Branches and ecological states of the gut microbiome

Different factors shapes the human gut microbiome

Diet

  Branches and ecological states of the gut microbiome -Journées Polepharma de Microbiomique -Rouen 2023 22.11.23 / Julien Tap / fme.micalis.fr 3 Branches and ecological states of the gut microbiome -Journées Polepharma de Microbiomique -Rouen 2023 22.11.23 / Julien Tap / fme.micalis.fr Branches and ecological states of the gut microbiome -Journées Polepharma de Microbiomique -Rouen 2023 22.11.23 / Julien Tap / fme.micalis.fr Several attempts to decipher the microbiome structure Limits: Few association with diet Few stability assessment (e.g. with timeseries) Branches and ecological states of the gut microbiome -Journées Polepharma de Microbiomique -Rouen 2023 22.11.23 / Julien Tap / fme.micalis.fr Branches and ecological states of the gut microbiome -Journées Polepharma de Microbiomique -Rouen 2023 22.11.23 / Julien Tap / fme.micalis.fr Branches and ecological states of the gut microbiome -Journées Polepharma de Microbiomique -Rouen 2023 22.11.23 / Julien Tap / fme.micalis.fr Low-diversity tips of branches display functional shifts oxidative stress response increase in Prevotella branch tip amino acid catabolism increase in Bacteroides branch tip p. 10 Branches and ecological states of the gut microbiome -Journées Polepharma de Microbiomique -Rouen 2023 22.11.23 / Julien Tap / fme.micalis.fr Altered states showed functional differences within species Bifidobacterium bifidum strains may be enriched in phage-related genes in subjects with altered gut microbiomes states Ladeira et al, 2023 400 B. bifidum MAGS phylogenomics

Microbiome states

Acknowledgments Rob Knight's lab Daniel McDonald Franck Lejzerowicz … Muriel Derrien Patrick Veiga Boris Le Nevé … Chays Mannichanh Fernando Azpiroz Iñigo Oyarzun …

Probiotics clinical effect is gut microbiome baseline dependent

After FMP intake (TP5), higher responder group showed higher methanogenesis activity than low-responders


J Lacaze

S Armendariz
email: bsarmendariz@tsfundiciones.com

P Larrañaga
email: cplarranaga@azterlan.es@d

I Asenjo
email: iasenjo@azterlan.es

J Sertucha
email: ejsertucha@azterlan.es

R Suárez
email: frsuarez@azterlan.es

Effect of Carbon Equivalent on Graphite Formation in Heavy-Section Ductile Iron Parts

Keywords: Ductile iron, Heavy castings, Chunky graphite, Carbon equivalent, Cooling curve

The influence of post-inoculation and of cerium and antimony additions on the solidification process and on the formation of chunky graphite in ductile iron heavy-section parts have been studied previously in the case of near-eutectic alloys. It appeared of interest to complement these works by analysing the effect of carbon equivalent on graphite degeneracy. In the present work, hypo-, hyper-and near-eutectic melts have been cast in large blocks and standard cups. Analysis of the corresponding cooling curves recorded during solidification as well as microstructure observations on these casts have been carried out. A clear effect of carbon equivalent as promoter of chunky graphite formation is observed. The results have been added to the set of data already available and various correlations are discussed.

Introduction

The solidification process of heavy-section parts made of nodular cast iron attracts interest for heavy automotive parts and for wind-mill hubs. The large sections of such castings lead to very slow solidification so that the material is prone to present degenerate graphite [1,2] that lowers the mechanical properties of the parts. One of the most critical forms of degenerate graphite is chunky graphite, denoted CHG in the following. It consists of large cells, typically 0.5 to 2 mm in diameter, of austenite embedded in a three-dimensional interconnected network of graphite [3]. In a series of previous papers [4][5][6][7], metallographic observations on large cast blocks and analysis of the cooling curves recorded during their solidification were presented. For the near-eutectic melts investigated, it has been found that post-inoculation increases the tendency for chunky graphite formation [4], and it was postulated that the opposite trend sometimes reported in the literature [2] was related to the use of chills. Analysis of the cooling curves recorded during solidification of these blocks showed that the solidification proceeds totally below the stable eutectic temperature and may be described in three successive steps [5,6]: i) formation of primary graphite nodules that grow freely in the melt; ii) precipitation of austenite that grows as dendrites encapsulating the primary nodules, thus leading to a so-called initial eutectic reaction; and iii) bulk eutectic reaction where the remaining melt solidifies giving the usual nodular structure. This is during this latter step that CHG cells may appear. Casting of blocks with or without addition of Sb confirmed the positive effect of this element in avoiding CHG formation [7]. In the present work, castings made with hypo-, hyperand eutectic melts were carried out in order to investigate the role of the carbon equivalent that has been reported to increase the tendency for CHG formation. The results obtained are compared to previous data.

Experimental

In the present study, three different melts (denoted as trials #9, #14 and #15) were prepared with the aim to obtain hypo-, hyper and near-eutectic alloys. From each of these melts, three cubic blocks 30 cm in size were cast, together with thermal analysis (TA) standard cups and medals for chemical analysis. The methodology followed for preparing all melts, casting blocks and sampling has been fully described previously [4]. The nodularizing treatment has been performed using a FeSiMg ferroalloy (45% Si, 0.9% Al, 2.8% Ca, 9% Mg and 1% rare earths, in weight). All three blocks of each trial were simultaneously cast in the same mould. In Table 1, the blocks are labelled as M5-X-Y, where M5 is for the so called thermal modulus i.e. the ratio between the area and the volume of the block (5 cm), X is the melt number and Y relates to additions made in the gating system of each block (N for no post-inoculation, I for post inoculation). Post-inoculation was performed by adding 0.18% of the weight of the block of a commercial inoculant (70-75% Si, 3.2-4.5% Al, 0.3-1.5% Ca, ~0.3% RE, in weight). Possible additions of pure Sb or of Si by means of a FeSi master alloy (76% Si, 0.5% Al, 0.5%Ca, 0.05% rare earth, in weight) are indicated as Sb and Si respectively at the end of the blocks label. The chemical analyses of the materials are reported in Table 1, together with the carbon equivalent C eq and the stable eutectic temperature T eut calculated according to formula given elsewhere [8]. Cooling curves were recorded from standard TA cups and from a thermocouple located at the centre of each block. After casting, all the blocks were cut and then analysed so as to evaluate the fraction V V of the blocks affected by CHG and to quantify the nodule count, N block , and the local surface area of chunky graphite, A A , at the centre of the blocks as previously described [4]. The occurrence of graphite flotation was also checked by metallographic observations. * Chemical composition was determined on the blocks.

Results

Figure 1 shows a micrograph from the central part of the hyper-eutectic casting M5-14I. The microstructure shows the same features than those observed in near-eutectic alloys with: i) large (primary) nodules embedded in dendrite-like ex-austenite; ii) CHG cells that present very fine graphite precipitates; and iii) an usual structure of nodular iron with (secondary) nodules much smaller than the primary ones. The microstructure of the hypo-eutectic alloy differed only by the larger amount of dendrite-like austenite. Table 2 lists the results of the values of V V , A A and N block and indicates if flotation was observed or not. As for near-eutectic alloys previously studied, it is seen that post-inoculation favours the formation of chunky graphite in all cases. The addition of FeSi to the melt (M5-9ISi) increases also the amount of CHG, and this may be possibly because of the associated marked increase in nodule count. As expected, addition of Sb decreases, or even suppresses in the present case, graphite degeneracy. It is seen that the highest V V value are obtained in the case of the hypereutectic melt, for both post-inoculated and non post-inoculated blocks. As already observed [4], the A A values do not follow the evolution of V V , generally changing more erratically. This effect of the carbon equivalent may be tentatively related to the fact that the fraction of austenite that precipitates as dendrites decreases when C eq increases. Finally, flotation appeared mainly in blocks made of hypereutectic melt. Figure 2 compares the cooling curves obtained with TA cups for non post-inoculated melts cast during trials #14 (hyper-eutectic) and #15 (hypo-eutectic). The record for the former alloy shows clearly the three arrest temperatures, the liquidus, the initial eutectic reaction and the bulk eutectic plateau. As described previously [6] and illustrated in Fig. 2, the corresponding characteristic temperatures are defined as T L and T EN for the first two arrests, T EU and T ER for the minimum and maximum bulk eutectic temperatures. When the first arrests show recalescence, superscripts 1 and 2 are added for labelling the respective minimum and maximum temperatures. The temperatures evaluated on the TA cups and blocks are listed in Table 3. In case the T L or T EN arrests could be identified but do not show recalescence, only the temperature at the inflexion point is given. In case these arrests were too faint to be observed, no temperature is reported. Recalescence during the eutectic reaction may be defined either as bulk eutectic recalescence ∆T=(T ER -T EU ) or as the maximum eutectic recalescence ∆T max =(T ER -T min ) where T min is the lowest of T EN and T EU values.

T T EU T ER

The curves recorded on the blocks did show the same overall features than the corresponding TA cup records, with however T EU and T ER values significantly higher in the former case than in the latter as already reported [6]. In the case of the hypo-eutectic alloy, the low carbon equivalent leads to an important primary deposition of austenite, and thus to a eutectic plateau that is much shorter than for the eutectic and hyper-eutectic alloys. Accordingly, bulk eutectic recalescence was lower for the hypo-eutectic alloy. The main difference between the thermal records for near-eutectic and hyper-eutectic alloys was that the temperature of the bulk eutectic reaction was slightly lower for the former that for the latter ones.

Discussion

The data from the present experimental study is now added to the results obtained previously from near-eutectic melts. Emphasis is put on the relationships discussed previously with the aim at checking if they may be extended to hypo-and hyper-eutectic alloys.

For the near-eutectic melts investigated so far, it has been possible to show experimentally that chunky graphite cells nucleate and grow during the bulk eutectic reaction [5,7]. Owing to the higher growth rate of these cells as compared to the kinetics of spheroidal graphite eutectic [9], it was expected that the formation of CHG should lead to an increase of bulk eutectic recalescence ∆T. The data available is plotted in Fig. 3 as V V values versus ∆T, where the results from the present study are shown with larger symbols than the previous ones. It is seen that the whole set of data gives a large cloud of scattered points, though post-inoculated alloys without Sb addition (solid circles) do clearly relate with higher V V values, as stressed with the full line. This graph could be understood as showing an increase of ∆T with V V , though with a different sensitivity for post inoculated alloys than for the other ones (non post-inoculated and Sb treated). The trend shown in Fig. 3 is somehow in contradiction with the expected one. Surprisingly, it was possible to find a much less scattered evolution of the data in plotting V V versus ∆T max , as shown in Fig. 4. The correlation obtained showed an increase of ∆T max from zero to a maximum value of 6-8°C at V V equal to 10-15%, then a continuous decrease at higher values of V V . This has been understood as due to the fact that the temperature of the blocks is homogeneous during most of the solidification, so that the effects of early formation of CHG are smoothed out by the larger volume of metal to be reheated [6]. Accordingly, it is thought that a quantitative description of the solidification of the blocks through micro-macroscopic numerical simulation could explain the graph in Fig. 3 because the T EU temperature is lower for non post-inoculated blocks. It should be noted also in Fig. 4 that the correlation does not apply to highly hypo-eutectic alloys as stressed with the arrow that points the data from the inoculated low-carbon alloy. Another interesting result was that a correlation could be made between V V values on blocks and ∆T measured on the TA cups. It is illustrated in Fig. 5 where it is seen a decrease of V V with ∆T for post-inoculated alloys, and the opposite trend for non post-inoculated ones. The two points marked with arrows are from hypo-and hyper-eutectic alloys, their position off the cloud of points shows that the correlation applies better to near-eutectic alloys. Such a correlation, if confirmed by further data, should be of significant help for melt preparation before casting.

Formation of chunky graphite is known to be strongly affected by low level additions, such as Sb that is used to get rid of it and Ce which has the opposite effect. Accordingly, the V V values were reported as function of the ratio Sb/Ce in a previous work [5] showing that a ratio higher than 0.8 could ensure castings free of CHG. This result agrees with previous conclusion by Tsumura et al. [10]. Going further in this line, Fig. 6 shows that the ratio (Sb+La)/(Mg+Ce) may be more relevant, it defines an envelop that shows a clear decrease of V V when the value of the ratio increases. This feature is similar to what was obtained with the ratio Sb/Ce, but the number of points defining the envelop was observed to be higher with this new ratio than with the previous one. It is seen however that there are still quite a number of points showing a large range of V V results at the minimum value (Sb+La)/(Mg+Ce)≈0.05. This is seen as the indication that better control of melt chemistry should be performed for a full mastering of chunky graphite formation.

Conclusions

This work has demonstrated the detrimental effect of increasing the carbon equivalent of nodular cast irons on chunky graphite formation. Such an effect may be essentially "mechanistic", in that hypoeutectic alloys do contain less eutectic, while hypereutectic alloys present less austenite dendrites and show higher nodule counts. This latter feature agrees with previous reports that have shown that increase of nodule count through post-inoculation does lead to higher tendency for chunky graphite formation. Some correlations between eutectic recalescence in cast blocks and in TA cups that have been described previously for near-eutectic alloys appear to be less appropriate for hypo-or hyper-eutectic alloys. The sensitivity of the alloys to the presence of low level elements and impurities has been stressed again, and this is in this line, i.e. a better control of "active" elements, that further studies should be performed for an improved understanding of the conditions leading to chunky graphite formation.

Fig. 1 .

 1 Fig. 1. Micrography at the centre of block M5-14I.

Fig. 2 .

 2 Fig. 2. Cooling curves recorded from the TA cups cast with non post-inoculated #14 and #15 melts.The characteristic temperatures considered in this work are indicated. As no recalescence showed up during the bulk eutectic reaction of M5-14N TA-cup, the T EN and T EU values were taken equal to the temperature at the inflexion point at the very beginning of the eutectic plateau.

Fig. 3 .

 3 Fig. 3. Correlation between recalescence during bulk eutectic reaction and V V values.

Fig. 4 .

 4 Fig. 4. Plot of V V versus ∆T max measured on blocks (the arrow indicates the point related to the post-inoculated hypo-eutectic alloy).

Fig. 5 .Fig. 6 .

 56 Fig. 5. Plot of V V versus ∆T from TA cups. The arrows show points related to hypo-and hypereutectic alloys.

Table 1 .

 1 Chemical composition (wt. %) of the alloys, carbon equivalent (wt. %) and stable eutectic temperature (°C).

	block	C	Si	P	S	Cu Mg	Ce	La	Sb* C eq	T eut
	M5-9N M5-9I	3.65	1.92 0.018 0.014 0.02 0.039 0.0029 0.0020	---	4.19 1162.1
	M5-9ISi		2.33*							4.30 1163.9
	M5-14N M5-14I	3.82 2.41 0.035	---	0.03 0.055 0.0028 0.0024	---	4.49 1164.2
	M5-14ISb								0.008
	M5-15N M5-15I	3.13 2.08 0.030 0.002 0.02 0.042 0.0026 0.0021	---	3.71 1162.8
	M5-15ISb								0.008

Table 2 .

 2 Microstructure data.

	Block	A A [%]	V V [%]	N block [mm -2 ]	Flotation
	M5-9N	10	1.9	47	No
	M5-9I	28	4.3	75	Yes
	M5-9ISi	39	6.1	135	No
	M5-14N	10 11.1	50	Yes
	M5-14I	10 17.6	88	Yes
	M5-14ISb 0	0	100	Yes
	M5-15N	0	0	31	No
	M5-15I	5	7.2	75	No
	M5-15ISb 0	0	75	No

Table 3 .

 3 Characteristic temperatures (°C) evaluated on TA cups and blocks.

				TA cups					Blocks
	Trial	1 L T	2 L T	1 EN T	2 EN T	T EU	T ER	1 L T	2 L T	1 EN T	2 EN

Acknowledgements

This paper is based on work supported by the Industry Department of the Spanish Government (PROFIT FIT-030000-2007-94).


Paul Ervé

Tchoupé Ngnekou

Marie-Christine C Lafont
email: bmariechristine.lafont@ensiacet.fr

François Senocq
email: cfrançois.senocq@ensiacet.fr

Jacques Lacaze
email: jacques.lacaze@ensiacet.fr

Bernard Viguier
email: ebernard.viguier@ensiacet.fr

P E T Ngnekou

Short Term Oxidation of a Ti-46Al-8Nb Alloy at 700°C in Air

Keywords: Titanium aluminides based on TiAl, Oxidation, Rutile, Amorphous oxide

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Owing to their low density and good mechanical properties, TiAl-based intermetallics have been considered to have a high potential for applications as structural materials at intermediate and high temperatures. Their oxidation resistance above 800°C is, however, a concern but this resistance may be improved by alloying, e.g. with niobium. In their pioneering work, Choudhury et al. [START_REF] Choudhury | Properties of high temperature alloys[END_REF] noticed that TiAl is an alumina former when oxidized in oxygen and titania former when oxidised in air, at 950°C. Later, Meier et al. [START_REF] Meier | Oxidation of High-Temperature Intermetallics[END_REF] stressed the very peculiar oxidation behaviour of binary Al-rich TiAl alloys that form a continuous alumina layer in air at high temperature (1100-1350°C) and in oxygen at much lower temperature (800°C), but do not form such a continuous alumina layer at high temperature in oxygen.

Since then, the so-called nitrogen effect associated to oxidation under air has been extensively studied as reviewed by Brady et al. [START_REF] Brady | Materials Science and Technology, Corrosion and Environmental Degradation[END_REF]. According to Rakowski et al. [START_REF] Rakowski | [END_REF], oxidation of a Ti-50Al (at. %) alloy at 800-900°C in air leads to a scale of intermixed Al 2 O 3 and TiN grains after 1 h. These TiN precipitates transform to TiO 2 with further oxidation while new TiN precipitates appear at the metal/scale interface. The scale becomes layered with an outer TiO 2 layer and an inner layer made of TiO 2 and Al 2 O 3 . Within the inner layer, the concentration of alumina is higher close to the outer layer. The same scale structure was found by Locci et al. [5] after long-term (9000 h) oxidation of Ti-48Al-2Cr-2Nb at 704°C in air. At higher temperatures, Taniguchi and Shibata [6] observed a similar layered oxide although there was no TiN. These authors also reported that when TiAl is alloyed, the alloying elements may form specific oxides that precipitate close to the substrate/oxide interface [6].

Because of this limited oxidation resistance associated with the nitrogen effect, present applications of TiAl alloys are foreseen for intermediate temperatures, in the 600 to 800°C range. However, even short-time exposure to oxygen at such intermediate temperatures leads to dramatic alloy embrittlement, the possible mechanisms of which have been discussed by Thomas et al. [7]. This suggested studying the very first stages of oxidation at a temperature representative of possible service temperatures, namely 700°C.

Experimental details

Small parallelepiped-shape specimens (10x10x1 mm 3 ) were machined from a cylindrical rod 20 mm in diameter and 70 mm in length provided by the Interdisciplinary Research Centre (IRC) in Materials for High Performance Applications (Birmingham U.K.). The composition of the alloy is Ti-45.1Al-8.5Nb (at. %). The specimens were polished using SiC papers up to the grade 4000, then carefully washed in an ultrasonic bath containing acetone. Isothermal oxidation experiments were conducted in a SETARAM TAG24 thermo-balance, which has a precision of 1 g. For reproducibility checking, two specimens were successively submitted to the same oxidation test. After placing the coupon in the TGA cell, the reaction and reference tubes were flushed with argon and then filled with synthetic air (20% O 2 and 80% N 2 ). The gas flow rate was then kept constant to about 0.4 L.h -1 during the tests for continuous renewal of the oxidizing environment. The specimens were brought to 700°C by continuous heating at a rate of 1°C.s -1 . The weight change was recorded every 30 seconds in the first experiment and every 15 seconds in the second. After 50 h, the specimens were allowed to cool down to room temperature at a rate of 1°C.s -1 and were then removed for X-Ray and TEM analysis.

XRD data were collected in glancing incidence configuration on a Seifert XRD 3000TT apparatus fitted with a diffracted beam graphite monochromator. CuK  radiation was used in the present study. TEM observations of the thin oxide layer were performed using a JEOL JEM 2010 microscope at the "Toulouse Paul Sabatier University Temscan Service". This TEM operates at 200 kV and is equipped with a TRACOR EDX spectrometer. Cross-sections of the oxide layer were prepared by cutting the specimens into thin slices (in a direction normal to the oxide/substrate interface) with a diamond wire saw. Two such slices were then glued together, oxide to oxide, and embedded in a 3mm diameter brass tube with epoxy resin. After curing, the tube was sectioned into approximately 300m thick discs that were then polished on both sides and dimpled before ionmilling to transparency with a low angle (0-10°) GATAN precision ion-beam polishing system (PIPS).

Results

The results of the isothermal oxidation experiments are shown in Fig. 1, where the square of the mass gain per unit area (∆m/A) 2 is plotted as a function of time for linearity testing that would correspond to diffusion-controlled oxidation kinetics. Both curves present exactly the same features, and the final mass gains are very similar. At the very beginning of the experiments and up to about 30 min, the mass gain increases very rapidly, as previously observed by Lidong Teng et al. [8] for the same alloy. These authors considered this initial oxidation stage to be a chemical reaction controlled stage at the surface of the material. They noticed that its duration increases from 150 s at 1200°C to 600s at 900°C. The present results confirm this tendency with a much longer reaction period of about 1800 s at 700°C. After this initial period, the mass-gain curves present three consecutive linear domains, labelled 1, 2, and 3 in Fig. 1, with a gradual reduction of the slope. This behaviour shows that oxidation kinetics at 700°C certainly follow three different parabolic-like regimes in the first 50 h, after a very short initial transient stage. It is worth noting that the slope changes occur exactly at the same time in both experiments underscoring their reproducibility. The mass gains corresponding to those transitions are slightly different, mainly because of different mass gains during the initial transition stage. Glancing-angle (4°) X-Ray diffraction was carried out before and after oxidation. The records are compared in Fig. 2, on which mainly appear the peaks related to the phases  2 -Ti 3 Al and -TiAl of the substrate. However, upon close examination, some additional faint peaks can be observed. One peak at 2 = 36.97° appears on both records and is thus not related to the oxidation experiment. Several peaks also appear on the oxidized sample: i) three peaks at 2and 62.75° that could be indexed according to the lines of the rutile; and ii) two peaks at 2 and corresponding to titanium nitride. It should therefore be stressed that no peak of any form of crystalline alumina (eta, gamma, theta or alpha) could be identified in the diffractogram.

----TiAl ----Ti Fig. 3 shows a TEM image of the oxidized specimen surface and four diffraction patterns (labelled 1 to 4) corresponding to various selected areas using a 250 nm selected area aperture. For each of the patterns, the actual position of the aperture was superimposed on the image with a dotted circle. The scale appears bi-layered, with the outer and inner layers being, respectively, about 100 nm and 150 nm thick. The surface of the substrate in contact with the scale has a composition that differs from the bulk material; this is called the intermediate zone, as described below in Fig. 3. The diffraction pattern #1 shows no spots, meaning that the outer layer of the oxide scale is amorphous. Pattern #2 was obtained selecting an area containing both layers of the scale and shows multiple ring-like spots indicating the presence of fine crystals in the inner layer. These rings could be indexed according to the tetragonal rutile TiO 2 with the a=b=0.45933 nm and c=0.29592 nm lattice parameters [START_REF] Villars | Pearson's handbook[END_REF] using the CaRine Crystallography software [START_REF] Boudias | [END_REF]. As with pattern #1, no trace of crystallised alumina could be found in this figure. Pattern #3 shows a superposition of the ringlike spots obtained from the inner scale and the spots of the -TiAl phase. Finally, the area selected for pattern #4 gives only spots corresponding to -TiAl.

In Fig. 3, the internal layer of the scale shows many dark crystallites which range in size from 20 to 50 nm. Fig. 4-a shows a bright field image from another location where the scale presents similar characteristics. In the imaged area, the outer layers of the scale were abraded during TEM thin foil preparation, so that the inner layer is now at the edge of the hole. The diffraction pattern recorded on one of these crystallites is shown in Fig. 4-b and was identified as the diffraction of the   111 zone axis of the tetragonal rutile. The ) 0 1 1 ( reflection was used to obtain the dark field image shown in Fig. 4-c. Many crystallites are thus highlighted, suggesting that a similar orientation may arise from an epitaxial nucleation and growth on the same support that could be one of the phases of the substrate.

Table 1 lists the average values of the semi-quantitative EDS analyses that were performed to estimate the composition of the layers in various locations (shown in Fig. 3). Analyses of the bulk material gave Nb contents that were too high when compared to the nominal composition of the alloy, thus indicating that the present results allow, at best, for qualitative comparison. In the intermediate zone, some Nb-rich areas were found, but most of them appear to be slightly depleted in Al with respect to the substrate. Measurements on the dark crystallites in the inner oxide layer showed higher Ti contents and lower Al contents than on the remaining areas, confirming the indexing as TiO 2 . Finally, the outer oxide layer appears as an Al-rich oxide.

Discussion

Based on the above results, the external layer is considered to be an amorphous aluminium rich oxide layer, and the inner layer consists of the same amorphous oxide with fine titania crystallites.

The fact that Ti is present in alumina is in line with data reported in the literature, e.g. by Lu Wei et al. [11]. The observation of an outer alumina rich scale layer agrees with reports on the same alloy by Godlewska et al. [12] and Lidong Teng et al. [8] for short term oxidation at 800°C [12] and in the 900 to 1200°C range [8]. In agreement with results by Wei Lu et al. [11] on a Ti-46.5Al-5Nb alloy oxidized in air at 900°C, Lu Wei et al. [11] and Godlewska et al. [12] observed the usual scale structure with an outer TiO 2 -rich layer, an intermediate Al 2 O 3 enriched layer and an inner layer containing nitrides and niobium-rich precipitates after longer term exposure to air. While the known beneficial effect of niobium on the oxidation resistance of TiAl alloys may be complex, the above results suggest that its efficiency is increased at alloying levels higher than 5 at. % when it leads to the formation of a continuous alumina layer. While it may, at first, be surprising to find that amorphous oxide may develop during oxidation at intermediate temperature, a review of the literature showed that this does indeed occur. Wei Lu et al. [11] have reported that after 5 min of oxidation, the scale formed on Ti-46.5Al-5Nb alloy consisted of TiO 2 crystals and amorphous alumina, the latter transforming to κ-alumina upon further oxidation. Similarly, Guo et al. [13] also reported the formation of an amorphous alumina layer in the scale during the early stages of oxidation of a Ti 42 Al 48 Cr 8 Ag 2 alloy at 900°C. In this case, they found an outermost layer of fine TiO 2 crystals followed by an inner amorphous alumina layer and thirdly a layer of mixed titanium and aluminium nitrides. Indeed, amorphous alumina has previously been observed in a large range of oxidised alloys. Podgursky et al. [14] and Rose et al. [15] showed detailed observations of alumina changes in the amorphous alumina sequence on the surface of a CoAl intermetallic compound exposed to oxygen. They reported that the oxidation at room temperature with subsequent annealing leads to a phase transition from amorphous to a well-ordered aluminium oxide at about 727°C. Haerig et al. [16] and Shimizu et al. [17] also reported initial amorphous Al 2 O 3-rich scales during the initial oxidation stages in air of a Ni 3 Al alloy at 500 and 700°C and of an Al-0.5wt%Mg alloy at 450 and 550°C.

Conclusion

When alloyed with a sufficient amount of niobium, TiAl alloys lead to the formation of an aluminium rich outer oxide layer during air oxidation, for temperatures in the 700 to 1200°C range. This layer appears to be amorphous after 50 hours at 700°C. The inner layer consists of the same amorphous oxide with fine titania crystallites. Further work is in progress to locate the TiN precipitates observed via X-Ray but not with TEM. An attempt is being carried out to look for any relationship between the slope changes in the oxidation kinetics curves and the layered structure of the resulting scale.

Figure 1 .

 1 Figure 1. Isothermal oxidation curves of the Ti-46Al-8Nb alloy at 700°C for 50 h. The square of the mass gain per unit area, (∆m/A) 2 , is plotted as a function of time.

  Figure 2. X-Ray diffractograms recorded on asreceived and oxidised samples.

Figure 3 .

 3 Figure 3. Cross section TEM bright field image of the oxide scale and diffraction patterns (labelled 1 to 4) from four different locations indicated with dotted circles.

Figure 4 .

 4 Figure 4. Bright field (a) and dark field (c) images of the cross section of an oxidized surface, and diffraction pattern of a well-defined crystal (b).

Table 1 .

 1 Semi-quantitative cross section EDS analyses of the scale and subscale (at. %).

	zone	Element	O	Al	Ti	Nb	Observation
	As received material	----	38 48	49 39	13 13	 2 -Ti 3 Al lamellae -TiAl lamellae
	Intermediate zone	--	28	55	17	 2-Ti 3 Al: Al depleted (and/or Ti enriched) and Nb enriched areas.
			--	43	41	16	-TiAl: Al depleted (and/or Ti enriched) and Nb enriched areas.
	Internal oxide	59 50	23 16	14 28	4 6	Al-rich areas Ti-rich areas near dark crystallites
	External oxide	62	20	14	4	Al-rich oxide

Acknowledgements

The present work was carried out within the framework of the European project IMPRESS (Intermetallic Materials Processing in Relation to Earth and Space Solidification), contract FP6-500635.


Serhii Babenko

Michael Defoort

Mohamed Djemai

Serge Nicaise

Valerii Hrytsenko

Stability analysis of quasilinear systems on time scale based on a new estimation of the upper bound of the time scale matrix exponential function

Keywords: Matrix exponential function, time scale theory, stability analysis, quasilinear systems

In this paper, we analyze the stability of quasilinear systems on time scale based on a new estimation of the upper bound of the time scale matrix exponential function. First, some upper bounds for the norm of the matrix exponential function e A (t, t 0 ) are derived, where A is a regressive square matrix, t, t 0 ∈ T, T being an arbitrary nonempty closed subset of the set of real numbers R, which is called a time scale. The matrix exponential function generalizes the usual matrix exponential as well as the integer power of a matrix: if T = R and A is a constant matrix, then e A (t, t 0 ) = e A(t-t 0 ) , while if T = Z and I + A is invertible, then e A (t, t 0 ) = (I + A) (t-t 0 ) . Using Putzer algorithm and estimating the maximum value of a function defined and bounded on a time scale T, we derive an upper bound for e A (t, t 0 ) in the form: ∥e A (t, t 0 )∥ ≤ Ke α (t, t 0 ), where t, t 0 ∈ T, K is a positive constant, α(t) is a function defined on T and e α (t, t 0 ) is a scalar exponential function. One of the upper bounds is used for stability investigation of quasilinear systems on a time scale.

Introduction

Time scale theory, initiated by Stefan Hilger in his PhD thesis in 1988, is a promising area of mathematical science whose topic of study is functions defined on an arbitrary nonempty closed subset of the set of real numbers R. In this theory, this subset is called a time scale. The main tasks of this theory are to unify concepts and results from continuous and discrete analysis and generalize the ones for any time scale. Differential equations from continuous analysis and difference equations from discrete analysis can be unified by time scale dynamic equations using time scale theory.

Many famous results from the theory of ordinary differential equations, calculus of variations, theory of difference equations have been generalized using the time scale theory in [START_REF] Agarwal | Dynamic inequalities on time scales[END_REF][START_REF] Bohner | Dynamic equations on time scales: An introduction with applications[END_REF][START_REF] Bohner | Advances in dynamic equations on time scales[END_REF][START_REF] Dacunha | Stability for time varying linear dynamic systems on time scales[END_REF][START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF]. Recently, many experts in the time scale theory are interested in the study of the stability of solutions of dynamic equations on a time scale as well as the stabilization problem on a time scale. In particular, in [START_REF] Malik | Controllability of singular dynamic systems on time scales[END_REF][START_REF] Xiao | Positivity and stability of delayed timescale-type differential-difference equations[END_REF][START_REF] Xiao | Positivity and stability of coupled differential-difference equations with time-varying delay on time scales[END_REF] various properties of solutions of dynamic equations on time scales, such as stability, positivity, controllability, are investigated. The monograph [START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF] has investigated different approaches for stability analysis of time scale dynamic equations: i) approaches based on dynamic integral inequalities; ii) approaches based on a generalization of the direct Lyapunov method for equations on time scales; iii) approaches which are an adaptation of the comparison method which is well-known for continuous-time and discrete-time systems. In [START_REF] Lu | Stability analysis of switched systems on time scales with all modes unstable[END_REF][START_REF] Taousser | Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions[END_REF][START_REF] Taousser | Stability analysis of a class of switched linear systems on non-uniform time domains[END_REF][START_REF] Kumar | Finite-time stability and stabilization results for switched impulsive dynamical systems on time scales[END_REF] the stability of switched systems on a time scale was studied. In [START_REF] Bohner | The linear quadratic regulator on time scales[END_REF][START_REF] Bohner | The linear quadratic tracker on time scales[END_REF][START_REF] Poulsen | Optimal control on stochastic time scales[END_REF], the control design for dynamic equations on time scales was studied. In the papers [START_REF] Babenko | On the consensus tracking investigation for multi-agent systems on time scale via matrix-valued Lyapunov functions[END_REF][START_REF] Babenko | Distributed leaderfollower consensus for a class of semilinear second-order multiagent systems using time scale theory[END_REF][START_REF] Guo | Prescribed-time bipartite consensus for signed directed networks on time scales[END_REF][START_REF] Lu | An improved stability theorem for nonlinear systems on time scales with application to multi-agent systems[END_REF][START_REF] Poulsen | Mean square consensus of doubleintegrator multi-agent systems under intermittent control: A stochastic time scale approach[END_REF] the consensus problem for multi-agent systems using time scale theory was investigated.

When investigating the stability of solutions of dynamic equations on a time scale, it is often necessary to estimate the norm of the matrix exponential function e A (t, t 0 ), which unifies the usual matrix exponential e A(t-t 0 ) from continuous analysis and the integer power of (I + A) from discrete analysis.

In the literature, one can find solutions for the problem of estimating the norm e A (t, t 0 ) for some particular cases. For instance, in [START_REF] Pötzsche | A spectral characterization of exponential stability for linear time-invariant systems on time scales[END_REF], when A is diagonal, the upper bound of the norm of e A (t, t 0 ) is obtained in the form: ∥e A (t, t 0 )∥ ≤ Ke α (t, t 0 ), ∀t, t 0 ∈ T(t ≥ t 0 ), [START_REF] Agarwal | Dynamic inequalities on time scales[END_REF] where T is an arbitrary time scale, K, α are known parameters and e α (t, t 0 ) is a scalar exponential function. If A is a Jordan block, a similar estimation is also obtained. However, for this case, the formula for calculating K in explicit form is not given. In [START_REF] Taousser | Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions[END_REF], for an arbitrary matrix A, another estimation for ∥e A (t, t 0 )∥ of the form (1) is obtained and the formula for calculating the constant K is proposed. According to that formula, one has to solve a maximum problem on the unbounded subset of the time scale T, which can be a very difficult task in practice. In the recent paper [START_REF] Russo | Matrix measures, stability and contraction theory for dynamical systems on time scales[END_REF] the estimation [START_REF] Agarwal | Dynamic inequalities on time scales[END_REF] with K = 1 and α = m(A, t) is obtained, where m(A, t) denotes a specific matrix measure (in the paper, it is called "initial growth rate") of A.

In the present paper, a new upper bound for ∥e A (t, t 0 )∥ of the form (1) is obtained and new formulas for calculating constant K are proposed. The new bound is applied for stability investigation of quasilinear systems on a time scale. The novelties of this paper are as follows:

• The estimation holds for an arbitrary matrix A and any time scale T.

• The proposed formulas allow to calculate K by solving a maximum problem on the bounded or even finite subset of T.

• The proposed estimation of ∥e A (t, t 0 )∥ is more accurate than the one from the paper [START_REF] Russo | Matrix measures, stability and contraction theory for dynamical systems on time scales[END_REF].

• It is shown that the proposed estimation can be used for stability analysis of dynamic equations on a time scale.

The paper is organized as follows. Section 2 contains mainly basic facts from the calculus on time scale. In Section 3, we deduce three upper bounds for the norm of the matrix exponential function. The first one has the form of a product of the exponential function and some ∆-differentiable function. The two other bounds have the form of the right hand part of inequality [START_REF] Agarwal | Dynamic inequalities on time scales[END_REF]. In Section 4, one of the bounds is used for stability investigation of quasilinear systems on a time scale. Some mathematical notations are used throughout this paper. For a square matrix A of size n × n, let Sp A be the spectrum of A and let λ i (A), i = 1, • • • , n be the eigenvalues of A repeated according to its multiplicity. ∥ • ∥ denotes the spectral norm of a square matrix, i.e. ∥A∥ = λ max (A T A), where λ max (A

T A) = max i=1,••• ,n λ i (A T A). As usual, I n ∈ R n×n is the identity matrix.

Preliminary results

In this section, we give some facts from calculus on time scales and propose few concepts which will be needed to state our main result. More information on this topic can be found in [START_REF] Bohner | Dynamic equations on time scales: An introduction with applications[END_REF][START_REF] Bohner | Advances in dynamic equations on time scales[END_REF][START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF].

According to [START_REF] Bohner | Dynamic equations on time scales: An introduction with applications[END_REF], a time scale T is an arbitrary nonempty closed subset of the set of real numbers R. For any t ∈ T, the forward jump operator σ is defined by σ(t) = inf{s ∈ T : s > t}. For each t ∈ T, µ(t) = σ(t) -t is called the graininess of time scale T. Let µ * and µ * denote sup t∈T µ(t) and inf t∈T µ(t), respectively. Further, let us recall the definitions of ∆-derivative, regressivity and exponential function.

Definition 1. [5, Def. 1.2.1] A function f : T → R
is said to be ∆differentiable at the point t ∈ T if there exists α ∈ R such that for any ε > 0, there exists a neighborhood U of t such that the inequality

| [f (σ(t)) -f (s)] -α[σ(t) -s] |< ε | σ(t) -s | holds for all s ∈ U . In this case, we denote f ∆ (t) = α. If f : T → R is ∆- differentiable at any t ∈ T, then it is called ∆-differentiable.
In [START_REF] Bohner | Dynamic equations on time scales: An introduction with applications[END_REF][START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF] and other monographs on time scales calculus one can find the properties of ∆-derivative. Based on them, the following formula for calculating of ∆-derivative can be obtained:

f ∆ (t) = lim τ →µ(t)+0 f (t + τ ) -f (t) τ . (2)
Definition 2. [5, Def. 1.4.1] A rd-continuous function p : T → R is said to be regressive and we write p ∈ R if 1 + µ(t)p(t) ̸ = 0 for all t ∈ T. Moreover, p is called positively regressive and we write p ∈ R + if 1 + µ(t)p(t) > 0 for all t ∈ T.

If a real number a satisfies the condition 1 + µ(t)a > 0, for all t ∈ T, then we will write similarly: a ∈ R + .

Denote by R(T; C) the space of rd-continuous, regressive mappings from T to C, where C denotes the set of complex numbers.

The generalized exponential function of scalar function p ∈ R is defined hereafter.

e p (t, s) = exp t s ξ µ(τ ) (p(τ ))∆τ , t ∈ T,
where ξ h is a cylindric transformation defined by the formula

ξ h (z) = 1 h Log(1 + zh), if h > 0, z, if h = 0, ( 3
)
where Log is the principal logarithm function.

Here, b a f (t)∆t denotes the Cauchy integral of f , defined on a time scale.

Further, the matrix exponential function is defined. If T = R, then e A (t, t 0 ) = e A(t-t 0 ) , while if T = Z and I + A is invertible, then e A (t, t 0 ) = (I + A) (t-t 0 ) .

Definition 4. [5, Def. 1.5.2] A rd-continuous n × n-matrix-valued function A(t) is called regressive if I n + µ(t)A(t) is invertible for all t ∈ T. The class of such matrices A is denoted by R = R(T) = R(T, R n×n ).
The next theorem gives us an algorithm to compute the matrix exponential function:

Theorem 1. [2, Thm 5.35] Let A ∈ R be a constant n × n-matrix. Suppose t 0 ∈ T. If λ 1 , λ 2 , ..., λ n are the eigenvalues of A, then e A (t, t 0 ) = n-1 i=0 r i+1 (t)P i , (4)
where r(t) := (r 1 (t), r 2 (t), ..., r n (t)) T is the solution of the initial value problem

r ∆ =         λ 1 0 0 • • • 0 1 λ 2 0 . . . . . . 0 1 λ 3 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 1 λ n         r, r(t 0 ) =        1 0 0 . . . 0        , (5)
and the P -matrices P 0 , P 1 , ..., P n are recursively defined by P 0 = I n and

P k+1 = (A -λ k+1 I n )P k , for 0 ≤ k ≤ n -1.
The following lemma gives us the estimation of the norm of e At .

Lemma 2. [23, Lemma 10.2.1] If ν i are the eigenvalues of a matrix A and

Λ = max i ℜ(ν i ), then ∥e At ∥ ≤ e Λt n-1 k=0 (2t∥A∥) k k! , t ≥ 0. (6)
The following functions, called generalized polynomials, are defined in [START_REF] Bohner | Dynamic equations on time scales: An introduction with applications[END_REF] recursively as follows:

h 0 (t, s) ≡ 1, ∀s, t ∈ T, (7)
and given h k for k ∈ N 0 , the function h k+1 is

h k+1 (t, s) = t s h k (τ, s)∆τ, s, t ∈ T. (8)
In [START_REF] Bohner | Dynamic equations on time scales: An introduction with applications[END_REF] the functions h k (t, s) are explicitly given for the cases T = R and T = Z, namely:

• if T = R, then h k (t, s) = (t-s) k k!
for all s, t ∈ R;

• if T = Z, then h k (t, s) = C k t-s for all s, t ∈ Z, where C k m with k ∈ N 0 , m ∈ R, denotes the general binomial coefficient m(m-1)•...•(m-k+1) k! .
Following [START_REF] Bohner | Dynamic equations on time scales: An introduction with applications[END_REF], let us introduce a notion, which plays a significant role to obtain our main result. Definition 6. Given a function λ ∈ R(T; C) such that 1 + µ(t)λ(t) ̸ = 0, for all t ∈ T, then

ℜ µ(t) (λ) = lim s→µ(t)+0 |1 + sλ(t)| -1 s (9)
for t ∈ T is the Hilger real part of λ.

Note, that for any complex number z, ℜ 0 (z) = ℜ(z), where ℜ(z) denotes the real part of z.

Estimation of the matrix exponential function on a time scale

In this section, we deduce the upper bounds for the norm of the matrix exponential function. This will be done according to the following steps: 1) using formula (4) from Theorem 1 (Putzer algorithm), we estimate the norm of the matrix exponential function e A (t, t 0 ) by the product e η 2 (t, t 0 )f (t, t 0 ), where η 2 characterizes the spectrum of matrix A and f (•, t 0 ) is a ∆-differentiable function.

2) considering an arbitrary ε > 0 and rewriting the obtained estimation in the form e η 2 ⊕ε (t, t 0 )e ⊖ε (t, t 0 )f (t, t 0 ), we prove that for positive values of ε the function F (t) = e ⊖ε (t, t 0 )f (t, t 0 ) is bounded on [t 0 , +∞). We estimate from above the maximum value of F (t) and, as a result, obtain an upper bound for ∥e A (t, t 0 )∥.

The next theorem investigates the first point of this scheme. In Theorem 4, the second point is deduced. Theorem 5 gives a modification of the result obtained in Theorem 4. Theorem 3. Let A ∈ R n×n be a matrix. If there exist real numbers η 1 , η 2 ∈ R + such that η 1 < η 2 and the following conditions are satisfied:

(i) η 1 ≤ ℜ µ(t) (λ i (A)) ≤ η 2 , ∀t ∈ T, for each eigenvalue λ i (A); (ii) η := min {1 + µ * η 1 , 1 + µ * η 1 } > 0.
Then the following estimation holds for all t 0 , t ∈ T

∥e A (t, t 0 )∥ ≤ e η 2 (t, t 0 ) n-1 k=0 2∥A∥ η k h k (t, t 0 ), ( 10
)
where t ≥ t 0 and the functions h k are defined by formulas [START_REF] Xiao | Positivity and stability of delayed timescale-type differential-difference equations[END_REF] and (8).

Proof. In order to prove Theorem 3, we propose to use the representation of the matrix exponential function from Theorem 1 and the idea of deducing the estimation (6) from Lemma 2. From formula (4) we get the estimation:

∥e A (t, t 0 )∥ ≤ n-1 k=0 |r k+1 (t, t 0 )| ∥P k ∥, (11)
where r k+1 (t, t 0 ) denotes r k+1 (t) from formula (4). To deduce [START_REF] Taousser | Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions[END_REF]

|e λ k (t, t 0 )| ≤ e η 2 (t, t 0 ), ∀t ≥ t 0 , k = 1, ..., n. (12)
According to Definition 3 of the exponential function, the following formula holds for all t ≥ t 0 and k = 1, 2, ..., n:

|e λ k (t, t 0 )| = exp t t 0 ξ µ(τ ) (λ k )∆τ = exp t t 0 ℜ ξ µ(τ ) (λ k ) ∆τ , (13)
where ξ µ(τ ) (λ k ) is the cylindric transformation defined by [START_REF] Bohner | Advances in dynamic equations on time scales[END_REF]. From (3) we get the formulas:

ℜ ξ µ(τ ) (λ k ) = ℜ Log(1+µ(τ )λ k ) µ(τ ) , if µ(τ ) > 0, ℜ(λ k ), if µ(τ ) = 0. = ln |1+µ(τ )λ k | µ(τ ) , if µ(τ ) > 0, ℜ(λ k ), if µ(τ ) = 0. ( 14
)
which hold for all τ ∈ T and k = 1, 2, ..., n. According to condition (i) of the present theorem ℜ µ(t) (λ k ) ≤ η 2 for all t ∈ T and for any k = 1, 2, ..., n. Using this property, Definition 6 of the Hilger real part ℜ µ(t) (λ) and the positive regressivity of η 2 ∈ R + , we obtain from ( 14) the following estimation:

ℜ ξ µ(τ ) (λ k ) ≤ ln(1+µ(τ )η 2 ) µ(τ ) , if µ(τ ) > 0, η 2 , if µ(τ ) = 0. = Log(1+µ(τ )η 2 ) µ(τ ) , if µ(τ ) > 0, η 2 , if µ(τ ) = 0. = ξ µ(τ ) (η 2 ), (15)
for all τ ∈ T and k = 1, 2, ..., n. Consequently, using properties of an integral on time scale (see Theorem 1.3.3 in [START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF]), from ( 13) and ( 15) we get [START_REF] Kumar | Finite-time stability and stabilization results for switched impulsive dynamical systems on time scales[END_REF].

For the next step we show that (12) implies the following estimation of the function |r k (t, t 0 )|:

|r k (t, t 0 )| ≤ e η 2 (t, t 0 ) h k-1 (t, t 0 ) η k-1 , (16)
which holds for all t ≥ t 0 from T and k = 1, 2, ..., n. The proof of ( 16) is made by induction on k. Let k = 1, then according to formula (5) the function r 1 (t, t 0 ) is a solution of the initial value problem:

r ∆ 1 = λ 1 r 1 , r 1 (t 0 ) = 1. ( 17
)
As it is known (see Theorem 2.33 from [START_REF] Bohner | Dynamic equations on time scales: An introduction with applications[END_REF]), this solution can be presented in the form r 1 (t, t 0 ) = e λ 1 (t, t 0 ). Then, ( 12) implies the estimation:

|r 1 (t, t 0 )| = |e λ 1 (t, t 0 )| ≤ e η 2 (t, t 0 ) = e η 2 (t, t 0 )h 0 (t, t 0 ) 1 η 0 ,
which means that estimation ( 16) holds for k = 1. Now, we assume that ( 16) holds for some arbitrary natural number l (1 ≤ l ≤ n -1), i.e.

|r l (t, t 0 )| ≤ e η 2 (t, t 0 ) h l-1 (t, t 0 ) η l-1 , (18)
and prove that estimation (16) holds for l + 1. According to formula (5) the function r l+1 (t, t 0 ) is a solution of the initial value problem:

r ∆ l+1 = r l + λ l+1 r l+1 , r l+1 (t 0 ) = 0. ( 19
)
The solution of problem [START_REF] Lu | An improved stability theorem for nonlinear systems on time scales with application to multi-agent systems[END_REF] is found by the formula of variation of constants (see Theorem 2.77 from [START_REF] Bohner | Dynamic equations on time scales: An introduction with applications[END_REF]):

r l+1 (t, t 0 ) = t t 0 e λ l+1 (t, σ(τ ))r l (τ, t 0 )∆τ.
Taking into account property (vii) from Theorem 1.4.2 given in [START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF], the last identity can be written in the form:

r l+1 (t, t 0 ) = t t 0 e λ l+1 (t, τ ) 1 + µ(τ )λ l+1 r l (τ, t 0 )∆τ. ( 20
)
From condition (i) and Definition (6) of the Hilger real part, it follows that

|1 + µ(τ )λ l+1 | ≥ 1 + µ(τ )η 1 ,
for all τ ∈ T. Using this estimation and condition (ii), we obtain the following estimation for the denominator of the integrand of the last integral in [START_REF] Poulsen | Mean square consensus of doubleintegrator multi-agent systems under intermittent control: A stochastic time scale approach[END_REF]:

|1 + µ(τ )λ l+1 | ≥ η > 0, ∀τ ∈ T. (21)
Then, using ( 12), ( 18) and ( 21) we deduce from (20) the following estimation:

|r l+1 (t, t 0 )| ≤ t t 0 e λ l+1 (t,τ ) 1+µ(τ )λ l+1 r l (τ, t 0 ) ∆τ ≤ t t 0 eη 2 (t,τ ) η e η 2 (τ, t 0 ) h l-1 (τ,t 0 ) η l-1 ∆τ = eη 2 (t,t 0 ) η l t t 0 h l-1 (τ, t 0 )∆τ = eη 2 (t,t 0 ) η l h l (t, t 0 ),
which proves that the estimation ( 16) holds for all t ≥ t 0 and k = 1, 2, ..., n. Now, let us estimate ∥P k ∥ from above. As it is known, max

i=1,n |λ i | ≤ ∥A∥.
Consequently,

∥P k ∥ = ∥(A -λ k I n )P k-1 ∥ ≤ ∥A -λ k I n ∥∥P k-1 ∥ ≤ (∥A∥ + |λ k |)∥P k-1 ∥ ≤ 2∥A∥∥P k-1 ∥, (22)
for all k = 1, 2, ..., n. Since ∥P 0 ∥ = ∥I n ∥ = 1, then [START_REF] Russo | Matrix measures, stability and contraction theory for dynamical systems on time scales[END_REF] implies:

∥P k ∥ ≤ (2∥A∥) k , (23)
for all k = 1, 2, ..., n. Thereby, using estimations ( 16) and ( 23) we obtain from inequality [START_REF] Taousser | Stability analysis of a class of switched linear systems on non-uniform time domains[END_REF] the estimation [START_REF] Taousser | Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions[END_REF]. The proof is completed.

Example 2. It is easy to check that Lemma 2 is a corollary of Theorem 3.

Let us set

T = R, η 1 = min i ℜ(λ i (A)), η 2 = max i ℜ(λ i (A)), t 0 = 0. ( 24
)
Then, all conditions of Theorem 3 hold. Using Example 1, we get from these assumptions:

e A (t, t 0 ) = e At , e η 2 (t, t 0 ) = e η 2 t , h k (t, t 0 ) = t k k! , η = 1. ( 25
)
Finally, from (10), we get (6).

Example 3. Let us set

T = Z, η 1 = min i |1 + λ i (A)| -1, η 2 = max i |1 + λ i (A)| -1, (26)
and assume that 1+η 1 > 0. Then, all conditions of Theorem 3 hold. Denoting t -t 0 = m and using Example 1, we get from these assumptions:

e A (t, t 0 ) = (I + A) m , e η 2 (t, t 0 ) = (max i |1 + λ i (A)|) m , η = min i |1 + λ i (A)|, h k (t, t 0 ) = C k m . ( 27
)
Taking into account the last formulas, we get from (10) the following estimation:

∥(I + A) m ∥ ≤ max i |1 + λ i (A)| m × n-1 k=0 C k m 2∥A∥ min i |1+λ i (A)| k , ∀m ∈ N. (28)
The next theorem gives another estimation for the norm of the matrix exponential function. This estimation has a form: ∥e A (t, t 0 )∥ ≤ M ε e η 2 ⊕ε (t, t 0 ), where ε and M ε are positive constants. Before formulating the next theorem, let us introduce the function ω ε (µ):

ω ε (µ) = 1 ε n-2 k=0 (1 + µε) k n-1 , ∀ε > 0, (29)
and the functions defined for all t 0 ∈ T by the following formulas:

t 1 (t 0 ) = inf s ∈ T : s ≥ t 0 + (n -1)η 2∥A∥ , (30)
t(t 0 ) = sup {s ∈ T : s ≤ t 0 + ω ε (µ * )} , t(t 0 ) = inf {s ∈ T : s ≥ t 0 + ω ε (µ * )} , (31)
φ(t) = n-1 k=0 2∥A∥ η k h k (t, t 0 ), ∀t ∈ T, t ≥ t 0 . ( 32
)
Theorem 4. Let A ∈ R n×n be a matrix such that there exist real numbers η 1 , η 2 ∈ R + with η 1 < η 2 such that the conditions (i) and (ii) from Theorem 3 are satisfied. Then, for an arbitrary t 0 ∈ T and a real number ε > 0 such that:

ω ε (µ * ) ≥ t 1 (t 0 ) -t 0 , (33)
and for any t ≥ t 0 from the scale T, the following estimate holds:

∥e A (t, t 0 )∥ ≤ L(ε)e η 2 ⊕ε (t, t 0 ), ( 34
)
where the positive constant L(ε) is given by:

L(ε) = max φ(t 1 (t 0 )), n(2∥A∥/η) n-1 (n-1)! × max s∈[t(t 0 ),t(t 0 )] ((s -t 0 ) n-1 e ⊖ε (s, t 0 )) . ( 35
)
Proof. As all conditions of Theorem 3 are satisfied, estimation (10) holds. Let us take an arbitrary t 0 ∈ T and consider ε > 0 which satisfies condition (33) (it is not difficult to see that such a number exists, indeed from (29) we see that for any µ ≥ 0, lim ε→0+ ω ε (µ) = +∞, consequently, for an arbitrary t 1 (t 0 ) -t 0 there exists ε > 0 such that ω ε (µ) ≥ t 1 (t 0 ) -t 0 ). Then, [START_REF] Taousser | Stability analysis of a class of uncertain switched systems on time scale using Lyapunov functions[END_REF] can be written in the following form: ∥e A (t, t 0 )∥ ≤ e η 2 (t, t 0 )e ε (t, t 0 )e ⊖ε (t, t 0 )φ(t) = e η 2 ⊕ε (t, t 0 )e ⊖ε (t, t 0 )φ(t).

(

) 36
The main idea is to estimate the function e ⊖ε (t, t 0 )φ(t) from the right-hand side of (36), by a constant from above. As we will show, this function is bounded, so such a constant exists. The best constant is clearly the maximum value of the mentioned function on the interval [t 0 , +∞), which appears to be very difficult to compute. Indeed, for the specific time scale T = R, e ⊖ε (t, t 0 )φ(t) is a real-valued function of a real variable. Hence, to find its maximum value one can apply the theory of extrema from classical analysis. But in practice, the maximum point of that function can be found only approximately by numerically solving the algebraic equation of the n-th degree.

In the case of an arbitrary time scale, we will be faced with more difficulties in finding the maximum value of e ⊖ε (t, t 0 )φ(t). These difficulties are caused by the following facts. Firstly, Ferma's theorem (also known as interior extremum theorem) does not hold for functions defined on a time scale. Namely, in general, if a function has a local extremum at some point of a time scale and is ∆-differentiable there, then its ∆-derivative at that point is not necessarily equal to zero1 . Second, for an arbitrary time scale the function φ(t) is not, in general, a polynomial, but has a more complex structure.

In order to overcome these difficulties, we estimate the function φ(t) from above by the simplest power function of the form C(t -t 0 ) n-1 , where C is a constant. The advantage of this estimate is that the maximum problem of the function C(t -t 0 ) n-1 e ⊖ε (t, t 0 ) on the interval [t 0 , +∞), as it will be shown, can be reduced to the problem of maximizing the same function on a subset which is finite or at least bounded.

Following the steps above, let us show by induction that the functions h k (t, t 0 ) satisfy the following estimation:

h k (t, t 0 ) ≤ (t -t 0 ) k k! , ∀t ≥ t 0 , t ∈ T, (37)
for all k = 0, 1, 2, ..., n -1.

As h 0 (t, t 0 ) ≡ 1 ≤ (t-t 0 ) 0 0! , (37) clearly holds for k = 0. Now, we assume that (37) holds for some arbitrary natural number l (0 ≤ l ≤ n -2), i.e.:

h l (t, t 0 ) ≤ (t -t 0 ) l l! , ∀t ≥ t 0 , t ∈ T, (38)
and prove that (37) holds for l+1. According to this assumption and formula (8), we have:

h l+1 (t, t 0 ) = t t 0 h l (s, t 0 )∆s ≤ t t 0 (s -t 0 ) l l! ∆s. ( 39
)
It is easy to show that

(s -t 0 ) l l! ≤ (s -t 0 ) l+1 (l + 1)! ∆ , ∀s ∈ [t 0 , t] ∩ T. (40)
Indeed, using the formula (2) for calculating the ∆-derivative, we obtain:

(s-t 0 ) l+1 (l+1)! ∆ = lim τ →µ(s)+0 (s+τ -t 0 ) l+1 -(s-t 0 ) l+1 τ (l+1)! = (l+1)(s-t 0 ) l + l+1 k=2 C k l+1 (s-t 0 ) l+1-k µ k-1 (s) (l+1)! = (s-t 0 ) l l! + l+1 k=2 C k l+1 (l+1)! (s -t 0 ) l+1-k µ k-1 (s) ≥ (s-t 0 ) l l!
, for all s ∈ [t 0 , t] ∩ T. Thus, using (40) in estimate (39) we get:

h l+1 (t, t 0 ) ≤ t t 0 (s-t 0 ) l+1 (l+1)! ∆ ∆s = (t-t 0 ) l+1 (l+1)! , (41)
which is (37) for l + 1. We then conclude that estimation (37) holds for all k = 0, 1, 2, ..., n -1. Then (37) implies the following estimation for φ(t):

φ(t) ≤ n-1 k=0 2∥A∥ η k (t-t 0 ) k k! = n-1 k=0 2∥A∥(t-t 0 ) η k 1 k! . (42)
Let us continue to estimate the function φ(t) from above. Obviously, for a fixed k ∈ N, the inequality

2∥A∥(t -t 0 ) η k 1 k! ≤ 2∥A∥(t -t 0 ) η k+1 1 (k + 1)! holds for t ≥ t 0 + (k+1)η 2∥A∥ .
Hence, for all t ≥ t 0 + (n-1)η 2∥A∥ , the inequality

2∥A∥(t -t 0 ) η k 1 k! ≤ 2∥A∥(t -t 0 ) η n-1 1 (n -1)!
holds for all k = 0, 1, ..., n -1. Using this estimate, we obtain from (42):

φ(t) ≤ n (n -1)! 2∥A∥(t -t 0 ) η n-1 , (43)
for all t ≥ t 1 (t 0 ), t ∈ T (recalling that t 1 (t 0 ) is defined by (30)). At the same time, since

φ ∆ (t) = n-1 k=0 2∥A∥ η k h ∆ k (t, t 0 ) = n-1 k=0 2∥A∥ η k h k-1 (t, t 0 )
and h k (t, t 0 ) > 0 for all k ∈ N 0 and t ≥ t 0 from the time scale, φ(t) increases on the set [t 0 , +∞) ∩ T. It implies the following estimation of φ(t) on the interval [t 0 , t 1 (t 0 )] of the time scale:

φ(t) ≤ φ(t 1 (t 0 )). (44)
Estimations ( 43) and ( 44) can be combined into one formula, in which the functions on the right are defined on T:

φ(t) ≤    φ(t 1 (t 0 )), t ∈ [t 0 , t 1 (t 0 )], n (n-1)! 2∥A∥(t-t 0 ) η n-1 , t ≥ t 1 (t 0 ). ( 45
)
This estimation will allow us to estimate the function f (t) = e ⊖ε (t, t 0 )φ(t) appropriately.

Since ε > 0, then according to the property (x) of the exponential function from Theorem 1.4.2 given in [START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF], we obtain e ε (t, t 0 ) > 0. Next, according to the property (viii) from the same Theorem, we get that e ∆ ε (t, t 0 ) = εe ε (t, t 0 ) > 0, i.e. the function e ε (t, t 0 ) increases with respect to variable t on the interval [t 0 , +∞) of the time scale T. Therefore e ε (t, t 0 ) ≥ e ε (t 0 , t 0 ) = 1 and e ⊖ε (t, t 0 ) = 1 eε(t,t 0 ) ≤ 1, for all t ≥ t 0 from the time scale. Taking into account this estimation as well as the estimation (45), the function f (t) can be estimated from above as follows:

f (t) ≤ φ(t 1 (t 0 )), (46)
for all t ∈ [t 0 , t 1 (t 0 )] from the time scale T. Let us estimate f (t) from above on the interval [t 1 (t 0 ), +∞) of the time scale T. From (45), we obtain:

f (t) ≤ n (n -1)! 2∥A∥(t -t 0 ) η n-1 • e ⊖ε (t, t 0 ), (47)
for all t ∈ [t 1 (t 0 ), +∞) from the time scale T. Consider the function f 1 (t) = (t -t 0 ) n-1 e ⊖ε (t, t 0 ). Using formula (2), property (3) from Theorem 1.2.2 and properties (iii) and (viii) from Theorem 1.4.2 given in [START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF], we can find its ∆-derivative:

f ∆ 1 (t) = (t -t 0 ) n-1 ∆ e ⊖ε (σ(t), t 0 ) + (t -t 0 ) n-1 e ∆ ⊖ε (t, t 0 ) = e ⊖ε (t,t 0 ) 1+µ(t)ε (t -t 0 ) lim s→µ(t)+0 1-n-1 √ 1+sε s + 1 × n-2 k=0 (t -t 0 + µ(t)) n-2-k (t -t 0 ) n-1 1 + µ(t)ε k .
One can calculate the limits from the last expression: lim

s→µ(t)+0 1-n-1 √ 1+sε s = -1 ωε(µ(t)) . Consequently, f ∆ 1 (t) = e ⊖ε (t,t 0 ) 1+µ(t)ε t 0 -t ωε(µ(t)) + 1 × n-2 k=0 (t -t 0 + µ(t)) n-2-k (t -t 0 ) n-1 1 + µ(t)ε k .
Let us investigate increasing and decreasing intervals of the function f 1 . From formula (29), we obtain:

ω ε (µ * ) ≤ ω ε (µ(t)) ≤ ω ε (µ * ), ∀t ∈ T. ( 48
) If t ∈ [t 1 (t 0 ), t 0 + ω(µ * )] ∩ T, then: -t-t 0 ωε(µ(t)) ≥ -t 0 +ωε(µ * )-t 0 ωε(µ(t)) = -ωε(µ * ) ωε(µ(t)) ≥ -ωε(µ * ) ωε(µ * ) = -1. Hence, f ∆ 1 (t) ≥ 0, for all t ∈ [t 1 (t 0 ), t 0 + ω ε (µ * )] ∩ T, and: f ∆ 1 (t) ≥ 0, ∀t ∈ [t 1 (t 0 ), t(t 0 )] ∩ T.
Due to properties of ∆-derivative and integral on the time scale, the last inequality means that the function f 1 (t) is increasing on [t 1 (t 0 ), t(t 0 )]:

f 1 (t) ≤ f 1 (t(t 0 )), ∀t ∈ [t 1 (t 0 ), t(t 0 )] ∩ T. ( 49
)
It is similarly proved, that:

f ∆ 1 (t) ≤ 0, ∀t ∈ [t(t 0 ), +∞) ∩ T, which means that f 1 (t) is decreasing on t ∈ [t(t 0 ), +∞): f 1 (t) ≤ f 1 (t(t 0 )), ∀t ∈ [t(t 0 ), +∞) ∩ T. ( 50
)
From ( 49) and ( 50) we obtain:

f 1 (t) ≤ max s∈[t(t 0 ),t(t 0 )] f 1 (s), ∀t ∈ [t 1 (t 0 ), +∞) ∩ T. (51)
Hence, taking into account (47), we get:

f (t) ≤ n (n-1)! 2∥A∥ η n-1 × max s∈[t(t 0 ),t(t 0 )] ((s -t 0 ) n-1 e ⊖ε (s, t 0 )) , (52)
for all t ≥ t 1 (t 0 ) from the time scale T. Combining estimations ( 46) and ( 52), from inequality (36) we get the estimation (34), where L(ε) is defined by (35). Theorem 4 is proved.

Theorem 5. Let A ∈ R n×n be a matrix. If there exist real numbers η 1 , η 2 ∈ R + with η 1 < η 2 such that conditions (i) and (ii) from Theorem 3 are satisfied. Then, for an arbitrary t 0 ∈ T and a real number ε > 0 such that (33) is fulfilled and for any t ≥ t 0 from the scale T, the following estimate holds:

∥e A (t, t 0 )∥ ≤ L ′ (ε)e η 2 ⊕ε (t, t 0 ), ( 53
)
where the positive constant L ′ (ε) is given by:

L ′ (ε) = max n-1 k=0 2∥A∥(t 1 (t 0 )-t 0 ) η k 1 k! , n (n-1)! 2∥A∥ η n-1 × max s∈[t(t 0 ),t(t 0 )] (s -t 0 ) n-1 e ⊖ε (s, t 0 ) . ( 54
)
Proof. The proof of this theorem is the same as the one of Theorem 4. The only difference is to obtain the constant L ′ (ε), instead of estimation (44) the following estimation is proposed:

φ(t) ≤ n-1 k=0 2∥A∥(t 1 (t 0 ) -t 0 ) η k 1 k! ,
for all t ∈ [t 0 , t 1 (t 0 )] from T, which holds due to (42) and (44).

The following proposition gives a specific way to find η 1 , η 2 , which satisfy conditions (i) and (ii) of Theorem 3.

Proposition 1. For a matrix A ∈ R n×n let us set:

η 1 = min i=1,n ℜ µ * (λ i (A)), η 2 = max i=1,n ℜ µ * (λ i (A)). ( 55
)
If η 1 , η 2 ∈ R + and, besides, η 1 satisfies condition (ii) from Theorem 3, then for an arbitrary t 0 ∈ T and real ε > 0 such that (33) is fulfilled, estimates (34)-( 35) and (53)-( 54) hold for any t ≥ t 0 from T.

Proof. Let us check condition (i) of Theorem 3 for the values η 1 , η 2 defined by (55). For this purpose, let us prove for any real numbers µ 1 , µ 2 ≥ 0 and z ∈ C, the following statement is true:

ℜ µ 1 (z) ≤ ℜ µ 2 (z) whenever µ 1 < µ 2 . ( 56
)
Suppose 0 ≤ µ 1 < µ 2 and consider arbitrary z ∈ Z and real numbers s 1 , s 2 such that s 2 > s 1 > 0. Then, we obtain

|1+zs 1 |-1 s 1 = 1 s 1 + z -1 s 1 = 1 s 1 + z + 1 s 2 -1 s 2 -1 s 1 ≤ 1 s 2 + z + 1 s 1 -1 s 2 -1 s 1 = 1 s 2 + z -1 s 2 = |1+zs 2 |-1 s 2 .
Taking in both sides of the inequality the limit as s 1 → µ 1 +0 and s 2 → µ 2 +0, we obtain (56).

From inequality (56), we obtain the following inequalities:

η 1 = min i=1,n ℜ µ * (λ i (A)) ≤ ℜ µ * (λ j (A)) ≤ ℜ µ(t) (λ j (A)) ≤ ℜ µ * (λ j (A)) ≤ max i=1,n ℜ µ * (λ i (A)) = η 2 ,
for all j = 1, n. Consequently, condition (i) of Theorem 3 holds. Now, we can conclude that all conditions of Theorems 4 and 5 are satisfied, i.e. estimates (34)-( 35) and ( 53)-( 54) hold for any t ≥ t 0 from T. Theorem is proved.

Remark 1. It is easy to check that L(ε) ≤ L ′ (ε), for all ε > 0, i.e. the estimation (53) is rougher than (34). However, for specific time scales estimating L ′ (ε) is simpler than L(ε), because in (54), we do not need to calculate the value of the function h k (t, t 0 ). In such cases, it is better to use the estimation (53).

Remark 2. In the cases T = R or T = hZ (h > 0), the time scale interval [t(t 0 ), t(t 0 )] from the formula (35) degenerates to one or two points. In such cases, the problem of calculating the maximum of this formula becomes trivial.

Indeed, if T = R, then µ(t) ≡ 0 on T and µ * = µ * = 0, so ω ε (µ * ) = ω ε (µ * ) = ω ε (0) = n-1 ε . Then t(t 0 ) = t(t 0 ) = t 0 + ω ε (0) = t 0 + n-1 ε .
Hence, in the formula (35) we get:

max s∈[t(t 0 ),t(t 0 )] (s -t 0 ) n-1 e ⊖ε (s, t 0 ) = (t(t 0 ) -t 0 ) n-1 e ⊖ε (t(t 0 ), t 0 ) = (t(t 0 ) -t 0 ) n-1 e ⊖ε (t(t 0 ), t 0 ). If T = hZ, then µ(t) ≡ h on T and µ * = µ * = h, so ω ε (µ * ) = ω ε (µ * ) = ω ε (h). Further two cases are possible. If t 0 + ω ε (h) ∈ hZ, then t(t 0 ) = t(t 0 ) = t 0 + ω(h). If t 0 + ω ε (h) /
∈ hZ, then t(t 0 ) < t(t 0 ), and between t(t 0 ) and t(t 0 )

there is no point from the scale hZ. In the last case, in the formula (35) we get: max

s∈[t(t 0 ),t(t 0 )] ((s -t 0 ) n-1 e ⊖ε (s, t 0 )) = max (t(t 0 ) -t 0 ) n-1 e ⊖ε (t(t 0 ), t 0 ), (t(t 0 ) -t 0 ) n-1 e ⊖ε (t(t 0 ), t 0 ) . Remark 3.
If T is a time scale, which has bounded graininess function µ(t) and consists of isolated points only, then the segment [t(t 0 ), t(t 0 )] consists of a finite set of points of the scale. Therefore, in this case the maximum problem from (35) is simple.

Based on Remark 2, we obtain the following corollaries from Theorem 5.

Corollary 1. For any matrix A ∈ R n×n and positive real number ε > 0 such that ε < 2∥A∥, the following estimation holds for all t ∈ R + :

∥e At ∥ ≤ max n-1 k=0 (n-1) k k! , n (n-1)! 2∥A∥(n-1) εe n-1 ×e (max i
ℜ(λ i (A))+ε)t . (57)
Proof. Making assumptions [START_REF] Babenko | Nonlinear dynamic inequalities and stability of quasilinear systems on time scales[END_REF], we obtain (25) and, besides,

η = 1, t 1 = (n-1) 2∥A∥ , t = t = n-1 ε , φ(t) = n-1 k=0 (2∥A∥t) k k! .
Then, the estimation (34) becomes (57).

Corollary 2. For any matrix A ∈ R n×n and positive real number ε > 0 such that min

i=1,n |1 + λ i (A)| > 0 and ω 1 < ω ε (1)
, the following estimation holds for all m ∈ N:

∥(I + A) m ∥ ≤ max i=1,3 L i • (1 + ε) max i |1 + λ i (A)| m , (58)
where

L 1 = n-1 k=0 C k ω 1 2∥A∥ min i |1+λ i (A)| k , L 2 = n (n-1)!(1+ε) ω 2∥A∥ω min i |1+λ i (A)| n-1 , L 3 = n (n-1)!(1+ε) ω 2∥A∥ω min i |1+λ i (A)| n-1 , ω 1 = inf s ∈ Z : s ≥ (n-1) min i |1+λ i (A)| 2∥A∥ , ω = sup {s ∈ Z : s ≤ ω ε (1)} , ω = inf {s ∈ Z : s ≥ ω ε (1)} .
Proof. Making assumptions [START_REF] Nasser | Sufficient conditions for uniform exponential stability and h-stability of some classes of dynamic equations on arbitrary time scales[END_REF] and denoting t -t 0 = m, we obtain (27) and, besides,

t 1 = inf s ∈ Z : s ≥ t 0 + (n-1) min i |1+λ i (A)| 2∥A∥ = t 0 + ω 1 , t = sup {s ∈ Z : s ≤ t 0 + ω ε (1)} = t 0 + ω, t = inf {s ∈ Z : s ≥ t 0 + ω ε (1)} = t 0 + ω, φ(t) = n-1 k=0 C k m 2∥A∥ min i |1 + λ i (A)| k .
Then the estimation (34) becomes (58).

The following example shows the advantage of the estimations we propose over the ones proposed in [START_REF] Russo | Matrix measures, stability and contraction theory for dynamical systems on time scales[END_REF].

Example 4 Let us consider the matrix A, which is a 4-dimensional Jordan block, whose diagonal is filled with 2. The objective is to derive estimations of ∥e A (t, t 0 )∥ for the cases T = R and T = Z using Corollaries 1 and 2.

It is easy to check, that ∥A∥ ≈ 2.85. Setting ε = 0.2, one can check that conditions of Corollaries 1 and 2 are satisfied. Hence, calculating the constants in formulas (57) and (58), we obtain the following estimations for ∥e A (t, 0)∥ for time scales T = R (formula (59)) and T = Z (formula (60)):

∥e A (t, 0)∥ = ∥e At ∥ ≤ 20772 • e 2.2t , ∀t ∈ R + , ( 59
) ∥e A (t, 0)∥ = ∥(I + A) t ∥ ≤ 1015 • 3.6 t , ∀t ∈ N. (60)
It should be emphasized that these estimations are particular cases of the same formula (34).

As it was noted in the Introduction, in the paper [START_REF] Russo | Matrix measures, stability and contraction theory for dynamical systems on time scales[END_REF] another approach to estimate ∥e A (t, t 0 )∥ is proposed. Applying this approach, we obtain the following estimation of ∥e A (t, t 0 )∥ in case T = R (formula (61)) and T = Z (formula (62)):

∥e A (t, 0)∥ = ∥e At ∥ ≤ e 2.81t , ∀ t ∈ R + . ( 61
) ∥e A (t, 0)∥ = ∥(I + A) t ∥ ≤ ∥I + A∥ t ≤ 3.84 t , (62)
for all t ∈ N ∪ {0}.

In Figure 1, the graphs of the common logarithms of the estimations from ( 59) and (61), as well as ∥e At ∥ are presented. Such a representation via logarithm is used in order to show the behaviour of these functions on a large time interval As we can see, for time scales T = R and T = Z, our estimations are more precise than estimations obtained from the approach proposed in [START_REF] Russo | Matrix measures, stability and contraction theory for dynamical systems on time scales[END_REF].

Stability analysis of quasilinear systems on a time scale

There are many publications devoted to stability investigation of quasilinear systems on a time scale. A review of the papers [START_REF] Babenko | Nonlinear dynamic inequalities and stability of quasilinear systems on time scales[END_REF][START_REF] Ben Nasser | On stability and stabilization of perturbed time scale systems with gronwall inequalities[END_REF][START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF][START_REF] Nasser | Sufficient conditions for uniform exponential stability and h-stability of some classes of dynamic equations on arbitrary time scales[END_REF][START_REF] Russo | Matrix measures, stability and contraction theory for dynamical systems on time scales[END_REF] on the stability investigation of quasilinear systems by the method of integral inequalities shows that all stability conditions obtained in these papers contain so-called growth constants K and α from estimation [START_REF] Agarwal | Dynamic inequalities on time scales[END_REF]. This fact demonstrates the importance of the estimation [START_REF] Agarwal | Dynamic inequalities on time scales[END_REF]. In this section we show how the estimations (34)-( 35) and ( 53)-(54) that we have proposed, allow to investigate exponential stability of quasilinear systems on a time scale.

Let us consider the following system:

x ∆ (t) = Ax(t) + f (t, x), (63)
where x : T → R n is the unknown, A is n × n regressive matrix, the vector function f : T × R n → R n is an rd-continuous function on T, satisfying f (t, x) = 0 if and only if x = 0 for all t ∈ T.

According to Definition 2.2.1 from [START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF] the solution x = 0 of system (63) is exponentially stable, uniformly with respect to t if there exist constants γ, λ > 0 (-λ ∈ R + ) such that for any t 0 ∈ T and x 0 ∈ R n , for the solutions x(t; t 0 , x 0 ) of system (63) the following holds for all t ≥ t 0 ∈ T: ∥x(t; t 0 , x 0 )∥ ≤ ∥x(t 0 )∥γe -λ (t, t 0 ). In [START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF] it is proved (see Theorem 2.4.3) that as soon as system (63) satisfies the following conditions: 1) the state x = 0 of the system x ∆ (t) = Ax(t) is exponentially stable, i.e. there exist numbers L, λ > 0 (-λ ∈ R + ) such that ∥e A (t, t 0 )∥ ≤ Le -λ (t, t 0 ), for all t ≥ t 0 ;

2) there exists a constant γ > 0 such that ∥f (t, x)∥ ≤ γ∥x∥, for all (t, x) ∈ T × S, where S is defined as S = {x ∈ R n : ∥x∥ < H, H = const > 0};

3) λ -γL > 0, then the solution x = 0 of system (63) is exponentially stable on T. Since our Proposition 1 provides an appropriate procedure to check condition 1) and estimate the involved constants L and λ, it is a powerful tool to obtain some exponential stability results for system (63). Let us illustrate this tool by the following example. x = 0 is exponentially stable. For this aim let us first find estimation of ∥e A (t, t 0 )∥, using Theorem 5 and Proposition 1. Indeed, let us set ε = 0.1 and find some bounds on the constants ∥A∥, µ * , µ * , η, ω ε (µ * ), t 1 (t 0 ), t(t 0 ), t(t 0 ), which is necessary to get the estimation. Since

µ(t) = k 2k+1 if t = k, 0, if t ∈ k + k 2k+1 , k + 1 , then µ * = inf t∈T µ(t) = 0, µ * = sup t∈T µ(t) = 1 2 .
Besides, ∥A∥ ∈ [3.5366, 3.5368] and Sp A = {-0.4, -1 -i, -1 + i}. According to formulas (55):

η 1 = min i=1,3 ℜ(λ i (A)) = -1, η 2 = max i=1,3 ℜ 0.5 (λ i (A)) = -0.4.
Using η 1 , we obtain that η = 0.5 > 0. According to formula (29),

ω ε (µ) = 1+ √ 1+µε ε . Then: ω ε (µ * ) = 20, ω ε (µ * ) = 20 + δ,
where δ = 10( √ 1.05 -1). Let us go on with t 1 (t 0 ). Due to formula (30)

t 1 (t 0 ) = inf {s ∈ T : s ≥ t 0 + η/∥A∥}. If t 0 ∈ [k + 1 -η ∥A∥ , k + 1] for k ∈ N 0 , then t 0 + η ∥A∥ ∈ [k + 1, k + 1 + η ∥A∥ ]
. Due to the fact that η ∥A∥ ≈ 0.5 3.5367 ≈ 0.14, and the length of each "gap" of T is larger than 1 3 , we conclude that

t 0 + η ∥A∥ ∈ [k + 1, k + 1 + k+1 2k+3 ], which implies that t 1 (t 0 ) = k + 1 + k+1 2k+3 .
For the remaining values of t 0 it can be seen that t 0 + η ∥A∥ is an interior point for one of the segments k + k 2k+1 , k + 1 , which implies that in this case t 1 (t 0 ) = t 0 + η ∥A∥ . Based on the analysis of t 1 (t 0 ), we obtain the following estimation:

t 1 (t 0 ) -t 0 ≤ 1 2 + η ∥A∥ . (64)
In order to calculate L we have also to estimate t(t 0 ) as well as t(t 0 ). Since ω ε (µ * ) = 20 and ω ε (µ * ) = 20 + δ, then according to formula (31) we get:

t(t 0 ) = sup {s ∈ T : s ≤ t 0 + 20} , t(t 0 ) = inf {s ∈ T : s ≥ t 0 + 20 + δ} . (65)
Using formulas (65), we can calculate t(t 0 ) and t(t 0 ) and then use these numbers for calculating L ′ (ε) by formula (54). Since t(t 0 ) and t(t 0 ) depend on t 0 , L ′ (ε) also depends on t 0 . This is an obstacle to the application of conditions 1)-3), which guarantee the exponential stability of the solution x = 0, since in 1) and 3) L is a constant. Let us consider segments i, i + i 2i+1 and estimate e ε i + i 2i+1 , i . Based on the structure of the time scale T from the present example, we can conclude that only ends of this segment belong to T and besides i+ i 2i+1 = σ(i). Hence, e ε i + i 2i+1 , i = e ε (σ(i), i) = 1 + µ(i)ε = 1 + iε 2i+1 > 1, for all i ∈ N. Similarly, let us consider segments i + i 2i+1 , i + 1 and estimate e ε i + 1, i + i 2i+1 . All points of that segment are the points from T. Consequently, e ε i + 1, i + i 2i+1 = e (1-i 2i+1 )ε = e ( i+1 2i+1 )ε > e ε 2 , for all i ∈ N. From the estimate (68) and the calculations that follow it, we obtain the following estimation: Finally, using the estimations (66), ( 67) and (69), from ( 53) and (54), we obtain the following estimation:

∥e A (t, t 0 )∥ ≤ 56192 • e -0.3 (t, t 0 ).

From this inequality one can conclude that condition 1) from above holds with L = 56192 and λ = -0.3. Since f (t, x) = 5 • 10 -6 • x sin t, then ∥f (t, x)∥ ≤ 5 • 10 -6 ∥x∥. It is now easy to verify that conditions 2)-3) also hold. Hence, the solution x = 0 of system (63) is exponentially stable.

Conclusion

In this paper, we have studied the matrix exponential function on time scale. This function generalizes the usual matrix exponential e A(t-t 0 ) as well as the m-th power (I + A) m for an arbitrary time scale. Using the time scale theory, we have proposed new estimations of the norm of the matrix exponential function. It is shown that such an estimation can be used for stability investigation of dynamic equations on a time scale.

Acknowledgements

This work is supported by grants of Ministry of Education and Science of Ukraine, project number is 0119U100421.

Definition 3 .

 3 [START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF] Def. 1.4.3] For p ∈ R and s ∈ T, the exponential function e p (•, s) is defined by

Figure 1 :

 1 Figure 1: Logarithms of the function f 1 (t) = ∥e At ∥ (red line) and its estimations f 2 (t) from (59) (blue line) and f 3 (t) from (61) (black line).

Example 5 .f

 5 Let us consider system (63), where A = (t, x) = 5•10 -6 •x sin t, x = (x 1 , x 2 , x 3 ) T , with the time scaleT = P {tσ k ,t k+1} = ∞ k=0 k + k 2k+1 , k + 1 (see Example 3.11 in [10]), and check if the solution

e

  ⊖ε (t(t 0 ), t 0 ) ≤ e ⊖ε (k + 20, k + 1). (68) Now, let us estimate e ⊖ε (k + 20, k + 1). Due to properties of the exponential function and specifics of the time scale T, we obtain the following estimation:e ⊖ε (k + 20, k + 1,i+ i 2i+1 )eε(i+ i 2i+1 ,i) .

e

  ⊖ε (t(t 0 ), t 0 ) <

  from (11) let us estimate |r k (t, t 0 )| and ∥P k ∥ from above. To estimate |r k (t, t 0 )|, let us prove the following estimate:

  To avoid this obstacle let us estimate L ′ (ε) from above. Firstly, we estimate the maximum from the formula (54):(s -t 0 ) 2 • max s∈[t(t 0 ),t(t 0 )] e ⊖ε (s, t 0 ) ≤ (t(t 0 ) -t 0 ) 2 e ⊖ε (t(t 0 ), t 0 ).Here we used the fact that the function e ⊖ε (•, t 0 ) decreases on the set [t 0 , +∞)∩ T. Now we estimate each of the multipliers from the last line of (66). For this aim we have to find estimations for t(t 0 ) and t(t 0 ). Due to δ < 0.25 we obtain that ift 0 ∈ [k + k 2k+1 , k + 1] for some k ∈ N 0 , then t 0 + 20 + δ < k + 21 + k+21 2k+43 ∈ T,which allows us to conclude that t(t 0 ) ≤ k + 21 + k+21 2k+43 in accordance with (65). Similarly, from the inequality t 0 + 20 > k + 20 the following estimation is obtained: t(t 0 ) ≥ k + 20. Now we are able to estimate each multiplier in the inequality (66). For the first one we get:(t(t 0 ) -t 0 ) 2 ≤ k + 21 + k+21 2k+43 -k + kNow let us estimate e ⊖ε (t(t 0 ), t 0 ) from above. Using Theorem 1.4.2 from[START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF] on properties of the exponential function, we obtain: e ⊖ε (t(t 0 ), t 0 ) = e ⊖ε (t(t 0 ), 0)e ⊖ε (0, t 0 ). Then, according to those properties and Remark 1.3.1 from[START_REF] Martynyuk | Stability theory for dynamic equations on time scales[END_REF], we conclude that function e ⊖ε (t, s) increases with respect to the first argument t and decreases with respect to the second one s. Since t 0 ≤ k + 1 and t(t 0 ) ≥ k + 20, the following estimation holds:

	max s∈[t(t 0 ),t(t 0 )]	((s -t 0 ) 2 e ⊖ε (s, t 0 ))
	≤ max			(66)
	s∈[t(t 0 ),t(t 0 )]
		2k+1	2 < 484.	(67)

Preprint submitted to Journal of the Franklin InstituteDecember 25, 2022

To see this, consider the function F (t) = (t -0.1)

defined on a time scale T = Z. At the points t = 0 and t = 1 from the time scale the function has a minimum on T, however f ∆ (0) = f (0 + 1) -f (0) = 0.8 ̸ = 0 as well as f ∆ (1) = f (1 + 1) -f (1) = 2.8 ̸ = 0.