(t),\\ w(0,x)=w_0.\end{array}$$

$$w\in Z_T:=C([0,T];L^2(0,L))\cap L^2(0,T;H^1(0,L)),$$

$${\partial }_x^{j}w\in L_x^{\infty }(0,L;H^{\frac{1-j}{3}}(0,T)),j=0,1,2.$$

$$||w|{|}_{Z_T}+\sum _{j=0}^2||{\partial }_x^{j}w|{|}_{L_x^{\infty }(0,L;H^{\frac{1-j}{3}}(0,T))}\le C\{||w_0|{|}_{L^2(0,L)}+||(k_1,k_2,k_3)|{|}_{\mathcal{H}}+||f|{|}_{L^1(0,T;L^2(0,L))}\}.$$

$$w(t,x)=S(t)w_0,$$

$$A=-{\partial }_{xxx}$$

$$D(A)=\{g\in H^3(0,L):g(0)=g^{\prime }(L)=g^{\prime \prime }(L)=0\}.$$

$$\{\begin{array}{l}{u}_t+u_{xxx}+a{v}_{xxx}=0,\\ c{v}_t+r{v}_x+ab{u}_{xxx}+v_{xxx}=0,\end{array}$$

$$u(0,x)=u_0(x)\text{and}v(0,x)=v_0(x),$$

$$\{\begin{array}{l}u(t,0)=h_0(t),u_x(t,L)=h_1(t),u_{xx}(t,L)=h_2(t),\\ v(t,0)=g_0(t),v_x(t,L)=g_1(t),v_{xx}(t,L)=g_2(t).\end{array}$$

$$\{\begin{array}{l}{u}_t+u_{xxx}+a{v}_{xxx}=p(t,x),\\ c{v}_t+ab{u}_{xxx}+v_{xxx}=q(t,x),\end{array}$$

$$\{\begin{array}{l}{\bar{u}}_t+{\alpha }^{+}{\bar{u}}_{xxx}=\bar{p},\\ \text{vspace}2mm{\bar{v}}_t+{\alpha }^{-}{\bar{v}}_{xxx}=\bar{q},\end{array}$$

$$\bar{u}(0,x)={\bar{u}}_0(x),\bar{v}(0,x)={\bar{v}}_0(x),$$

Full text

1. Introduction

In this paper we are concerned with the study of a nonlinear coupled system derived by Gear and Grimshaw arising in nonlinear dispersive media [10]. The model describes the strong interactions of two long internal gravity waves in a stratified fluid, where the two waves characterize two different models of the linearized equations of motion. The general structure of the model is a pair of Korteweg-de Vries (KdV) equations, with linear and nonlinear coupling terms:

[TABLE]

The parameters $a,a_{1},a_{2},b,b_{1}$ and $r$ are real constants, the unknowns $u$ and $v$ are real-valued functions of the variables $x$ and $t$ and subscripts indicate partial differentiation. The applicability of the system (1.1) in a particular context depends on many factors. Among the more universal of these is that the waves be unidirectional and essentially two-dimensional in character. In this case, the study of an initial boundary value problem arises naturally to model waves generated by a wavemaker mounted at one end of a uniform stretch of the medium.

Our main goal is to study an initial boundary value problem associated with (1.1) when $x\in[0,L]$ and $t\in\mathbb{R}^{+}$ with the following boundary conditions:

[TABLE]

The boundary functions $h_{i}$ and $g_{i}$, for $i=0,1,2$, are considered as control inputs acting on the boundary conditions. Their choice was motivated by the previous studies on the boundary controllability properties for the KdV equation developed in [6]. In fact, since the late 1990s [7], such boundary conditions has been mainly studied for the well-posedness of the KdV equation in the classical Sobolev spaces $H^{s}(0,L)$ (see, for instance, [6, 7, 13, 17]). The results obtained in these works play an important role in our analysis.

The question to be answered here is the following: Can one drive the solutions of the problem (1.1)-(1.2) to have certain desired properties by choosing appropriate boundary functions? More precisely, we have the following definition:

Definition 1.1**.**

Let $T>0$. The system (1.1)-(1.2) is exactly controllable in time $T$ if, for any initial state $(u_{0},v_{0})\in(L^{2}(0,L))^{2}$ and for any final state $(u_{T},v_{T})\in(L^{2}(0,L))^{2}$, there exist control functions $h_{0},g_{0}\in H^{\frac{1}{3}}(0,T)$, $h_{1},g_{1}\in L^{2}(0,T)$ and $h_{2},g_{2}\in H^{-\frac{1}{3}}(0,T)$, such that the solution $(u,v)$ of (1.1)-(1.2) satisfies

[TABLE]

The study of the controllability properties for system (1.1) was first addressed in [15] with the following set of boundary conditions:

[TABLE]

In order to prove the results, the authors combine the controllability of the linearized system and a fixed point argument. For the linear case, the controllability was obtained through the classical duality arguments [9], [14], [16], where the controllability problem is equivalent to establishing an observability inequality for the solutions of the corresponding adjoint system. In this case, a nonstandard unique continuation principle for the eigenfunctions of the differential operator associated with the model is required. As in [16], this is done by combining Fourier analysis and Paley-Winer theorem. Later on, in [3], the authors proved that, in some situations, it is possible to get the controllability of the linearized system by using only two controls on the Neumann boundary conditions, $h_{2}$ and $g_{2}$. Indeed, to prove the observability inequality they used a direct approach, based on the multiplier technique, which gives the observability inequality for small values of the length $L$ and large time of control $T$. In both works, the controllability of the nonlinear system runs exactly in the same way, using a fixed point argument.

More recently, combining the approach developed in [16] with some hidden regularity properties of the boundary terms, the exact controllability property for (1.1) was studied in [4] by considering the following set of boundary conditions:

[TABLE]

Depending on the controls configuration, the controllability property of (1.1)-(1.4) is obtained with some restriction over the size of the interval. More precisely, the controllability holds whenever $L$ does not belongs to a countable set of critical lengths. Later on, by considering the boundary conditions given by (1.3), the control problem for (1.1) was addressed in [5]. Proceeding as in [4], the authors proved that, with another configuration of four controls, the controllability of the system may depends on the length of the spatial domain.

In the works mentioned above [4, 5, 15], the controllability problem is addressed by following closely the ideas introduced in [16] while studying the boundary controllability of the KdV equation posed on a bounded interval $(0,L)$. Being different from other systems, the length $L$ of the spatial domain may play a crucial role in determining the controllability of the system, specially when some configurations of the controls input are allowed to be used. This phenomenon, the so-called critical length phenomenon, was observed for the first time in [16]. Roughly speaking, it was proved the existence of a finite dimensional subspace $M$ of $L^{2}(0,L)$, which is not reachable by the linearized KdV equation, when starting from the origin, if $L$ belongs to a countable set of lengths.

In this paper, we proceed as in [15], [16] to extend the analysis of the controllability properties for system (1.1)-(1.2). By considering different combinations of the boundary controls $g_{i}$ and $h_{i}$, with $i=0,1,2$, we prove that the controllability properties hold for any lenght $L$ of the spatial domain. This is done by combining the controllability of the linearized system and a fixed point argument in an appropriate function space. As in the previous works, the controllability of the linearized system is equivalent to the proof of an observability inequality for the solutions of the corresponding adjoint system. To prove such inequality, we derived linear estimates, including hidden regularities (sharp trace regularities) results for the solutions of the adjoint system. In some cases, however, this is not enough due to some technical difficulties we have to deal with. Indeed, proceeding as in [16], the arguments described above reduces the problem to show a unique continuation result for the state operator. To prove the result, we extend the solution under consideration by zero in $\mathbb{R}\setminus[0,L]$ and take the Fourier transform. Then, the issue is to establish when a certain quotient of entire functions still turns out to be an entire function, which leads to the study of the transcendental equation

[TABLE]

Equation (1.5) has been studied by a large number of authors (see, for instance, [8], [18]), and its solutions are given by the Lambert function $W(z)$ which satisfies

[TABLE]

Moreover, if $\alpha\neq 0$, equation (1.5) has infinitely many solutions, which are given by the zeros of the holomorphic complex functions

[TABLE]

where $\log(z)$ denotes de principal branche of the log-function. This approach allows us to conclude that, in some cases, the controllability holds for any $L>0$.

Our analysis is done under additional assumptions on some coefficients of the system. We remark that, according to [1], [19], the parameters $b$ and $c$ are automatically positive and $r$ is a nondimensional parameter that could be assumed very small. However, as it will become clear throughout the paper, in order to provide the tools to handle this problem, we assume that

[TABLE]

Our main results can be summarized as follows.

Theorem 1.1**.**

Let $T>0$, $L>0$ and $r\neq 0$. Then, there exists $\delta=\delta(t,L)>0$, such that, for any $(u_{0},v_{0})$, $(u_{T},v_{T})\in\mathcal{X}:=(L^{2}(0,L))^{2}$ verifying

[TABLE]

there exist controls $(h_{0},h_{1},h_{2})$ and $(g_{0},g_{1},g_{2})$ in $\mathcal{H}:=H^{\frac{1}{3}}(0,T)\times L^{2}(0,T)\times H^{-\frac{1}{3}}(0,T)$, with the following configurations for four controls:

1. (i)

$h_{0}\neq 0,h_{1}\neq 0,h_{2}\neq 0\quad\text{and}\quad g_{0}=0,g_{1}\neq 0,g_{2}=0$, 2. (ii)

$h_{0}=0,h_{1}\neq 0,h_{2}=0\quad\text{and}\quad g_{0}\neq 0,g_{1}\neq 0,g_{2}\neq 0$, 3. (iii)

$h_{0}\neq 0,h_{1}\neq 0,h_{2}=0\quad\text{and}\quad g_{0}\neq 0,g_{1}\neq 0,g_{2}=0$, 4. (iv)

$h_{0}=0,h_{1}\neq 0,h_{2}\neq 0\quad\text{and}\quad g_{0}=0,g_{1}\neq 0,g_{2}\neq 0$,

such that the system (1.1)-(1.2) under condition (1.6), admits a unique solution

[TABLE]

satisfying

[TABLE]

Moreover, if the parameters $a,b,c$ satisfy (1.6) and

[TABLE]

where $C_{1}=C_{1}(T,L)>0$ is the hidden regularity constant given in (2.19), the same result is obtained with the following configuration for three controls:

1. (v)

$h_{0}\neq 0,h_{1}\neq 0,h_{2}=0\quad\text{and}\quad g_{0}\neq 0,g_{1}=0,g_{2}=0$, 2. (vi)

$h_{0}\neq 0,h_{1}=0,h_{2}=0\quad\text{and}\quad g_{0}\neq 0,g_{1}\neq 0,g_{2}=0$.

The additional condition of the parameters $a,b$ and $c$ to prove the exact controllability for three controls allows to absorb some boundary terms leading to linear a priori estimates that play an important role in the proofs.

Following the ideas involved in the proof of Theorem 1.1, it is possible to obtain similar results with different configuration of three and four controls. We also remark that the corresponding linear system can be written in an equivalent diagonal form and, when the parameter $r=0$, we obtain two independent KdV equations. In this case, the model can be analyzed by using the same theory, which also leads to the desired results for the corresponding linear system. We address both issues in the next sessions.

In addition, Theorem 1.1 establishes as a fact that system (1.1)-(1.2) inherits the interesting controllability properties initially observed in [6] for the KdV equation. As far as we know, the problem we address here has not been addressed in the literature yet. In fact, since the system is very sensitive the choice of the boundary conditions, the existing results do not give an immediate answer to it, including [4] and [5].

The remainder of this paper is organized as follows. Section 2 covers the study of well-posedness for the linear system associated to (1.1), its adjoint and the full system. Additionally, we present various linear estimates, among them some hidden regularities results, which are crucial tool to prove our results. Section 3 is devoted to the observability conditions related to the adjoint system when different controls configuration are considered. In Section 4 we give a positive answer to the linear controllability problem and extend the result to the full system via contraction mapping theorem. Finally, Section 5 is dedicated to further comments and open problems.

2. Well-posedness

We first introduce the notation that will be used in the whole paper. From now on, we consider the product of Sobolev spaces

[TABLE]

endowed with their natural inner products, for the boundary data. We also introduce the space

[TABLE]

for the initial data space, endowed with the inner product

[TABLE]

2.1. Linear Systems

In this section, we analyze the well-posedness of both, the linear system associated to (1.1)-(1.2) and the corresponding adjoint system ($h_{1}=g_{i}\equiv 0$, $i=0,1,2$). We also derive some hidden (sharp trace) regularities results together with linear estimates for the solutions.

In the sequel, $C$ denotes a generic positive constant; $C_{0},C_{1}$, etc. are other positive (specific) constants.

The following results obtained for the KdV equation will be used in our proofs.

2.1.1. The KdV equation

In [6], the authors study the well-posedness of the following initial boundary value problem for the KdV equation posed on a bounded interval:

[TABLE]

Following the same approach (see, for instance, Propositions 2.5, 2.6 and 2.8 in [6]), a similar result can be derived for the problem

[TABLE]

More precisely,

Proposition 2.1**.**

Let $T>0$ be given. Then, for any ${w}_{0}\in L^{2}(0,L)$, $f\in L^{1}(0,T;L^{2}(0,L))$ and boundary data $(k_{1},k_{2},k_{3})\in\mathbf{\mathcal{H}}$, system (2.3) admits a unique solution

[TABLE]

which, in addition, has the hidden (or sharp trace) regularities

[TABLE]

Moreover, there exists $C>0$, such that

[TABLE]

We note that, if $f\equiv 0$ and $k_{i}\equiv 0$, for $i=0,1,2$, the solution of (2.3) can be written as

[TABLE]

where $\{S(t)\}_{t\geq 0}$ is the $C_{0}$ semigroup on the $L^{2}(0,L)$ space generated by the operator

[TABLE]

with the domain

[TABLE]

2.1.2. The direct system

We consider the direct system

[TABLE]

with the following initial and boundary data

[TABLE]

[TABLE]

When $h_{i}=g_{i}=0$, $\forall i=0,1,2$, the operator associated to the space variable is dissipative with the norm defined in (2.1). Nevertheless, it is not clear that the corresponding adjoint operator is also dissipative. Hence, the classical semigroup theory does not work directly over the system (2.4)-(2.6). In order to overcome this difficult, we apply a fixed point argument. Therefore, we first introduce the following nonhomogeneous system associated to (2.4)

[TABLE]

with initial and boundary data given by (2.5) and (2.6), respectively.

Remark 2.1**.**

Proceeding as [4, Proposition 2.1], we can make a change of variables to transform the system (2.7) into

[TABLE]

with $\alpha^{+},\alpha^{-}\in\mathbb{R}$, initial conditions

[TABLE]

and boundary conditions

[TABLE]

It will be useful to prove the well-posedness results for the linearized system.

Taking the considerations above into account, the well-posedness of the system (2.7), (2.5)–(2.6) can be stated as follows:

Proposition 2.2**.**

Let $T>0$ be given. Then, for any $({u}_{0},{v}_{0})\in\mathcal{X}$, ${p},{q}\in L^{1}(0,T;L^{2}(0,L))$ and boundary data $\vec{h},\vec{g}\in\mathbf{\mathcal{H}}$, system (2.7), (2.5)–(2.6) admits a unique solution

[TABLE]

which, in addition, has the hidden (or sharp trace) regularities

[TABLE]

Moreover, there exists $C>0$, such that

[TABLE]

Proof.

Observe that, according to Remark 2.1, system (2.7), (2.5)–(2.6) can be decoupled into two independent initial boundary value problem for the KdV equation as follows

[TABLE]

Moreover, the terms on right hand side of the equation, as well as, the boundary functions and the initial conditions, lie in the functional spaces $L^{1}(0,T;L^{2}(0,L))$, $\mathbf{\mathcal{H}}$ and $L^{2}(0,L)$, respectively. Then, the result follows from Propositions 2.1. We also remark that, if $\bar{p}=\bar{u}^{0}\equiv 0$, the solution $\bar{u}$ of (2.11) can be written as

[TABLE]

where $\vec{h}:=(\bar{h}_{0},\bar{h}_{1},\bar{h}_{2})$. The operator $S_{bdr}(t)$ is the called boundary integral operator, whose explicit representation can be found in [12], [13] (see also [4], [5]). For the solution $\bar{v}$ we have a similar representation. ∎

Having Proposition 2.2 in hands, the global well-posedness of the system (2.4)–(2.6) is obtained using a fixed-point argument.

Proposition 2.3**.**

Let $T>0$ be given. Then, for any $({u}_{0},{v}_{0})\in\mathcal{X}$ and boundary data $\vec{h},\vec{g}\in\mathbf{\mathcal{H}}$, system (2.4)–(2.6) admits a unique solution

[TABLE]

which, in addition, has the hidden (or sharp trace) regularities

[TABLE]

Moreover, there exists $C>0$, such that

[TABLE]

Proof.

Let $K_{T}:=\left\{(u,v)\in X_{T}:(u,v)\in L^{\infty}_{x}(0,L;(H^{\frac{1-k}{3}}(0,T))^{2}),k=0,1,2\right\}$ equipped with the norm

[TABLE]

Let $0<\beta\leq T$ to be determined later. For each $u,v\in K_{\beta}$, consider the problem

[TABLE]

According to Proposition 2.2, we can define the operator

[TABLE]

where $(\omega,\eta)$ is the solution of (2.13). Moreover,

[TABLE]

where the positive constant $C$ depends only on $T$. Since

[TABLE]

we obtain a positive constant $C>0$, such that

[TABLE]

Let $(u,v)\in B_{r}(t,0):=\left\{(u,v)\in K_{\beta}:\|(u,v)\|_{K_{\beta}}\leq r\right\}$, with

[TABLE]

Choosing $\beta>0$, satisfying

[TABLE]

from (2.15) we obtain

[TABLE]

The above estimate allows us to conclude that

[TABLE]

On the other hand, note that $\Gamma(u_{1},v_{1})-\Gamma(u_{2},v_{2})$ solves the following system

[TABLE]

Again, from Proposition 2.2 and (2.16), we have

[TABLE]

Hence, $\Gamma:B_{r}(t,0)\rightarrow B_{r}(t,0)$ is a contraction and, by Banach fixed point theorem, we obtain a unique $(u,v)\in B_{r}(t,0)$, such that

[TABLE]

and (2.14) holds, for all $t\in(0,\beta)$. Since the choice of $\beta$ is independent of $(u^{0},v^{0})$, the standard continuation extension argument yields that the solution $(u,v)$ belongs to $K_{T}$. The proof is complete. ∎

2.2. The adjoint system

The second system to be analyzed is the adjoint system associated to (2.4), that is,

[TABLE]

with data

[TABLE]

and boundary conditions

[TABLE]

Observe that the two last conditions of (2.18) can be summarized into

[TABLE]

Then, the main result of this subsection reads as follows:

Proposition 2.4**.**

Let $T>0$ be given. Then, for any $(\varphi_{T},v_{T})\in\mathcal{X}$ system (2.17)-(2.18) admits a unique solution $(\varphi,\psi)\in X_{T}$. Moreover,

[TABLE]

for some $C_{j}>0$, where $j=0,1,2$. Moreover, we have that

[TABLE]

Proof.

If we consider the change of variables $x\mapsto L-x$ and $t\mapsto T-t$, system (2.17)-(2.18) become

[TABLE]

with initial conditions

[TABLE]

and boundary data

[TABLE]

whit $\nu(t):=-a\psi_{xx}(t,0)$ and $\xi(t):=-ac\varphi_{xx}(t,0)-r\psi(t,0)$.

The proof can be done by using a fixed point argument as in Proposition 2.3. On the other hand, multiplying the first equation of (2.17) by $-t\varphi,$ the second one by $-\frac{b}{c}t\psi$ and integrating by parts over $(0,T)\times(0,L),$ we obtain

[TABLE]

and

[TABLE]

Adding the above identities, it yields

[TABLE]

Thus,

[TABLE]

Applying the boundary conditions, we have that

[TABLE]

∎

2.3. Nonlinear System

Following the ideas of [4, Theorem 2.7] and [5, Theorem 4.1], we can establish the well-posedness for the full system.

Theorem 2.1**.**

Let $T>0$ be given. Then, for any $({u}_{0},{v}_{0})\in\mathcal{X}$ and boundary data $\vec{h},\vec{g}\in\mathbf{\mathcal{H}}$, there exists $T^{*}>0$, depending on $\|({u}_{0},{v}_{0})\|_{\mathcal{X}}$, such that the system (1.1)–(1.2) admits a unique solution

[TABLE]

which, in addition, has the hidden (or sharp trace) regularities

[TABLE]

Moreover, the corresponding solution map is Lipschitz continuous.

Proof.

Let us consider

[TABLE]

which is a Banach space equipped with the norm

[TABLE]

Let $0<T^{*}\leq T$ to be determined later. For each $u,v\in\mathcal{F}_{T^{*}},$ consider the problem

[TABLE]

where the nonlinear terms given by

[TABLE]

and

[TABLE]

Proceeding as in [2, Lemma 3.1], we deduce that $f(u,v)$ and $s(u,v)$ belong to $L^{1}\left(0,T^{*};L^{2}(0,L)\right)$. Then, according to Proposition 2.2, we can define the operator

[TABLE]

where $(\omega,\eta)$ is the solution of (2.23). Moreover from [2, Lemma 3.1], we obtain a positive constant $C$ depending only on $T^{*},$ such that

[TABLE]

Let $(u,v)\in B_{r}(t,0):=\left\{(u,v)\in\mathcal{F}_{T^{*}}:\|(u,v)\|_{X_{T^{*}}}\leq r\right\}$, where

[TABLE]

From the estimate above, we get

[TABLE]

Then, we can choose $T^{*}>0,$ such that

[TABLE]

to obtain

[TABLE]

Thus, we conclude that $\Gamma:B_{r}(t,0)\subset\mathcal{F}_{T^{*}}\rightarrow B_{r}(t,0)$. On the other hand, $\Gamma\left(u_{1},v_{1}\right)-\Gamma\left(u_{2},v_{2}\right)$ solves the system

[TABLE]

Then, Proposition 2.2 and [2, Lemma 3.1] imply that

[TABLE]

for some positive constant $C_{3}.$ Thus, choosing $T^{*}>0,$ such that (2.24) holds and

[TABLE]

it follows that

[TABLE]

Hence, $\Gamma:B_{r}(t,0)\rightarrow B_{r}(t,0)$ is a contraction and, by Banach fixed-point theorem, we obtain a unique $(u,v)\in B_{r}(t,0),$ such that $\Gamma(u,v)=(u,v)\in\mathcal{F}_{T^{*}}$. ∎

2.3.1. Integral form

If we denote by \left(\begin{array}[]{cc}u\\ v\end{array}\right) the transpose of $(u,v)$, from the above results we deduce that the solution of (1.1)-(1.2) can be written as

[TABLE]

In the above formula,

[TABLE]

where $\left\{S^{\pm}(t)\right\}_{t\geq 0}$ are the $C_{0}$ semigroup on the space $L^{2}(0,L)$ generated by the linear operators

[TABLE]

with

[TABLE]

and domains

[TABLE]

The operators $\left\{S_{bdr}^{\pm}(x)\right\}_{t\geq 0}$ are the so-called boundary integral operators, defined in the proof of Proposition 2.2.

3. Observability Condition

We study the exact controllability property when the system (2.4)-(2.6) starts from zero, i.e., when $u_{0}\equiv v_{0}\equiv 0$. For another initial state $(u_{0},v_{0})\in\mathcal{X}$ the linearity of system together with the translation $(u,v)+(u_{0},v_{0})$ will let us to recover the solution.

In order to understand the interaction between the number of controls and the controllability properties, we consider the more general case $\mathbf{r\neq 0}$ in the linear system (2.17)-(2.18). Some comments about the case $r=0$ are presented in section 5 below.

A positive answer to the exact control problem is obtained by using the duality classical approach [9, 14]. In this case, the observability inequality of a dual problem plays a fundamental role. Thus, proceeding as in [16], we show that the observability inequality holds when several configurations of boundary controls are considered.

3.1. Case A: Observability inequality with four controls configurations

In this subsection, we consider the following boundary control configuration ($g_{0}=g_{2}=0$ and $h_{0}\neq 0,h_{1}\neq 0,h_{2}\neq 0$, $g_{1}\neq 0$):

[TABLE]

In this case, we establish the following observability inequality:

Proposition 3.1**.**

Let $T>0$ and $L>0$. Then, there exists $C=C(L,T)>0$, such that, if $(\varphi,\psi)\in C([0,T];L^{2}(0,L))^{2}$ is a solution of (2.17)-(2.18), the inequality

[TABLE]

holds for any $\varphi_{T},\psi_{T}\in L^{2}(0,L)$.

Proof.

The proof follows the ideas developed in [16]. Arguing by contradiction, we obtain a sequence o functions $\{(\varphi^{n}_{T},\psi^{n}_{T})\}_{n\in\mathbb{N}}$, such that

[TABLE]

Then,

[TABLE]

as $n\rightarrow\infty$. On the other hand, Proposition 2.4 guarantees that $\{(\varphi^{n},\psi^{n})\}_{n\in\mathbb{N}}$ is bounded in $L^{2}(0,T;(H^{1}(0,L))^{2})$ and, using the equations of the system, we deduce that $\{(\varphi_{t}^{n},\psi_{t}^{n})\}_{n\in\mathbb{N}}$ is bounded in $L^{2}(0,T;(H^{-2}(0,L))^{2})$. From the compact embedding $H^{1}(0,L)\hookrightarrow L^{2}(0,L)\hookrightarrow H^{-2}(0,L)$ and by using compactness arguments, it follows that $\{(\varphi^{n},\psi^{n})\}_{n\in\mathbb{N}}$ is relatively compact in $L^{2}(0,T;\mathcal{X})$. Hence, we obtain a subsequence, still denoted by the same index $n,$ satisfying

[TABLE]

Moreover, the hidden regularity of the solutions given by Proposition 2.4 and the boundedness given by (3.1) imply that the boundary terms $\left\{\varphi^{n}(L,\cdot)\right\}_{n\in\mathbb{N}}$ and $\left\{\psi^{n}(L,\cdot)\right\}_{n\in\mathbb{N}}$ are bounded in $H^{\frac{1}{3}}(0,T).$ Then, the compact embedding

[TABLE]

guarantees that the sequences above are relatively compact in $L^{2}(0,T)$. Hence, we obtain a subsequence, still denoted by the same index $n,$ satisfying

[TABLE]

In particular,

[TABLE]

Then, from (3.2) and (3.4), we deduce that

[TABLE]

In order to prove that $\{(\varphi^{n}_{T},\psi^{n}_{T})\}_{n\in\mathbb{N}}$ is a Cauchy sequence in $X$, we use inequality (2.20):

[TABLE]

From the hidden regularity given in Proposition 2.4 and by using Young inequality, we obtain a constant $C>0$, such that

[TABLE]

Thus, the convergences (3.2), (3.3) and (3.5) guarantee that $\{(\varphi_{T}^{n},\psi_{T}^{n})\}_{n\in\mathbb{N}}$ is a Cauchy sequence in $\mathcal{X}$. If we denote by $(\varphi_{T},\psi_{T})$ its limit, from (3.1) we have

[TABLE]

Moreover, from Proposition 2.4 it follows that

[TABLE]

are Cauchy sequences in $L^{2}(0,T)$ and $H^{-\frac{1}{3}}(0,T)$, respectively. Consequently,

[TABLE]

and (3.2) implies that

[TABLE]

Finally, from (3.6) and (3.9) we obtain that $(\varphi,\psi)$ is a solution of

[TABLE]

with additional boundary conditions

[TABLE]

and

[TABLE]

Notice that the solutions cannot be identically zero. However, as it will be shown in the following lemmas, one can conclude that $(\varphi,\psi)=(0,0)$, which drives us to a contradiction. ∎

Lemma 3.1**.**

For any $T>0$, let us denote by $N_{T}$ the space of final state $\{(\varphi_{T},\psi_{T})\in(L^{2}(0,L))^{2},$  such that the mild solution of (2.17)-(2.18) satisfies the additional boundary condition (3.10)}. Then, for $L\in(0,+\infty)$, $N_{T}=\{(0,0)\}$.

Proof.

This result follows directly from [Lemma 3.4, [16]] and the following lemma. ∎

Lemma 3.2**.**

Let $L\in(0,+\infty)$. Then, does not exist $\lambda\in\mathbb{C}$ and $\varphi,\psi\in H^{3}(0,L)\setminus\{0\}$, such that

[TABLE]

satisfying the boundary conditions

[TABLE]

Proof.

We follow the ideas used in the proof of [Lemma 3.5, [16]] (see also Lema 3.11 in [4]). Arguing by contradiction, we suppose that there exists $\varphi,\psi$ in $H^{3}(0,L)\setminus\{0\}$ solution of (3.11)-(3.12) and denote $\hat{\varphi}(\xi)=\int_{0}^{L}\mathrm{e}^{-ix\xi}\varphi(x)\mathrm{d}x$ and $\hat{\psi}(\xi)=\int_{0}^{L}\mathrm{e}^{-ix\xi}\psi(x)\mathrm{d}x$. Then, multiplying the equations in (3.11) by $e^{-ix\xi}$, integrating by parts over $(0,L)$ and using the boundary conditions (3.12), we have

[TABLE]

Setting $\lambda=ip$, with $p\in\mathbb{C}$, we obtain

[TABLE]

where

[TABLE]

and

[TABLE]

If we use the characterization of Fourier transform for function in $H^{2}(\mathbb{R})$ and Paley-Wiener Theorem [[20], page 161], we deduce that the existence of nontrivial solutions of the problem (3.11)-(3.12) is equivalent to the existence of $p\in\mathbb{C}$ and $\alpha,\beta,\gamma\in\mathbb{C}\setminus\{0\}$, such that

1. (1)

$\upsilon$ and $\nu$ are entire functions, 2. (2)

$\int_{\mathbb{R}}|\upsilon(\xi)|^{2}(1+|\xi|^{2})^{2}d\xi<\infty$ and $\int_{\mathbb{R}}|\nu(\xi)|^{2}(1+|\xi|^{2})^{2}d\xi<\infty$, 3. (3)

$|\upsilon(\xi)|\leq C(1+|\xi|)^{N}e^{L|Im(\xi)|}$ and $|\nu(\xi)|\leq C(1+|\xi|)^{N}e^{L|Im(\xi)|}$ for all $\xi\in\mathbb{C}$,

where $\upsilon$ and $\nu$ are given by

[TABLE]

with $\beta=-\left(ab\varphi(L)+\psi(L)\right)$ and $\gamma=ab\varphi_{xx}(0)+\psi_{xx}(0)$ are nonzero real numbers. On the other hand, note that if $\upsilon_{1}(\xi)$ and $\upsilon_{2}(\xi)$ are entire functions, then the functions given by

[TABLE]

where

[TABLE]

are entire functions as well. Moreover, $\mu_{1}$ and $\mu_{2}$ are entire functions if and only if the roots of $P(\xi)$ are also roots of $ia\xi^{3}\left(\beta\xi^{2}e^{iL\xi}-\gamma\right)$ and $i(p+\xi^{3})\left(\beta\xi^{2}e^{iL\xi}-\gamma\right)$. Hence, if we consider the roots $\xi_{0},\xi_{1},\xi_{2},\xi_{3},\xi_{4}$ and $\xi_{5}$ of $P(\xi)$, the polynomial $P(\cdot)$ can be written as

[TABLE]

where

[TABLE]

By using the Newton-Girard relations, we obtain that

[TABLE]

Note that every root of ${P(\xi)}$ is also root of

[TABLE]

In this case, we have that $B(\xi_{j})=0$, for all $j\in\{0,1,2,3,4,5\}$, i. e.,

[TABLE]

which implies that

[TABLE]

In order to obtain a contradiction, we proceed in several steps. More precisely, we will consider the following cases: $p\in\mathbb{C}\setminus\{\mathbb{R}\cup i\mathbb{R}\}$, $p\in\mathbb{R}\setminus\{0\}$, $p\in i\mathbb{R}\setminus\{0\}$ and $p=0$.

• •

Case 1: Initially, suppose that $\mathbf{p\in\mathbb{C}\setminus\{\mathbb{R}\cup i\mathbb{R}\}}.$ Observe that, in this case, zero is not the root of $P(\xi)$. Moreover, from (3.18), we deduce that each $\frac{iL}{2}\xi_{j}$ is solution of the equation

[TABLE]

where $\alpha=\pm\frac{iL}{2}\left(\frac{\gamma}{\beta}\right)^{1/2}.$ This peculiar transcendental equation has been studied by a large number of authors (see, for instance [8], [18]). The solutions are given by the Lambert function $W(z)$, which has the property

[TABLE]

Moreover, if $\alpha\neq 0$, the equation (3.19) has infinitely many solutions. Indeed, the solutions $z_{k}$ of the equation (3.19) are given by the zeros of the holomorphic complex function given by

[TABLE]

where $\log(z)$ denotes the principal branch of the log-function. Thus, form (3.18), we deduce that, there exist $k_{j}\in\mathbb{Z}$, for each $j\in\{0,1,2,3,4,5\}$ with $k_{j}\neq k_{i}$, such that

[TABLE]

i.e.,

[TABLE]

Here, we consider the case $\alpha=\frac{iL}{2}\left(\frac{\gamma}{\beta}\right)^{1/2}$, the case $\alpha=-\frac{iL}{2}\left(\frac{\gamma}{\beta}\right)^{1/2}$ is similar and we omit it. Thus, observe that

[TABLE]

Then, from relations (3.16), we obtain

[TABLE]

where $m=\sum_{j=0}^{5}k_{j}\in\mathbb{Z}$. Consequently,

[TABLE]

Since

[TABLE]

we get from the definition of $\log(\alpha)$

[TABLE]

or

[TABLE]

Hence,

[TABLE]

Finally, (3.21) allows us to conclude that

[TABLE]

On the other hand,

[TABLE]

Therefore, from (3.16) we have

[TABLE]

Combining the above identities, we obtain the following relation

[TABLE]

which implies that

[TABLE]

Finally, from (3.22) it follows that the coefficient $p$ for the polynomial $P(\xi)$ satisfies

[TABLE]

Moreover, it implies that $\mathcal{I}m\left(\frac{p^{2}}{|p|^{2}}\right)=0$. Hence $\frac{2\mathcal{R}e(p)\mathcal{I}m(p)}{\mathcal{R}e(p)^{2}+\mathcal{I}m(p)^{2}}=0$, and therefore, we deduce that

[TABLE]

which drives us to a contradiction since $p\in\mathbb{C}\setminus(\mathbb{R}\cup i\mathbb{R})$.

• •

Case 2: Suppose that $\mathbf{p\in\mathbb{R}\setminus\{0\}}$. Then, if $P(\cdot)$ has a root $\xi_{0}\in\mathbb{C}\setminus\mathbb{R}$, then $\bar{\xi}_{0}$ also is root. Then, there exist integers $k$ and $m$, such that

[TABLE]

Then, by (3.20)

[TABLE]

which are equivalent to

[TABLE]

Adding the above equations, we obtain $\arg(\alpha)=-(k+m-1)\pi$. From (3.21), we deduce that

[TABLE]

If $\frac{\gamma}{\beta}>0$, we have a contradiction. On the other hand, if $\frac{\gamma}{\beta}<0$, from (3.23) we deduce that $\frac{p^{2}}{|p|^{2}}=-1$, but since $p\in\mathbb{R}$, this is a contradiction as well. Hence, we have that all roots of $P(\cdot)$ are real, so from (3.18), it follows that $L\xi_{j}=n_{j}\pi$ for some $n_{j}\in\mathbb{Z}$ and $j\in\{0,1,2,3,4,5\}$, then

[TABLE]

Let $J=\{j\in\mathbb{N}:\xi_{j}>0\}$ and $K=\{k\in\mathbb{N}:\xi_{k}<0\}$. Remark that $J\cup K=\{0,1,2,3,4,5\}$. If $J=\emptyset$ or $K=\emptyset$, then

[TABLE]

which contradicts (3.16). Thus, $J\neq\emptyset$ and $K\neq\emptyset$. Whit out loss of generality, suppose that $0\in J$ and $1\in K$. Thus, we have that

[TABLE]

Now, if $J=\{0\}$, then $K=\{1,2,3,4,5\}$. Thus, $\sum_{j=2}^{5}\xi_{j}<0$. Moreover, from $\eqref{girard}$ and (3.24), we have that $\sum_{j=2}^{5}\xi_{j}=0$, which drives us to contradiction. Thus, again without loss of generality, suppose that $2\in J$. Then, it follows that

[TABLE]

Hence, $\xi_{j}<0$ for some $j\in\{3,4,5\}$. Thus, we can suppose that $3\in K$. In this case, we obtain that

[TABLE]

Similarly, we obtain that $\xi_{4}+\xi_{5}=0$. Finally, using (3.16) again, it follows that

[TABLE]

In this case, $p=0$, which is a contradiction.

• •

Case 3: Suppose that $\mathbf{p=iq,q\in\mathbb{R}\setminus\{0\}}$. In this case, note that the function $f(\xi)=(1-a^{2}b)\upsilon_{1}(i\xi),$ for $v_{1}$ defined by (3.14), is entire and it is given by

[TABLE]

where the polynomial $R(\cdot)$ has real coefficients and it is given by

[TABLE]

Let $\xi_{j}$, $j\in\{0,1,2,3,4,5\}$ the roots of $R(\cdot)$. Proceeding as in case 2, we also obtain that all the roots of $R(\cdot)$ are real numbers. Moreover, since the polynomial $R(\cdot)$ does not have a term of degree five, the Newton-Girard relations also imply that

[TABLE]

Since $\xi_{j}^{2}e^{L\xi_{j}}=-\frac{\gamma}{\beta}$, for all $j\in\{0,1,2,3,4,5\}$, implying that $\frac{\gamma}{\beta}<0$ and

[TABLE]

Hence, from (3.25), it follows that

[TABLE]

which drives us to the contradiction.

• •

Case 4: Suppose that $\mathbf{p=0.}$ Thus, we have that $\xi_{0}=0$ is a root of $P(\xi)$ and the polynomial $P$ can be write as

[TABLE]

Hence, from (3.15), the rational function

[TABLE]

is entire if zero is a root of $\beta\xi^{2}e^{iL\xi}-\gamma$, which contradicts the fact that $\gamma$ is a nonzero real number.

In any case, we have a contradiction. Thus, the lemma holds. This completes the proof of Lemma 3.1 and, consequently, the proof of Proposition 3.1. ∎

3.1.1. Further Configurations of Four Controls

Following the steps of the proof of Proposition 3.1, we obtain similar results for the following configuration with four controls:

[TABLE]

[TABLE]

and

[TABLE]

Indeed, with the controls configuration (3.26), the observability inequality associated with the control problem is given by

[TABLE]

Arguing by contradiction and proceeding as in the proof of Proposition 3.1, the problem is reduced to prove a unique continuation property for the solutions of a spectral system. In this case, the Fourier and Paley-Wiener approach leads to the study of the following entire functions

[TABLE]

whose properties are similar to those of the functions given in (3.14). Therefore, they can be analyzed as in the proof of Lemma 3.2 and, consequently, the controllability property holds for any $L>0$.

For the controls configuration (3.27), we can prove the following observability inequality:

[TABLE]

As in the previous cases, a contradiction argument reduces the problem to a analyze the unique continuation for the system

[TABLE]

satisfying the boundary conditions

[TABLE]

We claim that, for all $L\in\mathbb{R}^{+}$, the unique solution of (3.31)-(3.32) is $(\varphi,\psi)\equiv(0,0)$. Indeed, we suppose that there exists $\varphi,\psi$ in $H^{3}(0,L)\setminus\{0\}$, solution of (3.31)-(3.32), and let us denote $\hat{\varphi}(\xi)=\int_{0}^{L}\mathrm{e}^{-ix\xi}\varphi(x)\mathrm{d}x$ and $\hat{\psi}(\xi)=\int_{0}^{L}\mathrm{e}^{-ix\xi}\psi(x)\mathrm{d}x$. Then, multiplying the equations in (3.31) by e ${}^{-ix\xi},$ integrating by parts over $(0,L)$ and setting $\lambda=ip$, with $p\in\mathbb{C}$, from Paley-Wiener Theorem it follows that $\hat{\varphi}$ and $\hat{\psi}$ are entire functions given by

[TABLE]

with

[TABLE]

and the coefficients $\alpha_{1}=ab\varphi(L)+\psi(L)$ and $\alpha_{2}=\varphi(L)+a\psi(L).$ It drives us to contradiction since $\frac{B(\xi)}{P(\xi)}$ and $\frac{C(\xi)}{P(\xi)}$ can not be entire functions. Since $B(\xi)$ and $C(\xi)$ are polynomials of degree five, and $P(\xi)$ is a polynomial of degree six.

Finally, for the controls configuration (3.28), the observability inequality associated with the control problem is given by

[TABLE]

The approach applied in the previous cases reduces the problem to prove a unique continuation property for the system

[TABLE]

satisfying the boundary conditions

[TABLE]

Then, entire functions associated with the unique continuation property mentioned above are $\hat{\varphi}$ and $\hat{\psi}$, given by

[TABLE]

and

[TABLE]

It drives us to a contradiction since the numerators of $\hat{\varphi}$ and $\hat{\psi}$ are polynomials of degree three and the corresponding denominators are polynomials of degree six. Thus, $\hat{\varphi}$ and $\hat{\psi}$ can not be entire functions.

3.2. Case B: Observability inequality with three controls configuration

In this subsection, we consider the following boundary control configuration ($g_{0}\neq 0,g_{1}=g_{2}=0$ and $h_{0}\neq 0,h_{1}\neq 0,h_{2}=0$):

[TABLE]

As it is well known, the control problem should be more difficult if one removes controls. However, in the case of three controls, we obtain a positive answer for the exact controllability problem by using the approach introduced by Rosier in [16], provided that some relation between the coefficients, the length of the interval and the time $T$ holds. More precisely, under the above configuration of controls, we establish the following observability inequality:

Proposition 3.2**.**

Let $T>0$ and $L>0$. Assume that the parameters $a,b,c\in\mathbb{R}$ satisfy

[TABLE]

where $C_{1}$ is the hidden regularity constant given in (2.19). Then, there exists $C=C(L,T)>0$, such that, if $(\varphi,\psi)\in C([0,T];L^{2}(0,L))^{2}$ is a solution of (2.17)-(2.18), the inequality

[TABLE]

holds for any $\varphi_{T},\psi_{T}\in L^{2}(0,L)$.

Proof.

We argue by contradiction and following the steps of the proof of Proposition 3.1. The main difference between them is the absence of the condition $ab\varphi^{n}_{x}(t,L)+\psi^{n}_{x}(t,L)\rightarrow 0$ in $L^{2}(0,T)$, as $n\rightarrow\infty$, therefore we omit some details. So, we obtain a sequence of functions $(\varphi^{n}_{T},\psi^{n}_{T})$, such that

[TABLE]

From (3.37), we get

[TABLE]

as $n\rightarrow\infty$. In order to prove that $\{(\varphi^{n}_{T},\psi^{n}_{T})\}$ is a Cauchy sequence in $X$, we use the inequality (2.20) to obtain

[TABLE]

Then, from the hidden regularity given in Proposition 2.4, we obtain a constant $C_{1}>0$ satisfying

[TABLE]

which allows to conclude that

[TABLE]

where $C=\max\{K,\frac{1}{T},\frac{1}{c},\frac{b}{c}\}$, with $K^{-1}=\min\left\{1,\left(1-\left(\frac{1-a^{2}b}{c}\right)C_{1}\right)\right\}>0$. Proceeding as in the proof of Proposition 3.1 we obtain the convergence of terms on the right-hand side of (3.41) and conclude that $\{(\varphi_{T}^{n},\psi_{T}^{n})\}_{n\in\mathbb{N}}$ is a Cauchy sequence in $\mathcal{X}$. If we denote its limit by $(\varphi_{T},\psi_{T})$, from (3.37) we get

[TABLE]

Moreover, due to (3.38), Proposition 2.4 and compactness argument, we can prove that the corresponding solution $(\varphi,\psi)$ of

[TABLE]

satisfies the following additional boundary conditions

[TABLE]

Notice that due to (3.42), the solutions of (3.43)-(3.44) cannot be identically zero. However, arguing as ithe proofs of Lemmas 3.1 and 3.2, we can conclude that $(\varphi,\psi)=(0,0)$, which drives us to a contradiction. Indeed, in this case, the problem is reduced to prove that solution of the stationary problem

[TABLE]

satisfying the boundary conditions

[TABLE]

for some $\lambda\in\mathbb{C}$, has to be the trivial one. Arguing by contradiction, we suppose that there exists $\varphi,\psi$ in $H^{3}(0,L)\setminus\{0\}$, solution of (3.45)-(3.46), and let us denote $\hat{\varphi}(\xi)=\int_{0}^{L}\mathrm{e}^{-ix\xi}\varphi(x)\mathrm{d}x$ and $\hat{\psi}(\xi)=\int_{0}^{L}\mathrm{e}^{-ix\xi}\psi(x)\mathrm{d}x$. Then, multiplying the equations by e ${}^{-ix\xi},$ integrating by parts over $(0,L)$, using the boundary conditions, taking $\lambda=ip$ with $p\in\mathbb{C}$ and by using the Paley-Wiener Theorem, it follows that $\hat{\varphi}$ and $\hat{\psi}$ are entire functions given by

[TABLE]

where

[TABLE]

and the coefficients $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ are given by

[TABLE]

Note that $\frac{B(\xi)}{P(\xi)}$ and $\frac{C(\xi)}{P(\xi)}$ can not be entire functions. Indeed, $B(\xi)$ and $C(\xi)$ are polynomials of degree five, and $P(\xi)$ is a polynomial of degree six. Then, it drives us to a contradiction and the proof ends. ∎

3.2.1. Further Configuration of Three Controls

Following the steps of the proof of Proposition 3.2, we also obtain similar results for the configuration of the following controls:

[TABLE]

For the case of controls configuration (3.47), the observability inequality is given by

[TABLE]

provided that relation (3.35) holds. The proof of (3.48) follows closely the proof of (3.36).

4. Exact Controllability

In this section, we first use the observability inequalities proved in the previous section to study the linear controllability of the system (2.4)-(2.6) with different combinations of boundary controls. Next, the local controllability of the full system is derived by means of a fixed point argument.

We initiate our study assuming that the paramenter $r\neq 0$. This case involves a higher difficulty, since we need to consider new strategy for dealing with the whole system. Indeed, as point out in the proof of Proposition 2.2, when $r=0$ it is possible to investigate the controllability-observability of the linear system by studying a single KdV equation. We discuss the case where $r=0$ in section 5.

4.1. Linear System

In this subsection we study the controllability of the linearized system,

[TABLE]

with the boundary conditions

[TABLE]

Multiplying the system (2.4)-(2.6) by a solution of the adjoint system (2.17)-(2.18), we find an equivalent condition for the exact controllability property:

Lemma 4.1**.**

For any $\left(u^{1},v^{1}\right)$ in $\mathcal{X},$ there exist four controls $\vec{h}=\left(h_{0},h_{1},h_{2}\right)$ and $\vec{g}=\left(g_{0},g_{1},g_{2}\right)$ in $\mathcal{H}$ such that the solution $(u,v)$ of (2.4)-(2.6) satisfies $(u(T,x),v(T,x))=(u_{1}(x),v^{1}(x))$ if and only if

[TABLE]

for any $\left(\varphi_{T},\psi_{T}\right)$ in $\mathcal{X},$ where $(\varphi,\psi)$ is the solution of the backward system (2.17)-(2.18) with initial data $\left(\varphi_{T},\psi_{T}\right)$.

The next result gives a positive answer to the linear control problem associated to (1.1).

Theorem 4.1**.**

Let $T>0$ and $L>0.$ Then, system (4.1)-(4.2) is exactly controllable in time $T$ under the following four controls configurations

1. (i)

$h_{0}\neq 0,h_{1}\neq 0,h_{2}\neq 0\quad\text{and}\quad g_{0}=0,g_{1}\neq 0,g_{2}=0$, 2. (ii)

$h_{0}=0,h_{1}\neq 0,h_{2}=0\quad\text{and}\quad g_{0}\neq 0,g_{1}\neq 0,g_{2}\neq 0$, 3. (iii)

$h_{0}\neq 0,h_{1}\neq 0,h_{2}=0\quad\text{and}\quad g_{0}\neq 0,g_{1}\neq 0,g_{2}=0$, 4. (iv)

$h_{0}=0,h_{1}\neq 0,h_{2}\neq 0\quad\text{and}\quad g_{0}=0,g_{1}\neq 0,g_{2}\neq 0$.

Moreover, if the parameters $a,b,c$ satisfy

[TABLE]

where $C_{1}=C_{1}(T,L)>0$ is the hidden regularity constant given in (2.19), the same result is archived for the following configuration for three controls:

1. (i)

$h_{0}\neq 0,h_{1}\neq 0,h_{2}=0\quad\text{and}\quad g_{0}\neq 0,g_{1}=0,g_{2}=0$, 2. (ii)

$h_{0}\neq 0,h_{1}=0,h_{2}=0\quad\text{and}\quad g_{0}\neq 0,g_{1}\neq 0,g_{2}=0$.

Proof.

We will prove the theorem for the case of four controls with the configuration $h_{0}\neq 0,h_{1}\neq 0,h_{2}\neq 0\quad\text{and}\quad g_{0}=0,g_{1}\neq 0,g_{2}=0$. The other cases are similar, therefore we omit it. In fact, let us denote by $\Gamma$ the bounded linear map defined by

[TABLE]

where $(u,v)$ is the solution of (4.1)-(4.2) with

[TABLE]

and $(\varphi,\psi)$ the solution of the backward system (2.17)-(2.18) with initial data $\left(\varphi_{T},\psi_{T}\right)$. According to Lemma 4.1 and Proposition 3.1, we obtain

[TABLE]

Thus, by the Lax-Milgram theorem, $\Gamma$ is invertible. Consequently, for given $\left(u_{T},v_{T}\right)\in\mathcal{X},$ we can define $\left(\varphi_{T},\psi_{T}\right):=\Gamma^{-1}\left(u_{T},v_{T}\right)$ to solve the system (2.17)-(2.18) with initial data $\left(\varphi_{T},\psi_{T}\right)$ and get $(\varphi,\psi)\in X_{T}.$ Then, if $(h_{0},h_{1},h_{2})$ and $(g_{0},g_{1},g_{2})$ are given by (4.4), the corresponding solution $(u,v)$ of the system (4.1)-(4.2) satisfies

[TABLE]

The result follows from Lax–Milgram theorem. ∎

As a consequence of the previous analysis, we have the following result:

Proposition 4.1**.**

For any $L>0$ (when the linear controllability is guaranteed), there exists a bounded linear operator

[TABLE]

such that, for any $(u_{0},v_{0})\in\mathcal{X}$ and $(u_{T},v_{T})\in\mathcal{X}$,

[TABLE]

where $\vec{h}=(h_{0},h_{1},h_{2})$ and $\vec{g}=(g_{0},g_{1},g_{2})$ are the controls configurations given by Theorem 4.1. Then, the system (2.4)-(2.6) admits a unique solution $(u,v)\in X_{T}={C\left([0,T];\mathcal{X}\right)}\cap L^{2}(0,T;(H(0,L))^{2})$, such that

[TABLE]

4.2. Nonlinear System

In this section, we study the controllability of the full system,

[TABLE]

with boundary conditions

[TABLE]

According to Proposition 2.2 and the comments in subsection 2.3.1, the solution of (1.1)-(1.2) can be written as

[TABLE]

where $\{S(t)\}_{t\geq 0}$ and $\{S_{bdr}(t)\}_{t\geq 0}$ are a $C_{0}$-semigroup and a boundary differential operator, respectively.

Proof of Theorem 1.1.

We will treat the nonlinear problem (1.1)-(1.2) by using a classical fixed point argument. Indeed, for $u,v\in X_{T}$, let us introduce the notation

[TABLE]

Then, for any $u_{0},v_{0}\in L^{2}(0,T)$, Proposition 4.1 guarantees that there exists a mapping $\Psi$ which allows us to choose

[TABLE]

Thus, if we define the operator $\Gamma:X_{T}\to X_{T}$ as follows

[TABLE]

it is clear that

[TABLE]

We claim that the operator $\Gamma$ has a unique fixed point. To do that, we will prove that $\Gamma$ is a contraction in the space $X_{T}$. Indeed, let us consider

[TABLE]

for some $r>0$ to be chosen later. Proposition 4.1 and Theorem 2.1 imply that there exist constants $C_{1},C_{2}>0$, such that

[TABLE]

and

[TABLE]

From Lemma [2, Lemma 3.1], we deduce that

[TABLE]

where $C_{4}$ is a constant depending only on $T$. Thus, choosing $r$ and $\delta$ satisfying

[TABLE]

and

[TABLE]

the operator $\Gamma$ maps $B_{r}$ into itself for any \left(\begin{array}[]{cc}u\\ v\end{array}\right)\in\mathcal{X}_{T}. Now, proceeding as in the proof of Theorem 2.1, we obtain that

[TABLE]

for any \left(\begin{array}[]{cc}u\\ v\end{array}\right),\left(\begin{array}[]{cc}\widetilde{u}\\ \widetilde{v}\end{array}\right)\in B_{r}, where $C_{5}>0$ depends only on $T.$ Thus, taking $\delta>0,$ such that

[TABLE]

we have

[TABLE]

Therefore, the map $\Gamma$ is a contraction. Thus, Banach fixed-point theorem guarantees that $\Gamma$ has a fixed point in $B_{r}$ and its fixed point is the desired solution. ∎

5. Further Comments and Open Problems

This section is devoted to discussing the controllability of the system when the parameter $r=0$ and to present some open problems.

5.1. Controllability aspect when the parameter $\mathbf{r=0}$

Assume that $r=0$ in (1.1). According to Remark 2.1 the study of the linearized system associated to (2.4)-(2.6) can be replaced by the study of the systems (2.11) with $\bar{p},\bar{q}=0$. Without loss of generality, we consider $\alpha^{+}=\alpha^{-}=1$, i.e.,

[TABLE]

with boundary conditions

[TABLE]

The adjoint system corresponding to (5.1)-(5.2) is given by

[TABLE]

with boundary conditions

[TABLE]

Observe that the solution of (5.3)-(5.4) can be written as

[TABLE]

where $\{S_{1}(t)\}_{t\geq 0}$ is the $C_{0}$ semigroup on the space $L^{2}(0,L)$ generated by the operator

[TABLE]

with the domain

[TABLE]

If we multiply the equation in (5.1) by the solution of (5.3) and integrating in time and space, we obtain the equation

[TABLE]

which is key for establishing controllability by proceeding as in the previous sections. Let us analyze the system (5.1)-(5.2) with different combinations of controls.

1. (1)

Controls acting on the Neumann condition and on the second order derivative:

[TABLE]

Following the previous ideas, we can concentrate on two steps. First, the study of the observability inequality

[TABLE]

which is proved by using a contradiction argument. If (5.7) does not hold, there exists a sequence of functions $\{\varphi^{n}_{T}\}_{n\in\mathbb{N}}\in L^{2}(0,L)$, (without loss of generality, we assume that the sequence is normalized), such that, for all $n\in\mathbb{N}$,

[TABLE]

If we multiply the equation (5.3) by $t\varphi$ and integrate by parts, we obtain

[TABLE]

Combining (5.8), (5.9) and Proposition 2.8 in [6], it is possible to prove that, at least for a subsequence, $\varphi^{n}_{T}\rightarrow\varphi_{T}$ in $L^{2}(0,T)$ and $\varphi^{n}\rightarrow\varphi$ in $L^{2}(0,T;L^{2}(0,T))$, where $\varphi$ is the solution of the adjoint system with final data $\varphi_{T}$ and the additional conditions

[TABLE]

Moreover. due to (5.8), $\|\varphi_{T}\|_{L^{2}(0,L)}=1.$ The following claim shows that the above not is possible:

Claim 5.1**.**

For $T>0$ and $L>0$, we define the set $N_{T}$ of all functions $\varphi_{T}\in L^{2}(0,L)$ such that $\varphi\in C\left(0,T;L^{2}(0,L)\right)$ is a mild solution of (5.3)-(5.4) such that $\varphi_{x}(t,L)=0$ and $\varphi_{xx}(t,0)=0$. Then $N_{T}=\{0\}.$

Proof.

Let us assume that $N_{T}\neq 0$. Then the map $\varphi\in\mathbb{C}N_{T}\to A(\varphi_{T})\in\mathbb{C}N_{T}$, where $A=\partial_{xxx}$, has at least one eigenvalue ($\mathbb{C}N_{T}$ denotes the complexification of $N_{T}$). Let $s\in\mathbb{C}$ and $\phi\in H^{3}(0,L)\setminus\{0\}$ the corresponding eigenvalue and eigenfunction of $A$ which solves the following eigenvalue problem

[TABLE]

By direct computations, it is easy to see that $\phi\equiv 0$ is the unique solution of (5.10), for all $L>0$. ∎ 2. (2)

Controls acting on the Dirichlet condition:

[TABLE]

The null controllability should be expected. Indeed, it follows from the study of the system

[TABLE]

with the boundary conditions given in (5.2). This result, which has been proved in [11] using Carleman estimates, can be adapted to the system (5.1) without difficulty since the difference between the systems is one term of lower order.

5.2. Open problems

- Case $\mathbf{r\neq 0}$

The ideas contained in this work suggest that one can remove controls in several situations. For instance, if we consider the following configuration

[TABLE]

the observability associated with this case is given by

[TABLE]

Arguing by contradiction and proceeding as in the proof of Proposition 3.2, the problem is reduced to prove a unique continuation property for the solutions of a spectral system. In this case, the Fourier and Paley-Wiener approach leads to the study of the following entire functions

[TABLE]

and the nature of the roots of $e^{-iL\xi}\left(i\alpha\xi+\beta\xi^{2}\right)-\gamma$ is more complicated than the roots of (3.14). Therefore, the following question arises:

Open Problem 5.1**.**

Is the full system (1.1)-(1.2) locally exactly controllable, when three or fewer control inputs act on boundary conditions, without any restriction on the parameters $a$, $b$, and $c$?

On the other hand, the controllability results have been established when $r\neq 0$ are of local nature. It means that one can only guide a small amplitude initial state to a small amplitude terminal state, by choosing appropriate boundary control inputs. Thus, the following question remains open:

Open Problem 5.2**.**

(Global Controllability Problem) Is the full system (1.1)-(1.2) globally exactly controllable?