Di Liu
email: di.liu@rug.nl

Senior member, IEEE Simone Baldi
email: s.baldi@tudelft.nl

Vishrut Jain
email: v.j.jain@tudelft.nl

Senior member, IEEE Wenwu Yu
email: wwyu@seu.edu.cn

Senior member, IEEE Paolo Frasca
email: paolo.frasca@gipsa-lab.fr

Cyclic communication in adaptive strategies to platooning: the case of synchronized merging

Keywords: Platooning, merging maneuvers, automated vehicles, uncertain vehicle parameters, adaptive control

I. INTRODUCTION

I N automated driving, the so-called platooning is a recog- nized idea aiming to form groups of automated vehicles keeping a desired string formation [START_REF] Dey | A review of communication, driver characteristics, and controls aspects of cooperative adaptive cruise control (CACC)[END_REF]- [START_REF] Larsson | The vehicle platooning problem: Computational complexity and heuristics[END_REF]. This can be realized via the so-called Cooperative Adaptive Cruise Control (CACC), using the feedback from on-board sensors (laser or radar) and inter-vehicle (wireless) communication to control acceleration and braking [START_REF] Kayacan | Multiobjective h∞ control for string stability of cooperative adaptive cruise control systems[END_REF]- [START_REF] Guo | String stability of heterogeneous leader-following vehicle platoons based on constant spacing policy[END_REF]. To maximize performance, CACC strategies should guarantee synchronized behavior, i.e. the coordination of actions via inter-vehicle communication [START_REF] Yu | Distributed adaptive control of synchronization in complex networks[END_REF]- [START_REF] Kazemi | A learning-based stochastic MPC design for cooperative adaptive cruise control to handle interfering vehicles[END_REF]: while the typical synchronized behavior studied in CACC is the formation keeping task [START_REF] Ploeg | lp string stability of cascaded systems: Application to vehicle platooning[END_REF]- [START_REF] Kianfar | A control matching model predictive control approach to string stable vehicle platooning[END_REF], recent surveys on the challenges of CACC [START_REF] Dey | A review of communication, driver characteristics, and controls aspects of cooperative adaptive cruise control (CACC)[END_REF]- [START_REF] Larsson | The vehicle platooning problem: Computational complexity and heuristics[END_REF] show that a relevant challenge is including synchronized merging maneuvers in CACC protocols.

Communication protocols currently used for merging maneuvers are sequential, usually based on state machines that define the procedure for opening a gap between vehicles. Proposed merging protocols include [START_REF] Amoozadeh | Platoon management with cooperative adaptive cruise control enabled by VANET[END_REF] (joining and leaving platoons using state machines); [START_REF] Scarinci | Definition of a merging assistant strategy using intelligent vehicles[END_REF] (creating gaps to allow on-ramp vehicles to merge); [START_REF] Chien | Regulation layer controller design for automated highway system: Platoon merge and split controller design[END_REF] (merging into platoons and splitting from platoons); [START_REF] Rai | Real and virtual leader-follower strategies in lane changing, merging and overtaking maneuvers[END_REF], [START_REF] Bengtsson | Interaction protocol for highway platoon merge[END_REF] (lane changing and overtaking); sequential game-theoretic approaches [START_REF] Abdelghaffar | Development and testing of a novel game theoretic de-centralized traffic signal controller[END_REF], [START_REF] Elhenawy | An intersection game-theory-based traffic control algorithm in a connected vehicle environment[END_REF] (merging and lane-changing behavior). Two observations follow:

• Driveline: state machines define the state of the vehicle, but not the actual CACC command (acceleration/braking) to bring the vehicle in that state. Defining such a command is complex, especially in the presence of uncertain vehicle parameters (driveline time constants). Most CACC protocols ideally assume homogeneous drivelines or exactly known driveline constants [START_REF] Acciani | Cooperative adaptive cruise control over unreliable networks: an observer-based approach to increase robustness to packet loss[END_REF], [START_REF] Harfouch | Adaptive control of interconnected networked systems with application to heterogeneous platooning[END_REF].

• Sequentiality: The phases in state machines occur sequentially rather than synchronously. Take the state machine merging procedure in [START_REF] Tas | Making Bertha cooperate-team annieWAY's entry to the 2016 Grand Cooperative Driving Challenge[END_REF] as a representative example: a vehicle makes a gap; when the gap is fully open, the vehicle that opened the gap activates the Wait For Merge flag and sends the Safe To Merge flag to the merging vehicle; the other vehicle starts the merging; once the maneuver is over, the new platoon can switch the Pace Making flag.

A. The issue of cyclic communication

Sequentiality means that the communication phases in a state-machine protocol are unidirectional: in fact, designing bidirectional state machines is not trivial, as pointed out in [START_REF] Zheng | Safety rule-based cellular automaton modeling and simulation under V2V environment[END_REF]. However, if we think about how humans perform merging, they look at each other in a bidirectional (thus cyclic) interaction. More specifically, the vehicle that wants to merge from another lane must look at both the following and the preceding vehicle of the adjacent line. At the same time, the preceding vehicle in the adjacent line must look at the vehicle that wants to merge. This establishes a bidirectional interaction and poses the problem of designing merging maneuvers for automated vehicles under such cyclic interaction. Works that recently studied cycles in CACC protocols are [START_REF] Baldi | Adaptive synchronization of unknown heterogeneous agents: An adaptive virtual model reference approach[END_REF], [START_REF] Baldi | Adaptive state-feedback synchronization with distributed input: the cyclic case[END_REF]: because the input of a vehicle is affected by the input of the neighboring vehicles via the platooning protocol, the presence of cycles creates algebraic loops. In order to avoid that an algebraic loop makes the input not well posed, a standard idea is to remove the cycles [START_REF] Wang | Hierarchical decomposition based consensus tracking for uncertain interconnected systems via distributed adaptive output feedback control[END_REF]. Two open problems arise:

• Uncertainty: how to handle driveline uncertainty in CACC protocols during merging maneuvers? • Well-posedness: how to make the CACC input well posed in the presence of cyclic communication?

Fig. 1: Merging scenario: the presence of roadworks requires the platoon in yellow to merge in the other platoon (edited from [START_REF] Ploeg | Cooperative automated maneuvering at the 2016 Grand Cooperative Driving Challenge[END_REF], five or more vehicles can be considered).

B. Motivational scenario and contributions

This work will focus on a merging benchmark among two platoons in different lanes due to roadworks. This benchmark was proposed within the Grand Cooperative Driving Challenge (GCDC), a competition aiming at real-life testing of cooperative driving solutions [START_REF] Ploeg | Cooperative automated maneuvering at the 2016 Grand Cooperative Driving Challenge[END_REF], [START_REF] Englund | The Grand Cooperative Driving Challenge 2016: boosting the introduction of cooperative automated vehicles[END_REF]. Notably, the merging protocols proposed all teams participating to the most recent GCDC (held in 2016) overlooked synchronization and uncertainty issues, cf. the designs published in [START_REF] Tas | Making Bertha cooperate-team annieWAY's entry to the 2016 Grand Cooperative Driving Challenge[END_REF], [START_REF] Aramrattana | Team Halmstad approach to cooperative driving in the Grand Cooperative Driving Challenge 2016[END_REF], [START_REF] Alonso | The experience of DRIVERTIVE-DRIVERless cooperaTIve VEhicle-team in the 2016 GCDC[END_REF]. Even when relying on the CACC protocol as in [START_REF] Dolk | Cooperative automated driving for various traffic scenarios: Experimental validation in the GCDC 2016[END_REF] (gap creation by slowly increasing the standstill distance) no parametric uncertainty nor well-posed control inputs are studied. In line with most literature, consider longitudinal dynamics only (gap creation and gap closing), since synchronization for lateral dynamics poses problems that are still unsolved, such as nonholonomic constraints, corner cutting during turns, disturbance propagation in the lateral direction, lateral behavior modeling [START_REF] Loof | Automated lane changing with a controlled steering-wheel feedback torque for low lateral acceleration purposes[END_REF], [START_REF] Kang | Modeling driver merging behavior: A repeated game theoretical approach[END_REF]. Note that even the winning team of GCDC 2016, team Halmstad, had no automated solution for lateral control [START_REF] Aramrattana | Team Halmstad approach to cooperative driving in the Grand Cooperative Driving Challenge 2016[END_REF].

Despite the relatively low complexity of the GCDC traffic scenarios, its experience highlights several unsolved issues: handling heterogeneity and driveline uncertainty, and embedding the merging maneuver in a synchronization protocol with possibly bidirectional (cyclic) communication. This work contributes to tackling these issues by

• showing that merging maneuvers can be formulated as an adaptive synchronization protocol with driveline uncertainty; • guaranteeing well posedness of the CACC inputs at all times despite the communication being acyclic or not; • solving the well-posedness issue in a distributed way with information exchange with neighboring vehicles. The organization of the paper is as follows: Sect. II describes the CACC structure, and Sect. III explains the synchronization protocol. Well-posed inputs are studied in Sect. IV, while Sect. V provides simulations and Sect. VI concludes the work. Part of this work has been included in the thesis of one of the authors [START_REF] Jain | Longitudinal control for heterogeneous vehicle platooning with uncertain dynamics: Vehicle platooning[END_REF].

II. CACC SYSTEM STRUCTURE

  =   0 1 0 0 0 1 0 0 -1 τi   Ai   d i v i a i   xi +   0 0 1 τi   bi u i , i ∈ S N (1)
where a i and u i are respectively the acceleration (m/s 2 ) and input (m/s 2 ) of the i th vehicle, and τ i (s) represents the driveline time constant. The difference between a i and u i is that the first is the actual acceleration of the vehicle, whereas the second one is the desired acceleration as imposed by the pedal: in between these two quantities (both in m/s 2 ) are the driveline dynamics represented by the time constant τ i [START_REF] Ploeg | lp string stability of cascaded systems: Application to vehicle platooning[END_REF]. We say that we have uncertainty in driveline when the exact value of τ i in (1) is not known and cannot be used to design the controller (cf. Proposition 1). The uncertain driveline scenario was initially overlooked in CACC literature [START_REF] Ploeg | lp string stability of cascaded systems: Application to vehicle platooning[END_REF]: this problem was considered quite recently through adaptive control [START_REF] Harfouch | An adaptive switched control approach to heterogeneous platooning with intervehicle communication losses[END_REF]. The idea for solving this problem amounts to defining target (model reference) dynamics

  ḋ0 v0 ȧ0   =   0 1 0 0 0 1 0 0 -1 τ0   A0   d 0 v 0 a 0   +   0 0 1 τ0   b0 u 0   ḋ0 v0 ȧ0   =   0 1 0 0 0 1 a 01 a 02 a 03   Am   d 0 v 0 a 0   xm +   0 0 b 00   bm w (2)
where τ 0 is a nominal driveline time constant: such dynamics (2) represent a virtual leading vehicle, defining the platoon's desired (reference) behavior. The index i = 0 is assigned to the virtual leading vehicle, which is not shown in Fig. 1 because is to be thought as part of the control. The virtual leading dynamics can be thought as playing a similar role as reference dynamics in model reference adaptive control [START_REF] Baldi | Adaptive synchronization of unknown heterogeneous agents: An adaptive virtual model reference approach[END_REF]. In fact, the second equation in (2) comes from considering that the controller u 0 = k * ′ 0 x m + l * 0 w is designed for vehicle 0 (w is the desired acceleration of the leader). This controller makes the reference dynamics A m stable, i.e. a 01 , a 02 , a 03 are design parameters that make the matrix A m Hurwitz.

The main goal of a vehicle i, is to maintain a desired formation (i.e. distance) with other vehicles j (where the indexes i and j arise from the communication graph among vehicles, to be defined later). The desired distance is determined by the constant time headway (CTH) spacing policy

r j,i (t) = rj,i (t) + hv i (t) , i, j ∈ S N (3)
where rj,i is the standstill distance (m) between vehicles i and j and h is the time headway (s). Differently from the standard CACC, rj,i in (3) depends on time because it can change as a consequence of the time-varying gap during merging. The stability analysis proposed in this paper holds for constants r j,i , even though adaptive methods can maintain stability also for the slowly-time varying parameters [START_REF] Ioannou | Robust Adaptive Control[END_REF]. Define the state error between the j th and the i th vehicle as the collection of spacing distance, relative velocity, and relative acceleration

e j,i (t) =   d j (t) v j (t) a j (t)   -   d i (t) v i (t) a i (t)   +   r j,i (t) 0 0   . ( 4
)
The control objective is:

Problem 2.1: Given the CTH strategy (3), the leading dynamics (2), and the vehicle dynamics (1) with uncertain driveline time constants, find an adaptive well-posed strategy for u i such that the errors e j,i in (4) resulting from the benchmark of Fig. 1 are regulated to zero for all links instantiated during all phases of the merging (i.e. synchronization is achieved across all links instantiated during all phases of the merging).

III. THE ADAPTIVE SYNCHRONIZATION PROTOCOL

We now present how to change r j,i in (3) during the merging maneuver according to the varying communication links: consider the graphs in Fig. 2. An arrow from one vehicle to another denotes a communication link from the vehicle where the arrow starts to the vehicle where the arrow ends. Cyclic communication arises in graph 2 due to the presence of bidirectional arrows (vehicles 2 -3 and vehicles 4 -5).

Remark 1 (Human-like interaction): Uni-or bi-directional arrows in Fig. 2

emulate how human interaction with other vehicles. When driving in a formation (graphs 1 and 3), human drivers keep the distance by only looking at the preceding vehicle (they can look at the following vehicle via rear window but not for keeping the formation). However, during merging (graph 2), the driver that wants to merge (e.g. vehicle 2) must look at both the following and the preceding vehicle of the

adjacent line (e.g. vehicles 1 and 3). At the same time, the driver of vehicle 3 must look at the vehicle that wants to merge. This establishes a bidirectional interaction between vehicle 2 and vehicle 3 (similarly for vehicle 4 and vehicle 5).

Referring to the CTH in (3), the spacing policies are designed as, i = 1, 2, . . .:

• Graph 1: r 2i,2i+1 = 0 and r 2i+1,2i-1 = hv 2i+1 + r (r denotes standstill distance and h time headway); • Graph 2: r 2i,2i+1 decreases gradually from 0 to -(hv 2i+1 +r), while r 2i+1,2i-1 increases gradually from hv 2i+1 + r to hv 2i+1 + hv 2i + 2r;

• Graph 3: r i+1,i = hv i+1 + r. The term "gradually" means a slowly time-varying interpolation policy between the initial and the final value (cf. the detail in Fig. 3). Graph 1 considers r 2i,2i+1 = 0, indicating that the merging begins when the control makes adjacent lane vehicles reach the same longitudinal location. This is just to the purpose of simplifying analysis and different starting states to begin the maneuver could be considered.

Remark 2 (Changing links): Vehicles interact with neighboring vehicles differently depending if they must open the gap or merge into the gap. Consider vehicles 2 and 3 in Fig. 2 for simplicity (similar reasoning applies to vehicles 4 and 5).

• Before the merging (graph 1), vehicle [START_REF] Larsson | The vehicle platooning problem: Computational complexity and heuristics[END_REF]

The unexplored challenge in CACC literature is handling in a well-posed way the cyclic communication instantiated by graph 2. The problem of well posedness of the input is absent in acyclic graphs.

We introduce a compact notation for the vehicle uncertain dynamics in [START_REF] Dey | A review of communication, driver characteristics, and controls aspects of cooperative adaptive cruise control (CACC)[END_REF], which simplifies the subsequent derivations

ẋi = A i x i + b i u i , i ∈ S N (5)
where A i and b i are unknown matrices in the form of [START_REF] Dey | A review of communication, driver characteristics, and controls aspects of cooperative adaptive cruise control (CACC)[END_REF]. For the virtual leading dynamics we have

ẋm = A 0 x m + b 0 u 0 = A m x m + b m w (6)
where A m and b m are known matrices in the form of (2).

Being the system matrices in (5) unknown, the synchronization task has to be achieved in an "adaptive" fashion, meaning that the problem is solved by appropriate control gains that are unknown and must be estimated (cf. discussion in Remark 3).

The following result from literature [START_REF] Baldi | Adaptive synchronization of unknown heterogeneous agents: An adaptive virtual model reference approach[END_REF], [START_REF] Ioannou | Robust Adaptive Control[END_REF] justifies that the unknown gains solving the problem exist. Proposition 1: [Distributed matching conditions] For dynamics in the form ( 1) and ( 2), there exist k * i and l * i such that

A m = A i + b i k * ′ i , b m = b i l * i , i ∈ S N (7)
(the symbol ′ denotes transpose operation since k * i are vectors). Furthermore, the signs of l * i are positive, and there exist

k * 2i+1,2i-1 = k * 2i+1 -k * 2i-1 l * 2i+1 /l * 2i-1 , k * 2i,2i-1 = k * 2i - k * 2i-1 l * 2i /l * 2i-1 , k * 2i,2i+1 = k * 2i -k * 2i+1 l * 2i /l * 2i+1 , k * 2i+1,2i = k * 2i -k * 2i l * 2i+1 /l * 2i and scalars l * 2i+1,2i-1 = l * 2i+1 /l * 2i-1 , l * 2i,2i-1 = l * 2i /l * 2i-1 , l * 2i,2i+1 = l * 2i /l * 2i+1 , l * 2i+1,2i = l * 2i+1 /l * 2i such that A 2i-1 = A 2i+1 + b 2i+1 k * ′ 2i+1,2i-1 b 2i-1 = b 2i+1 l * 2i+1,2i-1 , A 2i-1 = A 2i + b 2i k * ′ 2i,2i-1 b 2i-1 = b 2i l * 2i,2i-1 , A 2i+1 = A 2i + b 2i k * ′ 2i,2i+1 b 2i+1 = b 2i l * 2i,2i+1 , A 2i = A 2i+1 + b 2i+1 k * ′ 2i+1,2i b 2i = b 2i+1 l * 2i+1,2i (8)
where i = 1, 2, . . .. It is shown in [START_REF] Baldi | Adaptive synchronization of unknown heterogeneous agents: An adaptive virtual model reference approach[END_REF] that (8) implies

A 0 = A i + b i k * ′ i,0 , b 0 = b i l * i,0 (9)
with

k * i,0 = k * i -k * 0 l * i /l * 0 and l * i,0 = l * i /l * 0 ,
where k * 0 and l * 0 are the control gains defined after [START_REF] Bevly | Lane change and merge maneuvers for connected and automated vehicles: A survey[END_REF]. Remark 3 (Solutions to matching conditions): Inspecting the structure of A i and b i in [START_REF] Dey | A review of communication, driver characteristics, and controls aspects of cooperative adaptive cruise control (CACC)[END_REF], reveals that for indexes i and j

k * i = [a 01 a 02 a 03 + 1 τ i ]τ i , l * i = b 00 τ i k * j,i = [0 0 ( 1 τ j - 1 τ i )]τ j , l * j,i = τ j τ i
from which some observe that the solutions to the matching conditions are unknown as they depend on the driveline time constants: in the absence of such information, we propose adaptive laws to estimate the solutions to the matching conditions.

Hereafter, we will show that Proposition 1 can be used for the adaptive controllers solving Problem 2.1.

A. The adaptive controller: acyclic graphs 1 and 3

Because graphs 1 and 3 are acyclic, the control and adaptive laws stem from the approach in [START_REF] Baldi | Adaptive synchronization of unknown heterogeneous agents: An adaptive virtual model reference approach[END_REF], and are given without proof.

Graph 1: For vehicle 1, design the controller

u 1 (t) = k ′ 1,0 (t)x m (t) + k ′ 1 (t)e 1,0 (t) + l 1,0 (t)u 0 (t) (10)
where k 1,0 (t), k 1 (t) and l 1,0 (t) are the estimates of k * 1,0 , k * 1 and l * 1,0 adapted by

k′ 1,0 (t) = -γ k b ′ m P e 1,0 (t)x ′ m (t) k′ 1 (t) = -γ k b ′ m P e 1,0 (t)e ′ 1,0 (t) l1,0 (t) = -γ l b ′ m P e 1,0 (t)u 0 (t) (11)
For 'even' vehicles 2, 4, . . ., design the controller

u 2i (t) = k ′ 2i,2i+1 (t)x 2i+1 (t) + k ′ 2i (t)e 2i,2i+1 (t) + l 2i,2i+1 (t)u 2i+1 (t) (12)
where i = 1, 2, . . ., and k 2i,2i+1 (t), k 2i (t) and l 2i,2i+1 (t) are the estimates of k * 2i,2i+1 , k * 2i and l * 2i,2i+1 adapted by

k′ 2i,2i+1 (t) = -γ k b ′ m P e 2i,2i+1 (t)x ′ 2i+1 (t) k′ 2i+1 (t) = -γ k b ′ m P e 2i,2i+1 (t)e ′ 2i,2i+1 (t) l2i,2i+1 (t) = -γ l b ′ m P e 2i,2i+1 (t)u 2i+1 (t) (13)
For 'odd' vehicles 3, 5, . . ., consider the controller

u 2i+1 (t) = k ′ 2i+1,2i-1 (t)x 2i-1 (t) + k ′ 2i-1 (t)e 2i+1,2i-1 (t) + l 2i+1,2i-1 (t)u 2i-1 (t) (14)
where i = 1, 2, . . ., and

k 2i+1,2i-1 (t), k 2i-1 (t) and l 2i+1,2i-1 (t) are the estimates of k * 2i+1,2i-1 , k * 2i-1 and l * 2i+1,2i-1 adapted by k′ 2i+1,2i-1 (t) = -γ k b ′ m P e 2i+1,2i-1 (t)x ′ 2i-1 (t) k′ 2i-1 (t) = -γ k b ′ m P e 2i+1,2i-1 (t)e ′ 2i+1,2i-1 (t) l2i+1,2i-1 (t) = -γ l b ′ m P e 2i+1,2i-1 (t)u 2i-1 (t) (15)
The scalars γ k , γ l > 0 are adaptive gains, and P is a positive definite matrix solution to the Lyapunov equation

P A m + A ′ m P = -Q, Q > 0. ( 16
)
Graph 3: The structure of the control law is the same as graph 1, albeit the index of the neighboring vehicle changes. For vehicle 1, design the same controller [START_REF] Ploeg | lp string stability of cascaded systems: Application to vehicle platooning[END_REF] and the adaptive laws [START_REF] Li | Performance enhanced predictive control for adaptive cruise control system considering road elevation information[END_REF]. For all other vehicles 2, 3, 4, . . ., design the controller

u i+1 (t) = k ′ i+1,i (t)x i (t) + k ′ i+1 (t)e i+1,i (t) + l i+1,i (t)u i (t) (17)
where i = 1, 2, . . .; the gains are adapted in a similar fashion as ( 11), ( 13), [START_REF] Chien | Regulation layer controller design for automated highway system: Platoon merge and split controller design[END_REF] except the different indexes for the neighboring vehicles.

Remark 4 (Communication and on-board measurements): Remark 3 shows that the first two components of k * j,i are zero. As a result, no absolute position/velocity are necessary, since the first two entries of x i multiplying k j,i have no effect in the control law. This is in line with CACC protocols, where on-board measurements of relative position and velocity (via laser devices) are used together with wireless communication of acceleration and inputs. In the absence of wireless communication e.g. due to communication failure, the literature has proposed to switch from a CACC protocol to an ACC protocol with only on-board measurements [START_REF] Harfouch | An adaptive switched control approach to heterogeneous platooning with intervehicle communication losses[END_REF], [START_REF] Ploeg | Graceful degradation of cooperative adaptive cruise control[END_REF], [START_REF] Van Arem | The impact of cooperative adaptive cruise control on traffic-flow characteristics[END_REF]. However it was shown that, as compared to CACC, ACC has degraded performance.

B. The adaptive controller: cyclic graph 2

An adaptive controller to handle graph 2 has not been proposed in literature: its design is explained and proved.

Theorem 1: For vehicle 1, consider controller (10) with adaptive laws [START_REF] Li | Performance enhanced predictive control for adaptive cruise control system considering road elevation information[END_REF]. For 'odd' vehicles 3, 5, . . ., consider controller

u 2i+1 (t) = k ′ 2i+1,2i (t) x 2i (t) 2 + k ′ 2i+1,2i-1 (t) x 2i-1 (t) 2 + k ′ 2i+1 (t) e 2i+1,2i (t) + e 2i+1,2i-1 (t) 2 + l 2i+1,2i (t) u 2i (t) 2 + l 2i+1,2i-1 (t) u 2i-1 (t) 2 (18)
(i = 1, 2, . . .) and the adaptive laws

k′ 2i+1,2i (t) = -γ k b ′ m P (e 2i+1,2i (t) + e 2i+1,2i-1 (t))x ′ 2i (t) k′ 2i+1,2i-1 (t) = -γ k b ′ m P (e 2i+1,2i (t) + e 2i+1,2i-1 (t))x ′ 2i-1 (t) k′ 2i+1 (t) = -γ k b ′ m P (e 2i+1,2i (t) + e 2i+1,2i-1 (t)) (e 2i+1,2i (t) + e 2i+1,2i-1 (t)) ′ l2i+1,2i (t) = -γ l b ′ m P (e 2i+1,2i (t) + e 2i+1,2i-1 (t))u 2i (t) l2i+1,2i-1 (t) = -γ l b ′ m P (e 2i+1,2i (t) + e 2i+1,2i-1 (t))u 2i-1 (t) (19)
For 'even' vehicles 2, 4, . . ., consider the controller

u 2i (t) = k ′ 2i,2i-1 (t) x 2i-1 (t) 2 + k ′ 2i,2i+1 (t)
x 2i+1 (t) 2

+ k ′ 2i (t) e 2i,2i-1 (t) + e 2i,2i+1 (t) 2
+ l 2i,2i-1 (t) u 2i-1 (t) 2 + l 2i,2i+1 (t) u 2i+1 (t) 2 (20)
(i = 1, 2, . . .) and the adaptive laws

k′ 2i,2i-1 (t) = -γ k b ′ m P (e 2i,2i-1 (t) + e 2i,2i+1 (t))x ′ 2i-1 (t) k′ 2i,2i+1 (t) = -γ k b ′ m P (e 2i,2i-1 (t) + e 2i,2i+1 (t))x ′ 2i+1 (t) k′ 2i (t) = -γ k b ′ m P (e 2i,2i-1 (t) + e 2i,2i+1 (t)) (e 2i,2i-1 (t) + e 2i,2i+1 (t)) ′ l2i,2i-1 (t) = -γ l b ′ m P (e 2i,2i-1 (t) + e 2i,2i+1 (t))u 2i-1 (t) l2i,2i+1 (t) = -γ l b ′ m P (e 2i,2i-1 (t) + e 2i,2i+1 (t))u 2i+1 (t) (21)
where all variables/parameters have a similar meaning as Section III-A. Then, provided that the inputs are well defined at very time instant, Problem 2.1 is solved in graph 2. Proof 1: See Appendix.

IV. WELL-POSEDNESS OF THE INPUT

Let us explicitly use 5 vehicles present the issues of wellposedness and its solution clear. In Theorem 1 we have assumed that u i , u i+1 , ..., are well defined at very time instant: however, in the presence of cycles, one u i on the left-hand side of its control law can re-appear on the right-hand side of another control law, so that inputs might not be wellposedness. To clarify this point, let us consider graph 2 in Fig. 2 and the control laws in Theorem 1. Let us write the inputs compactly as

     1 0 0 0 0 -l 2,1 2 -l 2,3 0 0 -l 3,1 -l 3,2 2 0 0 0 0 -l 4,3 2 -l 4,5 0 0 -l 5,3 -l 5,4 2      U      u 1 u 2 u 3 u 4 u 5      =      (k 1,0 -k 1 ) ′ x 0 + k ′ 1 x 1 + l 1,0 u 0 (k 2,1 -k 2 ) ′ x 1 + 2k ′ 2 x 2 + (k 2,3 -k 2 ) ′ x 3 (k 3,1 -k 3 ) ′ x 1 + (k 3,2 -k 3 ) ′ x 2 + 2k ′ 3 x 3 (k 4,2 -k 4 ) ′ x 2 + 2k ′ 4 x 4 + (k 4,5 -k 4 ) ′ x 5 (k 5,2 -k 5 ) ′ x 2 + (k 5,4 -k 5 ) ′ x 4 + 2k ′ 5 x 5      .
Even though no explicit inversion of U is used to obtain the inputs, it is clear that we need the matrix U to be invertible to guarantee well-posedness of u i at all time instants. Invertibility of U can be assessed via its determinant det

      1 0 0 0 0 -l 2,1 2 -l 2,3 0 0 -l 3,1 -l 3,2 2 0 0 0 0 -l 4,3 2 -l 4,5 0 0 -l 5,3 -l 5,4 2
      = l 2,3 l 3,2 l 4,5 l 5,4 -4l 4,5 l 5,4 -4l 2,3 l 3,2 + 16 = (4 -l 2,3 l 3,2 )(4 -l 4,5 l 5,4 ). ( 22
)
If we use the actual (but unknown) parameters from Proposition 3.1 to calculate the determinant, we obtain (4 -

l * 2,3 l * 3,2 )(4 -l * 4,5 l * 5,4
), giving a determinant equal to 9 (because l * 2,3 l * 3,2 = 1 and l * 4,5 l * 5,4 = 1). However, since the parameters from Proposition 3.1 are unknown, we have to use the estimated parameters to get the determinant. Therefore, the determinant of U can be different than 9 and in particular nothing forbid the critical case of 0 determinant from happening. Note that the event mentioned in Remark 4 of communication failure requiring to switch to an ACC with only on-board measurements, makes the well-posedness issue disappear, since inputs from other vehicles will not be used in the control law. As a result, well-posedness analysis refers to CACC with active communication.

The idea to guarantee well posedness at all time instants is simple: we allow vehicles 2 and 3 to communicate l 2,3 (t) and l 3,2 (t); similarly, we allow vehicles 4 and 5 to communicate l 4,5 (t) and l 5,4 (t). These vehicles can communicate their own estimates across the communication links. Then, we can constrain l 2,3 l 3,2 and l 4,5 l 5,4 to be positive and not greater than the critical value 4 that makes the determinant 0: which such constraints, the matrix U can always be inverted invertible. Therefore, one assumption is introduced.

Assumption 1.

The exact values of l * 2,3 , l * 3,2 , l * 4,5 and l * 5,4 unknown, but these parameters reside in convex compact sets Ω l (that describe an uncertainty set function of l 2,3 , l 3,2 , l 4,5 , l 5,4 ). These sets do not contain the set (4 -l 2,3 l 3,2 )(4 -l 4,5 l 5,4 ) = 0.

The choice of Ω l is not unique, and an example (cf. Fig. 4a) is given by l 2,3 ≥ 0, l 3,2 ≥ 0, l 4,5 ≥ 0, l 5,4 ≥ 0, l 2,3 ≤ -l 3,2 +3.99, l 4,5 ≤ -l 5,4 +3.99. Interestingly, the factorization in [START_REF] Tas | Making Bertha cooperate-team annieWAY's entry to the 2016 Grand Cooperative Driving Challenge[END_REF], allows Ω l to be decoupled in two planes, parametrized by (l 2,3 , l 3,2 ) and (l 4,5 , l 5,4 ). According to Assumption 1, Ω l is constructed to avoid the intersection with the critical lines l 2,3 l 3,2 = 4 and l 4,5 l 5,4 = 4 (along these lines the matrix U would not be invertible). Let us adopt a general description of sets Ω l as Ω l = {l 2,3 , l 3,2 , l 4,5 , l 5,4 | g(l 2,3 , l 3,2 , l 4,5 , l 5,4 ) ≤ 0} [START_REF] Zheng | Safety rule-based cellular automaton modeling and simulation under V2V environment[END_REF] for some appropriate vector function g(l 2,3 , l 3,2 , l 4,5 , l 5,4 ).

Remark 5 (Size of uncertainty): Assumption 1 limits the level of uncertainty inside a triangular region as in Fig. 4a. This region covers an uncertainty of 100% with respect to the nominal driveline time constant (such region is drawn in Fig. 4b with a circle). Therefore, the proposed methods accounts for a quite large uncertainty.

Remark 6 (Effect of number of vehicles): To study how the expression [START_REF] Tas | Making Bertha cooperate-team annieWAY's entry to the 2016 Grand Cooperative Driving Challenge[END_REF] scales with the number of vehicles, let us consider only the first three vehicles in Fig. 1: it is possible to obtain

det   1 0 0 -l 2,1 2 -l 2,3 -l 3,1 -l 3,2 2   = 4 -l 2,3 l 3,2 . ( 24
)
By comparing [START_REF] Baldi | Adaptive synchronization of unknown heterogeneous agents: An adaptive virtual model reference approach[END_REF] with [START_REF] Tas | Making Bertha cooperate-team annieWAY's entry to the 2016 Grand Cooperative Driving Challenge[END_REF], one can notice that [START_REF] Tas | Making Bertha cooperate-team annieWAY's entry to the 2016 Grand Cooperative Driving Challenge[END_REF] is the product of two 'smaller' determinants calculated for three vehicles. It is left to the interested reader to verify that, in the general case, the determinant takes the form

det(U ) = (N -1)/2 i=1 (4 -l 2i,2i+1 l 2i+1,2i ) (25)
where indicates the multiplication. This shows that the uncertainty bounds keeps the same structure even with increasing number of vehicles. Assumption 2. The actual parameters l * 2i,2i+1 , l * 2i+1,2i , i = 1, 2, . . . are known to reside in some convex compact sets (call them Ω l ) does not contain the set (4 -l 2i,2i+1 l 2i+1,2i ) = 0.

Theorem 2: Consider the merging phase described by graph 2 in Fig. 2. Under Assumption 2, consider the vehicles described by ( 5) and the leading vehicle described by [START_REF] Guo | String stability of heterogeneous leader-following vehicle platoons based on constant spacing policy[END_REF], the controllers (10), ( 18), [START_REF] Acciani | Cooperative adaptive cruise control over unreliable networks: an observer-based approach to increase robustness to packet loss[END_REF] and the adaptive laws ( 11), ( 19), [START_REF] Harfouch | Adaptive control of interconnected networked systems with application to heterogeneous platooning[END_REF] with the following modifications

l2i+1,2i (t) = P Ω l [-γ l b ′ m P (e 2i+1,2i (t) + e 2i+1,2i-1 (t))u 2i (t) δ l 2i+1,2i (t)
]

=    δ l2i+1,2i (t) if l 2i+1,2i (t) ∈ Ω l , or l 2i+1,2i (t) ∈ ∂Ω l with δ l2i+1,2i ∇g ≤ 0 0 otherwise l2i,2i+1 (t) = P Ω l [-γ l b ′ m P (e 2i,2i+1 (t) + e 2i,2i-1 (t))u 2i+1 (t) δ l 2i,2i+1 (t) ] =    δ l2i,2i+1 (t) if l 2i,2i+1 (t) ∈ Ω l , or l 2i,2i+1 (26)
∈ ∂Ω l with δ l2i,2i+1 ∇g ≤ 0 0 otherwise where i = 1, 2, . . ., and P Ω l denotes the projection operator inside the set Ω l . In particular, ∂Ω l is the border of Ω l and ∇g is the derivative of g with respect to l 2i,2i+1 or l 2i+1,2i . Then, problem 2.1 is solved in graph 2. Proof 2: See Appendix. Remark 7 (Distributed strategy): The importance of Theorem 2 is to guarantee well posed control input in a rigorous way, a point overlooked in CACC literature. Well-posedness is guaranteed in a distributed way (using information from neighboring vehicles) and the expression [START_REF] Baldi | Adaptive state-feedback synchronization with distributed input: the cyclic case[END_REF] shows that the adaptive/projection laws do not have to be modified if the number of vehicles N changes.

V. RESULTS ON GCDC SCENARIO

Our protocol is tested on the benchmark of Fig. 1 Delays are important and they have been studied for example in [START_REF] Ploeg | lp string stability of cascaded systems: Application to vehicle platooning[END_REF], [START_REF] Ploeg | Graceful degradation of cooperative adaptive cruise control[END_REF] (in the absence of parametric uncertainty). From a theory point view, delays be handled via a robust adaptive control approach, similarly to what considered in [START_REF] Baldi | Adaptive synchronization of unknown heterogeneous agents: An adaptive virtual model reference approach[END_REF]: this has not been addressed in this work in order to streamline the proposed methodology. However, to validate the approach in a more realistic scenario, all simulations include communication delays of 0.15 s, in line with typical delays reported in CACC literature [START_REF] Ploeg | lp string stability of cascaded systems: Application to vehicle platooning[END_REF], [START_REF] Ploeg | Graceful degradation of cooperative adaptive cruise control[END_REF], [START_REF] Xing | Padé approximation of delays in cooperative acc based on string stability requirements[END_REF].

A. Mixing architecture

During the merging phases, each vehicle has a different number of neighboring vehicles: for example, vehicle 2 has one neighbor in graph 1 (vehicle 3); two neighbors in graph 2 (vehicles 1 and 3); one neighbor in graph 3 (vehicle 1). The different number of neighboring vehicles requires a different controller for each phase. One reasonable idea could be a switching architecture [START_REF] Liberzon | Switching in Systems and Control, ser. Systems & Control: Foundations & Applications[END_REF], [START_REF] Yuan | Adaptive asymptotic tracking control of uncertain time-driven switched linear systems[END_REF]: however, a switching control action can produce undesirable peaks due to non-smooth transitions. Therefore, with the intent to reduce such transients, we explore a mixing architecture [START_REF] Baldi | Multiple model adaptive mixing control: The discrete-time case[END_REF] that interpolates among different control actions.

This idea is represented in Fig. 5 and formalized by considering the following control action for vehicle 2, 4, . . .

u 2i (t) = M 1 (t)(k ′ 2i,2i-1 (t) x 2i-1 (t) 2 + l 2i,2i-1 (t) u 2i-1 (t) 2 + k ′ 2i (t) e 2i,2i-1 (t) 2 ) + M 2 (t)(k ′ 2i,2i+1 (t)
x 2i+1 (t) 2

+ l 2i,2i+1 (t) u 2i+1 (t) 2 + k ′ 2i (t) e 2i,2i+1 (t) 2 ) (27)
where the time-varying M 1 (t) and M 2 (t) interpolate like this: in graph 1, M 1 (t) = 2 and M 2 (t) = 0; in graph 2, M 1 (t) = 1 and M 2 (t) = 1; in graph 3, M 1 (t) = 0 and M 2 (t) = 2. In between, we can use any interpolation strategy, such as linear interpolation with a user-defined transition time. Note that [START_REF] Ploeg | Cooperative automated maneuvering at the 2016 Grand Cooperative Driving Challenge[END_REF] embeds in one single (time-varying) strategy all controls u 2 during the three phases. In our simulations, the transition time is taken to be 5 s: in the special case in which the transition is abrupt (step change) we have the switching scenario. For the mixing case, the adaptive gains for vehicle 2, 4, . . . also implement a smooth interpolation:

k′ 2i,2i-1 (t) = -γ k b ′ m P (M 1 (t)[e 2i,2i-1 (t)] + M 2 (t)[e 2i,2i+1 (t)])x ′ 2i-1 (t) k′ 2i,2i+1 (t) = -γ k b ′ m P (M 1 (t)[e 2i,2i-1 (t)] + M 2 (t)[e 2i,2i+1 (t)])x ′ 2i+1 (t) k′ 2i (t) = -γ k b ′ m P (M 1 (t)[e 2i,2i-1 (t)] + M 2 (t)[e 2i,2i+1 (t)]) (M 1 (t)[e 2i,2i-1 (t)] + M 2 (t)[e 2i,2i+1 (t)]) ′ l2i,2i-1 (t) = -γ l b ′ m P (M 1 (t)[e 2i,2i-1 (t)] + M 2 (t)[e 2i,2i+1 (t)])u 2i-1 (t) l2i,2i+1 (t) = -γ l b ′ m P (M 1 (t)[e 2i,2i-1 (t)] + M 2 (t)[e 2i,2i+1 (t)])u 2i+1 (t). (28)
The same reasoning applies to the other vehicles.

B. Simulations

For the merging maneuver, Figs. 6a, 6b and 6c show the response of velocities, accelerations and inputs. Fig. 6d shows the relative distances in the frame of vehicle 1 (i.e. taking vehicle 1 as the origin). In the interval 0-40 seconds (graph 1), Fig. 6d shows that vehicles 3 and 2 achieve a desired distance from vehicle 1 while aligning with each other: their relative distance is zero but the vehicles are on different lanes. During this time, the velocity of vehicle 2 is larger in order to perform the alignment. Similarly, vehicles 4 and 5 achieve a desired distance from vehicle 2, while vehicle 4 (with larger velocity) aligns to vehicle 5. In the interval 40-60 seconds (graph 2), Fig. 6d shows that vehicle 3 creates a gap with vehicle 1, while vehicle 5 creates a gap with vehicle 3: by doing this, vehicles 2 and 4 start and achieve the merging. In the interval 60-80 seconds (graph 3), Fig. 6d shows that the formation of five vehicles is finally achieved.

The effectiveness of the mixing architecture with respect to the switching one can be assessed from Figs. 7a, 7b and 7c. These figures show, for a switching architecture, the response of velocities, accelerations and inputs. Let us compare Fig. 6b (proposed mixing) with Fig. 7b (switching): in the latter, undesirable large transients from -3 to 3 m/s 2 in acceleration appear. Similar comments apply to the comparisons between the inputs (desired acceleration as provided by the pedal) in Fig. 6c (proposed mixing) with Fig. 7c (switching). Summarizing, the mixing architecture proposed in our work has the merit to remove transient peaks and make the response smooth.

VI. CONCLUSIONS

This work has proposed a merging framework addressing some aspects overlooked in standard literature on pla- toon merging: coping with vehicle driveline uncertainty; handling bidirectional (cyclic) communication; guaranteeing wellposedness of the control input. To address these aspects, the protocol we proposed respectively relies on: adaptive laws to estimate the control gains solving the stability problem; analyze the control input in both acyclic and cyclic graphs; exploiting the graph structure to implement appropriate parameter projections for well posed inputs. The proposed approach holds scalability features, such as its distributed nature (well posedness is guaranteed using information from neighboring vehicles) and its structure of the uncertainty region used for projection that does not depend on the number of vehicles. These features do not cover all scalability aspects, such as scalability to traffic conditions or downstream conditions that might prevent some vehicles from creating a gap. Therefore, a promising future work is to further extend the proposed framework to study the impact of merging maneuvers on traffic and the effects of traffic conditions on merging. Studying such scenarios require to embed the proposed approach in a microscopic simulator (e.g. SUMO, AIMSUN) and to appropriately model merging behavior using field data [START_REF] Kang | Modeling driver merging behavior: A repeated game theoretical approach[END_REF], [START_REF] Van Arem | The impact of cooperative adaptive cruise control on traffic-flow characteristics[END_REF], [START_REF] Gunter | Model based string stability of adaptive cruise control systems using field data[END_REF]. We expect that a supervisor must be designed to determine the cruising speed of the platoon that make the merging possible depending on traffic conditions: all these aspects are worth of future research.

Fig. 1

 1 Fig. 1 represents the GCDC 2016 benchmark. Define v i and d i to be the velocity (m/s) and position (m) of vehicle i, respectively. We will use two sets of indexes, S o N = {i ∈ N| i = 1, 3, 5, . . .} is the 'odd' vehicle set, and S e N = {i ∈

Fig. 2 :

 2 Fig. 2: Graphs before/during/after merging. The dashed arrow denoted the platoon on the other lane before merging is initiated.

Fig. 3 :

 3 Fig. 3: Spacing policy during and after merging (only vehicles 1, 2, 3 are shown). The spacing is calculated with respect to the vehicle center of mass for simplicity.

  (a) Singular sets (red solid curve) and projection sets (shaded blue area) for vehicles 2, 3 and for vehicles 4, 5, respectively. The choice of the projection set is not unique. (b) The blue ball around with radius 1 represents a 100% uncertainty around nominal homogeneous driveline parameters (1, 1).

Fig. 4 :

 4 Fig. 4: Sets used to guarantee well-posed inputs

  . The reference model has: a 01 = -5, a 02 = -15, a 03 = -1.5, and b 00 = 1; the values of τ i in (1) are in Table I but they are known only for simulation (they are not used for control design). Initial states are also in Table I. The reference w corresponds to a steady-state speed of 20m/s while the merging is performed. The other parameters are: Q = diag(1, 1, 5), γ k = 1 • 10 -4 , γ l = 5 • 10 -4 , h = 0.7. initial control gains are chosen based on the (incorrect) a priori knowledgeTABLE I: Vehicles parameters and initial conditions.

  ,2,1] Vehicle 3 0.33 [-15,2,1] Vehicle 4 0.14 [-30,2,1] Vehicle 5 0.17 [-25,2,1]that all vehicles have homogeneous driveline τ 0 = 0.28: this amounts to taking k ij (0) = 0, l ij (0) = 1 and k i (0) = τ 0 [a 01 a 02 (a 03 +1/τ 0 )] ′ . Note that a 100% uncertainty around the nominal value of τ 0 = 0.28 corresponds to the range [0.14, 0.56]. Note that this uncertainty range covers the values of τ i in TableI. The maneuver works as:-0-40 s: vehicles 2 and 4 align with vehicles 3 and 5, respectively, while in the meantime vehicles 1, 3 and 5 form the initial platoon of three vehicles. -40-60s: vehicles 3 and 5 form a gap for vehicles 2 and 4, respectively, while in the meantime vehicles 2 and 4 start the merging. -60-80s: the final platoon of five vehicles is formed.

Fig. 5 :

 5 Fig. 5: Mixing architecture for vehicle k: the big knob represents the mixing action smoothly going from one controller to the next one according to the merging phase.

  Relative distances with vehicle 1. Note that vehicles 2 and 4 align with vehicles 3 and 5 on different lanes before merging.

Fig. 6 :

 6 Fig. 6: Merging with mixing architecture (proposed).

Fig. 7 :

 7 Fig.7: Merging with switching architecture (undesirable). Switching causes a large transients from -3 to 3 m/s 2 in acceleration. The mixing results in Fig.6remove such transients.

This work was partly funded by the Special Guiding Funds Double First-class no. 3307012001A, the Natural Science Foundation of China no. 62073074, and the ANR (French National Science Foundation) no. ANR-18-CE40-0010 (HANDY).

The proof is given for 5 vehicles only to the purposes of making the idea clear. We want to prove that: vehicle 3 synchronizes to vehicles 1 and 2 (e 3,1 , e 3,2 → 0); vehicle 2 synchronizes to vehicles 1 and 3 (e 2,1 , e 2,3 → 0); vehicle 5 synchronizes to vehicles 3 and 4 (e 5,3 , e 5,4 → 0); vehicle 4 synchronizes with vehicle 5 and vehicle 3 (e 4,5 , e 4,3 → 0). Extension to arbitrarily long platoons follows by properly using the 'even' and 'odd' indexes. We use the Lyapunov function

and the error dynamics, as depicted in Fig. 8 are where k = k -k * , l = l -l * (with appropriate subscripts) are the parameter estimation errors. For compactness, we have defined e 3,2,1 = e 3,1 + e 3,2 , e 2,3,1 = e 2,1 + e 2,3 , e 5,4,3 = e 5,3 + e 4,3 and e 4,5,3 = e 4,3 + e 5,3 . At this point, one can use standard Lyapunov theory for adaptive control [START_REF] Ioannou | Robust Adaptive Control[END_REF], and we can show V1 + V3,2,1 + V2,3,1 + V5,4,3 + V4,5,3 → 0 as t → ∞. Convergence of all errors to zero can be obtained via Barbalat's lemma.

For proving convergence with a general number of vehicles, the interested readers can exploit the Lyapunov function

and the error dynamics

along with similar steps as above.


Hajer Bahouri
email: hajer.bahouri@ljll.math.upmc.fr

AND Trevor M Leslie
email: lesliet@usc.edu

Galina Perelman
email: galina.perelman@u-pec.fr

H s BOUNDS FOR THE DERIVATIVE NONLINEAR SCHR ÖDINGER EQUATION

We study the derivative nonlinear Schrödinger equation on the real line and obtain global-intime bounds on high order Sobolev norms.

1 2 threshold and the critical space L 2 (R) by working in more general Fourier-Lebesgue spaces, c.f. Grünrock [6] and references therein.

INTRODUCTION

We consider the Cauchy problem for the derivative nonlinear Schrödinger equation (DNLS) on the real line R:

(1)

   i∂ t u + ∂ 2 x u = -i∂ x (|u| 2 u), u t=0 = u 0 ∈ H s (R), s ≥ 1 2 .
We remark right away that the DNLS is L 2 critical, as it is invariant under the scaling

(2) u(t, x) → u µ (t, x) := √ µu(µ 2 t, µx), µ > 0.

The DNLS equation was introduced by Mio-Ogino-Minami-Takeda and Mjølhus [START_REF] Mio | Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas[END_REF][START_REF] Mjølhus | On the modulational instability of hydromagnetic waves parallel to the magnetic field[END_REF] as a model for studying magnetohydrodynamics, and it has received a great deal of attention from the mathematics community after being shown to be completely integrable by Kaup-Newell [START_REF] Kaup | An exact solution for a derivative nonlinear Schrödinger equation[END_REF]. The infinitely many conserved quantities admitted by the DNLS equation play an important role in the wellposedness theory.

The first three-the mass, momentum, and energy-are as follows.

M (u) := R |u| 2 dx, (3)
P (u) := Im R uu x dx + 1 2 R |u| 4 dx, (4)
E(u) := R |u x | 2 - 3 2 Im(|u| 2 uu x ) + 1 2 |u| 6 dx. (5)
Before stating our main result, let us give a very brief review of what is known about the wellposedness of the DNLS equation. More detailed overviews can be found, for example, in the introductions of [START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF] and [START_REF] Killip | On the well-posedness problem for the derivative nonlinear schrödinger equation[END_REF]. Local wellposedness in H s (R) for s ≥ 1 2 was proven by Takaoka [START_REF] Takaoka | Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity[END_REF], improving earlier work [START_REF] Ozawa | On the nonlinear Schrödinger equations of derivative type[END_REF] by Ozawa. On the other hand, for s < 1 2 , the uniform continuity of the data-to-solution map fails in H s (R) [START_REF] Biagioni | Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations[END_REF][START_REF] Takaoka | Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in loworder Sobolev spaces[END_REF]. One can, however, close the 1 2 -derivative gap between the H A line of results, due to Hayashi-Ozawa [START_REF] Hayashi | On the derivative nonlinear Schrödinger equation[END_REF], Colliander-Keel-Staffilani-Takaoka-Tao [START_REF] Colliander | A refined global well-posedness result for Schrödinger equations with derivative[END_REF], Wu [START_REF] Wu | Global well-posedness on the derivative nonlinear Schrödinger equation[END_REF], and Guo-Wu [START_REF] Guo | Global well-posedness for the derivative nonlinear Schrödinger equation in H 1 2 (R)[END_REF], establishes global well-posedness of the DNLS equation in H s (R) for s ≥ 1 2 , for initial data having mass less than 4π. Another line (Pelinovsky-Saalmann-Shimabukuro [START_REF] Dmitry | The derivative NLS equation: global existence with solitons[END_REF], Pelinovsky-Shimabukuro [START_REF] Dmitry | Existence of global solutions to the derivative NLS equation with the inverse scattering transform method[END_REF], and Jenkins-Liu-Perry-Sulem [START_REF] Jenkins | Global well-posedness for the derivative non-linear Schrödinger equation[END_REF][START_REF] Jenkins | Global existence for the derivative nonlinear Schrödinger equation with arbitrary spectral singularities[END_REF][START_REF] Jenkins | The derivative nonlinear Schrödinger equation: global well-posedness and soliton resolution[END_REF]) uses inverse scattering techniques to establish global wellposedness under stronger regularity and decay assumptions on the initial data, but without a smallness requirement on the mass.

The first and third authors proved in [START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF] that the DNLS equation is globally well-posed in H s (R) for s ≥ 1 2 and that solutions generated from H 1 2 initial data remain bounded in H 1 2 (R) for all time. There have also been some recent works below the aforementioned s = 1 2 threshold of uniform H s continuity with respect to initial data [START_REF] Biagioni | Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations[END_REF][START_REF] Takaoka | Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in loworder Sobolev spaces[END_REF]. Klaus-Schippa [START_REF] Klaus | A priori estimates for the derivative nonlinear schrödinger equation[END_REF] gave H s a priori estimates for 0 < s < 1 2 in the case of small mass, Killip-Ntekoume-Vis ¸an [START_REF] Killip | On the well-posedness problem for the derivative nonlinear schrödinger equation[END_REF] improved the small mass assumption to 4π and furthermore proved a global wellposedness result in H s (R), 1 6 ≤ s < 1 2 , for initial data with mass less than 4π. Very recently, Harrop-Griffiths, Killip, and Vis ¸an [START_REF] Harrop-Griffiths | Large-data equicontinuity for the derivative NLS[END_REF] have removed the small mass assumption both from their H s a priori bounds, 0 < s < 1 2 , as well as from their global wellposedness result in H s (R) with 1 6 ≤ s < 1 2 . In this paper, we are concerned with the global-in-time boundedness of solutions to the DNLS equation in H s spaces. We prove that a uniform-in-time bound in H s (R) holds for all s ≥ 1 2 . Theorem 1.1. Suppose u is a solution to the DNLS equation with initial data u 0 ∈ H s (R), s ≥ 1 2 . There exists a finite positive constant C = C(s, u 0 H s (R) ), such that

sup t∈R u(t) H s (R) ≤ C(s, u 0 H s (R) ).
The main idea is to build off of the H s bounds with 0 < s < 1 2 from [START_REF] Harrop-Griffiths | Large-data equicontinuity for the derivative NLS[END_REF] and to take advantage of the complete integrability of the equation. As in [START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF], [START_REF] Harrop-Griffiths | Large-data equicontinuity for the derivative NLS[END_REF], the present work relies heavily on the conservation of the transmission coefficient for the spectral problem associated to the DNLS equation. This property has already been used in many other works; of particular relevance to us are the papers of Gérard [START_REF] Gérard | On the Conservation Laws of the Defocusing Cubic NLS Equation[END_REF], Killip-Vis ¸an-Zhang [START_REF] Killip | Low regularity conservation laws for integrable PDE[END_REF], Killip-Vis ¸an [START_REF] Killip | KdV is well-posed in H -1[END_REF], and Koch-Tataru [START_REF] Koch | Conserved energies for the cubic nonlinear Schrödinger equation in one dimension[END_REF], on the cubic NLS and KdV equations.

Note that by continuity of the flow, and the preservation of the Schwartz class under the flow, we lose nothing by restricting attention to the Schwartz class; we will thus work exclusively with Schwartz functions for the remainder of the manuscript. We will also suppress the time dependence when it does not play a role.

One can easily prove Theorem 1.1 in the special case s = 1, using the conserved quantity E(u). Indeed, simply rearranging (5) yields

u 2 Ḣ1 (R) = E(u) - 1 2 u 6 L 6 (R) + 3 2 R Im(|u| 2 uu x ) dx.
Clearly, the last term can be bounded above in absolute value by

1 2 u 2 Ḣ1 (R) + C u 6 L 6 (R)
, whence the desired bound follows by Sobolev embedding.

The higher-order Sobolev norms of integer order can be dealt with similarly, once we have a formula for the corresponding higher-order conserved quantities. We will show that for any nonnegative integer , one of the conserved quantities is equal to a constant multiple of u 2 Ḣ (R) , plus terms which are of lower order. For noninteger s, we will use a sort of 'generalized energy', comparable to u 2 Ḣs (R) , that will be defined in terms of the transmission coefficient of the DNLS spectral problem. We sketch presently the background necessary to define these objects precisely; for more details, see, for example, [START_REF] Ablowitz | Solitons and the inverse scattering transform[END_REF][START_REF] Jenkins | Global well-posedness for the derivative non-linear Schrödinger equation[END_REF][START_REF] Jenkins | Global existence for the derivative nonlinear Schrödinger equation with arbitrary spectral singularities[END_REF][START_REF] Jenkins | The derivative nonlinear Schrödinger equation: global well-posedness and soliton resolution[END_REF][START_REF] Kaup | An exact solution for a derivative nonlinear Schrödinger equation[END_REF][START_REF] Lee | Global solvability of the derivative nonlinear Schrödinger equation[END_REF][START_REF] Dmitry | Existence of global solutions to the derivative NLS equation with the inverse scattering transform method[END_REF][START_REF] Tsuchida | Complete integrability of derivative nonlinear Schrödinger-type equations[END_REF].

The DNLS equation can be obtained as a compatibility condition of the following system [START_REF] Kaup | An exact solution for a derivative nonlinear Schrödinger equation[END_REF]:

∂ x ψ = U(λ)ψ, ∂ t ψ = Υ(λ)ψ. (6)
Here λ ∈ C is a spectral parameter, independent of t and x, and ψ = ψ(t, x, λ) is C 2 -valued. The operators U(λ) and Υ(λ) are defined by

U(λ) = -iσ 3 (λ 2 + iλU ), Υ(λ) = -i(2λ 4 -λ 2 |u| 2 )σ 3 + 0 2λ 3 u -λ|u| 2 u + iλu x -2λ 3 u + λ|u| 2 u + iλu x 0 , (7)
where

σ 3 = 1 0 0 -1 , U = 0 u u 0 .
To be more specific about the sense in which the DNLS is a compatibility condition, we note that u satisfies the DNLS equation if and only if U and Υ satisfy the so-called 'zero-curvature' representation

∂U ∂t - ∂Υ ∂x + [U, Υ] = 0.
The first equation of ( 6) can be written in the form

(8) L u (λ)ψ := (iσ 3 ∂ x -λ 2 -iλU )ψ = 0,
which defines the scattering transform associated to the DNLS. Let us denote

Ω + := {λ ∈ C : Im λ 2 > 0}.
Then given u ∈ S(R) and λ ∈ Ω + , there are unique solutions to (8) (the "Jöst solutions") exhibiting the following behavior at ±∞:

ψ - 1 (x, λ) = e -iλ 2 x 1 0 + o(1) , as x → -∞, ψ + 2 (x, λ) = e iλ 2 x 0 1 + o(1) , as x → +∞. (9)
Finally, we denote by a u (λ) the Wronskian of the Jöst solutions defined above: 1 (10) a u (λ) = det(ψ - 1 (x, λ), ψ + 2 (x, λ)). Using the second equation in [START_REF] Grünrock | Bi-and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS[END_REF], it can be shown that a u (λ) is time-independent if u is a solution of (1). Furthermore, a u is a holomorphic function of λ in Ω + , and one may determine the behavior of a u at infinity by transforming (8) into a Zakharov-Shabat spectral problem, linear with respect to the spectral parameter, c.f. [START_REF] Kaup | An exact solution for a derivative nonlinear Schrödinger equation[END_REF], [START_REF] Dmitry | The derivative NLS equation: global existence with solitons[END_REF]. The equivalence between the two problems allows us to write [START_REF] Jenkins | Global existence for the derivative nonlinear Schrödinger equation with arbitrary spectral singularities[END_REF] lim

|λ|→∞, λ∈Ω + a u (λ) = e -i 2 u 2 L 2 (R) .
For fixed u, we can thus define the logarithm so that ( 12)

lim |λ|→∞,λ∈Ω + ln a u (λ) = - i 2 u 2 L 2 (R) .
Moreover, ln a u (λ) admits an asymptotic expansion of the following form:

(13) ln a u (λ) = ∞ j=0 E j (u) λ 2j as |λ| → ∞, λ ∈ Ω + .
Since a u (λ) is time-independent, the quantities E j (u) are conservation laws. They are all polynomial in u, u, and their derivatives. Furthermore, the E j (u)'s inherit scaling properties from a u (λ). That is, for µ > 0, the fact that a uµ (λ) = a u ( λ √ µ ) implies that E j (u µ ) = µ j E j (u), for each j ∈ N. The first 1

The transmission coefficient mentioned earlier is the inverse of a u (λ).

several of the E j (u)'s are (up to multiplicative constants) the conserved quantities (3)-( 5) mentioned earlier:

E 0 (u) = - i 2 u 2 L 2 (R) = - i 2 M (u), E 1 (u) = i 4 P (u), E 2 (u) = - i 8 E(u).
For each ∈ N * , the quantity E 2 (u) can be used to control u 2 Ḣ (R) . Let us define, for ρ positive sufficiently large and L ∈ N, [START_REF] Killip | On the well-posedness problem for the derivative nonlinear schrödinger equation[END_REF] ϕ L (u, ρ) = Im ln a u ( iρ) -

2L+1 j=0 E j (u) (iρ) j .
If u is a solution of the DNLS equation, then ϕ L (u, ρ) is time-independent, being a sum of timeindependent quantities.

In order to establish bounds on the H s norm of u, for s ≥ 1 2 , we will show that ∞ R ρ 2s-1 ϕ [s] (u, ρ)dρ with R > 0 large enough controls the Ḣs seminorm of u, in a sense to be made precise later. Here and below, we use [s] to denote the integer part of a real number s.

Our proof of Theorem 1.1 relies on a good understanding of the structure of the remainder associated to the expansion [START_REF] Kaup | An exact solution for a derivative nonlinear Schrödinger equation[END_REF]. Note that when λ 2 = iρ, the imaginary part of this remainder (which is what we really use) is simply ϕ L (u, ρ). In Section 2, we will introduce a determinant characterization of a u (λ); we use this characterization to formulate a technical statement (Lemma 2.1 below) on the size of the remainder. Assuming the result of Lemma 2.1, we will prove Theorem 1.1 at the end of Section 2. Then, in Section 3, we will prove our technical Lemma, completing the circle of ideas. Most of the work is contained in this last section.

Before moving on, let us establish a few notational conventions that we wish to add to the ones introduced above. First of all, we use the following normalization for the Fourier transform:

f (ζ) = 1 √ 2π R e ixζ f (x) dx.
The symbol N will denote the nonnegative integers, and N * = N\{0}. We will use • 2 to denote the Hilbert-Schmidt norm, and • will denote the operator norm on L 2 (R). And we will use the following shorthand for derivatives:

D = -i∂ x , L 0 = iσ 3 ∂ x . Whenever 2 ≤ p < ∞, we will use s * (p) to denote the Sobolev exponent s * (p) = 1 2 -1 p such that the embedding H s * (p) (R) → L p (R) holds.
Finally, we set notation for the following subset of Ω + :

Γ δ = {λ ∈ Ω + : δ < arg(λ 2 ) < π -δ}.
This notation will be useful in some of the intermediate steps we use to prove Theorem 1.1, as our estimates will frequently depend on |λ| 2 Im λ 2 (which is ≤ C(δ) on Γ δ ). However, the value of δ > 0 will be inconsequential for our final steps, where we will take λ 2 to be pure imaginary. Therefore, for simplicity of presentation, we will fix δ > 0 once and for all and suppress dependence on δ in all bounds below.

PROOF OF THE MAIN RESULT

2.1. The Determinant Characterization of a u (λ). An important property of a u (λ) is the fact that it can be realized as a perturbation determinant: [START_REF] Killip | KdV is well-posed in H -1[END_REF] a u (λ

) 2 = det(I -T u (λ) 2 ),
where

T u (λ) = iλ(L 0 -λ 2 ) -1 U, λ ∈ Ω + .
The operator T u (λ) is Hilbert-Schmidt, with

(16) T u (λ) 2 2 = |λ| 2 Im(λ 2 ) u 2 L 2 (R) .
As a consequence of (15), we may write 2 (17)

ln a u (λ) = - ∞ k=1 Tr(T u (λ) 2k ) 2k , if T u (λ) < 1.
This series will converge whenever λ ∈ Γ δ has large enough modulus; indeed, using the explicit kernel of (L 0 -λ 2 ) -1 , it can easily be shown that for any p > 2, we have

(18) T u (λ) |λ| u L p (R) Im(λ 2 ) 1-1 p , λ ∈ Ω + , u ∈ L p (R).
In particular, we can find R 0 = R 0 ( u

H 1 3 (R) ) such that T u (λ) ≤ 1 2 for all λ ∈ Γ δ satisfying |λ| 2 ≥ R 0 .
We will fix the notation R 0 for use below.

As we shall see later, each term of the series ( 17) can be expanded in powers of λ -2 :

(19) - Tr(T u (λ) 2k ) 2k = ∞ j=k-1 µ j,k (u) λ 2j .
According to ( 13) and ( 17), the E j (u)'s should then satisfy

(20) E j (u) = j+1 k=1 µ j,k (u)
.

We will use the following notation for the remainders after truncation of the expansions ( 17) and ( 19):

(21) ln a u (λ) = - 2L+2 k=1 Tr(T u (λ) 2k ) 2k + τ * L (u, λ), L ∈ N; (22) - Tr(T u (λ) 2k ) 2k = 2L+1 j=k-1 µ j,k (u) λ 2j + τ k L (u, λ), k ∈ {1, . . . , 2L + 2}, L ∈ N.
The primary difficulty of the proof of Theorem 1.1-and indeed, the subject of Lemma 2.1-is the understanding of the size and structure of the remainder terms τ k L (u, λ), and to a lesser extent, the µ j,k (u)'s. On the other hand, for λ ∈ Γ δ with large enough modulus, it is easy to bound the τ *

L (u, λ)'s. For example, if T u (λ) ≤ 1 2 , then |τ * L (u, λ)| = ln a u (λ) + 2L+2 k=1 Tr T 2k u (λ) 2k ≤ ∞ k=2L+3 T u (λ) 2k-2 T u (λ) 2 2 T u (λ) 4L+4 T u (λ) 2 2 u 4L+4 H s * (p) (R) u 2 L 2 (R) |λ| (4L+4)(1-2 p ) , 2 < p < ∞, s * (p) = 1 2 - 1 p . (23)
The following table summarizes the various relationships among the quantities introduced above and will be helpful to keep track of the numerology. More precise information about the µ j,k (u)'s and τ k L (u, λ)'s will be provided below.

2

This series expansion of ln a u (λ) is consistent with the definition [START_REF] Jenkins | Global well-posedness for the derivative non-linear Schrödinger equation[END_REF].

- Tr T 2 u (λ) 2 - Tr T 4 u (λ) 4 - Tr T 6 u (λ) 6 • • • - Tr T 4L+2 u (λ) 4L + 2 - Tr T 4L+4 u (λ) 4L + 4 ln au(λ) = µ 0,1 (u) E 0 (u) + µ 1,1 (u) λ 2 + µ 1,2 (u) λ 2 E 1 (u) λ 2 + µ 2,1 (u) λ 4 + µ 2,2 (u) λ 4 + µ 2,3 (u) λ 4 E 2 (u) λ 4 . . . . . . . . . . . . . . . + µ 2L,1 (u) λ 4L + µ 2L,2 (u) λ 4L + µ 2L,3 (u) λ 4L + • • • + µ 2L,2L+1 (u) λ 4L E 2L (u) λ 4L + µ 2L+1,1 (u) λ 4L+2 + µ 2L+1,2 (u) λ 4L+2 + µ 2L+1,3 (u) λ 4L+2 + • • • + µ 2L+1,2L+1 (u) λ 4L+2 + µ 2L+1,2L+2 (u) λ 4L+2 E 2L+1 (u) λ 4L+2 + τ 1 L (u, λ) + τ 2 L (u, λ) + τ 3 L (u, λ) + • • • + τ 2L+1 L (u, λ) + τ 2L+2 L (u, λ) + τ * L (u, λ)
2.2. Structure of the Traces. In this section, we record all the information about the traces that we need in order to prove our main result. We deal first with the easy case of Tr T u (λ) 2 , about which we need more explicit information. A straightforward computation gives us

(24) Tr T 2 u (λ) = 2iλ 2 R | u(ζ)| 2 ζ + 2λ 2 dζ.
We determine the expansion of Tr T 2 u (λ) by simply substituting into (24) the identity

2λ 2 ζ + 2λ 2 = 2L+1 j=0 - ζ 2λ 2 j + ζ ζ + 2λ 2 ζ 2λ 2 2L+1 , L ∈ N, to obtain (25) - Tr T 2 u (λ) 2 = 2L+1 j=0 1 λ 2j • i (-2) j+1 R ζ j | u(ζ)| 2 dζ =:µ j,1 (u) - i 4 L+1 λ 4L+2 R ζ 2L+2 | u(ζ)| 2 ζ + 2λ 2 dζ =:τ 1 L (u,λ) , L ∈ N.
Now we state our main Lemma, which describes the structure of the other µ j,k (u)'s and τ k L (u, λ)'s.

Lemma 2.1. For any k ∈ N * , L ∈ N, the traces Tr T 2k u (λ) admit the decomposition [START_REF] Ozawa | On the nonlinear Schrödinger equations of derivative type[END_REF]. The µ j,k (u)'s and τ k L (u, λ)'s satisfy the properties below, where for any n ∈ N we denote σ(n) = max{n, 1 3 }.

• Each µ j,k (u) is a homogeneous polynomial of degree 2k in u, u, and their derivatives; it is homogeneous with respect to the natural scaling. We have

|µ 2 ,2 (u)| u 3 H σ( -1) (R) u H (R) , ∈ N * , |µ 2 ,k (u)| u 2k H σ( -1) (R) , ∈ N * , k ∈ {3, . . . , 2 + 1}, |µ 2 +1,k (u)| u 2k H (R) , ∈ N * , k ∈ {2, . . . , 2 + 2}. (26)
• For |λ| 2 > R 0 , λ ∈ Γ δ , we have the following bounds:

|τ 2 L (u, λ)| α u 3 H σ(L) (R) u H L+α (R) |λ| 4L+2+2α , L ∈ N, 0 ≤ α < 1; (27) |τ k L (u, λ)| u 2k H L (R) |λ| 4L+4 , L ∈ N * , k ∈ {3, . . . , 2L + 2}. (28)
We postpone the proof of the Lemma until Section 3.

2.3.

Proof of Theorem 1.1. In this section, we will prove Theorem 1.1, assuming the result of Lemma 2.1. For s ∈ N * , the conclusion follows easily from Lemma 2.1, together with [START_REF] Mio | Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas[END_REF], [START_REF] Takaoka | Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity[END_REF], and an induction argument; we provide the details presently. Actually, the case s = 1 was already proved in the Introduction. Therefore, let us turn to our inductive hypothesis. For k = 1, . . . , -1, we assume that the following bound holds. ( 29)

sup t∈R u(t) H k (R) ≤ C(k, u 0 H k (R) ).
We will prove that the same bound holds with k = ≥ 2.

First of all, for any integer ≥ 2, and any time t, we have

u(t) 2 Ḣ (R) = C( )µ 2 ,1 (u(t)) by (25) = C( ) E 2 (u(t)) - 2 +1 k=2 µ 2 ,k (u(t))
by ( 20)

≤ C( )E 2 (u 0 ) + 1 2 u(t) 2 Ḣ (R) + C( , u 0 H -1 (R) ).
To pass to the last line, we used time-independence of E 2 (u(t)), the bounds [START_REF] Takaoka | Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in loworder Sobolev spaces[END_REF], and our inductive hypothesis (29) (with k = -1). Finally, using that

E 2 (u 0 ) = 2 +1 k=1 µ 2 ,k (u 0 ) ≤ C( , u 0 H (R) ), we get sup t∈R u(t) Ḣ (R) ≤ C( , u 0 H (R) ),
which finishes the induction argument, and thus the proof of Theorem 1.1 for s ∈ N * . It remains to consider the situation where s / ∈ N * . We start by recording the characterization of ϕ L (u, ρ) in terms of the remainders τ k L (u, √ iρ), and we also set notation for the quadratic part of ϕ L (u, ρ). We also note that the case L = 0 is included in the definition.

ϕ L (u, ρ) = Im ln a u ( iρ) - 2L+1 j=0 E j (u) (iρ) j = Im 2L+2 k=1 τ k L (u, iρ) + τ * L (u, iρ) , L ∈ N, (30) ϕ L,0 (u, ρ) = Im τ 1 L (u, iρ) = (-1) L 2 2L+1 ρ 2L R ζ 2L+2 | u(ζ)| 2 ζ 2 + 4ρ 2 dζ, L ∈ N. (31)
The conclusion of Theorem 1.1 for noninteger s ≥ 1 2 will be deduced from the following two Lemmas. Lemma 2.2. Suppose u ∈ S(R), s > 0, s / ∈ N * , and R > 0. Then the following comparison holds.

(32)

R + ρ 2s-1 |ϕ [s],0 (u, ρ)|dρ s u 2 Ḣs (R) s ∞ R ρ 2s-1 |ϕ [s],0 (u, ρ)|dρ + R 2(s-[s]) u 2 Ḣ[s] (R) .
Proof. Let us define the function

f ν : R → R, for 0 < ν < 1, by f ν (z) = |z| 2ν-1 1+z 2 . Note that f ν ∈ L 1 (R) for this range of ν.
We make a direct substitution of the formula (31) for ϕ [s],0 (u, ρ) into the left side of (32), then we switch the order of integration. Continuing the computation yields

∞ R ρ 2s-1 |ϕ [s],0 (u, ρ)|dρ = 1 2 2[s]+1 R ζ 2[s]+2 | u(ζ)| 2 ∞ R ρ 2(s-[s])-1 ζ 2 + 4ρ 2 dρ dζ = 1 2 2s+1 R |ζ| 2s | u(ζ)| 2 ∞ 2R |ζ| f s-[s] (z) dz dζ = 1 4 s+1 f s-[s] L 1 (R) u 2 Ḣs (R) - 1 2 R ζ 2 2s | u(ζ)| 2 2R |ζ| 0 f s-[s] (z) dz dζ.
We estimate the second term on the right by means of the trivial replacement 1 1+z 2 ≤ 1:

1 2 R ζ 2 2s | u(ζ)| 2 2R |ζ| 0 f s-[s] (z) dz dζ ≤ 1 2 R ζ 2 2s | u(ζ)| 2 2R |ζ| 0 z 2(s-[s])-1 dz dζ = R 2(s-[s]) s -[s] • u 2 Ḣ[s] (R) 4 [s]+1 .
The comparison (32) follows.

Lemma 2.3. Suppose u ∈ S(R), s > 0, s / ∈ N * . Denoting β = max{[s], s+[s]+1 4([s]+1) , 1 3 }, we have (33) |ϕ [s] (u, ρ) -ϕ [s],0 (u, ρ)| ≤ C(s, u H β (R) ) ρ s+[s]+1 ( u H s (R) + 1), ∀ρ ≥ R 0 . Proof. Choose p > 2 to solve 2([s] + 1)(1 -2 p ) = s + [s] + 1. (Note that s * (p) = s+[s]+1 4([s]+1
) for this choice of p.) Then for ρ > R 0 , we have

|ϕ [s] (u, ρ) -ϕ [s],0 (u, ρ)| ≤ 2[s]+2 k=2 |τ k [s] (u, iρ)| + |τ * [s] (u, iρ)| by (30), (31) ≤ C(s) u 3 H β (R) u H s (R) ρ s+[s]+1 + 2[s]+2 k=3 u 2k H [s] (R) ρ 2[s]+2 + u 4[s]+4 H s * (p) (R) u 2 L 2 (R) ρ (2[s]+2)(1-2 p )
by ( 27), ( 28), ( 23)

≤ C(s, u H β (R) ) ρ s+[s]+1 ( u H s (R) + 1).
In the second line, we understand the sum over k to be empty if [s] = 0.

The conclusion of Theorem 1.1 for noninteger s ≥ 1 2 follows from Lemmas 2.2 and 2.3, the timeindependence of the quantity ϕ [s] (u, ρ) for solutions of the DNLS equation, and the bound (34)

sup t∈R u(t) H β (R) ≤ C(β, u 0 H β (R) )
, Let us give the remaining details of the proof of Theorem 1.1 presently. For any t ∈ R, we have

where β = max{[s], s+[s]+1 4([s]+1) ,
u(t) 2 Ḣs (R) s ∞ R 0 ρ 2s-1 |ϕ [s],0 (u(t), ρ)|dρ + R 2(s-[s]) 0 u(t) 2 H [s] (R) ≤ C(s) ∞ R 0 ρ 2s-1 |ϕ [s] (u(t), ρ)|dρ + C(s, R 0 , u(t) H β (R) )( u(t) H s (R) + 1) ≤ C(s) ∞ R 0 ρ 2s-1 |ϕ [s] (u 0 , ρ)|dρ + C(s, u 0 H s (R) )( u(t) H s (R) + 1) ≤ 1 2 u(t) 2 H s (R) + C(s, u 0 H s (R) ),
which establishes the desired conclusion. Note that the first line in the calculation above is simply the upper bound in Lemma 2.2. To pass from the first line to the second, we use Lemma 2.3, followed by the lower bound of Lemma 2.2. We use (34) and the time independence of ϕ [s] (u(t), ρ) to pass to the third line. Finally, we justify the last line by noting that

∞ R 0 ρ 2s-1 |ϕ [s] (u 0 , ρ)|dρ s C(s, u 0 H s (R) ),
which follows from an application of Lemma 2.3, followed by the lower bound in Lemma 2.2.

3. PROOF OF LEMMA 2.1 3.1. Outline of the Proof. In this section, we expand each Tr(T 2k u (λ)) in powers of λ -2 , up to a specified order, and we establish bounds on the remainders, in order to prove our key Lemma 2.1. In Section 3.2, we consider the case L = 0, which is easy to treat explicitly but does not fit naturally into our argument for the other cases. When L ≥ 1, we follow the strategy of [START_REF] Gérard | On the Conservation Laws of the Defocusing Cubic NLS Equation[END_REF], deducing the expansions of the traces from the expansion of the resolvent L u (λ) -1 . The relationship between T u (λ) and L u (λ) is the following:

(35) L u (λ) = (L 0 -λ 2 )(I -T u (λ)).
Therefore,

(36) L u (λ) -1 = (I -T u (λ)) -1 (L 0 -λ 2 ) -1 = ∞ n=0 T u (λ) n (L 0 -λ 2 ) -1 =:Rn , T u (λ) < 1.
The point is that

(37) T 2k u (λ) = iλR 2k-1 U. Thus, the part of L -1 u (λ) that is of relevance to us is R 2k-1 , i.e.
, the term in the expansion (36) that is homogeneous of degree 2k -1 in u, u. In particular, we seek an expansion of λR 2k-1 in powers of λ -2 , up to order λ 4L+2 for a given L ∈ N * , and a good understanding of the remainder term.

Our strategy will be to examine the symbol R(x, ζ) of the pseudodifferential operator L u (λ) -1 . In Section 3.3, we will expand the diagonal and antidiagonal parts R d (x, ζ) and R a (x, ζ) of R(x, ζ) in powers of λ -2 , determining recursively the form of each term of the expansion. Homogeneity considerations will then give us the desired expansion of λR 2k-1 (and thus of Tr T 2k u (λ)) in powers of λ -2 . In Section 3.4, we identify the µ j,k (u)'s from [START_REF] Ozawa | On the nonlinear Schrödinger equations of derivative type[END_REF] and separate them from the remainder term. In Section 3.5 we estimate the remainder term, finishing the proof of the Lemma. The final Section 3.6 consists of the proof by induction of a technical result stated in Section 3.3.1, on the form of the terms of the expansions for R d and R a .

3.2. Case L = 0. Let us note first of all that the desired decomposition in the case L = 0 reads

ln a u (λ) = [µ 0,1 (u) + λ -2 µ 1,1 (u) + τ 1 0 (u, λ) =-1 2 Tr T 2 u (λ) ] + [λ -2 µ 1,2 (u) + τ 2 0 (u, λ) =-1 4 Tr T 4 u (λ)
] + τ * 0 (u, λ).

(See the table in Section 2.1.) The only term which we have not already understood is τ 2 0 (u, λ); in order to treat it, we decompose T 4 u (λ) explicitly as follows. A computation (the details of which are contained, for instance, in [START_REF] Bahouri | Global well-posedness for the derivative nonlinear schrödinger equation[END_REF]) tells us that

Tr T 4 u (λ) = i(2λ 2 ) 2 R u(x) (D + 2λ 2 ) -1 u(x) 2 (D -2λ 2 ) -1 u(x) dx.
Then, making a few simple manipulations, we can bring the right side of the equation above into the following form.

Tr T 4 u (λ) = i -2λ 2 R u(x) u(x) -(D + 2λ 2 ) -1 Du(x) 2 u(x) -(D -2λ 2 ) -1 Du(x) dx = - i 2λ 2 R |u(x)| 4 dx - R |u| 2 u(x)(D -2λ 2 ) -1 Du(x) dx -2 R |u| 2 u(x)(D + 2λ 2 ) -1 Du(x) dx + 2 R |u(x)| 2 (D + 2λ 2 ) -1 Du(x)(D -2λ 2 ) -1 Du(x) dx + R ((D + 2λ 2 ) -1 Du(x)) 2 u(x) 2 dx - R u(x)((D + 2λ 2 ) -1 Du(x)) 2 ((D -2λ 2 ) -1 Du(x)) dx = - 4 λ 2 i 8 u 4 L 4 (R) =µ 1,2 (u)
-4τ 2 0 (u, λ).

To estimate τ 2 0 (u, λ), we use the following simple Lemma, the proof of which we omit. Lemma 3.1. The following estimates hold, for λ ∈ Γ δ .

• If 0 ≤ α 1 ≤ α 2 ≤ 1, then (38) (D ± 2λ 2 ) -1 Du Ḣα 1 (R) α 2 -α 1 u Ḣα 2 (R) (2 Im(λ 2 )) α 2 -α 1 , ∀ u ∈ H α 2 (R). • If 2 ≤ p < ∞, then (39)
(D ± 2λ 2 ) -1 Du L p (R) p u H s * (p) (R) , ∀ u ∈ H s * (p) (R).
We estimate one of the terms defining τ 2 0 (u, λ) explicitly; the others can be dealt with in an entirely similar way.

1 λ 2 R u(x)((D + 2λ 2 ) -1 Du(x)) 2 ((D -2λ 2 ) -1 Du(x)) dx ≤ 1 |λ| 2 u L 6 (R) (D + 2λ 2 ) -1 Du 2 L 6 (R) (D -2λ 2 ) -1 Du L 2 (R) α u 3 H 1 3 (R) u H α (R) |λ| 2+2α .
We conclude that τ 2 0 (u, λ) satisfies the required bound, finishing the case L = 0.

3.3. Expanding the Resolvent.

3.3.1. Formal Expansion of R a and R d . As stated above, for L ≥ 1 we seek an expansion of the symbol of L -1 u (λ), in powers of λ -2 . That is, we seek to understand R(x, ζ) in the expression

(40) L -1 u (λ)f = 1 √ 2π dζe ixζ R(x, ζ) f (ζ).
The identity L u (λ)R(x, D) = I implies (41)

iσ 3 ∂ x R(x, ζ) -(ζσ 3 + λ 2 )R(x, ζ) -iλU (x)R(x, ζ) = I.
Introducing the new variable p = ζ λ 2 , this reads (42)

iσ 3 ∂ x R(x, ζ) -λ 2 (pσ 3 + 1)R(x, ζ) -iλU (x)R(x, ζ) = I.
We split R into its diagonal and antidiagonal parts R d and R a , respectively,

R(x, ζ) = R d (x, ζ) + R a (x, ζ),
and we also split equation (42) accordingly:

(43)

iσ 3 ∂ x R d (x, ζ) -λ 2 (pσ 3 + 1)R d (x, ζ) -iλU (x)R a (x, ζ) = I; (44) iσ 3 ∂ x R a (x, ζ) -λ 2 (pσ 3 + 1)R a (x, ζ) -iλU (x)R (x, ζ) = 0.
Setting the notation

R d (x, ζ) = k≥0 1 λ 2+2k R d k (x, p), R a (x, ζ) = k≥0 1 λ 3+2k R a k (x, p),
we rewrite (43) and (44) in expanded form:

I = -(pσ 3 + 1)R d 0 + ∞ k=1 iσ 3 ∂ x R d k-1 -(pσ 3 + 1)R d k -iU R a k-1 λ 2k ; (45) 0 = -(pσ 3 + 1)R a 0 -iU R d 0 + ∞ k=1 iσ 3 ∂ x R a k-1 -(pσ 3 + 1)R a k -iU R d k λ 2k . ( 46
)
We thus obtain the recursive system (47)-(49) below.

(47)

R d 0 (x, p) = - pσ 3 -1 p 2 -1 , R a 0 (x, p) = - iU p 2 -1 , (48) R d k (x, p) = 1 p 2 -1 -iU R a k-1 (x, p) + i∂ x R d k-1 (x, p)σ 3 (pσ 3 -1), k ≥ 1, R a k (x, p) = 1 p 2 -1 iU R d k (x, p) + i∂ x R a k-1 (x, p)σ 3 (pσ 3 + 1) = 1 p 2 -1 U 2 R a k-1 (x, p) -U ∂ x R d k-1 (x, p)σ 3 + i∂ x R a k-1 (x, p)σ 3 (pσ 3 + 1) , k ≥ 1. (49)
Note that we used the formula for R d k (x, p) to pass to the second line in the formula for R a k (x, p). We also used several times the fact that σ 3 A = -Aσ 3 for any antidiagonal matrix.

We use the computations above to clarify the form of the R d k 's and R a k 's; the precise statement is contained in the following Lemma.

Lemma 3.2. The R d k 's and R a k 's take the following form:

(50) R d k (x, p) = k r=1 R d k,r (x, p), k ≥ 1, (51) R a k (x, p) = k r=0 R a k,r (x, p), k ≥ 0,
where the entries of the R d k,r 's and R a k,r 's are homogeneous polynomials of degrees 2r and 2r + 1, respectively, in u, u, and their derivatives. More specifically, setting

Q γ = ∂ γ 1 x U • • • ∂ γn x U , for γ ∈ N n , we have R d k,r (x, p) = 1 (p 2 -1) k+1 γ∈N 2r |γ|=k-r Q γ (x)P |γ| (p)(pσ 3 -1), (52) R a k,r (x, p) = 1 (p 2 -1) k+1 γ∈N 2r+1 |γ|=k-r Q γ (x)P |γ| (p). ( 53
)
Here and below we use the notation P n to denote any diagonal matrix whose diagonal entries are polynomials in p having degree at most n.

We postpone the proof of this Lemma until Section 3.6, so as not to interrupt the flow of ideas.

3.3.2.

The Truncated Expansion, and a Formula for R 2m-1 . For a fixed N ∈ N * , we set the following notation. (Later we will set N = 2L.) .

R (N ) (x, p) = N k=0 R d k (x, p) λ 2+2k =:R (
(

) 54
The symbol R (N ) (x, p) is a truncated expansion of R(x, p) in inverse powers of λ, having diagonal and antidiagonal parts R

(N ) d , R (N )
a , respectively. The point of this definition is that, using Lemma 3.2, we know that R (N ) d,r is homogeneous of degree 2r in u, u, and their derivatives, while R (N ) a,r is homogeneous of degree 2r + 1 in these quantities. Expanding R (N ) according to (54) and applying the recursive identities (45)-( 46), we see that R (N ) (x, p) satisfies

[iσ 3 ∂ x -λ 2 (pσ 3 + 1) -iλU (x)]R (N ) (x, p) = I + Y (N ) (x, p), where Y (N ) (x, p) = Y (N ) d (x, p) + Y (N ) a (x, p), Y (N ) d (x, p) = 1 λ 2+2N iσ 3 (∂ x R d N )(x, p), Y (N ) a (x, p) = - 1 λ 1+2N R a N (x, p)(pσ 3 -1).
This implies

(55) L -1 u (λ) = R (N ) (x, λ -2 D) -L -1 u (λ)Y (N ) (x, λ -2 D).
Recall that R 2m-1 is the term in the expansion (36) which is homogeneous of order 2m-1 in u, u, and their derivatives. On the other hand, the portion of R (N ) which is of this homogeneity is precisely R (N ) a,m-1 . Combining these considerations with (55), we see that R 2m-1 is the difference between R (N ) a,m-1 and the part of L -1 u (λ)Y (N ) (x, λ -2 D) that is homogeneous of degree 2m -1 in u, u, and their derivatives. Using the expansion (36) to isolate this part, we obtain:

R 2m-1 = R (N ) a,m-1 (x, λ -2 D) - 1 λ 2+2N k+r =m-1 k≥0, 1≤r ≤N T u (λ) 2k+1 (L 0 -λ 2 ) -1 (L 0 R d N,r )(x, λ -2 D) - 1 λ 1+2N k+r =m-1 k≥0, 0≤r ≤N T u (λ) 2k (L 0 -λ 2 ) -1 R a N,r (x, λ -2 D)(λ -2 L 0 + 1). (56)
3.4. Extracting the µ j,m (u)'s. Combining ( 56) with ( 37), (54), and pulling out inverse powers of λ, we easily find the following formula for Tr T 2m u (λ) with m ≥ 2, truncated at N = 2L.

Tr(T 2m u (λ)) = 2L-1 j=m-1 1 λ 2j+2 Tr[iU R a j,m-1 (x, λ -2 D)] + k+r=m-1 k≥0, 1≤r≤2L (-1) k λ 4L+4+2k Tr (U (λ -2 L 0 -1) -1 ) 2k+2 (L 0 R d 2L,r )(x, λ -2 D) + k+r=m-1 k≥0, 0≤r≤2L i(-1) k+1 λ 4L+2+2k Tr (U (λ -2 L 0 -1) -1 ) 2k+1 R a 2L,r (x, λ -2 D)(λ -2 L 0 + 1) . (57)
We will refer to the three sums above as I, II, and III, respectively. We now identify the coefficients µ j,m (u)'s and verify that they satisfy the properties claimed in Lemma 2.1. The claimed homogeneity properties will be clear from the formulas that we derive below; we will just need to verify the bounds [START_REF] Takaoka | Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in loworder Sobolev spaces[END_REF]. The latter are also straightforward to verify but will require us to use the structure of the R d k,r 's and R a k,r 's from (52)-(53). The first sum has the form

-2m 2L-1 j=m-1 µ j,m (u) λ 2j with µ j,m (u) = - i 2mλ 2 Tr U R a j,m-1 (x, λ -2 D) = - i 4mπ γ∈N 2m-1 |γ|=j-(m-1) Tr U (x)Q γ (x) dx P |γ| ( ζ λ 2 ) (( ζ λ 2 ) 2 -1) j+1 dζ λ 2 . ( 58
)
Note that 'Tr' denotes an operator trace in the first line, whereas it refers to the 2×2 matrix trace in the second and third lines. We will use the notation 'Tr' similarly in what follows without further comment.

Since λ is presumed to lie in Γ δ , a comparison of the degrees in the numerator and denominator ensures that the integrals over ζ are finite and their values are independent of λ. The total number of derivatives in the x-integrals is j -(m -1), and we distribute these so that the highest order of the derivatives that fall on a single U is as small as possible. We list the bounds on µ j,m (u) according to m and the parity of j. In each case below, is a strictly positive integer.

• If j = 2 is even and m = 2, then there are 2 -1 derivatives; thus

|µ 2 ,2 (u)| u H (R) u 3 H σ( -1) (R) ,
where we recall the notation σ(n) = max{n, 1 3 }. • If j = 2 is even and m ≥ 3, then there are at most 2( -1) total derivatives. This establishes the following bounds:

|µ 2,3 (u)| u 6 H 1 3 (R) , |µ 2 ,m (u)| u 2m H -1 (R) , ≥ 2, m ∈ {3, . . . , 2 + 1}.
• If j = 2 + 1 is odd and m ≥ 2, then there are at most 2 derivatives, so |µ 2 +1,m (u)| u 2m H (R) . Let us remark that the formula (58) determines µ j,m (u) for all m ≥ 2 and all j ≥ m -1, not just for those µ j,m (u)'s that appear in the sum I. Therefore, the above bounds on the µ j,m (u)'s complete our proof of the estimates [START_REF] Takaoka | Global well-posedness for Schrödinger equations with derivative in a nonlinear term and data in loworder Sobolev spaces[END_REF]. However, in order to determine the remainders τ m L (u, λ), we still need to extract µ 2L,m (u) and µ 2L+1,m (u) from the sums II and III. To this end, we remove the parts of II and III which are of order λ -4L and λ -4L-2 ; these will correspond to µ 2L,m (u) and µ 2L+1,m (u), respectively. We deal first with µ 2L,m (u); the only term expected to be relevant is the k = 0 term in III, namely

(59) - i λ 4L+2 Tr U (λ -2 L 0 -1) -1 R a 2L,m-1 (x, λ -2 D)(λ -2 L 0 + 1) .
To extract the part of this expression that is really of order λ -4L , we commute the operator (λ

-2 L 0 -1) -1 with R a 2L,m-1 (x, λ -2 D
). We will have to do something similar several times below, so let us pause to write down a more general formula. Let A denote an antidiagonal operator with symbol A(x, ζ) and similarly let B denote a diagonal operator with symbol B(x, ζ). Then a simple application of the product rule gives the following operator identities.

A(L 0 + λ 2 ) -1 = -(L 0 -λ 2 ) -1 A + (L 0 -λ 2 ) -1 (L 0 A)(L 0 + λ 2 ) -1 ; (60) B(L 0 -λ 2 ) -1 = (L 0 -λ 2 ) -1 B + (L 0 -λ 2 ) -1 (L 0 B)(L 0 -λ 2 ) -1 . (61) Using (60) with A = R a 2L,m-1 , the expression (59) becomes i λ 4L+2 Tr U R a 2L,m-1 (x, λ -2 D) -2mµ 2L,m λ -4L - i λ 4L+4 Tr U (λ -2 L 0 -1) -1 (L 0 R a 2L,m-1 )(x, λ -2 D) . (62)
To extract µ 2L+1,m (u), we need to determine the part of II and III that is of order 2L + 1 in λ -2 . There are three quantities we need to consider:

• The second term in (62):

(63) - i λ 4L+4 Tr U (λ -2 L 0 -1) -1 (L 0 R a 2L,m-1 )(x, λ -2 D)
• The k = 0 term in II:

(64) 1 λ 4L+4 Tr (U (λ -2 L 0 -1) -1 ) 2 (L 0 R d 2L,m-1 )(x, λ -2 D)
• The k = 1 term in III:

(65) i λ 4L+4 Tr (U (λ -2 L 0 -1) -1 ) 3 R a 2L,m-2 (x, λ -2 D)(λ -2 L 0 + 1) .
A short calculation involving (49) confirms that the three quantities identified above sum to µ 2L+1,m as defined in (58): µ

2L+1,m + µ

2L+1,m + µ

2L+1,m = µ 2L+1,m .

3.5. Estimating the Remainder. The final step of the proof is to estimate the remainder terms, which we group together into τ m L (u, λ). This expression is a sum of the following terms: • The k ≥ 1 terms of II and the k ≥ 2 terms of III, where II and III denote (as above) the second and third sums in the decomposition (57). We refer to these as the 'Type 1' remainder terms. • The terms in (66), (68), and (69) where λ -4L-6 appears (six terms total). We refer to these as the 'Type 2' remainder terms.

3.5.1. Type 1 Remainder Terms. We begin with the two sums. We want to show that the following expression is bounded by u 2m H L (R) :

k+r=m-1 k≥1, 1≤r≤2L (-1) k λ 2k Tr (U (λ -2 L 0 -1) -1 ) 2k+2 (L 0 R d 2L,r )(x, λ -2 D) + k+r=m-1 k≥2, 0≤r≤2L i(-1) k+1 λ 2(k-1) Tr (U (λ -2 L 0 -1) -1 ) 2k+1 R a 2L,r (x, λ -2 D)(λ -2 L 0 + 1) . (70)
By virtue of (52), we can write

(L 0 R d 2L,r )(x, p) = 1 (p 2 -1) 2L+1 γ∈N 2r |γ|=2L-r+1 Q γ (x)P |γ| (p).
Thus, the traces in the first sum in (70) may be written as a sum of terms of the form

Tr (U (λ -2 L 0 -1) -1 ) 2k+2 Q γ (x)P 2L-r+1 (λ -2 D)(λ -4 D 2 -1) -2L-1 (71) with γ ∈ N 2r , |γ| = 2L -r + 1 ≤ 2L.
Integrating by parts repeatedly in the above expression until no derivative of order larger than L falls on any single U , we rewrite the expression (71) as a sum of terms of the form

Tr (∂ η 1 x U )(x)(λ -2 L 0 -1) -1 . . . (∂ η 2k+2 x U )(x)(λ -2 L 0 -1) -1 Q γ (x)P 2L-r+1 (λ -2 D)(λ -4 D 2 -1) -2L-1 , with η = (η 1 , . . . η 2k+2 ) ∈ N 2k+2 , γ = (γ 1 , . . . , γ 2r ) ∈ N 2r satisfying |η| + |γ| = 2L -r + 1 and max p,q
(η p , γ q ) ≤ L. Thus, the expression (71) can be bounded by |λ| 2 u 2m H L (R) , and therefore the first sum in (70) by u 2m H L (R) . We deal with the second sum in (70) in essentially the same way. Invoking (53), we may write

R a 2L,r (x, λ -2 D)(λ -2 L 0 + 1) = γ∈N 2r+1 |γ|=2L-r Q γ (x)P |γ|+1 (λ -2 D)(λ -4 D 2 -1) -2L-1 . Thus Tr (U (λ -2 L 0 -1) -1 ) 2k+1 R a 2L,r (x, λ -2 D)(λ -2 L 0 + 1) is a sum of terms of the form (72) Tr (U (λ -2 L 0 -1) -1 ) 2k+1 Q γ (x)P 2L-r+1 (λ -2 D)(λ -4 D 2 -1) -2L-1 ,
where γ ∈ N 2r+1 with |γ| = 2L -r ≤ 2L. As before, we integrate by parts repeatedly to rewrite the expression (72) as a sum of terms of the form

Tr (∂ η 1 x U )(x)(λ -2 L 0 -1) -1 . . . (∂ η 2k+1 x U )(x)(λ -2 L 0 -1) -1 Q γ (x)P 2L-r+1 (λ -2 D)(λ -4 D 2 -1) -2L-1 ,
with η = (η 1 , . . . η 2k+1 ) ∈ N 2k+1 , γ = (γ 1 , . . . , γ 2r+1 ) ∈ N 2r+1 satisfying |η| + |γ| = 2L -r and max p,q (η p , γ q ) ≤ L. This allows us to bound (72) by |λ| 2 u 2m H L (R) , thus completing the desired estimates on the Type 1 remainder terms.

3.5.2. Type 2 Remainder Terms. We now deal with the Type 2 remainder terms (the terms in (66), (68), and (69) where λ -4L-6 appears). When m ≥ 3, the total number of derivatives falling on the U 's is 2L; therefore we can bound all these terms by |λ| -4L-4 u 2m H L (R) by arguing exactly as we did for the Type 1 terms. To complete the proof of Lemma 2.1, it thus remains to consider the Type 2 remainder terms with m = 2. In this case, some of the U 's appear to be overloaded with derivatives, and we need an additional estimate. We state the following Lemma in terms of the 'overloaded' part of the Type 2 remainder term from (66), but the same manipulations will yield the bound we need for the other Type 2 remainders.

Lemma 3.3. The following estimate holds, for any j ∈ N and all α ∈ [0, 1].

(73) (L 0 + λ 2 ) -1 D j+1 U (L 0 -λ 2 ) -1 2 ≤ C(α)|λ| -1-2α u H j+α (R) . Proof. Denoting T = (L 0 + λ 2 ) -1 D j+1 U (L 0 -λ 2 ) -1
, we readily compute as follows:

T 2 2 = 1 π | D j+1 u(ζ 1 -ζ 2 )| 2 |ζ 1 -λ 2 | 2 |ζ 2 -λ 2 | 2 dζ 1 dζ 2 = 1 π |ζ 1 | 2 | D j u(ζ 1 )| 2 dζ 2 |ζ 1 + ζ 2 -λ 2 | 2 |ζ 2 -λ 2 | 2 dζ 1 = 2 Im λ 2 |ζ 1 | 2 | D j u(ζ 1 )| 2 |ζ 1 + 2i Im λ 2 | 2 dζ 1 ≤ C(α)|λ| -2-4α u 2 H j+α (R) .
This completes the proof.

With the above Lemma at our disposal, we now return to the estimation of the Type 2 remainder terms for m = 2; we deal first with the one that appears in (66). Omitting the prefactor -i λ 4L+6 , the quantity under consideration is

Tr U (λ -2 L 0 -1) -1 (D 2 R a 2L,1 )(x, λ -2 D)(λ -2 L 0 + 1) -1 , which we write as (74) Tr (λ -2 L 0 + 1) -1 (D 2 U )(λ -2 L 0 -1) -1 R a 2L,1 (x, λ -2 D
) . Proceeding as we did for the Type 1 terms, we rewrite this expression as a sum of terms of the form (75)

Tr (λ -2 L 0 + 1) -1 (D η+2 U )(λ -2 L 0 -1) -1 Q γ (x)P 2L-1 (λ -2 D)(λ -4 D 2 -1) -2L-1 ], with η ∈ N, γ = (γ 1 , γ 2 , γ 3 ) ∈ N 3 , η + |γ| = 2L -1, η ≤ L -1, and max(γ 1 , γ 2 , γ 3 ) ≤ L.
The expression (75) can be bounded by

(λ -2 L 0 + 1) -1 (D η+2 U )(λ -2 L 0 -1) -1 2 Q γ (x)P 2L-1 (λ -2 D)(λ -4 D 2 -1) -2L-1 2
. By virtue of Lemma 3.3, the above can bounded in turn by

C(α)|λ| 4-2α u H L+α (R) u 3 H L (R) , for any α ∈ [0, 1].
The next quantity we treat is the third term in (68); we want to estimate

Tr (λ -2 L 0 -1) -1 (L 0 U 2 )(λ -2 L 0 -1) -1 (L 0 R d 2L,1 )(x, λ -2 D)(λ -2 L 0 + 1) -1 .
After an integration by parts we are left with the expression

-Tr (λ -2 L 0 -1) -1 (D 2 U 2 )(λ -2 L 0 -1) -1 R d 2L,1 (x, λ -2 D)(λ -2 L 0 + 1) -1
. that can be treated in exactly the same way as (74). We note only the modification to Lemma 3.3 that we use, namely

(L 0 -λ 2 ) -1 (D L+1 U 2 )(L 0 -λ 2 ) -1 2 α |λ| -1-2α U 2 H L+α (R) α |λ| -1-2α u H L+α (R) u H L (R) .
We next consider the second term in (68):

Tr (λ -2 L 0 + 1) -1 (L 0 U )(λ -2 L 0 -1) -1 U (λ -2 L 0 -1) -1 (L 0 R d 2L,1 )(x, λ -2 D)
, where as usual we have suppressed the prefactor λ -4L-6 . We start by rewriting it as the sum

-Tr (λ -2 L 0 + 1) -1 (D 2 U )(λ -2 L 0 -1) -1 U (λ -2 L 0 -1) -1 R d 2L,1 (x, λ -2 D) + Tr (λ -2 L 0 + 1) -1 (L 0 U )(λ -2 L 0 -1) -1 (L 0 U )(λ -2 L 0 -1) -1 R d 2L,1 (x, λ -2 D) . (76)
For the first term here we proceed exactly as before: substituting (52) and integrating by parts we rewrite it as a sum of expressions of the form

Tr (λ -2 L 0 + 1) -1 (D η 1 +2 U )(λ -2 L 0 -1) -1 (D η 2 U )(λ -2 L 0 -1) -1 Q γ (x)P 2L (λ -2 D)(λ -4 D 2 -1) -2L-1 ], with η = (η 1 , η 2 ) ∈ N 2 , γ = (γ 1 , γ 2 ) ∈ N 2 , |η|+|γ| = 2L-1, max(η 1 , η 2 ) ≤ L-1, and max(γ 1 , γ 2 ) ≤ L.
We estimate the above by

(λ -2 L 0 + 1) -1 (D η 1 +2 U )(λ -2 L 0 -1) -1 2 (D η 2 U )(λ -2 L 0 -1) -1 Q γ (x)P 2L (λ -2 D)(λ -4 D 2 -1) -2L-1
2 , which can in turn be bounded by C(α)|λ| 4-2α u H L+α (R) u 3 H L (R) . To treat the second term in (76) we distinguish the cases L = 1 and L ≥ 2. In the case of L = 1 we estimate this expression by 1 2 [ and choosing p such that σ = 1 2 -1 p , we get the bound |λ| 3-2σ u 3 H 1 (R) u H 1+σ (R) ≤ |λ| 4-2α u 3 H 1 (R) u H 1+α (R) . If L ≥ 2, we can proceed as for the Type 1 remainder terms and bound the second term in (76) by |λ| 2 u 4 H L (R) . This finishes our considerations of Type 2 remainder terms coming from (68). The last group of Type 2 remainder terms comes from (69). Omitting the common prefactor, the quantity of interest is Tr (L 0 U )(λ -2 L 0 -1) -1 U (λ -2 L 0 -1) -1 U (λ -2 L 0 -1) -1 R a 2L,0 (x, λ -2 D) -Tr (λ -2 L 0 -1) -1 (L 0 U 2 )(λ -2 L 0 -1) -1 U (λ -2 L 0 -1) -1 R a 2L,0 (x, λ -2 D) -Tr (λ -4 D 2 -1) -1 (L 0 U 3 )(λ -2 L 0 -1) -1 R a 2L,0 (x, λ -2 D) , We check that the inner sums (together with the common factors of (p 2 -1) -k-1 and (pσ 3 -1)) can be absorbed into R As above, we perform the routine verifications of the numerology as follows.

(λ -2 L 0 + 1) -1 (L 0 U ) 2 (λ -2 L 0 -1) -1 (L 0 U ) (λ -2 L 0 -1) -1 R d 2L,1 (x, λ -2 D) 2 |λ| 2+ 2 p u 3 H 1 (R) Du L p (R) , 2 ≤ p ≤ ∞. Putting σ = α 2 ∈ [0,
• When γ ∈ N 2r+1 and |γ| = (k-1)-r, the term U 2 Q γ (x)P |γ| (x) can be absorbed into R a k,r+1 (x, p): -First, U 2 Q γ = Q (0,0,γ) , with (0, 0, γ) ∈ N

  R a 2L,0 (x, p) = ∂ 2Lx U (x)P 2L (p)(p 2 -1) -2L-1 . No new ideas are involved in the estimation of these terms; we simply integrate by parts L -1 times to keep L + 1 derivatives on U coming from R a 2L,0 and then apply Lemma 3.3. We omit the remaining details for these terms. Having now established the required bounds on the remainder |τ m L (u, λ)|, we have completed the proof of Lemma 2.1, modulo the proof of Lemma 3.2 below.

3. 6 . 2 - 1 ) 2 •

 6212 Proof of Lemma 3.2. We argue by induction. The base cases are easy to verify explicitly:U 2 • (-1) • (pσ 3 -1) = R d 1,1 (x, p).Using (48)-(49) along with our inductive hypothesis, we may write, for k ≥ 1, i∂ x Q γ (x)P |γ| (p)(pσ 3 -1)σ 3 (pσ 3 -1).

U 2 -

 2 d k,r+1 (x, p) and R d k,r (x, p), respectively.• When γ ∈ N 2r+1 and |γ| = (k-1)-r, the term iU Q γ (x)P |γ| (p) can be absorbed into R d k,r+1 (x, p): -First, U Q γ = Q (0,γ) , with (0, γ) ∈ N 2(r+1) and |(0, γ)| = |γ| = k -(r + 1). -Second, deg P |γ| (p) ≤ (k -1) -r = k -(r + 1). • When γ ∈ N 2r and |γ| = (k -1) -r, the term ∂ x Q γ (x)P |γ| (p) can be absorbed into R d k,r (x, p): -First, ∂ x Q γ is a sum of Q γ 's, with |γ | = |γ| + 1 = k -r; -Second, deg P |γ| (p)(pσ 3 -1) ≤ [(k -1) -r] + 1 = k -r.The formula for R a k (x, p) may be verified in exactly the same way. We write Q γ (x)P |γ| (p) + i∂ x Q γ (x)P |γ| (p)σ 3 (pσ 3 + 1) |η|=(k-1)-r U ∂ x Q η (x)P |η| (p)(pσ 3 -1)σ 3 .

  2(r+1)+1 , |(0, 0, γ)| = |γ| = k -(r + 1); -Second, deg P |γ| (p) ≤ (k -1) -r = k -(r + 1). • When γ ∈ N 2r+1 and |γ| = (k -1) -r, the term i∂ x Q γ (x)P |γ| (p)σ 3 (pσ 3 + 1) can be absorbed into R a k,r (x, p). When η ∈ N 2r and |η| = (k-1)-r, the same is true of U ∂ x Q η (x)P |η| (p)(pσ 3 -1)σ 3 . -First, both ∂ x Q γ and U ∂ x Q η are sums of Q γ 's, with γ ∈ N 2r+1 and |γ | = k -r.-Second, P |γ| (p)σ 3 (pσ 3 + 1) and P |η| (p)(pσ 3 -1)σ 3 each have degree at most k -r.

July This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2021 semester.

We deal with each of these in turn, denoting their contributions to µ 2L+1,m by µ

2L+1,m , and µ

(3) 2L+1,m , respectively. To put (63) in the desired form, we simply apply (60) again, this time with

Thus

Next, we look at (64). We perform two commutations, using

Substituting this into (64) yields

We take

Finally, we look at (65). Proceeding as in (67) but commuting one more time, we get

Substituting the above into (65

2L,m-2 (x, λ -2 D) .