Éric Lachance-Tremblay
email: eric.lachance-tremblay@euroviaqc.ca

Michel Vaillancourt

Daniel Perraton

Hervé Di Benedetto

H Di

Comparison of the moisture damage of bituminous binder coupled with glass and limestone substrate using pull-off test

Keywords: bituminous binder, glass, limestone, moisture damage, pull-off test liant bitumineux, verre, calcaire, dommages par l'humidité, essai en arrachement

In this paper, the moisture susceptibility of different bituminous binders with two substrates (glass and limestone) was investigated. To that end, the tensile strength of different combinations of bituminous binder-substrate bond was measured using a pull-off test. This test was adapted from the pneumatic adhesion tensile testing instrument (PATTI) test to improve repeatability. Samples were tested in dry condition and after a 7-day conditioning in hot water bath (60 °C). An analysis of variance (ANOVA) was performed on the test results. Overall, the results show that in dry condition, the pull-off strength is a function of the bituminous binder type rather than of the substrate type. After water conditioning, an increase in the pull-off strength was observed for the bituminous binder without polymers and coupled with glass substrate. This was associated with an increase in binder stiffness. For the limestone substrate, the effect of water conditioning was significant only for one type of binder.

Introduction

Durability is one of the most important properties when designing materials for road structures. Moisture damage is commonly recognized as one of the most severe distresses for pavement materials [START_REF] Baldi-Sevilla | Influence of bitumen and aggregate polarity on interfacial adhesion[END_REF][START_REF] Chaturabong | Effect of moisture on the cohesion of asphalt mastics and bonding with surface of aggregates[END_REF][START_REF] Hamedi | Evaluating the effect of asphalt binder modification using nanomaterials on the moisture damage of hot mix asphalt[END_REF]Mehrara and Khodaii 2013;[START_REF] Porot | Bitumen/aggregate affinity-Rilem Round Robin Test on Rolling Bottle Test[END_REF]. Moisture damage is defined as a degradation of material mechanical properties due to the presence of moisture in a liquid or vapor state [START_REF] Caro | Moisture susceptibility of asphalt mixtures, Part 1: mechanisms[END_REF]. For bituminous materials, the result of moisture damage is expressed as a loss of the adhesive bond between bituminous binder and aggregate and (or) loss of cohesion of the mastic [START_REF] Caro | Moisture susceptibility of asphalt mixtures, Part 1: mechanisms[END_REF][START_REF] Chaturabong | Effect of moisture on the cohesion of asphalt mastics and bonding with surface of aggregates[END_REF][START_REF] Hamedi | Evaluating the effect of asphalt binder modification using nanomaterials on the moisture damage of hot mix asphalt[END_REF].

Failure due to moisture damage in bituminous materials can be separated into three categories: (1) cohesive failure in the binder film, (2) adhesive failure at the binder-aggregate interface, and (3) cohesive failure of the aggregate [START_REF] Pinzon | Contribution de méthodes non destructives à l'évaluation de l'effet de l'eau sur les enrobés bitumineux[END_REF]. Cohesive failure in the binder film occurs when water reacts chemically with the binder and reduces its cohesion [START_REF] Kanitpong | Role of adhesion and thin film tackiness of asphalt binders in moisture damage of HMA[END_REF]. In the case of adhesive failure, it occurs when water weakens the bond between the binder and the aggregate surface. This causes a separation of the binder film from the aggregate [START_REF] Bahia | The effect of bitumen stiffness on the adhesive strength measured by the bitumen bond strength test[END_REF]). In the literature, it is generally agreed that moisture damage is partly cohesive and partly adhesive [START_REF] Bahia | The effect of bitumen stiffness on the adhesive strength measured by the bitumen bond strength test[END_REF][START_REF] Kanitpong | Role of adhesion and thin film tackiness of asphalt binders in moisture damage of HMA[END_REF]. For asphalt mixture, the moisture damage intensity is influenced by the characteristics of the materials (binder and aggregates) and the quality of the binder-aggregate adhesion. Adhesion between the binder and aggregates is a very complex phenomenon that cannot be explained by one theory, but rather, by a combination of multiple theories, including the following [START_REF] Caro | Moisture susceptibility of asphalt mixtures, Part 1: mechanisms[END_REF]Hefer and Little 2005;[START_REF] Kanitpong | Role of adhesion and thin film tackiness of asphalt binders in moisture damage of HMA[END_REF]):

• Mechanical theory: liquid asphalt binder penetrates the aggregate pores and irregularities which provide a mechanical interlock;

• Chemical reaction theory: adsorbed asphalt binder may chemically react with the aggregate surface components;

• Electrostatic theory: adhesion is the result of attraction of an electropositive and an electronegative surface;

• Interfacial energy theory: molecules at the surface of a material usually have an excess of energy compared to those from the bulk of the material. Due to this excess of energy, when two materials come into contact, the adhesion is energetically favorable because the system tends to reduce the interfacial free energy.

The effect of water on bituminous materials is described by six mechanisms: (1) detachment or debonding, (2) displacement, (3) dispersion, (4) film rupture and (or) microcracks, (5) desorption, and (6) spontaneous emulsification [START_REF] Caro | Moisture susceptibility of asphalt mixtures, Part 1: mechanisms[END_REF]). An extensive description of these mechanisms can be found in the literature [START_REF] Caro | Moisture susceptibility of asphalt mixtures, Part 1: mechanisms[END_REF][START_REF] Kringos | Raveling of asphaltic mixes due to water damage: computational identification of controlling parameters[END_REF][START_REF] Little | Chemical and mechanical mechanisms of moisture damage in hot mix asphalt pavements[END_REF]. Moreover, physical solicitations such as stress caused by traffic loading and (or) temperature variation, as well as water entrance in the pavement structure, can potentially impact bituminous mechanical properties (Mehrara and Khodaii 2013).

Multiple test methods are used to evaluate the moisture susceptibility of bituminous materials. The AASTHO T283 test is based on indirect tensile strength measurements on asphalt samples before and after water conditioning (AASHTO 2014). However, this test method has multiple disadvantages: it is time consuming, test results are highly variable due to saturation level differences, a visual examination of the stripping is subjective, and the conditioning procedure does not simulate repeated generations of pore pressure under loading [START_REF] Liang | Refine AASHTO T283 resistance of compacted bituminous mixture to moisture induced damage for Superpave[END_REF]. Another commonly used test is the Hamburg wheel track test, which is very popular for its simplicity, but does not directly relate with pavement field performances, and is susceptible to misinterpreting moisture susceptibility [START_REF] Buss | Evaluation of moisture susceptibility tests for warm mix asphalts[END_REF]. Moreover, those test methods measure bulk properties of the mixtures rather than any specific change in the adhesion and (or) cohesion properties of the binder-aggregate system [START_REF] Canestrari | Adhesive and cohesive properties of asphalt-aggregate systems subjected to moisture damage[END_REF]. Traditional testing methods cannot be considered to be sufficient predictors of the cause of stripping. A better understanding of moisture susceptibility can be achieved if the tests are performed directly on the binder-aggregate system [START_REF] Canestrari | Adhesive and cohesive properties of asphalt-aggregate systems subjected to moisture damage[END_REF]. To that end, a practical method to measure the adhesive and cohesive properties of a binder aggregate, the pneumatic adhesion tensile testing instrument (PATTI), was developed by [START_REF] Kanitpong | Role of adhesion and thin film tackiness of asphalt binders in moisture damage of HMA[END_REF] and modified to improve its repeatability [START_REF] Canestrari | Adhesive and cohesive properties of asphalt-aggregate systems subjected to moisture damage[END_REF][START_REF] Kanitpong | Role of adhesion and thin film tackiness of asphalt binders in moisture damage of HMA[END_REF][START_REF] Meng | Affinity of asphalt to mineral aggregate: pull-off test evaluation[END_REF]Youtcheff and Aurilio 1997). Recently, many authors have published papers evaluating bituminous material moisture susceptibility through binder-aggregate interface bond strength measurements [START_REF] Aashto ; Washington | Standard method of test for resistance of compacted asphalt mixtures to moisture-induced damage[END_REF][START_REF] Apeagyei | Influence of aggregate absorption and diffusion properties on moisture damage in asphalt mixtures[END_REF][START_REF] Chaturabong | Effect of moisture on the cohesion of asphalt mastics and bonding with surface of aggregates[END_REF][START_REF] Figueroa | Effect of water conditioning for extended periods on the properties of asphalt binders[END_REF][START_REF] Zhang | Moisture damage evaluation of aggregate-bitumen bonds with the respect of moisture absorption, tensile strength and failure surface[END_REF].

The work presented in this paper is part of a major research program looking at the use of glass aggregates in hot mix asphalt. The paper thus aims to evaluate the adhesion between different bituminous binders and aggregate types. To that end, a pull-off test was used to measure the bond strengths of different bituminous binder and aggregate combinations. A total of six bituminous binders and two substrates (limestone and glass) were studied. Pull-off tests were performed both in dry condition and after a 7-day hot water curing (60 °C).

Experimental

Tested materials

In this research, six binders and two substrates were used. The two substrates were glass and limestone plate. The limestone plate was sawn from boulder-size limestone aggregates and polished to ensure a smooth surface comparable to a glass plate. Six binders were chosen with different performance grades (PG), crude sources, and polymer contents (Table 1).

Pull-off strength test procedure

The pull-off test is often used to measure the adhesion strength between a binder and an aggregate substrate. Youtcheff and Aurilio (1997) successfully used the PATTI to evaluate the binderaggregate bond strength. This test was initially developed to measure the adhesion between a paint, coating or other adhesives with a rigid substrate [START_REF] Canestrari | Adhesive and cohesive properties of asphalt-aggregate systems subjected to moisture damage[END_REF]. For the bituminous binder, a pull-stub is glued with bituminous binder to an aggregate substrate. Then, a pulling tensile force is applied through a pneumatic system until failure occurs. The pull-off strength (kPa) of a binder-aggregate system is expressed as the force required to peel the pull-stub from the substrate surface. This test has the advantage of allowing water conditioning of samples prior to testing, as well as the interpretation of the failure mode (adhesive failure or cohesive failure). However, its main disadvantage is that the applied stress rate cannot be controlled [START_REF] Zhang | Moisture damage evaluation of aggregate-bitumen bonds with the respect of moisture absorption, tensile strength and failure surface[END_REF]. Moreover, the stress-strain relationship cannot be established.

In this project, the PATTI pull-off test method was adopted. To apply and measure the traction force, an automatic portable Elcometer pull-off adhesion device was used (Fig. 1). This device has a hydraulic pump that allows a continuous tensile force rate application.

Custom-made 50 mm diameter brass pull-stubs were used (Fig. 2). The 300 m edge along the perimeter allows for control of the binder film thickness, which has been shown to be important for repeatability [START_REF] Kanitpong | Role of adhesion and thin film tackiness of asphalt binders in moisture damage of HMA[END_REF]. Furthermore, four cuts were made along the perimeter edge to allow excess binder to flow out of the stub. For each binder-substrate combination, one set of samples was prepared for each of the two testing conditions (in dry condition and after 7-day conditioning in hot water bath at 60 °C, or wet condition). The temperature, as well as the water level of the water bath, were monitored daily to ensure that samples were always covered by water at the target temperature. It is important to note that for the tests done in wet condition, the samples were put in a 20 °C water bath after the 7-day hot water curing for thermal conditioning.

Sample preparation

For each bituminous binder, a specific heating temperature was chosen to reach the same viscosity (0.17 Pa•s) (Table 2). To prepare the samples, the pull-off stub, the binder, and the substrate were heated at a specific heating temperature. Then, a small quantity of binder (approximately 0.9 g) was dropped onto the substrate. A scale with a precision of 0.001 g was used to measure the amount of binder used. Immediately following that, the pull-off stub was placed on the binder such as to prevent air from being trapped between the stub and the binder. Samples were then cooled at room temperature for 24 h.

Pull-off strength test results and analysis

For all pull-off tests performed in this project, a direct tensile force rate of 370 ± 20 kN/s was applied. Three replicate tests were performed in dry condition, and three replicate tests after water conditioning (wet condition) for each binder-aggregate combination. Source 1 3.5

Test results for dry condition

The average pull-off strength test results, as well as the coefficient of variation (CV %) in dry condition for all binder-substrate combinations, are presented in Table 3. Regarding the glass substrate, it can be seen that all three PG64-22 binders have higher pull-off strengths than the PG58-28 binder and the polymer modified binders (PG70-28 and PG76-28). In the case of the limestone substrate, a similar trend is observed: PG64-22 # 1 and PG64-22 # 2 have higher pull-off strengths than the other binders.

To try to explain why the three PG64-22 binders have different pull-off strengths, their respective complex shear modulus (G*) was taken into consideration. [START_REF] Zhang | Moisture damage evaluation of aggregate-bitumen bonds with the respect of moisture absorption, tensile strength and failure surface[END_REF] recently showed that in dry condition, the tensile strength of a binder-aggregate bond is dominated by the binder itself rather than by the aggregate type. Moreover, it was shown that the higher the G* value, the higher the pull-off strength. Regarding the three different PG64-22 binders in this project, their G* at 64 °C are 1.30 kPa (PG64-22 # 1), 1.378 kPa (PG64-22 # 2), and 1.22 kPa (PG64-22 # 3), respectively. PG64-22 # 1 has the highest G* and the highest pulloff strength. PG64-22 # 3 has the lowest G* and the lowest pull-off strength, which is in agreement with what has been observed in other research studies.

Test results for wet condition

Each binder-aggregate combination was tested after water conditioning (7 days in a hot water bath at 60 °C). As a reminder, after water conditioning, samples were put in a water bath at 20 °C for thermal conditioning. Table 4 shows the average pull-off strength and the coefficients of variation (CV %) in wet condition. Significant differences can be seen when comparing the results of the two substrates; all pull-off strength values of the binder-limestone combinations are lower than those of the glass substrate.

A recent research study showed that the moisture damage of a given aggregate-binder bond is not solely affected by the substrate type (i.e., mineralogical composition), but also by the moisture content of the aggregate [START_REF] Zhang | Moisture damage evaluation of aggregate-bitumen bonds with the respect of moisture absorption, tensile strength and failure surface[END_REF]. The authors mentioned that for moisture-susceptible aggregates, the moisture content is a relevant factor to take into consideration. The authors tested the pull-off strength of one binder, combined with a substrate, while varying the moisture content. Their results showed a strong correlation with the moisture content and the tensile strength reduction, and the higher the moisture content, the higher the decrease in pull-off strength. Another research study also showed that the degradation of the binder-aggregate bond is controlled in part by the amount of water that can diffuse from the aggregate into the binder-aggregate interface [START_REF] Aashto ; Washington | Standard method of test for resistance of compacted asphalt mixtures to moisture-induced damage[END_REF]. [START_REF] Apeagyei | Influence of aggregate absorption and diffusion properties on moisture damage in asphalt mixtures[END_REF] identified the following characteristics for a susceptible aggregate-binder bond: high porosity, high moisture absorption, and high diffusion coefficient.

In the present research, the limestone absorption value is 0.5% and glass absorption is nil. In the case of limestone aggregate, water can reach the binder-aggregate interface not only through the sample boundaries, but can also diffuse through the limestone. In the case of glass, the water can reach the binder-aggregate interface only through the sample boundaries. Therefore, it can be assumed that a higher amount of water can reach the binderlimestone interface than in comparison with the glass aggregate. This could explain why limestone substrate pull-off strength values are lower than those of the glass substrate.

Variation of the pull-off strength after water conditioning

The variation of the pull-off strength after water conditioning was calculated with eq. ( 1) for all binder substrates (Fig. 3). A positive value means that there was an increase in the bond strength after water conditioning and a negative value means there was a decrease in the bond strength after water conditioning.

(1)

ͩ WET pull-off strength Ϫ DRY pull-off strength DRY pull-off strength ͪ × 100
In Fig. 3, it can be seen that all binder-glass combinations have a positive value. This means that binder-glass bond strengths are higher after water conditioning. However, for polymer modified binders (PG70-28 and PG76-28), variation values are quite low. For the neat binders (PG58-28 and PG64-22), variation values range from 19.2% to 43.9%. For the limestone-binder combination, only the PG58-28 binder has a positive difference value (PG76-28 variation value of 0.2% is considered non-significant). For the other binders, the variation values range from -10.3% to -41.5%.

Effect of water conditioning on failure surface

In addition to the pull-off strength variation, the effect of water conditioning can be evaluated on the failure surface. The failure surface photographs of all binder-substrate combinations tested in dry and wet conditions are presented in Table 5. The failure surface can be interpreted either as the results of a cohesive failure, of an adhesive failure, or of a combination of cohesiveadhesive failures. In Table 5, it is clear that all binder-substrate combinations tested in dry condition show a cohesive failure. PG70-28 and PG76-28 binders with limestone pictures are not presented because of a computer data loss. Regarding the test done in wet condition, all failure surfaces show a combination of cohesive-adhesive failures.

As proposed by [START_REF] Zhang | Moisture damage evaluation of aggregate-bitumen bonds with the respect of moisture absorption, tensile strength and failure surface[END_REF], the retained cohesive surface percentages were calculated using a gray analysis in an image software application. This allows the quantification of the level of cohesive and adhesive failure mode [START_REF] Zhang | Moisture damage evaluation of aggregate-bitumen bonds with the respect of moisture absorption, tensile strength and failure surface[END_REF]. Figure 4 shows the calculated retained cohesive surface percentage for all binder-substrate combinations tested in wet condition. For the glass substrate, the retained percentage values for polymer modified binder (PG-70-28 and PG76-28) are the highest (over 90%). This means that the surface failure in wet condition was primarily cohesive. This is also the case for PG58-28 and PG64-22 # 3, where the retained cohesive values are around 80%. Looking at PG64-22 # 1 and PG64-22 # 2, both had very low retained cohesive values (under 30%), which means that the principal mode of failure was adhesive. For the limestone substrate, the principal mode of fail- ure in wet condition was cohesive (i.e., retained cohesive values higher than 70%), except for PG64-22 # 2. In this case, the retained cohesive value was very low (approximately 35%).

Measurement of the stress-strain evolution

To explain the increase in pull-off strength observed after water conditioning, the pull-off test apparatus was modified to allow a measurement of the displacement. A plastic head was fabricated to receive two of 12.5 mm LVDTs (Fig. 5). The displacement measurements were done directly on the pull-stub. For each test, the average of the displacement measurements of the two LVDT was considered. Figure 6 presents an example of the relation between displacement and time. There are two distinct displacement evolutions. The rate of increase of the displacement (i.e., the slope of a linear regression) for phase 1 is significantly lower than for phase 2. The transition between phase 1 and phase 2 is characterized by an inflection point. Because the displacement measurements were carried out from a separate unit from the pull-off device, we could not establish the exact moment at which failure occurs. Therefore, the assumption that failure occurs at the transition between phase 1 and phase 2 was made. From the displacement evolution slope of phase 1, the stress-strain relation was calculated.

Figure 7 shows the stress-strain relation for the PG58-28 binder tested with the glass substrate both in dry and wet conditions. Further, the strain amplitude for a stress of 0.8 kPa is indicated. In this case, the strain values for 0.8 kPa in wet condition are lower than those in dry condition. We can assume that PG58-28 is more rigid in wet condition than in dry condition. This observation can be related with the results previously presented; the pull-off strength of the PG58-28 binder with glass substrate is higher wet condition than in dry condition (Fig. 3) even though stripping occurred (Table 5). Therefore, there is an increase in the binder stiffness due to hot water conditioning.

The displacement measurements were also done on the PG64-22 # 2 binder with glass and limestone substrate in dry and wet conditions. There is also an increase in the binder stiffness for the glass substrate in wet condition as compared to the dry condition, as shown in Fig. 8. The opposite trend is observed for the limestone substrate. There is a decrease in the binder stiffness in wet condition, as compared with the dry condition. This is in agreement with the test results previously presented. For the glass substrate, the pull-off strengths in wet condition are higher than in dry condition. For the limestone substrate, the pull-off strengths in wet condition are lower than in dry condition.

Statistical analysis of the test results

To verify the effect of the substrate type as well as the water conditioning on the pull-off strength, a statistical analysis was performed. Using a one-way analysis of variance (ANOVA) with a 95% confidence level, three hypotheses were verified: (1) the type of substrate has an effect on the pull-off strength in dry condition, (2) wet conditioning has an effect on pull-off strength for glass substrate, and (3) wet conditioning has an effect on pull-off strength for limestone substrate. The ANOVA is generally presented as a statistical test to compare the means of two or more populations. As such, it is possible to determine if there are statistically significant differences between two populations.

Prior to performing the ANOVA analysis, two basic hypotheses must be verified: (1) each population is normally distributed, and (2) the variances of the populations are equal [START_REF] Bourges | Modélisation statistiques des données[END_REF]. In this case, because the size of the population is small (n = 3 for each binder-substrate combination), it could be assumed that each population was normally distributed. Moreover, small differences from normality of a population do not significantly affect the ANOVA analysis [START_REF] Ryan | Moisture sensitivity of asphalt binders: evaluation and modeling of the Pneumatic Adhesion test results[END_REF]. The equality of the variances was verified with the Bartlett test for each of the three hypotheses. One Bartlett test gave a negative result, meaning that this combination cannot be included in the ANOVA analysis (PG64-22 # 1 for dry test). The results of the ANOVA analysis performed are presented in Table 6, where it is indicated whether each effect analyzed is significant; as well, the p-value is shown. For a 95% confidence level, the error margin is 5% (0.05). If the p-value is lower than 0.05, it means that there are significant differences between the population analyzed, and the hypothesis is thus verified.

With respect to the effect of the type of substrate on the pull-off strength, no significant differences between glass substrate and limestone substrate for all tested binders were observed. It is fair to say that in dry condition, the pull-off strength is independent of the substrate type, and thus that the bonding strength is rather affected by the binder type. This observation confirms what has been observed in other research works [START_REF] Zhang | Moisture damage evaluation of aggregate-bitumen bonds with the respect of moisture absorption, tensile strength and failure surface[END_REF].

Regarding the glass substrate, the ANOVA analysis showed that water conditioning has a significant effect on PG58-28 and on the three PG64-22 binders (all four being neat binders). Both binders modified with polymers (PG70-28 and PG76-28) with glass were not affected by water conditioning, according to the ANOVA analysis. For the binder-limestone substrate combination, the statistical analysis showed that the water conditioning effect is significant for PG58-28 and PG64-22 # 1. In this case, the effect of water conditioning for PG58-28 is expressed as an increase in the pull-off strength and a decrease for PG64-22 # 1.

Conclusion

In this research, the pull-off strength of various binders (six) with glass and limestone substrates were measured. All bindersubstrate combinations were tested in dry condition and after water conditioning (7 days in a 60°water bath). From the analysis conducted on the test results, the following conclusions were drawn:

• In dry condition, no significant differences were observed between the pull-off strength of glass and limestone substrates. This suggests that in dry condition, the pull-off strength is binder-dependent rather than substrate-dependent. Moreover, all tested binder-substrate combinations in dry condition showed cohesive failure;

• After water conditioning, the pull-off strength values are higher for the glass substrate combination than for the lime- • Significant differences were observed for the glass substrate between the pull-off strength of the dry condition versus the wet condition for the neat binders (PG58-28 and all three PG64-22). There was an increase in the pull-off strength after water conditioning. On the other hand, the effect of water conditioning was not significant for polymer-modified binders with glass substrate;

• For the limestone substrate, two binders were significantly affected by the water conditioning. In those cases, the effect of water is expressed as an increase in the pull-off strength for one binder (PG58-28) and a decrease for the other one (PG64-22 # 1).

Fig. 1 .

 1 Fig. 1. Pull-off test adhesion device used: automatic portable Elcometer adhesion tester. [Colour online.]

Table 5 .

 5 Failure surface pictures of all tested binder-substrate combinations in dry condition and after water conditioning. /A: Non-available due to data loss.

Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .

 45678 Fig. 4. Retained cohesive surface percentage of all binder-substrate combinations tested after water conditioning. [Colour online.]

Table 1 .

 1 Performance grade (PG), crude source, and polymer content of the tested bituminous binder.

		Crude	Polymer
	PG grade	source	content (%)
	PG58-28	Source 1	0
	PG64-22	Source 1	0
	PG64-22	Source 2	0
	PG64-22	Source 3	0
	PG70-28	Source 1	2.2
	PG76-28

Table 2 .

 2 Heating temperature used to prepare each binder-substrate sample.

		Heating temperature for
	Binder	170 mPa•s viscosity (°C)
	PG58-28	151
	PG64-22 # 1	160
	PG64-22 # 2	160
	PG64-22 # 3	158
	PG70-28	170
	PG76-28	170

Table 3 .

 3 Average pull-off strength and coefficient of variation for all binder-aggregate combinations tested in dry condition (three replicated per combination).

		Glass		Limestone
		Avg. pull-off		Avg. pull-off
	Binder type	strength (kPa)	CV (%)	strength (kPa)	CV (%)
	PG58-28	1145	9.0	1016	10.4
	PG64-22 # 1	1907	10.0	1541	2.0
	PG64-22 # 2	2198	6.8	2045	3.3
	PG64-22 # 3	1614	4.7	1347	13.8
	PG70-28	1476	6.2	1650	11.2
	PG76-28	1291	7.0	1283	1.5

Table 4 .

 4 Average pull-off strength and coefficient of variation for all binder-aggregate combinations tested in wet condition (three replicates per combination).

		Glass			Limestone
		Avg. pull-off			Avg. pull-off
	Binder type	strength (kPa)	CV (%)	strength (kPa)	CV (%)
	PG58-28	1648	3.0		1295	6.5
	PG64-22 # 1	2337	9.1		1382	3.4
	PG64-22 # 2	2621	2.0		1813	4.9
	PG64-22 # 3	2009	5.8		972	1.0
	PG70-28	1552	9.6		965	14.0
	PG76-28	1450	3.1		1285	5.6
	60%
	43.9%
	20% 40%	27.4%	22.6%	19.2%	24.5%	12.3%
						5.1%	0.2%
	0%
	-20%		-10.3%	-11.3%
	-40%					-27.8%
						-41.5%
	-60%
		PG58-28 PG64-22 # 1 PG64-22 # 2 PG64-22 # 3 PG70-28	PG76-28

Fig. 3. Variation of the pull-off strength after wet conditioning for all binder-substrate combinations. [Colour online.] Ratio between wet and dry pull-off strength (%) Binder type Glass Limestone Published by NRC Research Press

Table 6 .

 6 Results of the ANOVA analysis performed at a 95% confidence level. . A possible explanation for this observation is the difference in terms of moisture absorption. Glass has no absorption, while limestone does;

		Type of substrate for	Wet conditioning for	Wet conditioning for
		dry tests		glass substrate		limestone substrate
	Binder	Significant	p-value	Significant	p-value	Significant	p-value
	PG58-28	NO	0.1688	YES	0.0001	YES	0.0337
	PG64-22 # 1	Bartlett test negative	YES	0.0370	YES	0.0028
	PG64-22 # 2	NO	0.2806	YES	0.0349	NO	0.0537
	PG64-22 # 3	NO	0.0982	YES	0.0178	NO	0.1029
	PG70-28	NO	0.2379	NO	0.4942	NO	0.0513
	PG76-28	NO	0.9151	NO	0.0522	NO	0.9752


Long Hu

Guillaume Olive

Minimal null control time of some 1D hyperbolic balance laws with constant coecients and properties of related kernel equations

Keywords: Hyperbolic systems, Minimal control time, Backstepping method, Nonlinear recurrence relation 2010 Mathematics Subject Classication. 35L40, 93B05, 93D15, 11B

In this work, we study the null controllability by one-sided boundary controls of onedimensional hyperbolic balance laws with constant coecients. Our rst result shows that, when the system has only one negative or positive speed, the minimal null control time of such systems depends on some orthogonality conditions for a particular sequence. This sequence is explicit in function of the coecients of the system but it is dened by a nonlinear recurrence relation. Our second result then completes the previous one by giving explicit bounds on the number of orthogonality conditions that have to be checked in two nontrivial situations. The proofs rely on a careful analysis of the so-called kernel equations associated with the system, including a new well-posedness result. Our results are also valid for the nite-time stabilization property.

1 Introduction and main results

Problem description

In this paper, we are interested in the null controllability properties of a class of one-dimensional (1D) hyperbolic system with constant coecients (see e.g. [BC16, Chap. 1] for applications).

The equations describing such phenomenons are ∂y ∂t (t, x) + Λ ∂y ∂x (t, x) = M y(t, x).

(1a) Above, t ∈ (0, T ) is the time variable, T > 0, x ∈ (0, 1) is the space variable and the state is y : (0, T ) × (0, 1) → R n (n ≥ 2). The matrix Λ ∈ R n×n will always be assumed diagonal Λ = diag (λ 1 , . . . , λ n ), with m ≥ 1 negative speeds and p ≥ 1 positive speeds (m + p = n):

λ 1 < • • • < λ m < 0 < λ m+1 < • • • < λ m+p . (1b)
* School of Mathematics, Shandong University, Jinan, Shandong 250100, China. E-mail: hul@sdu.edu.cn

Faculty of Mathematics and Computer Science, Jagiellonian University, ul. ojasiewicza 6, 30-348 Kraków,

Poland. E-mail: math.golive@gmail.com or guillaume.olive@uj.edu.pl

The matrix M ∈ R n×n couples the equations of the system inside the domain and will be called the internal coupling matrix. We will consider an initial condition at time t = 0: y(0, x) = y 0 (x).

(1c)

Let us now discuss the boundary conditions. The structure of Λ induces a natural splitting of the state into components corresponding to negative and positive speeds, denoted respectively by y -and y + . For the above system to be well-posed in (0, T ) × (0, 1) with an initial condition at time t = 0, we then need to add boundary conditions at x = 1 for y -and at x = 0 for y + . We will consider the following type of boundary conditions, motivated by its numerous applications (see again [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]):

y -(t, 1) = u(t), y + (t, 0) = Qy -(t, 0).

(1d)

The function u is the so-called control, it will be at our disposal. It only acts on one part of the boundary and, on the other part of the boundary, the equations are coupled by the matrix Q ∈ R p×m . This matrix will be called the boundary coupling matrix. In what follows, (1a), (1c) and (1d) together will be referred to as system (1).

We recall that system (1) is well-posed in the following functional setting: for every T > 0, y 0 ∈ L 2 (0, 1) n and u ∈ L 2 (0, T ) m , there exists a unique solution y to system (1) with y ∈ C 0 ([0, T ]; L 2 (0, 1) n ) ∩ C 0 ([0, 1]; L 2 (0, T ) n ).

By solution we mean solution along the characteristics. We refer for instance to [START_REF] Coron | Boundary stabilization in nite time of onedimensional linear hyperbolic balance laws with coecients depending on time and space[END_REF] for a proof of this well-posedness result in such a setting (see also [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF] Appendix A] when u = 0).

The regularity C 0 ([0, T ]; L 2 (0, 1) n ) of the solution allows us to consider control problems in the space L 2 (0, 1) n : Denition 1.1. Let T > 0 be xed. We say that system (1) is null controllable in time T if, for every y 0 ∈ L 2 (0, 1) n , there exists u ∈ L 2 (0, T ) m such that the corresponding solution y to system (1) satises y(T, •) = 0.

Since controllability in time T 1 implies controllability in any time T 2 ≥ T 1 , it is natural to try to nd the smallest possible control time, the so-called minimal control time.

Denition 1.2. For any Λ, M and Q as above, we denote by T inf (Λ, M, Q) ∈ [0, +∞] the minimal null control time of system (1), that is

T inf (Λ, M, Q) = inf {T > 0 | System (1) is null controllable in time T } .
(2)

The time T inf (Λ, M, Q) is named minimal null control time according to the current literature, despite it is not always a minimal element of the set. We keep this naming here, but we use the notation with the inf to avoid eventual confusions. The goal of this article is to characterize T inf (Λ, M, Q) in some new situations.

In order to state our results and those of the literature, we need to introduce the following times:

T i = 1 -λ i if i ≤ m, T i = 1 λ i if i ≥ m + 1.
The time T i is the time needed for the controllability of a single equation (the transport equation)

with speed λ i . Note that the assumption (1b) implies in particular the following order relation:

T 1 ≤ • • • ≤ T m , T n ≤ • • • ≤ T m+1 .
(3)

Literature

Here, we briey describe the results of the literature about the null controllability of system (1).

All the results below are also valid for space-dependent versions of this system.

It was rst proved in the celebrated survey [START_REF] Russell | Controllability and stabilizability theory for linear partial dierential equations: recent progress and open questions[END_REF] that system (1) is null controllable in any time T ≥ T m+1 + T m . A strength of this result is that it is valid for any M and Q.

However, it was also observed in that paper that the minimal control time can be smaller than T m+1 + T m . Finding the minimal control time even in the simpler case M = 0 was then left as an open problem.

For M = 0, the minimal null control time was eventually found in [START_REF] Weck | A remark on controllability for symmetric hyperbolic systems in one space dimension[END_REF]. The author gave an explicit expression of this time in terms of some indices related to Q.

Finding the minimal null control time for arbitrary M and Q is still an open challenging problem. There has been a recent resurgence on the characterization of this time. A rst result in this direction was obtained in [START_REF] Coron | Optimal time for the controllability of linear hyperbolic systems in one-dimensional space[END_REF] and then completed in [START_REF] Coron | Null-controllability of linear hyperbolic systems in one dimensional space[END_REF]. Therein, the authors introduced a class of boundary coupling matrices Q for which they showed that the minimal null control time is smaller than T m+1 + T m , whatever the internal coupling matrix M is.

For full row rank boundary coupling matrices (rank Q = p), the minimal null control time was found in [START_REF] Hu | Minimal time for the exact controllability of one-dimensional rst-order linear hyperbolic systems by one-sided boundary controls[END_REF]. In this case, it has been shown that this time is the same as for the system without internal coupling (M = 0).

For systems of n = 2 equations, the minimal null control time was found in [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF] and [START_REF] Hu | Null controllability and nite-time stabilization in minimal time of one-dimensional rst-order 2 × 2 linear hyperbolic systems[END_REF]. In particular, it has been shown in the second reference that this time depends on the internal coupling matrix M when the boundary coupling matrix is zero. This is a feature that was not observed nor highlighted in all the other works and that the results of the present paper will also share.

Finally, the smallest and largest values that the minimal null control time can take with respect to the internal coupling matrix M were found in [START_REF] Hu | Equivalent one-dimensional rst-order linear hyperbolic systems and range of the minimal null control time with respect to the internal coupling matrix[END_REF].

Other related works include for instance [Cor+21; CN21b; MA22] about time-dependent versions of system (1) and [Li10; LR10; Hu15; CN20; CN22] for quasilinear versions of this system (in a C 1 framework).

Notations and important denitions

To state the main results of this article we rst need to introduce some notations.

All along this article, we denote by A T the transpose of a matrix A. For any integer N ≥ 1, R N ×N 0 denotes the set of matrices of size N ×N with diagonal entries all equal to zero. The matrix Id N denotes the N × N identity matrix. A matrix (or matrix-valued function) of size N 1 × N 2 will simply be denoted using the corresponding lowercase letter when N 2 = 1 (e.g. Q ∈ R p×m will be denoted by q ∈ R p when m = 1). The inner product of two vectors v 1 , v 2 ∈ R n-1 will be denoted by v 1 , v 2 .

Let us now introduce a sequence that will play a key role throughout this paper. For any i ∈ {1, . . . , n}, we rst dene the following quantities.

For every r, j ∈ {1, . . . , n}, we denote by

α rj = m rj λ i -λ j if j = i, α ri = m ri λ i . A = (α rj ) r,j =i , D = diag λ i + λ j λ i -λ j j =i ,
and let ψ 0 , w ∈ R n-1 be the vectors dened by

ψ 0 = (α ij ) j =i , v = - 1 2 α ji j =i
.

Let (ψ r ) r∈N ⊂ R n-1 be the sequence dened by ψ 0 and

ψ 1 = A T ψ 0 , ψ r = A T ψ r-1 + D r-2 =0 v, ψ r-2-ψ , ∀r ≥ 2. (4)
Finally, for q ∈ R n-1 , let b ∈ R n-1 be the nonzero vector dened by b = (β j ) j =i , β = -Λ 1 q .

(5)

All the above quantities depend on the index i but we omit it for clarity.

With the previous notations, we introduce the following sets. For any k ∈ {2, . . . , n + 1}, we denote by C k the set of (M, q) ∈ R n×n 0 × R n-1 such that, for every 2 ≤ i < k, we have q i-1 = 0 and b, ψ r = 0, ∀r ∈ N.

Here, we use the convention that C 2 = R n×n 0 × R n-1 . Additionally, we will denote by C n+2 = ∅.

Note that we then have

C 2 ⊃ C 3 ⊃ • • • ⊃ C n+1 ⊃ C n+2 .

Main results and comments

The rst result of this article is the following characterization of the minimal null control time in the case of one negative speed.

Theorem 1.3. Assume that m = 1. Let us denote by

τ i = max {T 1 + T i , T 2 } if 2 ≤ i ≤ n, τ n+1 = max {T 1 , T 2 } , (7)
(we have

τ 2 ≥ τ 3 ≥ • • • ≥ τ n+1 from (3)).
Then, for any M ∈ R n×n 0 and q ∈ R n-1 , we have:

1. T inf (Λ, M, q) ∈ {τ 2 , . . . , τ n+1 }.
Moreover, the inmum is always reached (in (2)).

2. For any k ∈ {2, . . . , n + 1}, we have

T inf (Λ, M, q) = τ k ⇐⇒ (M, q) ∈ C k \ C k+1 .
We recall that the set C k is dened at the end of Section 1.3. Remark 1.5. Theorem 1.3 remains valid if we replace everywhere in this article the null controllability property by the nite-time stabilization property by L 2 bounded feedbacks (that is when the control u is looked under the more particular form u(t) = 1 0 r(ξ) T y(t, ξ) dξ with r ∈ L 2 (0, 1) n ). This easily follows from the proof below.

Even though the set C k is explicit in function of the parameters of the system, the orthogonality conditions (6) that dene this set can be dicult to study in general because the sequence (ψ r ) r∈N is dened by a nonlinear recurrence relation. Note however that there always exists an integer N ≥ 1 such that

ψ r ∈ span {ψ s | s ≤ N -1} , ∀r ≥ N, so that b, ψ r = 0, ∀r ∈ N ⇐⇒ b, ψ r = 0, ∀r ∈ {0, . . . , N -1} . (8)
This means that (6) only needs to be checked for the rst N values of r. However, such a N depends on the sequence and it is a priori unknown, so that, in practice, we do not know when we have to stop checking the orthogonality conditions. Our second result provides information on this issue in two particular cases:

Theorem 1.6. Let i ∈ {1, . . . , n} be xed. Dene

N ψ = min {N ≥ 1 | (8) holds} .
We have N ψ ≤ 3 for n = 3 and N ψ ≤ 6 for n = 4.

Remark 1.7. It would be interesting to nd a bound of N ψ with respect to n for arbitrary n.

Our main results can for instance be combined to deduce a very explicit characterization of the minimal null control time in the following particular case:

Corollary 1.8. Assume that m = 1 and p = 2. Then, for any M ∈ R 3×3 0 and q ∈ R 2 , we have:

1. T inf (Λ, M, q) = max {T 1 , T 2 } if, and only if, (M, q) satises q = 0, m 21 = m 31 = 0.

2. T inf (Λ, M, q) = max {T 1 + T 3 , T 2 } if, and only if, (M, q) satises q = 0, m 21 = m 23 = 0, m 31 = 0, or q 1 = 0, q 2 = 0 and

  m21 = m 23 = 0 or      m 21 = rsm 23 , m 31 = r 2 sm 13 , m 32 = -rm 12 ,    ,
where r = -λ3q2 λ1 and s = λ2-λ1 λ2-λ3 .

3. In all the other situations, T inf (Λ, M, q) = T 1 + T 2 .

For p = 3, there is no simple presentation as for p = 2, even though the orthogonality conditions are explicit (see also Remark 1.10 below) and we know that we only have to check the rst six conditions. Therefore, we only give a nontrivial example:

Example 1.9. Let σ ∈ R \ {0} be arbitrary and consider system (1) with

Λ = diag (-1, 1, 2, 3), M =     0 -3 1/(2σ) 0 2 0 0 -2 3σ 0 0 -σ 0 2 1/(2σ) 0     , q =   0 0 1/3   .
For i = 2, we have b = 1 0 -1 T and

ψ 0 =   1 0 1   , ψ 1 = - 1 σ   0 1 0   , ψ 2 = - 3 2 ψ 0 , ψ 3 = -3ψ 1 , ψ 4 = 9 2 ψ 0 , ψ 5 = 45 4 ψ 1 .
For i = 3, we have the same b and

ψ 0 = σ   1 0 1   , ψ 1 = -σ   0 1 0   , ψ 2 = - 3 4 ψ 0 , ψ 3 = - 3 2 ψ 1 , ψ 4 = 9 8 ψ 0 , ψ 5 = 45 16 ψ 1 .
From Theorems 1.3 and 1.6, we deduce that T inf (Λ, M, q) = τ 4 = 4/3.

Remark 1.10. For arbitrary n, we will see that the orthogonality conditions (6) are satised if one of the following three conditions holds:

(C1) Kal (A, v) T ψ 0 = Kal (A, b) T ψ 0 = 0, where Kal (A, h) = (h|Ah| • • • |A n-2 h) ∈ R (n-1)×(n-1)
denotes the Kalman matrix of (A, h), for any h ∈ R n-1 .

(C2) There exists ∅ = J {1, . . . , n -1} such that ψ 0 j = a rj = b r = 0 for every j ∈ J and r ∈ J.

(C3) b, ψ 0 = 0 and there exists j 0 ∈ {1, . . . , n -1} such that b j0 = 0 and rank ∆ j0 = 1, where

∆ j0 = (D -d j0 )ψ 0 A T e j0 A T ψ 0 v j0 ψ 0 -v, ψ 0 e j0 ∈ R 2(n-1)×2 ,
where d j0 is the j 0 -th diagonal entry of D and e j0 is the j 0 -th canonical vector of R n-1 .

We will also see that, for n = 3 (resp. n = 4), it is necessary that one of the conditions (C1), (C2) (resp. (C1), (C2), (C3)) holds (it is however preferable to use Theorem 1.6 in these cases).

The rest of this article is organized as follows. In Section 2, we use the equivalence between the controllability of system (1) and that of a simpler system to obtain a characterization of this property in terms of some orthogonality conditions for the derivatives at the origin of any solution to the so-called kernel equations. In Section 3, we compute these derivatives for a particular solution and we obtain a general formula for this solution. In Section 4, we study the orthogonality conditions associated with the previous solution and we deduce our main results. In Section 5, we supplement ours results by studying the structure of the solution associated with the orthogonality conditions. Finally, in Appendix A, we give a simple proof of the characterization of the controllability properties for the equivalent system and, in Appendix B, we prove the existence of a solution to the kernel equations by a new approach.

2 An equivalent system and the kernel equations

The rst step in the proof of our results is to consider a system which is equivalent to our initial system from a control point of view.

Lemma 2.1. For any T > 0, system (1) is null controllable in time T if, and only if, so is the system

       ∂ ỹ ∂t (t, x) + Λ ∂ ỹ ∂x (t, x) = F (x)ỹ -(t, 0), ỹ-(t, 1) = ũ(t), ỹ+ (t, 0) = Qỹ -(t, 0), ỹ(0, x) = ỹ0 (x), (9)
where F ∈ C 0 ([0, 1]) n×m is dened by

F (x) = -K(x, 0)Λ Id m Q , (10)
and K ∈ C 0 (T ) n×n is any solution to

   Λ ∂K ∂x (x, ξ) + ∂K ∂ξ (x, ξ)Λ + K(x, ξ)M = 0, ΛK(x, x) -K(x, x)Λ = M, (11)
in the closure of the triangle

T = (x, ξ) ∈ R 2 0 < ξ < x < 1 .
By solution to (11) we mean solution along the characteristics. This result is by now wellknown: it consists in using the invertible spatial transformation ỹ(t, x) = y(t, x) -

x 0 K(x, ξ)y(t, ξ) dξ,
in order to transform a solution of system (1) into a solution of system (9) (see e.g. [Hu+19, Section 2.2]). This idea is the starting point of the so-called backstepping method for partial dierential equations and introduced more specically for hyperbolic systems of two equations in [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF]. Equations (11) are thus called the kernel equations. The dicult point is not so much the result of the previous lemma but rather to prove that (11) actually has at least a solution.

It follows from the results of [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF] that there are many solutions to the kernel equations (11) in T .

Remark 2.2. The choice of solution to the kernel equations (11) does not aect the controllability properties of system (1) because all the corresponding systems (9)-( 10) are equivalent from a control point of view. Now, two problems naturally arise:

1. Can we characterize the null controllability of the equivalent system (9) in function of Λ, Q and F ? 2. If so, can this characterization be presented explicitly in terms of Λ, Q and M ?

These problems are still open in general. One particular case where the rst problem has been completely solved is the case m = 1 (one negative speed). This was done in [HO21b, Section 5].

Theorem 2.3. Assume that m = 1. Then, system (9) is null controllable in time T if, and only if,

T ≥ max 2≤i≤n {T 1 + T * i , T 2 } ,
where

T * i = T i if q i-1 = 0, T i (1 -(f i )) if q i-1 = 0,
where

(f i ) = sup I(f i ) with I(f i ) = { ∈ (0, 1) | f i = 0 in (0, )}, if I(f i ) = ∅, and (f i ) = 0 otherwise.
The second problem could not be solved though because, even if the conditions for f are explicit, the map M -→ f (dened by ( 11)-( 10), with m = 1) is quite complicated. It was left as an open problem in the same paper. This is precisely where our main results step in.

From the above result of [START_REF] Hu | Null controllability and nite-time stabilization in minimal time of one-dimensional rst-order 2 × 2 linear hyperbolic systems[END_REF] we see that the values at x = 0 of f and its derivatives (assuming it is smooth) can aect the minimal null control time T inf (Λ, -, q, f ) of the system

(T * i = T i if f (N ) i
(0) = 0 for some N ≥ 0). Our idea is to show that these values in fact completely characterize T inf (Λ, -, q, f ) because M is constant and that we can explicitly relate them to M thanks to the kernel equations.

It is clear that T inf (Λ, -, q, f ) is solely characterized by f (0), f (0), f (0), etc. if we have f is analytic in a neighborhood of [0, 1). ( 12
)
Under such an assumption, Theorem 2.3 takes a simpler form:

Corollary 2.4. Assume that m = 1, let q ∈ R n-1 be given and assume (12). Then, we have:

1. T inf (Λ, -, q, f ) ∈ {τ 2 , . . . , τ n+1 } (recall (7)).
Moreover, the inmum is always reached.

2. For any k ∈ {2, . . . , n + 1}, we have

T inf (Λ, -, q, f ) = τ k ⇐⇒ (q, f ) ∈ S k \ S k+1 ,
where, for every k ∈ {2, . . . , n + 1}, S k is the set of (q, f ) ∈ R n-1 × C 0 ([0, 1]) n such that

q i-1 = f i = 0 for every 2 ≤ i < k (we use the convention that S 2 = R n-1 × C 0 ([0, 1]) n ),
and

S n+2 = ∅.
This result is immediate from the previous theorem but we give a simple and direct proof in Appendix A. Note that it is the complete analogue of Theorem 1.3 for system (9). By Lemma 2.1, the minimal null control time for the initial system (1) is thus also completely determined by the sets S k . However, apart from S n+1 , these sets are not explicit in terms of M , which is unsatisfactory.

Assumption ( 12) is indeed satised in our framework because we can always nd an analytic solution to the kernel equations since M is constant. More precisely, we have the following result:

Theorem 2.5. Let m, p ≥ 1 be arbitrary. Assume that

M ∈ R n×n 0 . For any δ ∈ R with δ = 1, there exists a unique K ∈ C ∞ (R 2 ) n×n that satises (11) for every (x, ξ) ∈ R 2 and the condition diag K(x, δx) = 0, ∀x ∈ R. (13)
Moreover, it satises the estimate

∀ bounded V ⊂ R 2 , ∃C > 0, K C s (V ) n×n ≤ C s , ∀s ∈ N. (14)
The kernel equations (11) have been extensively studied in the literature (see e.g. [Cor+13;

DVK13; HD15; Hu+16; Hu+19; CN19]) but Theorem 2.5 does not follow from the results contained therein. The most important dierence is that, in Theorem 2.5, the kernel exists on a larger set than the triangle T . This is crucial since we want x -→ f (x) = -K(x, 0)Λ 1 q T to be analytic in an interval of the form (-ε, 1), ε > 0, which does not lie entirely in T . This yields nontrivial issues in the standard xed point approach, notably because we now have to consider points that are on the other side of the diagonal (x, x), that is the condition imposed in the kernel equations at (x, x) cannot be consider as a boundary condition anymore. We have developed in Appendix B a new approach to solve the kernel equations that encompasses in particular the proof of Theorem 2.5.

Remark 2.6. Estimate ( 14) and Taylor's theorem show that the solution K to (11)-( 13) is in fact a power series.

As a consequence of Theorem 2.5, we see that, if

q i-1 = 0, then f = -K(•, 0)Λ 1 q T satises f i = 0 in (0, 1) if, and only if, b, ∂ r k c ∂x r (0, 0) = 0, ∀r ∈ N, (15)
where b ∈ R n-1 is dened in (5) and k c = (k ij ) j =i . It remains to relate the derivatives of the kernel at the origin with M . This is the purpose of the next section. This will be done only for a very well chosen particular solution to the kernel equations (i.e. for one δ = 1) but this will be enough for our purposes as already underlined in Remark 2.2.

Remark 2.7. We emphasize that, in all the sections below and unless specically mentioned, the number of negative speeds m is arbitrary (the orthogonality conditions (15) are studied for

any nonzero b ∈ R n-1 ).
3 The derivatives of the kernel at the origin

Normalization of the equations

Let us rst observe that a feature of the kernel equations (11)-( 13) is that it does not couple dierent rows of K:

     λ i ∂k ij ∂x (x, ξ) + ∂k ij ∂ξ (x, ξ)λ j + n r=1 k ir (x, ξ)m rj = 0, λ i k ij (x, x) -k ij (x, x)λ j = m ij (j = i), k ii (x, δx) = 0. (16)
Therefore, all along Section 3, i ∈ {1, . . . , n} is xed and we will drop the dependence on i for clarity.

Let us now introduce some important notations. Some of them have already been introduced in Section 1.3 but they are recalled here for the sake of the presentation.

It is convenient to normalize the kernel equations by λ i -λ j for j = i and by λ i otherwise.

The kernel equations (16) become

     µ j ∂k j ∂x (x, ξ) + ∂k j ∂ξ (x, ξ)ν j + n r=1 k r (x, ξ)α rj = 0, k j (x, x) = α ij (j = i), k i (x, δx) = 0, (17)
where

k = k i1 • • • k in T and µ j = λ i λ i -λ j , ν j = λ j λ i -λ j , (j = i), µ i = ν i = 1, α rj = m rj λ i -λ j , (j = i), α ri = m ri λ i .
Note that, with this normalization, we have in particular µ j -ν j = 1 for j = i.

Since the component k i plays a dierent role than all the other components k j with j = i, we rewrite (17) in a matrix form separating both quantities. Let us denote by n c = n -1 and introduce k c = (k j ) j =i . Then, system (17) can be written as

             D µ ∂k c ∂x (x, ξ) + D ν ∂k c ∂ξ (x, ξ) + A T k c (x, ξ) + k i (x, ξ)ψ 0 = 0, ∂k i ∂x (x, ξ) + ∂k i ∂ξ (x, ξ) + w, k c (x, ξ) = 0, k c (x, x) = ψ 0 , k i (x, δx) = 0, (18)
where

D µ , D ν , A ∈ R n c ×n c
are the matrices dened by

D µ = diag (µ j ) j =i , D ν = diag (ν j ) j =i , A = (α rj ) r,j =i , and ψ 0 , w ∈ R n c
are the vectors dened by

ψ 0 = (α ij ) j =i , w = (α ji ) j =i .
Note that we used that α ii = 0 (since M ∈ R n×n 0

). Finally, it will also be convenient to use the matrix D ∈ R n c ×n c and the vector v ∈ R n c dened by

D = D µ + D ν , v = - 1 2 w. (19)

Computation of the derivatives

The main result of this section is the following.

Theorem 3.1. For the solution to (18) with δ = -1, we have

∂ γ+σ k c ∂x γ ∂ξ σ (0, 0) = γ r=0 σ s=0 (-1) r γ r σ s ψ γ+σ-(r+s),r+s , ∀γ, σ ∈ N, (20)
where (ψ r,s ) r,s∈N is the sequence dened by

ψ r,0 = ψ r , ψ r,s = 0 if r < s, ψ r,s = r-s =0 v, ψ r-1-,s-1 ψ ,0 if r ≥ s ≥ 1, (21)
where (ψ r ) r∈N is the sequence dened in (4).

Combining this result with the estimates (14) and Taylor's theorem, we obtain an explicit formula for the solution to (18) when δ = -1:

Corollary 3.2. For δ = -1, the solution to (18) is given by

k c (x, ξ) = +∞ r=0 +∞ s=0 (-1) r r!s! ψ r,s (x -ξ) r (x + ξ) s , k i (x, ξ) = - x x-ξ 2 w, k c (σ, σ -x + ξ) dσ,
for every (x, ξ) ∈ R 2 , where (ψ r,s ) r,s∈N is the sequence dened by ( 21) and the series is normally convergent on any compact set of R 2 .

Remark 3.3. Explicit solutions to the kernel equations were also obtained in [VK14, Section 3.4] for systems of n = 2 equations.

Proof of Theorem 3.1.

1. To explain the special role played by δ = -1, we start the computations with an arbitrary δ = 1. The rst idea is to form a system involving only k c by expressing k i as a function of k c :

k i (x, ξ) = - x x-ξ 1-δ w, k c (σ, σ -x + ξ) dσ. As a result, k c solves      D µ ∂k c ∂x (x, ξ) + D ν ∂k c ∂ξ (x, ξ) + A T k c (x, ξ) - x x-ξ 1-δ w, k c (σ, σ -x + ξ) dσ ψ 0 = 0, k c (x, x) = ψ 0 .
We now transform this system into a Cauchy problem by introducing the transformation

h(t, θ) = k c -t + θ 2 , t + θ 2 .
Using that D µ -D ν = Id n c , we can check that h satises the system

     ∂h ∂t (t, θ) = D ∂h ∂θ (t, θ) + A T h(t, θ) + θ -1+δ 1-δ t v, h(t, η) dη ψ 0 , h(0, θ) = ψ 0 , (22)
where we recall that D and v are dened in (19). Note as well that

∂ γ+σ k c ∂x γ ∂ξ σ (x, ξ) = γ r=0 σ s=0 (-1) r γ r σ s ∂ γ+σ h ∂t γ+σ-(r+s) ∂θ r+s (-x + ξ, x + ξ), ∀γ, σ ∈ N,
so that the derivatives of k c at (0, 0) can be deduced from those of h. They will be computed from ( 22) and we see that the computations considerably simplify if the lower bound of the integral vanishes, that is if we choose δ = -1. For this choice, we dene ψ r,s = ∂ r+s h ∂t r ∂θ s (0, 0).

We are going to show that it satises (21).

2. All along the rest of the proof, we will use the notation c r,s = v, ψ r,s . First observe that system (22) (with δ = -1) yields the following identities:

     ψ r+1,0 = Dψ r,1 + A T ψ r,0 , ψ r+1,s = Dψ r,s+1 + A T ψ r,s + c r,s-1 ψ 0,0 , ψ 0,s = 0, (23)
for every r ≥ 0 and s ≥ 1. The second property in ( 21) is easily proved by induction on r ≥ 1. To establish the two other identities, it is sucient to prove the following statement for any N ≥ 1:

             ψ s+q,s = q =0 c s-1+q-,s-1 ψ ,0 , ∀s ≥ 1, ∀0 ≤ q ≤ N, ψ r,0 = A T ψ r-1,0 + D r-2 =0 c r-2-,0 ψ ,0 , ∀2 ≤ r ≤ N + 1. ( 24
)
We prove it by induction. For N = 1, this is clear. Indeed, for any s ≥ 1, we have Assume now that (24) holds for N ≥ 1 and let us prove it for N + 1. We rst show that

ψ s,s = Dψ s-1,s+1 + A T ψ s-1,s + c s-1,s-1 ψ 0,0 (
ψ s+N +1,s = N +1 =0 c s+N -,s-1 ψ ,0 , ∀s ≥ 1. (26)
For any s ≥ 1, we have

ψ s+N +1,s = Dψ s+N,s+1 + A T ψ s+N,s + c s+N,s-1 ψ 0,0 (by (23)), = D N -1 =0 c s+N -1-,s ψ ,0 + A T N =0 c s-1+N -,s-1 ψ ,0
+ c s+N,s-1 ψ 0,0 (by assumption (24)),

= D N -1 =0 c s+N -1-,s ψ ,0 + N +1 r=2 c s+N -r,s-1 A T ψ r-1,0 + c s-1+N,s-1 ψ 1,0 + c s+N,s-1 ψ 0,0 .
Therefore, if we show the identity

N -1 =0 c s+N -1-,s ψ ,0 = N +1 r=2 c s+N -r,s-1 r-2 =0 c r-2-,0 ψ ,0 , (27)
then we can use the second condition in (24) to obtain the desired identity (26). To establish (27), we use the rst condition in (24) to deduce that, for every 0

≤ ≤ N -1, v, ψ s+N -1-,s = N -1- j=0 c s-1+N -1--j,s-1 c j,0 = N +1 r= +2 c s+N -r,s-1 c r-2-,0 .
Finally, a simple change of order of summation leads to (27). It remains to show the second identity in (24) for r = N + 2, namely

ψ N +2,0 = A T ψ N +1,0 + D N =0 c N -,0 ψ ,0 .
We have

ψ N +2,0 = Dψ N +1,1 + A T ψ N +1,0 (by (23)), = D N =0 c N -,0 ψ ,0 + A T ψ N +1,0 (by (24)).
Remark 3.4. Theoretically, we can also compute all the derivatives at (0, 0) of the solution to (18) for arbitrary δ = 1. This can be done by taking derivatives and inverting some matrix.

However, the size of this matrix grows with the order of derivatives and computations rapidly become more and more complicated. It seems dicult with such a strategy to obtain a suitable formula for arbitrary δ. At the same time, we see from (22) that a dierent choice of δ means more derivatives to be computed, as for instance with δ = 0 which leads to an integral of the form θ -t

. In addition to that, we recall that one choice of δ is actually sucient for the purposes of this paper (Remark 2.2).

Study of the orthogonality conditions

In this section, we use the computations obtained in the previous section to study the orthogonality conditions (15). We start with the conclusion of the proof of our rst main result.

Proof of Theorem 1.3. We recall that, from the results of the previous sections, we only have to show the equivalence between the orthogonality conditions (15) and (6). First observe that, from the denition (21) of the sequence (ψ r,s ) r,s∈N , it is clear that (6) is equivalent to b, ψ r,s = 0, ∀r, s ∈ N.

We can check that this condition is equivalent to (15) using ( 20) and (21).

Remark 4.1. The proof above and the analyticity of k c in R 2 shows that the following four properties are in fact equivalent:

1. b, k c (x, ξ) = 0 for every (x, ξ) ∈ R 2 .

2. b, k c (x, 0) = 0 for every x ∈ R.

3. b, ψ r,0 = 0 for every r ∈ N.

4. b, ψ r,s = 0 for every r, s ∈ N.

We are now going to study the orthogonality conditions (6) for the sequence (ψ r ) r∈N and prove our second main result. For the rest of Section 4, the matrices A, D ∈ R n c ×n c and the vectors ψ 0 , v, b ∈ R n c can in fact be arbitrary. We emphasize that n is also arbitrary, it is only during the proof of Theorem 1.6 that we will assume that n = 3 or n = 4.

Some invariant subspaces of the sequence

We start the general study of the orthogonality conditions (6) with the description of two simple invariant subspaces of the sequence (ψ r ) r∈N .

Proposition 4.2. Assume that ψ 0 ∈ E for some E ⊂ R n c satisfying one of the following two conditions:

A T (E) ⊂ E, E ⊂ ker v T . ( 28
) A T (E) ⊂ E, D(E) ⊂ E. (29)
Then, ψ r ∈ E for every r ∈ N.

Proof. We prove the result by induction on r. For r = 0 this is trivial and for r = 1 this follows from the denition ψ 1 = A T ψ 0 and the property A T (E) ⊂ E. Assume then that ψ ∈ E for every 0 ≤ ≤ r for some r ≥ 1 and let us show that ψ r+1 ∈ E. Since r + 1 ≥ 2, we have

ψ r+1 = A T ψ r + r-1 =0 v, ψ r-1-Dψ .
Clearly, the rst part A T ψ r belongs to E since ψ r ∈ E and A T (E) ⊂ E. The remaining part also belongs to E since either v, ψ = 0 for every 0 ≤ ≤ r

(if E ⊂ ker v T ) or Dψ ∈ E for every 0 ≤ ≤ r (if D(E) ⊂ E).
If we can nd a subspace E such as in the previous proposition and which is in addition included in ker b T , then we see that the whole sequence will be guaranteed to stay in ker b T .

Characterization of rank one sequences

In this section, we characterize when the rank of (ψ r ) r∈N is equal to one, and we use it to prove Theorem 1.6 in the case n = 3. We recall that, by denition, rank (ψ r ) r∈S = dim span {ψ r | r ∈ S} for any S ⊂ N.

From now on, it will be convenient to use the following notation:

E s = span {ψ r | r ≤ s} , ∀s ∈ N.
First of all, it is clear that rank (ψ r ) r∈N = 1 ⇐⇒ ψ 0 = 0, ψ r ∈ E 0 , ∀r ≥ 1 .

We have the following characterization:

Proposition 4.3. The following three conditions are equivalent:

1. ψ r ∈ E 0 for every r ≥ 1.

2. ψ r ∈ E 0 for every r ∈ {1, 2}.

3. E 0 satises (28) or (29).

Proof. The implication 1 =⇒ 2 is trivial. The implication 3 =⇒ 1 follows from Proposition 4.2.

Let us show that 2 =⇒ 3. We write

ψ r = α r ψ 0 , r = 1, 2, for some α r ∈ R. The condition for r = 1 gives A T ψ 0 = α 1 ψ 0 , which is equivalent to A T (E 0 ) ⊂ E 0 . The condition for r = 2 gives A T ψ 1 + v, ψ 0 Dψ 0 = α 2 ψ 0 , which implies v, ψ 0 Dψ 0 ∈ E 0 , that is either E 0 ⊂ ker v T or D(E 0 ) ⊂ E 0 .
This establishes the desired equivalences.

Proof of Theorem 1.6 (case n = 3). Assume that b, ψ r = 0, ∀r ∈ {0, 1, 2} .

Since n c = 2, we have dim ker b T = 1 and thus rank (ψ r ) r∈{0,1,2} ≤ 1.

As a result, ψ r ∈ E 0 for r ∈ {1, 2} and it follows that ψ r ∈ E 0 for every r ≥ 1 by Proposition 4.3.

Characterization of rank two sequences

In this section, we characterize when the rank of (ψ r ) r∈N is equal to two, and we use it to prove Theorem 1.6 in the case n = 4. The study of rank two sequences is considerably more dicult than for rank one.

We start with the following simple observation.

Proposition 4.4. Let {0, 1, 2} ⊂ S ⊂ N. We have rank (ψ r ) r∈S = 2 if, and only if, we have one of the following two conditions:

1. rank (ψ 0 |ψ 1 ) = 1, rank (ψ 0 |ψ 2 ) = 2 and ψ r ∈ E 2 for every r ∈ S \ {0, 1, 2}.

2. rank (ψ 0 |ψ 1 ) = 2 and ψ r ∈ E 1 for every r ∈ S \ {0, 1}.

Proof. We only need to observe that the situation rank (ψ 0 |ψ 1 ) = rank (ψ 0 |ψ 2 ) = 1 does not happen. Indeed, in that situation we have ψ 1 , ψ 2 ∈ E 0 and thus ψ r ∈ E 0 for every r ≥ 1 by Proposition 4.3, which shows that the sequence cannot be of rank two.

We now characterize both conditions of Proposition 4.4.

Proposition 4.5. Assume that rank (ψ 0 |ψ 1 ) = 1 and rank (ψ 0 |ψ 2 ) = 2. Then, the following three conditions are equivalent:

1. ψ r ∈ E 2 for every r ≥ 3.

2. ψ r ∈ E 2 for every r ∈ {3, 4}.

3. E 2 satises (29).

Proof. The implication 1 =⇒ 2 is trivial. The implication 3 =⇒ 1 follows from Proposition 4.2. Let us show that 2 =⇒ 3. From the denition of ψ 2 and the rank assumptions, we have v, ψ 0 = 0 and Dψ 0 ∈ span ψ 0 , ψ 2 . Using these facts, we can rst check that the condition

ψ 3 ∈ E 2 gives A T (E 2 ) ⊂ E 2 and then that the condition ψ 4 ∈ E 2 yields D(E 2 ) ⊂ E 2 .
The second condition in Proposition 4.4 is more dicult to characterize.

Proposition 4.6. Assume that rank (ψ 0 |ψ 1 ) = 2. Then, the following two conditions are equivalent:

1. ψ r ∈ E 1 for every r ≥ 2.

2. ψ r ∈ E 1 for every r ∈ {2, . . . , 5}.

Before proving Proposition 4.6, we prove our second main result.

Proof of Theorem 1.6 (case n = 4). Assume that b, ψ r = 0, ∀r ∈ {0, . . . , 5} .

Since n c = 3, we have dim ker b T = 2 and thus rank (ψ r ) r∈{0,...,5} ≤ 2.

If the rank is in fact less than or equal to 1, then we conclude as in the proof of Theorem 1.6 in the case n = 3. If the rank is exactly equal to 2, then the conclusion follows from Propositions 4.4, 4.5 and 4.6.

We now turn to the proof of the key proposition.

Proof of Proposition 4.6. If E 1 satises (28) or (29), then the result follows from Proposition 4.2.

Therefore, from now on, we assume that E 1 does not meet any of these conditions.

1. Let N ≥ 4 be arbitrary and consider the property: for every r ∈ {2, . . . , N }, there exist α r , β r ∈ R such that

ψ r = α r ψ 0 + β r ψ 1 . (30)
Let us also set

α 0 = 1, β 0 = 0, α 1 = 0, β 1 = 1, (31)
so that the previous identity is always true for r = 0, 1. Using the denition (4) of the sequence we see that, for r ≥ 2, identity (30) is equivalent to

α r ψ 0 + β r ψ 1 =α r-1 A T ψ 0 + β r-1 A T ψ 1 + r-2 =0 c r-2-α Dψ 0 + r-2 =0 c r-2-β Dψ 1 ,
where we introduced, for every s ∈ {0, . . . , N },

c s = α s v, ψ 0 + β s v, ψ 1 .
Let us eliminate the terms A T ψ 0 , A T ψ 1 . By denition of the sequence, we have A T ψ 0 = ψ 1 .

On the other hand, condition (30) for r = 2 yields

A T ψ 1 = α 2 ψ 0 + β 2 ψ 1 -c 0 Dψ 0 . ( 32
)
As a result, for r ≥ 3, identity (30) is equivalent to

(ψ 0 |ψ 1 )u r = (Dψ 0 |Dψ 1 )f r-1 , (33)
where u r , f r-1 ∈ R 2 are the vectors dened by

u r = α r -β r-1 α 2 β r -(α r-1 + β r-1 β 2 ) , f r-1 = β r-1 (-c 0 ) + r-2 =0 c r-2-α r-2 =0 c r-2-β . ( 34
)
Let us make some observations.

For any r 1 , r 2 , if λf r1-1 + µf r2-1 = 0 for some λ, µ ∈ R, then λu r1 + µu r2 = 0. Indeed, denoting by u = λu r1 + µu r2 , we have (ψ 0 |ψ 1 )u = 0 and thus u = 0 since we assumed that ψ 0 , ψ 1 are linearly independent.

f r1-1 and f r2-1 are linearly dependent for any r 1 , r 2 . Indeed, otherwise we obtain (ψ 0 |ψ 1 )U = (Dψ 0 |Dψ 1 ) for some matrix U ∈ R 2×2 . This means that D(E 1 ) ⊂ E 1 .

It then follows from (32) that A T (E 1 ) ⊂ E 1 as well. Therefore, E 1 satises (29), but this situation has been excluded from the beginning of the proof.

f 2 and f 3 are linearly dependent only if c 0 = 0. Indeed, if c 0 = 0, then A T ψ 1 ∈ E 1 by (32), so that A T (E 1 ) ⊂ E 1 . If c 0 = 0, we also have det(f 2 |f 3 ) = c 2
1 , which cannot be zero since otherwise c 0 = c 1 = 0, that is E 1 ⊂ ker v T , and thus E 1 satises (28) (excluded).

The necessary condition det(f 2 |f r-1 ) = 0 is equivalent to the identity

r-2 =0 c r-2-β f 2 -c 0 f r-1 = 0,
which, by the rst observation above, in turn implies that

r-2 =0 c r-2-β u 3 -c 0 u r = 0.
Since c 0 = 0, this gives the following formulas:

             f r-1 = r-2 =0 cr-2-β f 2 , u r = r-2 =0 cr-2-β u 3 , (35)
where we introduced cs = cs c0 . Conversely, it is clear that if we have (35), then (33) will also hold for r ≥ 3, provided that it holds for r = 3.

Finally, observe that the second formula in (35), combined with the denition (34) of u r , uniquely determines all the α r , β r for r ≥ 4 as a function of α 2 , β 2 and α 3 , β 3 :

             α r = β r-1 α 2 + r-2 =0 cr-2-β (α 3 -β 2 α 2 ) , β r = α r-1 + β r-1 β 2 + r-2 =0 cr-2-β β 3 -(α 2 + β 2 2 ) , (36)
and that the rst formula in ( 35) is equivalent to

β r-1 = r-2 =0 cr-2-(α + (β 2 -c1 )β ) . ( 37
)
In summary, we have shown that the property considered is equivalent to: c 0 = 0 and there exist α 2 , β 2 and α 3 , β 3 such that (30) holds for r = 2, 3 and such that the sequence dened by (36) (with (31)) satises (37) for every r ∈ {4, . . . , N }.

2. Let us now study the sequence (36). The proof of the result will be complete after we show that the following three conditions are equivalent:

(a) Condition (37) holds for every r ≥ 4.

(b) Condition (37) holds for r = 4, 5.

(c) α 3 and β 3 are given by

β 3 = (α 2 + β 2 2 ) -(β 2 -c1 ) 2 + α 2 + (β 2 -c1 )β 2 , α 3 = β 2 α 2 -(β 2 -c1 )(β 3 -(α 2 + β 2 2 )). (38)
We start with the implication 2b =⇒ 2c. Condition (37) for r = 4 immediately gives β 3 as a function of α 2 , β 2 :

β 3 = 2 =0 c2-(α + (β 2 -c1 )β ) .
On the other hand, condition (37) for r = 5 gives

β 4 = 2 =1 c3-(α + (β 2 -c1 )β ) + c0 (α 3 + (β 2 -c1 )β 3 ) + c3 (α 0 + (β 2 -c1 )β 0 ) = 2α 3 + β 2 β 3 + 2 =1 c3-(α + (β 2 -c1 )β ) ,
whereas, by denition (36),

β 4 = α 3 + β 3 β 2 + 2 =0 c2-β β 3 -(α 2 + β 2 2 ) .
Identifying both expressions gives α 3 as a function of α 2 , β 2 :

α 3 = - 2 =1 c3-(α + (β 2 -c1 )β ) + 2 =0 c2-β β 3 -(α 2 + β 2 2 ) .
We can check that the previous formulas are equivalent to (38) (we prefer the expressions in (38) because they make appear some coecients involved in (36)).

Let us now prove the implication 2c =⇒ 2a. We prove it by induction on r. For r = 4, this holds by very denition of β 3 as we have seen above. Assume now that (37) holds for some arbitrary r ≥ 4, and let us prove it for r + 1, that is to prove that we have

β r = r-1 =0 cr-1-(α + (β 2 -c1 )β ) . ( 39
)
By denition (36) of β r , we have

β r =α r-1 + β r-1 c1 + (β 2 -c1 ) β r-1 -(β 2 -c1 ) r-2 =0 cr-2-β + β 3 -(α 2 + β 2 2 ) + (β 2 -c1 ) 2 r-2 =0 cr-2-β .
Using the induction hypothesis (37) and the denition (38) of β 3 , we obtain

β r = cr-1 + r-2 =0 cr-2-((β 2 -c1 )α + (α 2 + (β 2 -c1 )β 2 )β ) .
Now observe that, using the denition of α 3 , we have

α +1 + (β 2 -c1 )β +1 = β α 2 + (β 2 -c1 )(α + β β 2 ) = (β 2 -c1 )α + (α 2 + (β 2 -c1 )β 2 )β .
Formula (39) easily follows this identity and the previous one.

Kernel associated with the orthogonality conditions

In this section, we supplement our results by giving a more explicit characterization of the conditions found in the previous section and that guaranteed the orthogonality conditions. Then, we discuss the structure of the associated kernel.

Kernel associated with the invariant subspaces

Here we discuss properties related to the invariant subspaces of Section 4.1. We recall that, for any vector h ∈ R n c , we denote the Kalman matrix of (A, h) by

Kal (A, h) = (h|Ah|A 2 h| • • • |A n c -1 h) ∈ R n c ×n c .
Proposition 5.1. 1. There exists E satisfying (28), ψ 0 ∈ E and E ⊂ ker b T , if, and only if,

Kal (A, v) T ψ 0 = Kal (A, b) T ψ 0 = 0. ( 40
)
2. Assume that Kal (A, v) T ψ 0 = 0. Then, for any δ = 1, the solution to the kernel equations ( 18) is

k i (x, ξ) = 0, k c (x, ξ) = e -A T (x-ξ) ψ 0 , ∀(x, ξ) ∈ R 2 .
If, moreover, Kal (A, b) T ψ 0 = 0, then the orthogonality conditions (15) are satised.

Proof.

1. Assume that (40) holds. Let us dene

E = ker Kal (A, v) T ∩ ker Kal (A, b) T .
By assumption, ψ 0 ∈ E and it is clear that E ⊂ ker v T and E ⊂ ker b T . Finally, E is stable by A T thanks to Cayley-Hamilton theorem. Conversely, assume that (28) holds for some

E ⊂ ker b T with ψ 0 ∈ E. Since ψ 0 ∈ E and E is stable by A T , we have (A T ) k ψ 0 ∈ E for every k ∈ N. Since E ⊂ ker v T and E ⊂ ker b T , we obtain (40).
2. We see from the kernel equations (18) that k i = 0 if, and only if, we have Let us now address the second type of invariant subspaces introduced in Section 4.1.

         D µ ∂k c ∂x (x, ξ) + D ν ∂k c ∂ξ (x, ξ) + A T k c (x, ξ) = 0, v, k c (x, ξ) = 0, k c (x, x) = ψ 0 . Using that D µ -D ν = Id n c , it is clear that k c (x, ξ) = e -A T (x-ξ) ψ 0 satises
Proposition 5.2. Assume that ψ 0 , b = 0.

1. There exists E satisfying (29), ψ 0 ∈ E and E ⊂ ker b T if, and only if, there exists a nonempty J {1, . . . , n c } such that

ψ 0 j = a rj = b r = 0, ∀j ∈ J, ∀r ∈ J.
2. Assume that there exists a nonempty J {1, . . . , n c } such that ψ 0 j = a rj = 0 for every j ∈ J and r ∈ J. Then, for any δ = 1, the solution to the kernel equations (18) satises

k c j = 0, ∀j ∈ J. (41)
If, moreover, b r = 0 for every r ∈ J, then the orthogonality conditions (15) are satised.

Proof.

1. Since D is a diagonal matrix with distinct entries, its invariant subspaces are of the form

E = span {e r | r ∈ J} ,
for some J ⊂ {1, . . . , n c }, where e 1 , . . . , e n c are the canonical vectors of R n c . Since ψ 0 = 0 (resp. b = 0), we have J = ∅ (resp. J = {1, . . . , n c }). Then, we easily check that such a subspace is invariant by A T if, and only if, a rj = 0 for every r ∈ J and j ∈ J and that it is included in ker b T if, and only if, b r = 0 for every j ∈ J.

2. Property (41) is a consequence of the uniqueness of the solution to the kernel equations.

The orthogonality conditions are clearly satised under the additional assumption that b r = 0 for every r ∈ J.

Remark 5.3. The rst item in the above propositions gives explicit conditions that guarantee that the orthogonality conditions (15) hold (when combined with the results of the previous sections). We found these conditions with an algebraic approach. On the other hand, once these conditions are known, the second item of the above propositions show how to use them to obtain an analytic proof of the orthogonality conditions. Observe in addition that these dierent proofs are valid for arbitrary δ = 1.

Kernel associated with nontrivial rank two sequences

In the same spirit as in the previous section, we now we discuss the following property, related to Proposition 4.6:

E 1 ⊂ ker b T , ψ r ∈ E 1 , ∀r ∈ {2, . . . , 5} . (42)
Below, we denote by c 0 = v, ψ 0 .

Proposition 5.4. 1. Assume that rank (ψ 0 |ψ 1 ) = 2, E 1 satises neither (28) nor (29), c 0 = 0 and b, ψ 0 = 0. Then, condition (42) holds if, and only if, there exists j 0 ∈ {1, . . . , n c } such that

b j0 = 0, rank ∆ j0 = 1,
where ∆ j0 ∈ R 2n c ×2 is given by

∆ j0 = (D -d j0 )ψ 0 A T e j0 A T ψ 0 v j0 ψ 0 -c 0 e j0 ,
where d j0 is the j 0 -th diagonal entry of D and e j0 is the j 0 -th canonical vector of R n c .

Assume that

n = 4, i = 2, b =   1 0 -ρ   , ρ = 0,
c 0 = 0, α 24 = 0, b, ψ 0 = 0 and rank ∆ 2 = 1. Then, the solution to the kernel equations (18) with δ = -1 is given, for some σ ∈ R \ {0}, by

             k 21 = k 24 ρ, k 22 = 1 -σα 24 σµ 3 ∂k 24 ∂x + σν 3 ∂k 24 ∂ξ -α 32 k 24 , k 23 = 1 -σα 24 ∂k 24 ∂x + ∂k 24 ∂ξ -σα 23 k 24 , (43)
where

k 24 ∈ C ∞ (R 2 ) is the solution to              µ 3 ∂ 2 k 24 ∂x 2 (x, ξ) + (µ 3 + ν 3 ) ∂ 2 k 24 ∂x∂ξ (x, ξ) + ν 3 ∂ 2 k 24 ∂ξ 2 (x, ξ) = -2c 0 k 24 (x, ξ), µ 3 ∂k 24 ∂x (x, -x) + ν 3 ∂k 24 ∂ξ (x, -x) = α 32 σ k 24 (x, -x), k 24 (x, x) = α 24 . (44)
To see that (44) indeed has a solution, we can introduce h = µ 3 ∂k24 ∂x +ν 3 ∂k24 ∂ξ and observe that (k 24 , h) satises a rst-order hyperbolic system similar to the kernel equations ( 16) and whose well-posedness can be established as in the proof of Theorem 2.5.

Proof of Proposition 5.4.

1. Under the assumptions of the proposition and from the proof of Proposition 4.6, we have ψ r ∈ E 1 for every r ∈ {2, . . . , 5} if, and only if, there exist

α 2 , β 2 ∈ R such that ψ 2 = α 2 ψ 0 + β 2 ψ 1 , ψ 3 = α 3 ψ 0 + β 3 ψ 1 ,
where α 3 , β 3 are given by (38). We can check that this is equivalent to the existence of ρ, θ ∈ R such that

(A T -ρ)ψ 1 = -c 0 (D -θ)ψ 0 + c1 (ψ 1 -ρψ 0 ), (D -θ)(ψ 1 -ρψ 0 ) = 0, (45)
where we recall that c1 = v,ψ 1 c0 . Since D is diagonal with distinct entries and ψ 0 , ψ 1 are linearly independent, the second condition in (45) is equivalent to the existence of some

j 0 ∈ {1, . . . , n c } and r ∈ R \ {0} such that θ = d j0 , ψ 1 = ρψ 0 + re j0 . (46)
Plugging the second identity in the rst condition in (45) and recalling that ψ 1 = A T ψ 0 , we see that this condition simply becomes

rA T e j0 = -c 0 (D -d j0 )ψ 0 + rc 1 e j0 .
Comparing the j 0 -th components of both quantities, using that the diagonal of A is zero and r = 0, we see that c1 = 0. Recalling (46), the condition c1 = 0 is equivalent to ρ = -rv j0 with r = r c0 . In summary, there exist ρ, θ ∈ R such that (45) holds if, and only if, there exist j 0 ∈ {1, . . . , n c } such that ∃r ∈ R \ {0} , (D -d j0 )ψ 0 + rA T e j0 = 0,

ψ 1 + r(v j0 ψ 0 -c 0 e j0 ) = 0. (47)
We can check that this condition is equivalent to rank ∆ j0 = 1. Finally, it is clear that E 1 ⊂ ker b T , i.e. b, ψ 1 = 0, if, and only if, b j0 = 0.

2. Since i = 2, we have (for the notations, see Section 3.1)

A T =   0 α 31 α 41 α 13 0 α 43 α 14 α 34 0   , D = 2diag (µ 1 , µ 3 , µ 4 )-Id 3 , ψ 0 =   α 21 α 23 α 24   , v = - 1 2   α 12 α 32 α 42   .
Clearly, b, ψ 0 = 0 is equivalent to

α 21 = α 24 ρ. (48)
On the other hand, using the characterization (47), we see that rank ∆ 2 = 1 if, and only if,

α 31 = σ(µ 1 -µ 3 )ρα 24 , (49)
α 34 = σ(µ 4 -µ 3 )α 24 , (50)
α 41 = - ρ σ α 32 + σ 2 (µ 1 -µ 3 )α 23 , (51)
α 42 + α 12 ρ = σ(α 43 + α 13 ρ),

α 14 = - 1 ρσ α 32 + σ 2 (µ 4 -µ 3 )α 23 , (52)
where σ = -2 r . Using these conditions we easily check that ((

k c ) T , k 2 ) = (k 21 , k 23 , k 24 , k 22 )
dened by ( 43)-( 44) satises the kernel equations ( 18) with δ = -1.

To conclude this section we will present a method which shows how conditions ( 48)-(53) can also be found from an analytic point of view.

Another proof of Proposition 5.4, item 2.

1. For every j, let us denote by P j the rst-order linear partial dierential operator

P j = µ j ∂ ∂x + ∂ ∂ξ ν j .
Since we want the orthogonality condition b, k c (•, 0) = 0, we look for a solution satisfying Assume that ω = 0 (this will follow a posteriori from (49), (50), using also that σ, ρ, α 24 = 0). (55)

Plugging these relations in the second and third equations in (54) give the following two second-order partial dierential equations for k 24 :

Qk 24 = 0, Qk 24 = 0,

where Q = Q (2) + Q (1) + Q (0) , with Q (2) = P 2 (-α 34 ρP 1 + α 31 P 4 ) , Q (1)
= (-α 34 α 41 + α 31 α 14 ρ)P 2 + (α 24 ρP 1 -α 21 P 4 ) α 32 , Q (0) = (α 24 α 41 -α 21 α 14 ρ)α 32 + ω(α 42 + α 12 ρ),

and Q = Q(2) + Q(1) + Q(0) , with Q(2) = P 3 (α 24 ρP 1 -α 21 P 4 ) , Q(1) = (α 24 α 41 -α 21 α 14 ρ)P 3 + (-α 34 ρP 1 + α 31 P 4 ) α 23 , Q(0) = (-α 34 α 41 + α 31 α 14 ρ)α 23 + ω(α 43 + α 13 ρ).

2. We are going to nd conditions to guarantee that these two equations are compatible. To this end, it is for instance sucient to have

Q (r) = σ Q(r) , r = 0, 1, 2, (56)
for some σ ∈ R. We rst look at the operators of highest order. Using the identities

α 21 = α 24 ρ, µ j -ν j = 1 (j = 2), µ 2 = ν 2 = 1, (57)
we have α 24 ρP 1 -α 21 P 4 = α 24 ρ (P 1 -P 4 ) = α 24 ρ(µ 1 -µ 4 )P 2 .

(

) 58
It follows that

Q(2) = α 24 ρ(µ 1 -µ 4 )P 3 P 2 . (59)
Consequently, we see that (56) holds for r = 2 if we have -α 34 ρP 1 + α 31 P 4 = σα 24 ρ(µ 1 -µ 4 )P 3 .

(

) 60
This identity holds if (α 34 , α 31 ) satises -ρµ 1 µ 4 -ρν 1 ν 4 α 34 α 31 = σα 24 ρ(µ 1 -µ 4 ) µ 3 ν 3 , which is equivalent to (50)-(49) (using ρ = 0 and (57)).

3. Let us now compute the rst-order dierential operators. We have Q (1) = (-α 34 α 41 + α 31 α 14 ρ + α 24 ρ(µ 1 -µ 4 )α 32 ) P 2 (by (58)), Q(1) = (α 24 α 41 -α 21 α 14 ρ + σα 24 ρ(µ 1 -µ 4 )α 23 ) P 3 (by ( 60)).

As a result, we have (56

) for r = 1, if Q (1) = Q(1) = 0, that is, if -α 34 α 41 + α 31 α 14 ρ = -α 24 ρ(µ 1 -µ 4 )α 32 ,
α 24 α 41 -α 21 α 14 ρ = -σα 24 ρ(µ 1 -µ 4 )α 23 .

(

) 61
This holds if (α 41 , α 14 ) satises -α 34 α 31 ρ α 24 -α 21 ρ α 41 α 14 = -α 24 ρ(µ 1 -µ 4 ) α 32 σα 23 , which is equivalent to (51) and (53) (using (48), (49), (50) and α 24 , ρ = 0).

4. Let us now compute the zero order terms. Using (61), we immediately see that Q(0) = -α 24 ρ(µ 1 -µ 4 )α 32 α 23 + ω(α 43 + α 13 ρ),

Q (0) = -σα 24 ρ(µ 1 -µ 4 )α 23 α 32 + ω(α 42 + α 12 ρ).
As a result, we see that (56) holds for r = 0 if we have condition (52). Moreover, using (48), ( 49) and (50), we have

ω = -σα 2 24 ρ(µ 1 -µ 4 ), (62)
so that, using again (48) and the denition of c 0 , we obtain

Q (0) = σα 24 ρ(µ 1 -µ 4 )(2c 0 ).
It follows that k 24 indeed satises the rst equation in (44) (recall that Q (2) = σ Q(2) with (59) and Q (1) = 0).

5. Using (60), ( 61), ( 58) and (62), we can simplify the expressions in (55) to obtain

       k 22 = 1 -σα 24 (σP 3 -α 32 ) k 24 , k 23 = 1 -σα 24 (P 2 -σα 23 ) k 24 .
In addition, it follows from these formula that the remaining conditions are satised. Indeed, the condition k 22 (x, -x) = 0 is exactly the condition that we require for k 24 at (x, -x) in (44) and the condition k 23 (x, x) = α 23 follows from the above expression since k 24 (x, x) = α 24 and (P 2 k 24 )(x, x) = d dx k 24 (x, x) = 0.

Remark 5.5. In [VK14, Section 3.3], the authors showed that we can solve a kernel system of two equations of the form

             ∂k 21 ∂x - ∂k 21 ∂ξ + k 22 α 21 = 0, ∂k 22 ∂x + ∂k 22 ∂ξ + k 21 α 12 = 0, k 21 (x, x) = α 21 , k 22 (x, 0) = 0,
with α 21 = 0 by rst expressing k 22 from the rst equation and then showing that the resulting second order equation for k 21 indeed has a solution. The method we introduced in the second proof of Proposition 5.4, item 2, can be seen as an extension of the method of [START_REF] Vazquez | Marcum Q-functions and explicit kernels for stabilization of 2×2 linear hyperbolic systems with constant coecients[END_REF] where, instead of dividing by a scalar (namely, α 21 ), we invert a matrix.

A Controllability of the equivalent system

In this appendix, we give a simple and direct proof of Corollary 2.4. We recall that it can be deduced from Theorem 2.3 but this result is based on the Titchmarsh convolution theorem (see [START_REF] Hu | Null controllability and nite-time stabilization in minimal time of one-dimensional rst-order 2 × 2 linear hyperbolic systems[END_REF]) and we show here how to directly prove the corollary without resorting to this dicult result.

Proof of Corollary 2.4. It is enough to show that, if (q, f ) ∈ S k \S k+1 for some k ∈ {2, . . . , n + 1}, then system (9) (with m = 1) is null controllable in time T if, and only if, T ≥ τ k .

1. We rst observe that system (9) is equivalent to the same system with f 1 = 0. This follows from the invertible spatial transformation

ŷ1 (t, x) = ỹ1 (t, x) - x 0 h(x -ξ)ỹ 1 (t, ξ) dξ,
where the kernel h is the solution to

h(x)λ 1 + x 0 h(x -ξ)f 1 (ξ) dξ = f 1 (x), 0 < x < 1.
Therefore, for the rest of the proof, we assume that f 1 = 0.

2. Assume now that (q, f ) ∈ S k \ S k+1 for some k ∈ {2, . . . , n} (the result for k = n + 1 is trivial). It will be convenient to use the notation qi = q i-1 for 2 ≤ i ≤ n. Let us write system (9) (with f 1 = 0) component-wise:

       ∂ ỹ1 ∂t (t, x) + λ 1 ∂ ỹ1 ∂x (t, x) = 0, ỹ1 (t, 1) = ũ(t), ỹ1 (0, x) = ỹ0 1 (x),        ∂ ỹi ∂t (t, x) + λ i ∂ ỹi ∂x (t, x) = f i (x)v(t), ỹi (t, 0) = qi v(t), ỹi (0, x) = ỹ0 i (
ỹk (T, x) = 0 is equivalent to qk α(τ ) + τ 0 β(τ -σ)α(σ) dσ = 0, 0 < τ < T k , (63)
where α

(θ) = v(-θ + T ) and β(θ) = f k (λ k θ) for 0 < θ < T k .
4. We now have two possibilities for (63).

(a) Case qk = 0. Then, by uniqueness of the solution to the Volterra equation of the second kind (63), we obtain α = 0 in (0, T k ). This means that v = 0 in (T -T k , T ).

Since this is true for any ỹ1 0 , it is possible only if T 1 ≤ T -T k , which is the desired condition.

(b) Case qk = 0. Since (q, f ) ∈ S k \ S k+1 , we necessarily have f k = 0. Since f k is analytic in a neighborhood of [0, 1), this implies in particular that there exists N ≥ 1 such that

f (N -1) k (0) = 0, f ( ) k (0) = 0, ∀ < N -1.
Then, taking N times the derivative with respect to τ in (63) (with qk = 0), we obtain the new Volterra equation

cα(τ ) + τ 0 β (N ) (τ -σ)α(σ) dσ = 0, 0 < τ < T k , where c = β (N -1) (0) = f (N -1) k (0)λ N -1 k
. Therefore, c = 0 and the situation is now identical to the previous case.

B Solution to the kernel equations

In this appendix, we present a new approach to solve the kernel equations that encompasses in particular the proof of Theorem 2.5. We recall that, when considering the kernel equations in the triangle T = (x, ξ) ∈ R 2 0 < ξ < x < 1 , the approach used in all current results in the literature ([Cor+13; DVK13; HD15; Hu+16; Hu+19; CN19], etc.) consists in adding articial boundary conditions to close the system of kernel equations. In our approach, we will not consider the condition at (x, x) as a boundary condition but rather as an initial condition.

We will simply let propagate this condition along the characteristics and nd the corresponding so-called domain of determinacy, much in the spirit of the reference books [LY85; Bre00]. Then, another idea of our method is also to solve the equation for j = i and plug it into the other equations of the system to obtain a new system with initial conditions at (x, x) only (as in the proof of Theorem 3.1). Moreover, this gives a natural bound in |x -ξ| for the estimates needed to prove the contraction of the mapping dening the integral equations corresponding to the new system (rather than |x -(1 -ε)ξ| as in [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF][START_REF] Hu | Boundary exponential stabilization of 1-dimensional inhomogeneous quasi-linear hyperbolic systems[END_REF]).

All along this appendix, i ∈ {1, . . . , n} is xed and we continue using the notation k = (k ij ) 1≤j≤n to denote the transpose of the i-th row of K. We also emphasize that m ≥ 1 is arbitrary.

First of all, it will be more convenient to work with the kernel equations normalized by λ i :      ∂k j ∂x (x, ξ) + λj ∂k j ∂ξ (x, ξ) + n r=1 k r (x, ξ) mrj = 0, k j (x, x) = f j (j = i), k i (x, δx) = 0,

where λj = λ j λ i , mrj = m rj λ i , f j = m ij λ i -λ j .

From now on, we will assume for instance that i ≥ m + 1, so that λ i > 0 and thus, from (1b), Let us now consider the more general condition k j (x, x) = f j (x) (j = i),

λ1 < • • • < λi-1 < 1 < λi+1 < • • • < λn .
where f j is a function dened on an interval of the form [a, b] with a < 0 < b. Even if f j is constant in (64), we will need to consider space-dependent data to deduce the existence of smooth solutions by an inductive argument. We will describe the largest domain D ⊂ R 2 where the system can then be solved along the characteristics. We rst take care of the characteristics for j = i. Recalling the ordering (65), we introduce (see Figure 1). Above, we use the usual conventions for i = 1 and i = n. We now take care of the characteristic for j = i.

D c = (x, ξ) ∈ R 2 ζ i-1 (x;
for some C > 0 that does not depend on any f j .

For s = 0, by solution we mean solution along the characteristics, see below.

The rst part of Theorem 2.5 follows from the previous result and the following simple observation:

∀V ⊂ R 2 , ∃a < 0 < b, V ⊂ D.

(68)

On the other hand, using that the coecients of the system are constant and arguing as in the proof of [CN19, Lemma 6.2], we can show that (67) holds with C = R s for some R > 0 that does not depend on s. This establishes the estimate in Theorem 2.5.

Let us now prove Theorem B.1. We start with a description of the key properties satised by the point where the j-th characteristic intersects the corresponding data line.

Lemma B.2. For every j ∈ {1, . . . , n}, there exists σ j ∈ C ∞ (D) such that, for every (x, ξ) ∈ D,

we have: For every j = i, we have

|σ j (x, ξ) -x| ≤ C |x -ξ| , (69)
for some C > 0 that does not depend on j, x, ξ.

We point out that ζ j and σ j are explicit. In particular, this is how we prove estimate (69). Now, instead of writing (66) along all the characteristics (as it is usually done), we rst replace k i by formally solving the corresponding equation (recall that mii = 0): (70)

k i (x, ξ) = f i (σ i (x, ξ)) -
Let us introduce the following notations to exclude the i-th components: k c = (k j ) j =i , f c = (f j ) j =i , σ c = (σ j ) j =i , ζ c = (ζ j ) j =i , M c = ( mrj ) r,j =i , ψ = ( mij ) j =i and w = ( mji ) j =i . Then, plugging the previous expression of k i in (66) and integrating along the characteristics, we can transform this system into the following system of integral equations for k c : k c (x, ξ) =f c (σ c (x, ξ)) -

Remark 1. 4 .

 4 Theorem 1.3 solves the open problem raised at the end of [HO21b, Section 5] for systems with constant coecients.

  by (23)), = c s-1,s-1 ψ 0,0 (by the second property in (21)), (25) ψ s+1,s = Dψ s,s+1 + A T ψ s,s + c s,s-1 ψ 0,0 (by (23)), = A T ψ s,s + c s,s-1 ψ 0,0 (by the second property in (21)), = c s-1,s-1 ψ 1,0 + c s,s-1 ψ 0,0 (by (25)), and ψ 2,0 = Dψ 1,1 + A T ψ 1,0 (by (23)), = c 0,0 Dψ 0,0 + A T ψ 1,0 (by (25)).

  the rst equation. The second condition follows from the assumption Kal (A, v) T ψ 0 = 0 and Cayley-Hamilton theorem. The third condition is trivial. Finally, the orthogonality conditions are clearly satised under the additional assumption Kal (A, b) T ψ 0 = 0.

kk

  21 = k 24 ρ, (recall also Remark 4.1). In particular, we assume (48). Then, the problem is to nd a 22 α 21 + k 23 α 31 = (-ρP 1 -α 41 )k 24 , P 2 k 22 + k 23 α 32 + k 24 (α 42 + α 12 ρ) = 0, P 3 k 23 + k 22 α 23 + k 24 (α 43 + α 13 ρ) = 0, k 22 α 24 + k 23 α 34 = (-P 4 -α 14 ρ)k 24 . (54) Let us denote by ω = det α 21 α 31 α 24 α 34 .

  every (x, ξ) ∈ R × R, we denote by s -→ ζ j (s; x, ξ) the solution to s; x, ξ) = λj , ∀s ∈ R, ζ j (x; x, ξ) = ξ.

  a, a) < ξ < ζ i-1 (x; b, b) ζ i+1 (x; b, b) < ξ < ζ i+1 (x; a, a),

Figure 2 :

 2 Figure 1: Domain D c

  ζ j (σ j (x, ξ); x, ξ) = σ j (x, ξ) with σ j (x, ξ) ∈ [a, b] for j = i and ζ i (σ i (x, ξ); x, ξ) = δσ i (x, ξ) with σ i (x, ξ) ∈ I. (s, ζ j (s; x, ξ)) ∈ D for every s ∈ [σ j (x, ξ), x].

  x σi(x,ξ) r =i k r (η, ζ i (η; x, ξ)) mri dη.

  , ζ c (s; x, ξ))m c r ds x σ c (x,ξ) f i (σ c i (s, ζ c (s; x, ξ)))ψ ds ζ i (η; s, ζ c (s; x, ξ)))w r dη ψ ds, . . . , n -1} and (x, ξ) ∈ D.

  Then, the rst and fourth equations in (54) give 34 ρP 1 + α 31 P 4 -α 34 α 41 + α 31 α 14 ρ) k 24 , 24 ρP 1 -α 21 P 4 + α 24 α 41 -α 21 α 14 ρ) k 24 .

	    	k 22 = (-α k 23 = 1 ω 1 ω (α

Acknowledgements

This project was supported by National Natural Science Foundation of China (Nos. 12122110 and 12071258) and National Science Centre, Poland UMO-2020/39/D/ST1/01136. For the purpose of Open Access, the authors have applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

All the quantities in (70) and (71) are well dened thanks to Lemma B.2. It remains to prove the existence and uniqueness of a C s solution k c to this system of integral equations. We start with s = 0. As usual, we use the Banach xed point theorem and suitable estimates. A solution to this system is a xed point of the map F (k c ) = u 0 + Φk c , where

and Φ is the linear map Φ = Φ 1 + Φ 2 with

and Lemma B.3. There exists C > 0 such that, for every N ∈ N * , we have

for every k c ∈ B, ∈ {1, . . . , n -1} and (x, ξ) ∈ D.

Proof. We prove the property by induction on N . Let us rst consider N = 1. We have

for some C 3 > 0 that does not depend on , x, ξ (see ( 69)). This proves the property for N = 1.

Let us now assume that the property holds for N and let us prove it for N + 1. We have

Using the induction assumption, we get

Similarly, noting that η -ζ i (η; s, ζ c (s; x, ξ)) = s -ζ c (s; x, ξ), we get

N + 1 .

We conclude thanks to the estimate (72).

Finally, the estimate (67) can be deduced from the identities

Φ r u 0 , combined with the fact that Φ N is a contraction and that u 0 can be estimated by the right-hand side of (67) (with s = 0). This concludes the proof of Theorem B.1 for s = 0.

To prove the result for s ≥ 1 we can argue as in the proof of [Bre00, Theorem 3.6] and then use an induction argument.

Remark B.4. The proof above can be adapted to deal with space-dependent systems, i.e. when λ j and m rj depend on x. The additional condition for k i has to be modied though, but we can for instance consider k i (x, 0) = f i (x). Note that we still have explicit formulas for the corresponding ζ j and σ j .

Remark B.5. Our approach can be used to recover existence results in the triangle T . To this end, we simply extend the parameters λ j and m rj to [a, b] ⊃ [0, 1] in a smooth way. Then, for a, b large enough, the domain D will contain the triangle T (recall (68)) and we apply Theorem B.1 in this D. This approach is dierent from all the previous ones in the literature, which consisted in adding articial boundary conditions at some parts of the boundary of T . Note in addition that extending λ j and m rj outside [0, 1] in a smooth way is easier than building articial boundary conditions that satisfy compatibility conditions associated with the kernel equations.