Morgan Le Delliou
email: morgan.ledelliou@uam.es

L   E Merger as Intermittent Accretion

Keywords: Cosmology, theory -Dark Matter -Galaxies, formation -galaxies, halos -gravitation

published or not. The documents may come

Introduction

Structure formation in the Cold Dark Matter (CDM, or more simply DM) paradigm is dominated by the hierarchical picture of repeated mergers. This picture was emphasised by Syer & White (1998), explaining the dynamical formation of halo density profile with a feedback mechanism provided by repeated mergers. Whereas it is now believed that isotropisation of the velocity dispersion (angular momentum; see Le Delliou & Henriksen 2003; Barnes et al. 2005;MacMillan et al. 2006) via the radial-orbit instability (also viewed as adiabatic variability of self-similarity, Henriksen 2007) is responsible for the density profile formation, their picture remains a widely accepted description of the merger digestion mechanism. Despite its simple spherical symmetry and apparent lack of compliance with the merger paradigm, some studies have shown that the Secondary Infall Model (SIM) is a viable model to predict the structure and density profile evolutions of DM haloes as compared to N-body simulations (Ascasibar et al. 2007;Salvador-Solé et al. 2007).

This letter proposes to understand this paradox by examining the merger paradigm within the SIM and studying how merger events impact on the relaxation and structure of a CDM halo.

The SIM stems from the seminal work of Gunn & Gott (1972), and the SSIM (Self-similar SIM) started when Fillmore & Goldreich (1984) and Bertschinger (1984) independently found self-similar solutions to the SIM. It was later shown that those solutions can be reached from non-self-similar initial conditions (e.g. in Hoffman & Shaham 1985;White & Zaritsky 1992;Ryden 1993;Henriksen & Widrow 1995, 1997;Avila-Reese et al. 1999;Henriksen & Widrow 1999;del Popolo et al. 2000;Henriksen & Le Delliou 2002;Le Delliou & Henriksen 2003) and a systematic approach to the SSIM was used in Henriksen & Widrow (1995, 1997, 1999); Henriksen & Le Delliou (2002); Le Delliou & Henriksen (2003), derived from the Carter-Henriksen formalism (Carter & Henriksen 1991, hereafter CH). Some extensions to the SIM were proposed that included the effects of angular momentum to explain flat halo cusps (Hiotelis 2002;Le Delliou & Henriksen 2003;Ascasibar et al. 2004;Williams et al. 2004;Lu et al. 2006), but no funda-mental attempt was made before Le Delliou (2002) to confront the SIM with the merger paradigm.

The following section (Sec. 2) will describe how and why the SSIM can be extended to model a merger event. Then Sec. 3 will discuss how the symmetry of the SSIM still allows for a form of tidal stripping and dynamical friction, before presenting the consequences of such a merger in the SSIM in Sec. 4, and to make some concluding remarks in Sec. 5.

Merger in an Infall

Modelling a merger event in a spherical geometry may appear contradictory but it is possible to a certain extent. To understand this it is important to realise the following: a very small amount of substructures are seen in N-body simulations; Diemand et al. (2007) find that only 5.3% of the total mass fraction of haloes lie in subhaloes. In the Syer & White (1998) picture, incoming satellite haloes merge with their parent, fall in the centre and contribute to the density profile and to the parent's relaxation and virialisation. However, in simulations, subobjects swing back and forth several times in their parents before being digested. That process can be modelled in a simpler way: on average, spherical symmetry is not bad (Ascasibar et al. 2007) as it reproduces the correct time scales and density profiles. Shell codes are much simpler than N-body codes and therefore provide with robust tests of certain aspects of their results. Other simplifying approaches have been used to understand halo formation, such as phase-space coarse graining (Le Delliou & Henriksen 2003;Henriksen 2004Henriksen , 2006) ) or in the one dimensional slab model used in Binney (2004), where it was shown to explain the formation of cosmic web sheets through the interplay of phase mixing and violent relaxation, also present in spherical models. Henriksen & Widrow (1999) have shown that relaxation is moderately violent (in their figure 9) and induced by a phase space instability (Henriksen & Widrow 1997). Section 3 will detail how another perspective of phase mixing and moderately violent relaxation through phase space instability can be interpreted as some sort of tidal stripping and dynamical friction.

In 1. Density, mass and mass perturbation ratios defining the satellite initial OD for the mergers in the SSIM. The mass perturbation measures the perturbation of the OD compared to the background halo region it spans, just before entering the core. First column gives parent initial power law seed and panel order in reference to figures 1, 2 and 3.

namical Lagrangian treatment of infall using the CH (Carter & Henriksen 1991) self-similar variables that reveals when the system reaches naturally a self-similar regime. A halo is modelled from a radial power law perturbation δρ/ρ ∝ r -ǫ on an Einsteinde Sitter homogeneous background, that is evolved to reach its quasi-stationary self-similar regime in its core1 (Henriksen & Widrow 1999). The SIM is known to establish a self-similar infall phase (Henriksen & Widrow 1997), which then leads to a semi-universal power law density profile (Fillmore & Goldreich 1984;Bertschinger 1984): for initial power index ǫ ≤ 2, the isothermal sphere (ρ ∝ r -µ with µ = 2) is the semi-universal attractor, whereas with ǫ > 2, there is a continuum of attractors with µ = 3ǫ/(1+ǫ). Positive overdensity and the requirement of a finite initial core mass in the centre limit the range to 0 ≤ ǫ < 3. The cores explored here were chosen, as presented in Table 1, according to their SSIM behaviour defined by their initial power index: typical shallow (ǫ = 3/2) and steep (ǫ = 5/2) profiles, with the addition of an extreme steep case (ǫ = 2.9) to test the behaviour of a highly concentrated parent halo. The steep and shallow denominations refer to the comparison relative to the isothermal sphere.

In this geometry, an overdensity (hereafter OD, or satellite), representing a spherically averaged satellite halo, is a region of overdense shells close to the edge of the core, the parent halo (hereafter core, or parent).

The OD is evolved dynamically from an initial gaußian density profile added on top of the background halo profile over a finite region. That evolution runs long enough to observe the signature of the OD's own stationary regime in phase space. This is manifested in the mixing of its Liouville sheet during the OD's dynamical mass accretion of halo shells from its environment. The OD's definition as a set of particles (shells) is frozen when the core swallows it.

At that point are recorded the ratios of OD-over-core masses, M ratio , of their densities, D ratio , and the measure of the perturbation provided by the OD on its background surroundings, in mass, M OD /M BG . For each case, three different satellites were chosen, trying to obtain various types of mass and density ratios between satellites and parents.

Since they were allowed to accrete mass dynamically from their environment, ODs were laid close to the edge of the core to maintain some control over the final frozen mass and density ratios. Some configurations of those ratios were too difficult to obtain: in the shallow case, with high M ratio , lower values for D ratio were prevented by the high density background the OD accretes from, while for the steep cases, also with high M ratio , higher D ratio couldn't be obtained because of their cores' poor density backgrounds which tended to spread the ODs (see Sec. 4's tidal effect). Virial ratio and phase space diagrams at the end of the self-similar phase for three sets of ODs in the ǫ = 3 2 = 1.5 case. Overdensity shells are emphasised in green in phase space. Digestion time is defined from OD core entry (pre-spike, see Sec. 4) to virial recovery (measured on left panels). Zoomed encapsulation of those spikes in middle and lower left panels show their measure in troughless cases. T, X and Y are respectively the self-similar time, radius and radial velocity, which units are set by G = M(∞) = 1 (Henriksen & Widrow 1997).

The ratios indicated are measured at the time of core entry. The explored values are presented in Table 1.

It is crucial to point out that the numerical implementation of the SSIM entails a shell code where finite size shells model the continuous system. That will play a role in the discussion of the results.

Merger paradigm and SSIM

Syer & White (1998) have attempted to define the singularity of mergers in an effort, at the time, to explain the universality of the density profile found in N-body simulation by Navarro et al. (1996, hereafter NFW): their key feature is the feedback mechanism between dynamical friction from the parent halo and tidal stripping of the satellite. Even though this is not anymore considered to hold the key to the formation of the density profile, their merger digestion mechanisms is still widely accepted to describe the behaviour of satellites. I argue that both mechanisms can be modelled within the SSIM despite its one-dimensional nature.

Tidal acceleration on an infinitesimal shell of mass dm = 4πρr 2 dr -located at radius r, containing the system mass M and with thickness dr -can be defined as the differential gravity between its boundaries. Defining the cumulative average density profile

ρ r = M(r) 4πr 3 /3 , (1)
the inward oriented elementary tidal acceleration reads, to leading order,

dT =4πGdr ρ - 2 3 ρ r . (2)
It is thus clear that regions of peak density below the cumulative average (ρ < 2 3 ρ r ) will experience a net disruptive tidal acceleration spreading apart shells in those regions, in the radial direction. In this spherically averaged study of a merger, this models tidal stripping. Dynamical friction classically is defined as the creation of a wake by a moving mass in a gravitating medium which back reaction entails a net drag force upon the moving mass. In the SSIM, a massive shell is crossing the core's shell in its travelling inwards or outwards. We will see that a radial drag force, with the correct orientation, is experienced as a result of this motion in the spherically averaged model. This crossing of shells by the OD's results in shells just outside of it feeling more or less mass pulling inwards, depending on the direction of the motion of the massive OD shells. That leads to a differential tightening or spreading of core's shell behind the moving mass, in the fashion of a wake. However in spherical symmetry, an outer wake does not contribute to the pull on the OD. Nevertheless, its mass corresponds to shells that defected from inside because of OD motion, and their effect can be seen in the dynamics (see Appendix A).

In a similar fashion, the dynamical effect on the OD from its motion can be described in terms of a drag force: the crossing of core shells by massive OD shell lead to a decrease, or increase, of the resulting inner mass of the moving OD, depending on the direction of motion. Thus, with inner mass goes the inner pull, which can be interpreted a dragging force that adds to the total force, that should be experienced in the opposite direction of the motion.

Therefore, the SSIM with an outer overdensity can be interpreted to model the main features of the merger paradigm.

Digestions

Indeed, it is possible to keep track, in the Lagrangian shell model, of the defined satellite's (OD's) components once they have been absorbed by the parent (core). The core can be considered isolated at the end of the accretion phase (Henriksen & Widrow 1997). The phase space configurations of simulated merged haloes are displayed on Figs. 1,2, and 3's right panels, distinguishing between the core and OD's accreted shells. This reveals how the different ODs, in their various (shallow or steep) environments, either retain some degree of coherence after being ingested by the core or have been digested and scattered over the core's phase space.

The left panels of Figs. 1, 2, and 3 examine the Virial ratios of the corresponding cores, and show a remarkable robustness in the SSIM: the quasi-stable self-similar phase 2 is shown to be either marginally or strongly disturbed by the OD absorption, but to return to the original undisturbed level of the parent after a digestion time T digestion , provided a mass flow still fuels the self-similar equilibrium. Digestion is manifested by a more or less pronounced initial decrease (entry of extra mass in core increases W), followed by a spike (first crossing of centre gives m OD high velocities, thus peaks K) and then, for stronger disturbance, a trough (energy exchanges from phase space instability, shells spend on average more time at low velocities, thus 2 with Virial markedly different from usual value of 1! 2. There, they are compared with the OD's free fall dynamical time through the core, T dynamical , also indicated on the figures. T dynamical is defined as the free fall time to the centre of a test shell across a constant density distribution, equivalent to the core, in self-similar variables. From Table 2, without Fig. 1's two lowest panels, where the definition of T digestion is problematic, the average T digestion /T dynamical = 3.33, with a standard deviation of 0.77, can be computed. It shows the core digests the OD in 2 to 4 passages in the central relaxation region of phase space. This is comparable to the number of distinguishable Lagrange-Liouville streams present in the core's outer phase space regions, as seen from Figs. 1, 2, and 3's right panels.

From the OD's point of view, the mergers display their effects in phase spaces, represented on Figs. 1, 2, and 3's right panels, on which two features are crucial: the spread (or compactness) of the OD over the core at the end of the infall phase and the presence of some, or all, of its shells in the centre of the core's phase space. This reflects the digestion mechanisms adopted by Syer & White (1998). Their proposal aimed at a dynamical explanation of the NFW profile. Although this explanation is not anymore considered (see Sec. 1), it is interesting to note that the presently discussed single merger model in the SSIM shows signs of inflections (central flattening and edge steepening) from its semi-universal, almost isothermal, density profile. However this is not the focus of this paper.

The OD's compactness resists to tidal stripping while its final presence in the centre is driven by dynamical friction. The fate of a model satellite in the SSIM displays behaviour well in agreement with the merger digestion mechanisms proposed by Syer & White: in the SSIM a combination of density and mass ratios leads to emphasise each effect. High D ratio s seem to be the dominant factor for OD's compactness, while high M ratio s promote the sinking of the OD to the centre of the core's phase space.

All possible qualitative types of behaviour are present: if both ratios, M ratio and D ratio , are strong enough, the OD survives almost intact to the centre of phase space (Figs. 2's lower and 3's middle right panels). If only M ratio is high while D ratio is low, the OD is scattered at the centre (Figs. 1, 2 and 3's upper right panels). Conversely, a high D ratio and low M ratio lead to a compact Virial ratio and phase space diagrams at the end of the self-similar phase, for three sets of ODs in the ǫ = 2.9 case, including an emphasis on digested overdensity shells in phase space and a measure of digestion time. Same units as in Fig. 1.

OD around, but not reaching, the centre of phase space (Fig. 1's lower right panel). Finally if both ratios are too low, the OD is scattered without reaching the centre of phase space (Figs. 1 and 2's middle and 3's lower right panels).

A step further in this phenomenology would be to note that a combination of both ratios should be taken (M ratio .D ratio , see Table 2), for which a threshold can be defined for reaching the centre and another for compactness of the OD. However this classification seems to require an additional dependency with the steepness of the initial profile. Indeed the available data offer different ranges for each initial profile case. The shallow case calls for higher values for the M ratio .D ratio thresholds than the steep cases. This reflects the shallow case's wider spread of material, compared with the steep cases, that the OD has to cross on its journey towards the centre of phase space.

As an illustration of our model, we can assume the Milky Way (hereafter MW) to have a shallow profile and use the corresponding reliable digestion time model, that is with ǫ = 1.5, M ratio = 0.751 and T digestion = 2.50. The corresponding satellite S would have a mass M S ≃ 44M LMC compared to the Large Magellanic Cloud (hereafter LMC), which is huge. The model then yields a very short digestion time, also compared with the age of the oldest stars in the MW T MW = 13.2Gyr, as

T digestion ≃584Myr ≃ T MW 22.6 . (3)
Its dynamical time T dynamical ≃ 234Myr indicates that at the end of digestion, this satellite's shells would be lined between the second incoming and second outgoing inner streams of the core and the model suggests it to then sink to the centre by the end of the MW formation as seen on Fig. 1's upper right panel.

Discussion and conclusions

The SSIM has proven its capacity to model a merger event. Its simplicity allows one to probe the dynamics of the merger and the most remarkable result of this work shows that the selfsimilar quasi-stable regime of quasi-Virial equilibrium is extremely robust to perturbations that can be of comparable size to the core (equal mass mergers): the Virial ratio, after a more or less long period of digestion returns to its stabilised original undisturbed level, after only 2 to 4 passages in the centre, and continues its usual evolution. The spreading and sinking of the satellite's particles across the parents and towards its centre agree with the tidal stripping and dynamical friction picture from Syer & White (1998), provided some adaptation to the language of the SSIM's symmetry. Finally, and this is the claim of this paper, the numerical implementation of the model requiring discretisation, the rapid oscillations of the Virial ratio in the accretion phase offer a novel interpretation in the light of the SSIM merger model: instead of a continuous stream of mass, the model presents a repeated bombardment of finite mass shells that can be understood as small overdensities; Fig. 1's zoomed two lowest right panels show a spike to manifest the weakest mergers digestion; thus the wiggles in the Virial ratio can be interpreted as manifestation of repeated mergers that are at this level indistinguishable from accretion. Therefore there is no fundamental difference between mergers and accretion, the latter being a series of repeated merger with vanishing mass, while the latter is just intermittent accretion. This reconciles approaches such as Salvador-Solé et al. (2007) where accretion was presented as a memory loss mechanism, eliminating the need to refer to mergers.

Figure 2 .

 2 Figure 2. Steep case:Virial ratio and phase space diagrams at the end of the self-similar phase, for three sets of ODs in the ǫ = 5 2 = 2.5 case, including an emphasis on digested overdensity shells in phases space and a measure of digestion time. Same units as in Fig.1.

Figure 3 .

 3 Figure3. Extreme steep case: Virial ratio and phase space diagrams at the end of the self-similar phase, for three sets of ODs in the ǫ = 2.9 case, including an emphasis on digested overdensity shells in phase space and a measure of digestion time. Same units as in Fig.1.

  this paper the SSIM is implemented in a shell code (see details in Le Delliou 2002, and references therein) with fully dy-

			Morgan Le Delliou: Merger as Intermittent Accretion
	ǫ, panel	M ratio	D ratio	M OD /M BG
	3/2, upper panel	0.751	0.282	1.173
	3/2, middle panel 4.25×10 -2 7.10×10 -2 9.38×10 -2
	3/2, lower panel	6.92×10 -2	0.168	1.453
	5/2, upper panel	0.889	5.51×10 -2	0.319
	5/2, middle panel	0.439	5.54×10 -2	0.290
	5/2, lower panel	0.178	0.454	1.133
	2.9, upper panel	0.753	9.19×10 -2	0.416
	2.9, middle panel	0.407	0.641	1.118
	2.9, lower panel	0.301	9.71×10 -2	0.344
	Table

Table 2 .

 2 Digestion and dynamical times and strength parameter of the OD for the mergers in the SSIM. Again, first column gives parent initial power law seed and panel order in figures 1, 2 and 3.

	ǫ, panel	T digestion T dynamical	T digestion T dynamical	M ratio .D ratio
	3 2 , upper p. 3 2 , middle p. 3 2 , lower p. 5 2 , upper p. 5 2 , middle p. 5 2 , lower p. 2.9, upper p.	2.50 0.13 0.13 4.21 3.07 2.11 4.83	0.70 0.73 0.71 1.21 1.12 0.98 1.17	3.57 0.178 0.183 3.48 2.74 2.15 4.13	0.212 3.017×10 -3 1.163×10 -2 4.989×10 -2 2.432×10 -2 8.081×10 -2 6.920×10 -2
	2.9, middle p.	4.94	1.10	4.49	2.609×10 -1
	2.9, lower p.	3.07	1.11	2.77	2.923×10 -2

The core, or self gravitating system, is defined as the set of shells in the multiple flow region. Its edge's radius is that of the outermost shell that has passed only once through the centre, as seen in phase space.

Acknowledgements. The work of MLeD is supported by CSIC (Spain) under the contract JAEDoc072, with partial support from CICYT project FPA2006-05807, at the IFT, Universidad Autónoma de Madrid, Spain, and was also supported by FCT (Portugal) under the grant SFRH/BD/16630/2004, at the CFTC, Lisbon University, Portugal. Evidently, thanks should go the R.N.Henriksen for discussions and comments and for directing MLeD's thesis work from which these results are extracted.

Appendix A: Spherical model of dynamical friction

A thin OD shell in the inward/outward direction, adding mass ±dm OD , crossing shells at r creates a differential acceleration w.r.t. the state without OD which induces an infinitesimal displacement, thus a wake,

This wake of mass dm W = ρr 2 dr induces on the OD an acceleration (backreaction)

Morgan Le Delliou: Merger as Intermittent Accretion 5 opposite to the direction of motion. In addition, the amplitude of the drag force is shown proportional to dm OD ρ, related to M ratio .D ratio .


Aurélien Duret
email: aurelien.duret@ifsttar.fr

Yufei Yuan
email: y.yuan@tudelft.nl

Traffic State Estimation Based on Eulerian and Lagrangian observations in a Mesoscopic Modeling Framework

Keywords: traffic state estimation, data assimilation, LWR model, mesoscopic model, Eulerian observation, loop data, Lagrangian observation, probe data, traffic monitoring, traffic forecasting

The paper proposes a model-based framework for estimating traffic states from Eulerian (loop) and/or Lagrangian (probe) data. Lagrangian-Space formulation of the LWR model adopted as the underlying traffic model provides suitable properties for receiving both Eulerian and Lagrangian external information. Three independent methods are proposed to address Eulerian data, Lagrangian data and the combination of both, respectively. These methods are defined in a consistent framework so as to be implemented simultaneously. The proposed framework has been verified on the synthetic data derived from the same underlying traffic flow model. Strength and weakness of both data sources are discussed. Next, the proposed framework has been applied to a freeway corridor. The validity has been tested using the data from a microscopic simulator, and the performance is satisfactory even for low rate of probe vehicles around 5%.

Introduction

State of the art

Traffic state estimation (TSE) is crucial in real-time dynamic traffic management and information applications. The essence of TSE is to reproduce traffic conditions based on available observation data. One class of available estimation methods does not make use of traffic flow dynamics, but relies on basic statistics and interpolation. These are referred to as data-driven methods. Another class of estimation methods relies on dynamic traffic flow models. These are referred to as model-based methods. The focus of this article is on the latter because it potentially provides better results than the former class in non-recurrent situations (work zones, accidents, social events, etc.), regarding the monitoring-forecasting capabilities.

Model-based TSE relies on two components : a model-based component and a data assimilation algorithm. The model-based component consists of two parts : (i ) a dynamic traffic flow model to predict the evolution of the state variables ; and (ii ) a set of observation equations relating sensor observations to the system state. Thereafter, a data-assimilation technique is adopted to combine the model predictions with the sensor observations. For example, the Kalman filter (KF) [START_REF] Nanthawichit | Application of probevehicle data for real-time traffic-state estimation and short-term traveltime prediction on a freeway[END_REF][START_REF] Chu | Adaptive kalman filter based freeway travel time estimation[END_REF][START_REF] Herrera | Incorporation of lagrangian measurements in freeway traffic state estimation[END_REF] and it advanced relatives, such as Extended KF [START_REF] Wang | Real-time freeway traffic state estimation based on extended kalman filter : A general approach[END_REF], Unscented KF [START_REF] Ngoduy | Applicable filtering framework for online multiclass freeway network estimation[END_REF], Ensemble KF [START_REF] Work | An ensemble kalman filtering approach to highway traffic estimation using gps enabled mobile devices[END_REF] have been widely applied in the field of traffic state estimation.

The same traffic flow model can be formulated in three two-dimensional coordinates regarding space x, time t and vehicle number n. Laval and Leclercq [START_REF] Laval | The hamilton-jacobi partial differential equation and the three representations of traffic flow[END_REF] have presented three equivalent variational formulations of the first-order traffic flow models, namely N (x, t) model, X(t, n) model, T (n, x) model respectively, under the theory of Hamilton-Jacobi partial differential equations. Under such defined coordinate systems, sensor observations from road networks can be defined into two categories : (i ) Eulerian sensing data -observations (e.g., aggregated speeds, flows) from spatially-fixed sensors (such as inductive loops, video sensors, and radar sensors) over a fixed report frequency, this type is dominating the information sources in the field of transportation research for decades ; and (ii ) Lagrangian sensing datainformation from probe samples at a fixed time interval (such as position and speed information of individual vehicles [START_REF] Herrera | Incorporation of lagrangian measurements in freeway traffic state estimation[END_REF], and/or probe spacing and position information [START_REF] Seo | Probe vehicle-based traffic state estimation method with spacing information and conservation law[END_REF]), this class is becoming an increasingly popular source. In literature, most of TSE applications are based on the traditional space-time (Eulerian) formulation. Aggregated traffic quantities (e.g., flows, densities or speeds [START_REF] Nanthawichit | Application of probevehicle data for real-time traffic-state estimation and short-term traveltime prediction on a freeway[END_REF][START_REF] Chu | Adaptive kalman filter based freeway travel time estimation[END_REF][START_REF] Herrera | Incorporation of lagrangian measurements in freeway traffic state estimation[END_REF][START_REF] Wang | Real-time freeway traffic state estimation based on extended kalman filter : A general approach[END_REF][START_REF] Work | An ensemble kalman filtering approach to highway traffic estimation using gps enabled mobile devices[END_REF]) are usually considered as system states, but no individual vehicle tracking is involved. The popularity of this formulation is due to the fact that incorporating Eulerian data is straightforward and intuitive. Recent studies have shown that a first-order (LWR) traffic flow model [START_REF] Lighthill | On kinematic waves ii : A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF] can be formulated and solved more efficiently and accurately in vehicle number-time (Lagrangian-time) coordinates [START_REF] Leclercq | The lagrangian coordinates and what it means for first order traffic flow models[END_REF]. And its related Lagrangian formulation of TSE enables more accurate and efficient application of data assimilation methods, due to the solution to the mode-switching problem (traffic information travels in one direction), less non-linearity of the system model, and the nature set of observation equations to deal with Lagrangian data [START_REF] Yuan | Mesoscopic traffic state estimation based on a variational formulation of the lwr model in lagrangian-space coordinates and kalman filter[END_REF][START_REF] Yuan | Real-time lagrangian traffic state estimator for freeways[END_REF]. However, the computation cost depends on the discretized platoon size (set to 1 vehicle classically) and time grid (often set around 1 second), which might be time consuming. The vehicle numberspace formulation inherits both the numerical benefits and modeling flexibility of the other two formulations.

Many previous studies have investigated the data assimilation problems on freeway networks with both Eulerian and Lagrangian observations under various modeling paradigms. Early studies in [START_REF] Chu | Adaptive kalman filter based freeway travel time estimation[END_REF][START_REF] Nanthawichit | Application of probevehicle data for real-time traffic-state estimation and short-term traveltime prediction on a freeway[END_REF] have applied the conservation law equations of traffic flow in Eulerian coordinates (LWR type [START_REF] Lighthill | On kinematic waves ii : A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF] and Payne's type [START_REF] Payne | Models of freeway traffic and control[END_REF], respectively) as the underlying traffic systems, to assimilate both simulated fixed-detector data and probe-vehicle data. Falling in this modeling category, the applications in [START_REF] Work | An ensemble kalman filtering approach to highway traffic estimation using gps enabled mobile devices[END_REF][START_REF] Herrera | Incorporation of lagrangian measurements in freeway traffic state estimation[END_REF] have also performed data assimilation with isolated Eulerian and Lagrangian data. Work et al.

[30] use a speed-based conservation law equation, while Herrera and Bayen [START_REF] Herrera | Incorporation of lagrangian measurements in freeway traffic state estimation[END_REF] use an extended LWR model in the Eulerian formulation. Alternatively, the variational (HJ-PDE) formulation can be used as an equivalence to describe the same underlying traffic systems [START_REF] Aubin | Dirichlet problems for some hamilton-jacobi equations with inequality constraints[END_REF][START_REF] Laval | The hamilton-jacobi partial differential equation and the three representations of traffic flow[END_REF]. This modeling approach is considered to be much simpler to compute and numerically more accurate under same conditions, compared with the conservation law approach. However, only a few studies have applied such formulations for state estimation purposes. And the existing applications mainly adopt the aforementioned N (x, t) model, where it considers the evolution of cumulated vehicle counts.

For example, Claudel and Bayen [START_REF] Claudel | Convex formulations of data assimilation problems for a class of hamilton-jacobi equations[END_REF] have applied the Hamilton-Jacobi formulation and generalized Lax-Hopf formula [START_REF] Aubin | Dirichlet problems for some hamilton-jacobi equations with inequality constraints[END_REF] for data assimilation and reconciliation utilizing loop and probe data ; Newell's three-detector model [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part i : General theory[END_REF] has also been extented for state estimation using heterogeneous data sources [START_REF] Clairais | Calibration of the fundamental diagram based on loop and probe data[END_REF][START_REF] Deng | Traffic state estimation and uncertainty quantification based on heterogeneous data sources : A three detector approach[END_REF] : loop detector data, Bluetooth travel time data and probe GPS data, respectively.

To the best of our knowledge, none of previous research has applied the HJ-PDE formulation using the X(t, n) model or the T (n, x) model for traffic data assimilation. As discussed, in comparison to the X(t, n) model, the simulation efficiency of the T (n, x) model is independent on the discretized platoon size (set to 1 vehicle classically) and the spatio-temporal grid.

Meanwhile, observation data are located along the vehicle number -space grids. Hence it is more advanced and convenient for data assimilation. This article presents a complete TSE framework for assimilating both Eulerian and Lagrangian data under a vehicle number -space (Lagrangian-space := L-S) formulation (namely, the T (n, x) model). It inherits the classic modeling approach that has been widely applied for the data assimilation problem in the field of traffic management as well as meteorology, oceanography, image processing, etc. [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF][START_REF] Kordic | Kalman Filter. Intech[END_REF][START_REF] Talagrand | Assimilation of observations, an introduction[END_REF]. This modeling paradigm consists of system equations that capture the evolution of the state vector over time, and observation equations that capture the mapping of the state vector on the observations.

Objectives and contributions

This paper presents a generic data assimilation framework based on a mesoscopic-LWR model formulated in Lagrangian-space coordinates, using both Lagrangian and Eulerian observations. The term mesoscopic is in response to the two other counterparts, since the Lagrangian-time coordinates can apply in a microscopic simulation framework and the Eulerian coordinates can accommodate in a macroscopic one. In this work, the system model is the Lagrangian-space formulation of the LWR model. It individually represents vehicles but only tracks their states at cell boundaries. We will develop algorithms and observation models to incorporate data from both Eulerian and Lagrangian sensors, as well as considering the observation noise in both data sources. And we do not apply specific data assimilation techniques (e.g., KF-based approaches) ; instead we try to demonstrate the sequential data assimilation concepts via reasonable assumptions. The algorithms on how to estimate network traffic states under the proposed model-based framework from the two data sources will be the main contribution of this work.

Contents of the paper

This paper is organized as follows. Section 2 presents the underlying traffic dynamics, including its formulation, solutions and properties. Section 3 describes the methodology of the proposed TSE framework, including how to assimilate Eulerian data, Lagrangian data, and the combination of the two sources, respectively. Sections 4 and 5 illustrate the model validation and an application to a freeway corridor. Discussion and conclusions are drawn in Sections 6 and 7.

LS-LWR model

This section defines the underlying process model in the state estimation framework, where the model formulation, numerical solution and its properties are discussed.

Conservation law and variational theory

This section first presents a mesoscopic formulation of the LWR model as the process model in the estimation framework. The LWR model is formulated in vehicle platoon and space (n, x) coordinates. The current mesoscopic formulation combines a vehicular description with macroscopic behavioral rules. It relaxes the temporal coordinate, and this entitles a transformation of a temporal progressing approach (e.g., in Eulerian or Lagrangian-time simulation framework) to an event-progressing approach (trigger event can be the change of time headway or pace, and/or a correction procedure based on an observation from fixed loops or probe vehicles).

The formulation follows the principle of the Hamilton-Jacobi (HJ) theory, to find an expression of the LWR model in Lagrangian-space coordinates.

This model is also referred to as the T -model.

The LWR model can be described by a hyperbolic equation under the conservation law :

∂ x h -∂ N (1/V (h)) = 0 (1)
Here, h denotes the time headway. The inverse speed 1/v (or called pace) can be derived from the fundamental diagram 1/V (h).

Previous authors have proposed to apply variational theory in Eulerian coordinates (x, t) [START_REF] Daganzo | A variational formulation of kinematic waves : basic theory and complex boundary conditions[END_REF] and Lagrangian coordinates (n, t) [START_REF] Leclercq | The lagrangian coordinates and what it means for first order traffic flow models[END_REF]. Here, we transpose the demonstration in Lagrangian-space coordinates (n, x), following the same rationale in [START_REF] Leclercq | The lagrangian coordinates and what it means for first order traffic flow models[END_REF]. The problem can be expressed as the Hamilton-Jacobi derived from the fundamental diagram :

∂ x T = 1 V (∂ N T ) (2)
Here, the function 1/V represents the flux function of the problem.

Numerical solutions in Lagrangian-space coordinates

Here, a Godunov scheme [START_REF] Godunov | A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF] is applied to solve the conservation law equation (hyperbolic equation) above with an upwind method. This would preserve the numerical benefit of Lagrangian traffic flow models. Figure 1 illustrates the mesoscopic numerical grid (see grey area) for the Godunov scheme. On the mesoscopic grid, the time headway is determined by :

h x+∆x n = h x n + ∆x ∆n .( 1
V (h x n ) - 1 V (h x n-1 ) ) (3)
The CFL condition that guarantees the convergence of the Godunov scheme [START_REF] Courant | On the partial difference equations of mathematical physics[END_REF] is :

∆n ≥ max h ∂ h ( 1 V ) ∆x (4)
Alternatively, the problem can also be expressed in terms of T (n, x) considering the 'passage time' flux that crosses the boundary of the cell n, regarded as a variational formulation of the T -model :

T (n, x) -T (n, 0) ∆x = 1 V ( T (n,x)-T (0,x) ∆n ) (5)
In this expression, V depends on the fundamental diagram. Here, we consider a triangular fundamental with three parameters : the free-flow speed v m ,

q k 1/V h v m 1/v m -w 1/k x 1/w.k x Fundamental diagram (Eulerian coordinates) Fundamental diagram (Mesoscopic coordinates) k x
1 V (h) = max( 1 v m , -k x ( 1 wk x -h)) (6)
The numerical solution to the problem is simplified as [START_REF] Laval | The hamilton-jacobi partial differential equation and the three representations of traffic flow[END_REF] :

T (n, x) = max(T (n, 0) + x v m , T (0, x + n k x ) + n w.k x ) (7)
Finally, the origin n = 0, x = 0 could be shifted to n -∆n, x -∆x and we find :

T (n, x) = max(T d , T s ), (8)
where T d = T (n, x -∆x) + ∆x vm and T s = T (n -∆n, x -∆n kx ) + ∆n w.kx represents the demand and the supply term, respectively. The demand term defines the arrival time of a vehicle from upstream in non-constrained (free-flowing) conditions. The resulting passage time of a vehicle is at least equal to its arrival time but could be delayed due to the downstream conditions. Thus, the supply time provides such information in constrained (congested) conditions.

This numerical solution indicates traffic flow is divided into vehicle platoons of certain size ∆n, and road stretch is discretized into spatial cells of certain length ∆x. Note that, the cell length ∆x in the simulation is not necessarily to be equal. The state in this formulation is the passage time T (n, x) of vehicle platoons at cell boundaries. This state is always determined by the maximum of two uncorrelated terms : the demand (arrival) time and the supply time. For an elaborate description we refer to [START_REF] Laval | The hamilton-jacobi partial differential equation and the three representations of traffic flow[END_REF].

Properties

The current mesoscopic formulation is based on the notions from the with only one expression to consider all traffic conditions. Hence, it does not require memory and it is more flexible and time-efficient for data assimilation (no complex matrix inversion and multiplication). Moreover, the numerical scheme allows for long cells and cell boundaries can be located at network discontinuities only (merges, diverges, and lane-drops). In this way, the spatial discontinuities can address easily. The computation cost depends on the number of cell boundaries (x-dimension) in the network and the number of vehicles (n-dimension) to propagate during the simulation. Therefore, this would improve computational efficiency for large scale applications.

More importantly, this mesoscopic scheme is particularly convenient for data assimilation. In reality, the flow characteristics are mostly observed at fixed points (e.g., spatial fixed loop data) or along vehicle trajectories (e.g., vehicle-number fixed probe data). As discussed in literature that the Eulerian formulation is suitable for incorporating loop data and the Lagrangian-time formulation is suitable for probe data assimilation, the Lagrangian-space formulation is considered to be well-compatible for assimilating both types of observations. Because these observations are located on cell boundaries of the mesoscopic grid, which makes any traffic state estimation method convenient with this approach/formulation. This formulation can be easily coupled with any data assimilation techniques to perform state estimation. Due to the nature of the mesoscopic system model, the TSE might be not restricted to discretized mesoscopic x -n grids. If we know any two boundaries in the network and an observation at a certain location or of a certain vehicle, we can generalize TSE for this specific assimilation problem.

Data assimilation methodology

Traffic flow prediction is an initial/boundary value problem, where a traffic model forecasts the evolution of traffic states on the network given initial/boundary states. Such data assimilation model-based approaches have been developed and widely applied in other fields, notably for atmospheric modeling and forecast [START_REF] Dimet | Variational algorithms for analysis and assimilation of meteorological observations : theoretical aspects[END_REF][START_REF] Talagrand | Assimilation of observations, an introduction[END_REF]. Because predictions become more accurate when uncertainties of initial values are reduced, the data assimilation problem first combines observed data and the "first guess" provided by the model to estimate the mostly possible initial states at observation locations, also called analysis states. When the model is updated accordingly, it becomes ready to forecast the evolution of the system. [START_REF] Kalnay | Atmospheric modeling, data assimilation and predictability[END_REF] presents the typical data assimilation framework for atmospheric modeling (see chapter 5), where a sequential loop is run every time new data are observed. The composition of the loop returns an overarching theory of sequential data assimilation with four fundamental steps :

-

. Reminder on the existing methodology

A data assimilation method using sole loop data first proposed in [START_REF] Duret | Data assimilation using a mesoscopic lighthill-whitham-richards model and loop detector data[END_REF],

has been validated on synthetic scenarios and tested on a large-scale network.

Here the basic concepts are reviewed and the reader is referred to the paper for more details.

It requires the numerical scheme to be set as follows : ∆n to 1 and cells boundaries at each loop location. It considers flow and speed time series collected by loop sensors at locations {X loop } with a given frequency ∆T .

Then it is implemented as a sequential procedure, for which each sequence is divided into 4 successive steps (see Figure 3). As mentioned by the authors in [START_REF] Duret | Data assimilation using a mesoscopic lighthill-whitham-richards model and loop detector data[END_REF], the update of the model has to be implemented so that the CFL stability condition is respected [START_REF] Courant | On the partial difference equations of mathematical physics[END_REF]. The solution proposed is a parsimonious adjustment of the demand and/or the supply terms at cell boundaries.

Focus on step 3 : Update of arrival and supply times at cell boundaries

The a-state proposed by the Global Analysis consists of a regime r a and a headway h a , at observation location and over the period P . 8 ( 23 ) situations can be met (see Table 1) and the model needs to be updated accordingly.

The update consists of (i) adding, deleting, advancing or delaying vehicles at the cell boundaries and (ii) then updating the passing times of vehicles accordingly. In this paper, the update is identical to the one presented in [START_REF] Duret | Data assimilation using a mesoscopic lighthill-whitham-richards model and loop detector data[END_REF] except that the passing times are managed slightly differently to be better suited for combining with the assimilation of probe data (see the next section) while keeping the same performance. To do so the passing times are not updated directly. The method updates the demand and the supply terms at cell boundaries over the period P If the analysis regime is free-flowing (combination 3, 4, and 7 in Table 1) :

T s (n, X) = -∞ T d (n, X) = T (n -1, X) + h a (9) 270 271
If the analysis regime is congested (combination 1, 2 and 6 in Table 1) :

T s (n, X) = T (n -1, X) + h a T d (n, X) = -∞ (10)
Note that combinations 5 and 8 in Table 1 correspond to errors on both the local demand and supply. No update is required and the problem has to be addressed at global level (demand, model parameters, etc.).

TSE based on probe data

The data assimilation framework presented above is limited to Eulerian In this paper, the TSE estimator enables to assimilate positions and times without any aggregation process, which allow for using most of the details of probe data. The method is divided into 4 steps (see Figure 4).

- and N b (t, x down ) can then be considered for estimating local n-index of the probe based on variational principles applied for the three-detector problem [START_REF] Daganzo | A simple traffic analysis procedure[END_REF][START_REF] Daganzo | A variational formulation of kinematic waves : basic theory and complex boundary conditions[END_REF][START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part i : General theory[END_REF] as illustrated in Figure 5. Local errors on the estimated n-index may induce global inconsistencies on the resulting arrival/supply times. To tackle this problem, a global optimization is developed and it consists of two steps. The first step aims at building the variational proximity matrix, which returns the variational cost (in veh.) between each of the time-space observations from probes (with respect to the variational principles [START_REF] Daganzo | A variational formulation of kinematic waves : basic theory and complex boundary conditions[END_REF]). Based on that, the second step calculates the optimal n-index, denoted n * . The optimal solution minimizes the entropy of the system [START_REF] Ansorge | What does the entropy condition mean in traffic flow theory ?[END_REF][START_REF] Velan | A note on the entropy solutions of the hydrodynamic model of traffic flow[END_REF] while keeping a constant n-index along probe trajectories. The optimization procedure searches in the range of all possible n-indices, and this search range is defined by the minimum and maximum values from the variational principles and the range of local n-index estimation. The For each probe vehicle, only the latest triplet {n a p , x o p,i , t o p,i } is considered for updating the arrival time at the cell downstream, as illustrated in Figure 7.

n local p,i = min N b up,u , N b down,w + k x .(x down -x o p,i ) (11)
t a a,n a p = t o p,i + x down -x o p,i u (13)
Upstream : update of supply times. The upstream cell boundary is influenced by probe vehicles located in a time window with a length P and that moves with a maximum jam speed w, see Figure 8.

For each probe vehicle, triplets {n a p,i , x o p,i , t o p,i } are considered as internal boundary conditions to revise supply times at the cell boundary upstream.

Within the influencing area, the updated supply times respect as illustrated in Figure 9 : that each cell boundary has to be updated over a time period ∆T U , which is bounded as a wave cannot travel through a whole cell during this time period. Consequently, if ∆T ≥ ∆T U then the updating process must proceed step by step (as described in the previous section) with a maximum time step ∆T U .

t a s,n a p +(x o p,i -xup).kx = t o p,i + x o p,i -x up w , ∀i ∈ S p (14

Assimilating both loop and probe data

Loop and probe data provide information of different nature, it is therefore impossible to fuse the two data sources to perform a one-shot assimilation process. Reviewing their respective actions, the two TSE estimators act in a complementary manner. On one hand, TSE based on loop data allows for an adjustment of the flow by adding -deleting -advancing -delaying vehicles at loop sensors locations. From a physical point of view, it acts as a 'flow regulator' at cell boundaries. On the other hand, the TSE based on probe data adjusts arrival and supply times at cell boundaries considering probe trajectories as internal cell-boundary conditions. From a physical point of view, it acts as a 'travel time regulator' along cells travelled by probe vehicles.

To make the best potential use of both data, we propose first to estimate trafic states from loop data at loop sensors locations and then to estimate traffic states from probe data everywhere else. The main reason for this se-quence is the following : TSE based on loop data improves the flow estimation at cell boundaries and therefore enhance the TSE based on probe data along cells. It results in a 7 steps methodology illustrated on Figure 10 : -

Model validation

This section aims to analyze and validate the performance of the TSE methodology with loop sensors and probe sensors (separately and jointly).

Experimental validation setup

The ground truth is emulated based on a microscopic LWR model (Newell's car-following model [START_REF] Newell | A simplified theory of kinematic waves in highway traffic, part i : General theory[END_REF], equivalent to the LWR model at a macroscopic scale). The model has been run on a homogeneous road stretch (L = 2000m, single lane) with a demand-supply scenario so that a congestion propagates through the network, see vehicle trajectories in Figure 11. A loop sensor located in the middle of the network (x = 1000m) collects flows and speeds with an aggregation period of 1-minute. Moreover, 10% of the vehicles are considered as probe sensors for which time-position information is reported at every 30s. The traffic flow model is a LS-LWR model. The network is composed of 2 cells of 1000m in length, upstream and downstream of the loop sensor location (namely the upstream and the downstream cells). The demandsupply scenario has also been predefined with an approximative demand and a high supply so that traffic conditions are always free-flowing on the network. In summary, when a congestion occurs, loop sensors can estimate travel times providing that congestion states have passed over the loops. The result shows that travel times might be underestimated over the network level. And this underestimation will become significant when traffic congestion is triggered far downstream the loop sensor. We conclude that for operational purposes loop sensors have to be located as close as the triggering location of a jam/bottleneck to provide accurate estimation. In addition, the complementary information from downstream loop sensors can improve the performance of data assimilation.

TSE based on loop observation model

TSE based on probe observation model

Figure 13 provides the estimated traffic states considering probe data only. The performance of TSE based on sole probe data provides similar performance over the two cells. It is noteworthy that TSE is very responsive as the congestion phenomenon occurs, mainly due to the probe data with a homogeneous coverage of the network both in time and space. It should also be noted that travel times are underestimated in this validation scenario due to the experimental setup. The traffic model considers a low demand versus high supply scenario. Information from probes allows for an adjustment of the supply times at the intercell boundary, but it does not rectify the underestimated flow demand (from downstream) and thus underestimate travel times. Note that the result depends on the experimental setup as travel times will become overestimated if the demand from the upstream boundary is overestimated.

We conclude that for operational purposes, the knowledge of the demand at any point of the network is decisive and critical when probe data are used for estimating traffic states. The estimation can be enhanced with an accurate prior estimation of the demand ; or combining probe data with loop data, as proposed in section 3.3.

TSE based on loop and probe observation model

Application to a freeway corridor

The previous section demonstrates the exactness of the estimator when applying to a network with FIFO conditions and homogeneous driving behavior. These assumptions are restrictive and not reflective of reality. This section aims at evaluating the performance of the estimator considering a multi-lane corridor with on-and off-ramps, with a relaxed FIFO assumption and distributed driving behavior.

Preparation of the observational model

Ground truth data have been emulated based on a microscopic traffic simulator (FOSIM [START_REF] Dijker | Fosim (freeway operations simulation[END_REF]). This simulator is developed at the Delft University of Technology, specially designed for the detailed analysis in freeway networks.

All the parameters in terms of driving behaviors have been calibrated and validated based on data from Dutch freeways. A three-lane freeway with one on-ramp and one off-ramp is designed, as illustrated in Figure 15 (the first 500 m as the warming-up section in Simulation, the last 1000m as the cooling-down section). Based on FOSIM simulation results, Eulerian and Lagrangian observation models have been built. First, loop sensors have been located on the main road : loop 1 -100m after the entrance of the network, and loop 2 -100m upstream of on-ramp. Second, 10% equipped probe vehicles return their exact positions every 20s.

Preparation of the traffic model

The traffic model is the mesoscopic LWR model applied on a network

Sensitivity to the penetration rate of probe vehicles

We expect that the performance of the data assimilation process can be improved when the amount of available data increase (n-, x-and t-frequency) and data are disaggregated. Here, we investigate the sensitivity of the proposed framework regarding the penetration rates of probe vehicles.

The sensitivity has been tested on the scenario with cars only, with various penetration rates range from 0% to 10%. Figure 18 illustrates the sensitivity of the method regarding the penetration rates of probe vehicles (over The performance of the TSE is improved with only one percent of the observed probe vehicles. We also observe that the performance becomes marginal as the percentage of probe vehicle exceeds 6%. It can be interpreted as follow : the gain for data assimilation is marginal and the remaining improvement needs to be found elsewhere, for instance, with the three other aspects : First, the model relies on assumptions and limitations (triangular fundamental diagram, FIFO traffic stream, etc.). Second, the definition of the demand on the network, which determines the traffic volumes that highly impacts the performance of the n-index estimation. And last but not least, the calibration of the model. We recall that the model has been poorly calibrated (default parameters) to emphasize the benefit of data assimilation procedure. Substantial gain is expected with a proper calibration procedure during the preparation of the model. We conclude that low penetration rates, around 5%, are sufficient for a realistic traffic state estimation.

Discussion

Based on the previous results, we conclude that Eulerian observations can update arrival and supply times, by adjusting the demand (flow) via adding or deleting vehicles locally. However, since loop sensors are spatially fixed, they only catch supply information as information propagates upstream. Hence, travel time estimation might be inconsistent (see section 4.2, the case of the downstream cell). It is therefore critical to locate loop sensors at the spots of a jam and a bottleneck to provide accurate estimation.

In contrast, Lagrangian observations spread over the network in space and time. Given a reasonable resolution (above 10%), they can update both arrival times and supply times without any latency. However, two limitations exist in this method. First, the n-index estimation relies on the assumption of the FIFO condition, which is unrealistic. This might lead to poor estimation of travel times when probe information deviates from the average traffic conditions (see section 5.3). Second, the demand/flow at cell boundaries cannot be adjusted. A prior estimation of the demand will improve the performance of TSE with probe data (see section 4.2).

The combination of the two data sources compensates the limitations of each other. The experiment results demonstrate that TSE with data combination outperforms the estimation with a single source.

Conclusion

Main findings

A TSE estimator based Eulerian observations combined with a mesoscopic LWR model has been proposed and validated in [START_REF] Duret | Data assimilation using a mesoscopic lighthill-whitham-richards model and loop detector data[END_REF]. This paper complements the methodology with Lagrangian observations. Now both Eulerian and Lagrangian observations can be used for TSE in a unique framework.

Eulerian observations provide comprehensive observations in time and vehicle for a discrete set of locations in the network. At those locations, model states are successfully revised, which provide good performance when observations are located near the head of congestion. The update acts as a 'flow regulator' at cell boundaries by adding, deleting, advancing or delaying vehicles.

Lagrangian observations provide a homogeneous coverage of the network in time and space for a discrete set of (probe) vehicles. Probe vehicles allow for an revision of demand-supply times at neighboring cell boundary of the network. The update acts as a cell 'travel time regulator' that yields good results under the condition that the demand on the network is known. Note that the n-index values of probe vehicles are critical and essential in the proposed approach. The calculation of this variable is application-specific (under FIFO or non-FIFO condition) regarding estimation performance, and it is subjective for further investigation.

In the TSE framework with data combination as proposed in section The methodology has been verified on the synthetic data derived from the same underlying traffic flow model. Meanwhile, the proposed TSE framework has been applied to a freeway corridor with a relaxed FIFO condition and distributed driving behavior. The validity has been tested using the data from a microscopic simulator, and the performance is satisfactory even for low rate of probe vehicles around 5%. With increasing estimation accuracy and computational efficiency, the proposed TSE framework will be beneficial for decision support traffic management.

Further research

The robustness of data assimilation methodology is demonstrated in the paper. However, we propose here some avenues to improve the present methodology. On the model side, assumptions are to be relaxed to enhance the model and its ability to reproduce well-documented traffic flow phenomenons (non-FIFO condition, multi-class traffic, capacity drop). On the observation side, the exponential growth data sources (e.g. bluetooth -mobile -information from connected vehicles) will provide massive additional Lagrangian and Eulerian information, which can be assimilated based on the proposed framework providing their reliability and the proposition of adequate transformation operators.

The data assimilation framework represents a solid base for on-line estimating the reliability of both, again, traffic model and observed data. On the model side, during the assimilation procedure, the discrepancy between the background states from the model and the analysis states can be analyzed for detecting incident on road networks and adjusting the model (demand, parameters) accordingly. On the observed data side, the discrepancy between the observed states and the analysis states can be help for estimating the observation reliability or detecting a problem with the data collection system.

And finally, we should recall that data assimilation has two main objectives :

to provide a continuous estimation of traffic states at the present time based on discrete and aggregate observations ; and to propose the most possible short term evolution of traffic states. The present paper is mainly focused on the first item. Its capability to provide robust short term forecasts should be explored, for instance, by implementing the proposed framework on a real large-scale network.

Figure 1 :

 1 Figure 1: Numerical solutions in Lagrangian-space coordinates

Figure 2 :

 2 Figure 2: Fundamental diagrams in two coordinate systems

  variational theory. It can incorporate the numerical benefits and modeling flexibility of both Eulerian and Lagrangian-time models. Simultaneously, this formulation allows state distinction on both cell class and vehicle class, combining a vehicular description with macroscopic behavioral rules. It individually represents vehicles (platoons) but only tracks their passage times at cell boundaries. Therefore, travel times can be easily derived from the model, which is more convenient compared to other (e.g., Eulerian or Lagrangian) formulations of state estimation. This discrete model evolves state by state,

  Step 1 : the transformation operation makes observed data and model predictions comparable -Step 2 : the global analysis provides the analysis states -Step 3 : the model is updated accordingly -Step 4 : the model forecasts the system evolution, by propagating information from data rich to data poor areas. In the paper, methods for estimating traffic states based on loop and probe data are presented in sections 3.1 and 3.2, respectively. Then a method that combines both data sources is presented in section 3.3. Three definitions with respect to different traffic states are given in the following : -an observation (o-) state is a traffic state measured by a sensor -a background (b-) state is a state forecasted by a traffic flow model -an analysis (a-) state is the result of an analysis procedure (or algorithm) that provides the most likely state regarding o-and b-states 3.1. TSE based on loop data 3.1.1

-Step 1 :

 1 the o-state and b-state are collected and transformed -Step 2 : a Global Analysis is performed to (a-) state -Step 3 : the state of the model is updated accordingly, by adjusting arrival and supply times at cells boundaries -Step4 : the model is run to provide a background state for the next sequence

Figure 3 :

 3 Figure 3: TSE based on loop data : methodology

(

  loop) data while nowadays increasing amount of traffic data are collected by Lagrangian (probe) sensors. Thus a TSE estimator based on Lagrangian observations becomes essential for real applications. Probe sensors collect positions of equipped vehicles at a given time frequency. They are usually processed for providing aggregated indicators, for instance the mean speed per link. However, most of the wealth of probe data is lost during the aggregating process.

Step 1 :

 1 the o-state and the b-state are collected and transformed -Step 2 : Global Analysis, which consists of estimating the n-index of probe vehicles -Step 3 : the model is updated accordingly, which consists of adjusting arrival and supply times at cell boundaries of the model -Step 4 : the model is run over the next sequence to provide a new background The two following sections elaborate steps 2 and 3, which are the keys to successfully update traffic states. 3.2.1. Focus on step 2 : Estimating the n-index of probe vehicles Let us consider the probe vehicle p that provides a set S p of observed timepositions denoted {t o p,i , x o p,i }, i ∈ S p . Simultaneously the model provides a background state T b (n, x) at cell boundaries, from which analogous function N b (t, x) can be easily defined (T is a monotonically increasing function) upstream (x = x up ) and downstream (x = x down ) probe positions. N b (t, x up )

Figure 4 :

 4 Figure 4: TSE based on probe data : methodology

Figure 5 Figure 6 :

 56 Figure 5: n-index estimation

)Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: Update of the arrival time at the downstream cell boundary

  Step 1 : collection and transformation of the loop data and the model background states -Step 2 : Global Analysis, which consists of estimating headway-regime pairs (a-states) at each loop location -Step 3 : the model is updated accordingly (see section 3.1). At this stage, the updated model provides the best possible estimated traffic states at cell boundaries. This version of the model is considered as a new model background to be combined with probe data -Step 4 : collection and transformation of the probe data and the (updated) model background -Step 5 : Global Analysis, which consists of estimation the n-index of probes along cells -Step 6 : update of the model accordingly, by revising arrival and supply times at every cell boundaries, except those already updated during the step 3. -Step 7 : run the model over the next sequence Here again, this sequence has to be implemented respecting CFL stability condition mentioned in section 3.2.2.

Figure 10 :

 10 Figure 10: TSE based on loop and probe data : methodology

Figure 11 :

 11 Figure 11: Observational model

Figure 12

 12 Figure 12 provides the estimated traffic states considering data from the loop sensor. In this figure, traffic states have been rearranged to provide travel times over the two cells. The red line provides the reference (ground truth) travel times and the blue line returns the reconstructed travel times. Upstream cell. Until the time t = 2000s, the traffic conditions are freeflowing. Between the time period t = 2200s and t = 2800s, a congestion propagates through the upstream cell. The estimated traffic states comply with the observed travel times, which validates the ability of the TSE estimator to adjust the network supply at the loop sensor location. Downstream cell. Downstream the loop sensor, the estimated traffic states are free-flowing until the end of the simulation, whereas the ground truth indicate that a congestion occurs. Indeed, the loop sensor data only indicate

Figure 14

 14 Figure 14 provides the estimated traffic states considering both loop and probe data. The results show the travel time estimation here outperforms and cumulates the benefits mentioned for loop and probe observation model considered separately.

Figure 14 :

 14 Figure 14: Travel time estimation from loop and probe data assimilation

Figure 15 :

 15 Figure 15: Networks

with 7 Figure 16 :

 716 Figure 16: FOSIM observation models, seed 1 : cars only (a and c) and mixed traffic (b and d)

5. 3 .

 3 Results with different observation modelsThree observation models have been tested : 'loop only', 'probe only', and 'combined loop and probe'. Travel time estimation based on the three observation models are illustrated in Figure17, ground truth (in red) and default model travel times (in black) are also displayed.During the free-flowing period, travel times are properly estimated regardless of observation models or traffic composition. However, significant differences are observed when a congestion occurs. The results analysis only focuses on the period t = [20 -45]min when the congestion is onset.TSE based on the loop observation model underestimates travel timesduring the congestion period, regardless of traffic composition. This can be caused by an underestimation of the upstream demand and/or an overestimation of the supply. Loop 1 located at the entrance of the network is supposed to update the demand according to the ground truth, so the overestimation of the supply is the cause : loop 2 is located 100 meters upstream the head of the congestion, which cannot detect immediately after its onset.TSE based on the probe observation model presents a better performance. However, it tends to overestimate travel times. It can be caused by a poor prior estimation of demand, which skews the n-index estimation of probe vehicle and leads to poor estimation of arrival/supply times. It can also be caused by poorly calibrated traffic parameters in the traffic model and/or non FIFO observations, which is confirmed in Figure17(b) that shows the overestimation is enhanced for a mixed traffic (ranging from t = [30 -40]min). By analyzing FOSIM trajectories, it is observed that during congestion trucks are stuck on the right-most lane (over congested) while most of the cars travel faster on left-most lanes. The FIFO assumption is not fulfilled and the consequence on the performance of TSE can be explained as follows. When a probe vehicle (for instance a truck) returns its position, arrival and supply times are estimated in the (FIFO) mesoscopic model thereof. We conclude that when the characteristics of probe vehicles are distant from the mean traffic stream, this induces bias in traffic state estimation which tends to overestimate travel times.

Figure 17 :

 17 Figure 17: Comparison of estimated travel times

Figure 18 :

 18 Figure 18: Travel time estimation for different probe penetration rates (10 replications)

Figure 19 :

 19 Figure 19: MoEs for different probe penetration rates (10 replications)

3. 3 ,

 3 Eulerian and Lagrangian observations become highly complementary. Eulerian observations successfully update traffic states (especially the flow) at loop locations of the network while Lagrangian observations successfully update cell travel times along the network. The estimation outperforms the scenarios considering loop and probe observations separately.

Table 2 :

 2 Scenario with a homogeneous stretch of road

		model model + loop model + probe model + (loop and probe)
	RMSE (s)	229	58	28	28
	MAPE (%)	34	25	12	11
	MPE (%)	-34	-25	-12	-11

Table 3 :

 3 Scenario with cars only -performance of the different observational models(10 replications)

		model model + loop model + probe model + (loop and probe)
	RMSE (s)	72	57	25	25
	MAPE (%)	24	20	9	8
	MPE (%)	-24	-19	0.65	0.43

Table 4 :

 4 Scenario with cars and trucks -performance of the different observational models(10 replications)

		model model + loop model + probe model + (loop and probe)
	RMSE (s)	234	117	112	109
	MAPE (%)	50	27	20	19
	MPE (%)	-50	-27	8	14

Acknowledgment

The authors sincerely thank L. Leclercq (University of Lyon, ENTPE, IFSTTAR, LICIT, UMR-T9401) for the fruitful discussions and assistance on the methodology presented in the paper. We would also like to thank anonymous reviewers for their valuable suggestions.


Mo Chen

Lionel Rosier
email: lionel.rosier@univ-littoral.fr

Reachable states for the distributed control of the heat equation

Keywords: 2010 Mathematics Subject Classification: 35K40, 93B05 heat equation, distributed control, reachable states, analytic extensions

We are concerned with the determination of the reachable states for the distributed control of the heat equation on an interval. We consider either periodic boundary conditions or homogeneous Dirichlet boundary conditions. We prove that for a L 2 distributed control, the reachable states are in the Sobolev space H 1 and that they have complex analytic extensions on squares whose horizontal diagonals are regions where no control is applied.

Introduction

The null controllability of the heat equation has been investigated for a long time [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF], and sharp results in dimension N were obtained in the nineties by using Carleman estimates [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Fursikov | Controllability of Evolutions Equations[END_REF]. By contrast, sharp results for the exact controllability of the heat equation were obtained only recently. In [START_REF] Martin | On the reachable states for the boundary control of the heat equation[END_REF], the authors noticed that for the boundary control of the heat equation on a real interval, the reachable space was sandwiched between the set of analytic functions on a ball and the set of analytic functions on a square. These results were improved in [START_REF] Dardé | On the reachable set for the one-dimensional heat equation[END_REF][START_REF] Hartmann | From the reachable space of the heat equation to Hilbert spaces of holomorphic functions[END_REF], where the reachable space was sandwiched between two spaces of analytic functions on squares. The sharp result, derived in [START_REF] Orsoni | Reachable states and holomorphic function spaces for the 1-D heat equation[END_REF][START_REF] Hartmann | Separation of singularities for the Bergman space and application to control theory[END_REF], tells that the reachable space for the heat equation on the interval (-1, 1) with two boundary controls at x = ±1 taken in the space L 2 (0, T ), is exactly the Bergman space of the functions that are both analytic and square integrable on the square Ω = {x+iy; |x|+|y| < 1}. See also [START_REF] Laurent | Exact controllability of semilinear heat equations in spaces of analytic functions[END_REF] for the reachable spaces of semilinear parabolic equations, and [START_REF] Martin | Exact controllability of a linear Korteweg-de Vries equation by the flatness approach[END_REF] and [START_REF] Chen | Exact controllabilty of the linear Zakharov-Kuznetsov equation[END_REF] for the reachable spaces of the linear Korteweg-de Vries equation and the linear Zakharov-Kuznetsov equation, respectively. The present paper is concerned with the determination of the reachable space for the distributed control of the heat equation on an interval. Roughly, it is proved that when using L 2 distributed controls, the reachable states are H 1 in the control region and analytic elsewhere. At the same time, we shall provide a short proof of the main result in [START_REF] Dardé | On the reachable set for the one-dimensional heat equation[END_REF] by using the method introduced in [START_REF] Lutz | On the Borel summability of divergent solutions of the heat equation[END_REF] to construct backward solutions of the heat equation.

Let us review the main results in this paper. Consider first periodic boundary conditions. Let T = R/2πZ and let χ (l 1 ,l 2 ) (x) = 1 if x ∈ (l 1 , l 2 ), 0 otherwise. We consider the control problem:

w t -w xx = χ (l 1 ,l 2 ) u w(x, 0) = 0 (x, t) ∈ T × (0, T ), x ∈ T.
(1.1)

For 0 < L 1 < L 2 < 2π, we introduce the open set S(L 1 , L 2 ) = {x + iy ∈ C; |y| < x -L 2 and |y| < L 1 + 2π -x}
which is a square with the interval (L 2 , L 1 + 2π) as a diagonal. Define

H(L 1 , L 2 ) = {f ∈ H 1 (T); f | (L 2 ,L 1 +2π) can be extended as an analytic function in S(L 1 , L 2 )}, A(L 1 , L 2 ) = {f ∈ H(L 1 , L 2 ); S(L 1 ,L 2 ) |f (x + iy)| 2 dxdy < ∞}.
Note that any function f ∈ H(L 1 , L 2 ) can be extended as an analytic function on ∪ k∈Z S(L 1 , L 2 )+ 2kπ, by 2π-periodicity of f . The following result is the first main result in this paper.

Theorem 1.1. Let T > 0 and 0 < l 1 < l 2 < 2π. Then (i) for any u ∈ L 2 (0, T ; L 2 (T)), the solution w of system (1.1) satisfies w(•, T ) ∈ A(l 1 , l 2 );

(ii) for any 0 < ε < (l 2 -l 1 )/2 and any w T ∈ H(l 1 + ε, l 2 -ε), there exists a control input u ∈ L 2 (0, T ; L 2 (T)) such that the solution w of system (1.1) satisfies w(•, T ) = w T in T.

Remark 1.1. 1. The above result is "almost sharp", for ε can be taken as small as desired in the inclusion

H(l 1 + ε, l 2 -ε) ⊂ A(l 1 , l 2 ).
2. Having in mind the characterisation of the reachable space for the boundary control of the heat equation in [START_REF] Hartmann | Separation of singularities for the Bergman space and application to control theory[END_REF], it is natural to conjecture that the reachable space for system (1.1) is the Bergman space A(l 1 , l 2 ).

Next, we consider distributed control systems on the interval (0, 1) with homogeneous boundary conditions. 1 Let 0 < l 1 < l 2 < 1. For any given u ∈ L 2 (0, T, L 2 (0, 1)), let w denote the solution of the control system

   w t -w xx = χ (l 1 ,l 2 ) u w(0, t) = w(1, t) = 0 w(x, 0) = 0 (x, t) ∈ (0, 1) × (0, T ), t ∈ (0, T ), x ∈ (0, 1). (1.2)
1 Homogeneous Neumann boundary conditions could be treated in a similar way. The following result is the second main result in this paper. Theorem 1.2. Let T > 0 and 0 < l 1 < l 2 < 1. Then (i) for any u ∈ L 2 (0, T ; L 2 (0, 1)), the solution w of system (1.2) satisfies w(•, T ) ∈ H 1 0 (0, 1), w(•, T ) ∈ A(l 1 ) and w(1 -•, T ) ∈ A(1 -l 2 );

(ii) for any 0 < ε < (l 2 -l 1 )/2, for any w T ∈ H 1 0 (0, 1) with w T ∈ H(l 1 + ε) and w T (1 -•) ∈ H(1 -l 2 + ε), there exists a control function u ∈ L 2 (0, T ; L 2 (0, 1)) such that the solution w of system (1.2) satisfies w(•, T ) = w T in (0, 1). Remark 1.2. The result is again "almost sharp". We conjecture that for 0 < l 1 < l 2 < 1, the reachable space for system (1.2) is the set of functions w T ∈ H 1 0 (0, 1) such that w T ∈ A(l 1 ) and w

T (1 -•) ∈ A(1 -l 2 ).
The paper is outlined as follows. The proof of Theorem 1.1 (resp. Theorem 1.2) is given in Section 2 (resp. in Section 3). We provide in appendix a short proof of the main result in [START_REF] Dardé | On the reachable set for the one-dimensional heat equation[END_REF] which is used to prove Theorem 1.1.

2 Proof of Theorem 1.1 (i) Pick any u ∈ L 2 (0, T, L 2 (T)) and let w denote the solution of (1.2). Let us first check that w ∈ C([0, T ], H 1 (T)). Expand χ (l 1 ,l 2 ) u as a Fourier series:

χ (l 1 ,l 2 ) (x)u(x, t) = n∈Z u n (t)e inx with T 0 ( n∈Z |u n (t)| 2 )dt < ∞. Then one has by Duhamel formula for all t ∈ [0, T ] w(x, t) = t 0 n∈Z e -n 2 (t-τ ) u n (τ )e inx dτ in L 2 (T).
Since for n = 0 and t ∈ [0, T ] we have, by Cauchy-Schwarz inequality,

t 0 |e -n 2 (t-τ ) u n (τ )|dτ ≤ ( t 0 e -2n 2 (t-τ ) dτ ) 1 2 ( t 0 |u n (τ )| 2 dτ ) 1 2 ≤ 1 √ 2n 2 ( T 0 |u n (τ )| 2 dτ ) 1 2 ,
we infer that

n∈Z * t 0 |e -n 2 (t-τ ) u n (τ )|dτ ≤ n∈Z * 1 2n 2 1 2 n∈Z * T 0 |u n (τ )| 2 dτ 1 2 < ∞ (2.1) so that w(x, t) = n∈Z ( t 0 e -n 2 (t-τ ) u n (τ ) dτ )e inx for (x, t) ∈ T × [0, T ], and that n∈Z * n t 0 e -n 2 (t-τ ) u n (τ )dτ 2 ≤ 1 2 n∈Z * T 0 |u n (τ )| 2 dτ < ∞. (2.2) Thus w ∈ C([0, T ], H 1 (T)) (2.3) and w(., T ) ∈ H 1 (T). Introduce the function v(x, t) = w(x, t) for (x, t) ∈ [l 2 , l 1 + 2π] × [0, T ].
Then v satisfies the following system (ii) We introduce a partition of unity. Let ψ 1 , ψ 2 ∈ C ∞ (T) be such that

   v t -v xx = 0 v(l 2 , t) = w(l 2 , t), v(l 1 + 2π, t) = w(l 1 + 2π, t) v(x, 0) = 0 (x, t) ∈ (l 2 , l 1 + 2π) × (0, T ), t ∈ (0, T ), x ∈ (l 2 , l 1 + 2π),
ψ 1 + ψ 2 = 1, 0 ≤ ψ i ≤ 1, i = 1, 2 ψ 1 (x) = 0, x ∈ T \ (l 1 , l 2 ), ψ 2 (x) = 0, x ∈ (l 1 + ε 2 , l 2 - ε 2 ).
Figure 3: partition of unity Consider a control problem with a distributed control supported in T:

w 1t -w 1xx = u 1 w 1 (x, 0) = 0 (x, t) ∈ T × (0, T ), x ∈ T, (2.4)
where u 1 is the control.

Lemma 2.1. For any T > 0 and any w T ∈ H 1 (T), there exists a function

u 1 ∈ L 2 (0, T ; L 2 (T))
such that the solution of (2.4) satisfies

w 1 (x, T ) = w T (x) ∀x ∈ T.
Proof. Expand the control input as a Fourier series

u 1 (x, t) = n∈Z u n (t)e inx .
Then by Duhamel formula

w 1 (x, t) = t 0 n∈Z e -n 2 (t-τ ) u n (τ )e inx dτ.
Since w T ∈ H 1 (T), we can expand w T as w

T (x) = n∈Z a n e inx with n∈Z n 2 |a n | 2 < ∞. Pick u n (t) =        a 0 T for n = 0, 2n 2 e -n 2 (T -t) 1 -e -2n 2 T a n for n ∈ Z * .
Then we obtain for

n ∈ Z * T 0 |u n (t)| 2 dt = 2n 2 1 -e -2n 2 T 2 |a n | 2 T 0 e -2n 2 (T -t) dt = 2n 2 1 -e -2n 2 T |a n | 2 ∼ 2n 2 |a n | 2 as n → +∞. It follows that u 1 2 L 2 (0,T ;L 2 (T)) = 2π n∈Z T 0 |u 2 n (t)|dt < ∞.
On the other hand, we have

T 0 e -n 2 (T -τ ) u n (τ )dτ = a n ∀n ∈ Z,
and hence w 1 (•, T ) = w T . The proof of Lemma 2.1 is complete.

It follows from (2.4) that the function ψ 1 w 1 satisfies the system:

   (ψ 1 w 1 ) t -(ψ 1 w 1 ) xx = u 1 (ψ 1 w 1 )(x, 0) = 0 (ψ 1 w 1 )(x, T ) = (ψ 1 w T )(x) (x, t) ∈ T × (0, T ), x ∈ T, x ∈ T, (2.5)
with

u 1 =ψ 1 (w 1t -w 1xx ) -2ψ 1x w 1x -ψ 1xx w 1 =ψ 1 u 1 -2ψ 1x w 1x -ψ 1xx w 1 .
By (2.3) (still valid for l 1 = 0 and l 2 = 2π), we have that w 1 ∈ C([0, T ], H 1 (T)). By construction of ψ 1 , we have that ∂ n x ψ 1 (x) = 0 for all x ∈ T \ (l 1 , l 2 ) and all n ≥ 0, and hence

u 1 = χ (l 1 ,l 2 ) u 1 in L 2 (0, T, L 2 (T)).
Let Hol(Ω) denote the space of (complex) analytic functions in

Ω. A function h ∈ C ∞ ([0, T ]
) is said to be Gevrey of order 2, and we write h ∈ G 2 ([0, T ]), if there exist some positive constants C, R such that

|∂ p t h(t)| ≤ C (p!) 2 R p ∀t ∈ [0, T ], ∀p ≥ 0.
The following result is needed.

Proposition 2.1. Let L > 1, T > 0, and ψ ∈ Hol(S(L)). Then there exist h

-1 , h 1 ∈ G 2 ([0, T ])
such that the solution w = w(x, t) of the control system

w t -w xx = 0, (x, t) ∈ (-1, 1) × (0, T ), (2.6) w(-1, t) = h -1 (t), w(1, t) = h 1 (t), t ∈ (0, T ), (2.7)
w(x, 0) = 0, x ∈ (-1, 1), (2.8)

satisfies w ∈ C ∞ ([-1, 1] × [0, T ]) and w(x, T ) = ψ(x) for x ∈ [-1, 1].
If, in addition, ψ is odd, then we can require that w(., t) be odd for all t ∈ [0, T ], so that h -1 (t) = -h 1 (t) and w(0, t) = 0 for all t ∈ [0, T ].

Note that a similar result with

h -1 , h 1 ∈ C ∞ ([0, T ]) was derived in [2, Theorem 5.2].
Here, we provide in Appendix a very short proof of Proposition 2.1 which is interesting in itself.

By Proposition 2.1, if

w T ∈ H(l 1 + ε, l 2 -ε), we can find h 1 , h 2 ∈ G 2 ([0, T ]
) such that the solution w 2 of the following system

   w 2t -w 2xx = 0 w 2 (l 2 -ε 2 , t) = h 1 (t), w 2 (l 1 + ε 2 + 2π, t) = h 2 (t) w 2 (x, 0) = 0 (x, t) ∈ (l 2 -ε 2 , l 1 + ε 2 + 2π) × (0, T ), t ∈ (0, T ), x ∈ (l 2 -ε 2 , l 1 + ε 2 + 2π) (2.9) satisfies w 2 ∈ C ∞ ([l 2 -2 , l 1 + ε 2 + 2π] × [0, T ]
) and

w 2 (x, T ) = w T (x), ∀x ∈ (l 2 - ε 2 , l 1 + ε 2 + 2π).
Extend w 2 as a function in C ∞ (T × [0, T ]) (i.e. as a function smooth in (x, t) and 2π-periodic in x) and still denote this function by w 2 . Then we have

   (ψ 2 w 2 ) t -(ψ 2 w 2 ) xx = u 2 (ψ 2 w 2 )(x, 0) = 0 (ψ 2 w 2 )(x, T ) = (ψ 2 w T )(x) (x, t) ∈ T × (0, T ), x ∈ T, x ∈ T, (2.10)
where

u 2 =ψ 2 (w 2t -w 2xx ) -2ψ 2x w 2x -ψ 2xx w 2 .
It follows from the definition of ψ 2 and the first equation in (2.9) that u 2 (x, t) = 0 for x ∈ [l 2 , l 1 + 2π] and t ∈ [0, T ]. Thus we have

u 2 = χ (l 1 ,l 2 ) u 2 in L 2 (0, T, L 2 (T)). Since w 2 ∈ C ∞ (T × [0, T ]), we have u 2 ∈ C ∞ (T × [0, T ]
). Combining (2.5) and (2.10), if we take

u = χ (l 1 ,l 2 ) u 1 + χ (l 1 ,l 2 ) u 2 , then w = ψ 1 w 1 + ψ 2 w 2 satisfies (1.1) and w(x, T ) = (ψ 1 w T )(x) + (ψ 2 w T )(x) = w T (x), ∀x ∈ T.
The proof of Theorem 1.1 is complete.

Dirichlet boundary conditions

In this section, we prove Theorem 1.2. The necessary conditions in (i) are obtained as in the proof of Theorem 1.1. Indeed, introducing v 1 = w |(0,l 1 )×(0,T ) and v 2 = w |(l 2 ,1)×(0,T ) and applying [5, Proposition 5.1], we obtain the desired result.

To prove (ii), we use again a partition of unity. We pick some functions ψ 1 , ψ 2 , ψ 3 ∈ C ∞ ([0, 1]) such that

ψ 1 + ψ 2 + ψ 3 = 1, 0 ≤ ψ i ≤ 1, i = 1, 2, 3, ψ 1 (x) = 0 x ∈ [0, l 1 ] ∪ [l 2 , 1], ψ 2 (x) = 0 x ∈ [l 1 + ε 2 , 1], ψ 3 (x) = 0 x ∈ [0, l 2 - ε 2 ].
As in the proof of Lemma 2.1, we can show that for any w T ∈ H 1 0 (0, 1), there exists a function u 1 ∈ L 2 (0, T ; L 2 (0, 1)) such that the solution w 1 of the system

   w 1t -w 1xx = u 1 w 1 (0, t) = w 1 (1, t) = 0 w 1 (x, 0) = 0 (x, t) ∈ (0, 1) × (0, T ), t ∈ (0, T ), x ∈ (0, 1), satisfies w 1 (x, T ) = w T (x), ∀x ∈ (0, 1).
Proceeding as for system (2.5), we see that there exists a control function u 1 ∈ L 2 (0, T ; L 2 (0, 1)) such that ψ 1 w 1 is the solution of the following system:

       (ψ 1 w 1 ) t -(ψ 1 w 1 ) xx = χ (l 1 ,l 2 ) u 1 (ψ 1 w 1 )(0, t) = (ψ 1 w 1 )(1, t) = 0 (ψ 1 w 1 )(x, 0) = 0 (ψ 1 w 1 )(x, T ) = (ψ 1 w T )(x)
(x, t) ∈ (0, 1) × (0, T ), t ∈ (0, T ), x ∈ (0, 1), x ∈ (0, 1).

Since w T ∈ H(l 1 + ε), there exists by Proposition 2.1 a function

h 1 ∈ G 2 ([0, T ]) such that the solution w 2 of    w 2t -w 2xx = 0 w 2 (0, t) = 0, w 2 (l 1 + ε 2 , t) = h 1 (t) w 2 (x, 0) = 0 (x, t) ∈ (0, l 1 + ε 2 ) × (0, T ), t ∈ (0, T ), x ∈ (0, l 1 + ε 2 ) satisfies w 2 ∈ C ∞ ([0, l 1 + ε 2 ] × [0, T ]
) and

w 2 (x, T ) = w T (x), ∀x ∈ (0, l 1 + ε 2 ).
We still denote by w 2 a smooth extension of w 2 to [0, 1] × [0, T ]. Then there exists a function

u 2 ∈ L 2 (0, T ; L 2 (0, 1)) such that ψ 2 w 2 solves        (ψ 2 w 2 ) t -(ψ 2 w 2 ) xx = χ (l 1 ,l 1 + ε 2 ) u 2 (ψ 2 w 2 )(0, t) = (ψ 2 w 2 )(1, t) = 0 (ψ 2 w 2 )(x, 0) = 0 (ψ 2 w 2 )(x, T ) = (ψ 2 w T )(x) (x, t) ∈ (0, 1) × (0, T ), t ∈ (0, T ), x ∈ (0, 1), x ∈ (0, 1). Similarly, since w T (1 -•) ∈ H(1 -l 2 + ε), there exists a function h 2 ∈ G 2 ([0, T ]) such that the solution w 3 of the system    w 3t -w 3xx = 0 w 3 (0, t) = 0, w 3 (1 -l 2 + ε 2 , t) = h 2 (t) w 3 (x, 0) = 0 (x, t) ∈ (0, 1 -l 2 + ε 2 ) × (0, T ), t ∈ (0, T ), x ∈ (0, 1 -l 2 + ε 2 ) satisfies w 3 ∈ C ∞ ([0, 1 -l 2 + ε 2 ] × [0, T ]
) and

w 3 (x, T ) = w T (1 -x), x ∈ (0, 1 -l 2 + ε 2 ). Let w 3 (x, t) = w 3 (1 -x, t), x ∈ (l 2 - ε 2 , 1), t ∈ (0, T ).
Then we have

       w 3t -w 3xx = 0 w 3 (l 2 -ε 2 , t) = h 2 (t), w 3 (1, t) = 0 w 3 (x, 0) = 0 w 3 (x, T ) = w T (x) (x, t) ∈ (l 2 -ε 2 , 1) × (0, T ), t ∈ (0, T ), x ∈ (l 2 -ε 2 , 1), x ∈ (l 2 -ε 2 , 1
). We still denote by w 3 a smooth extension of w 3 to [0, 1] × [0, T ]. It follows that there exists a control function u 3 ∈ L 2 (0, T ; L 2 (0, 1)) such that ψ 3 w 3 solves

       (ψ 3 w 3 ) t -(ψ 3 w 3 ) xx = χ (l 2 -ε 2 ,l 2 ) u 3 (ψ 3 w 3 )(0, t) = (ψ 3 w 3 )(1, t) = 0 (ψ 3 w 3 )(x, 0) = 0 (ψ 3 w 3 )(x, T ) = (ψ 3 w T )(x) (x, t) ∈ (0, 1) × (0, T ), t ∈ (0, T ), x ∈ (0, 1), x ∈ (0, 1).
Finally, if we take

u = χ (l 1 ,l 2 ) u 1 + χ (l 1 ,l 1 + ε 2 ) u 2 + χ (l 2 -ε 2 ,l 2 )
u 3 , we infer that w = ψ 1 w 1 + ψ 2 w 2 + ψ 3 w 3 is the solution of (1.2). On the other hand, we have that

w(x, T ) = (ψ 1 w T )(x) + (ψ 2 w T )(x) + (ψ 3 w T )(x) = w T (x), ∀x ∈ (0, 1).
The proof of Theorem 1.2 is complete.

Appendix. Proof of Proposition 2.1

The proof is inspired by [START_REF] Lutz | On the Borel summability of divergent solutions of the heat equation[END_REF] where backward solutions of the heat equation were obtained by integrating the heat kernel along lines different from the real line. Introduce the following notations borrowed from [START_REF] Lutz | On the Borel summability of divergent solutions of the heat equation[END_REF]. For θ ∈ R and R > 0, let

O(θ, R) := {z ∈ C; |z -Re iθ | < R}, Ω(θ, R) := {z ∈ C; dist(z, e iθ R) < R}.
(See Figure 4.) Note that

S(1) = Ω( π 4 , 1 √ 2 ) ∩ Ω( 3π 4 , 1 √ 2 ).
The first lemma is concerned with separation of singularities. Lemma 3.1. Let 1 < l < L and ψ ∈ Hol(S(L)). Then there exist

θ 1 ∈ (π, 3π 2 ), θ 2 ∈ ( π 2 , π), r ∈ ( 1 √ 2 , +∞), ψ 1 ∈ Hol Ω( θ 1 2 , r) and ψ 2 ∈ Hol Ω( θ 2 2 , r) such that S(l) ⊂ Ω( θ 1 2 , r) ∩ Ω( θ 2 2 , r), ∂ j z ψ i ∈ L ∞ (Ω( θ i 2 , r)), i = 1, 2, j ∈ N, ψ = ψ 1 + ψ 2 in S(l).
Proof of Lemma 3.1. Pick any l ∈ (l, L). Let γ 1 (t) = (1 -t + it) l for t ∈ [0, 1], and let

γ 2 (t), γ 3 (t), γ 4 (t) = (iγ 1 (t), -γ 1 (t), -iγ 1 (t)) for t ∈ [0, 1].
Let γ : [0, 4] → C be defined by

γ(t) = γ i (t -i + 1) for i ∈ {1, ..., 4}, t ∈ [i -1, i].
Note that γ([0, 4]) = ∂S( l). We infer from Cauchy formula that for any z ∈ S( l)

ψ(z) = 1 2πi γ ψ(ζ) ζ -z dζ = 1 2πi γ 1 ∪γ 3 ψ(ζ) ζ -z dζ + 1 2πi γ 2 ∪γ 4 ψ(ζ) ζ -z dζ =: ψ 1 (z) + ψ 2 (z)
where But if τ ∈ O(θ, R), | τ e iθ -R| < R, so that Re e iθ τ = Re τ e iθ > 0. On the other hand

ψ 1 ∈ Hol(C \ (γ 1 ∪ γ 3 )) and ψ 2 ∈ Hol(C \ (γ 2 ∪ γ 4 )). Since S(l) ⊂ S( l) = Ω π 4 , l √ 2 ∩ Ω 3π 4 , l √ 2 , lim τ →0 - v(z, τ ) = ψ(z), z ∈ Ω( θ 2 , r). (3.2) If, in addition, ∂ j z ψ ∈ L ∞ (Ω( θ 2 , r)) for all j ∈ N and if S(1) ⊂ Ω( θ 2 , r), then v ∈ C ∞ ([-1, 1] × [-T, 0]) for all T > 0 with -T ∈ O(θ, R).
|ψ(z -ζ)| ≤ ψ L ∞ (Ω( θ 2 ,r)) , for z ∈ Ω( θ 2 , r) and ζ ∈ Re i θ 2 .
Straightforward calculations show that

v zz = 1 √ 4πτ ∞e i θ 2 -∞e i θ 2 e -ζ 2 4τ ψ (z -ζ)dζ = 1 √ 4πτ ∞e i θ 2 -∞e i θ 2 d 2 dζ 2 [e -ζ 2 4τ ]ψ(z -ζ)dζ = 1 √ 4πτ ∞e i θ 2 -∞e i θ 2 - 1 2τ + ζ 2 4τ 2 e -ζ 2 4τ ψ(z -ζ)dζ, while v τ = 1 √ 4πτ (- 1 2τ
)

∞e i θ 2 -∞e i θ 2 e -ζ 2 4τ ψ(z -ζ)dζ + 1 √ 4πτ ( 1 4τ 2 ) ∞e i θ 2 -∞e i θ 2 ζ 2 e -ζ 2 4τ ψ(z -ζ)dζ
and hence (3.1) holds in Ω( θ 2 , r) × O(θ, R). It remains to show that (3.2) is fulfilled. First, we notice that for π 2 < θ < 3π 2 and R > 0, there exists T > 0 such that O(θ, R) ∩ R = (-T, 0) (see Figure 4). Therefore, taking the limit of v(z, τ ) as τ → 0 -is meaningful.

Claim 1. For θ ∈ ( π 2 , 3π
2 ) and τ ∈ (-∞, 0), we have 1

√ 4πτ ∞e i θ 2 -∞e i θ 2 e -ζ 2 4τ dζ = 1. Indeed, if θ ∈ ( π 2 , π) (resp. θ ∈ (π, 3π 2 )), we have for κ ∈ ( θ 2 , π 2 ) (resp. κ ∈ ( π 2 , θ 2 )), s ∈ R, τ ∈ (-∞, 0), and ζ := se iκ , |e -ζ 2 4τ | = |e -s 2 4τ cos(2κ) | ≤ e -s 2 4|τ | | cos(θ)| . Thus lim |s|→+∞ π 2 θ 2 e -(se iκ ) 2 4τ sie iκ dκ = 0 if θ ∈ ( π 2 , π) (resp. lim |s|→+∞ θ 2 π 2 e -(se iκ ) 2 4τ sie iκ dκ = 0 if θ ∈ (π, 3π 2
).)

It follows from the residue theorem that 1 √ 4πτ

∞e i θ 2 -∞e i θ 2 e -ζ 2 4τ dζ = 1 √ 4πτ ∞e i π 2 -∞e i π 2 e -ζ 2 4τ dζ = 1 i 4π|τ | ∞ -∞ e -s 2 4|τ | ids = 1,
which completes the proof of Claim 1. Finally, letting ζ = |τ |ξ, we see that for any z ∈ Ω( θ 2 , r)

v(z, τ ) -ψ(z) = 1 4π τ |τ | ∞e i θ 2 -∞e i θ 2 e -ξ 2 4 τ |τ | ψ(z -|τ |ξ -ψ(z) dξ
tends to 0 as τ → 0 -, by dominated convergence. Finally, assume that ∂ j z ψ ∈ L ∞ (Ω( θ 2 , r)) for all j ∈ N and that S(1) ⊂ Ω( θ 2 , r). Claim 2. For all j ∈ N, we have

∂ j z [v(z, τ ) -ψ(z)] → 0 as τ → 0 -uniformly for z ∈ [-1, 1]
. Indeed, for all j ∈ N, we can write

∂ j z [v(z, τ ) -ψ(z)] = 1 4π τ |τ | ∞e i θ 2 -∞e i θ 2 e -ξ 2 4 τ |τ | ψ (j) (z -|τ |ξ -ψ (j) (z) dξ.
For given ε > 0, we pick a number A > 0 such that

|s|>A e - (se i θ 2 ) 2 4 τ |τ | ds < ε. Then 1 4π τ |τ | ξ ∈ (-∞, ∞)e i θ 2 , |ξ| > A e -ξ 2 4 τ |τ | ψ (j) (z -|τ |ξ -ψ (j) (z) dξ ≤ 2 ψ j L ∞ (Ω( θ 2 ,r)) ε √ 4π •
On the other hand, it follows from the uniform continuity of ψ (j) on some open neighborhood of

[-1, 1] that 1 4π τ |τ | ξ ∈ (-∞, ∞)e i θ 2 , |ξ| ≤ A e -ξ 2 4 τ |τ | ψ (j) (z -|τ |ξ -ψ (j) (z) dξ → 0 as τ → 0 -, uniformly for z ∈ [-1, 1]. Claim 2 is proved. Using the fact that ∂ k t ∂ j z v = ∂ 2k+j z v for k, j ∈ N, τ < 0, and z ∈ [-1, 1], we infer that v ∈ C ∞ ([-1, 1] × [-T, 0]) for any T > 0 such that -T ∈ O(θ, R). The proof of Lemma 3.2 is complete.
Let us go back to the proof of Proposition 2.1. Pick ψ 1 and ψ 2 as given by Lemma 3.1, and let To complete the proof, we proceed as in [START_REF] Strohmaier | Analytic properties of heat equation solutions and reachable sets[END_REF], combining the above construction with a null controllability result. Pick any s ∈ (1, 2) and any function ρ ∈ G s ([-T , 0]) such that ρ(t) = 1 for -T ≤ t ≤ -T 2 and ρ(t) = 0 for -T 4 ≤ t ≤ 0. Let g 0 (t) := ρ(t)v(0, t) and g 1 (t) := ρ(t)∂ τ v(0, t) for t ∈ [-T , 0]. Using the fact that v(0, .), ∂ τ v(0, .) ∈ Hol(O(θ 1 , R) ∩ O(θ 2 , R)) and [12, Lemma 3.7], we infer that g 0 , g 1 ∈ G s ([-T , 0]). Therefore, for any R > 1 there exists some constant C > 0 such that

v j (z, τ ) = 1 √ 4πτ ∞e i θ j 2 -∞e i θ j 2 e -ζ 2 4τ ψ j (z -ζ)dζ for j = 1, 2, z ∈ Ω( θ j 2 , r) and τ ∈ O(θ j , R). Then v 1,τ -v 1,zz = 0, in Ω( θ 1 2 , r) × O(θ 1 , R), v 2,τ -v 2,zz = 0, in Ω( θ 2 2 , r) × O(θ 2 , R).
|∂ j t g 0 (t)| + |∂ j t g 1 (t)| ≤ C (2j)
), x ∈ (-1, 1), so that v(., 0) = 0 and v(., -T ) = v(., -T ). Let h ±1 (t) := v(±1, t) -v(±1, t), for t ∈ (-T , 0), w(x, t) := v(x, t) -v(x, t), for t ∈ (-T , 0), x ∈ (-1, 1).

Then w satisfies

      
w t -w xx = 0, t ∈ (-T , 0), x ∈ (-1, 1), w(x, -T ) = 0,

x ∈ (-1, 1), w(±1, t) = h ±1 (t), t ∈ (-T , 0), w(x, 0) = ψ(x),

x ∈ (-1, 1).

Figure 1 :

 1 Figure 1: Reachable states for periodic boundary conditions.

Figure 2 :

 2 Figure 2: Reachable states for homogeneous Dirichlet boundary conditions.

  where the boundary controls w(l 2 , .) and w(l 1 + 2π, .) are in C([0, T ]), by(2.3). Then v(•, T ) can be extended as an analytic function in S(l 1 , l 2 ) by[START_REF] Martin | On the reachable states for the boundary control of the heat equation[END_REF] Theorem 2.1]. Furthermore, w(•, T ) ∈ L 2 (S(l 1 , l 2 )) by[START_REF] Hartmann | From the reachable space of the heat equation to Hilbert spaces of holomorphic functions[END_REF] Proposition 1.2]. Therefore w(•, T ) ∈ A(l 1 , l 2 ).

Figure 4 :

 4 Figure 4: The set O(θ, R) for π 2 < θ < 3π 2 .

Proof of Lemma 3 . 2 .-ζ 2 4τ= e -s 2 4 e iθ τ = e -s 2 4

 32242 For s ∈ R and ζ = e i θ 2 s, we have e Re e iθ τ .

  Let v(z, τ ) = v 1 (z, τ ) + v 2 (z, τ ). Then v is well-defined and analytic inD := [Ω( θ 1 2 , r) ∩ Ω( θ 2 2 , r)] × [O(θ 1 , R) ∩ O(θ 2 , R)],and it fulfillsv τ -v zz = 0 in D, lim τ →0 - v(z, τ ) = ψ 1 (z) + ψ 2 (z) = ψ(z) in Ω( T > 0 is such that [-T , 0) ⊂ O(θ 1 , R) ∩ O(θ 2 , R) and T ≤ T , then v(., -T ) is analytic in the open set Ω( θ 1 2 , r) ∩ Ω( θ 2 2, r) which contains S(1).

  ! R j ∀t ∈ [-T , 0], ∀j ∈ N.According to[START_REF] Martin | On the reachable states for the boundary control of the heat equation[END_REF] Proposition 3.1], the problemvtvxx = 0, t ∈ (-T , 0), x ∈ (-1, 1), v(0, t) = g 0 (t), ∂ x v(0, t) = g 1 (t), t ∈ (-T , 0) possesses a solution v ∈ C ∞ ([-1, 1] × [-T , 0]). It follows then from the definition of ρ and Holmgren's theorem that v(x, t) = 0, t ∈ (-T 4 , 0), x ∈ (-1, 1), v(x, t) = v(x, t), t ∈ (-T , -

We pick a branch of the argument function defined inC \ R-i (resp. in C \ R+i) if π 2 < θ ≤ π (resp. if π < θ ≤

3π 2 ).

Acknowledgements. The authors thank Camille Laurent for useful remarks. Most of this work was done when Mo Chen visited Université du Littoral Côte d'Opale in 2019. Mo Chen is supported by NSFC Grant (11701078) and China Scholarship Council. Lionel Rosier is partially supported by CAPES: n o 88881.520205/2020-01 and MATH AMSUD: 21-MATH-03.

(see Figure 5). The proof of Lemma 3.1 is complete.

Figure 5: Separation of singularities.

The second lemma yields a backward solution of the heat equation for the initial data ψ 1 (resp. ψ 2 ).

is well-defined and analytic in z and τ for z ∈ Ω( θ 2 , r) and τ ∈ O(θ, R) for any R > 0. 2 Furthermore, v satisfies v τ -v zz = 0, z ∈ Ω( θ 2 , r), τ ∈ O(θ, R), (3.1)