Selima Bennaceur
email: selima.bennaceur@iup.univ-evry.fr

Naoufel Azouz
email: azouz@iup.univ-evry.fr

Djaber Boukraa
email: boukraa@iup.univ-evry.fr

AN EFFICIENT MODELLING OF FLEXIBLE AIRSHIPS

Keywords: flexible airships, small deformations, incremental scheme, modal synthesis

This paper presents an efficient modelling of airships with small deformations moving in an ideal fluid. The formalism is based on the Updated Lagrangian Method (U.L.M.).This formalism proposes to take into account the coupling between the rigid body motion and the deformation as well as the interaction with the surrounding fluid. The resolution of the equations of motion is incremental. The behaviour of the airship is defined relatively to a virtual non-deformed reference configuration moving with the body. The flexibility is represented by a deformation modes issued from a Finite Elements Method analysis. The increment of rigid body motion is represented similarly by rigid modes. A modal synthesis is used to solve the general system equations of motion. Time constant matrices appears (i.e. mass and structural stiffness matrices), and we show a convenient technique to actualise the time dependant matrices.

Introduction

The interest for the modelling and control of airships increases significantly in the last years. The complexity and capability of airships are expanding rapidly and the range of missions they designed to support is growing. However in order for airships to reach this potential, significant technical issue must be overcome. One main point of this challenge is the modelling of the structural flexibility. It is important to note that several kinds of airships, usually called blimps, are mainly constituted of a balloon filled of gas. The only solid parts are the careen and the tail fins. The integration of the structural flexibility in the dynamic analysis is then useful, but it is now in an embryonic state and is only just emerging. Several researches were done using the assumption of rigid body behaviour for airships [START_REF] Fossen | Guidance and control of ocean vehicles[END_REF][START_REF] Bestaoui | Dynamic modelling of small autonomous blimps[END_REF]. The flexibility effects are sometimes modelled as a perturbation. However in other flying objects, such as light aircrafts, the introduction of the flexibility in the dynamic model becomes essential [START_REF] Calise | Adaptive compensation for flexible dynamics[END_REF][START_REF] Bianchin | State space reduced order models for static aeroelasticity and flight mechanics of flexible aircrafts[END_REF]. We try by this study to contribute to the analysis of the deformation of the airships by introducing the effect of flexibility as non controlled supplementary degrees of freedom. The deformation of the blimp is not considered as a perturbation but rather acting on the motion of the airship.

The influence of structural flexibility on dynamics of mechanical systems has become increasingly important in classical robotics [START_REF] Simo | The role of non-linear theories in transient dynamic analysis of flexible structures[END_REF][START_REF] Pascal | Vibrations analysis of flexible multibody systems[END_REF] and recently in flying robots (i.e. Airships, UAVs…).

Several approaches, to study the problems of flexible bodies, have been proposed in the literature. These approaches can be classified into two groups. The first group uses the Newton-Euler description [START_REF] Boyer | Generalization of Newton-Euler model for flexible manipulators[END_REF] which is an interesting method in regard of the time computation. However it is not easy to use it in the case of the flexible body and it is sometimes sensitive to the numerical simulation. In the other hand the total Lagrangian method [START_REF] Shabana | Dynamics of multibody systems[END_REF], this consists to define the motion relatively to a fixed reference frame, leads also to complex relations when describing stresses and strains in the flexible body.

An Updated Lagrangian Method (U.L.M.) was proposed by Bathe & al. [9] and developed for deformable bodies that undergo large translational and rotational displacements. The resolution of the dynamic problem is incremental. The configuration and the motion of the body are identified using a moving reference configuration representing the position of the deformable body in the preceding step. Azouz & al. [10] propose as reference configuration a rigid body configuration which follows the motion of the body without coinciding with it. This approach is convenient for a body with small deformations. Using this approach, we develop an efficient formalism to describe the behaviour of airships with small deformations. The motion is given by coupled sets of rigid and elastic variables. The nonlinear equations are formulated in terms of a set of time invariant matrices expressed in a reference configuration (i.e. mass and stiffness matrices). Time-variant quantities appear in the nonlinear terms that represent the dynamic coupling between the rigid body modes and the elastic deformation. We show in this study a suitable technique to actualise these terms using matrix partitioning and canonical decomposition. The dynamic system is reduced and solved through a modal synthesis. The elastic deformation is represented by a set of generalised variables and natural modes issued from a Finite Element method (FEM) discretization.

Airships are also governed by the aerodynamic forces that have to be modelled. The basis to analyse the motion of a rigid body in a perfect fluid has been established in the 19 th century and has been described by Lamb [START_REF] Lamb | On the motion of solids through a liquid. Hydrodynamics[END_REF]. In his work, Lamb considered the case of simple displacement in a big infinite mass of fluid and where the movement of this last is entirely due to the motion of a solid, and it is irrotational and acyclic. He proved that the kinetic energy of the fluid can be expressed as a quadratic shape of the six velocities of translation and rotation of the vehicle. The derivations given by Lamb will be used in the description of the airship, in a stationary uniform atmosphere. The terms depending on the acceleration or the added masses come from the fact that the fluid considered perfect is accelerated. When an ellipsoid body moves in an incompressible and infinite inviscid fluid so that the external flow is everywhere irrotational and continuous, the kinetic energy of the fluid produces an effect equivalent to an important increase of the mass and of the moments of inertia of the body [START_REF] Thomasson | Eq. of motion of a vehicle in a moving fluid[END_REF][START_REF] Khoury | Airship Technology[END_REF]. In our model, the fluid effects have been introduced into the dynamical system through a modal synthesis to build the global dynamic system of the flexible airship.

Dynamic model of the airship

The dynamics of the airship with small deformation is highly non-linear. The non linearities are essentially due to the large rotations of the body and the interaction between rigid body motion and the deformation. In our formulation, we use an Update Lagrangian Method. The description of the airship behaviour is made relatively to a reference configuration which changes in the time accordingly to the airship configuration. This method is chosen because it takes into account the buckling between the rigid body motion and the deformation, and could be coupled with results of classical structural dynamics. In fact, many dynamic parameters will be kept unchanged during the simulation. In the case of a small deformation, the configuration at the time t (C t ) is not so different from the one obtained by the transformation of a rigid body in the initial configuration. We choose as a reference configuration, the configuration of a virtual rigid body following the motion. Hence, we can linearize the airship displacement around this configuration. We consider two frames in the derivation of the motion equations. These frames are: the fixed frame related to the earth R f and the moving frame R m related to the reference configuration (C ref ). The position and the orientation of the vehicle should be described relatively to the reference frame. The origin O' of R m coincides with the inertia centre of the undeformed vehicle in the reference frame. Its axes are the principal axes of symmetry.

Description of the reference configuration

Figure 2. Incremental scheme

In this model we use an incremental scheme. The displacement of each point L is given by: ()

() () () =+ ∆+ ref ref d Q UL U L U L U L (1)
Where U d is the small displacement related to the whole deformation, ; from the equation ( 1) we can obtain the following expression:

( )

2 ref U U HQ HQ HQ =+ + + (2)
, Q and Q represent the time derivative of the displacement Q .

Description of the strains

To describe the strains we use the symmetric Green-Lagrange tensor ε which is related to the reference configuration. We note J the gradient tensor relative at the reference configuration coordinates JQ =∇ and the Green tensor ε can be expressed as:

( ) 1 . 2 tt JJJ J ε ⎡⎤ =+ + ⎣⎦ .
In this case we have small deformations, consequently 1 J

; we can then neglect the non linear terms, hence the strain tensor becomes:

Where D is a differential operator, ij ε are the components of the strain tensor.

Description of the stresses

For the description of the stresses applied on the airship, we use the second tensor of Piola- The material of the careen is assumed elastic, homogeneous and isotropic such as we have a linear behaviour law between stresses and strains:

E σ ε = (4)
Here σ is the vector of the stress components and E the symmetric tensor of elastic properties of the material.

Dynamic equilibrium

Let's consider the dynamic equilibrium of an airship. According to the classical mechanics laws, the equilibrium is represented by the two following relations [START_REF] Ciarlet | Mathematical Elasticity[END_REF]:

v div f U σ ρ += on (C t ) (5)
Consequently, the equation ( 7) is similar to the dynamic equation of the deformable bodies in structural mechanics.

Resolution of the dynamic equation

To define the exact configuration of the deformable airship, we should use an infinite number of co-ordinates in order to calculate the location of every point of the body. It has been shown that the motion of the deformable airship is governed by a set of partial differential equations that are space and time dependent. The discretization consists to subdivide the airship in a number of sub-domains with simple shapes that could be studied easily. We will use approximate techniques namely Rayleigh-Ritz method coupled with Finite Elements discretization.

Discretization of the airship

We indicate that the discretization consists to subdivide the airship in a number of sub-domains with simple shape. The displacement of the airship is then described through some interpolations functions and nodal displacement. We must note that the number of elements required for an acceptable approximation of the displacement should be high. However when using modal synthesis, this number of elements play a part only in a preliminary computation. Just a few numbers of "useful" nodes are kept.

We should note that the nacelle is represented by three points in the bottom of the airship. In the central point we apply its actual heavy, in the other point we model the effect of the two propellers. where N i is an interpolation space function [START_REF] Zienckiewicz | The Finite Element Method[END_REF]. This relation is valid for any given displacement, especially for:

QN . q = (8)
We note M i and i L K the mass and stiffness matrices of the element i defined as follows:

i ii T i i ref ref V M. N . N . d V ρ = ∫ (9) i ii T T i i i i Lr e f V K N .D .E .D .N .dV = ∫ (10)
V i and E i are respectively the volume and the tensor of the elastic properties of the element i, D i is the differential operator of an element i.

The mass and stiffness matrices of the whole airship are built by an adequate assembly of elementary matrices of mass and stiffness:

ne i i1 MM = = ∑ ( 11
) ne i LL i1 KK = = ∑ (12)
ne is the number of elements of the airship.

Equation of motion of the discretized airship

Lets us now analyse the different terms of the equilibrium equation [START_REF] Boyer | Generalization of Newton-Euler model for flexible manipulators[END_REF]. According to the equation ( 2), the first left hand term could be expressed as:

Relations (3; 8-13) give :

ef r K.q B.q M.q F F F + += + - (15)
We define here the diagonal bloc matrix of rotation H which concerns the rotation of all the nodes:

[ ] [] [] H H . H . . H ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ ⎡⎤ = ⎜⎟ ⎣⎦ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ (16)
The stiffness matrix K is constituted by: K L the structural stiffness, and The structural stiffness and mass matrices are defined relatively to a moving reference frame R m in the "rigid" reference configuration. They are then constant. The dynamic equation ( 15) has a huge size and the number of degree of freedom (d.o.f) is tremendous, especially if we choose to model the airship with small elements; this leads to a very high computational cost and many problems to establish control laws. For this reason, we tried to solve this dynamic equation using a modal synthesis.

Modal synthesis

We decompose the increment Q into a rigid contribution r Q and a deformable part d Q such as in its discretized form we have: The displacement due to the deformation of the airship is:

nd dd i d i i1 qY X = = ∑ ( 19
)
nd: is the number of significant deformable modes kept for the study, Y di is a time dependent variable of the deformable mode i, X di is a free-free natural mode of rank i. The choice of these boundary conditions is in harmony with the real configuration of the airship. O' is the origin of R m , L is a given nodal point, ′ OL is the skew matrix associated with the vector ′ OL.

The deformable airship could be considered as low frequency structure. We can model its vibration accurately with the first modes of lower frequencies.

The projection of dynamical equation of motion [START_REF] Zienckiewicz | The Finite Element Method[END_REF] in the modal basis composed of rigid and flexible modes permits to reduce the number of d.o.f. The dynamic equation can then be written as

: ef R .Y .Y Y .Y + ++ = + - DL M B K K F F F (20)
M and K L are the constant mass and stiffness matrices projected in the modal basis (M = X T .M. X ), X is the matrix of the modal basis issued from the collection of all rigid and flexible modes. The other bold terms are terms projected in the modal basis: The mass matrix is defined in the "rigid" reference configuration. We have then as sub-matrix of translation (M TT ) a diagonal matrix corresponding to the mass of the airship. M RR is the inertia matrix of rotation; and according that the centre of gravity coincides with the origin of the reference frame, we have M TR = 0. The choice of free modes of deformation allows to say that M Td = M Rd = 0.

M = ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
The use of constant mass and stiffness matrices is important for the reduction of the computational time. In the other hand, we obtain two time-variant matrices (B and K D ) that we should actualise at each step. However we develop a computational technique which makes easier this task. For example a given matrix A can be written as:

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ . . .
The matrix K D can then be computed by this way:

33 TT T IJ ij i1 j1 2M H H 2 H H Z == ⎡⎤ == ⎣⎦ ∑∑ D K X . . X (22)
where

T IJ IJ ZM A = X X are constant
matrices defined before the beginning of the time varying simulation of motion.

Interaction fluid-airship

Consider a simple dynamical model of the action of a fluid on a body. To present this model, we assume that the flow is quasi-steady, i.e. the distribution of the velocities of particles of the medium coincides with the distribution corresponding to the steady motion of the body. Thus the medium responds only to the current motion of the body and forgets its initial conditions. Therefore, within the framework of this hypothesis, the resultant force and torque acting on the body can be represented in the form of a function of the instantaneous distribution of velocities in this body. Thus we arrive at the statement of the problem of the motion of a body in a dragging medium as a problem of classical dynamics.

Flow representation

To take into account the interaction of the airship with the surrounding fluid medium, a model of the flow is needed. Here, we rely on the potential flow theory corresponding to the following hypothesis: a. the air can be considered as a perfect fluid with uniform density ρ air , i.e. an incompressible gas with vanishing viscosity, b. the flow is irrotational c. Only the flow outside the airship contributes significantly to the aerodynamics forces.

Denoting by v the velocity field in the fluid domain Ω air , the incompressibility and irrotational assumptions leads to:

.v 0 ; v 0 ∇= ∇ ∧=

∧ is the vectorial product of two vectors. ∇ is the gradient symbol, and the flow field may be described in terms of a potential F such that:

v =∇Φ

From the incompressibility constraint, it is easy to show that the potential obeys to the homogeneous Laplace equation:

2 air 0i n ∇Φ = Ω (25)
with Newman boundary conditions:

[] ( ) t nq n Y n o n C ∇Φ = -= - ∂ .. ( X ) . (26)
[X ] is the matrix of all the modes (rigid and deformable modes), Y is the column matrix of all the time dependent variables defined in §3.4, n is a unit vector, normal to ( )

t C ∂ .
Thus, one of the important characteristic of this representation is that v only depends on the current boundary conditions, and not on the history of the flow: the model is quasi-steady. To solve the potential equation, we use the boundary integral representation of the Laplace equation, together with standard boundary element method. It consists in the determination of a piecewise constant distribution of singularities over ( ) t C ∂ (see [START_REF] Katz | Low-speed Aerodynamics, from Wings Theory to Panel Methods[END_REF] for details on the numerical treatment).

Fluid forces

For this assumption, the pressure at any point in the fluid domain (including ( ) The subscript ∞ denotes the undisturbed conditions far from the airship. This pressure distribution over the airship surface can be integrated to compute the resulting forces and torques which in turn are projected on the modal basis. At the end, and with the linear property of the Laplace equation, the generalised fluid forces vector can be rewritten as:

fa d f FM Y B Y =- - (28)
where M ad is the matrix of the added masses (virtual masses), and Y is the column matrix of all Y i , B f is a modal damping due to the flexibility of the hull. For a quasi-ellipsoid airship the extra-diagonal terms of M a d can be neglected (for more details about the constitutive terms of M a d , the reader can see [START_REF] Lamb | On the motion of solids through a liquid. Hydrodynamics[END_REF][START_REF] Fossen | Guidance and control of ocean vehicles[END_REF]).

Modal projection

Simulation results

To illustrate this incremental formalism we study the blimp belonging to the L.S.C having the following characteristics:

-The envelope:

Length: 6.25 m. Diameter: 1.52 m. Volume: 7.48 m 3 .

-Mass of the airship: 5.8 Kg.

-Payload: 1.58 Kg.

The blimp is thrusted by two contrarotating propellers. In the first step we apply to the blimp two opposite forces in the propellers to generate a yaw motion (Fig 7).The flexible airship should rotate about 90° around the z-axis. A P.I.D controller will impose this task, and we study the behaviour of the airship during this manoeuvre. The number of deformable modes kept is nd = 2. This number seems to give an acceptable approximation of the flexible behaviour. We were guided for that by the modal masses of these two modes which represent roughly 65% of the total mass of the airship. In this simulation one applied around the yaw angle a torque consigns with: One notices in figure 5 that the amplitude of deviation of the yaw angle decreases significantly in few seconds, it stabilizes oneself while merging with the instruction.

We show the position of the tip of the airship in figure 6. The deformations (in figure 8) are about 0.15m what is more or less significant considering one has small deformations. In this simulation one visualized the tip of the airship, and one sees well the oscillation of the latter with non-negligible amplitude.One classical motion of an airship is the helicoidally motion. This usually represents the optimal trim trajectories. This represents a complete aerial motion of the airship. The propellers were oriented adequately to assure a combined motion along the moving x-axis and z-axis and the tails were oriented to give a homogeneous yaw motion.

One can see (in figure 9) the influence of the deformation on the overall motion of the airship. This proves that in our model, the deformation is not a sample perturbation around the main rigid motion, but it acts on this motion. Experimental data will be available in few months to validate the results of our incremental scheme.

Finally we show the influence of the added masses on the total displacement of the airship. It is noticed (in figure 10) that the immersed airship has a delay compared to the body alone and this is because of the addition of the virtual masses representing the influence of the displacement of the mass of air around the airship when this last accelerates.

Conclusion

The study of the flexible structures in the space has different performance requirements which may lead to troublesome calculation and inextricable problems of controllability.

We introduced in this paper the flexibility of the airship as an extension of the classical structural dynamics taking into account the coupling between the rigid body motion and the deformation.

The Updated Lagrangian Method (U.L.M) applied on rigid reference configuration and the uses of an updated modal synthesis permit to diminish the dynamic equations of the flexible airship to a reduced set of degrees of freedom.

Simulation results prove that the integration of the flexibility in the dynamic system of the airship is very important and could not be neglected. The influence of surrounding air is also taken into account in our global model by the mean of added masses and through the same modal algorithm.

The effect of surrounding air with the flexibility of the hull will be treated in future works.

Figure 1 .

 1 Figure 1. Presentation of the different frames

  of the rigid body motion. Let us note H the rotation matrix between the initial frame and the local reference R m such as .

ρ

  σ is the stress tensor of Cauchy, d JI J =+ , I d is the 3x3 identity matrix, and ref ρ is the mass density of the reference configuration equal to its initial value 0 defined on the initial configuration (C 0 ).

Figure 3 .

 3 Figure 3. Discretization of the airship Let us now consider a displacement field U i of an element i of the body. It can be expressed in function of the column matrix u representing the displacement, relatively to the local reference frame, of all nodes as follows: ii UN u =

  here that, in the incremental scheme the reference acceleration ref U related to the reference configuration is known. Using the equations (3) and (7), we can write the second left hand term as: q . N . D . E . ( D . N )q . d V σε ∂= ∂ ∫∫

  matrix of residual inertia terms due to the dynamics of the reference configuration.

  being the modal amplitude of the rigid mode i. Hence for a 3D motion, we have six variables ( ) are three modes of the translation motion. X rot : are the three modes of rotation.

  note M '=M +M ad , B'=B+B f . The effect of the fluid on the structure is then represented mainly by the adjunction of the added masses matrix M a d to the mass matrix of the structure. I are moments of the inertia's body.

Figure 4 .

 4 Figure 4. The LSC airship To illustrate our incremental formalism, two kinds of motions are tested for this airship.

  position and ψ : angular position of the airship.

Figure 5 .

 5 Figure 5. The superposition of trajectories (yaw angle and desired).

Figure 6 .

 6 Figure 6. Position of the tip of the airship

Figure 7 .Figure 8 .

 78 Figure 7. Superposition of the motions of the flexible and rigid airship

Figure 9 .

 9 Figure 9. Rise of the airship.

Figure 10 .

 10 Figure 10. Superposition of the displacement of the airship with and without air environing


Guanghui Hu
email: hu@csrc.ac.cn

Yavar Kian
email: yavar.kian@univ-amu.fr

AND Peijun Li
email: lipeijun@math.purdue.edu

Yue Zhao

Inverse moving source problems in electrodynamics

may come

INTRODUCTION

Consider the time-dependent Maxwell equation for the electric field E in a homogeneous medium

∂ 2 t E(x, t) + ∇ × (∇ × E(x, t)) = F (x, t), x ∈ R 3 , t > 0, (1.1)
which is supplemented by the homogeneous initial conditions

E(x, 0) = ∂ t E(x, 0) = 0, x ∈ R 3 . (1.2)
We assume that the electrodynamic field is excited by a moving point source radiating over a finite time period. Specifically, the source function F is assumed to be given in the following form F (x, t) = J (x -a(t)) g(t),

where J : R 3 → R 3 is the source profile function, g : R + → R the temporal function, and a : R + → R 3 is the orbit function of the moving source. Hence the source term is assumed to be a product of the spatially moving source function J (x-a(t)) and the temporal function g(t). Physically, the spatially moving source function can be thought as an approximation of a pulsed signal which is transmitted by a moving antenna, whereas the temporal function is usually used to model the evolution of source magnitude in time. Throughout, we make the following assumptions:

(1) The profile function J (x) is compactly supported in B R := {x : |x| < R} for some R > 0;

(2) The source radiates only over a finite time period [0, T 0 ] for some T 0 > 0, i.e., g(t) = 0 for t ≥ T 0 and t ≤ 0; (3) The source moves in a bounded domain, i.e., |a(t)| < R 1 for all t ∈ R + and some R 1 > 0.

These assumptions imply that the source term F is supported in B R × (0, T 0 ) for R > R + R 1 . Unless otherwise stated, we always take T := T 0 + R + R 1 + R and set Γ R := {x ∈ R 3 : |x| = R}. Denote by ν the unit normal vector on Γ R and let Γ ⊂ Γ R be an open subset with a positive Lebesgue measure.

We study the inverse moving source problems of determining the profile function J (x) and the orbit function a(t) from boundary measurements of the tangential trace of the electric field over a finite time interval, E(x, t) × ν| Γ×[0,T ] . Specifically, we consider the following two inverse problems:

(i) IP1. Assume that a(t) is known, the inverse problem is to determine J from the measurement E(x, t) × ν, x ∈ Γ, t ∈ (0, T ). (ii) IP2. Assume that J is a known vector function, the inverse problem is to determine a(t), t ∈ (0, T 0 ) from the measurement E(x, t) × ν, x ∈ Γ, t ∈ (0, T ). The IP1 is a linear inverse problem, whereas the IP2 is a nonlinear inverse problem. The time-dependent inverse source problems have attracted considerable attention [START_REF] Yu | A uniqueness result in an inverse hyperbolic problem with analyticity[END_REF][START_REF] Klibanov | Inverse problems and Carleman estimates[END_REF][START_REF] Li | Carleman estimates for second order hyperbolic systems in anisotropic cases and an inverse source problem. Part II: an inverse source problem[END_REF][START_REF] Li | An inverse source problem for Maxwell's equations in anisotropic media[END_REF][START_REF] Ola | An inverse boundary value problem in electrodynamics[END_REF][START_REF] Ramm | Electromagnetic inverse problem with surface measurements at low frequencies[END_REF][START_REF] Yamamoto | On an inverse problem of determining source terms in Maxwell's equations with a single measurement[END_REF]. However, the inverse moving source problems are rarely studied for the wave propagation. We refer to [START_REF] Garnier | Super-resolution in time-reversal focusing on a moving source[END_REF] on the inverse moving 1 arXiv:submit/2509773 [math.AP] 15 Dec 2018 source problems by using the time-reversal method and to [START_REF] Stefanov | Inverse scattering problem for a class of moving obstacles[END_REF][START_REF] Stefanov | Inverse scattering problem for moving obstacles[END_REF] for the inverse problems of moving obstacles. numerical methods can be found in [START_REF] Nakaguchi | An algebraic reconstruction of a moving point source for a scalar wave equation[END_REF][START_REF] Wang | Mathematical design of a novel input/instruction device using a moving acoustic emitter[END_REF] to identify the orbit of a moving acoustic point source. To the best of our knowledge, the uniqueness result is not available for the inverse moving source problem, which is the focuse of this paper.

Recently, a Fourier method was proposed for solving inverse source problems for the time-dependent Lamé system [START_REF] Bao | Inverse source problems in elastodynamics[END_REF] and the Maxwell system [START_REF] Hu | Inverse source problems in electrodynamics[END_REF], where the source term is assumed to be the product of a spatial function and a temporal function. These work were motivated by the studies on the uniqueness and increasing stability in recovering compactly supported source terms with multiple frequency data [4-6, 11, 12, 23, 23]. It is known that there is no uniqueness for the inverse source problems with a single frequency data due to the existence of non-radiating sources [START_REF] Albanese | The inverse source problem for Maxwell's equations[END_REF]. In [START_REF] Bao | Inverse source problems in elastodynamics[END_REF][START_REF] Hu | Inverse source problems in electrodynamics[END_REF], the idea was to use the Fourier transform and combine with Huygens' principle to reduce the time-dependent inverse problem into an inverse problem in the Fourier domain with multi-frequency data. The idea was further extended in [START_REF] Hu | Uniqueness and stability for the recovery of a time-dependent source and initial conditions in elastodynamics[END_REF] to handle the time-dependent source problems in elastodynamics where the uniqueness and stability were studied.

In this paper, we use partial boundary measurements of dynamical Dirichlet data over a finite time interval to recover either the source profile function or the orbit function. In Sections 3 and 4.2, we show that the ideas of [START_REF] Bao | Inverse source problems in elastodynamics[END_REF][START_REF] Hu | Inverse source problems in electrodynamics[END_REF] and [START_REF] Hu | Uniqueness and stability for the recovery of a time-dependent source and initial conditions in elastodynamics[END_REF] can be used to recover the source profile function as well as the moving trajectory which lies on a flat surface. For general moving orbit functions, we apply the moment theory to deduce the uniqueness under a priori assumptions on the path of the moving source, see Section 4.1. When the compactly supported temporal function shrinks to a Dirac distribution, we show in Section 5 that the data measured at four discrete receivers on a sphere is sufficient to uniquely determine the impulsive time point and to the source location. This work is a nontrivial extension of the Fourier approach from recovering the spatial sources to recovering the orbit functions. The latter is nonlinear and more difficult to handle.

The rest of the paper is organized as follows. In Section 2, we present some preliminary results concerning the regularity and well-posedness of the direct problem. Sections 3 and 4 are devoted to the uniqueness of IP1 and IP2, respectively. In Section 5, we show the uniqueness to recover a Dirac distribution of the source function by using a finite number of receivers.

THE DIRECT PROBLEM

In addition to those assumptions given in the previous section, we give some additional conditions on the source functions:

J ∈ H 2 (R 3 ), div J = 0 in R 3 , g ∈ C 1 (R + ), a ∈ C 1 (R + ).
It follows from [START_REF] Albanese | The inverse source problem for Maxwell's equations[END_REF] that any source function can be decomposed into a sum of radiating and non-radiating parts. The non-radiating part cannot be determined and gives rise to the non-uniqueness issue. By the divergence-free condition of J , we eliminate non-radiating sources in order to ensure the uniqueness of the inverse problem. Since the source term J has a compact support in B R × (0, T ), we may show the following result by Huygens' principle.

Lemma 2.1. It holds that E(x, t) = 0 for all x ∈ B R , t > T .
The proof of Lemma 2.1 is similar to that of Lemma 2.1 in [START_REF] Hu | Inverse source problems in electrodynamics[END_REF]. It states that the electric field E over B R must vanish after time T . This property of the electric field plays an important role in the mathematical justification of the Fourier approach.

Noting ∇ • J = 0, taking the divergence on both sides of (1.1), and using the initial conditions (1.2), we have

∂ 2 t (∇ • E(x, t)) = 0, x ∈ R 3 , t > 0 and ∇ • E(x, 0) = ∂ t (∇ • E(x, 0)) = 0.
Therefore, ∇ • E(x, t) = 0 for all x ∈ R 3 and t > 0. In view of the identify ∇ × ∇× = -∆ + ∇∇, we obtain from (1.1)-(1.2) that

∂ 2 t E(x, t) -∆E(x, t) = J (x -a(t))g(t), x ∈ R 3 , t > 0, E(x, 0) = ∂ t E(x, 0) = 0, x ∈ R 3 . (2.1)
We briefly introduce some notation on functional spaces with the time variable. Given the Banach space X with norm || • || X , the space C([0, T ]; X) consists of all continuous functions f :

[0, T ] → X with the norm ||f || C([0,T ];X) := max t∈[0,T ] ||f (t, •)|| X .
The Sobolev space W m,p (0, T ; X), where both m and p are positive integers such that

1 ≤ m < ∞, 1 ≤ p < ∞, comprises all functions f ∈ L 2 (0, T ; X) such that ∂ k t f, k = 0, 1, 2, • • • , m
exist in the weak sense and belong to L p (0, T ; X). The norm of W m,p (0, T ; X) is given by

||f || W m,p (0,T ;X) := T 0 m k=0 ||∂ k t f (t, •)|| p X 1/p . Denote H m = W m,2
. Now we state the regularity of the solution for the initial value problem (2.1). The proof follows similar arguments to the proof of Lemma 2.3 in [START_REF] Hu | Inverse source problems in electrodynamics[END_REF] by taking p = 2.

Lemma 2.2. The initial value problem (2.1) admits a unique solution

E ∈ C(0, T ; H 3 (R 3 )) 3 ∩ H τ (0, T ; H 2-α+1 (R 3 )) 3 , α = 1, 2, 3, which satisfies E C(0,T ;H 3 (R 3 )) 3 + E H τ (0,T ;H 2-τ +1 (R 3 )) 3 ≤ C g L 2 (0,T ) J H 2 (R 3 ) 3 ,
where C is a positive constant depending on R.

Applying the Sobolev imbedding theorem, it follows from Lemma 2.2 that

E ∈ C([0, T ]; H 2 (R 3 )) 3 ∩ C 2 ([0, T ]; L 2 (R 3 )) 3 .
Denote by I the 3-by-3 identity matrix and by H the Heaviside step function. Recall the Green tensor G(x, t) to the Maxwell system (see e.g., [START_REF] Hu | Inverse source problems in electrodynamics[END_REF])

G(x, t) = 1 4π|x| δ (|x| -t)I -∇∇ 1 4π|x| H(|x| -t) ,
which satisfies

∂ 2 t G(x, t) + ∇ × (∇ × G(x, t)) = -δ(t)δ(x) I with the homogeneous initial conditions: G(x, 0) = ∂ t G(x, 0) = 0, |x| = 0.
Taking the Fourier transform of G(x, t) with respect to the time variable yields

Ĝ(x, κ) = g(x, κ)I + 1 κ 2 ∇∇ g(x, κ) , (2.2)
which is known as the Green tensor to the reduced time-harmonic Maxwell system with the wavenumber κ.

Here g is the fundamental solution of the three-dimensional Helmholtz equation and is given by

g(x, κ) = 1 4π e iκ|x| |x| .
It is clearly to verify that Ĝ(x, κ) satisfies

∇ × (∇ × Ĝ) -κ 2 Ĝ = δ(x)I, x ∈ R 3 , |x| = 0.

DETERMINATION OF THE SOURCE PROFILE FUNCTION

In this section we consider IP1. Below we state the uniqueness result. The idea of the proof is to adopt the Fourier approach of [START_REF] Hu | Inverse source problems in electrodynamics[END_REF] to the case of a moving point source.

Theorem 3.1. Suppose that the orbit function a is given and that T 0 0 g(t)dt = 0. Then the source profile function J (x) can be uniquely determined by the partial data set {E(x, t) × ν : x ∈ Γ, t ∈ (0, T )}.

Proof. Assume that there are two functions J 1 and J 2 which satisfy

∂ 2 t E 1 (x, t) + ∇ × (∇ × E 1 (x, t)) = J 1 (x -a(t)) g(t), x ∈ R 3 , t > 0, E 1 (x, 0) = ∂ t E 1 (x, 0) = 0, x ∈ R 3 ,
and

∂ 2 t E 2 (x, t) + ∇ × (∇ × E 2 (x, t)) = J 2 (x -a(t)) g(t), x ∈ R 3 , t > 0, E 2 (x, 0) = ∂ t E 2 (x, 0) = 0, x ∈ R 3 . It suffices to show J 1 (x) = J 2 (x) in B R if E 1 (x, t) × ν = E 2 (x, t) × ν for all x ∈ Γ, t ∈ (0, T ). Let E = E 1 -E 2 and f (x, t) = J 1 (x -a(t)) g(t) -J 2 (x -a(t)) g(t).
Then we have

     ∂ 2 t E(x, t) + ∇ × (∇ × E(x, t)) = f (x, t), x ∈ R 3 , t > 0, E(x, 0) = ∂ t E(x, 0) = 0, x ∈ R 3 , E(x, t) × ν = 0, x ∈ Γ, t > 0.
Denote by Ê(x, κ) the Fourier transform of E(x, t) with respect to the time t, i.e.,

Ê(x, κ) = R E(x, t)e -iκt dt, x ∈ B R , κ ∈ R + . (3.1)
By Lemma 2.1, the improper integral on the right hand side of (3.1) makes sense and it holds that

Ê(x, κ) = T 0 E(x, t)e -iκt dt < ∞, x ∈ B R , κ > 0. Hence Ê(x, κ) × ν = 0, ∀x ∈ Γ, κ ∈ R + .
Taking the Fourier transform of (1.1) with respect to the time t, we obtain

∇ × (∇ × Ê) -κ 2 Ê = T 0 f (x, t)e -iκt dt, x ∈ R 3 . (3.2) Since supp(J ) ⊂ B R and |a(t)| < R 1 ,
it is clear to note that Ê is analytic with respect to x in a neighbourhood of Γ R ⊇ Γ and Ê satisfies the Silver-Müller radiation condition:

lim r→∞ ((∇ × Ê) × x -iκr Ê) = 0, r = |x|,
for any fixed frequency κ > 0. In fact, the radiation condition of Ê can be straightforwardly derived from the expression of E in terms of the Green tensor G(x, t) together with the radiation condition of Ĝ(x ; κ). The details may be found in [START_REF] Hu | Inverse source problems in electrodynamics[END_REF]. Hence, we have Ê(x, κ) × ν = 0 on the whole boundary Γ R . It follows from (2.2) that

Ê(x, κ) = R 3 Ĝ(x -y, κ) T 0 f (y, t)e -iκt dt dy.
Let Ê ×ν and Ĥ ×ν be the tangential trace of the electric and the magnetic fields in the frequency domain, respectively. In the Fourier domain, there exists a capacity operator T :

H -1/2 (div, Γ R ) → H -1/2 (div, Γ R )
such that the following transparent boundary condition can be imposed on Γ R (see e.g., [START_REF] Nédélec | Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems[END_REF]):

Ĥ × ν = T ( Ê × ν) on Γ R . (3.3)
This implies that Ĥ × ν is uniquely determined by Ê × ν on Γ R , provided Ĥ and Ê are radiating solutions.

The transparent boundary condition (3.3) can be equivalently written as

(∇ × Ê) × ν = iκT ( Ê × ν) on Γ R . (3.4)
Next we introduce the functions Êinc and Ĥinc by

Êinc (x) = pe iκx•d and Ĥinc (x) = qe iκx•d , (3.5)
where d ∈ S 2 is a unit vector and p, q are two unit polarization vectors satisfying p • d = 0, q = p × d. It is easy to verify that Êinc and Ĥinc satisfy the homogeneous time-harmonic Maxwell equations in R 3 :

∇ × (∇ × Êinc ) -κ 2 Êinc = 0 (3.6) and ∇ × (∇ × Ĥinc ) -κ 2 Ĥinc = 0. (3.7) Let ξ = -κd with |ξ| = κ ∈ (0, ∞). We have from (3.5) that Êinc = pe -iξ•x and Ĥinc = qe -iξ•x .
Multiplying both sides of (3.2) by Êinc and using the integration by parts over B R and (3.6), we have from Ê(x, κ) × ν = 0 on Γ R and the transparent boundary condition (3.4) that

B R T 0 f (x, t)e -iκt • Êinc dt dx = B R (∇ × (∇ × Ê) -κ 2 Ê) • Êinc dx = Γ R ν × (∇ × Ê) • Êinc -ν × (∇ × Êinc ) • Êds = - Γ R iκT ( Ê × ν) • Êinc + ( Ê × ν) • (∇ × Êinc ) ds =0. (3.8)
Hence from (3.8) we obtain

B R T 0 pe -iξ•x • g(t)J 1 (x -a(t))e -iκt dtdx = B R T 0 pe -iξ•x • g(t)J 2 (x -a(t))e -iκt dtdx.
By Fubini's theorem, it is easy to obtain

p • Ĵ 1 (κd) T 0 g(t)e -iκd•a(t) e -iκt dt = p • Ĵ 2 (κd) T 0 g(t)e -iκd•a(t) e -iκt dt.
(3.9)

Taking the limit κ → 0 + yields

lim κ→0 T 0 g(t)e -iκd•a(t) e -iκt dt = T 0 g(t)dt > 0.
Hence, there exist a small positive constant δ such that for all κ ∈ (0, δ), T 0 g(t)e -iκd•a(t) e -iκt dt = 0, which together with (3.9) implies that

p • Ĵ 1 (κd) = p • Ĵ 2 (κd).
Similarly, we may deduce from (3.7) and the integration by parts that

q • Ĵ 1 (κd) = q • Ĵ 2 (κd) for all d ∈ S 2 , κ ∈ (0, δ).
On the other hand, since

J i , i = 1, 2 is compactly supported in B R and ∇ x • J i = 0 in B R, we have R 3 de -iκx•d • J i (x)dx = - 1 iκ B R ∇e -iκx•d • J i (x)dx = 1 iκ B R e -iκx•d ∇ • J i (x)dx = 0.
This implies that d • Ĵ i (κd) = 0. Since p, q, d are orthonormal vectors, they form an orthonormal basis in R 3 . It follows from the previous identities that

Ĵ 1 (κd) = p • Ĵ 1 (κd)p + q • Ĵ 1 (κd)q + d • Ĵ 1 (κd)d = p • Ĵ 2 (κd)p + q • Ĵ 2 (κd)q + d • Ĵ 2 (κd)d = Ĵ 2 (κd)
for all d ∈ S 2 and κ ∈ (0, δ). Noting that Ĵ i , i = 1, 2, are analytical functions in R 3 , we obtain Ĵ 1 (ξ) = Ĵ 2 (ξ) for all ξ ∈ R 3 , which completes the proof by taking the inverse Fourier transform.

DETERMINATION OF MOVING ORBIT FUNCTION

In this section, we assume that the source profile function J is given. To prove the uniqueness for IP2, we consider two cases:

Case (i): The orbit {a(t

) : t ∈ [0, T 0 ]} ⊂ B R 1 ∩ R 3 is a curve lying in three dimensions; Case (ii): {a(t) : t ∈ [0, T 0 ]} ⊂ B R 1 ∩ Π,
where Π is a plane in three dimensions.

The second case means that the path of the moving source lies on a bounded flat surface in three dimensions. Cases (i) and (ii) will be discussed separately in the subsequent two subsections.

4.1. Uniqueness to IP2 in case (i). Before stating the uniqueness result, we need an auxillary lemma.

Lemma 4.1. Let f 1 , f 2 , g ∈ C 1 [0, L] be functions such that f 1 > 0, f 2 > 0, g > 0 on [0, L]; f 1 (0) = f 2 (0).
In addition, suppose that

L 0 f n 1 (s)g(s)ds = L 0 f n 2 (s)g(s)ds (4.1)
for all integers n = 0, 1, 2 • • • . Then it holds that f 1 = f 2 on [0, L].
Proof. Without loss of generality we assume that f 1 (0) = f 2 (0) = 0. Otherwise, we may consider the functions s → f j (s) -f j (0) in place of f j . To prove Lemma 4.1, we first show f 1 (L) = f 2 (L) and then apply the moment theory to get

f 1 ≡ f 2 .
Assume without loss of generality that f 1 (L) > f 2 (L). Write f 1 (L) = c and sup x∈(0,L) g(x) = M . Since f 1 (s) > 0 and f 1 (0) = 0, we have c > 0. Therefore, there exists sufficiently small positive numbers > 0 and δ 1 , δ 2 > 0 such that

f 1 (s) ≥ c -δ 1 , f 2 (s) ≤ c -2δ 1 , g(s) ≥ δ 2 for all s ∈ [L -2 , L -], f 1 (s) > f 2 (s) for all s ∈ [L -2 , L].
Using the above relations, we deduce from (4.1) that

0 = L 0 f n 1 (s)g(s) -f n 1 (s)g(s)ds = L L- f n 1 (s)g(s) -f n 2 (s)g(s)ds + L- L-2 f n 1 (s)g(s) -f n 2 (s)g(s)ds + L-2 0 f n 1 (s)g(s) -f n 2 (s)g(s)ds ≥ L- L-2 f n 1 (s)g(s) -f n 2 (s)g(s)ds - L-2 0 f n 2 (s)g(s)ds ≥ δ 2 (c -δ 1 ) n -(c -2δ 1 ) n -(L -2 )M (c -2δ 1 ) n ≥ (c -δ 1 ) n δ 2 -( δ 2 + (L -2 )M ) c -2δ 1 c -δ 1 n , which means that (c -δ 1 ) n δ 2 -( δ 2 + (L -2 )M ) c -2δ 1 c -δ 1 n ≤ 0 for all integers n = 0, 1, 2 • • • . However, since c-2δ 1 c-δ 1 < 1, there exists a sufficiently large integer N > 0 such that δ 2 -( δ 2 + (L -2 )M ) c -2δ 1 c -δ 1 N > 0.
Then we obtain

(c -δ 1 ) N δ 2 -( δ 2 + (L -2 )M ) c -2δ 1 c -δ 1 N > 0,
which is a contradiction. Therefore, we obtain

f 1 (L) = f 2 (L). Denote c = f 1 (0) = f 2 (0) and d = f 1 (L) = f 2 (L). Since f j is monotonically increasing, the relation τ = f j (s) implies that s = f -1 j (τ ) for all s ∈ [0, L] and τ ∈ [c, d].
Using the change of variables, we get

L 0 f n j (s)g(s)ds = d c τ n g • (f -1 j (τ ))(f -1 j ) (τ )dτ, j = 1, 2.
Hence, it follows from (4.1) that

d c τ n dµ = d c τ n dν, (4.2)
where µ and ν are two Lebesgue measures such that

dµ = g • (f -1 1 (τ ))(f -1 1 ) (τ )dτ, dν = g • (f -1 2 (τ ))(f - 1
2 ) (τ )dτ. By the Stone-Weierstrass theorem, it is easy to note from (4.2) that dµ = dν, which means

g • (f -1 1 (τ ))(f -1 1 ) (τ ) = g • (f -1 2 (τ ))(f -1 2 ) (τ ) for all τ ∈ [c, d]. (4.3)
Introduce two functions

F 1 (τ ) = f -1 1 (τ ) 0 g(s)ds, F 2 (τ ) = f -1 2 (τ ) 0 g(s)ds.
Hence, from (4.3) we deduce

F 1 (τ ) = F 2 (τ ) for τ ∈ [c, d]. Moreover, since f -1 1 (c) = f -1 2 (c) = 0, we have F 1 (c) = F 2 (c) = 0 and then F 1 (τ ) = F 2 (τ ) for τ ∈ [c, d], i.e., f -1 1 (τ ) 0 g(s)ds = f -1 2 (τ ) 0 g(s)ds. (4.4)
From (4.4), it is easy to know that f -1

1 (τ ) = f -1 2 (τ ) for all τ ∈ [c, d]. Otherwise, suppose f -1 1 (τ 0 ) = f -1 2 (τ 0 ) at some point τ 0 ∈ [c, d]. Since g(s)
> 0 for all s ∈ (0, L), we obtain that

f -1 1 (τ 0 ) 0 g(s)ds = f -1 2 (τ 0 ) 0 g(s)ds,
which is a contradiction. Consequently, we obtain f -1 1 = f -1 2 and thus f 1 (s) = f 2 (s) for all s ∈ [0, L]. The proof is complete.

Our uniqueness result for the determination of a is stated as follows.

Theorem 4.2. Assume that a(0) = O ∈ R 3 is located at the origin and that each component a j , j = 1, 2, 3 of a satisfies |a i (t)| < 1 for t ∈ [0, T 0 ]. Then the function a(t), t ∈ [0, T 0 ] can be uniquely determined by the data set {E(x, t) × ν : x ∈ Γ, t ∈ (0, T )}.

Proof. Assume that there are two orbit functions a and b such that

∂ 2 t E 1 (x, t) + ∇ × (∇ × E 1 (x, t)) = J (x -a(t))g(t), x ∈ R 3 , t > 0, E 1 (x, 0) = ∂ t E 1 (x, 0) = 0, x ∈ R 3 ,
and

∂ 2 t E 2 (x, t) + ∇ × (∇ × E 2 (x, t)) = J (x -b(t))g(t), x ∈ R 3 , t > 0, E 2 (x, 0) = ∂ t E 2 (x, 0) = 0, x ∈ R 3 .
Here we assume that b(0) = O and |b j (t)| < 1 for t ∈ [0, t 0 ] and j = 1, 2, 3. We need to show

a(t) = b(t) in (0, T 0 ) if E 1 (x, t) × ν(x) = E 2 (x, t) × ν for x ∈ Γ, t ∈ (0, T ).
For each unit vector d, we can choose two unit polarization vectors p, q such that p • d = 0, q = p × d. Letting E = E 1 -E 2 and following similar arguments as those of Theorem 3.1, we obtain

p • Ĵ (κd) T 0 g(t)e -iκd•a(t) e -iκt dt = p • Ĵ (κd) T 0 g(t)e -iκd•b(t) e -iκt dt, (4.5)
q • Ĵ (κd) T 0 g(t)e -iκd•a(t) e -iκt dt = q • Ĵ (κd) T 0 g(t)e -iκd•b(t) e -iκt dt, (4.6)
and

d • Ĵ (κd) = 0, which means Ĵ (κd) = p • Ĵ (κd)p + q • Ĵ (κd)q.
Therefore, since J = 0, for each unit vector d there exists a sequence {κ j } +∞ j=1 such that lim j→0 κ j = 0 and for each κ j , either p • Ĵ (κ j d) = 0 or q • Ĵ (κ j d) = 0. Hence from (4.5)-(4.6) we have

T 0 e -iκ j d•a(t) e -iκ j t g(t)dt = T 0 e -iκ j d•b(t) e -iκ j t g(t)dt, j = 1, 2, • • • . (4.7)
Expanding e -iκ j d•a(t) e -iκ j t and e -iκ j d•a(t) e -iκ j t into power series with respect to κ j , we write (4.7) as

∞ n=0 α n n! κ n j = ∞ n=0 β n n! κ n j , (4.8)
where

α n := T 0 (d • a(t) + t) n g(t)dt, β n := T 0 (d • b(t) + t) n g(t)dt, n = 1, 2 • • • .
In view of the fact that supp(g) ⊂ [0, T 0 ], we get

α n = T 0 0 (d • a(t) + t) n g(t)dt, β n = T 0 0 (d • b(t) + t) n g(t)dt, n = 1, 2 • • • .
Since (4.8) holds for all κ j and lim j→∞ κ j = 0, it is easy to conclude that α n = β n for n = 0, 1, 2 • • • . Choosing d = (1, 0, 0), we have

(a 1 (t) + t) = 1 + a 1 (t) > 0, (b 1 (t) + t) = 1 + b 1 (t) > 0, a 1 (0) = b 1 (0).
It follows from α n = β n and Lemma 4.1 that a 1 (t) = b 1 (t) for t ∈ [0, T 0 ]. Similarly letting d = (0, 1, 0) and d = (0, 0, 1) we have a 2 (t) = b 2 (t) and a 3 (t) = b 3 (t) for t ∈ [0, T 0 ], respectively, which proves that

a(t) = b(t) for t ∈ [0, T 0 ].
Remark 4.3. In Theorem 4.2, it states that we can only recover the function a(t) over the finite time period [0, T 0 ] because the moving source radiates in this time period, i.e., supp(g) = [0, T 0 ]. The information of a(t) for t > T 0 cannot be retrieved. The monotonicity assumption a j ≥ 0 for j = 1, 2, 3 can be replaced by the following condition: there exist three linearly independent unit directions d j , j = 1, 2, 3 such that

|d j • a (t)| < 1, t ∈ [0, T 0 ], j = 1, 2, 3.
Note that this condition can always be fulfilled if the source moves along a straight line with the speed less than one.

4.2.

Uniqueness to IP2 in case (ii). For simplicity of notation, let x = (x 1 , x 2 , 0)

for x = (x 1 , x 2 , x 3 ) and R 2 = {x ∈ R 3 : x 3 = 0}. Let ã(t) ∈ R 2 for all t ∈ [0, T 0 ].
In this subsection, we assume that

F (x, t) = J (x -ã(t)) h(x 3 ) g(t), x ∈ R 3 , t ∈ R + ,
where

J (x) = (J 1 (x), J 2 (x), 0) ∈ H 2 (R 2 ) 3 depends only on x and h ∈ H 2 (R), supp(h) ⊂ (-R, R) √ 2/2
. Moreover, we assume that h does not vanish identically and

supp(J ) ⊂ {x ∈ R 2 : |x| < R√ 2/2}, ∇ xJ (x) = 0.
The temporal function g is defined the same as in the previous sections. The above assumptions imply that we still have supp(F ) ⊂ B R × [0, T 0 ] and div F = 0 in R 3 . We consider the inhomogeneous Maxwell system

∂ 2 t E(x, t) + ∇ × (∇ × E(x, t)) = J (x -ã(t)) h(x 3 ) g(t), x ∈ R 3 , t > 0, E(x, 0) = ∂ t E(x, 0) = 0, x ∈ R 3 . (4.9)
Since the equation (4.9) is a special case of (1.1), the results of Lemmas 2.1 and 2.2 also apply to this case. For our inverse problem, it is assumed that J ∈ A is a given source function, where the admissible set

A = {J = (J 1 , J 2 , 0) : J i (0) > J i (x) for i = 1 or i = 2 and all x = 0}.
The x 3 -dependent function h is also assumed to be given. We point out that these a priori information of J and h are physically reasonable, while J and h can be regarded as approximation of the Dirac functions (for example, Gaussian functions) with respect to x and x 3 , respectively. Our aim is to recover the unknown orbit function

ã(t) ∈ C 1 ([0, T 0 ]) 2 which has a upper bound |ã(t)| ≤ R 1 for some R 1 > 0 and for all t ∈ [0, T 0 ]. Let R > R + R 1 and T = T 0 + R + R + R 1 .
Below we prove that the tangential trace of the dynamical magnetic field on Γ R × (0, T ) can be uniquely determined by that of the electric field. It will be used in the subsequent uniqueness proof with the data measured on the whole surface Γ R . Lemma 4.4. Assume that the electric field

E ∈ C([0, T ]; H 2 (R 3 )) 3 ∩ C 2 ([0, T ]; L 2 (R 3 )) 3 satisfies ∂ 2 t E(x, t) + ∇ × (∇ × E(x, t)) = 0, |x| > R, t ∈ (0, T ), E(x, 0) = ∂ t E(x, 0) = 0, x ∈ R 3 . If E × ν = 0 on Γ R × (0, T ), then (∇ × E) × ν = 0 on Γ R × (0, T ).
Proof. Let us assume that E × ν = 0 on Γ R × (0, T ) and consider V defined by

V (x, t) = t 0 E(x, s)ds, (x, t) ∈ R 3 × (0, T ).
In view of (4.4) and the fact that E(x, t) × ν = 0 on Γ R × (0, T ), we find

     ∂ 2 t V (x, t) + ∇ × (∇ × V (x, t)) = 0, |x| > R, t ∈ (0, T ), V (x, 0) = ∂ t V (x, 0) = 0, x ∈ R 3 , ∂ t V (x, t) × ν(x) = 0, (x, t) ∈ Γ R × (0, T ).
(4.10)

We define the energy E associated to V on

Ω := {x ∈ R 3 : |x| > R} E(t) := Ω (|∂ t V (x, t)| 2 + |∇ x × V (x, t)| 2 )dx, t ∈ [0, T ]. Since E ∈ C([0, T ]; H 2 (R 3 )) 3 ∩ C 2 ([0, T ]; L 2 (R 3 )) 3 , we have V ∈ C([0, T ]; H 2 (R 3 )) 3 ∩ C 1 ([0, T ]; H 1 (R 3 )) 3 ∩ C 2 ([0, T ]; L 2 (R 3 )) 3 .
It follows that E ∈ C 1 ([0, T ]). Moreover, we get

E (t) = 2 Ω [∂ 2 t V (x, t) • ∂ t V (x, t) + (∇ x × V (x, t)) • (∇ x × ∂ t V (x, t))] dx.
Integrating by parts in x ∈ Ω and applying (4.10), we obtain

E (t) = 2 Ω [∂ 2 t V + ∇ x × (∇ x × V )] • ∂ t V (x, t) dx + 2 Γ R (∇ x × V ) • (ν × ∂ t V (x, t))ds = 0.
This proves that E is a constant function. Since

E(0) = Ω (|∂ t V (x, 0)| 2 + |∇ x × V (x, 0)| 2 )dx = 0,
we deduce E(t) = 0 for all t ∈ [0, T ]. In particular, we have

Ω |E(x, t)| 2 dx = Ω |∂ t V (x, t)| 2 dx ≤ E(t) = 0, t ∈ [0, T ]. This proves that E(x, t) = 0, |x| > R, t ∈ (0, T ),
which implies that (∇ × E) × ν = 0 on Γ R × (0, T ) and completes the proof.

In the following lemma, we present a uniqueness result for recovering ã from the tangential trace of the electric field measured on Γ R . Our arguments are inspired by a recent uniqueness result [START_REF] Hu | Uniqueness and stability for the recovery of a time-dependent source and initial conditions in elastodynamics[END_REF] to inverse source problems in elastodynamics. Compared to the uniqueness result of Theorem 4.2, the slow moving assumption of the source is not required in the following Theorem 4.5.

Theorem 4.5. Assume that J ∈ A and the non-vanishing function h are both known. Then the function ã(t), t ∈ [0, T 0 ] can be uniquely determined by the data set {E(x, t) × ν : x ∈ Γ R , t ∈ (0, T )}.

Proof. Assume that there are two functions ã and b such that

∂ 2 t E 1 (x, t) + ∇ × (∇ × E 1 (x, t)) = J (x -ã(t))h(x 3 )g(t), x ∈ R 3 , t > 0, E 1 (x, 0) = ∂ t E 1 (x, 0) = 0, x ∈ R 3 , (4.11)
and

∂ 2 t E 2 (x, t) + ∇ × (∇ × E 2 (x, t)) = J (x -b(t))h(x 3 )g(t), x ∈ R 3 , t > 0, E 2 (x, 0) = ∂ t E 2 (x, 0) = 0, x ∈ R 3 . (4.12) It suffices to show that ã(t) = b(t) in (0, T 0 ) if E 1 (x, t) × ν = E 2 (x, t) × ν for x ∈ Γ R , t ∈ (0, T ). Denote E = E 1 -E 2 and f (x, t) = J (x -ã(t))g(t) -J (x -b(t))g(t).
Subtracting (4.11) from (4.12) yields

∂ 2 t E(x, t) + ∇ × (∇ × E(x, t)) = f (x, t)h(x 3 )g(t), x ∈ R 3 , t > 0, E(x, 0) = ∂ t E(x, 0) = 0, x ∈ R 3 . (4.13)
Since h does not vanish identically, we can always find an interval Λ = (a -, a

+ ) ⊂ R + such that R√ 2/2 - R√ 2/2 e λx 3 h(x 3 )dx 3 = 0, ∀λ ∈ Λ. (4.14)
Set

H := {(x 1 , x 2 ) : a 2 -< x 2 2 -x 2 1 < a 2 + , x 1 > 0, x 2 > 0}, which is an open set in R 2 . We choose a test function F (x, t) of the form F (x, t) = pe -iκ 1 t e -iκ 2 d•x e √ κ 2 2 -κ 2 1 x 3 ,
where d = (d 1 , d 2 , 0) is a unit vector, p = (p 1 , p 2 , 0) is a unit vector orthogonal to d, and κ 1 , κ 2 are positive constants such that (κ 1 , κ 2 ) ∈ H. It is easy to verify that

∂ 2 t F (x, t) + ∇ × (∇ × F (x, t)) = 0. (4.15)
Since E(x, t) × ν = 0 on Γ R , from Lemma 4.4, we also have (∇ × E(x, t)) × ν = 0 on Γ R . Consequently, multiplying both sides of the Maxwell system by F and using integration by parts over [0, T ] × B R , we can obtain from (4.15) that

T 0 B R f (x, t)h(x 3 ) • F (x, t)dxdt = T 0 B R ∂ 2 t E(x, t) + ∇ × (∇ × E(x, t)) • F (x, t)dxdt = T 0 Γ R ν × (∇ × E(x, t)) • F (x, t) -ν × (∇ × F (x, t)) • E(x, t)dsdt = T 0 Γ R ν × (∇ × E(x, t)) • F (x, t) -(E(x, t) × ν) • (∇ × F (x, t))dsdt = 0.
Note that in the last step we have used Lemma 4.4. Recalling the definition of F and f , we obtain from the previous identity that

R√ 2/2 - R√ 2/2 e √ κ 2 2 -κ 2 1 x 3 h(x 3 )dx 3 p • T 0 B R f (x, t)e -iκ 1 t e -iκ 2 d•x dxdt = 0.
In view of (4.14) and the choice of κ 1 , κ 2 , we get

p • T 0 B R f (x, t)e -iκ 1 t e -iκ 2 d•x dxdt = 0.
For a vector v(x, t) ∈ R 3 , denote by v(ξ), ξ ∈ R 3 the Fourier transform of v with respect to the variable (x, t), i.e.,

v(ξ) = R 3 v(x, t)e -iξ•(x,t) dxdt.
Consequently, it holds that

p • f (κ 2 d, κ 1 ) = 0
for all κ 2 > κ 1 > 0 and | d| = 1.

On the other hand, since ∇ x • J = 0, we have

∇ x • f = 0. Hence, d • T 0 B R f (x, t)e -iκ 1 t e -iκ 2 d•x dxdt = - 1 iκ 2 T 0 B R f (x, t) • ∇ xe -iκ 2 d•x dxdt = 1 iκ 2 T 0 B R ∇ x • f (x, t)e -iκ 2 d•x dxdt = 0, which means d • f (κ 2 d, κ 1 ) = 0 for all (κ 1 , κ 2 ) ∈ H and | d| = 1.
Since both d and p are orthonormal vectors in R 2 , they form an orthonormal basis in R 2 . Therefore we have

f (κ 2 d, κ 1 ) = d • f (κ 2 d, κ 1 ) d + p • f (κ 2 d, κ 1 )p = 0 for all (κ 1 , κ 2 ) ∈ H and | d| = 1. Since f is analytic in R 3 and {(κ 1 , κ 2 d) : (κ 1 , κ 2 ) ∈ H, | d| = 1}
is an open set in R 3 , we have f (ξ) = 0 for all ξ ∈ R 3 , which means f (x, t) ≡ 0 and then

J (x -ã(t))g(t) = J (x -b(t))g(t)
for all x ∈ R 2 and t > 0. This particulary gives J (x -ã(t)) = J (x -b(t)) for all t ∈ (0, T 0 ), x = (x 1 , x 2 , 0).

(4.16)

Assume that there exists one time point t 0 ∈ (0, T 0 ) such that ã(t 0 ) = b(t 0 ). By choosing x = ã(t 0 ) we deduce from (4.16) that

J (0) = J (ã(t 0 ) -b(t 0 )),
which is a contradiction to our assumption that J ∈ A. This finishes the proof of ã(t) = b(t) for t ∈ [0, T 0 ].

Remark 4.6. The proof of Theorem 4.5 does not depend on the Fourier transform of the electromagnetic field in time, but it requires the data measured on the whole surface Γ R . However, the Fourier approach presented in the proof of Theorems 3.1 and 4.2 straightforwardly carries over to the proof of Theorem 4.5 without any additional difficulties. Particulary, the result of Theorem 4.5 remain valid with the partial data {E(x, t) × ν : x ∈ Γ ⊂ Γ R , t ∈ (0, T )}.

Remark 4.7. In the case of the scalar wave equation,

∂ 2 t u(x, t) + ∇ × (∇ × u(x, t)) = J(x -ã(t)) h(x 3 ) g(t), x ∈ R 3 , t > 0, u(x, 0) = ∂ t u(x, 0) = 0, x ∈ R 3 ,
where J : R 2 → R + is a scalar function compactly supported on {(x 1 , x 2 ) ∈ R 2 : x 2 1 + x 2 2 < R2 }. Then, following the same arguments as in the proof of Theorem 4.5, one can prove that ã(t), t ∈ [0, T 0 ] can be uniquely determined by the data set {E(x, t) × ν : x ∈ Γ ⊂ Γ R , t ∈ (0, T )}.

INVERSE MOVING SOURCE PROBLEM FOR A DELTA DISTRIBUTION

As seen in the previous sections, when the temporal function g is supported on [0, T 0 ], it is possible to recover the moving orbit function a(t) for t ∈ [0, T 0 ]. In this section we consider the case where the temporal function shrinks to the Dirac distribution g(t) = δ(t -t 0 ) with some unknown time point t 0 > 0. Our aim is to determine t 0 and a(t 0 ) from the electric data at a finite number of measurement points.

Consider the following initial value problem of the time-dependent Maxwell equation

∂ 2 t E(x, t) + ∇ × (∇ × E(x, t)) = -J (x -a(t))δ(t -t 0 ), x ∈ R 3 , t > 0, E(x, 0) = ∂ t E(x, 0) = 0, x ∈ R 3 . (5.1)
Since ∇ • J = 0, the electric field E(x) in this case can be expressed as

E(x, t) = ∞ 0 R 3 G(x -y, t -s)J (y -a(s))δ(s -t 0 )dyds = ∞ 0 R 3 1 4π|x -y| δ(|x -y| -(t -s))J (y -a(s))δ(s -t 0 )dyds - ∞ 0 R 3 ∇ x ∇ x 1 4π|x -y| H(|x -y| + s -t) J (y -a(s))δ(s -t 0 )dyds = ∞ 0 R 3 1 4π|x -y| δ(|x -y| -(t -s))J (y -a(s))δ(s -t 0 )dyds - ∞ 0 R 3 ∇ y ∇ y 1 4π|x -y| H(|x -y| + s -t) J (y -a(s))δ(s -t 0 )dyds = R 3 1 4π|x -y| δ(|x -y| -(t -t 0 ))J (y -a(t 0 ))dy. (5.2)
Before stating the main theorem of this section, we describe the strategy for the choice of four measurement points (or receivers) on the sphere Γ R . The geometry is shown in Figure 1. First, we choose arbitrarily three different points x 1 , x 2 , x 3 ∈ Γ R . Denote by P the uniquely determined plane passing through x 1 , x 2 and x 3 , and by L the line passing through the origin and perpendicular to P . Obviously the straight line L has two intersection points with Γ R . Choose one of the intersection points with the longer distance to plane P as the fourth point x 4 . If the two intersection points have the same distance to P , we can choose either one of them as x 4 . By our choice of x j , j = 1, 2, 3, 4, they cannot lie on one side of any plane passing through the origin, if the plane P determined by x j , j = 1, 2, 3 does not pass through the origin.

Theorem 5.1. Let the measurement positions x j ∈ Γ R , j = 1, • • • , 4 be given as above and let J be specified as in the introduction part. We assume additionally that supp(J ) = B R and there exists a small constant δ > 0 such that |J i (x)| > 0 for all R -δ ≤ |x| ≤ R and i = 1, 2, 3. Then both t 0 and a(t 0 ) can be uniquely determined by the data set {E(x j , t)

: j = 1, • • • , 4, t ∈ (0, T )}, where T = t 0 + R + R 1 + R.
Proof. Analogously to Lemma 2.1, one can prove that E(x, t) = 0 for all x ∈ B R and t > T . Taking the Fourier transform of E(x, t) in (3.1) with respect to t and making use of the representation of E in (5.2), we obtain

Ê(x, κ) = R 3 e iκ(t 0 +|x-y|) |x -y| J (y -a(t 0 ))dy = e iκt 0 ∞ 0 e iκρ 1 ρ Γρ(x) J (y -a(t 0 ))dydρ, (5.3)
where Γ ρ (x) := {y ∈ R 3 : |y -x| = ρ}. Assume that there are two orbit functions a and b and two time points t 0 and t0 such that

∂ 2 t E 1 (x, t) + ∇ × (∇ × E 1 (x, t)) = -J (x -a(t))δ(t -t 0 ), x ∈ R 3 , t > 0, E 1 (x, 0) = ∂ t E 1 (x, 0) = 0, x ∈ R 3 , and
∂ 2 t E 2 (x, t) + ∇ × (∇ × E 2 (x, t)) = -J (x -b(t))δ(t -t0 ), x ∈ R 3 , t > 0, E 2 (x, 0) = ∂ t E 2 (x, 0) = 0, x ∈ R 3 .
We need to prove t 0 = t0 and a(t 0 ) = b( t0 ) under the condition E 1 (x j , t) = E 2 (x j , t) for t ∈ [0, T ] and j = 1, 2, 3, 4. Below we denote by x ∈ Γ R one of the measurement points x j (j = 1, • • • , 4). Introduce the functions F , F a , F b : R + → R as follows: Physically, the right and left hand sides of the above identity represent the difference of the flight time between x and a(t 0 ), b( t0 ). Note that the wave speed has been normalized to one for simplicity. Finally, we prove that the identity (5.6) cannot hold simultaneously for our choice of measurement points x j ∈ Γ R (j = 1, • • • , 4). Obviously, the set {x ∈ R 3 : |x -b( t0 )| -|x -a(t 0 )| = t 0 -t0 } represents one sheet of a hyperboloid. This implies that x j (j = 1, 2, 3, 4) should be located on one half sphere of radius R excluding the corresponding equator, which is a contradiction to our choice of x j . Then we have t 0 = t0 and (5.6) then becomes |x -b(t 0 )| -|x -a(t 0 )| = 0. This implies that x 1 , x 2 , x 3 , x 4 should be on the same plane. This is also a contradiction to our choice of x i , i = 1, • • • , 4. Then we have a(t 0 ) = b(t 0 ). Remark 5.2. If the source term on the right hand side of (5.1) takes the form F (x, t) = -J (x -a(t)) m j=1 δ(t -t j ), with the impulsive time points

F (ρ) = 1 ρ Γρ(x) J (
t 1 < t 2 < • • • < t m , |t j+1 -t j | > R.
One can prove that the set {(t j , a(t j )) : j = 1, 2, • • • , m} can be uniquely determined by {E(x j , t) : j = 1, • • • , 4, t ∈ (0, T )}, where T = t m + R + R 1 + R. In fact, for 2 ≤ j ≤ m, one can prove that (t j , a(t j )) can be uniquely determined by {E(x j , t) : j = 1, • • • , 4, t ∈ (T j-1 , T j )}, where T j = T j-1 + t j and T 1 := t 1 + R + R 1 + R.

FIGURE 1 .

 1 FIGURE 1. Geometry of the four measurement points.

1 ρ

 1 y)dy, F a (ρ) = Γρ(x)J (y -a(t 0 ))dy,F b (ρ) = 1 ρ Γρ(x)J (y -b( t0 ))dy.Since supp(J ) = B R and by our assumption, each component J j (x) (j = 1, 2, 3) is either positive or negative in a small neighborhood of Γ R, we can obtain thatinf{ρ ∈ supp(F )} = |x| -R, sup{ρ ∈ supp(F )} = |x| + R, inf{ρ ∈ supp(F a )} = |x -a(t 0 )| -R, sup{ρ ∈ supp(F a )} = |x -a(t 0 )| + R, inf{ρ ∈ supp(F b )} = |x -b( t0 )| -R, sup{ρ ∈ supp(F b )} = |x -b( t0 )| + R. (5.4) Since E 1 (x, t) = E 2 (x, t), t ∈ [0, T ] for some point x ∈ ∂B R , from (5.3) we have e iκt 0 F a (κ) = e iκ t0 F b (κ)for all κ > 0, which meansF a (κ) = e -iκ(t 0 -t0 ) F b (κ). (5.5)Recalling the property of the Fourier transform,F b (ρ -(t 0 -t0 ))(κ) = e -iκ(t 0 -t0 ) F b (κ),we deduce from (5.5) thatF b (ρ -(t 0 -t0 )) = F a (ρ), ρ ∈ R + . Particularly, inf{supp(F b (• -(t 0 -t0 )))} = inf{supp(F a (•))}, sup{supp(F b (• -(t 0 -t0 )))} = sup{supp(F a (•))}.Therefore, we derive from (5.4) that|x -b( t0 )| -R + (t 0 -t0 ) = |x -a(t 0 )| -R,|x -b( t0 )| + R + (t 0 -t0 ) = |x -a(t 0 )| + R, which means |x -b( t0 )| -|x -a(t 0 )| = t0 -t 0 .(5.6)

ACKNOWLEDGEMENT

The work of G. Hu is supported by the NSFC grant (No. 11671028) and NSAF grant (No. U1530401). The work of Y. Kian is supported by the French National Research Agency ANR (project MultiOnde) grant ANR-17-CE40-0029.