Antoine Hollebecque

Emilie Lanoy

Frederic Troallen

Laurie Soulat-Dufour

Christophe Massard

Rastislav Bahleda

Andrea Varga

Anas Gazzah

Sophie Postel-Vinay

Vincent Ribrag

Eric Deutsch

Eric Angevin

Franck Boccara

Ariel Cohen

Jean-Charles Soria

Stephane Ederhy
email: stephane.ederhy@aphp.fr

Cardiac troponin I elevation and overall survival among cancer patients receiving investigational compounds during phase I trials

Keywords: Troponin, Cardio oncology, Cardiotoxicity, Phase I trial, Drug development

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A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T

Introduction

The evaluation of cardiovascular safety is an important step in the development of new treatments for cancer [START_REF] Postel-Vinay | Towards new methods for the determination of dose limiting toxicities and the assessment of the recommended dose for further studies of molecularly targeted agents--dose-Limiting Toxicity and Toxicity Assessment Recommendation Group for Early Trials of Targeted therapies, an European Organisation for Research and Treatment of Cancer-led study[END_REF]. In phase I trials, cardiovascular evaluation is frequently mandated by protocol and is based on clinical examination, QT measurement, left ventricular systolic function, and biomarker measurement [START_REF] Johnson | Evaluation, prevention and management of cancer therapyinduced cardiotoxicity: a contemporary approach for clinicians[END_REF]. B-type natriuretic peptide and troponin play an important role in the evaluation of cardiotoxicity [START_REF] Ky | Early increases in multiple biomarkers predict subsequent cardiotoxicity in patients with breast cancer treated with doxorubicin, taxanes, and trastuzumab[END_REF]. Troponin is a marker of myonecrosis and was first validated as a diagnostic and prognostic marker in patients with suspected acute coronary syndromes [START_REF] Thygesen | Third universal definition of myocardial infarction[END_REF]. In the context of patients with cancer, cardiac troponin was proposed as an indicator of cardiotoxicity related to cancer treatments [START_REF] Plana | Expert consensus for multimodality imaging evaluation of adult patients during and after cancer therapy: a report from the American Society of Echocardiography and the European Association of Cardiovascular Imaging[END_REF]. Elevated troponin has also been reported in patients treated with moleculartargeted drugs [START_REF] Cardinale | Trastuzumab-induced cardiotoxicity: clinical and prognostic implications of troponin I evaluation[END_REF], but the incidence, mechanisms, and prognostic significance of a troponin rise in patients with cancer are unknown [START_REF] Ederhy | Frequency and management of troponin I elevation in patients treated with molecular targeted therapies in phase I trials[END_REF]. In this study, we investigated the incidence of and risk factors associated with troponin elevation, and its effect on overall survival.

Methods

Patients

Clinical, biological, and therapeutic data from patients treated in phase I trials between January 2010 and November 2012 at the Drug Development Department (DITEP) of Gustave Roussy Cancer campus were reviewed. Patients with a history or current condition of uncontrolled cardiovascular disease including unstable angina, myocardial infarction, or congestive heart failure class II-IV within the preceding 12 months, cerebrovascular accident, transient ischemic attack within the preceding 6 months, pulmonary embolism within the preceding 3 months, and patients presenting a QTc prolongation (Fridericia; QTc >450 milliseconds in men or >470 milliseconds in women) were excluded from the phase I trials and thus from this analysis.

Patients were eligible for this analysis if they had at least one cardiac troponin I (cTnI) measurement taken before study-drug initiation and if they had a subsequent repeat troponin evaluation during the trial. Patients with a troponin measurement >0.06 ng/mL during the screening period were excluded from the analysis. If a patient had been successively recruited in different phase I trials, only the first trial in which the patient participated was considered. The study was approved by the local ethics committee.

A C C E P T E D M A N U S C R I P T

Cardiac troponin I measurement

During the screening period, cTnI was evaluated for each patient and then at each visit, which was planned on a weekly basis until the end of the clinical trial. cTnI was measured in blood serum samples using the same immunoassay system (Beckman-Coulter, Fullerton, USA) with the AccuTnI © (ref A78803) assay and AccuTnI © (ref 33345) calibrators.

Both external and internal quality controls were used to verify the accuracy of the results. The 99th percentile cut-off value of cTnI (from a healthy reference population) was set at 0.06 ng/mL.

Statistical analysis

For each patient, cardiovascular risk factors, and clinical and biological data were recorded and reviewed. Phase I drugs were categorized according to their mechanism of action. To identify factors associated with troponin elevation, the incidence and relative risk of troponin elevation according to clinical characteristics were estimated using univariable Poisson regression models. All variables with a p-value of <0.20 were entered in a multivariable Poisson model. The final model was retained using a backward procedure, including variables with a p-value of <0.05. To estimate the effect of troponin elevation on survival, a matched case-control design was used. One patient with troponin elevation after x weeks of treatment was matched with one patient without troponin elevation after x weeks of treatment (control) recruited in the same protocol. Matching was performed with replacement.

In the overall survival analysis, patients were followed from the week of troponin elevation (cases) or the week of troponin elevation in the matched case (controls) to death or to the end of follow-up. Overall survival from troponin elevation was estimated using the Kaplan-Meier method. A

Cox proportional hazards model stratified on matched pairs was used to assess the effect of troponin elevation on overall survival adjusted for age, sex, body mass index, history of hypertension, history of hypercholesterolemia, and human epidermal growth factor receptor 2 (HER2) inhibitor treatment. All variables with a p-value of <0.20 were entered in a model, and a backward procedure retaining variables with a p-value of <0.05 was used to select the best model.

All analyses were done using SAS 9.3 software (SAS Institute, Cary, North Carolina).

A C C E P T E D M A N U S C R I P T

Results

Patient baseline characteristics

Overall, 649 consecutive patients were recruited in the phase I trials, of which 499 patients had at least one troponin evaluation before study-drug initiation. Thirty-two patients were secondarily excluded because they had a history of troponin elevation at the screening evaluation and four because of missing data. The study population therefore comprised 463 patients (Fig. 1), who had been treated for a total of 1159 months. Most patients (95%) had metastatic disease; the remaining 5% had locally advanced cancer.

The median age of the study population was 57 years (interquartile range [IQR]: 49-64 years), the median body mass index was 24 kg/m 2 (IQR: 22-26), and median fibrinogen concentration was 4.8 g/L (IQR: 3.8-6.1). Seventy-eight patients (17%) had a history of hypertension, 28 patients (6%) had a history of diabetes, and 28 patients (6%) had a history of dyslipidemia. The median number of previous lines of treatment was 2 (IQR: 0-2). Molecular-targeted agents and compounds administered during the phase I trials and their main mechanisms of action are detailed in Table 1.

Incidence of troponin elevation

During phase I drug exposure, troponin elevation occurred in 42 (9%) patients. The crude incidence was 36/1000 person-months (95% confidence interval [CI]: 25-47). The median delay between drug introduction and troponin elevation was 5 weeks (IQR: 3-13) and the median elevated troponin concentration was 0.12 ng/mL (IQR: 0.10-0.15). Among the 42 patients with elevated troponin, the elevation was transient in 29 (69%) patients, persistent in 4 (10%) patients, and diagnosed from the last available troponin measurement with no further values in 9 (21%) patients.

When considering higher cut-offs for troponin elevation, the incidence of troponin >0.12 ng/mL was 11/1000 person-months (95% CI: 5-17) and for >0.15 ng/mL was 9/1000 person-months (95% CI: 2-

14).

None of the patients who had elevated troponin elevation had symptoms suggesting acute coronary syndrome, electrocardiographic changes, or a decrease in left ventricular ejection fraction >5 units compared with baseline (data not shown).

A C C E P T E D M A N U S C R I P T

Coronary angiography was performed in 32 patients. No cases of coronary occlusion or coronary artery stenosis >70% were found. Pulmonary embolism was ruled out using computed tomography scanning in 28 of the 42 patients with elevated troponin.

Risk factors for troponin elevation during phase I trials

The incidence of elevated troponin according to the patients' baseline characteristics and phase I treatment is described in Table 2. In univariate analysis, antiangiogenic drugs and fibrinogen elevation ≥6 g/L were associated with troponin elevation, whereas sex, age, body mass index, history of diabetes, hypertension, hypercholesterolemia, and use of HER2 inhibitors were not. The relative risk of troponin elevation was 1.9 (95% CI: 1.1-3.3; p=0.01) when patients received an antiangiogenic agent during the phase I trials. Fibrinogen concentration ≥6 g/L was associated with a relative risk of troponin elevation of 1.9 (95% CI: 0.95-3.6; p=0.035) whereas fibrinogen values between 4 g/L and 6 g/L were not (relative risk: 0.89, 95% CI: 0.44-1.8). In multivariable analysis, the use of antiangiogenic drugs was the only independent predictive predictor of troponin elevation. The incidence of elevated troponin was 60/1000 person-months (95% CI: 35-85) in patients exposed to an angiogenesis inhibitor.

Prognosis value of troponin elevation

The median overall survival of the 463 patients included in the analysis was 9 months (95% CI: 8-10). In the 42 patients with elevated troponin, the median overall survival after troponin elevation was 8 months (95% CI: 2-13).

In the case-control analysis, 42 patients who experienced a troponin elevation were matched for age with 36 controls (Table 3). The risk of death was higher in patients who experienced troponin elevation (Fig. 2) with a hazard ratio for death of 2.9 (95% CI: 1.2-6.8).

Discussion

The three mains findings of this study, involving patients with cancer participating in phase I trials, are as follows. First, 9% of patients experienced a troponin elevation, defined as a value >0.06 ng/mL. Second, the use of antiangiogenic agentsand not cardiovascular risk factorswas

A C C E P T E D M A N U S C R I P T

significantly associated with troponin elevation and was the main predictor of troponin rise. Third, a slight rise in troponin was associated with a poor outcome and an increased risk of death.

The recent introduction of molecular-targeted agents has led to a significant improvement in the management and treatment of solid tumors with a poor prognosis [START_REF] Wang | Broad targeting of angiogenesis for cancer prevention and therapy[END_REF]. However, their use is associated with several unexpected and undesirable cardiac and vascular side-effects, including left ventricular systolic dysfunction and hypertension, considered "on-target" side-effects, and QT interval prolongation, considered an "off-target" side-effect [START_REF] Schmidinger | Cardiac toxicity of sunitinib and sorafenib in patients with metastatic renal cell carcinoma[END_REF][START_REF] Chu | Cardiotoxicity associated with tyrosine kinase inhibitor sunitinib[END_REF]. Troponin rise in patients receiving a chemotherapy regimen containing anthracycline or trastuzumab is associated with left ventricular systolic dysfunction and adverse cardiac events [START_REF] Plana | Expert consensus for multimodality imaging evaluation of adult patients during and after cancer therapy: a report from the American Society of Echocardiography and the European Association of Cardiovascular Imaging[END_REF][START_REF] Cardinale | Trastuzumab-induced cardiotoxicity: clinical and prognostic implications of troponin I evaluation[END_REF]. We extend these findings, showing that antiangiogenic agents are potentially associated with troponin elevation [START_REF] Schmidinger | Cardiac toxicity of sunitinib and sorafenib in patients with metastatic renal cell carcinoma[END_REF].

We did not find any association between troponin rise and left ventricular systolic dysfunction in our study. Several potentials explanations could be proposed. First, our study was not designed to study the link between troponin elevation and left ventricular systolic dysfunction. Not all of the patients in our cohort were explored with echocardiography at baseline and during the course of the phase I trial. All patients were fully evaluated at the time of troponin rise, including echocardiography in all patients and coronary angiography in some patients. We used conventional troponin, which could have limited our ability to detect a slight rise in troponin and thus cardiotoxicity. The total duration of follow-up was relatively short due to the advanced stage of cancer in our patients; this factor may have limited our ability to detect a link between slight troponin elevation and left ventricular systolic dysfunction that could occur later after administration of the cardiotoxic chemotherapy. Finally, it could be hypothesized that the mechanism implicated in troponin rise due to an angiogenic agent is different from anthracycline or trastuzumab and does not lead in all cases to left ventricular systolic dysfunction

The mechanism underlying a slight troponin rise in the context of anticancer therapy remains poorly understood and probably differs according to the study drug used. In patients receiving anthracycline, it has been postulated that troponin rise is due to the generation of reactive oxygen species [START_REF] Chu | Cardiotoxicity associated with tyrosine kinase inhibitor sunitinib[END_REF]. In patients receiving HER2, the mechanisms are not fully understood [START_REF] Cardinale | Trastuzumab-induced cardiotoxicity: clinical and prognostic implications of troponin I evaluation[END_REF]. In a murine model, Chu et al. found that cardiotoxicity associated with sunitinib was due to mitochondrial injury without apoptosis or fibrosis [START_REF] Chu | Cardiotoxicity associated with tyrosine kinase inhibitor sunitinib[END_REF].

A C C E P T E D M A N U S C R I P T

In phase I trials, the frequency, mechanisms, and the prognostic significance of troponin elevation remains largely unknown, and is insufficiently described and understood. We have previously shown that patients receiving molecular-targeted agents are prone to develop a troponin rise and that the mechanisms underlying troponin elevation are not associated with coronary occlusion or myocarditis [START_REF] Ederhy | Frequency and management of troponin I elevation in patients treated with molecular targeted therapies in phase I trials[END_REF]. In the present study we found that 9% of the cohort exhibited a slight troponin rise, and this was associated with important prognostic implications.

The spontaneous prognosis of patients in phase I trials remains poor despite having a good performance status at study entry [START_REF] Ploquin | Life-expectancy of patients enrolled in phase 1 clinical trials: a systematic review of published prognostic models[END_REF]. Several models have been developed to predict mortality; demographic variables (functional status), oncologic variables (number of metastases), and biological factors (lactate dehydrogenase, white blood cells) are robust predictors of mortality in phase I trials [START_REF] Olmos | Patient selection for oncology phase I trials: a multi-institutional study of prognostic factors[END_REF]. Recent studies demonstrated that cardiac biomarkers are elevated in patients with cancer and are related to all-cause mortality as well as alteration of echocardiographic variables [START_REF] Pavo | Cardiovascular biomarkers in patients with cancer and their association with all-cause mortality[END_REF][START_REF] Rhea | Incremental prognostic value of echocardiographic strain and its association with mortality in cancer patients[END_REF]. These results, in conjunction with ours, suggest that cardiac involvement in patients with cancer evaluated with biomarkers or by echocardiography could be considered as a potential marker of overall survival and disease progression, and not only as a marker of cardiotoxicity.

Limitations

Our study has several limitations. Our study was restricted to conventional troponin only, and did not include high-sensitivity troponin or other biomarkers. The specific cause of death in our cohort could not be addressed precisely in approximately 65% of patients, and cardiovascular events could not be identified or well characterized. Pulmonary embolism was not systematically ruled out and coronary angiography was not performed in all patients with a troponin rise.

In conclusion, troponin elevation is a relatively frequent event among patients with cancer participating in phase I trials. Patients exposed to antiangiogenic drugs present a higher risk of a rise in cardiac troponin, which is associated with a higher risk of death. The results of our study suggest that patients presenting with troponin elevation deserve closer clinical and cardiologic follow-up. Abbreviations: HER2 = human epidermal growth factor receptor 2; MAP = mitogen-activated protein kinase; PIK = phosphatidylinositol-4,5-bisphosphate 3-kinase. Abbreviations: BMI = body mass index; HER2 = human epidermal growth factor receptor 2.

Legends

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A C C E P T E D M A N U S C R I P T Table 3

Description of patients included and not included in the matched case-control analysis.

Fig. 1 .

 1 Fig. 1. Patient flow chart.

Fig. 2 .

 2 Fig. 2. Overall survival according to troponin elevation (matched population).

Figure 1 649

 1 Figure 1

Figure 2

 2 Figure 2

Table 1

 1 Mechanism of treatment and occurrence of troponin >0.06 ng/mL (1 line per therapy).

	N (%)	Patient-months on	Incident case of
		phase I drug	troponin elevation while
		treatment	on phase I therapy
		Sum

Table 2

 2 Incidence of troponin elevation under phase I treatment according to baseline characteristics at phase

	I drug initiation.
					N (%)	Patient-months	Incident	Crude incidence	p
						under phase I	cases of	per 1000 person-	Value
						drug treatment	troponin	months (95% CI)
							elevation
	All				463 (100) 1159	42	36 (25-47)
	Sex						0.82
	Male				253 (55)	654	25	38 (23-53)
	Female				210 (45)	505	17	34 (18-50)
	Age at baseline				0.55
	<50 years		124 (27)	323	16	49 (25-74)
	50 to <65 years	247 (53)	564	17	30 (16-44)
	≥65 years		92 (20)	271	9	33 (12-55)
	BMI at baseline				0.21
	Missing				38 (8)	76	3	39 (0-84)
	<20 kg/m	2		49 (11)	99	5	51 (6-95)
	20 to <25 kg/m	2	209 (45)	498	21	42 (24-60)
	≥25 kg/m	2		167 (36)	486	13	27 (12-41)
	Hypertension				0.83
	No				385 (83)	964	36	37 (25-50)
	Yes				78 (17)	195	6	31 (6-55)
	Diabetes
	No				436 (94)	1077	42	39 (27-51)	0.07
	Yes				27 (6)	81	0	0 (0-4)
	Hypercholesterolaemia			0.38
	No				434 (94)	1107	39	35 (24-46)
	Yes				29 (6)	52	3	58 (0-123)

<4 g/L 7 (19) 11 (31) 3 (50)

Acknowledgements

Sophie Rushton-Smith, PhD, provided editorial assistance on the last version of the article and was funded by the authors.

Source of funding

None received. This study was conducted independently of any industry or other grant support.

Disclosure of conflicts of interest

All authors had full access to all of the data in this study and take responsibility for the integrity of the data and the accuracy of the data analysis. All authors contributed to the design, analysis, interpretation of data, drafting the article, or revising it critically for important intellectual content and approved the final version to be published.


P Gabriel

F Salvarani

Exponential relaxation to self-similarity for the superquadratic fragmentation equation

Keywords: fragmentation equation, self-similarity, exponential convergence, spectral gap, longtime behavior AMS Class. No. Primary: 35B40, 35P15, 45K05; Secondary: 82D60, 92D25

We consider the self-similar fragmentation equation with a superquadratic fragmentation rate and provide a quantitative estimate of the spectral gap.

Introduction

The fragmentation equation

∂ t f (t, x) = Ff (t, x), t ≥ 0, x > 0, f (0, x) = f in (x), x > 0 (1)
is a model that describes the time evolution of a population structured with respect to the size x of the individuals.

The key term of the model is the fragmentation operator F, defined as

Ff (x) := ∞ x b(y, x)f (y) dy -B(x)f (x). (2)
The fragmentation operator quantifies the generation of smaller individuals from a member of the population of size x: the individuals split with a rate B(x) and generate smaller individuals of size y ∈ (0, x), whose distribution is governed by the kernel b(x, y). Models involving the fragmentation operator appear in various applications. Among them we can mention crushing of rocks, droplet breakup or combustion [START_REF] Banasiak | Conservative and shattering solutions for some classes of fragmentation models[END_REF] which are pure fragmentation phenomena, but also cell division [START_REF] Perthame | Transport equations in biology[END_REF], protein polymerization [START_REF] Gabriel | Équations de Transport-Fragmentation et Applications aux Maladies à Prions [Transport-Fragmentation Equations and Applications to Prion Diseases[END_REF] or data transmission protocols on the web [START_REF] Bardet | Total variation estimates for the TCP process[END_REF], for which the fragmentation process occurs together with some "growth" phenomenon. * University of Versailles St-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, 45 Avenue de États-Unis, 78035 Versailles cedex, France. Email: pierre.gabriel@uvsq.fr † Dipartimento di Matematica F. Casorati, Universita degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy. Email: francesco.salvarani@unipv.it 1 In order to ensure the conservation of the total mass of particles which may occur during the fragmentation process, the coefficients B(x) and b(y, x) must be linked through the relation y 0 xb(y, x) dx = yB(y).

(

This assumption ensures, at least formally, the mass conservation law

∀t > 0, ∞ 0 xf (t, x) dx = ∞ 0 xf in (x) dx := ρ in . (4)
Moreover, it is well known that xf (t, x) converges to a Dirac mass at x = 0 when t → +∞. Usually, the various contributions that are available in the literature restrict their attention to coefficients which satisfy the homogeneous assumptions (see [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF] for instance)

B(x) = x γ , γ > 0, and b(y, x) = y γ-1 p x y , (5)
where dµ(z) := p(z) dz is a positive measure supported on [0, 1] which satisfies

1 0 z dµ(z) = 1.
These hypotheses guarantee that the relation ( 3) is verified.

From a mathematical point of view, it is convenient to perform the (mass preserving) selfsimilar change of variable

f (t, x) = (1 + t) 2/γ g 1 γ log(1 + t), (1 + t) 1/γ x ,
or, by writing g in terms of f , g(t, x) := e -2t f (e γt -1, e -t x).

It allows to deduce that g(t, x) satisfies the self-similar fragmentation equation

∂ t g + ∂ x (xg) + g = γFg, t ≥ 0, x > 0, g(0, x) = f in (x), x > 0. (6)
Equation ( 6) belongs to the class of growth-fragmentation equations and it admits -unlike Equation ( 1) -positive steady-states [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF]. Denote by G the unique positive steady-state of Equation ( 6) with normalized mass, i.e. the solution of (xG

) ′ + G = γFG, G > 0, ∞ 0 xG(x) dx = 1.
Then it has been proved (see [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]) that the solution g(t, x) of the self-similar fragmentation equation ( 6) converges to ρ in G(x) when t → +∞.

Coming back to the fragmentation equation [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF], this result implies the convergence of f (t, x) to the self-similar solution (t, x) → ρ in (1 + t) 2/γ G((1 + t) 1/γ x) and hence the convergence of xf (t, x) to a Dirac mass δ 0 .

In order to obtain more precise quantitative properties of the previous equation, one can wonder about the rate of convergence of g(t, x) to the asymptotic profile G(x). Many recent works are dedicated to this question and prove, under different assumptions and with different techniques, an exponential rate of convergence for growth-fragmentation equations [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF][START_REF] Bardet | Total variation estimates for the TCP process[END_REF][START_REF] Cáceres | Rate of convergence to self-similarity for the fragmentation equation in L 1 spaces[END_REF][START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF][START_REF] Cloez | Wasserstein decay of one dimensional jump-diffusions[END_REF][START_REF] Laurençot | Exponential decay for the growth-fragmentation/celldivision equation[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF][START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF].

Nevertheless, to our knowledge the only results about the specific case of the self-similar fragmentation equation are those provided by Cáceres, Cañizo and Mischler [START_REF] Cáceres | Rate of convergence to self-similarity for the fragmentation equation in L 1 spaces[END_REF][START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF]. They prove exponential convergence in the Hilbert space L 2 ((x + x k ) dx) for a sufficiently large exponent k in [START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF], and in the Banach space L 1 ((x m + x M ) dx) for suitable exponents 1/2 < m < 1 < M < 2 in [START_REF] Cáceres | Rate of convergence to self-similarity for the fragmentation equation in L 1 spaces[END_REF]. For proving their results, the authors of the aforementioned articles require the measure p to be a bounded function (from above and below) and the power γ of the fragmentation rate to be less than 2.

The current paper aims to obtain a convergence result for super-quadratic rates, namely when γ ≥ 2. We obtain exponential convergence to the asymptotic state by working in the weighted Hilbert space L 2 (x dx), under the following assumptions:

γ ≥ 2 and p(z) ≡ 2. ( 7
)
The fact that p(z) is a constant means that the distribution of the fragments is uniform: the probability to get a fragment of size x or x ′ from a particle of size y > x, x ′ is exactly the same.

Then the condition 1 0 zp(z) dz = 1 imposes this constant to be equal to 2, meaning that the fragmentation is necessarily binary. Our assumption on p is more restrictive than in [START_REF] Cáceres | Rate of convergence to self-similarity for the fragmentation equation in L 1 spaces[END_REF][START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations[END_REF], but in return we get a stronger result in the sense that we obtain an estimate of the exponential rate. Now we state the main theorem of this paper.

Theorem 1.1. Let g in ∈ L 1 (x dx) ∩ L 2 (x dx) and let g ∈ C([0, ∞), L 1 (x dx
)) be the unique solution of the self-similar fragmentation equation ( 6) with initial condition g in and with fragmentation coefficients satisfying (5) and [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF], that is

B(x) = x γ , γ ≥ 2 and b(y, x) = 2y γ-1 .
Then the following estimate holds:

g(t, •) -ρ in G L 2 (x dx) ≤ g in -ρ in G L 2 (x dx) e -t , t ≥ 0.

Preliminaries

Define the suitable weighted spaces

Lp k := L p (R + , x k dx) for p ≥ 1, k ∈ R, and Ẇ 1,1 1 := W 1,1 (R + , x dx).
For u ∈ L1

Theorem [8]. If B(x) = x γ , γ ≥ 2 and b(y, x) = 2y γ-1 , for any f in ∈ L2 1 ∩ L1 1 , there exists a unique solution f ∈ C([0, ∞); L1 1 ) ∩ L 1 loc ([0, ∞); L1 1+γ ) to the fragmentation equation (1) such that the mass conservation (4) is satisfied. If, moreover, f in ∈ Ξ := L1 1+γ ∩ Ẇ 1,1 1
, the associated solution g to the self-similar fragmentation equation is such that (g(t, •)) t≥0 is uniformly bounded in Ξ.

In the following lemma we give some useful properties of the set

Ξ = L1 1+γ ∩ Ẇ 1,1 1
and of the subset

Ξ 0 := u ∈ Ξ, ∞ 0 xu(x) dx = 0 . Lemma 2.1. The set Ξ = L1 1+γ ∩ Ẇ 1,1 1 satisfies G ∈ Ξ, Ξ ⊂ C(0, ∞) ∩ L 1 ∩ L2 1 ∩ L2 1+γ and ∀u ∈ Ξ, lim x→0 xu(x) = lim x→+∞ xu(x) = 0.
Moreover for any function u ∈ Ξ 0 the following inequality holds:

∀x > 0, |M (x)| ≤ x -γ u L1 1+γ .
Proof. The fact that the steady-state G belongs to Ξ is a consequence of the estimates in [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF].

In the case when p(z) ≡ 2 it can also be deduced from the explicit formula (see [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF])

G(x) = γ Γ(2/γ) e -x γ
where Γ is the Euler Gamma function.

For γ ≥ 0 and u ∈ Ξ such that Proof. Using Lemma 2.1 we can deduce, for u ∈ Ξ 0 ,

(u, (xu) ′ ) = ∞ 0 xu(x) xu(x) ′ dx = 1 2 ∞ 0 (xu(x)) 2 ′ dx = 0 and (u, Fu) = 2 ∞ 0 xu(x) ∞ x y γ-1 u(y) dy dx - ∞ 0 x γ+1 u 2 (x) dx = 2 ∞ 0 x γ-1 u(x) x 0 yu(y) dy dx - ∞ 0 x γ+1 u 2 (x) dx = 2 ∞ 0 x γ-2 M ′ (x)M (x) dx - ∞ 0 x γ+1 u 2 (x) dx = -(γ -2) ∞ 0 x γ-3 M 2 (x) dx - ∞ 0
x γ+1 u 2 (x) dx ≤ 0.

Proof of Theorem 1.1. Assume first that g in ∈ Ξ. From Theorem 3.1 we obtain the differential inequality d dt g(t, •) -ρ in G L2

Conclusion

We have proved a spectral gap result for the self-similar fragmentation operator L with a superquadratic fragmentation rate B(x). More precisely we have obtained that this spectral gap is larger than 1. This is a new result concerning the long-time behaviour of the fragmentation equation [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF]. It also allows to extend the results obtained in [START_REF] Gabriel | Long-time asymptotics for nonlinear growth-fragmentation equations[END_REF] for non-linear growth-fragmentation equations.

  xu(x) which vanishes at x = 0. Now we recall the following existence and uniqueness result of a solution to the fragmentation equation, easily deduced from[START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF], Theorems 3.1-3.2 and Lemma 3.4:

∞0 0 yTheorem 3 . 1 . 2 L2 1 .

 03121 xu(x) dx = 0 we can write for x > 0|M (x)| = -∞ x yu(y) dy ≤ x -γ ∞ 1+γ |u(y)| dy.3 Proof of the main theorem Define the self-similar fragmentation operator Lu := -(xu) ′ -u + γFu and denote by (u, v) := ∞ 0 xu(x)v(x) dx the canonical scalar product in L2 1 . Theorem 1.1 is a consequence of the following result. Under Assumptions (5) and (7), i.e. for B(x) = x γ , γ ≥ 2 and b(y, x) = 2y γ-1 , we have ∀u ∈ Ξ 0 , (u, Lu) ≤ -u

1 ≤

 1 -g(t, •) -ρ in G L2 1which gives the result. Then we may remove the additional assumption g in ∈ Ξ.

Acknowledgement

This work has been performed in the framework of the ANR project KIBORD, headed by Laurent Desvillettes.


Mirko Brentari

Denis Arzelier
email: arzelier@laas.fr

Christophe Louembet
email: louembet@laas.fr

Sofia Urbina
email: lsurbina@laas.fr

Luca Zaccarian
email: zaccarian@laas.fr

L Zaccar- Ian

A hybrid control framework for impulsive control of satellite rendezvous

We focus on the problem of satellite rendezvous between two spacecraft in elliptic orbits. Using a linearized model of the relative dynamics, we first propose a periodic similarity transformation based on Floquet-Lyapunov theory, leading to a set of coordinates under which the free motion is linear time-invariant. Then we address the problem of impulsive control of satellite rendezvous as a hybrid dynamical system, and we show that the arising elegant representation enables designing impulsive control laws with different tradeoffs between computational complexity and fuel consumption. The adopted hybrid formalism allows us to prove suitable stability properties of the proposed controllers. The results are comparatively illustrated on simulation examples.

I. INTRODUCTION

Considering the increasing need for satellite servicing in space, the capability of operating an active spacecraft, the follower denoted by F, in close proximity of a satellite, the leader denoted by L will be crucial for fulfilling complex safe space missions objectives comprising inspection, repairing, refueling or monitoring [START_REF] Fehse | Automated rendezvous and docking of spacecraft[END_REF]. The whole relative spacecraft maneuvering process composes what is known as the rendezvous and proximity operations which mainly consists in getting the follower from one orbit to a box near the leader (close range rendezvous) [START_REF] Louembet | Robust rendezvous planning under maneuver execution errors[END_REF], [START_REF] Deaconu | A two-impulse method for stabilizing the spacecraft relative motion with respect to a periodic trajectory[END_REF] and then in beginning the proximity operations required by the mission objectives. When dealing with the preliminary planning phase of space missions, it is usual to approximate actual finite-thrust powered phases of finite duration by impulsive maneuvers. The impulsive approximation for the thrust means that instantaneous velocity increments are applied to the chaser when firing, whereas its position is continuous. This assumption, made in this paper, has proved to be very useful in reducing the complexity of guidance and control design and has been widely used in the literature dedicated to rendezvous (see [START_REF] Fehse | Automated rendezvous and docking of spacecraft[END_REF], [START_REF] Deaconu | Constrained periodic spacecraft relative motion using non-negative polynomials[END_REF], [START_REF] Deaconu | A two-impulse method for stabilizing the spacecraft relative motion with respect to a periodic trajectory[END_REF] and references therein).

In this article, we are mainly interested by the first phase for which it is highly recommended to design fuel efficient impulsive maneuvers guiding the follower, from one point to a specified tolerance region in the proximity of the leader where the relative motion of the follower will be periodic and bounded. It is well known that, under Keplerian assumptions, the relative motion between spacecraft is bounded [START_REF] Gurfil | Relative motion between elliptic orbits: Generalized boundedness conditions and optimal formationkeeping[END_REF]. Different conditions for the periodicity of the linearized equations of the nonlinear relative motion have been given in the literature. Here we consider the parametric expression for the relative motion proposed in [START_REF] Deaconu | Constrained periodic spacecraft relative motion using non-negative polynomials[END_REF] and used to characterize in a simple way periodic relative motions. In a linearized context, any relative periodic trajectory is defined by 6 constant parameters, one of them being equal to 0. In particular, building on the result presented in [START_REF] Deaconu | Constrained periodic spacecraft relative motion using non-negative polynomials[END_REF], the contribution of the present paper is twofold. First, we propose a new coordinate transformation which leads to a simplified characterization of periodic trajectories when applied to the Tschauner-Hempel equations of the elliptic linearized relative motion. Second, three different hybrid feedback-control laws are designed by taking advantage of the particular formulation of the rendezvous problem. The use of the hybrid framework [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] for representing nonlinear hybrid dynamical systems (whose solutions exhibit continuous evolution and impulsive behavior) allows us to state and prove suitable stability properties of the proposed impulsive control laws (including, e.g., the one originally presented in [START_REF] Deaconu | A two-impulse method for stabilizing the spacecraft relative motion with respect to a periodic trajectory[END_REF]) when applied to the linear timevarying dynamics of the closed-loop satellite rendezvous system. Due to lack of space, all statements and proofs are omitted but are given in the extended version [START_REF] Brentari | A hybrid control framework for impulsive control of satellite rendezvous[END_REF].

Notations: e, ν and T are respectively the eccentricity, the true anomaly and the period of the leader's orbit. f represents the derivation of the function f with respect to the true anomaly ν. I n is the identity matrix of dimension n. For a set S, S denotes the closure of the set S.

II. LTI STATE-SPACE FOR THE LINEARISED RELATIVE

EQUATIONS OF MOTION

The proximity operations between two spacecraft are characterized by the use of relative navigation since the separation between spacecraft is sufficiently small. In this framework, the relative motion of the follower is described in the Local-Vertical-Local-Horizontal (LVLH) frame attached to the leader body [START_REF] Louembet | Robust rendezvous planning under maneuver execution errors[END_REF]. The origin of the coordinate frame is located at the center of mass of the leader and the space is spanned by (x, y, z) where the z axis is in the radial direction (R-bar) oriented towards the center of the Earth, the y axis is perpendicular to the leader orbital plane and pointing in the opposite direction of the angular momentum (H-bar) while the x axis is chosen such that x = y × z. Under Keplerian assumptions (no orbital perturbations are considered) and an elliptic reference orbit, the equations of motion for the relative motion in the LVLH frame may be linearized for close separation between the leader and the follower and are known as the Tschauner-Hempel (TH) equations [START_REF] Tschauner | The elliptic orbit rendezvous[END_REF], [START_REF] Schaub | Analytical mechanics of space systems[END_REF]:

ζ = A 0 (t)ζ free motion ζ + = ζ + 0 3×3 I 3 t + t - f (t) m F dt
when applying impulsive thrusts, [START_REF] Alfriend | Dynamics and control of spacecraft formations: Challenges and some solutions[END_REF] where state ζ = (x, y, z, dx/dt, dy/dt, dz/dt) represents positions and velocities in the three fundamental axes of the LVLH frame, matrix A 0 is a suitable periodic function of time t [START_REF] Tschauner | The elliptic orbit rendezvous[END_REF], f (t) is the thrust vector, ζ + is the state vector right after the jump and m F is the mass of the follower. We may define the impulsive control input (essentially equivalent to velocity jumps in the three axes) as:

∆v(t k ) := t + k t - k 1 m F f x (t) f y (t) f z (t) dt, (2)
where t k is a generic firing time, which is directly associated to the fuel consumption.

In order to simplify the TH equations, classical derivations correspond to applying a change of independent variable from time t to true anomaly ν, noting that:

dν dt = n (1 -e 2 ) 3/2 (1 + e cos ν ρ(ν) ) 2 =: k 2 ρ(ν) 2 , (3)
where n = 2π/T is the mean motion of the leader orbit, satisfying for any fixed ν 0 , t 0 :

ν -ν 0 = 2π ⇒ n(t -t 0 ) = 2π. (4)
This leads to the following hybrid representation for a new state ξ(ν), replacing ζ(t):

ξ = A(ν)ξ free motion ξ + = ξ + 1 k 2 ρ(ν) 2 0 3×3 I 3 =:B(ν)
u when applying impulsive thrusts, [START_REF] Fehse | Automated rendezvous and docking of spacecraft[END_REF] where u = ∆v represents the applied impulse, state ξ = (x, y, z, dx/dν, dy/dν, dz/dν) represents positions and velocities with respect to ν, and A(ν [START_REF] Tschauner | The elliptic orbit rendezvous[END_REF], [START_REF] Schaub | Analytical mechanics of space systems[END_REF]), where the dependence on ν has been omitted for brevity's sake and where we used c ν := cos(ν) and s ν := sin(ν).

) :=     0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ecν ρ 0 -2 esν ρ 2 esν ρ 0 2 0 -1 ρ 0 0 2 esν ρ 0 2 esν ρ 0 2+ρ ρ -2 0 2 esν ρ     (see
Previous work [START_REF] Tschauner | The elliptic orbit rendezvous[END_REF], showed the usefulness of the following coordinate change for simplifying expression of dynamics (5):

T (ν) := ρ(ν)I 3×3 0 3×3 ρ(ν) I 3×3 ρ(ν)I 3×3 . (6)
In this article, we propose the two following additional transformations, the first one arising from similar observations to those in [START_REF] Deaconu | A two-impulse method for stabilizing the spacecraft relative motion with respect to a periodic trajectory[END_REF], and the second one arising from a suitable Floquet-Lyapunov derivation:

C(ν) := (7)          0 c ν 0 0 -s ν 0 0 s ν 0 0 c ν 0 1 0 -3esν (1+ρ)
ρ(e 2 -1)

esν (1+ρ) e 2 -1 0 ρ 2 -ecν -3 e 2 -1 e 0 -3s ν s ν (1 + ρ) 0 c ν ρ 0 0 3(cν +e) e 2 -1 -cν (1+ρ)+e e 2 -1 0 s ν ρ 0 0 -3(3ecν +e 2 +2) e 2 -1 3ρ 2 e 2 -1 0 -3esν ρ e 2 -1          S(ν) :=          1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 σ(ν) (1 -e 2 ) 3/2 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1          , (8)
where we introduced the function:

σ(ν) :=(ν -ν) -n(t -t) =(ν -ν) -(M -M ) =∆ν -∆M, (9)
based on an arbitrary true anomaly value ν (in our simulations we select ν = 0) corresponding to a specific time value t (in our simulations we select t = 0). Function σ in ( 9) is clearly periodic and bounded, due to the geometric dependence of ν and M when the leader follows a Keplerian elliptic motion.

The following result establishes a first contribution of this paper showing that the dynamics (5) can be transformed to a convenient linear time-invariant form by exploiting transformations ( 6), [START_REF] Gurfil | Relative motion between elliptic orbits: Generalized boundedness conditions and optimal formationkeeping[END_REF], and [START_REF] Inalhan | Relative dynamics and control of spacecraft formations in eccentric orbits[END_REF]. The proof of the lemma is omitted as it follows from straightforward (even though lengthy) mathematical derivations.

Lemma 1: Consider matrices in ( 6), [START_REF] Gurfil | Relative motion between elliptic orbits: Generalized boundedness conditions and optimal formationkeeping[END_REF], and [START_REF] Inalhan | Relative dynamics and control of spacecraft formations in eccentric orbits[END_REF]. Then the following operation:

ξ = R(ν)ξ := S(ν) C(ν) T (ν)ξ , (10)
is a linear time-varying coordinates change, namely R(ν) is invertible for all ν and R and R -1 are uniformly bounded. Moreover, R is periodic and transforms A(ν) into the following time-invariant form:

A = R (ν)R -1 (ν) + R(ν)A(ν)R -1 (ν) =     0 0 | 0 0 0 0 0 0 | 0 0 0 0 0 0 | 0 0 0 (1-e 2 ) -3/2 0 0 | 0 0 0 0 0 0 | 0 0 0 0 0 0 | 0 0 0 0     .
From the peculiar structure (quasi Jordan form) of the dynamic matrix Â, it is observed (as already emphasized for similar coordinate changes in [START_REF] Deaconu | A two-impulse method for stabilizing the spacecraft relative motion with respect to a periodic trajectory[END_REF]), that any periodic free motion of the original dynamics (1) can be suitably parametrized by specific selections of the state ξ with the last component being zero. It is therefore convenient representing the system in terms of the error with respect to a desired motion ξ ref :

ξ ref := [ξ ref 1 ξ ref 2 ξ ref 3 ξ ref 4 ξ ref 5 0] T , (11)
so that one may analyze the dynamics of the mismatch ξ = ξ -ξ ref between the coordinate ξ in [START_REF] Ross | 6 space trajectory optimization and l 1-optimal control problems[END_REF] and a constant reference value in [START_REF] Schaub | Analytical mechanics of space systems[END_REF], representing a desired target periodic motion.

III. IMPULSIVE CONTROL OF THE RELATIVE DYNAMICS

The coordinate transformation presented in Lemma 1 of the previous section is a useful means for suitably designing an impulsive control law assigning the firing times t k in (2) and also the corresponding selections of ∆v(t k ). Among others, one objective is to reduce as much as possible the fuel consumption well characterized in [START_REF] Ross | 6 space trajectory optimization and l 1-optimal control problems[END_REF] when firing is achieved by 6 identical thrusters rigidly mounted to the satellite, and corresponding to the cost function

J = t k ∈T |∆v x (t k )| + |∆v y (t k )| + |∆v z (t k )| = t k ∈T |∆v(t k )| 1 ,
where T is the set of the times when impulsive thrusts are applied and | • | 1 denotes the 1-norm. A more convenient expression of J is given in terms of the representation in [START_REF] Fehse | Automated rendezvous and docking of spacecraft[END_REF] and corresponds to:

J = ν k ∈V |u(ν k )| 1 , (13)
B(ν) = 1 k 2 ρ 2 (1 -e 2 )                  0 -(1 -e 2 )ρsν 0 0 (1 -e 2 )ρcν 0 -e(1 + ρ)ρsν - 3σρ 3 (1 -e 2 ) 3/2 0 3σeρ 2 sν (1 -e 2 ) 3/2 -ρ 3 + ρ 2 + 2ρ (1 -e 2 )(1 + ρ)ρsν 0 (1 -e 2 )ρ 2 cν (1 + ρ)ρcν + eρ 0 -ρ 2 sν -3ρ 3 0 3eρ 2 sν                  (12)
with V being the set of firing instants. Based on (13) we may formulate our control design problem as follows: Problem 1: Given plant (1) and its equivalent form (5), design a state feedback impulsive control law selecting the firing instants ν k , k ∈ N and the corresponding inputs u(ν k ) such that (i) for any selection of reference [START_REF] Schaub | Analytical mechanics of space systems[END_REF], the point ξ = ξ ref is globally asymptotically stable for the closed loop dynamics; (ii) the cost J is minimized over a family of possible input selections. To solve Problem 1, in this section we will propose hybrid control laws relying on the presence of a timer τ in charge of the sequencing of the impulsive control actions. Then, using the hybrid systems notation in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] and state ξ = ξ -ξ ref , we may write the following general dynamic description of the closed loop, enjoying the desirable property that timers ν and τ evolve in the compact set [0, 2π] and that the flow equation for ξ is ξ = A ξ, because Aξ ref = 0:

   ξ = A ξ, ν = 1, τ = -1, ( ξ, ν, τ ) ∈ C, (14a)    ξ+ = ξ, ν + = 0, τ + = τ, ( ξ, ν, τ ) ∈ D ν , (14b)
       ξ+ = ξ + B(ν)γ u ξ, ν , ν + = ν, τ + = γ τ ξ, ν , ( ξ, ν, τ ) ∈ D u . (14c)
In equation ( 14), the impulsive control law has been selected as a feedback controller:

u = γ u ( ξ, ν), τ + = γ τ ( ξ, u). (15)
Moreover, according to the coordinate change given in [START_REF] Ross | 6 space trajectory optimization and l 1-optimal control problems[END_REF], and to the results of Lemma 1, matrix B(ν) = R(ν)B(ν), is a periodic function of ν arising from combining the similarity transformation in [START_REF] Ross | 6 space trajectory optimization and l 1-optimal control problems[END_REF] with the input matrix in [START_REF] Fehse | Automated rendezvous and docking of spacecraft[END_REF], and corresponds to the matrix reported in [START_REF] Tschauner | The elliptic orbit rendezvous[END_REF] at the top of this page.

Equation ( 14) is a compact representation of the impulsive feedback control action as a set of dynamical constraints that solutions should satisfy for their correct evolution. In particular, using an overall state η = ( ξ, ν, τ ), this dynamics falls into the general class of systems studied in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]:

η ∈ C, η = F (η), η ∈ D, η + ∈ G(η). (16)
For our model, the following selections are made:

D ν := R 6 × {2π} × [0, 2π], (17a)
D u := R 6 × [0, 2π] × {0}, ( 17b
) D := D ν ∪ D u , (17c)
C := (R 6 × [0, 2π] × [0, 2π]) \ D, (17d)
which, due to (17c), is a choice that prioritizes jumps. In particular, based on (14), functions F and G in (16) are selected as:

F (η) =   A ξ 1 -1   ; G(η) = i∈{u,ν} s.t. η∈Di G i (η); (18a) G ν (η) =   ξ 0 τ   ; G u (η) =     ξ + B(ν)γ u ξ, ν ν γ τ ξ, ν     . (18b)

IV. CONTROL LAWS

In this section, we propose three different selections for the impulsive control law (15) solving Problem 1. These selections are different in terms of performance trade-offs and are comparatively illustrated on the example studies of Section V.

A. Periodic norm-minimizing control

While formulation (14), (15) is general enough to allow for aperiodic optimized sampling, the simplest possible selection of function γ τ in (15) is given by periodic thrusters firing, corresponding to a certain period ν ∈ [0, 2π] fixed a priori. This is equivalent to the following constant selection:

γ τ ( ξ, ν) = ν, (19)
encoding the fact that each pair of consecutive jumps has a fixed angular distance of ν.

Regarding the selection of the stabilizer γ u , to be evaluated periodically, we make here a conservative selection leading to the useful feature that after each impulse, the state ξ6 = ξ 6 is driven to zero, so that in the absence of noise the spacecraft evolves through periodic (therefore bounded) motions. In particular, the following optimal selection is chosen:

u = arg min u | ξ+ | 2 , subject to: ξ+ = ξ + B(ν)u, ξ+ 6 = 0. (20)
Due to the specific structure of matrix function B in (12) at the top of the page, we may provide an explicit form of the minimizer in (20) after defining the following quantities:

b6 (ν) = 1 k 2   3ρ e 2 -1 0 3e sin(ν) 1-e 2   , B⊥ 6 (ν) = e sin(ν) 0 0 1 ρ(ν) 0 , (21a)
which clearly satisfy b6 (ν) T B⊥ 6 (ν) = 0 because matrix B⊥ 6 (ν) generates the orthogonal complement of b6 (ν). With these definitions in place, we may write the explicit expression of the proposed control law as:

γ u ( ξ, ν) = u 6 -B⊥ 6 (ν)( B(ν) B⊥ 6 (ν)) -L ( ξ + B(ν)u 6 ) with u 6 = - b6 (ν) | b6 (ν)| 2 ξ6 , (21b)
where 21) is that it instantaneously minimizes the norm of ξ constrained to the fact that the subsequent motion be periodic. Since the norm of b6 in (21a) is never zero, then clearly, equation (21b) is always well-posed and ensures that ξ+ 6 = 0. In addition to this, instantaneously minimizing the norm of ξ also ensures the best possible decrease at the specific fixed instant of time enforced by the rigid periodic selection. With this logic in place, we can guarantee stability of the closed-loop but not convergence. Indeed, we can guarantee non-increase of | ξ| across jumps but there is no guarantee of obtaining a strict decrease. As a result, we anticipate a slow convergence (if any) in our simulation section when using this controller. Despite this fact, the choice (19), ( 21) is still an interesting one because it ensures that approaching between the two satellites is performed through periodic (bounded) motions, leading to some degree of fault tolerance (in case of malfunctioning, the satellite is on a stable orbit).

M -L = (M T M ) -1 M T denotes
• It can be proven that the proposed controller solves part of item (i) of Problem 1. See [2, Thm 1] for details.

B. Periodic bi-impulsive control

A second selection that we propose for the controller in ( 15) is once again periodic, thereby corresponding to selection (19) for γ τ . However, it corresponds to a wiser selection of γ u (in terms of envisioned fuel consumption), performed in similar ways to what is proposed in [START_REF] Deaconu | A two-impulse method for stabilizing the spacecraft relative motion with respect to a periodic trajectory[END_REF], by focusing on the overall effect on the state ξ of two impulses performed at a distance of ν from one another. In particular, using straightforward computations, if two impulses u 1 and u 2 happen at times µ 1 and µ 2 = µ 1 + ν, we obtain, along the corresponding solution:

Φ(-ν) ξ(µ 2 , j + 2) = ξ(µ 1 , j) + B(ν1)Φ(-ν) B(ν1 + ν) M (ν1,ν):= u1 u2 , (22)
where µ 1 , µ 2 denote the (angular) times, j denotes the number of elapsed thruster firings, Φ(µ) = e Aµ =   1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 µ(1-e 2 ) -3/2 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1   is the state transition matrix of the (LTI) flow dynamics in (14), and ν 1 = ν(µ 1 , j).

Based on relation ( 22), and to the end of selecting u 1 , u 2 in such a way that ξ(µ 2 , j + 2) is zero, it is important to study the invertibility properties of matrix M (ν, ν), which is done in the following conjecture. The result of the conjecture restricts the set of possible selections of ν in (19).

Conjecture 1: For any value of ν ∈ [0, 2π], matrix M (ν, ν) in ( 22) is invertible if and only if ν = kπ, k ∈ Z.

Conjecture 1 is motivated by numerical experience and its formal proof is regarded as future work. If Conjecture 1 holds, for any selection ν ∈ (0, 2π) \ {π}, equation ( 22) can be inverted to compute the unique pair of inputs u 1 , u 2 ensuring ξ(µ 2 , j + 2) = 0 (namely that the state ξ is driven to zero after two impulses separated by ν times). Then, following a receding horizon type of paradigm, we may apply the first pulse and re-evaluate the control law at the next pulse. The above control design paradigm leads to the following selection:

γ u ( ξ, ν) = -[ I 0 ] M (ν, ν) -1 ξ. (23)
Then, the overall control strategy ( 19), (23) guarantees item (i) of Problem 1 as established formally in [2, Thm 2].

C. Non-periodic bi-impulsive control

With the goal of minimizing the fuel consumption, a main drawback of the strategies proposed in the two previous sections is the fact that the (angular) time elapsed between two consecutive impulses is fixed and equal to ν. This is indeed a direct consequence of the simplified selection (19). In this section, we propose a different control paradigm, corresponding to an increased computational complexity, where the bi-impulsive control paradigm in (23) is combined with an optimized selection of the waiting time τ + before the next impulse. In particular, the overall control law corresponds to:

γ τ ( ξ, ν) = arg min ν∈[0,2π] M (ν, ν) -1 ξ 1 = arg min ν∈[0,2π] J, (24a) γ u ( ξ, ν) = -[ I 0 ] M (ν, γ τ ( ξ, ν)) -1 ξ. (24b)
The new control law (24) corresponds to selecting an optimal elapsed time µ before the next impulse, according to the fuel consumption index (13). In particular, the corresponding optimization is nonlinear and, because of Conjecture 1, it corresponds to minimizing the argument in the two intervals (0, π) and (π, 2π) where matrix M (ν, •) is invertible. Then, the minimum can be selected as the minimum of the two minimizers.

Similar to the previous solution (19), (23), it can be formally established that controller (24) solves Problem 1. See [2, Thm 2] for details.

V. SIMULATIONS

In this section, we present the simulation results obtained with the control laws designed in Section IV and system (14) (equivalently ( 16)-( 18)) using a linearized model simulator. In these simulations, an orbit with a semi-major axis of 7011 km and eccentricity e = 0.4 will be considered for the leader vehicle. The simulations aim at stabilizing a periodic trajectory specified by a suitable selection of ξ ref in [START_REF] Schaub | Analytical mechanics of space systems[END_REF], which is free to evolve inside a tolerance box B. In [START_REF] Deaconu | Constrained periodic spacecraft relative motion using non-negative polynomials[END_REF], a method is introduced to obtain the value of ξ ref in [START_REF] Schaub | Analytical mechanics of space systems[END_REF]

where in (26) the first three components are meters and the last three are meters per second. The initial relative velocity has been selected to be zero to account for the fact that the starting point of our trajectory may be a holding point arising from a previous station keeping along the space mission. The initial leader true anomaly values ν 0 vary from 0 to 360 degrees in equally spaced intervals of 10 degrees. All our simulations have been performed from the initial true anomaly ν 0 up to a final true anomaly ν f = ν 0 +20π, namely ten orbital periods later.

Two performance indexes are considered: the fuel consumption J in (13) and the true anomaly ν B or angular time in terms of orbital periods (time elapsed when the follower first reaches the bounding box B). Figures 1 and2 represent respectively these two indexes fixing each one of the initial conditions in (26) and using the three control laws in Section IV, where:

-the orange line (-× -) refers to controller (19), (21) of Section IV-A, with the selection ν = π/2 in (19); -the blue line (-• -) refers to controller (19), (23) of Section IV-B, with the same firing interval ν = π/2; -the green line (--) refers to the response using controller (24) discussed in Section IV-C. Figure 1 shows the cost J for each control law when the leader's initial true anomaly ν 0 varies. From this figure it is possible so see that for every initial state ζ 0 and every control law, the minimum cost always appears in the proximities of 180 (deg), corresponding to the apogee of the orbit. It would be interesting therefore to perform the satellite maneuver when the leader's initial true anomaly ν 0 lies in the neighborhood of this value in order to minimize the amount of dispensed fuel. As expected, controller (24) always leads to the minimum value of J. 1 and2.

All Figures from 1 to 6 highlight that the speed of convergence to the bounding box B is relatively slow when using the control law of Section IV-A. In fact, from Table I, it is possible to observe that parameter ν B is much larger for this control law than the one obtained in the two other cases. Figures 3,4, 5 and 6 represent the trajectories of the follower satellite for an initial leader true anomaly of 180 (deg), and it can be seen that the trajectories corresponding to the first control law are much longer than the ones of the two other controls.

The periodic norm-minimizing control presents a slow convergence, while associated to an increased fuel consumption. On the other hand, an advantage of this controller is that the solution evolves along periodic motions, thereby being fault tolerant. Instead, the two bi-impulsive solutions show a reduced fuel consumption with a strong preference for the non-periodic one, always leading to a more than substantial fuel saving.

Both the bi-impulsive control laws discussed in Sections IV-B and IV-C behave better than the previous one in terms of convergence speed. In fact the reference trajectory is tracked after the firing period ν with zero error, by design.

VI. CONCLUSIONS

In this article, a new model, based on Floquet-Lyapunov theory, is developed in order to obtain a linear time-invariant free motion representation of the rendezvous problem. This problem is then recast as a stabilization problem for a periodic trajectory in a hybrid dynamical system framework. Two different control laws are developed and compared to a control scheme given in [START_REF] Deaconu | A two-impulse method for stabilizing the spacecraft relative motion with respect to a periodic trajectory[END_REF], which is re-interpreted in this hybrid context. Alternative trade-offs should be investigated that adjust ν within optimized fuel consumption under some constraints on the maximum distance reached during the transient.

  The tolerance box B is centered at point ζ f (expressed in the target's position frame used in (1)) and has positive and negative widths ζ tol in the three LVLH directions, where ζ f = [100 0 0] T m and ζ tol = [150 25 25] T m.

  ensuring that the corresponding periodic evolution in the LVLH coordinates is always contained in the tolerance box B. The resulting values of the reference state ξ ref correspond to: ξ ref = [7.68 17.68 87.78 33.04 -15.77 0] T , (25) where we emphasize that the last element is zero (a necessary and sufficient condition for periodic motion). For each one of the three control laws in Section IV, four different initial conditions ζ 0 for state ζ in (1) have been used, selected as: ζ 01 = [500 400 10 0 0 0] T , ζ 02 = [-200 100 200 0 0 0] T , ζ 03 = [100 -350 -20 0 0 0] T , ζ 04 = [320 0 -64 0 0 0] T ,

  Note that ζ 04 is chosen to lie in the orbital plane of the target (the second component is zero). The four initial conditions ζ 01 , ζ 02 , ζ 03 and ζ 04 in (26) correspond to four different initial distances from the tolerance box B: |ζ 01 | B = 512.9571 m |ζ 02 | B = 314.2451 m |ζ 03 | B = 325.0000 m |ζ 04 | B = 326.3372 m with the distance from the bounding box B being defined as: |ζ| B = |dz ζ tol (ζ 1..3 -ζ f )| , where ζ 1..3 = (x, y, z) contains the first three components of state ζ in (1) and dz ζ tol (x) is the decentralized vector dead-zone function with limits ζ tol .

Fig. 1 :

 1 Fig. 1: Comparison of the cost J among the three control laws for the initial states in (26).

Fig. 2 :

 2 Fig. 2: Comparison of the number of orbital periods ν B /2π among the three control laws for the initial states in (26).

Fig. 3 :Fig. 4 :

 34 Fig. 3: Evolution of ζ 1..3 with the three control laws from the initial state ζ 01 and ν 0 = 180 o .

Fig. 5 :Fig. 6 :

 56 Fig. 5: Evolution of ζ 1..3 with the three control laws from the initial state ζ 03 and ν 0 = 180 o .

TABLE I :

 I Minimum values of fuel consumption J (m/s) and angular time ν B (no. of orbits) for all the simulations presented in Figures

Work supported in part by ANR under project LimICoS, contract number 12 BS03 005 01, by the iCODE institute, research project of the Idex Paris-Saclay, and by grant OptHySYS funded by the University of Trento. Work supported in part by CNES under convention 117826.