Manish Kumar †

Subrata Manish Majumdar

Kumar

ON THE CONTROLLABILITY OF A SYSTEM COUPLING KURAMOTO-SIVASHINSKY-KORTEWEG-DE VRIES AND TRANSPORT EQUATIONS

Keywords: 2020 Mathematics Subject Classification. 93B05, 93C20, 35M33, 30E05 Parabolic-hyperbolic coupled system, Kuramoto-Sivashinsky-Korteweg-de Vries equation, transport equation, moment method, biorthogonal family, null controllability

In this paper, we study the null controllability of a coupled parabolic-hyperbolic system in one dimension with a single control using the moment method. More precisely, we consider a system coupling Kuramoto-Sivanshinsky-Korteweg-de Vries equation and transport equation through first order derivatives. We explore the null controllability of four different control systems with the control acting either on the periodic boundary or in some open subset of the interior of the domain with periodic boundary conditions. Depending on the position of the control, we get some regular periodic Sobolev space as the space of initial data for which the null controllability holds, provided the time is sufficiently large.

Introduction and main results

1.1. Setting of the problem. The study of coupled systems containing different types of equations has gained immense interest among the researchers throughout years in the theory of control of the partial differential equations (PDEs). Among them, infinite dimensional systems containing parabolic and hyperbolic PDEs are one of the important and well-studied model in the literature, see for example, [START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF], [START_REF] Zhang | Control, observation and polynomial decay for a coupled heat-wave system[END_REF], [START_REF] Zhang | Polynomial decay and control of a 1 -d hyperbolic-parabolic coupled system[END_REF], [START_REF] Beauchard | Null-controllability of linear parabolic transport systems[END_REF], [START_REF] Chowdhury | Approximate controllability for linearized compressible barotropic Navier-Stokes system in one and two dimensions[END_REF], [START_REF] Hansen | Boundary control of a one-dimensional linear thermoelastic rod[END_REF]. Coupled parabolic-hyperbolic model describes many physical phenomenon such as predator-prey population models, biological chemotaxis (such as tumor and fungal growth) and a particular part of EUV lithography involving a liquid metal droplet stream convecting through plasma.

This paper studies the controllability aspect of the following parabolic-hyperbolic mixed class system coupling a fourth order parabolic system, linear Kuramoto-Sivashinsky-Korteweg-de Vries (KS-KdV in short) equation with a transport equation through first order derivatives in one dimensional periodic domain (0, 2π):

u t + γu xxxx + u xxx + νu xx = η x , (t, x) ∈ (0, ∞) × (0, 2π), η t + Γ η x = u x , (t, x) ∈ (0, ∞) × (0, 2π), (1.1)
where γ, ν > 0 are coefficients accounting for the long-wave instabilities and the short wave dissipation respectively and Γ ∈ R \ {0} is the transport velocity. Using the following transformation

u(t, x) → 1 α u t β , x δ , η(t, x) → η t β , x δ in (0, ∞) × (0, 2π),
one can recast the above system in the following form:

u t + u xxxx + u xxx + Au xx = Bη x , (t, x) ∈ (0, ∞) × (0, 2πδ), η t + η x = u x , (t, x) ∈ (0, ∞) × (0, 2πδ), (1.2)
with α = Γ, β = Γ γ , δ = 1 γ , A = νγ, B = γ 2 Γ . For the simplicity of expressions involved in the analysis, we take the constants A and B equal to 1.

Let us assume T > 0. We first consider the following system for our study

u t + u xxxx + u xxx + u xx = η x , (t, x) ∈ (0, T ) × (0, 2π), η t + η x = u x , (t, x) ∈ (0, T ) × (0, 2π), (1.3)
with the periodic boundary conditions

    
u(t, 0) = u(t, 2π), u x (t, 0) = u x (t, 2π), t ∈ (0, T ), u xx (t, 0) = u xx (t, 2π), u xxx (t, 0) = u xxx (t, 2π), t ∈ (0, T ), η(t, 0) = η(t, 2π), t ∈ (0, T ), (1.4) and the initial conditions u(0, x) = u 0 (x), η(0, x) = η 0 (x), x ∈ (0, 2π).

(

The aim of the present paper is to investigate the control properties of the system (1.3)-(1.4)- (1.5). For that, let us consider the following generalized control system:

                   u t + u xxxx + u xxx + u xx = η x + h 1 , (t, x
) ∈ (0, T ) × (0, 2π), η t + η x = u x + h 2 , (t, x) ∈ (0, T ) × (0, 2π), u(t, 0) = u(t, 2π) + q 1 (t), u x (t, 0) = u x (t, 2π), t ∈ (0, T ), u xx (t, 0) = u xx (t, 2π), u xxx (t, 0) = u xxx (t, 2π), t ∈ (0, T ), η(t, 0) = η(t, 2π) + q 2 (t), t ∈ (0, T ), u(0, x) = u 0 (x), η(0, x) = η 0 (x),

x ∈ (0, 2π), (1.6) where h 1 , h 2 are interior controls and q 1 , q 2 are boundary controls. Let us first define the notion of null controllability for the system (1.3)-(1.5).

Definition 1.1. Let X be any Hilbert space. We say that the system (1.3) is null controllable in space X at time T > 0 if for any (u 0 , η 0 ) ∈ X there exist at least one nonzero control function among h 1 , h 2 ∈ L 2 (0, T ; L 2 (0, 2π)), q 1 , q 2 ∈ L 2 (0, T ) such that the solution of (1.6) satisfies (u(T, •), η(T, •)) = (0, 0). ❖ When the above holds with nonzero controls h 1 ( or h 2 ) ∈ L 2 (0, T ; L 2 (ω)) for ω ⊂ (0, 2π) and h 2 ( or h 1 ) = q 1 = q 2 ≡ 0, then we say the system is localized interior null controllable. ❖ When the above holds with nonzero controls q 1 ( or q 2 ) ∈ L 2 (0, T ) and q 2 ( or q 1 ) = h 1 = h 2 ≡ 0, then we say the system is boundary null controllable.

In this paper, we establish the null controllability results for the system (1.3)-(1.4)-(1.5) using a single control, i.e., only one control function among h 1 , h 2 , q 1 and q 2 will act on the system (1.6) and rest will be assumed to be identically 0. Depending on which control function acts on the system, we get a space, X in which the system (1.3)-(1.5) is null controllable. All these null controllability results are precisely stated in Section 1.4.1. To the best of authors' knowledge, this is the first work to consider the above parabolic-hyperbolic coupled system and study its control properties.

1.2. Relevant Literature and Motivation. The Kuramoto-Sivashinsky (KS) equation, u t + γu xxxx + νu xx + uu x = 0 was first proposed independently by Kuramoto and Tsuzuki as model for Beluozov-Zabotinskii reaction patterns in reaction-diffusion system in [START_REF] Kuramoto | On the formation of dissipative structures in reaction-diffusion systems: Reductive perturbation approach[END_REF] and by Sivashinsky as model for unstable flame fronts in [START_REF] Sivashinsky | On flame propagation under conditions of stoichiometry[END_REF]. Later this equation has been studied in a series of papers [START_REF] Kuramoto | Persistent propagation of concentration waves in dissipative media far from thermal equilibrium[END_REF], [START_REF] Chen | Nonlinear waves on liquid film surfaces-ii. bifurcation analyses of the long-wave equation[END_REF], [START_REF] Hooper | Nonlinear instability at the interface between two viscous fluids[END_REF], [START_REF] Nicolaenko | Some global dynamical properties of the kuramoto-sivashinsky equations: Nonlinear stability and attractors[END_REF], [START_REF] Robinson | Inertial manifolds for the kuramoto-sivashinsky equation[END_REF], [START_REF] Giacomelli | New bounds for the kuramoto-sivashinsky equation[END_REF], [START_REF] Goodman | Stability of the kuramoto-sivashinsky and related systems[END_REF], [START_REF] Panagiotis | Global stabilization of the Kuramoto-Sivashinsky equation via distributed output feedback control[END_REF], [START_REF] Armaou | Feedback control of the Kuramoto-Sivashinsky equation[END_REF] and references therein. The following KS-KdV equation was introduced by Benney in [START_REF] Benney | Long waves on liquid films[END_REF] to include dispersive effects by adding the KdV term u xxx u t + γu xxxx + u xxx + νu xx + uu x = 0 (1.7) which is used to study a wide range of nonlinear dissipative waves. The controllability of parabolic partial differential equations is being studied extensively by many researchers for a long time. As a result, many methods have been developed and employed to study the controllability of parabolic PDEs, in particular for heat equation, like Moment method (see [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]), Transmutation method (see [START_REF] Miller | The control transmutation method and the cost of fast controls[END_REF]), Flatness method (see [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF]), Backstepping approach (see [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF]) and Carleman estimates (see [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], [START_REF] Fursikov | Controllability of evolution equations[END_REF]). Among these, the following two methods have been mainly explored for the controllability of the KS equation u t + γu xxxx + νu xx + uu x = 0 and linear KS equation u t + γu xxxx + νu xx = 0 so far:

• Moment method (see [START_REF] Cerpa | Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation[END_REF], [START_REF] Cerpa | On the control of the linear Kuramoto-Sivashinsky equation[END_REF], etc)

• Carleman estimates (see [START_REF] Cerpa | Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation[END_REF], [START_REF] Cerpa | On the control of the linear Kuramoto-Sivashinsky equation[END_REF], etc).

Study of controllability of KS system is not restricted to single equation only. Numerous works regarding controllability of system coupling KS-KdV equation (1.7) and second order parabolic equation, namely heat equation have been done. One can refer to the works [START_REF] Cerpa | On the boundary control of a parabolic system coupling KS-KdV and heat equations[END_REF], [START_REF] Carreño | Local controllability of the stabilized Kuramoto-Sivashinsky system by a single control acting on the heat equation[END_REF], [START_REF] Cerpa | Null controllability of the stabilized Kuramoto-Sivashinsky system with one distributed control[END_REF], [START_REF] Carreño | Boundary controllability of a cascade system coupling fourthand second-order parabolic equations[END_REF], [START_REF] Hernández-Santamaría | Boundary controllability of a simplified stabilized Kuramoto-Sivashinsky system[END_REF] for such study.

On the other hand, controllability of hyperbolic partial differential equation has also been extensively studied. Besides moment method [START_REF] Russell | Canonical forms and spectral determination for a class of hyperbolic distributed parameter control systems[END_REF] and Carleman approach [START_REF] Cannarsa | Observability inequalities for transport equations through Carleman estimates[END_REF], multiplier method [START_REF] Komornik | Exact controllability and stabilization[END_REF] is very useful for this type of model.

The control behavior of the hyperbolic and parabolic systems are very different. The former in general, is exactly controllable, whereas parabolic systems are not so due to regularity issues, but are null controllable and approximate controllable. In general, null controllability of parabolic PDEs hold at any time T > 0, contrary to which, the hyperbolic systems need a time T > T 0 , for some T 0 > 0.

Controllability of systems, coupling parabolic and hyperbolic equations has also fascinated many researchers. Let us briefly mention some important work among them. For the introductory study of the linear and nonlinear parabolic-hyperbolic model we refer to an excellent book [START_REF] Zheng | Nonlinear parabolic equations and hyperbolic-parabolic coupled systems, volume 76 of Pitman Monographs and Surveys in Pure and Applied Mathematics[END_REF]. Null controllability of Thermoelasticity system, which represents a coupling between heat and wave equation has been studied by Lebeau and Zuazua in [START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF]. In [START_REF] Martin | Null controllability of the structurally damped wave equation with moving control[END_REF], Martin, Rosier and Rouchon studied the null controllability of the structurally damped wave equation in the one dimensional torus T using the method of moments by means of distributed moving control with sufficiently regular initial data at sufficiently large time. If the support of the moving control evolves with a constant velocity V 0 , then the transformed system is closely related to the one dimensional compressible Navier-Stokes system in one dimension linearized around a constant steady state (Q 0 , V 0 ), with Q 0 > 0, V 0 > 0:

ρ t + V 0 ρ x + Q 0 u x = 0, u t -ν Q0 u xx + V 0 u x + aγQ γ-2 0 ρ x = 0. (1.8)
Chowdhury and Mitra have explored the moment method in [START_REF] Chowdhury | Null controllability of the linearized compressible Navier-Stokes equations using moment method[END_REF] to establish the null controllability of this system by means of a single control acting in the system either as bilinear interior control or boundary control for the parabolic component, where the existence of desired biorthogonal family has been followed from [START_REF] Martin | Null controllability of the structurally damped wave equation with moving control[END_REF]. It is worth to mention that, recently in [START_REF] Beauchard | Null-controllability of linear parabolic transport systems[END_REF], Beauchard, Koenig, and Le Balc'h adopted the method of [START_REF] Lebeau | Null-controllability of a system of linear thermoelasticity[END_REF] to give the best existing result for transport-second order parabolic system in T (which is the generalized version of the above-mentioned system (1.8)) regarding optimal time and regularity of initial data. The null controllability of the following Stabilized KS system

                   u t + u xxxx + u xxx + u xx = η x , (t, x) ∈ (0, T ) × (0, 2π), η t -ϵη xx + η x = u x , (t, x) ∈ (0, T ) × (0, 2π), u(t, 0) = u(t, 2π), u x (t, 0) = u x (t, 2π), t ∈ (0, T ), u xx (t, 0) = u xx (t, 2π), u xxx (t, 0) = u xxx (t, 2π), t ∈ (0, T ), η(t, 0) = η(t, 2π), η x (t, 0) = η x (t, 2π), t ∈ (0, T ), u(0, x) = u 0 (x), η(0, x) = η 0 (x),
x ∈ (0, 2π),

(1.9) with ϵ = 1 has been studied in [START_REF] Kumar | Null controllability of the linear stabilized kuramoto-sivashinsky system using moment method[END_REF], using moment method along with the construction of required biorthogonal family. The considered system (1.3)-(1.5) can be roughly thought of as limiting system of the above stabilized KS system as ϵ → 0. In the present work, we have followed the same approach to get the null controllability results.

1.3. Functional framework. We denote the space of complex valued square summable series over integers by l 2 (Z) and the space of periodic complex valued functions that are square integrable over [0, 2π] by L 2 (0, 2π). We introduce the state space Z = L 2 (0, 2π) × L 2 (0, 2π), with the usual Hermitian product

u 1 v 1 , u 2 v 2 Z = 2π 0 u 1 (x)u 2 (x)dx + 2π 0 v 1 (x)v 2 (x)dx, (1.10) ∀ u 1 , u 2 , v 1 , v 2 ∈ L 2 (0, 2π).
For s ≥ 0, we define the following classical periodic Sobolev space H s p (0, 2π) as

H s p (0, 2π) = ϕ = k∈Z ϕ k e -ikx : k∈Z 1 + |k| 2s |ϕ k | 2 < ∞ ⊆ L 2 (0, 2π)
with the norm

∥ϕ∥ H s p (0,2π) = k∈Z 1 + |k| 2s |ϕ k | 2 1 2
.

(1.11)

Note that H 0 p (0, 2π) = L 2 (0, 2π) and

ϕ k = 1 2π 2π 0 ϕ(x)e ikx dx, k ∈ Z. Moreover, for i ∈ N H i p (0, 2π) = {u ∈ H i (0, 2π) : u j (0) = u j (2π), j = 0, 1, 2, .., i -1}.
Now, let Ḣs p (0, 2π) be the subspace of H s p (0, 2π) containing the functions ϕ ∈ H s p (0, 2π), which satisfy

2π 0 ϕ(x)dx = 0. (1.
12)

The norm associated to Ḣs (0, 2π) can be given by

∥ϕ∥ Ḣs p (0,2π) = k∈Z |k| 2s |ϕ k | 2 1 2 , (1.13)
which is equivalent to the norm ∥ϕ∥ H s p (0,2π) . Further, for any Hilbert space H, we will use the notation ⟨u, η⟩ H and ⟨u, η⟩ H ′ ,H to denote the inner product and duality product, respectively, where H ′ denotes the dual space of H. Also, we decompose the initial data (u 0 , η 0 ) ∈ Z in Fourier series as:

u 0 (x) = k∈Z a k e -ikx , η 0 (x) = k∈Z b k e -ikx , (1.14)
where, {a k }, {b k } ∈ l 2 (Z). We further introduce the notation H s p for s ≥ 0 to define the Hilbert space:

H s p = (u 0 , η 0 ) ∈ H s p (0, 2π) × H s+3 p (0, 2π) : u 0 , e ±ix L 2 (0,2π) = η 0 , e ±ix L 2 (0,2π) = 0 , (1.15)
induced with the following inner product

u 1 η 1 , u 2 η 2 H s p = ⟨u 1 , u 2 ⟩ H s p + ⟨η 1 , η 2 ⟩ H s+3 p .
1.4. Main results. In this section, we state all the main results of this paper concerning null controllability of the system (1.3)-(1.5), assuming the well-posedness of the associated control system. The well-posedness results for the considered control systems can be found in Section 2.

1.4.1. Null controllability for KS-KdV-Transport system with bilinear interior control. Let us first consider the following control problem where the interior control acts on the KS-KdV equation:

     u t + u xxxx + u xxx + u xx = η x + h(x, t), (t, x) ∈ (0, T ) × (0, 2π), η t + η x = u x , (t, x) ∈ (0, T ) × (0, 2π), u(0, x) = u 0 (x), η(0, x) = η 0 (x),
x ∈ (0, 2π), (1.16) with the periodic boundary conditions (1.4).

Theorem 1.2. For any time T > 2π and any (u 0 , η 0 ) ∈ H s p with 2π 0 η 0 = 0 and s > 13 2 , there exists a localized interior control h ∈ L 2 ((0, T ); L 2 (0, 2π)) of the form h(x, t) = 1 ω f (x)g(t), where ω is any nonempty open subset of (0, 2π), such that the solution of (1.16) satisfies u(T, •) = 0 and η(T, •) = 0.

Next we state the null controllability result for the system (1.3) by means of a localized bilinear interior control acting in the transport equation. For that let us first write the corresponding control system: (1.17) with the periodic boundary condition (1.4).

     u t + u xxxx + u xxx + u xx = η x , (t, x) ∈ (0, T ) × (0, 2π), η t + η x = u x + h(t, x), (t, x) ∈ (0, T ) × (0, 2π), u(0, x) = u 0 (x), η(0, x) = η 0 (x), x ∈ (0, 2π),
Theorem 1.3. For any time T > 2π and any (u 0 , η 0 ) ∈ H s p with 2π 0 u 0 = 0 and s > 7 2 , there exists a localized interior control h ∈ L 2 ((0, T ); L 2 (0, 2π)) of the form h(x, t) = 1 ω f (x)g(t), where ω is any nonempty open subset of (0, 2π), such that the solution of (1.17) satisfies u(T, •) = 0 and η(T, •) = 0. 1.4.2. Boundary null controllability for KS-KdV-Transport system. In this section, we state the results concerning boundary controllability of the system (1.3). We first consider the system with control acting through the zeroth order derivative KS-KdV component. Let us write the corresponding control system:

                   u t + u xxxx + u xxx + u xx = η x , (t, x) ∈ (0, T ) × (0, 2π), η t + η x = u x , (t, x) ∈ (0, T ) × (0, 2π), u(t, 0) = u(t, 2π) + q(t), u x (t, 0) = u x (t, 2π), t ∈ (0, T ), u xx (t, 0) = u xx (t, 2π), u xxx (t, 0) = u xxx (t, 2π), t ∈ (0, T ), η(t, 0) = η(t, 2π), t ∈ (0, T ), u(0, x) = u 0 (x), η(0, x) = η 0 (x),
x ∈ (0, 2π), (1.18) q is the boundary control.

Theorem 1.4. Let T > 2π. For any initial data (u 0 , η 0 ) ∈ H s p with 2π 0 u 0 = 0 and s > 7 2 , there exists a control q ∈ L 2 (0, T ) such that the solution of (1.18) satisfies u(T, •) = 0 and η(T, •) = 0.

Our final result is devoted to the boundary null controllability of system (1.3) under the action of control acting on the transport component, and so consider the following control system:

         u t + u xxxx + u xxx + u xx = η x , (t, x) ∈ (0, T ) × (0, 2π), η t + η x = u x ,
(t, x) ∈ (0, T ) × (0, 2π), η(t, 0) = η(t, 2π) + q(t), t ∈ (0, T ), u(0, x) = u 0 (x), η(0, x) = η 0 (x), x ∈ (0, 2π), (1.19) with the periodic boundary conditions for u as in (1.4).

Theorem 1.5. Let T > 2π. For any initial data (u 0 , η 0 ) ∈ H s p with 2π 0 u 0 = 2π 0 η 0 = 0 and s > 3 2 , there exists a control q ∈ L 2 (0, T ) such that the solution of (1.19) satisfies u(T, •) = 0 and η(T, •) = 0. 1.5. The method of moment. As mentioned before, we have used the well-known method of moments to study the null controllability of system (1.3)- (1.5). In this section, we describe this method a bit and give some information relevant to our work.

The method of moments completely depends on the eigen-elements of the underlying spatial operator, describing the system in ODE setup as in (2.1). Roughly speaking about this method, one has to first find an identity, equivalent to the null controllability of the considered system using the main system and its adjoint system. Further, using the eigenvectors of the adjoint system (which should be complete in the concerned space) as the terminal data of adjoint system, one can derive a system of identities, called moment problem. Thus, the question of null controllability boils down to the existence of solution of the derived moment problem. Now solving the moment problem is immediate, once the existence of certain biorthogonal family with proper L 2 -estimate is known. Thus, the main difficulty in solving the moment problem and hence getting the null controllability result for a system is to get the existence of desired biorthogonal family.

This method along with its advantage of giving the explicit form of control, instead of just giving the existence of control, comes up with a limitation as well. Since this method completely depends on the spectral properties of the corresponding operator, it is in general limited to the study of controllability of systems in one dimensional space. This method of reducing the problem of null controllability to moment problem was initially done by H.O. Fattorini and D.L. Russel for linear parabolic equation in their work [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]. They further generalized the result regarding the existence of biorthogonal family of real exponentials in [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF]. The construction of biorthogonal family has been further investigated for complex exponentials as well, for e.g., see [START_REF] Fernández-Cara | Boundary controllability of parabolic coupled equations[END_REF], [START_REF] Benabdallah | Sharp estimates of the onedimensional boundary control cost for parabolic systems and application to the $n$-dimensional boundary null controllability in cylindrical domains[END_REF], [START_REF] Farid Ammar Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF], [START_REF] González | Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions[END_REF]. All of these results require some conditions to be satisfied by the power of exponentials, which is someway related to parabolic systems. But as our system involves hyperbolic equation as well, the power of exponentials (eigenvalues of A in our case) does not satisfy such conditions and so those results cannot be used. More precisely, we need to construct a family biorthogonal to {e -λ ± k t } k∈Z , where λ ± k denote the hyperbolic and parabolic branches of eigenvalues of the underlying spatial operator. It is worth to mention that, the biorthogonal for a exponential family involving a hyperbolic and parabolic branches of eigenvalues has been constructed in [START_REF] Martin | Null controllability of the structurally damped wave equation with moving control[END_REF]. We have followed this work to construct the desired biorthogonal family in our case.

1.6. Organization. At first, we state the well-posedness results for the concerned systems of this paper in Section 2, whose proofs can be found in the Appendix C (Section 8). In Section 3, we mainly find the eigenvalues and corresponding eigenvectors of adjoint of the underlying spatial operator of equation (3.1), and further reduce the null controllability problem to the corresponding moment problem. This section also contains statement of the theorem concerning the existence of desired biorthogonal family. Next assuming this result, concerning the existence of biorthogonal family, we prove all the aforementioned controllability results in Section 4. At last in Section 5, we give the construction of the required biorthogonal family.

Well-Posedness

In this section, we state the well-posedness results for the main system (1.3) as well as for all the considered control systems, whose proof can be found in the appendix (Section 8). At first, we write the system (1.3)-(1.4)-(1.5) in the abstract ODE set up:

U(t) = AU(t), t ∈ (0, T ), U(0) = U 0 , (2.1)
where, U = (u, η) T , U 0 = (u 0 , η 0 ) T ,

A =      - d 4 dx 4 - d 3 dx 3 - d 2 dx 2 d dx d dx - d dx      (2.2) with D(A) = H 4 p (0, 2π) × H 1 p (0, 2π). Proposition 2.1. Let h ∈ C 1 ([0, T ] × (0, 2π)), then for any (u 0 , η 0 ) ∈ D(A) system (1.16) has a unique solution (u, η) ∈ C([0, T ]; D(A)). Moreover, if (u 0 , η 0 ) ∈ Z, then the generalized solution (u, η) ∈ C([0, T ]; Z).
Remark 2.2. In particular, we have well-posedness of the homogeneous system (1.3).

Corollary 2.3. Let h ∈ C 1 ([0, T ] × (0, 2π)), then for any (u 0 , η 0 ) ∈ D(A) system (1.17) has a unique solution (u, η) ∈ C([0, T ]; D(A)). Moreover, if (u 0 , η 0 ) ∈ Z, then the generalized solution (u, η) ∈ C([0, T ]; Z). Let us denote X = C([0, T ]; L 2 (0, 2π) × L 2 (0, 2π)) ∩ L 2 (0, T ; H 2 (0, 2π) × L 2 (0, 2π)).
Proposition 2.4. Let h ∈ L 2 (0, T ); L 2 (0, 2π) . Then, for any (u 0 , η 0 ) ∈ Z, system (1.16) has unique solution (u, η) ∈ X. Moreover, we have the following estimate

||(u, η)|| X ≤ C ||h|| L 2 (0,T ;L 2 (0,2π)) + ||(u 0 , η 0 )|| L 2 (0,2π)×L 2 (0,2π) . (2.3)
Remark 2.5. Using similar analysis, we can get energy estimate for the solution of (1.17). Now, consider the following forward adjoint system of (1.18):

                   φ t + φ xxxx -φ xxx + φ xx + ψ x = 0, ψ t -ψ x + φ x = 0, φ(t, 0) = φ(t, 2π), φ x (t, 0) = φ x (t, 2π), φ xx (t, 0) = φ xx (t, 2π), φ xxx (t, 0) = φ xxx (t, 2π), ψ(t, 0) = ψ(t, 2π), ψ x (t, 0) = ψ x (t, 2π), φ(0, x) = φ 0 (x), ψ(0, x) = ψ 0 (x).
(2.4)

Using similar analysis as done in the proof of Proposition 2.1 and Proposition 2.4, we can get the following result:

Proposition 2.6. For any (φ 0 , ψ 0 ) ∈ Z, the adjoint system (2.4) has a unique solution (φ, ψ) ∈ X which satisfies:

||(φ, ψ)|| X ≤ C ||(φ 0 , ψ 0 )|| Z . (2.5)
Further, using trace regularity result, we get

||(φ x (•, 0), ψ(•, 0))|| L 2 (0,T )×L 2 (0,T ) ≤ C ||(φ 0 , ψ 0 )|| Z . (2.6)
Next, we state the result of well-posedness for the control system (1.18). We use the transposition scheme for proving it and so we first define what we mean by solution of system (1.18) in the sense of transposition.

Definition 2.7. For any given (u 0 , η 0 ) ∈ Z and q ∈ L 2 (0, T ), we say that (u, η) is a solution of system

(1.18), if for every t ∈ [0, T ] the function (u(t, •), η(t, •)) ∈ Z satisfies 2π 0 u(t, x)φ 0 (x) dx + 2π 0 η(t, x)ψ 0 (x) dx = 2π 0 u 0 (x)φ(t, x) dx + 2π 0 η 0 (x)ψ(t, x) dx - t 0 φ xxx (t -s, 0) -φ xx (t -s, 0) -φ x (t -s, 0) + ψ(t -s, 0) q(s) ds, (2.7)
where (φ, ψ) is solution of (2.4) posed in (0, t) × (0, 2π) with (φ(0, •), ψ(0,

•)) = (φ 0 , ψ 0 ) ∈ H 3 p (0, 2π) × L 2 (0, 2π).
Proposition 2.8. For any (u 0 , η 0 ) ∈ Z and q ∈ L 2 (0, T ), the control system (1.18) has a unique solution (u, η) ∈ C([0, T ]; Z) which satisfy

||(u, η)|| C([0,T ];Z) ≤ C ||q|| L 2 (0,T ) + ||(u 0 , η 0 )|| Z .
Remark 2.9. In a similar way, we can define solution for the control system (1.19) in sense of transposition and then by following the same steps of the proof of the above proposition, we can find an analogous result for (1.19), giving existence of the solution in appropriate space.

The method of Moment: Null controllability of KS-KdV-Transport system

In this section we demonstrate the method of moment to prove the null controllability of the aforementioned control systems. To employ this method for proving all the theorems 1.2 1.3, 1.4, 1.5, we need to construct a family biorthogonal to {1, e -λ ± k t } k∈Z\{-1,0} , where {λ ± k , λ 0 = 0} are eigenvalues of the underlying spatial operator corresponding to the system (1.3). But the method we use for constructing such biorthogonal family does not work here due to the form of eigenvalues, and so we shift the eigenvalues by -1 using the transformation, ũ = e -t u, η = e -t η in the system (1.3) to get the desired form of the eigenvalues (see Section 5 for details).

We shall make the abuse of notation in writing (u, η) for (ũ, η). Then the system satisfied by (ũ, η) becomes

                   u t + u xxxx + u xxx + u xx + u = η x , (t, x) ∈ (0, T ) × (0, 2π), η t + η x + η = u x , (t, x) ∈ (0, T ) × (0, 2π), u(t, 0) = u(t, 2π), u x (t, 0) = u x (t, 2π), t ∈ (0, T ), u xx (t, 0) = u xx (t, 2π), u xxx (t, 0) = u xxx (t, 2π), t ∈ (0, T ), η(t, 0) = η(t, 2π), t ∈ (0, T ), u(0, x) = u 0 (x), η(0, x) = η 0 (x),
x ∈ (0, 2π).

(3.1)

Let U = (u, η) T and let us denote the identity map by I, then the above system can be rewritten as:

U(t) = ÃU(t), t ∈ (0, T ), U(0) = U 0 = (u 0 , η 0 ), (3.2)
where,

à =      - d 4 dx 4 - d 3 dx 3 - d 2 dx 2 -I d dx d dx - d dx -I      (3.3) with D( Ã) = D(A) = H 4 p (0, 2π) × H 1 p (0, 2π
). The adjoint of à is given by:

à * φ ψ =      - d 4 dx 4 + d 3 dx 3 - d 2 dx 2 -I - d dx - d dx d dx -I      φ ψ , (3.4)
with

D( Ã * ) = D( Ã) = H 4 p (0, 2π) × H 1 p (0, 2π
). Thus, the adjoint system of (1.3) can be written as:

                   φ t -φ xxxx + φ xxx -φ xx -φ -ψ x = 0, x ∈ (0, 2π), t ∈ (0, T ) ψ t + ψ x -ψ -φ x = 0, x ∈ (0, 2π), t ∈ (0, T ), φ(t, 0) = φ(t, 2π), φ x (t, 0) = φ x (t, 2π), t ∈ (0, T ), φ xx (t, 0) = φ xx (t, 2π), φ xxx (t, 0) = φ xxx (t, 2π), t ∈ (0, T ), ψ(t, 0) = ψ(t, 2π), t ∈ (0, T ), φ(T, x) = φ T (x), ψ(T, x) = ψ T (x),
x ∈ (0, 2π).

(

Remark 3.1. Note that proving Theorem 1.2, Theorem 1.3, Theorem 1.4 and Theorem 1.5 is equivalent to proving those results for the corresponding control system of (3.1). This is due to the fact that exponential function is nonvanishing for any finite t. More precisely, (u(T, x), η(T, x)) = (0, 0) iff (ũ(T, x), η(T, x)) = (0, 0). So, from now on, our main goal would be to prove the above theorems for the transformed system (3.1).

3.1. Spectral decomposition of the adjoint operator à * . Let us write the eigen-equation corresponding to the adjoint operator à * ,

à * ξ ζ = λ ξ ζ , λ ∈ C,
which can be explicitly written as

         -ξ (iv) + ξ ′′′ -ξ ′′ -ξ -ζ ′ = λξ, ζ ′ -ζ -ξ ′ = λζ, ξ i (0) = ξ i (2π), i = 0, 1, 2, 3, ζ(0) = ζ(2π).
Combining these two equations we obtain

-ξ (v) + (2 + λ)ξ (iv) -(2 + λ)ξ ′′′ + λξ ′′ -(1 + λ)ξ ′ + (1 + λ) 2 ξ = 0. (3.6)
Expanding φ as a Fourier series ξ = k∈Z ξ k e ikx , we obtain that (3.6) is satisfied provided for each k ∈ Z the following holds

λ 2 + λ(k 4 + ik 3 -k 2 -ik + 2) + (-ik 5 + 2k 4 + 2ik 3 -ik + 1) = 0.
The solution of the above equation is given by

λ ± k = 1 2 -(k 4 + ik 3 -k 2 -ik + 2) ± k 8 + 2ik 7 -3k 6 -k 4 -2ik 3 -5k 2 , ∀ k ∈ Z,
λ + 0 = λ - 0 = -1, (3.7)
λ + -1 = λ + 1 = -1 + √ 2i, (3.8) λ - -1 = λ - 1 = -1 - √ 2i, (3.9)
λ + k = ik -1 + iO(|k| -1 ) + O(|k| -2 ), as |k| → ∞, (3.10)
λ - k = -k 4 -ik 3 + k 2 -1 + iO(|k| -1 ) + O(|k| -2 ), as |k| → ∞. (3.11)
that {λ ± k } k∈Z\{-1,0} is a set of distinct elements. Moreover, note that λ + m = λ - n = -1 iff m = n = 0 and so (1 + λ + m ) = (1 + λ - n ) = 0 iff m = n = 0. Let us denote λ 0 = λ + 0 = λ - 0 = -1.
Then, Φ 0 = 1 0 and Φ0 = 0 1 are two linearly independent eigenvectors corresponding to λ 0 = -1. For k ∈ Z \ {0}, the eigenfunctions can be written as

Φ + k = e ikx θ + k e ikx , Φ - k = e ikx θ - k e ikx , (3.12)
where,

θ ± k = ξ ± k (1+λ ± k ) is well defined as 1+λ ± k ̸ = 0 (see Remark 3.2) with ξ ± k = -k 4 -ik 3 +k 2 -ik-(1+λ ± k ).
θ + k < C|k| 3 and θ - k < c |k| 3 → 0, as |k| → ∞.
(3.13) 3.2. Reduction to the moment problem. The interior control systems for (3.1), analogous to (1. [START_REF] Chen | Nonlinear waves on liquid film surfaces-ii. bifurcation analyses of the long-wave equation[END_REF]) and (1.17) are respectively given as:

     u t + u xxxx + u xxx + u xx + u = η x + 1 ω h(t, x), (t, x) ∈ (0, T ) × (0, 2π), η t + η x + η = u x , (t, x) ∈ (0, T ) × (0, 2π), u(0, x) = u 0 (x), η(0, x) = η 0 (x), x ∈ (0, 2π), (3.14)
and

     u t + u xxxx + u xxx + u xx + u = η x , (t, x) ∈ (0, T ) × (0, 2π), η t + η x + η = u x + 1 ω h(t, x), (t, x) ∈ (0, T ) × (0, 2π), u(0, x) = u 0 (x), η(0, x) = η 0 (x),
x ∈ (0, 2π), (3.15) with the same periodic boundary condition as in (3.1). Following are the boundary control systems for (3.1) analogous to (1.18) and (1.19):

                   u t + u xxxx + u xxx + u xx + u = η x , (t, x) ∈ (0, T ) × (0, 2π), η t + η x + η = u x , (t, x) ∈ (0, T ) × (0, 2π), u(t, 0) = u(t, 2π) + q(t), u x (t, 0) = u x (t, 2π), t ∈ (0, T ), u xx (t, 0) = u xx (t, 2π), u xxx (t, 0) = u xxx (t, 2π), t ∈ (0, T ), η(t, 0) = η(t, 2π), t ∈ (0, T ), u(0, x) = u 0 (x), η(0, x) = η 0 (x), x ∈ (0, 2π), (3.16)
                   u t + u xxxx + u xxx + u xx + u = η x , (t, x) ∈ (0, T ) × (0, 2π), η t + η x + η = u x , (t, x) ∈ (0, T ) × (0, 2π), u(t, 0) = u(t, 2π), u x (t, 0) = u x (t, 2π), t ∈ (0, T ), u xx (t, 0) = u xx (t, 2π), u xxx (t, 0) = u xxx (t, 2π), t ∈ (0, T ), η(t, 0) = η(t, 2π) + q(t), t ∈ (0, T ), u(0, x) = u 0 (x), η(0, x) = η 0 (x),
x ∈ (0, 2π).

(3.17)

In this section, we derive a system of identities called moment problem, for the control systems (3.14), (3.15), (3.16) and (3.17). Further, we show that solving the obtained moment problems are equivalent to proving null controllability of the corresponding control systems, which is our main goal in this paper.

3.2.1.

Localized interior bilinear control acting in the KS-KdV equation. Let us consider the control system (3.14). Assume the initial data (u 0 , η 0 ) of (3.14), terminal data (φ T , ψ T ) of (3.5) and f, g to be smooth enough, so that we can ensure the required smoothness of solutions (u, v) and (φ, ψ) of systems (3.14) and (3.5), respectively. We first multiply (3.14) by φ, ψ , where h(x, t) = f (x)g(t) and integrate it on (0, T ) × (0, 2π). On performing integration by parts, we get

u(T, •) η(T, •) , φ T ψ T Z - u 0 η 0 , φ(0, •) ψ(0, •) Z = T 0 2π 0 f (x)g(t)φ(t, x)dxdt. (3.18)
Using density property one can show that the above identity holds even if (u 0 , η 0 ) ∈ Z, (φ T , ψ T ) ∈ Z, f ∈ L 2 (0, 2π) and g ∈ L 2 (0, T ).

We now vary the terminal data (φ T , ψ T ) over the eigenvectors of A * . Let us first take (φ T , ψ T ) as

φ T ψ T = e ikx θ ± k e ikx , k ∈ Z \ {0}
, then the solution of the adjoint system (3.5) is given by

φ k (t, x) ψ k (t, x) = e λ ± k (T -t) e ikx θ ± k e ikx , k ∈ Z \ {0}. Therefore the identity (3.18) reduces to u(T, •) η(T, •) , φ T ψ T Z -e λ ± k T 2 2π 0 u 0 (x)e -ikx + θ ± k 2π 0 η 0 (x)e -ikx = f k T 2 -T 2 e -λ ± k t g t + T 2 .
This can be rewritten in the following form

- u(T, •) η(T, •) , φ T ψ T Z + f k e λ ± k T 2 T 2 -T 2 e -λ ± k t g t + T 2 = -e λ ± k T γ ± k , (3.19)
where

f k = 2π 0 f (x)e -ikx , and γ ± k = 2π 0 u 0 (x)e -ikx + θ ± k 2π 0 η 0 (x)e -ikx = 2π a k + θ ± k b k , ∀k ∈ Z\{0}.
(3.20) Next we take the terminal data as

φ T ψ T = Φ 0 = 1 0 ,
then the solution of the adjoint system (3.5) is given by

φ 0 (t, x) ψ 0 (t, x) = e -(T -t) 1 0
and so identity (3.18) gives

- u(T, •) η(T, •) , φ T ψ T Z + e -T 2 f 0 T 0 g t + T 2 e t dt = -e -T 2π 0 u 0 . (3.21)
Finally, taking the terminal data as

φ T ψ T = Φ0 = 0 1 .
we get the solution of the adjoint system (3.

       f k T 2 -T 2 e -λ ± k t g t + T 2 = -e λ ± k T 2 γ ± k , f k , γ ± k are given by (3.20), f 0 T 2 -T 2 e t g t + T 2 dt = -e -T 2 γ 0 , γ 0 = 2π 0 u 0 , 2π 0 η 0 = 0. (3.23)
Conversely, assume (3.23) to be true for some f ∈ L 2 (0, 2π), g ∈ L 2 (0, T ), then for k ∈ Z \ {0} (3. [START_REF] Panagiotis | Global stabilization of the Kuramoto-Sivashinsky equation via distributed output feedback control[END_REF]) gives u(T, x), e ikx L 2 (0,2π) + θ ± k η(T, x), e ikx L 2 (0,2π) = 0 and (3.21), (3.22) gives

⟨u(T, x), 1⟩ L 2 (0,2π) = 0, ⟨η(T, x), 1⟩ L 2 (0,2π) = 0. As θ + k ̸ = θ - k (see Remark 3.3) for k ∈ Z \ {0} and {e ikx } k∈Z forms a basis of L 2 (0, 2π), we get u(T, •) = η(T, •) = 0.
Thus, we have the following lemma:

Lemma 3.4. The solution of control system (1.16) satisfies (u(T, •), η(T, •)) = (0, 0) if and only if for any initial data (u 0 , η 0 ) ∈ L 2 (0, 2π) × L2 (0, 2π), there exist a function h of form h(x, t) = f (x)g(t) with f ∈ L 2 (0, 2π) and g ∈ L 2 (0, T ) such that (3.23) holds.

3.2.2.

Localized interior bilinear control acting in the transport equation. In this subsection, we consider the system (3.15) where the localized interior control now acts on the transport component. Similar to the case of (3.14) we can formulate the corresponding moment problem for system (3.15) as well, which is stated precisely in the following lemma:

Lemma 3.5. The solution of control system (3.15) satisfies (u(T, •), η(T, •)) = (0, 0) if and only if for any initial data (u 0 , η 0 ) ∈ L2 (0, 2π) × L 2 (0, 2π), there exist function h ∈ L 2 (0, T ; L 2 (0, 2π)) of the form h(x, t) = f (x)g(t) which solves the following moment problem:

   θ ± k f k T 2 -T 2 e -λ ± k t g t + T 2 = -e λ ± k T 2 γ ± k , f 0 T 2 -T 2 e t g t + T 2 = -e -T 2 γ 0 , (3.24)
where

f k = 2π 0 f (x)e -ikx dx, γ ± k = 2π 0 u 0 (x)e -ikx + θ ± k 2π 0 η 0 (x)e -ikx , ∀k ∈ Z \ {0}, f 0 = 2π 0 f (x) dx and γ 0 = 2π 0 η 0 (x) dx.

3.2.3.

Boundary control acting on the KS-KdV component. In this subsection we derive moment problem for the system (3.16). Let us first assume that the control q and the initial data (u 0 , η 0 ) are smooth enough and let terminal data (φ T , ψ T ) ∈ D(A * ) then the solution (u, η) of (1.18) is sufficiently smooth and the solution (φ, ψ) of (3.5) belong to C([0, T ]; D(A * )). Now, multiply the first two equation of (3.16) by φ and ψ, respectively and then integrate over (0, T )×(0, 2π). Further performing integration by parts successively in the obtained identity, we get

u(T, •) η(T, •) , φ T ψ T D(A * ) ′ ,D(A * ) - u 0 η 0 , φ(0, •) ψ(0, •) Z + T 0 φ xxx (t, 2π) -φ xx (t, 2π) + φ x (t, 2π) + ψ(t, 2π) q(t)dt = 0. (3.25)
Using density argument, one can easily conclude about the above identity even if the initial data (u 0 , η 0 ) and terminal data (φ T , ψ T ) belong to Z.

Again following the same process as done in Section 3.2.1, we get a moment problem corresponding to the system (3.16) as stated in the following lemma: Lemma 3.6. The solution of control system (3.16) satisfies (u(T, •), η(T, •)) = (0, 0) if and only if for any initial data (u 0 , η 0 ) ∈ L2 (0, 2π) × L 2 (0, 2π), there exists function q ∈ L 2 (0, T ) which solves the following moment problem:

   T 2 -T 2 e -λ ± k t q t + T 2 = e λ ± k T 2 γ ± k , for k ∈ Z \ {-1, 0}, T 2 -T 2 e t q t + T 2 dt = e -T 2 γ 0 , (3.26)
where

γ ± k = 1 ik 3 + k 2 -ik + θ ± k 2π 0 u 0 (x)e -ikx + θ ± k 2π 0 η 0 (x)e -ikx and γ 0 = 2π 0 η 0 (x)dx.
From the asymptotic expression of λ ± k , one can easily conclude that (-

ik 3 + k 2 + ik)(1 + λ ± k ) + ξ ± k and hence ik 3 + k 2 -ik + θ ± k are nonzero.
Also, for the first few finite case, one can conclude from the graph (see Figure 4).

(-ik 3 + k 2 + ik)(1 + λ - k ) + ξ - k ̸ = 0 Figure 4. Plot of (-ik 3 + k 2 + ik)(1 + λ ± k ) + ξ ± k for first few k symmetric about 0 3.2.4.
Boundary control acting on the transport component. Following the same argument as done in the last Section 3.2.3, we derive moment problem for the control system (3.17) in this subsection. Let (u 0 , η 0 ) ∈ L 2 (0, 2π)×L 2 (0, 2π), (φ T , ψ T ) ∈ D(A * ) and q ∈ L 2 (0, T ). Then, the identity analogous to (3.25) is given as

u(T, •) η(T, •) , φ T ψ T D(A * ) ′ ,D(A * ) - u 0 η 0 , φ(0, •) ψ(0, •) Z + T 0 φ(t, 2π) -ψ(t, 2π) q(t) dt = 0. (3.27)
and so if we try to find the moment problem as done above, we get the following identities:

         T 2 -T 2 e -λ ± k t q(t + T 2 ) = e λ ± k T 2 γ ± k , k ∈ Z \ {0}, T 2 -T 2 e t q t + T 2 = e -T 2 γ 0 , T 2 -T 2 e t q t + T 2 = -e -T 2 γ0 , (3.28)
where,

γ ± k = 1 1 -θ ± k 2π 0 u 0 (x)e -ikx dx + θ ± k 2π 0 η 0 (x)e -ikx dx , γ 0 = 2π 0 u 0 and γ0 = 2π 0 η 0 . (3.29) Note that (1 -θ ± k ) ̸ = 0 iff (1 -θ ± k ) ̸ = 0 iff k 4 + ik 3 -k 2 + ik + 2(1 + λ ± k ) ̸ = 0,
k 4 + ik 3 -k 2 + ik + 2(1 + λ - k ) ̸ = 0 Figure 5. Plot of k 4 + ik 3 -k 2 + ik + 2(1 + λ ± k ) ̸ = 0 for first few k symmetric about 0
Remark 3.7. Note that the second and third equation of (3.28) contradicts each other if γ 0 ̸ = -γ 0 , so we restrict the set of initial data to satisfy γ 0 = -γ 0 .

Thus, we have the following lemma:

Lemma 3.8. The solution of control system (3.15) vanishes at t = T if and only if for any initial data

(u 0 , η 0 ) ∈ L 2 (0, 2π) × L 2 (0, 2π) satisfying 2π 0 u 0 = - 2π 0 η 0 , there exists a function q ∈ L 2 (0, T ) which solves the moment problem T 2 -T 2 e -λ ± k t q t + T 2 = -e λ ± k T 2 γ ± k , k ∈ Z \ {0}. (3.30)
3.3. Biorthogonal family. We are now at the stage to solve the aforementioned moment problems. Before solving it, we need existence of a suitable biorthogonal family of {e t , e -λ ± k t } k∈Z\{-1,0} which essentially gives solution for each of the moment problem derived above. Proposition 3.9 (Biorthogonal family). Let T > 2π. Then there exists a family

{Θ ± k } k∈Z\{-1,0} ∪{Θ 0 } of functions in L 2 -T 2 , T 2 satisfying      T 2 -T 2 Θ ± k (t)e -λ ± l t dt = δ kl δ ± , l ∈ Z \ {-1}, k ∈ Z \ {-1, 0}, T 2 -T 2 Θ 0 (t)e -λ ± l t dt = δ 0l , l ∈ Z \ {-1},
where,

δ kl = 1, if k = l 0, if k ̸ = l and δ ± = 1, if sign on Θ k and λ k is same 0, otherwise .
Moreover, we have following estimates on the obtained family

||Θ + k || L 2 (R) ≤ C |k| 4 , k ∈ Z \ {-1, 0}, ||Θ - k || L 2 (R) ≤ C|k| 15 e -T 2 (k 4 -k 2 ) , k ∈ Z \ {-1, 0}, ||Θ 0 || L 2 (R) ≤ C,
where, C is a positive constant.

The proof of this proposition can be found in Section 5.

Proof of the Main theorems

As discussed in Section 3.2, it is clear that proving the main results of this paper is actually equivalent to solving the corresponding reduced moment problem.

Before going into the proof of Theorem 1.2 and Theorem 1.3, let us first state a result which gives existence of f ∈ L 2 (0, 2π) satisfying certain desirable estimates, so that we can define g ∈ L 2 (0, T ) which solves the corresponding moment problem. Proposition 4.1. Let ω ⊂ (0, 2π) be any nonempty open set. Then there exits f ∈ L 2 (0, 2π) supported in ω such that

f k = 2π 0 f (x)e -ikx dx ̸ = 0, for k ∈ Z, |f k | ≥ C k 2 , for some C > 0, for k ∈ Z \ {0}. (4.1)
The proof of this proposition is done after the proof of interior controllability theorems and so assuming this proposition for the time being, let us prove the interior controllability results.

4.1. Proof of Theorem 1.2.

Proof. From Remark 3.1, it is sufficient to prove the theorem for system (3.14) instead of (1.16), which is equivalent to solving the moment problem (3.23). Let us define g as

g(t) = - k∈Z\{0,±1} f -1 k e λ + k T 2 γ + k Θ + k t - T 2 - k∈Z\{0,±1} f -1 k e λ - k T 2 γ - k Θ - k t - T 2 -f -1 0 e -T 2 γ 0 Θ 0 t - T 2
Using proposition (3.9), one can easily conclude that g solves the moment problem (3.23). Now, we just need to show that g ∈ L 2 (0, T ).

||g|| L 2 (0,T ) ≤ k∈Z\{0,±1} e λ + k T 2 |f k | -1 |γ + k | ||Θ + k || L 2 (-T 2 , T 2 ) + k∈Z\{0,±1} e λ - k T 2 |f k | -1 |γ - k | ||Θ - k || L 2 (-T 2 , T 2 ) + e -T 2 |f 0 | -1 |γ 0 | ||Θ 0 || L 2 (-T 2 , T 2 ) ≤ C   k∈Z\{0,±1} |f k | -1 (|a k | + |k| 3 |b k |) |k| 4 + k∈Z\{0,±1} |f k | -1 |a k | + 1 |k| 3 |b k | |k| 15 e -T (k 4 -k 2 )   ≤ C   k∈Z\{0,±1} |k| 6 (|a k | + |k| 3 |b k |) + k∈Z\{0,±1} |k| 3 |a k | + |b k | |k| 14 e -T (k 4 -k 2 )   ,
where (4.1) has been used in the last estimate. Clearly, the second term of the last inequality is finite due to negative exponential and the fact that {a k }, {b k } ∈ l 2 (Z) ⊂ l ∞ (Z). Thus, we are only left to prove the finiteness of first term. Let (u 0 , η 0 ) ∈ H s p for s > 13 2 with 2π 0 η 0 = 0, then using the expansion of the initial conditions as in (1.14), we have:

  k∈Z\{0,±1} |k| 2s |a k | 2 + |k| 6 |b k | 2   1 2 ≤   k∈Z\{0,±1} (1 + |k| 2s ) |a k | 2 + |k| 6 |b k | 2   1 2 ≤ ||(u 0 , η 0 )|| H s p < ∞, for s > 13 2 . ( 4.2)
In fact for any ϵ > 0, we have:

k∈Z\{0,±1} |k| 6 (|a k | + |k| 3 |b k |) ≤ C k∈Z\{0,±1} |k| -(1+ϵ) 2 |k| 6+ (1+ϵ) 2 (|a k | + |k| 3 |b k |) ≤ C   k∈Z\{0,±1} |k| -1-ϵ   1 2   k∈Z\{0,±1} (|a k | 2 + |k| 6 |b k | 2 )|k| 13+ϵ   1
g(t) = - k∈Z\{0,±1} θ + k -1 f -1 k e λ + k T 2 γ + k Θ + k t - T 2 - k∈Z\{0,±1} θ - k -1 f -1 k e λ - k T 2 γ - k Θ - k t - T 2 -f -1 0 e -T 2 γ 0 Θ 0 t - T 2 .
Using the same argument as above, one can easily prove that g ∈ L 2 (0, T ) solves the moment problem (3.24) if (u 0 , η 0 ) ∈ H s p with 2π 0 u 0 = 0 and s > 7 2 , and hence proves the theorem. □

Proof of Proposition 4.1

Proof. Let α ∈ ω and ρ ∈ (0, 1) be a quadratic irrational (irrational number which is a root of quadratic equation with integral coefficients) such that [α, α + ρπ] is subset of ω. Define

f (x) = χ [α,α+ρπ] (x), ∀x ∈ (0, 2π)
Clearly, f ∈ L 2 (0, 2π) with support inside ω and also f 0 = ρπ ̸ = 0,

f k = 2π 0 f (x)e -ikx = e -ikα ik (1 -e -ikρπ ) ̸ = 0, k ∈ Z \ {0}.
Now, as ρ is quadratic irrational so it can be approximated by rational numbers to order 2 and to no higher order (see [START_REF] Bell | Gh hardy and em wright, an introduction to the theory of numbers[END_REF], Theorem 188), i.e., there exist C > 0 such that for any integers p and q, q ̸ = 0,

ρ - p q ≥ C q 2 . (4.3) Also, for k ∈ Z \ {0} |f k | = 1 |k| |1 -e -ikρπ | = 1 |k| 2
q(t) = k∈Z\{0,±1} e λ + k T 2 γ + k Θ + k t - T 2 + k∈Z\{0,±1} e λ - k T 2 γ - k Θ - k t - T 2 + e -T 2 γ 0 Θ 0 t - T 2 .
Using proposition (3.9), one can easily conclude that this q solves the moment problem (3.26). Now we just need to show that q ∈ L 2 (0, T ).

||q|| L 2 (0,T ) ≤ k∈Z\{0,±1} |e λ + k T 2 | |γ + k | ||Θ + k || L 2 (-T 2 , T 2 ) + k∈Z\{0,±1} |e λ - k T 2 | |γ - k | |Θ - k || L 2 (-T 2 , T 2 ) + |e -T 2 | |γ 0 | ||Θ 0 || L 2 (-T 2 , T 2 ) ≤ C   k∈Z\{0,±1} (|a k | + |k| 3 |b k |) |k| |k| 4 + k∈Z\{0,±1} |a k | |k| 3 + |b k | |k| 6 |k| 15 e -T (k 4 -k 2 )   ≤ C   k∈Z\{0,±1} |k| -1-ϵ   1 2   k∈Z\{0,±1} (|a k | 2 + |k| 6 |b k | 2 )|k| 7+ϵ   1 2
.

For the last inequality, we use Cauchy-Schwartz inequality for the first term and the second term is automatically finite due to the exponential term. Now as the initial conditions (u 0 , η 0 ) ∈ H s p with 2π 0 u 0 = 0 for s > 7 2 , so ||q|| L 2 (0,T ) < ∞ and hence the proof is complete. □ 4.4. Proof of Theorem 1.5.

Proof. Let us solve the moment problem (3.30) which is equivalent to proving the theorem. For that, let us define q as

q(t) = k∈Z\{0,±1} e λ + k T 2 γ + k Θ + k t - T 2 + k∈Z\{0,±1} e λ - k T 2 γ - k Θ - k t - T 2
Following the same steps, one can easily prove that q ∈ L 2 (0, T ) if (u 0 , η 0 ) ∈ H s p with 2π 0 u 0 = 2π 0 η 0 = 0, for s > 3 2 . □

Biorthogonal Constructions

This section is devoted to the proof of Proposition 3.9, which is inspired from [START_REF] Martin | Null controllability of the structurally damped wave equation with moving control[END_REF]. The entire study relies on the theory of logarithmic integral and complex analysis. One can refer to the books [START_REF] Young | An introduction to nonharmonic Fourier series[END_REF], [START_REF] Koosis | The logarithmic integral. I[END_REF] and [START_REF] Koosis | The logarithmic integral[END_REF] for more details. All the constants used in this section while finding the relevant estimates are generic, which may vary from line to line. Let us first define exponential type and sine type functions. The following proposition states some important properties of sine type functions: Proposition 5.3. Let f be a sine type function, and let {µ k } k∈I with I ⊂ Z be its sequence of zeros. Then, we have: (a) for any ϵ > 0, there exist constants K ϵ , Kϵ > 0 such that

K ϵ e π|y| ≤ |f (x + iy)| ≤ Kϵ e π|y| , if dist(x + iy, {µ k }) > ϵ, (b) there exist some constants K 1 , K 2 > 0 such that K 1 < |f ′ (µ k )| < K 2 , ∀k ∈ I.
The main goal of this section is to find a class of entire functions E = {Ψ ± k , Ψ 0 , } k∈Z\{-1,0} with the following properties:

1. The family E contains entire functions of exponential type T 2 . That means there exists a positive constant C such that

|Ψ 0 (z)| ≤ Ce T 2 |z| and Ψ ± k (z) ≤ Ce T 2 |z| , ∀z ∈ C. (5.1)
2. All the members of E are square integrable on the real line, i.e.

R Ψ ± k (x) 2 dx < ∞ and R |Ψ 0 (x)| 2 dx < ∞.
(5.2)

3. The following relation hold

Ψ ± k (iµ ± l ) = δ kl δ ± , k ∈ Z \ {-1, 0}, l ∈ Z \ {-1}, Ψ 0 (iµ ± l ) = δ 0l , l ∈ Z \ {-1}, (5.3)
where,

µ + 0 = µ - 0 = µ 0 = -1, µ ± k = λ ± k , ∀k ∈ Z \ {0}
B = {Θ ± k , Θ 0 } k∈Z\{-1,0} in L 2 (R) supported in [-T 2 , T 2
], such that the following representation holds

Ψ ± k (z) = T 2 -T 2 e izt Θ ± k (t)dt, z ∈ C. (5.4) Clearly, Θ ± k , Θ 0 are the Fourier transformations of Ψ ± k , Ψ 0 respectively, for k ∈ Z \ {-1, 0}. Also by Plancharel's Theorem, we have R Ψ ± k (x) 2 = 2π T 2 -T 2 Θ ± k (t) 2 dt.
Note that, (5.3) with the representation (5.4) imply the following

     T 2 -T 2 e -µ ± l t Θ ± k dt = δ kl δ ± , for k ∈ Z \ {-1, 0}, l ∈ Z \ {-1}, T 2 -T 2 e -µ ± l t Θ 0 dt = δ 0l , for l ∈ Z \ {-1},
which will essentially prove the Proposition (3.9) as µ ± k = λ ± k . Now we are at position of formulating the construction of the family E satisfying (5.1), (5.2) and (5.3). Let us first introduce the following entire function which has simple zeros exactly at iµ ± k :

P (z) = 1 + z i k∈Z\{-1,0} 1 - z iµ + k k∈Z\{-1,0} 1 - z iµ - k .
(5.5) 5.1. Estimating the canonical product P. In this subsection we derive few estimates on P and P ′ , which is very crucial for the construction of biorthogonal family:

Proposition 5.5. Let P be the canonical product defined in (5.5). Then P is an entire function of exponential type at most π, which satisfies the following estimates for some positive constants C, C 1 , C 2 independent of k:

|P (x)| ≤ C |x + i| -1 e 2π(cos( π 8 )+sin( π 8 ))|x| 1 4 , ∀x ∈ R, (5.6)
P ′ (iµ + k ) ≥ C 1 |k| -1 e 2π(cos( π 8 )+sin( π 8 ))|k| 1 4 , k ∈ Z \ {-1, 0} (5.7
)

P ′ (iµ - k ) ≥ C 2 |k| -7 e π(k 4 -k 2 +2|k|) , k ∈ Z \ {-1, 0}.
(5.8)

Proof. Let us recall the asymptotic expression for µ ± k :

µ + k = -ik -1 -iO(|k| -1 ) + O(|k| -2 ), as |k| → ∞, µ - k = -k 4 + ik 3 + k 2 -1 -iO(|k| -1 ) + O(|k| -2
), as |k| → ∞. Now, we define the product

P 1 (z) = k∈Z\{-1} 1 - z iµ + k .
(5.9)

Lemma 5.6 (Young [START_REF] Young | An introduction to nonharmonic Fourier series[END_REF], Koosis [START_REF] Koosis | The logarithmic integral. I[END_REF], Rosier [START_REF] Martin | Null controllability of the structurally damped wave equation with moving control[END_REF]).

Let Λ k = k + d k , where d k = d + O(k -1 ), for k ∈ Z\{0} as |k| → ∞ for some constant d ∈ C, and that Λ k ̸ = Λ l for k ̸ = l. Then f (z) = k∈Z 1 -z Λ k
is an entire function of type sine.

Note that µ + 0 = -1 and iµ

+ k = k + d k , ∀ k ∈ Z \ {-1, 0}, where d k = -i + O(k -1 ) as |k| → ∞. Also, as λ + k ̸ = λ + l (see Remark 3.2), so µ + k ̸ = µ + l for k, l ∈ Z \ {-1} such that k ̸ = l.
Thanks to Lemma 5.6, we have P 1 is an entire function of sine type (in particular, of exponential type π) and so for any ϵ > 0 there exist positive constants

C 1 , C 2 , C 3 , C 4 , C 5 where C 2 , C 3 depends on ϵ, such that |P 1 (z)| ≤ C 1 e π|z| , z ∈ C, (5.10)
C 2 e π|y| ≤ |P 1 (x + iy)| ≤ C 3 e π|y| , if dist x + iy, {iµ + k } > ϵ, (5.11)
C 4 < P ′ 1 (iµ + k ) < C 5 , ∀k ∈ Z \ {-1}.
(5.12) Now using the above inequality (5.11) and continuity of P 1 , we get a positive constant C such that

|P 1 (x)| ≤ C, ∀ x ∈ R. (5.13)
Let arg z denote the principal argument of a complex number z ∈ C \ R -, i.e., arg z ∈ (-π, π). Then, we write

log z = log |z| + i arg z, n √ z = n |z| e i arg z n . (5.14) For k ∈ Z \ {-1, 0}, we define δk = sgn(k) 4 -µ - k = k + b k , where b k = -i 4 + O(k -1
) as |k| → ∞ and let δ0 = 1. Next, we define

P 2 (z) = k∈Z\{-1} 1 - z iµ - k , (5.15)
Q 1 (z) = k∈Z\{-1} 1 - z δk , (5.16
)

Q 2 (z) = k∈Z\{-1} 1 + z 4 µ - k .
(5.17) Thus, we get the following relations:

Q 2 (z) = Q 1 (z)Q 1 (-z)Q 1 (iz)Q 1 (-iz), (5.18)
P 2 (z) = Q 2 (e i π 8 4 √ z), (5.19)
P (z) = P 1 (z)P 2 (z) 1 + z i . (5.20)
Now as λ - k ̸ = λ - l (see Remark 3.2), so δk ̸ = δl for k, l ∈ Z \ {-1} such that k ̸ = l. So, by Lemma 5.6 we conclude that Q 1 is an entire function of sine type and hence, for any ϵ > 0 there exist positive constants C 6 , C 7 , C 8 , C 9 , where C 7 , C 8 depends on ϵ, such that

|Q 1 (z)| ≤C 6 e π|z| , (5.21)
C 7 e π|y| ≤ |Q 1 (x + iy)| ≤ C 8 e π|y| , dist(x + iy, { δk }) > ϵ, (5.22)
Q ′ 1 ( δk ) ≥C 9 , k ∈ Z \ {-1}. (5.23)
Now using the estimates for Q 1 and the expression (5.18), we get

|Q 2 (z)| ≤ C 1 e 4π|z| , C 1 e 2π(|x|+|y|) ≤ |Q 2 (x + iy)| ≤ C 2 e 2π(|x|+|y|) , if dist({±(x + iy), ±(-y + ix)}, { δk }) > ϵ. (5.24)
Substituting e i π 8 4

√ z in the place of z = x + iy in (5.24), we obtain

Ce 2π |ℜ(e i π 8 4 √ z)|+|ℑ(e i π 8 4 √ z)| ≤ |P 2 (z)| ≤ Ce 2π |ℜ(e i π 8 4 √ z)|+|ℑ(e i π 8 4 √ z)| , (5.25)
if dist {±e i π 8 4 √ z, ±e i 5π 8 4
√ z}, { δk } > ϵ. Further, using (5.14) we get

|P 2 (z)| ≤ Ce 4π 4 √ |z| . (5.26)
Now by (5.10) and (5.26) it follows that

|P (z)| = |P 1 (z)P 2 (z)| 1 + z i ≤ C e (π+ϵ)|z| e 4π 4 √ |z|-ϵ|z| |z + i| ≤ C(ϵ)e (π+ϵ)|z| .
Thus, P is an entire function of exponential type at most π. Note that

±e i π 8 4 √ x = ±e i π 8 4 √ x, x > 0 ±e i 3π 8 4 √ -x, x < 0 , ±e i 5π 8 4 √ x = ∓e i -3π 8 4 √ x, x > 0, ∓e i -π 8 4 √ -x, x < 0.
Thus, (5.25) holds for large x, and so using continuity of P 2 , we have:

|P 2 (x)| ≤ Ce 2π(| cos( π 8 )|+| sin( π 8 )|)|x| 1 4 , ∀x ∈ R. (5.27)
Combining (5.13), (5.20) and (5.27) we get the estimate (5.6):

|P (x)| ≤ C |x + i| -1 e 2π(cos( π 8 )+sin( π 8 ))|x| 1 4 , ∀x ∈ R. (5.28)
We now establish the estimates (5.7) and (5.8). Performing differentiation on (5.20) we get

P ′ (z) = P ′ 1 (z)P 2 (z) + P 1 (z)P ′ 2 (z) 1 + z i - iP 1 (z)P 2 (z) 1 + z i 2 . (5.29)
Using the fact that P 1 (iµ + k ) = 0 in (5.29), we get

P ′ (iµ + k ) = P ′ 1 (iµ + k )P 2 (iµ + k ) (1 + µ + k ) , k ∈ Z \ {-1, 0}. (5.30)
As

λ + k ̸ = λ - l , ∀k ∈ Z \ {-1, 0}, l ∈ Z \ {-1}(see Remark 3.
2), and also for large values of k, l, the gap between λ + k , λ - l increases, thus there exists a δ > 0, such that

|µ + k -µ - l | > δ. Now, ℜ e i π 8 4 iµ + k = cos π 8 |k| 1/4 -sin π 8 |k| -3/4 + O(|k| -1 ) , (5.31)
and ℑ e i π 8

4 iµ + k = -cos π 8 |k| -3/4 + sin π 8 |k| 1/4 + O(|k| -1 ) . (5.32)
Therefore using (5.25), (5.31), (5.32), one can get a constant C > 0 such that

P 2 (iµ + k ) ≥ Ce 2π ℜ e i π 8 4 √ iµ + k + ℑ e i π 8 4 √ iµ + k ≥ Ce 2π(cos( π 8 )+sin( π 8 ))|k| 1 4 . (5.33)
Thus combining the estimates (5.12) and (5.33) in the equality (5.30), we get the estimate (5.7):

P ′ (iµ + k ) ≥ C |1 + µ + k | e 2π(cos( π 8 )+sin( π 8 ))|k| 1 4 ≥ C|k| -1 e 2π(cos( π 8 )+sin( π 8 ))|k| 1 4 , ∀k ∈ Z \ {-1, 0}. (5.34)
Finally, we estimate |P ′ (iµ - k )|. From (5.29) and using the fact that P 2 (iµ - k ) = 0, we have

P ′ (iµ - k ) = P 1 (iµ - k )P ′ 2 (iµ - k ) (1 + µ - k ) . (5.35)
Let us recall that δk = sgn(k)e iπ 8

4 iµ - k are zeros of Q 1 and so using the relations (5.18), (5.19), we get

|P ′ 2 (iµ - k )| = |ie iπ 8 | 4 3/4 iµ - k -Q ′ 1 ( δk )Q 1 (i δk )Q 1 (-δk )Q 1 (-i δk ) . Note that λ - k ̸ = λ - m for k ∈ Z \ {-1, 0}, m ∈ Z \ {-1}, (see Remark 3.
2) and also the distance between them increases for large values of k, m, thus we get existence of a δ > 0 such that

| δk + δm | > δ, | δk + i δm | > δ, | δk -i δm | > δ,
and hence the estimates (5.22) and (5.23) lead to

|P ′ 2 (iµ - k )| ≥ C 1 |4 3/4 iµ - k | e 2π|k| .
It remains to compute the estimate of the term |P 1 (iµ - k )|. By (5.11) we have

|P 1 (iµ - k )| ≥ Ce π|(k 4 -k 2 -1-O(k -2 ))| ≥ Ce π(k 4 -k 2 )
. Finally, combining the last two estimate in (5.35), we obtain (5.8):

|P ′ (iµ - k )| ≥ C 3/4 iµ - k |1 + µ - k | e 2π|k| e π(k 4 -k 2 ) ≥ C|k| -7 e π(k 4 -k 2 +2|k|) , ∀k ∈ Z \ {-1, 0}. (5.36)
□ 5.2. Construction of the multiplier. In this section we construct an appropriate Beurling and Malliavin's multiplier M (see [START_REF] Beurling | On Fourier transforms of measures with compact support[END_REF] for details) which compensates the growth of P (see (5.6)). More precisely, we construct a multiplier M such that P (x)M (x) becomes bounded for all x ∈ R. Here we mainly follow the works [START_REF] López | Uniform null controllability of a fourth-order parabolic equation with a transport term[END_REF] and [START_REF] Martin | Null controllability of the structurally damped wave equation with moving control[END_REF] in which the Beurling and Malliavin's multiplier approach have been adopted. Let us denote a = T 2π -1 and let b be a positive real number to be chosen later. Proposition 5.7 (Estimate for the multiplier M ). There exist an entire function, M of exponential type at most aπ which satisfies the following bound for some C, C, C > 0:

|M (x)| ≤ C |x| exp -2π cos π 8 + sin π 8 |x| 1 4 , ∀x ∈ R, (5.37)
|M (iµ + k )| ≥ C|k| -3 exp -2π cos π 8 + sin π 8 |k| 1 4 , ∀k ∈ Z \ {-1, 0}, (5.38)
|M (iµ - k )| ≥ C |k| -8 exp πa(k 4 -k 2 ) -c|k| , ∀k ∈ Z \ {-1, 0}. (5.39)
Proof. The proof of the Proposition 5.7 contains two parts. In the first part, we describe the construction of multiplier, M. In the second part, we find the above-mentioned estimates (5.37), (5.38) and (5.39) of M , which is crucial for the construction of required biorthogonal family. Construction of the multiplier. At first, consider the function defined on the ray (0, ∞)

s(t) = at -bt 1 4 , t > 0.
For γ ∈ (0, 2), we have (see [START_REF] López | Uniform null controllability of a fourth-order parabolic equation with a transport term[END_REF]):

∞ 0 log 1 - x 2 t 2 dt γ = |x| γ π cot πγ 2 , ∀x ∈ R,
Thus, the above expression and the definition of s gives the following

∞ 0 log 1 - x 2 t 2 ds(t) = -b|x| 1 4 π cot π 8 ∀x ∈ R. (5.40)
Note that:

• s(B) = 0 for B = b a 4 3 , • s(t) is increasing for t > 1 4 4 3 B. Let us define ν(t) = 0, t ≤ B, s(t), t ≥ B.
Then, ν is non-negative and non-decreasing function. Let us define

g(z) = ∞ 0 log 1 - z 2 t 2 dν(t) = ∞ B log 1 - z 2 t 2 ds(t), z ∈ C \ R.
(5.41)

U (z) = ∞ 0 log 1 - z 2 t 2 dν(t) = ∞ B log 1 - z 2 t 2 ds(t), z ∈ C. (5.42) Note that, • g is holomorphic function on C \ R and U is continuous on C. • U (z) = ℜ(g(z)) for z ∈ C \ R as log(z) = log |z| + i arg(z).
• If we atomize the measure dν (i.e., to make the measure as integer valued), then the integral g would become logarithm of an entire function of exponential type and hence taking exponential of g would give us entire function of exponential type. Thus, let us define Estimates on the multiplier. Now, we find desired upper and lower bounds for Ũ , which is done in two steps, to get the desired estimates of M .

g(z) = ∞ 0 log 1 - z 2 t 2 d[ν(t)] = ∞ B log 1 - z 2 t 2 d[s(t)], z ∈ C \ R, Ũ (z) = ∞ 0 log 1 - z 2 t 2 d[ν(t)] = ∞ B log 1 - z 2 t 2 d[s(t)], z ∈ C,
Step1. Estimating U : Lemma 5.8 (see [START_REF] López | Uniform null controllability of a fourth-order parabolic equation with a transport term[END_REF]). Let U be defined by (5.42). Then there exists θ ∈ L ∞ (R + ) such that the following identity holds

U (x) + b π cot π 8 4 |x| = -aBθ |x| B , x ∈ R. (5.43)
For the sake of completeness we give sketch of the proof of this lemma in the Appendix A (Section 6).

Lemma 5.9 (see [START_REF] López | Uniform null controllability of a fourth-order parabolic equation with a transport term[END_REF], Lemma 17). The function U defined in (5.42) satisfies the following relation

U (z) = |ℑ(z)| πa + 1 π ∞ -∞ U (t) |z -t| 2 , z ∈ C \ R.
(5.44)

Without loss of generality, we assume y = ℑ(z) > 0. Then plugging (5.43) in the second term of the right hand side of (5.44) we obtain

y π ∞ -∞ U (t) |z -t| 2 dt = - y π ∞ -∞ b π cot π 8 (x -t) 2 + y 2 |t| 1 4 dt - y π aB ∞ -∞ θ |t| B (x -t) 2 + y 2 dt. (5.45)
Then the second term of (5.45) can be estimated as

y π aB ∞ -∞ θ |t| B (x -t) 2 + y 2 dt ≤ ||θ|| ∞ aB y π ∞ -∞ ds y x y -s 2 + 1 = aB||θ|| ∞ . (5.46)
The first term of (5.45) can be simplified as

y π ∞ -∞ (-b π) cot π 8 (x -t) 2 + y 2 |t| 1 4 dt = -b y cot π 8 ∞ -∞ |y| 1 4 |s| 1 4 (x -ys) 2 + y 2 y ds = -b cot π 8 y 1 4 ∞ -∞ |s| 1 4 x y -s 2 + 1
ds.

(5.47)

We have the following identity (details can be found in Appendix B (Section 7))

T = y 1 4 ∞ -∞ |s| 1 4 x y -s 2 + 1 ds = π √ 2(y 2 + x 2 ) 1 4 + |x| + y 2 + x 2 2 √ 2 + sgn(x) π( √ 2 -1) (y 2 + x 2 ) -|x| 2 √ 2 √ 2(y 2 + x 2 ) 1 4 + |x| + y 2 + x 2 .
(5.48)

Elementary computation gives

π|x| 1 4 ≤ T ≤ π|x| 1 4 + π √ 2 + 1 2 √ 2 4 |y| + π( √ 2 -1) 2 √ 2 √ 2 + 1
-b π cot π 8 |x| 1 4 + C |y| 1 4 -aB||θ|| ∞ ≤ y π ∞ ∞ U (t) |z -t| 2 dt ≤ -b π cot π 8 |x| 1 4 + aB||θ|| ∞ . (5.50)
Finally, using the relation (5.44) of U (z) we get:

-C b π cot π 8 |y| 1 4 -aB||θ|| ∞ ≤ U (x + iy) -πa|y| + b π cot π 8 |x| 1 4 ≤ aB||θ|| ∞ .
(5.51)

Step2. Estimate for Ũ : We first find the bounds on Ũ -U . This estimate along with that of U will give the required estimates for Ũ and hence for the multiplier M .

Lemma 5.10 (see [START_REF] Martin | Null controllability of the structurally damped wave equation with moving control[END_REF]). Let ν : R + → R + be nondecreasing and null on (0, B). Then for z ∈ C \ R, we have:

-log + |x| |y| -log + x 2 + y 2 B 2 -log 2 ≤ Ũ (z) -U (z) ≤ log + |x| |y| . (5.52)
Using (5.51) and the above lemma, we can obtain the bounds for Ũ , which is given as: (5.54) Thus, we prove the desired estimate (5.37).

-b π cot π 8 |x| 1 4 + C |y| 1 4 -aB||θ|| ∞ + πa|y|-log + |x| |y| -log + x 2 + y 2 B 2 -log 2 ≤ Ũ (x + iy) ≤ -b π cot π 8 |x|
The remaining task is to find the upper bounds of M at iµ + k and iµ - k for k ∈ Z\{-1, 0} as mentioned in (5.38) and (5.39). Recall

µ + k = -1 -ik -iO(|k| -1 ) + O(|k| -2 ), µ - k = -k 4 + ik 3 + k 2 -1 -iO(|k| -1 ) + O(|k| -2 ). Plugging µ + k in (5.53), we write Ũ (-iµ + k -i) = Ũ ((-k -O(k -1 )) + i(-2 -O(k -2 ))
). and thus for k ∈ Z \ {-1, 0} using (5.54), we obtain

|M (-iµ + k )| = exp( Ũ (-iµ + k -i)) ≥ C exp -b cot π 8 π (|k| 1 4 + C|2 + O(k -2 )| 1 4 ) -aB||θ|| ∞ + aπ|2 + O(k -2 )| exp -log + c|k| -log + ck 2 , for some c > 0 ≥ C |k| -3 exp -2π cos π 8 + sin π 8 |k| 1 4
.

(5.55)

Next we estimate for the parabolic eigen-element part. We write Ũ (-iµ -

k -i) = Ũ ((k 3 -O(k -1 )) + i(k 4 -k 2 -2 -O(k -2 ))
). Inserting this expression in (5.54) for k ∈ Z \ {-1, 0}, we find the following

|M (-iµ - k )| = exp Ũ (-iµ - k -i) ≥ C exp -bπ cot π 8 (|k| 3 4 + C(k 4 -k 2 ) 1 4 ) + πa(k 4 -k 2 ) -log + (k 8 ) ≥ C exp πa(k 4 -k 2 ) -cπ|k| |k| -8
, for some c > 0.

(5.56) □ 5.3. Proof of Proposition 3.9. For k ∈ Z \ {-1, 0}, define

Ψ ± k (z) = P (z) P ′ (iµ ± k )(z -iµ ± k ) M (z) M (iµ ± k ) , and
Ψ 0 (z) = P (z) P ′ (-i)(z + i) M (z) M (-i) .
Clearly, Ψ k is an entire function of exponential type at most π + aπ = T 2 and satisfies

Ψ ± k (iµ ± l ) = δ kl δ ± , ∀ l ∈ Z \ {-1} and k ∈ Z \ {-1, 0}, Ψ 0 (iµ ± l ) = δ 0l , ∀ l ∈ Z \ {-1}. It can be seen that |P (x)M (x)| ≤ C.
For k ∈ Z \ {-1, 0} and x ∈ R, we obtain:

|Ψ + k (x)| ≤ C|k| 4 |x| |x -i| |x -k + i| ≤ C|k| 4 1 |x -k + i| , |Ψ - k (x)| ≤ C|k| 15 e -T 2 (k 4 -k 2 ) |x| |x -i|(|x + k 3 | + |k 4 -2k 2 |) , for some C, C > 0, which shows that Ψ + k , Ψ - k ∈ L 2 (R) with ||Ψ + k || L 2 (R) ≤ C |k| 4 , ||Ψ - k || L 2 (R) ≤ C|k| 15 e -T 2 (k 4 -k 2 ) .
Similarly,

|Ψ 0 (x)| ≤ C |x + i| ,
and so Ψ 0 ∈ L 2 (R) as well.

As discussed in the beginning of this section, existence of the biorthogonal family {Θ ± k , Θ 0 } k∈Z\{-1,0} are immediate as the Fourier transform of Ψ ± k and Ψ 0 . Then, Proof. First we write the expression of U (5.42) restricted on the real line :

{Θ ± k , Θ 0 } k∈Z\{-1,0} ⊂ L 2 (R) with ||Θ + k || L 2 (R) ≤ C |k| 4 , ||Θ - k || L 2 (R) ≤ C|k| 15 e -T 2 (k 4 -k 2 ) , ||Θ 0 || L 2 (R) ≤ C.
U (x) = ∞ 0 log 1 - x 2 t 2 ds(t) - B 0 log 1 - x 2 t 2 ds(t), x ∈ R.
Thus using (5.40), we obtain

U (x) + b π cot π 8 4 |x| = - B 0 log 1 - x 2 t 2 ds(t), x ∈ R.
Applying a change of variable t → t B , we have

B 0 log 1 - x 2 t 2 ds(t) = aB 1 0 log 1 - x 2 B 2 t 2 d(t -t 1/4 ).
Let us define the following function in (0, ∞)

θ(x) = 1 0 log 1 - x 2 t 2 d(t -t 1/4 ).
One can show θ ∈ L ∞ (0, ∞) (see [START_REF] López | Uniform null controllability of a fourth-order parabolic equation with a transport term[END_REF], Lemma 16 for details). A direct computation conclude the proof of Lemma 5.8. □ 7. Appendix B. Details of the integration 5.48

Let us denote

J = ∞ -∞ |s| 1 4 (x -s) 2 + 1 ds = 0 -∞ (-s) 1 4 (x -s) 2 + 1 ds + ∞ 0 (s) 1 4
(x -s) 2 + 1 ds.

Using change of variable u = (-s) 1 4 and u = (s) 1 4 in the integrals, we get

J = ∞ 0 4u 4 1 + (x + u 4 ) 2 ds + ∞ 0 4u 4 1 + (x -u 4 ) 2 ds =: 4 (I 1 + I 2 ).
Using Residue theorem, we get:

I 1 + e iπ/4 I 2 = 2πiR 1 , (7.1)
I 2 + e iπ/4 I 1 = 2πiR 2 , (7.2)
where

R 1 = lim z→(1+x 2 ) 1/8 e iα/4 z -(1 + x 2 ) 1/8 e iα/4 z 4 1 + (x + z 4 ) 2 = (1 + x 2 ) 1/8 8[x + (1 + x 2 ) 1/2 e iα ]e -iα/4 , R 2 = lim z→(1+x 2 ) 1/8 e iθ/4 z -(1 + x 2 ) 1/8 e iθ/4 z 4 1 + (x -z 4 ) 2 = - (1 + x 2 ) 1/8 8[x -(1 + x 2 ) 1/2 e iθ ]e -iθ/4 , with θ = π -α = arctan 1
x . Solving the above system of linear equations (7.1)-(7.2) in I 1 , I 2 , we get:

I 1 = π(1 + x 2 ) 1/8 4 x sin θ 4 + cos θ 4 + 1 + x 2 sin 3θ 4 -cos 3θ 4
I 2 = √ 2π(1 + x 2 ) 1/8 4 -x cos θ 4 + 1 + x 2 cos 3θ 4 .
with periodic boundary conditions. The following analysis has been adopted from [START_REF] Girinon | Quelques problèmes aux limites pour les équations de Navier-Stokes[END_REF], where the author has shown the maximality of the corresponding spatial operator for the linearized compresseble Navier-Stokes equation (1.8) with Dirichlet boundary conditions. We split the proof of the existence of such (u, η) in the following two steps:

Step(1): First, we consider the following system for ϵ > 0:

               u xxxx + u xxx + u xx + 2u -η x = f, -ϵη xx + η x + 2η -u x = g, u(0) = u(2π), u x (0) = u x (2π), u xx (0) = u xx (2π), u xxx (0) = u xxx (2π), η(0) = η(2π), η x (0) = η x (2π). (8.2) Let V = H 2 p (0, 2π) × H 1 p (0, 2π
). Then we define a continuous sesquilinear form S :

V × V → C as S u η , ρ σ = 2π 0 u xx ρ xx - 2π 0 u xx ρ x + 2π 0 u xx ρ + 2 2π 0 uρ - 2π 0 η x ρ +ϵ 2π 0 η x σ x + 2π 0 η x σ + 2 2π 0 ησ - 2π 0 u x σ.
Also, we define a continuous antilinear map F on V as

F u η = 2π 0 f u + 2π 0 gη. Now, we compute ℜ S u η , u η = 2π 0 |u xx | 2 -ℜ 2π 0 u xx u x + ℜ 2π 0 u xx u + 2 2π 0 |u| 2 -ℜ 2π 0 η x u + ϵ 2π 0 |η x | 2 + ℜ 2π 0 η x η + 2 2π 0 |η| 2 -ℜ 2π 0 u x η = 2π 0 |u xx | 2 + ℜ 2π 0 u xx u + 2 2π 0 |u| 2 + ϵ 2π 0 |η x | 2 + 2 2π 0 |η| 2 ≥ 2π 0 |u xx | 2 - 1 2 2π 0 |u xx | 2 - 1 2 2π 0 |u| 2 + 2 2π 0 |u| 2 + ϵ 2π 0 |η x | 2 + 2 2π 0 |η| 2 (8.3) ≥ α u η V
, for some α > 0.

In the last step, we have used the equivalence of the norm ||u|| H 2 with ||u|| L 2 + ||u xx || L 2 . Thus S is coercive and continuous, so by Lax -Milgram theorem (see [START_REF] Dautray | Mathematical analysis and numerical methods for science and technology[END_REF]: Theorem 1, Chapter VII), we get the existence of a unique u ϵ η ϵ ∈ V such that:

S u ϵ η ϵ , ρ σ = F ρ σ , ∀ ρ σ ∈ V. (8.4)
Also, we have

ℜ S u ϵ η ϵ , u ϵ η ϵ ≤ 1 2 2π 0 (|f | 2 + |u ϵ | 2 ) + 1 2 2π 0 (|g| 2 + |η ϵ | 2 ).
Further, using the above calculation (8.3) we get

1 2 2π 0 |u ϵ xx | 2 + 2π 0 |u ϵ | 2 + ϵ 2π 0 |η ϵ x | 2 + 3 2 2π 0 |η ϵ | 2 ≤ 1 2 2π 0 |f | 2 + 1 2 2π 0 |g| 2 < ∞,
which implies {u ϵ } is bounded in H 2 (0, 2π), and {η ϵ }, { √ ϵ η ϵ x } are bounded in L 2 (0, 2π). So, we get:

u ϵ ⇀ u in H 2 (0, 2π), η ϵ ⇀ η in L 2 (0, 2π), || √ ϵη ϵ x || L 2 (0,2π) → 0 as ϵ → 0.
Step

η ϵ x σ x - 2π 0 v ϵ σ x + 2 2π 0 η ϵ σ - 2π 0 u ϵ x σ = 2π 0 gσ. (8.8) Now, as u ϵ η ϵ ∈ V = H 2 p (0, 2π) × H 1 p (0, 2π), so we get u(0) = u(2π), u x (0) = u x (2π).
From (8.5), we have 2π 0

(u ϵ xx + u ϵ x )ρ xx = 2π 0 (f -u ϵ xx -2u ϵ + η ϵ x )ρ. Restricting ρ on C ∞ c (0, 2π), we get u ϵ xxxx + u ϵ xxx + u ϵ xx + 2u ϵ -η ϵ x = f (8.13)
in sense of distribution. As argued above for (8.12), we get u ϵ ∈ H 4 (0, 2π). On multiplying (8.13) by u ϵ xx and then integrating on (0, 2π), we get Proof. Let us assume that h, u 0 , η 0 to be smooth enough. Then we multiply the first and second equation of system (1.16) by u and η, respectively and then integrating over (0, 2π), we get: Proof. To prove this proposition, we will need the following result:

Lemma 8.1. For any (φ 0 , ψ 0 ) ∈ H 3 p (0, 2π)×L 2 (0, 2π), the solution of system (2.4), (φ, ψ) ∈ X satisfies: ||φ xx (•, 0)|| L 2 (0,T ) + ||φ xxx (•, 0)|| L 2 (0,T ) ≤ C ||(φ 0 , ψ 0 )|| H 3 p (0,2π)×L 2 (0,2π) . Assuming this lemma for the time being, let us first prove the proposition. For each t ∈ [0, T ], consider the system (2.4) in (0, t) × (0, 2π), with the initial data φ(0, •) ψ(0, •) = φ 0 ψ 0 ∈ H 3 p (0, 2π) × L 2 (0, 2π). By Proposition 2.6, there exist a unique solution φ(s, x) ψ(s, x) ∈ C([0, t]; Z) ∩ L 2 (0, t; H 2 p (0, 2π) × L 2 (0, 2π)). Next, for each t ∈ [0, T ] we define a linear functional F t : H 3 p (0, 2π) × L 2 (0, 2π) → C given by: φ 0 ψ 0 → . □

Figure 1 .

 1 Figure 1. Hyperbolic (λ + k ) and parabolic (λ - k ) branching of eigenvalues of A *

Figure 2 .

 2 Figure 2. A closer look at the eigenvalues

Figure 3 .

 3 Figure 3. Plot of ξ ± k for first few k symmetric about 0

  5) as φ0 (t, x) ψ0 (t, x) = e -(T -t) 0 1 and thus from identity (3.18) we get u(T, •) η(T, •) , φ T ψ T Z + e -T 2π 0 η 0 = 0. (3.22) Now assume u(T, •) = 0 and η(T, •) = 0, then (3.19), (3.21) and (3.22) becomes:

-

  (-ik 3 + k 2 + ik)(1 + λ + k ) + ξ + k ̸ = 0

  which is true for large |k|, and for first finite number of k ̸ = 0, one can check from the Figure5.

k 4 +

 4 ik 3 -k 2 + ik + 2(1 + λ + k )

Definition 5 . 1 (

 51 Entire functions of exponential type). An entire function f is said to be of exponential type A if there exist positive constant B such that|f (z)| ≤ Be A|z| , z ∈ C,and of exponential type at most A if for any ϵ > 0, there exits B ϵ > 0 such that |f (z)| ≤ B ϵ e (A+ϵ)|z| , z ∈ C. Definition 5.2. (Sine type function) An entire function f of exponential type π is said to be of sine type if (i) the zeros of f (z), say µ k satisfies gap condition, i.e., there exist δ > 0 such that |µ k -µ l | > δ for k ̸ = l, and (ii) there exist positive constants C 1 ,C 2 and C 3 such that C 1 e π|y| ≤ |f (x + iy)| ≤ C 2 e π|y| , ∀ x, y ∈ R with |y| ≥ C 3 .

  and λ ± k are given by (3.10) nd(3.11). Let us now state the celebrated Paley-Wiener theorem, from which we can conclude about the desired biorthogonal family using the family E.

Theorem 5 . 4 (

 54 Paley-Wiener). Let f be an entire function of exponential type A and suppose∞ -∞ |f (x)| 2 dx < ∞.Then there exists a function ϕ ∈ L 2 (-A, A) with the followingf (z) = A -A e izt ϕ(t)dt, z ∈ C.Thus applying the Paley-Wiener theorem for the class E of entire functions, one can get a family of functions

1 - z 2 τ 2 k,

 12 where the notation [x] is for the integral part of x. Then g is holomorphic function on C \ R and Ũ is continuous on C. Thus, we can write exp(g(z)) = k∈N for z ∈ C \ R. where τ k 's are the point of discontinuity of the map t → [v(t)]. Now, let us define the multiplier M as M (z) = exp(g(z -i)). Then, for z ∈ C we have |M (z)| = exp( Ũ (z -i)) as Ũ (z -i) = ℜ(g(z -i)).

1 4 ++ sin π 8 |x| 1 4 8

 418 aB||θ|| ∞ + log + |x| |y| + πa|y|. (5.53) Now, we are in position to find the estimate of M . Putting y = -1 in the above inequality and choosing |M (x)| = exp( Ũ (x -i)) ≤ C exp -2π cos π 8 + log + (|x|) ≤ C |x| exp -2π cos π

6 .

 6 Appendix A. Proof ofLemma 5.8

  2): In this step, we show u η ∈ D(A) and solves (8

0 u 15 )u 19 ) 8 . 2 .

 0151982 {u ϵ } is bounded in H 3 (0, 2π) and so u ϵ ⇀ u in H 3 (0, 2π). So, (8.5) can be rewritten as -Also, on multiplying (8.12) by ρ ∈ H 2 p (0, 2π) and then integrating on (0, 2π), we get:-2π xxx ρ x + u xxx ρ |On comparing (8.14) and (8.15), we get u xxx (0)ρ(0) = u xxx (2π)ρ(2π) and so u xxx (0) = u xxx (2π). Thus, xx ρ xx -ρ x u xx | Again comparing (8.18) and (8.19), we get u xx (0) = u xx (2π). Hence, u η ∈ D(A) and solves (8.1). □ Proof of Proposition 2.4.

2 2 8 . 3 .

 2283 Multiplying both sides of the above equation by e -3s and then integrating with respect to s over (0, t) ⊂ (0, T ), we obtain Taking supremum in t over [0, T ], we obtain the estimate (2.3) for smooth solution. Next using the classical density argument, we get the desired result.□ Proof of Proposition 2.8.

φ 3 p 8 . 3 . 1 . 0 φ xxt φ xx + 2π 0 |φ xxxx | 2 -ℜ 2π 0 φ xxx φ xxxx + ℜ 2π 0 φ xx φ xxxx -ℜ 2π 0 ψδ 2π 0 |φ xxxx | 2 - 0 |φ| 2 -|ψ x | 2 . 2 .H 3 pψ 0 H 3 p

 38310000002022233 xxx (t -s, 0) -φ xx (t -s, 0) -φ x (t -s, 0) + ψ(t -s, 0) q(s) ds Now, using (2.6) and above lemma 8.1, one can conclude that the functional F t is continuous and hence by Riesz representation theorem, there exist a unique u(s, •) η(s, •) ∈ L 2 (0, 2π) × L 2 (0, 2π) such that equation (2.7) holds and hence unique solution exists. Moreover, we have: (0,2π)×L 2 (0,2π)for each φ 0 ψ 0 ∈ H 3 p (0, 2π) × L 2 (0, 2π). This completes the proof of proposition. □ Proof of Lemma 8.1. Multiplying the first equation of (2.4) by φ xxxx and then integrating on (0, 2π), we getℜ 2π x φ xxxx = 0. Using ab > -a 2 2 -b 2 2 and -ab > -a 2 2 -b 22 the above inequality can be written as: H m (0, 2π) → H m-1 (0, 2π) compactly for m ∈ N, so by Ehrling's lemma we get existence c(ϵ) > 0 for any ϵ > 0 such that||φ xxx || 2 L 2 (0,2π) ≤ ϵ||φ xxxx || 2 L 2 (0,2π) + c(ϵ)||φ|| 2 L 2 (0,2π) . (8.21) Using (8.21) in (8.20), we get:d dt 2π 0 |φ xx | 2 + 2 -δϵ -3 δ 2π 0 |φ xxxx | 2 -δ c(ϵ) 2π 0 |φ| 2 -δ 2π 0 |φ xx | 2 -δ 2π 0 |ψ x | 2 ≤ 0.We then multiply the above inequality by e -δt to getd dt e -δt 2π 0 |φ xx | 2 + e -δt 2 -δϵ -3 δ c(ϵ)e -δt 2π δe -δt 2π 0 |ψ x | 2 ≤ 0.Integrating w.r.t t over (0, s) ⊂ (0, T ), we gete -δs2π 0 |φ xx (s, •)| 2 + e -δs 2 -δϵ -Now choosing ϵ = 3 δ 2 > 0, for large enough δ and taking supremum over s ∈ [0, T ], we get: Using Poincaré inequality and (2.5) together with the above inequality, one can easily conclude ||φ|| L 2 (0,T ;H 4 (0,2π)) ≤ C φ 0 ψ 0 (0,2π)×L 2 (0,2π) . Further, using the trace regularity we get the desired estimate ||φ xx (•, 0)|| L 2 (0,T ) + ||φ xxx (•, 0)|| L 2 (0,T ) ≤ C φ 0 (0,2π)×L 2 (0,2π)

  Proof. Considering the moment problem (3.24) with f as in Proposition 4.1 and noting the fact that θ ± k ̸ = 0 (see Remark 3.3), let us define g as

		2
		.
	As 13+ϵ 2	> 13 2 so by (4.2),
		k∈Z\{0,±1}

(|a k | 2 + |k| 6 |b k | 2 )|k| 13+ϵ < ∞, and also k∈Z\{0,±1} |k| -1-ϵ < ∞.

Thus, g ∈ L 2 (0, T ) and hence solves the moment problem

(3.23)

. □ 4.2. Proof of Theorem 1.3.

  Proof. Again proving this theorem is equivalent to solving the moment problem(3.26). Let us now define

					sin 2 kρπ 2	-i 2 sin	kρπ 2	cos	kρπ 2
		= 2	sin kρπ 2 |k|	.	(4.4)
	Note that,
		sin 2 x ≥	4x 2 π 2 , for x ∈ -	π 2	,	π 2	.	(4.5)
	Case I. If 0 ≤ kρπ 2 -pπ ≤ π 2 , then we have
	sin 2 kρπ 2	= sin 2 kρπ 2	-pπ ≥ k 2 ρ -	2p k	2	, by (4.5),
								≥ k 2 C k 4 =	C k 2 , by (4.3).
	Case II. If π 2 ≤ kρπ 2 -pπ ≤ π, i.e., -π 2 ≤ kρπ 2 -(p + 1)π ≤ 0, then we have
	sin 2 kρπ 2	= sin 2 kρπ 2	-(p + 1)π ≥	C k 2 , by last case.
	Combining the above two cases, we have
		sin 2 π 2	kρ ≥	C 2 k

For any fixed

k ∈ Z \ {0}, choose p ∈ Z such that 0 ≤ kρπ 2 -pπ ≤ π. 2 , ∀ k ∈ Z \ {0}. (

4

.6) Plugging the estimate (4.6) in (4.4), we obtain |f k | ≥ C |k| 2 for some C > 0, k ∈ Z \ {0}. Hence we have, |f k | ≥ C |k| 2 for k ∈ Z \ {0}. □ 4.3. Proof of Theorem 1.4.

Acknowledgements

We are indebted to Prof. Lionel Rossier for his suggestion regarding spectral behaviour of our system. Manish kumar acknowledges financial support from Prime Minister Research Fellowship, Govt. of India. Subrata Majumdar acknowledges financial supports from the Department of Atomic Energy and National Board for Higher Mathematics Fellowship, Grant No. 0203/16(21)/2018-R&D-II/10708.

, x ∈ R, θ ∈ -π 2 , π 2 , we have the following

and

Thus, from (7.3) we have

This section is devoted to the proofs of the well posedness results stated in the Section 2. We prove the results for the control systems corresponding to the transformed system (3.1), which is given as (3.2). Then, using the transformation (u, η) → e t (u, η), we get the well-posedness of the original control systems as considered in the propositions of Section 2.

8.1. Proof of Proposition 2.1.

Proof. We first prove A to be an infinitesimal generator of a continuous semigroup. To apply Hille-Yosida theorem it is enough to show that A is maximal dissipative.

Maximality of A: We need to show that for any given f g ∈ Z, there exists u η ∈ D(A) such that:

which is equivalent to the following

Let ϵ → 0 (8.7) and (8.8), then we have Now restricting σ on C ∞ c (0, 2π), we have

and so we have:

in the sense of distribution. Now, we multiply (8.11) by σ ∈ H 1 p (0, 2π) and then integrate on (0, 2π) to get

Using integration by parts in (8.9), we get:

On comparing the last two equations, we get η(0)σ(0) = η(2π)σ(2π) which gives η(0) = η(2π). Using η(0) = η(2π) and the fact that ρ ∈ H 2 p (0, 2π) in (8.9), we get

Now, from (8.11) we have η x ∈ L 2 (0, 2π) and so we get η ∈ H 1 (0, 2π). Also, from (8.12) we have

) which in particular means u xx + u x ∈ H 1 (0, 2π), and also

For m ∈ N, H m (0, 2π) -→ H m-1 (0, 2π) compactly and also H 1 (0, 2π) -→ C([0, 2π]) compactly, so u ϵ ⇀ u in H 2 (0, 2π) gives: u ϵ → u in H 1 (0, 2π) and u ϵ

x → u x in C([0, 2π]).

So, u ϵ x (0) u ϵ x (2π) → u x (0) u x (2π) . Also, u ϵ (0) u ϵ (2π) → u(0) u(2π) by continuity of trace operator.


Manya V Afonso
email: mafonso@lx.it.pt

José M Bioucas-Dias

Mário A T Figueiredo
email: mario.figueiredo@lx.it.pt

Image Restoration with Compound Regularization Using a Bregman Iterative Algorithm

Some imaging inverse problems may require the solution to simultaneously exhibit properties that are not enforceable by a single regularizer. One way to attain this goal is to use a linear combinations of regularizers, thus encouraging the solution to simultaneously exhibit the characteristics enforced by each individual regularizer. In this paper, we address the optimization problem resulting from this type of compound regularization using the split Bregman iterative method. The resulting algorithm only requires the ability to efficiently compute the denoising operator associated to each involved regularizer. Convergence is guaranteed by the theory behind the Bregman iterative approach to solving constrained optimization problems. In experiments with images that are simultaneously sparse and piece-wise smooth, the proposed algorithm successfully solves the deconvolution problem with a compound regularizer that is the linear combination of the ℓ 1 and total variation (TV) regularizers. The lowest MSE obtained with the (ℓ 1+TV) regularizer is lower than that obtained with TV or ℓ 1 alone, for any value of the corresponding regularization parameters.

I. INTRODUCTION

Linear inverse problems involve estimating an unknown signal/image with certain characteristics (such as sparseness or piece-wise smoothness) enforced by a suitable regularizer. In several problems such as image denoising, image restoration [START_REF] Figueiredo | An EM algorithm for wavelet-based image restoration[END_REF], [START_REF] Bioucas-Dias | A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration[END_REF], image reconstruction, and some formulations of compressed sensing [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF], [START_REF] Donoho | Compressed sensing[END_REF], [START_REF] Haupt | Signal reconstruction from noisy random projections[END_REF], the solution is defined as the minimizer of an objective function, leading to an optimization problem of the form û = arg min

u∈R n 1 2 Au -f 2 2 + τ Φ(u), (1)
where A : R n → R m is the linear observation (direct) operator, f ∈ R m is the observed data, Φ : R n → R is the regularizer function, and τ ∈ [0, +∞[ is the regularization parameter.

If the operator A is the identity, (1) is a denoising problem, the solution of which is unique (if Φ is convex) and called the Moreau proximal mapping (MPM) of Φ [START_REF] Combettes | Signal recovery by proximal forwardbackward splitting[END_REF]. For some choices of Φ, the MPM has a simple closed form; e.g., if Φ(u) = u 1 = i |u i |, the MPM is the well-known softthreshold function [START_REF] Donoho | De-noising by soft thresholding[END_REF]. In other cases, even if no closed form is available, the MPM can be efficiently computed [START_REF] Combettes | Signal recovery by proximal forwardbackward splitting[END_REF]; e.g., when Φ(u) is separable, that is, Φ(u) = i φ i (u i ).

For general non-diagonal operators, (1) has to be solved using iterative algorithms, such as iterative shrinkagethresholding (IST) [START_REF] Figueiredo | Majorizationminimization algorithms for wavelet-based image restoration[END_REF], also known as forward-backward splitting [START_REF] Combettes | Signal recovery by proximal forwardbackward splitting[END_REF], or faster versions of IST such as two-step IST (TwIST) [START_REF] Bioucas-Dias | A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration[END_REF], in which the MPM is iteratively applied.

In certain problems, it may be desirable to favor solutions that simultaneously exhibit properties that are enforced by two (or more) different regularizers. For example, total variation (TV) regularization applied to images [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF], encourages piecewise smooth solutions, while an l 1 (or l p , with p ≤ 1) regularizer favors sparse solutions; however, there is no "simple" regularizer that favors both these characteristics simultaneously, as may be important in certain problems. To achieve this goal, compound regularizers (i.e., linear combinations of "simple" regularizers [START_REF] Bioucas-Dias | An iterative algorithm for linear inverse problems with compound regularizers[END_REF]) must be used, leading to problems with the form û = arg min

u 1 2 Au -f 2 2 + τ 1 Φ 1 (u) + τ 2 Φ 2 (u), (2)
where Φ 1 : R n → R and Φ 2 : R n → R are the regularizers, with respective regularization parameters τ 1 > 0 and τ 2 > 0. An iterative algorithm for solving (2) (which is easily generalizable to more than two regularizers) has been recently proposed [START_REF] Bioucas-Dias | An iterative algorithm for linear inverse problems with compound regularizers[END_REF]; that approach involves a constrained optimization formulation of (2) followed by minimization of the associated Lagrangian using a block-coordinate descent (also known as alternating minimization) algorithm. A similar formulation, specifically tailored for regularizers which can be written as ℓ 1 norms (such as the ℓ 1 norm itself and anisotropic TV) was also very recently proposed [START_REF] Goldstein | The split Bregman method for L1 regularized problems[END_REF]; in that work, the constrained problem is attacked using a so-called split Bregman method.

In [START_REF] Malgouyres | Minimizing the total variation under a general convex constraint for image restoration[END_REF], a hybrid framework has been considered, uniting wavelet thresholding methods (focussing on the data fidelity term), and the total variation regularizer. A similar formulation has been considered in [START_REF] Durand | Reconstruction of wavelet coefficients using total variation minimization[END_REF], with the denoising problem solved by an algorithm based on a subgradient descent combining a projection on a linear space. A deconvolution problem formulated as the minimization of a convex functional with a data-fidelity term reflecting the noise properties, a non-smooth sparsity-promoting penalty over the image representation coefficients, and another term to ensure positivity of the restored image has been treated in [START_REF] Dupé | A proximal iteration for deconvolving Poisson noisy images using sparse representations[END_REF], with a forward-backward splitting algorithm used to solve the minimization problem. In [START_REF] Chaux | A forward-backward algorithm for image restoration with sparse representations[END_REF], a generic image deconvolution problem in Hilbert spaces using more than one regularizer terms has been analysed and a flexible forwardbackward algorithm for solving it has been presented.

In this paper, we propose an approach for solving problems of the form (2) involving any regularizers for which the Moreau proximal mappings are known (not just ℓ 1 norms). As in [START_REF] Bioucas-Dias | An iterative algorithm for linear inverse problems with compound regularizers[END_REF], the approach involves a constrained optimization formulation of (2), which is then directly addressed using a Bregman iterative method [START_REF] Yin | Bregman iterative algorithms for ℓ 1 -minimization with applications to compressed sensing[END_REF]. In this paper, we illustrate this approach in the problem of deconvolving an image which is known to have a few white blobs on a black background; such an image is characterized by having a low ℓ 1 norm (it's mostly black, i.e., sparse) and a low TV norm (it's piecewise flat). Using a combination of ℓ 1 and TV regularizers, we show the ability of the algorithm to solve the resulting problem and also that the resulting estimates have lower MSE than what can be achieved using each of the two regularizers alone.

II. PROPOSED METHOD

A. Constrained Formulation

A constrained optimization problem equivalent to the unconstrained problem (2) is min u,v 1 2 Au -f 2 2 + τ 1 Φ 1 (u) + τ 2 Φ 2 (v) subject to u -v 2 2 = 0. ( 3
)
One approach to handling this constrained problem is to consider its Lagrangian and minimize it using a blockdescent algorithm [START_REF] Bioucas-Dias | An iterative algorithm for linear inverse problems with compound regularizers[END_REF]. However, an extremely large value of the Lagrange multiplier is required for the minimizer of the Lagrangian to closely approximate the solution of (3), causing numerical difficulties. The alternative herein proposed is to use a split Bregman iterative method to directly solve (3).

The following subsections will briefly review the basic concepts of Bregman and split Bregman iterations, before we describe how they are used to solve (3).

B. Bregman Iterations

Consider a constrained problem of the form

min x E(x) subject to H(x) = 0, (4)
with E and H convex, H differentiable, and min x H(x) = 0 (see [START_REF] Goldstein | The split Bregman method for L1 regularized problems[END_REF], [START_REF] Yin | Bregman iterative algorithms for ℓ 1 -minimization with applications to compressed sensing[END_REF], for more details). The so-called Bregman divergence associated with the convex function E is defined as

D p E (x, y) ≡ E(x) -E(y) -p, x -y , (5)
where p belongs to the subgradient of E at y i.e., p ∈ ∂E(y) = {u : E(x) ≥ E(y)+ u, x-y , ∀x ∈ domE}.

Notice that by letting x = [u T v T ] T , we can write (3) in the form [START_REF] Donoho | Compressed sensing[END_REF], where H(x) = uv 2 2 and E(x) is the objective function in problem [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF].

The Bregman iteration is given by

x k+1 = arg min x D p k E (x, x k ) + µH(x) (6)
= arg min

x E(x) -p k , x -x k + µH(x),
where p k ∈ ∂E(x k ). It has been shown that, for any µ > 0, this procedure converges to a solution of (4) [START_REF] Goldstein | The split Bregman method for L1 regularized problems[END_REF], [START_REF] Yin | Bregman iterative algorithms for ℓ 1 -minimization with applications to compressed sensing[END_REF].

Concerning the update of p k , we have from [START_REF] Combettes | Signal recovery by proximal forwardbackward splitting[END_REF], that 0 ∈ ∂(D p E (x, x k )+µH(x)), when this sub-differential is evaluated at x k+1 , that is

0 ∈ ∂(D p E (x k+1 , x k ) + µH(x k+1
)). Since it was assumed that H is differentiable, and since p k+1 ∈ ∂E(x k ) at this point, p k+1 should be chosen as

p k+1 = p k -µ∇H(x k+1 ). (7)

C. Split Bregman Iterations

The split Bregman formulation for l 1 -regularized problems, proposed in [START_REF] Goldstein | The split Bregman method for L1 regularized problems[END_REF], separates the l 1 and l 2 portions of the energy in the problem

min u Φ(u) 1 + J(u) (8)
where J(.) and Φ(.) are convex functionals and Φ(.) is differentiable, by introducing an additional variable d ∈ R n and the constraint d = Φ(u). The constrained problem is formulated as min u,d

d 1 + J(u) subject to d -Φ(u) 2 2 = 0. (9)
Applying the Bregman iterations described in Subsection II-B, with

E([u, v]) = d 1 + J(u) and H([u, v]) = d - Φ(u) 2 2
, it can be shown that this problem is solved by the two-phase algorithm

(u k+1 , d k+1 ) = min u,d d 1 + J(u) + + λ 2 d -Φ(u) -b k 2 2 (10) b k+1 = b k + (Φ(u k+1 ) -d k+1 ). ( 11
)
The problem (10) can be minimized efficiently by iteratively minimizing with respect to u and d, in two steps

u k+1 = min u J(u) + λ 2 d k -Φ(u) -b k 2 2 , (12)
d k+1 = min d d 1 + λ 2 d -Φ(u k+1 ) -b k 2 2 . ( 13
)
The problem [START_REF] Figueiredo | Majorizationminimization algorithms for wavelet-based image restoration[END_REF] is thus reduced to a sequence of unconstrained problems and Bregman updates.

D. Applying Split Bregman to (3)

In ( 3), clubbing the data misfit term 1 2 Auf 2 2 and the regularizer term τ 1 Φ 1 (u) together, we can apply a similar approach, that is, iteratively minimizing with respect to u and v, separately.

After some algebraic manipulations, we can show that the split Bregman iteration for the constrained problem (3) has the form

u k+1 = arg min u 1 2 Au -f 2 2 + τ 1 Φ 1 (u) + + µ 2 u -v k -b k 2 2 = arg min u 1 2 Ku -g 2 2 + τ 1 Φ 1 (u) ( 14
)
v k+1 = arg min v τ 2 Φ 2 (v) + µ 2 u k -v -b k 2 2 = arg min v τ 2 Φ 2 (v) + µ 2 u k -b k -v 2 2 ( 15
)
where

K = A √ µ I n , g = f √ µ(v k + b k )
,

and b k+1 = b k -(u k -v k ), (16)
and the initial values are u 0 = 0, v 0 = 0, and b 0 = 0.

Since each of the problems ( 14) and ( 15) involves only one regularizer, for which the MPM is known, they can be efficiently solved using, e.g., the IST or TwIST algorithms [START_REF] Bioucas-Dias | A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration[END_REF]. As convergence is guaranteed for any value of µ > 0, we can choose it so as to make these problems well-conditioned. The iterations can be terminated when the constraint term u k -v k 2 2 falls below some threshold and the relative change in the objective function in (3) goes below some tolerance level. The final value of either u k or v k , after applying any inverse transform if applicable, is taken as the estimate of u, û = u final

III. RESULTS

For the purpose of demonstration, we will consider a deconvolution problem (i.e., A is the matrix representation of a convolution), with a synthetically generated image shown in Figure 1(a), which is both sparse and piecewise smooth. We therefore use a combination of the ℓ 1 and the TV regularizers. The MPM (denoising operator) for the TV regularizer is implemented by Chambolle's algorithm [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF], and the MPM for the ℓ 1 regularizer is the well-known soft threshold [START_REF] Donoho | De-noising by soft thresholding[END_REF].

The image blurred with a 7 × 7 uniform kernel and with added noise (σ = 0.1, SNR = -2.081 dB) is shown in Figure 1(b). The problems ( 14) and ( 15) at each Bregman iteration (outer loop) are solved with a single inner loop iteration, as it has empirically been found in [START_REF] Goldstein | The split Bregman method for L1 regularized problems[END_REF] that optimal efficiency is obtained with a single iteration in the inner loop. Each problem (at each iteration) is solved using the TwIST algorithm. The value of µ was taken as 0.15, which was found to make the two problems well conditioned, as well as have a reasonable speed of convergence. The optimal values of the regularization parameters which led to the lowest mean square error, were found by hand tuning to be τ T V = 0.05 and τ ℓ1 = 0.06 × A T g ∞ . The estimate obtained by the proposed method is shown in Figure 2(a) and is clearly better (sparser) than the one obtained using only the TV regularizer, shown in Figure 2(b). Figure 3 shows the plots of the objective term in [START_REF] Candès | Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information[END_REF]

at iteration k, E(k) = 1 2 Au k -f 2 2 + τ 1 Φ 1 (u k ) + τ 2 Φ 2 (v k ), the plot of the normalized distance u k -v k 2
2 , and that of the rate of change of u, u ku (k-1) / u k . The stopping criterion used to terminate the iterative procedure was convergence of the objective function.

The plot of the MSE obtained with the compound regularizer (ℓ 1 +TV) and with only TV regularization, for different values of the regularization parameter τ T V , is shown in figure

4(a)

. Figure 4(b) shows a similar MSE comparison between using only ℓ 1 regularization and ℓ 1 +TV regularization. The lowest MSE obtained with the ℓ 1 +TV regularizer is lower than that obtained with TV or ℓ 1 alone, for any value of the corresponding regularization parameters.

IV. CONCLUDING REMARKS

We have introduced a new algorithm for solving the optimization problems resulting from using more than one regularizer in imaging inverse problems. The algorithm only requires the ability to efficiently compute the denoising operator associated to each involved regularizer. It was illustrated on a problem of image deblurring, with encouraging results. The lowest MSE obtained using the ℓ 1 +TV regularizer was lower than that obtained with TV or ℓ 1 regularization alone, confirming the usefulness of using compound regularization.

Figure 1 .

 1 Figure 1. (a): Original image, (b): Image blurred (7 × 7 uniform blur) and corrupted with zero mean Gaussian noise with σ = 0.1, SNR = -2.081 dB).

Figure 3 .

 3 Figure 3. (a): Objective function evolving over iterations, (b): Distance uv 2 2 evolving over iterations, (c) Rate of change of u evolving over iterations.

Figure 4 .

 4 Figure 4. (a): MSE obtained with ℓ 1 +TV and only TV regularizers, for different values of the regularization parameter τ T V , (b): MSE obtained with ℓ 1 +TV and only ℓ 1 regularizers, for different values of the regularization parameter τ ℓ 1 .

Acknowledgment

The work of Manya Afonso is supported by the EU by a Marie-Curie Fellowship (EST-SIGNAL program : http://est-signal.i3s.unice.fr) under contract No MEST-CT-2005-021175.