Rémi M Bonetto

Pierre Castel

Stéphane P Robert

Virginie M Tassistro

Magalie Claeys-Bruno

Michelle Sergent

Camille A Delecourt

| Didier Cowen

| Xavier Carcopino

Thierry G Orsi Ere
email: thierry.orsiere@imbe.fr

Thierry G Orsière

Evaluation of PIG-A-mutated granulocytes and ex-vivo binucleated micronucleated lymphocytes frequencies after breast cancer radiotherapy in humans

Keywords: granulocyte, human, micronucleus assay, mutation, PIG-A gene, radiotherapy 1 |

Although the PIG-A gene mutation frequency (MF) is considered a good proxy to evaluate the somatic MF in animals, evidence remains scarce in humans. In this study, a granulocyte PIG-A-mutant assay was evaluated in patients undergoing radiation therapy (RT) for breast cancer. Breast cancer patients undergoing adjuvant RT were prospectively enrolled. RT involved the whole breast, with (WBNRT) or without (WBRT) nodal area irradiation. Blood samples were obtained from participants before (T0) RT, and T1, T2, and T3 samples were collected 3 weeks after the initiation of RT, at the end of RT, and at least 10 weeks after RT discontinuation, respectively. The MF was assessed using a flow cytometry protocol identifying PIG-A-mutant granulocytes. Cytokinesis-blocked micronucleated lymphocyte (CBML) frequencies were also evaluated. Thirty patients were included, and five of them had received chemotherapy prior to RT. The mean (±SD) PIG-A MFs were 7.7 (±12.1) per million at T0, 5.2 (±8.6) at T1, 6.4 (±8.0) at T2 and 3.8 (±36.0) at T3. No statistically significant increases were observed between the PIG-A MF at T0 and the MFs at other times.

RT significantly increased the CBML frequencies: 7.9 ‰ (±3.1‰) versus 33.6‰ (±17.2‰) (p < .0001). By multivariate analysis, the CBML frequency was correlated with age at RT initiation (p = .043) and irradiation volume at RT discontinuation (p = .0001) but not with chemotherapy. RT for breast cancer therapy failed to induce an increase in the PIG-A MF. The PIG-A assay in humans needs further evaluation, in various genotoxic exposures and including various circulating human cells.

optimized protocols related to human blood PIG-A mutation and micronucleated reticulocyte flow cytometric assays have been published [START_REF] Torous | Human blood PIG-A mutation and micronucleated reticulocyte flow cytometric assays: method optimization and evaluation of intra-and inter-subject variation[END_REF]. This highlights the importance of developing tests that can quantify DNA damage in humans.

Research on the PIG-A gene as a genotoxicity biomarker has emerged with studies on paroxysmal nocturnal hemoglobinuria (PNH) [START_REF] Takeda | Deficiency of the GPI anchor caused by a somatic mutation of the PIG-A gene in paroxysmal nocturnal hemoglobinuria[END_REF]. The enzyme encoded by the PIG-A gene is critical for the biosynthesis of the membrane anchor glycosylphosphatidylinositol (GPI). Phenotypically, in PNH patients, PIG-A-mutant cells lack the GPIanchored complement inhibitor, leading to complement-mediated hemolysis and clinical features of the disease [START_REF] Brodsky | Paroxysmal nocturnal hemoglobinuria[END_REF]. In parallel, mutation-carrying blood cells and the corresponding phenotype have been identified in healthy individuals at extremely low frequencies (between 10 and 30 GPI-deficient cells per 10 6 cells) (Araten et al., 1999;[START_REF] Hu | PIG-A mutations in normal hematopoiesis[END_REF]. The PIG-A gene has therefore been considered a potential sentinel gene, being highly conserved among species [START_REF] Kawagoe | Molecular cloning of murine pig-a, a gene for GPI-anchor biosynthesis, and demonstration of interspecies conservation of its structure, function, and genetic locus[END_REF], with a mutation frequency (MF) reflecting the rate of acquired somatic mutations [START_REF] Peruzzi | The use of PIG-A as a sentinel gene for the study of the somatic mutation rate and of mutagenic agents in vivo[END_REF]. Reciprocity between a cell with a mutation at the PIG-A locus and a GPI-deficient phenotype has been recognized by the International Workgroup on Genotoxicity Testing [START_REF] Gollapudi | The in vivo pig-a assay: a report of the International Workshop On Genotoxicity Testing (IWGT) Workgroup[END_REF]. Thus, it is now commonly accepted that GPIdeficient cells carry inactivating mutations within the PIG-A gene and are equally referred to as PIG-A mutants or PIG-A-mutated cells.

Using flow cytometry, several studies have enabled the development of the "PIG-A assay," which estimates the frequency of cells with a mutation in the PIG-A gene by measuring the frequency of GPI anchordeficient circulating cells [START_REF] Kawagoe | Molecular cloning of murine pig-a, a gene for GPI-anchor biosynthesis, and demonstration of interspecies conservation of its structure, function, and genetic locus[END_REF]Araten et al., 1999). Currently, this assay is commonly applied for genotoxicity testing in rodents.

A significant increase in the Pig-a MF has been reported in rodent reticulocyte and erythrocyte populations following exposure to chemotherapeutic agents, such as N-nitroso-N-ethylurea and cisplatin [START_REF] Miura | Accumulation and persistence of pig-a mutant peripheral red blood cells following treatment of rats with single and split doses of N-ethyl-N-nitrosourea[END_REF][START_REF] Phonethepswath | Erythrocyte-based pig-a gene mutation assay: demonstration of crossspecies potential[END_REF][START_REF] Phonethepswath | Pig-a mutation: kinetics in rat erythrocytes following exposure to five prototypical mutagens[END_REF][START_REF] Dertinger | Persistence of cisplatin-induced mutagenicity in hematopoietic stem cells: implications for secondary cancer risk following chemotherapy[END_REF][START_REF] Bhalli | Manifestation and persistence of pig-a mutant red blood cells in C57BL/6 mice following single and split doses of N-ethyl-N-nitrosourea[END_REF], or X and γ ionizing radiation [START_REF] Ohtani | The in vivo pig-a gene mutation assay is useful for evaluating the genotoxicity of ionizing radiation in mice[END_REF][START_REF] Ohtani | Persistence of red blood cells with pig-a mutation in p53 knockout mice exposed to X-irradiation[END_REF][START_REF] Graupner | Gamma radiation at a human relevant low dose rate is genotoxic in mice[END_REF]. In humans, GPI-deficient cell identification is mainly performed for the diagnosis and monitoring of PNH patients [START_REF] Parker | Diagnosis and management of paroxysmal nocturnal hemoglobinuria[END_REF][START_REF] Borowitz | Guidelines for the diagnosis and monitoring of paroxysmal nocturnal hemoglobinuria and related disorders by flow cytometry[END_REF]. In relation to gene mutations, the MF of the PIG-A gene was found to be spontaneously higher not only in patients with cancer predisposition due to defects in DNA repair but also in cancer cells, suggesting the MF of the PIG-A gene as a potential biomarker of genomic instability [START_REF] Olsen | The pig-a gene mutation assay in mice and human cells: a review[END_REF]. A recent study reported higher frequencies of GPI-deficient erythrocytes in patients with esophageal adenocarcinoma than in healthy donors or patients with a preneoplastic status and a major increase in this population in patients receiving chemotherapy [START_REF] Haboubi | Developing a blood-based gene mutation assay as a novel biomarker for oesophageal adenocarcinoma[END_REF]. However, data remain controversial, as such increases were not found in other studies on patients receiving different genotoxic chemotherapy regimens or on soldiers exposed to depleted uranium [START_REF] Dobrovolsky | Monitoring humans for somatic mutation in the endogenous PIG-a gene using red blood cells[END_REF][START_REF] Mcdiarmid | Measures of genotoxicity in gulf war I veterans exposed to depleted uranium[END_REF][START_REF] Olsen | The pig-a gene mutation assay in mice and human cells: a review[END_REF].

The PIG-A assay has been evaluated in humans following ionizing radiation exposure. According to United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) data, medical use of radiation accounts for 98% of the population dose contribution from all artificial sources and represents 20% of the total population exposure to ionizing radiation [START_REF] Unep | Radiation effects and sources: What is radiation? What does radiation do to us? Where does radiation come from?[END_REF]. The UNSCEAR estimated that a 1-Gy exposure yields up to 40 DNA double-strand breaks, 1,000 DNA singlestrand breaks, and 500 damaged bases, including 150 DNA cross-links, per cell (United Nations, 2000). Radiation therapy (RT), as part of cancer treatment, is the most ionizing medical radiation modality. As breast cancer is the most common cancer in women worldwide, with some patients receiving chemotherapy in addition to RT, breast cancer patients are likely to be directly affected by radiation genotoxicity and therefore represent a suitable human model for the evaluation of the PIG-A assay in humans following ionizing radiation exposure.

The aim of this work was to prospectively evaluate the PIG-A assay by quantifying the GPI-deficient granulocyte frequency in women receiving RT for the treatment of breast cancer as a controlled genotoxic environment.

| MATERIALS AND METHODS

| Patients

Patients with recently diagnosed breast cancer were prospectively enrolled over an 18-month period from January 2016 to June 2017.

Patients were included immediately after initial breast surgery, either conservative or radical (i.e., mastectomy). Only patients who were indicated with adjuvant RT were included. Patients with a history of previous RT were not included. Patients were differentiated according to the RT regimen they received: whole breast RT (WBRT; n = 14) and whole breast and nodal RT (WBNRT; n = 16) (Figure 1). Relevant data on personal or familial medical history, smoking habits, alcohol consumption, drug intake, and recent X-ray diagnostic examination were collected for each participant using a standard questionnaire prior to the first blood sampling time point. The protocol received approval from the Sud-Méditerranée 1 Committee for the Protection of Populations (ID RCB: 2015-A01753-46). Written informed consent was obtained from all participants.

| Radiation therapy

RT was delivered in three-dimensional conformational modalities (the usual technique recommended for adjuvant breast irradiation allowing a conformational beam for anatomic irradiation of the planning target volume consisting of the breast and lymph node area) with a Synergy accelerator (Elekta, Stockholm, Sweden), giving a dose rate set at 0.4 Gy/min. All patients received 50 Gy divided into 25 fractions of 2 Gy each. Patients who had a conservative surgery, that is, tumor removal only, received an additional overprint of 16 Gy divided into eight fractions. Patients who underwent total mastectomy, that is, removal of the whole mammary gland, received an additional overprint of 10 Gy divided into five fractions at the level of the scar, according to the risk of skin recurrence. Irradiation of the supraclavicular nodal area and the internal mammary chain was performed in patients with lymph node involvement or with tumors located in the inner part of the breast. Axillary irradiation was delivered in cases of axillary nodal macrometastasis when complementary nodal dissection had not been performed. For the internal mammary chain, the delivered dose was 50 Gy in 25 fractions of 2 Gy each in a combination of 20 Gy of photons and 30 Gy of electrons to spare mediastinal structures. For the supraclavicular and/or axillary area, the delivered dose was 46 Gy in 23 fractions of 2 Gy. The irradiated bone volume was estimated with dosimetry calculations performed using a Philips Pinnacle treatment planning system (Philips Healthcare, Amsterdam, The Netherlands). For each participant, target volumes, organs at risk and the bone volume fraction within the 0.5 Gy isodose volume were contoured. The irradiated bone volume was calculated as an estimation of the volume of the irradiated bone marrow.

| Blood samples

Four blood samples were collected from each patient. The initial sample (T0) was obtained immediately before the initiation of RT. The second (T1), third (T2), and last (T3) samples were collected 3 weeks after the initiation of RT (within 2 hr after the end of the last RT administration), at the end of RT, and at least 10 weeks after RT discontinuation, respectively. Blood samples were collected using anonymized EDTA tubes (Beckton-Dickinson vacutainer) for the PIG-A assay and lithium heparin tubes for the micronucleus assay. Tubes were immediately sent to the "Biogenotoxicology, Human Health & Environment" Laboratory (IMBE-Mediterranean Institute of Biodiversity and Ecology-Medicine School of Marseilles) for immediate processing.

| Quantification of the GPI-deficient granulocyte frequency: The PIG-A assay

Two milliliters of anticoagulant-treated fresh blood was mixed with 20 ml of erythrocyte lysis solution (154 mM ammonium chloride, 10 mM potassium hydrogen carbonate, and 0.1 mM EDTA) for 10 min at room temperature. After centrifugation at 300g for 5 min at room temperature, immunostaining was performed directly on the pellet by adding anti-CD15-PB, anti-CD33-PC7, anti-CD45-Krome Orange, and anti-CD66b-APC antibodies and FLAER-Alexa488 and incubating the mixture for 30 min in the dark at room temperature.

The pellet was washed with PBS and resuspended in PBS before immediate flow cytometry analysis.

Quantification of PIG-A-mutant granulocytes was performed using flow cytometry. Fluorochrome-conjugated antibodies were chosen according to good laboratory practice recommendations for the diagnosis and monitoring of PNH patients. For selection, the antibodies were selected according to their ability to identify granulocytes within whole blood: anti-CD15-PB, anti-CD33-PC7, and anti-CD45-KO. Reagents used for the identification of GPIdeficient cells within the granulocyte population were an anti-CD66b-APC antibody, a GPI-dependent granulocyte-specific antibody, and FLAER-Alexa488, a GPI-specific marker.

Analyses were carried out on a Gallios model flow cytometer (Beckman-Coulter, Villepinte, France). The results are expressed using cytograms. The phenotyping protocol was as follows: first, cell doublets were eliminated using the forward scatter area versus forward scatter height parameters. The positivity of CD45 expression allowed us to focus the analysis on immune cell populations. In this framework, two graphs (CD45 vs. cell structure and then CD45 vs. CD33) made it possible to limit the analysis to nonlymphoid immune cell populations. Two independent GPI myeloid markers, CD15 and CD33, were used to limit the analysis to granulocytes. Selected granulocytes were defined as CD15+/CD33dim, and granulocyte selection was limited to the highest 95% CD15 signal intensity to increase specificity.

Finally, among these granulocytes, FLAER and CD66b doublenegative cells were considered mutants. The analysis was performed on at least 1x10 6 granulocytes, and the acquisition rate was limited to 4,000 events per second to improve the accuracy of the analysis. The primary endpoint was the MF, defined as the number of GPI-deficient F I G U R E 1 Flowchart of the population and sampling schedule. Among the 30 patients, 5 received chemotherapy before enrollment. CBML, cytokinesisblocked micronucleated lymphocytes assay; PIG-A, PIG-A assay; WBNRT, whole breast and nodal area radiation therapy; WBRT, whole breast radiation therapy granulocytes among all granulocytes, determined by flow cytometry and expressed per million events.

The first step of the optimization process consisted of defining the incubation time, antibody concentrations, and flow cytometric parameters (data not shown). Optimization and validation of GPIdeficient granulocyte detection were performed using blood samples from a healthy donor and a PNH patient. This allowed for the validation of the detection strategy using PNH cells as the positive control.

The gating strategy is illustrated in Figure 2. The fluorescence thresholds for FLAER and CD66b below which an event was considered doubly negative were obtained using a blood sample from the PNH patient. The stability of the counting of GPI-deficient cells was confirmed by means of serial dilution producing a number of GPIdeficient cells lower than 50; this value is commonly considered a lower limit for flow cytometry (Figure S1).

| Micronucleus test

The frequencies of cytokinesis-blocked micronucleated lymphocytes (CBMLs) in culture were determined for T0 (before the initiation of radiotherapy) and T2 (at the end of radiotherapy, the mean delay between T0 and T2 was 212.4 [±84.2] days) samples. One aliquot of whole blood was transferred into flasks and incubated at 37 C with 5% CO2 for 44 hr in X-vivo 10 culture medium (Lonza S.p.r.l, Verviers, Belgium) supplemented with 1% heparin (Sigma Aldrich, St. Quentin Fallavier, France) and 1% phytohemagglutinin (Sigma Aldrich). The cells were then treated with cytochalasin B (Sigma Aldrich) and incubated for 24 hr at 37 C with 5% CO2. At the end of the incubation, the cells were centrifuged (220 g, 10 min, at 20 C), treated with 0.075 mM KCl and centrifuged again (150 g, 1 min, at 20 C).

The cells were then treated with fixative (methanol: acetic acid 3:1 v/v) for 20 min at 4 C and centrifuged (300 g, 10 min, 20 C). The cell pellets were resuspended in fixative and stored at 4 C. Lymphocytes in fresh fixative were dropped onto clean slides on ice, air-dried and stained with 2% Giemsa (Sigma Aldrich). Blinded and duplicate MN counts were performed by the same investigator on lymphocytes with well-preserved cytoplasm only. Two thousand binucleated (BN) cells were analyzed for each subject. Cells were cytologically scored using the cytome approach to evaluate the viability status (necrosis, apoptosis), mitotic status (mononucleated, BN, and multinucleated), and presence of micronuclei [START_REF] Fenech | Cytokinesis-block micronucleus cytome assay[END_REF]. The cytokinesis-block proliferation index (CBPI) was calculated as follows: CBPI = (M1 + 2 M2 + 3 M3)/N (where M1-M3 represent the numbers of cells with 1 to 3 or more nuclei and N is the total number of viable cells scored, that is, total cell number excluding necrotic and apoptotic cells). The results are reported as CBMLs per 1,000 cells. Note: Smoking status were described as non-smokers (NS), current (CS), and former smokers (FS) and quantified in pack-years unit (PAU). PIG-A MF were expressed per 10 6 . Average number of analyzed granulocytes in PIG-A assay was 2.7 x 10 6 . Abbreviations: C, Cyclophosphamide; CBML, cytokinesis-blocked micronucleated lymphocytes; E, Epirubicin; F, 5-fluoro-uracil; ND, not determined; PAU, pack-year unit; PTV, planning target volume; PTX, paclitaxel; TXT, docetaxel; WBNRT, whole breast and nodal radiation therapy; WBRT, whole breast radiation therapy.

| Statistical analysis

3 | RESULTS

| Population description and sampling protocol

Patient characteristics are summarized in Table 1. Fourteen (46.6%) patients were included in the WBRT group, and 16 (53.3%) were included in the WBNRT group. The mean age was 61.9 (±12.0) years. Five (16.7%) patients received chemotherapy within 5 months before the initiation of RT. All five received a combination of epirubicin and cyclophosphamide. In addition, three received 5-fluorouracil, two received docetaxel, and two received paclitaxel. Eight (26.7%) patients were smokers (five current smokers and three former smokers for over 2 years). None of the 30 patients were occupationally exposed to genotoxic agents. Within the 3 months preceding RT, 10 (33.3%) patients had at least one X-ray examination, 4 (13.3%) had a PET CT scan, 10 (33.3%) had a bone scan, and 3 (10.0%) had an infectious episode. No patient was occupationally exposed to ionizing radiation; one (3.3%) reported asbestos exposure. The mean delay between the T0 and T1 blood samples was 21.6 (±2.5) days, that between the T0 and T2 blood samples was 44.5 (±4.5) days, and that between the T0 and T3 blood samples was 212.4 (±84.2) days (Table 1).

| Delivered RT doses

Delivered RT doses ranged from 50.1 to 66.9 Gy, with a mean of 63.6 (±5.7) Gy. The proportion of the bone volume within the 0.5-Gy isodose volume was significantly higher in the WBNRT group than in the WBRT group: 589 ± 101 versus 233 ± 96 cm 3 (p < .001) (Table 2).

However, there was no significant difference in the mean RT dose delivered to the bone between the patient groups: 8.5 (±1.4) Gy versus 7.3 (±2.6) Gy, respectively.

| Repeatability of the PIG-A assay

T0 blood samples were tested in duplicate for each patient. Due to a technical issue in one duplicate, data were not available for four patients.

Using a Bland and Altman plot, the technique revealed good repeatability.

Only 2/26 (7.3%) values exceeded the threshold set at ±1.96 SD around the mean of differences between each duplicate (Figure S2).

| Stability of samples over time

T0 blood samples from four patients were stored in the dark at 4 C.

The PIG-A assay was carried out at 0, 24, 48, and 72 hr, and the Note: For each patient, irradiated bone volume and doses were estimated with dosimetry calculations performed by the dedicated software of the treatment planning system. Significant variation was noted in the bone volume contained in the isodose 0.5 Gy. Dose distribution was similar between groups.

Abbreviations: ns, non-significant; WBNRT: whole breast and nodal radiation therapy; WBRT, whole breast radiation therapy. ***p < .001.

F I G U R E 3 Overall GPI-deficient granulocyte frequencies before initiation of radiotherapy (T0), during (T1), at the end of treatment (T2) and during follow-up (T3) (n = 30, NS: non-significant; **: p < .01).

For each time point, lines are the median, boxes are the first and third quartile, whiskers are the minimum and maximum values and the plus sign is the mean. Despite very close data distributions (expressed as median [IQR], mean ± SD, per million granulocytes), statistically significant decrease was noted between T0 (3.9[2.3;8.2], 7.7 ± 12.3) and T1 (2.8[1.5;5.0], 5.1 ± 8.7, p = .0027) and T0 and T3 (2.1[0.4;2.9], 3.8 ± 6.2, p = .0049). No difference was observed between T0 and T2 (3.8[0.4;2.9], 6.4 ± 8.2) distributions results were compared to the T0 value generated at 0 hr (Figure S3).

No significant difference was found in the frequency of PIG-A mutant granulocytes between the 24, 48, and 72-hr time points compared to the 0-hr time point.

| Effect of RT on the PIG-A MF

The quantification of PIG-A-mutant granulocytes was performed in 30 patients before (T0), during (T1) and at the end of RT (T2).

| Effect of chemotherapy performed before the initiation of RT on the PIG-A MF

Compared to patients not treated with chemotherapy, the PIG-A MF at T0 was not significantly different in the five patients who had received chemotherapy before the initiation of RT (Table 1).

| Effects of tobacco and age on the PIG-A MF

| Effect of RT on CBML frequencies: The micronucleus test

This test was performed with the T0 and T2 samples from 16 patients.

The CBML frequency was significantly higher at T2 than at T0 for all F I G U R E 4 Impact of age (a) and smoking habits (b) on GPI-deficient granulocyte frequencies on T0 samples (n = 30). No relationship between GPI-deficient granulocyte frequencies and neither age nor tobacco consumption (expressed in pack year) was noted in the whole population F I G U R E 5 Distribution of CBML frequency per 1,000 cells before (T0) and at the end of radiotherapy (T2), for all patients (n = 16) and in WBRT (n = 7) or WBNRT (n = 9) groups. Radiotherapy significantly increased CBML frequencies in overall, WBRT and WBNRT groups. At T2, CBML frequencies were statistically higher in WBNRT group than in WBRT group. *: p < .05; **: p < .01; ***: p < .001. CBML, cytokinesis-blocked micronucleated lymphocytes; WBNRT, whole breast and nodal radiation therapy; WBRT, whole breast radiation therapy patients: 33.6 (±17.2) versus 7.9 (±3.2) (p < .001). The same differences were observed in the WBRT and WBNRT groups: 21.6 (±10.8) versus 7.4 (±2.2) (p = .0056) and 42.9 (±15.6) versus 8.4 (±3.7) (p < .001), respectively (Figure 5). The CBML frequency was significantly higher at T2 in the WBNRT group than in the WBRT group (p = .0083). The CBML frequency at T0 was not significantly influenced by previous administration of chemotherapy (p = .89). By multivariate analysis, the CBML frequency was significantly correlated with age at T0 (p = .043) and irradiation extent at T2 (p < .0001) but not with chemotherapy or the smoking status (Table 3). A significant association was found between age and the CMBL frequency at T0 (p = .02) and between the CBML frequency at T2 and irradiation extent (p = .02). No associations were found with chemotherapy or the smoking status (Table 3).

| DISCUSSION

To our knowledge, our study is the first to evaluate the value of the PIG-A gene mutation as a biomarker of genotoxicity in humans by detecting GPI-deficient human granulocytes before and after controlled genotoxic exposure. With a median PIG-A MF prior to any treatment of 3.9 x 10 -6 granulocytes (range: 0 to 60.5 x 10 -6 ), our results are similar to those published previously, supporting the validity of our method.

Specifically, for a population of 142 healthy donors, Rondelli et al.

reported a median frequency of PIG-A mutants of 4.9 x 10 -6 (range:

1 x 10 -6 to 37.5 x 10 -6 ) [START_REF] Rondelli | The frequency of granulocytes with spontaneous somatic mutations: a wide distribution in a normal human population[END_REF]. The basal granulocyte PIG-A MFs in our study were consistent with previously published data on erythrocyte PIG-A MFs by Dertinger et al. and Cao et al. (Dertinger et al., 2015;[START_REF] Cao | A population study using the human erythrocyte PIG-A assay: erythrocyte PIG-A assay-based population study[END_REF]. Additionally, Horibata found similar basal PIG-A MFs in total red blood cells among healthy donors and cancer patients and, as we also reported, high spontaneous mutation frequencies for two patients [START_REF] Horibata | Monitoring genotoxicity in patients receiving chemotherapy for cancer: application of the PIG-A assay[END_REF].

Our assay failed to demonstrate any increase in the PIG-A-mutant granulocyte population following RT in 30 patients recently diagnosed with breast cancer. The same lack of increase was also observed in the group with the largest irradiated bone volume (the WBNRT group). Several hypotheses may explain this absence of an increase in the PIG-A MF following RT. First, the bone marrow volume contained in the ribs and sternum represents 16.1 and 3.1% of the total hematopoietic bone marrow, respectively [START_REF] Cristy | Active bone marrow distribution as a function of age in humans[END_REF]. Since the irradiation administered in our RT protocol concerned only a part of the sternum and ribs, the probability of a granulocyte precursor gaining a mutation could have been too low to significantly impact the PIG-A MF observed in our population. Additionally, the high cytotoxic effect of RT should also be considered. Irradiations of more than 30 Gy, even fractionated, have been described as cytotoxic for the bone marrow [START_REF] Mauch | Hematopoietic stem cell compartment: acute and late effects of radiation therapy and chemotherapy[END_REF]. Due to damage to the medullary microenvironment, the recolonization of the bone marrow by hematopoietic stem cells after strong irradiation (>40 Gy) may take several years. Relatively high doses have been shown to cause irreversible injury to the microvasculature, leading to irrevocable bone marrow fibrosis (>50 Gy) [START_REF] Mauch | Hematopoietic stem cell compartment: acute and late effects of radiation therapy and chemotherapy[END_REF]. It is therefore possible that the irradiated bone marrow may have been destroyed, resulting in the lack of increase in the PIG-A MF observed.

Granulocytes were chosen as the target human blood cells because they are unlikely to undergo complement-mediated lysis due to their short intravascular lifetime [START_REF] Athens | Leukokinetic studies. IV. The total blood, circulating and marginal granulocyte pools and the granulocyte turnover rate in normal subjects[END_REF]. It is unlikely that the absence of PIG-A-mutant granulocyte detection was related to technical issues. Indeed, our assay was directly inspired by the one routinely used for the diagnosis and monitoring of PNH patients [START_REF] Sutherland | Practical guidelines for the high-sensitivity detection and monitoring of paroxysmal nocturnal hemoglobinuria clones by flow cytometry[END_REF]. We aimed to optimize the phenotyping technique for circulating cells, while managing throughput and accuracy constraints and good repeatability of our assay. One should also consider the possibility that granulocytes were not the proper type of blood cells to target.

The fact that the micronucleus test was positive supports breast RT as a valuable genotoxic exposure model. CBML frequency is known as a reference assay for the assessment of genotoxic exposure.

The increase in the CBML frequency observed after partial irradiation has been widely described in the literature [START_REF] Thierens | Micronucleus induction in peripheral blood lymphocytes of patients under radiotherapy treatment for cervical cancer or Hodgkin's disease[END_REF][START_REF] Lee | Persistence of micronuclei in lymphocytes of cancer patients after radiotherapy[END_REF][START_REF] Emamgholizadeh Minaei | Evaluation of cytogenetic alterations in peripheral blood lymphocytes of esophageal cancer patients treated with radiotherapy or Chemoradiotherapy using cytokinesis-blocked micronucleus assay[END_REF]. In our study, increases in the CBML frequency were noted in both the WBRT and WBNRT groups following irradiation, and the induction of micronuclei seems to be correlated with the irradiated bone volume. The confirmation of the genotoxic nature of mammary irradiation by the micronucleus test performed with lymphocytes suggests an inability of our assay to demonstrate an increase in the PIG-A MF in granulocytes.

However, the positivity of CBMLs in lymphocytes may not be inconsistent with the absence of PIG-A mutation induction in granulocytes.

Lymphocytes, being circulating cells, likely had a shorter exposure time to RT and therefore were exposed to a lower radiation dose than myeloid hematopoietic stem cells, which are precursors of granulocytes, contained in the irradiated medullary stroma. As mentioned above, high doses of radiotherapy (>30 Gy) could have led to bone marrow fibrosis and stem cell apoptosis.

Moreover, lymphocytes were sampled just before and within 2 hr after the end of the irradiation regimen and forced to divide ex vivo so that DNA double-strand breaks, which are known to be induced by ionizing radiation, induced micronuclei formation. In contrast, to produce an increase in the PIG-A MF, mutations must be fixed in the bone marrow via cell division, and the cells bearing mutations must transit to the peripheral blood. One cannot exclude that the turnover of granulocytes may not have been sufficient for a small cohort of et al., 2011). In contrast, Haboubi et al. recently found that the PIG-A assay performed with erythrocytes could help to detect esophageal adenocarcinoma and might be a useful predictor of disease progression [START_REF] Haboubi | Developing a blood-based gene mutation assay as a novel biomarker for oesophageal adenocarcinoma[END_REF]. It may thus be possible that the sensitivity of the PIG-A assay differs depending on the origin of genomic instability.

| CONCLUSION

Our assay failed to demonstrate an increase in the PIG-A-mutant granulocyte population following RT for breast cancer therapy, even in the group with the largest irradiated bone volume (WBNRT group).

Further studies are needed to evaluate the sensitivity of the PIG-A assay in humans subjected to genotoxic exposures with targeting of different blood cells.

  When appropriate, normality was assessed using the Kolmogorov-Smirnov test. Non-Gaussian data are presented as the median [interquartile range]. Corresponding means (±SDs) are also provided for the PIG-A F I G U R E 2 Gating strategy for GPI-deficient granulocyte analyses. (a) Cells initially identified by their forward scatter and side scatter profiles; (b) singlets identified by their forward scatter area and the forward scatter height; (c) CD45 positive cells were gated to select total leucocytes; (d) cells CD45-positive and CD33-dim were gated to coarsely select granulocytes among myeloid cells; (e) granulocytes were finally selected by gating the 95% brightest CD15 positive cells. Cells negative for both FLAER and CD66b fluorescence (C-quadrant) were considered GPI-deficient granulocytes whereas cells negative only for FLAER or only for CD66b were considered wild type granulocytes. F: PNH patient; G: healthy donor. H: cancer patient MF. Data concerning the repeatability criteria of the PIG-A assay during the T0 sampling were analyzed using the Bland and Altman method, in which thresholds were set at ±1.96 SD of the mean difference between duplicates. Exposure modalities were compared using Student's t test. Assay results during exposure were compared using statistical tests for paired samples (Student or Wilcoxon tests, as appropriate). A two-sided value of p < .05 was considered statistically significant. Statistical analyses were performed using R software v2018 ("R foundation for statistical computing," Vienna, Austria) and GraphPad Prism version 8.0.0 for Windows (GraphPad Software, San Diego, CA). T A B L E 1 Patient characteristics and sample results

  T3 samples were collected from 27 patients, as 3 patients declined to have the last sample taken. An average of 2.7 (±1.4) million granulocytes was counted per analysis. The mean (±SD) PIG-A MFs were 7.7 per million (±12.1) at T0, 5.2 (±8.6) at T1, 6.4 (±8.0) at T2 and 3.8 (±36.0) at T3 (Figure 3). Although the data distributions at the different time points were extremely close, the frequency of PIG-A-mutant granulocytes was significantly lower at T1 (p = .0035) and T3 (p = .006) than at T0. No statistically significant difference in the PIG-A MF was observed between T0 and T2. Similar observations were noted within the WBRT and WBNRT groups (Figure S4A,B).

  Confounding factors represented by age and smoking status (current/ past or quantified in pack-years) were analyzed. None of these factors were significantly associated with the PIG-A MF by univariate analysis (Figure 4a,b).

  mutated cells to significantly affect the PIG-A MF. Myeloid precursors perpetually undergo mitosis and thus are probably more sensitive to genotoxic stimuli than circulating lymphocytes, which are quiescently maintained in the G0 phase of the cell cycle. Indeed, because of the long circulating lifespan of lymphocytes, chromosomal damage can be revealed later by an ex vivo culture step. The persistence of CBMLs, which has been reported up to 5 years after irradiation, could reflect this clinical feature[START_REF] Lee | Persistence of micronuclei in lymphocytes of cancer patients after radiotherapy[END_REF]. Finally, ionizing radiation is among the most potent clastogens and therefore may be poorly detected by an X-linked reporter gene mutation. The CBML frequency is increased following chromosome breakage and/or losses, whereas PIG-A MFs should be mainly increased following gene mutation; thus, the nature of the DNA and/or chromosomal damage could have an impact on the sensitivities of the two assays. The accurate examination of the individual evolution of the PIG-A MF during RT showed three recognizable patterns: no modification of the PIG-A MF while the CBML frequency increased (12 patients), an increase in the PIG-A MF only (1 patient) and increases in both the PIG-A MF and the CBML frequency (3 patients). Even if the small sample size does not allow a definitive conclusion to be reached, interindividual differences in response to mutagen and clastogen exposures may explain these findings.To our knowledge, only four studies have already evaluated the consequences of ionizing radiation and/or chemotherapy on the frequency of PIG-A-mutant cells in human blood. MacDiarmid et al. did not observe any increase in the PIG-A-mutant lymphocyte frequency following depleted uranium exposure in military veterans[START_REF] Mcdiarmid | Measures of genotoxicity in gulf war I veterans exposed to depleted uranium[END_REF]. Horibata et al evaluated the PIG-A assay with human red blood cells using a cohort of 27 cancer patients treated with chemotherapy and/or RT. The authors did not report any increase in the PIG-A-mutant cell frequency among the patients treated with RT (n = 14)[START_REF] Horibata | Monitoring genotoxicity in patients receiving chemotherapy for cancer: application of the PIG-A assay[END_REF]. Nevertheless, this population was highly heterogeneous, with different types of cancers, various RT modalities and possibly unsuitable sampling times. Dobrovolsky et al evaluated the PIG-A assay with red blood cells in a cohort of 10 cancer patients treated with chemotherapy. Six of the patients had breast cancer and received chemotherapeutic agents similar to those used in our study. Using this cell type as a target, they found only an approximately three-fold increase in the frequency of red blood cells originating from a PIG-A-mutant precursor in a single patient treated with a combination of cisplatin and etoposide(Dobrovolsky

  T A B L E 3 Identification of factors contributing to CBML values

	Genotoxic	Independent
	endpoint	variables	Coefficient	p-value
	CBML	Radiotherapy group	9.2	.03
		Smoking status	8.2	.11
		Chemotherapy	5.9	.36
		Radiotherapy	25.7	<.0001
		(T0 vs. T2)
		Age	0.4	.043
	Note: The relative influence of the different independent variables
	(radiotherapy exposure categorized for all patients and according to
	WBRT or WBNRT group, smoking status categorized as smoking or non-
	smoking, chemotherapy categorized as chemotherapy or no
	chemotherapy) on the CBML levels tested by multivariate linear
	regression analysis. Coefficients corresponds to the additional number of
	CBML predicted when the corresponding variable shows a one-unit shift
	or a category switch.

ACKNOWLEDGMENTS

This work was completed with the support of (a) the A*MIDEX project "CREER" (No. ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir," French Government Program of the French National Research Agency (ANR); (b) SATT-SE (ID 870); and (c) ARS PACA (REVELA13; dossier N. 20120658). The authors would like to thank Dr. Romaric Lacroix and Dr. Sylvie Cointe for helpful discussion and Jocelyne Pompili for helpful technical assistance.

A*MIDEX, Grant/Award Number: ANR-11-IDEX-0001-02; ARS PACA, Grant/Award Number: REVELA13; 20120658; SATT-SE, Grant/Award Number: ID 870; French National Research Agency, Grant/Award Number: REVELA13


Andrey Polyakov
email: andrey.polyakov@inria.fr

Input-to-State Stability of Homogeneous Infinite Dimensional Systems with Locally Lipschitz Nonlinearities

Keywords: stability of distributed parameter systems, input-to-state stability, semigroup and operator theory

The Input-to-State Stability (ISS) of homogeneous evolution equations with unbounded linear operators and locally Lipschitz nonlinearities in Banach spaces is studied using a new homogeneous converse Lyapunov theorem. It is shown that, similarly to finite-dimensional models, uniform asymptotic stability of an unperturbed homogeneous system guarantees its ISS with respect to homogeneously involved exogenous inputs.

Introduction

Homogeneity is a dilation symmetry, which can be discovered for functions and vector fields in R n (see [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF], [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF], [START_REF] Bernuau | On homogeneity and its application in sliding mode control[END_REF]) as well as for functionals and operators in Banach and Hilbert spaces (see e.g. [START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF], [START_REF] Fischer | Quantization on Nilpotent Lie Groups[END_REF], [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF]). All linear and many nonlinear models of mathematical physics are homogeneous in a generalized sense or, at least, admit a locally homogeneous approximation. For example, the dilation symmetry can be discovered in heat, wave, Navier-Stokes, Burgers, Korteweg-de Vries and Saint-Venant equations (see [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]).

The homogeneity simplifies stability and robustness analysis of control systems (see e.g. [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Ryan | Universal stabilization of a class of nonlinear systems with homogeneous vector fields[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]) as well as nonlinear controllers/observers design (see [START_REF] Kawski | Homogeneous stabilizing feedback laws[END_REF], [START_REF] Coron | Adding an integrator for the stabilization problem[END_REF], [START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF], [START_REF] Lopez-Ramirez | Finite-time and fixed-time observer design: Implicit Lyapunov function approach[END_REF]). The so-called homogeneity degree specifies a convergence rate of any asymptotically stable homogeneous system (see e.g. [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF]). In particular, the negative homogeneity degree corresponds to finite-time stability.

Input-to-State Stability (ISS) is one of notion that has had the greatest impact in the study of control systems. Being originally introduced by E. Sontag for Ordinary Differential Equations (ODEs) [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF], it has been ex-

The author acknowledges the support of ANR DIGISLID project (ANR-18-CE40-0008) and the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.

tended to discrete-time systems [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF], [START_REF] Lazar | Lyapunov functions stability and input-tostate stability subtleties for discrete-time discontinuous systems[END_REF], impulsive and hybrid systems [START_REF] Haddad | Impulsive and Hybrid Dynamical Systems: Stability Dissipativity and Control[END_REF], [START_REF] Cai | Characterizations of inputto-state stability for hybrid systems[END_REF] and Partial Differential Equations(PDEs) [START_REF] Jayawardhana | Infinite-dimensional feedback systems: the circle criterion and input-to-state stability[END_REF], [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF], [START_REF] Dashkovskiy | Input-tostate stability of infinite-dimensional control systems[END_REF], [START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF], [START_REF] Mironchenko | Input-to-state stability of infinite dimensional systems: Recent results and open questions[END_REF]. The method of Lyapunov function is the main tool for stability analysis and stabilization of both finite and infinite dimensional nonlinear systems (see [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF], [START_REF] Deimling | Multivariable Differential Equations[END_REF], [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF] among others). Characterization (necessary and sufficient conditions) of stability in terms of a Lyapunov function is useful for ISS analysis (see [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF], [START_REF] Prieur | ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws[END_REF], [START_REF] Mironchenko | Non-coercive lyapunov functions for infinite-dimensional systems[END_REF], [START_REF] Jacob | Noncoercive lyapunov functions for input-to-state stability of infinite-dimensional systems[END_REF]). It is known, since [START_REF] Zubov | On systems of ordinary differential equations with generalized homogenous right-hand sides[END_REF] and [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], that any asymptotically stable homogeneous ODE admits a homogeneous Lyapunov function.

The key difference between Zubov's and Rosier's theorems is the regularity assumption to the homogeneous Lyapunov function. Vladimir Zubov proved only the existence of a continuous homogeneous Lyapunov function, while Lionel Rosier constructed a smooth one using the Kurzweil's converse Lyapunov theorem. The Rosier's theorem allows a rather simple ISS analysis of homogeneous finite-dimensional systems to be developed (see e.g. [START_REF] Ryan | Universal stabilization of a class of nonlinear systems with homogeneous vector fields[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]) showing that the asymptotic stability of an unperturbed homogeneous ODE implies its ISS with respect to homogeneously involved perturbations. This paper extends the mentioned result to a class of infinite dimensional models.

The theory of strongly continuous semigroups of linear bounded operators in Banach spaces proposes a unified approach to solution existence and uniqueness problem as well as to analysis of distributed parameter dynamical systems such as PDEs (see e.g. [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF][START_REF] Engel | One-Parameter Semigroups for Linear Evolution Equations[END_REF], [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]). In this paper we deal with dynamical models described by evo-lution equations with unbounded linear closed densely defined operators and locally Lipschitz continuous nonlinearities in Banach spaces. Such models always have the so-called mild solutions, which are unique in the regularity domain (see e.g. [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], Chapter 6, Theorem 1.4).

The extension of the results known for homogeneous ODEs to evolution equations in Banach spaces is complicated because of the impossibility to use the conventional differential calculus. Solutions of ODEs are, at least, absolutely continuous functions of time, which are differentiable almost everywhere. One-parametric curves in a Banach space B may be globally Lipschitz but nowhere differentiable (see e.g. [START_REF] Tabor | Generalized differential inclusions in banach spaces[END_REF]). The differentiability (in a strong sense) and even local Lipschitz/Hölder continuity of a mild solution x : [0, +∞) → B cannot be always guaranteed. It is not obvious if the composition t → V (x(t)) of a smooth Lyapunov function candidate V : B → R with the mild solution x is differentiable. The smoothness of a Lyapunov function for an infinitedimensional system seems to be a rather restrictive assumption in the general case. However, a successful ISS analysis, even in the finite-dimensional case (see [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF]), requires some regularity of ISS Lyapunov functions. The homogeneous converse Lyapunov theorem for infinite dimensional systems presented in [START_REF] Polyakov | On homogeneous Lyapunov function theorem for evolution equations[END_REF] does not guarantee even a continuity of the Lyapunov function. In this paper, we manged to construct a locally Lipschitz continuous homogeneous Lyapunov function, which allows the ISS results existing for homogeneous ODEs to be successfully extended to evolution systems in Banach spaces.

Notation.

• R is the field of real numbers; R + = [0, +∞); • B is a real Banach space with a norm • ; 0 ∈ B is the zero element; • L(B 1 , B 2 ) denotes the space of linear bounded operators B 1 → B 2 with the norm A = sup

x =0 Ax x ; • I ∈ L(B, B) -the identity operator; • f 1 • f 2 and f 1 f 2 denote a composition of operators f 1
and f 2 ; • S is the unit sphere in B; B(r) ⊂ B is the ball in B centered at 0 of the radius r > 0;

K(r) := {x ∈ B : 1/r < x < r}, r > 1; • C([t 1 , t 2 ], B) is the space of continuous functions x : [t 1 , t 2 ] → B with the uniform norm x C = sup t∈[t1,t2] x(t) with -∞ < t 1 < t 2 < +∞; • L 1 ((t 1 , t 2 ), B) is the space of Bochner integrable func- tions (t 1 , t 2 ) → B, where -∞ ≤ t 1 < t 2 ≤ +∞; • L ∞ ((t 1 , t 2 ), B
) is a space of uniformly essentially bounded Bochner measurable functions with the norm q ∞ := ess sup s∈(t1,t2) q(t) ;

• H p (Ω, R) is the Sobolev space of functions Ω → R,
where Ω ∈ R n is an open connected set with a smooth boundary;

• C ∞ c (Ω) is a set of infinitely smooth functions R n → R with compact supports in Ω; • H p 0 (Ω, R) is the closure of C ∞ c (Ω) in the norm of H p ; • K is a set of strictly increasing functions σ : [0, +∞) → [0, +∞) such that σ(0) = 0; • K ∞ := {σ ∈ K : σ(s) → +∞ as s → +∞}; • the function β : R + × R + → R + belongs to the class KL if β(•, t) ∈ K and t → β(s, t
) is strictly decreasing to zero;

• the symbol a.e = (resp.

a.e

≤ or

a.e ∈ ) means that the identity (resp. inequality or inclusion) holds almost everywhere;

• D + V (t) = lim sup h→0 + V (t+h)-V (t) h
denotes the right-hand upper Dini derivative of the function v : R → R;

• D + V (x; g) = lim sup h→0 + V (x+hg)-V (x) h
denotes the right-hand upper directional derivative of the functional V : B → R in the direction g ∈ B.

Problem Statement

Let us consider the nonlinear system

ẋ = Ax + f (x, q), t > t 0 , x(t 0 ) = x 0 (1)
where

x(t) ∈ B is a system state, B is a real Banach space; t 0 ∈ R is an initial instant of time; x 0 ∈ B is an initial state; q ∈ L ∞ (R, V) is an exogenous input;
V is a real Banach space; A : D(A) ⊂ B → B is a linear unbounded closed densely defined operator which generates a strongly continuous semigroup Φ of linear bounded operators on B; f : B × V → B is a non-linear mapping such that f (0, 0) = 0.

We assume that, on B\{0} × V, the nonlinear mapping f is locally Lipschitz continuous in the first argument and locally γ-Hölder continuous in the second one, i.e. for any r > 0 there exists L r > 0 such that

f (x 1 , q 1 )-f (x 2 , q 2 ) ≤ L r ( x 1 -x 2 + q 1 -q 2 γ ) , (2)
for all x 1 , x 2 ∈ K(r) and q 1 , q 2 ∈ B(r), where γ ∈ (0, 1].

The non-linear evolution equations are well-studied in the literature (see, for example, [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF][START_REF] Engel | One-Parameter Semigroups for Linear Evolution Equations[END_REF]), where the notion of solution is introduced using the theory of evolution semigroups.

Definition 1 Let a linear closed densely defined operator A : D(A) ⊂ B → B be an infinitesimal generator of a strongly continuous semigroup Φ of linear bounded operators on B. A continuous function x : [t 0 , t 1 ) → B is said to be a mild solution of (1) if f (x(•), q(•)) ∈ L 1 ((t 0 , t 1 ), B),

x(t) = Φ(t-t 0 )x 0 + t t0 Φ(t-s)f (x(s), q(s))ds, ∀t ∈ [t 0 , t 1 ).
If the mild solution satisfies (1) for (almost) all t ∈ (t 0 , t 1 ) then x is called a classical (strong) solution of (1).

The integral in the above definition is understood in the sense of Bochner. The assumption f (0, 0) = 0 guarantees the existence of the zero solution of the nonperturbed system [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. The regularity of f ensures the existence and uniqueness of mild solutions on B\{0} as well as their continuous dependence on initial conditions (see Appendix).

In this paper we deal with operators A and f , which are homogeneous (symmetric in a sense explained below).

It is known that the asymptotic stability of the nonperturbed (q = 0) homogeneous ODE (B = R n ) guarantees its ISS provided that the perturbation q is homogeneously involved into the system (see e.g. [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF] for more details). The first aim of the paper is to extend this result to the infinite dimensional model [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. The corresponding ISS analysis for ODEs was based on the homogeneous Lyapunov function theorem proven in [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. Therefore, the second aim is to extend the mentioned theorem to homogeneous evolution equations in Banach spaces.

3 Homogeneous systems

Dilations in Banach Spaces

Dilations in finite-dimensional and infinite-dimensional spaces are studied in the literature (see e.g. [START_REF] Husch | Topological Characterization of The Dilation and The Translation in Frechet Spaces[END_REF], [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF]). Here we just recall briefly some related definitions and properties.

d(s) = e sGd = ∞ i=0 s i G i d i! , where G d ∈ R n×n is an anti-Hurwitz matrix.
Nonlinear dilations in R n are studied in [START_REF] Khomenuk | On systems of ordinary differential equations with generalized homogenous righthand sides[END_REF], [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Rosier | Etude de quelques problèmes de stabilization[END_REF].

Examples of linear dilations in function spaces are 1) a uniformly continuous dilation in C([0, p], R) can be defined as (d(s)z)(y) = e s-0.5sy/p z(y), where s ∈ R, y ∈ [0, p].

2) a strongly continuous dilation in L 2 (R n , R) is defined as (d(s)z)(y) = e αs z(e βs y), where s ∈ R, y ∈ R n , α > nβ/2.

For more details about linear dilations in function spaces we refer the reader to [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Chapter 6].

Definition 4 A dilation d is strictly monotone in B if ∃β > 0 such that d(s) ≤ e βs , ∀s ≤ 0. ( 3
)
Monotone dilations in R n are studied in [START_REF] Polyakov | Sliding mode control design using canonical homogeneous norm[END_REF], where, in particular, it is shown that all linear dilations in R n are strictly monotone under a proper selection of the weighted Euclidean norm.

The canonical homogeneous norm

The dilation introduces a new norm topology in B using the so-called canonical homogeneous norm. The following theorem refines some results from [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF].

, σ ∈ K ∞ such that σ( u ) ≤ u d ≤ σ( u ), ∀u ∈ B.
In [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Theorem 7.1]it is also proven that • d is a norm in a Banach space B homeomorphic to B. This justifies the name "norm" for the functional • d .

Homogeneous operators

Homogeneous functionals and operators [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF] on B are defined similarly to homogeneous functions and vector fields (see e.g. [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], [START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF]) taking into account their possible unboundedness.

Definition 7 An operator f : D(f ) ⊂ B → B (a func- tional h : D(h) ⊂ B → R) is said to be d-homogeneous of degree ν ∈ R if the domain D(f ) (resp. D(h)) is a d-homogeneous cone and e νs d(s)f (u) = f (d(s)u), ∀s ∈ R, ∀u ∈ D(f ), (resp. h(d(s)u) = e νs h(u), ∀s ∈ R, ∀u ∈ D(h))
(5) where d is a group of linear operators in B.

We say that an evolution equation is d-homogeneous of degree µ ∈ R if its right-hand side is a d-homogeneous operator of degree µ. Notice that the identity (5) can be understood in the weak sense as shown below.

Example 1 (Homogeneity of Laplace operator)

Let us consider the Laplace operator ∆ :

D(∆) ⊂ L 2 → L 2 , D(∆) = u ∈ L 2 : ∃f ∈ L 2 , u • ∆φ = f • φ, ∀φ ∈ C ∞ c . Let d be dilation in L 2 given by (d(s)u)(x) = e αs u(e βs x), where s ∈ R, x ∈ R n , α > nβ/2. Let us show that the operator ∆ is d-homogeneous of degree 2β. Notice that if φ ∈ C ∞ c then, obviously, (∆ • d(s))φ)(x) = ∆e αs φ(e βs x) = e (α+2β)s (∆φ)(e βs x) = e 2βs ((d(s) • ∆)φ)(x). In other words, the Laplace operator is d-homogeneous as an operator C ∞ c → C ∞ c . Since C ∞ c is dense in L 2 , H 1 and H 2 then, it is expectable that, ∆ is d-homogeneous as an operator in L 2 . Let us prove this rigorously. Let u ∈ D(∆) and ∆u = f ∈ L 2 in the weak sense, i.e. u∆φ = f φ, ∀φ ∈ C ∞ c .
Since d(s)f ∈ L 2 then using the change-of-variable theorem in the Lebesgue integral we derive

e 2βs (d(s)f ) • φ = e (α+2β)s f (e βs x) • φ(x)dx = e (α+(2-n)β)s f (x) • φ(e -βs x)dx = e (2α+(2-n)β)s f • φ = e (2α+(2-n)β)s u•∆ φ = e (2α+(2-n)β)s u•(∆•d(-s))φ = e (α-nβ)s u(x)•∆φ(e -βs x)dx = e αs u(e βs x)•∆φ(x)dx. Hence, d(s)u ∈ D(∆) and (∆ • d(s))u = e 2βs d(s)f = e 2βs (d(s) • ∆)u in the weak sense for any s ∈ R, u ∈ D(∆).
Homogeneity allows local properties of nonlinear operators (such as regularity) to be extend globally [37, Chapter 7].

Symmetry of solutions of homogeneous systems

A semigroup generated by a closed densely defined linear homogeneous operator in B is homogeneous as well. Lemma 1 proves the symmetry of solutions of (1) for f ≡ 0:

x d(s)x0 (t) = d(s)x x0 (e µs t), s ∈ R, t ≥ 0, (7)
where x z denotes a solution of (1) with the initial data x(0) = z. This symmetry of solutions takes a place for a d-homogeneous evolution equation ( 1) even if f = 0.

The following theorem was originally presented in [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF] for more general class of evolution systems.

ẋ = Ax + f (x), t > 0 ( 8
)
then for any s ∈ R the function x s : [0, e -µs T ) → B given by x s (t) = d(s)x(e µs t), t ∈ [0, e -µs T ) is the mild solution of the evolution equation (1) with the scaled initial condition x(0) = d(s)x 0 .

The latter theorem expands globally any local property of solutions. For example, if the origin of (1) is locally stable then, from (7) and the limit property of d, we immediately derive global stability. Similarly, the existence of solutions for small initial data ensures the existence of solutions for large initial data.

Below we generalize Theorem 2 to homogeneous systems with disturbances (see Theorem 4).

Homogeneous Lyapunov function theorem

Recall that the origin of the system ( 8) is said to be

• globally uniformly Lyapunov stable if there exists ε ∈ K ∞ such that x x0 (t) ≤ ε( x 0 ), ∀t ≥ 0, ∀x 0 ∈ B (9)
for any mild solution x x0 of (8) with x(0) = x 0 ; • globally uniformly asymptotically stable if it is globally uniformly Lyapunov stable and ∀ε > 0, ∀R > ε, there exists T = T (R, ε) > 0 such that the inclusion

x 0 ∈ B(R) implies x x0 (t) ∈ B(ε) for all t > T .
The homogeneity degree specifies the convergence rate of a stable homogeneous system (see e.g. [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] for ODE models and [START_REF] Polyakov | On homogeneous distributed parameters equations[END_REF], [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Theorem 8.6] for evolution systems in Banach spaces). For instance, any uniformly asymptotically stable d-homogeneous system of negative (resp. zero) degree is globally uniformly finite-time (exponentially) stable.

Theorem 3 Let d be a strictly monotone strongly continuous dilation in B, a linear closed densely defined operator A : D(A) ⊂ B → B be a generator of a strongly continuous semigroup Φ of linear bounded operators on B, f : B → B be locally Lipschitz continuous on B\{0} and the evolution equation (8) be d-homogeneous of degree µ ∈ R. Let m > 0 be an arbitrary real number.

The origin of (8) is uniformly asymptotically stable if and only if there exists a continuous positive definite functional

V : B → R + such that 1) V is d-homogeneous of degree m and locally Lipschitz continuous on B\{0}; 2) ∃k, k > 0 satisfying k x m d ≤ V (x) ≤ k x m d , ∀x ∈ B; (10)
3) for any mild solution x x0 of (8) with x(0) = x 0 ∈ D(A)\{0} the inequality

D + V (x x0 (t))
a.e.

≤ -W (x x0 (t)) [START_REF] Deimling | Multivariable Differential Equations[END_REF] holds as long as x x0 (t) = 0, where W : B\{0} → R is a positive definite d-homogeneous functional of degree m + µ such that

∃c > 0 : W (x) ≥ c x m+µ d ∀x ∈ B\{0}.
Formally, the latter theorem requires a knowledge of trajectories of the system [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF] to check the condition [START_REF] Deimling | Multivariable Differential Equations[END_REF].

The Lyapunov function method is appreciated in practice if the asymptotic stability can be established using the right-hand side of the evolution system only. This can be done in the case of reflexive Banach spaces.

Corollary 1 For a reflexive Banach space B, Theorem 3 remains true even when the inequality (11) is replaced with

D + V (x; Ax+f (x)) ≤ -W (x), ∀x ∈ D(A)\{0}. ( 12
)
This corollary holds, in particular, for Hilbert spaces.

Example 2 Let us consider the following PDE

∂x ∂t = ∆x -γ x µ x, where γ > 0, µ ∈ R, x : R + × R n → R, ∆ : H 2 (R n , R) ⊂ L 2 (R n , R) → L 2 (R n , R)
is the Laplace operator on L 2 (R n , R) with the domain H 2 (R n , R) (Sobolev space); and

x = x, x , x, y = R n x • y dz.
The latter PDE admits the representation (1)

with B = L 2 (R n , R), A = ∆ and f : B → B given by f (x) = -γ x ν x.
It is well known (see e.g. [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], Chapter 7) that A is the generator of a strongly continuous semigroup of linear bounded operators on L 2 (R m , R).

In Hence, the norm

• is a d-homogeneous functional of degree α -nβ/2 > 0, f is d-homogeneous of degree ν(α - nβ/2) and x d = x 1 α-nβ/2
. Therefore, for 2β = ν(αnβ/2) the system is d-homogeneous of degree µ := 2β and the sign of the homogeneity degree is defined by the sign of ν. A d-homogeneous Lyapunov functional of degree m := 2(α -nβ/2) > 0 for the considered PDE model is given by V (x) =1 2 x 2 . Indeed, for x ∈ D(A)\{0} we derive

D + V (x; ∆x + f (x)) = x, ∆x + f (x) ≤ x, f (x) = -γ x ν+2
, where the dissipative property x, ∆x ≤ 0 of the Laplace operator is utilized on the indeterminate step. The functional x → γ x ν+2 is d-homogeneous of degree m + µ exactly as claimed in Theorem 3.

Other examples of homogeneous Lyapunov functions can be found in [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF].

The existence of the so-called coercive Lyapunov functional [START_REF] Mironchenko | Non-coercive lyapunov functions for infinite-dimensional systems[END_REF] for any linear exponentially stable evolution system is the trivial corollary of Theorem 3. For the Hilbert space H, the required Lyapunov functional V : H → R can always be selected in the form V (x 0 ) = P x 0 , x 0 + sup t≥0 x x0 (t), x x0 (t) , where x 0 ∈ H and P ∈ L(H, H) is defined in [8, Theorem 5.3.1].

Homogeneous systems with perturbations

A symmetry of solutions can also be discovered for the perturbed homogeneous system (1). Let x d(s)x0,q : [t 0 , t 0 + e -µs T ] → B denote a mild solution of the system (1) for x 0 ∈ B replaced with d(s)x 0 ∈ B and q ∈ L ∞ ((t 0 , t 0 + T ), V) replaced with q ∈ L ∞ ((t 0 , t 0 + e -µs T ), V) given by

q(t) = d(s)q(t 0 + e µs (t -t 0 )), ∀t ∈ t 0 , t 0 + T e µs ,
where d is a group of linear bounded operators on V.

f (d(s)x, d(s)q) = e µs d(s)f (x, q), ∀x ∈ B, ∀q ∈ V, ∀s ∈ R.
If x x0,q : [t 0 , t 0 + T ] → B is a mild solution of the system (1) then for any s ∈ R one has

x d(s)x0,q (t) = d(s)x x0,q (t 0 + e µs (t -t 0 )), (14)
for all t ∈ t 0 , t 0 + T e µs .

Theorem 4 becomes Theorem 2 for q = 0. The latter theorem allows a rather simple ISS analysis of homogeneous evolution equations to be developed. The definition of the input-to-state stability in this case can be introduced similarly to ODE models (see, for example, [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF], [START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF], [START_REF] Mironchenko | Non-coercive lyapunov functions for infinite-dimensional systems[END_REF]).

Definition 8

The system (1) is said to be input-to-state stable (ISS) if there exists β ∈ KL and γ ∈ K such that

x x0,q (t) ≤ β( x 0 , t) + γ q L ∞ ((0,t),V) , ∀t ≥ 0,
for any mild solution x x0,q of (1).

The local ISS 1 of the system (1) with f satisfying (2) is studied in [START_REF] Dashkovskiy | Input-tostate stability of infinite-dimensional control systems[END_REF]. The homogeneity allows a simple procedure for the global ISS analysis to be developed.

Corollary 2 Let d and d be strictly monotone strongly continuous dilations in B and V, respectively, and let all conditions of Theorem 4 be fulfilled. If the origin of the system (1) with q = 0 is uniformly asymptotically stable then this system is ISS.

The obtained corollary implies that the uniform asymptotic stability of the unperturbed homogeneous system (1) ensures its ISS with respect to homogeneously involved exogenous inputs.

Example 3 (ISS of Reaction Diffusion Equation)

Let us consider the following PDE model

∂x ∂t = ∆x + f (x, q), (15)
where x(t) ∈ L 2 (Ω, R) and ∆ :

D(∆) ⊂ L 2 (Ω, R) → L 2 (Ω, R) is the Laplace operator (see Example 2), Ω ⊂ R n is an open connected set with a smooth boundary, D(∆) = H 1 0 (Ω, R) ∩ H 2 (Ω, R), f : L 2 (Ω, R) × V → L 2 (Ω, R) is a nonlinear operator, q ∈ L ∞ (R + , V) and V is a Banach space of any kind.
In [START_REF] Dashkovskiy | Input-tostate stability of infinite-dimensional control systems[END_REF], [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF], the ISS analysis of the system (15) with f (x, q) = f (x)+q, f :

L 2 (Ω, R) → L 2 (Ω, R) is presented.
In [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF], the case of the time varying operator A is studied as well. Using Corollary 2, the ISS of (15) with f satisfying (2) can be studied provided that the system is homogeneous and uniformly asymptotically stable for q = 0.

Case 1). Let

Ω = R n . Let us consider f of the form f (x, q) = -γ x + q 1 2 (x + q 1 ) + | x, q 2 | µ xq 3 ,
where µ, γ > 0, x ∈ L 2 (R n , R) and q = (q 1 , q 2 , q 3 ) ∈ V,

V := L 2 (R n , R) × L 2 (R n , R) × L ∞ (R n , R),
and q = q 1 + q 2 + q 3 . The first term in f can be interpreted as a distributed non-linear feedback control with a measurement noise q 1 . The variables q 2 and q 3 can be treated as exogenous inputs.

Then for q = 0 we derive the system studied in Example 2, that is globally uniformly asymptotically stable and dhomogeneous of degree 2β provided that the dilation d in L 2 (R n , R) is defined as follows

(d(s)x)(z) = e αs x(e βs z), s ∈ R, z ∈ R n with α = (1 + n/2)β > 0.
The considered operator f satisfies the regularity assumption (2). For r ∈ R and

( d(s)q)(z) =    
e αs q 1 (e βs z) e (nβ-α+(2α-nβ-rβ)/µ)s q 2 (e βs z) e rβs q 3 (e βs z)

   
we have f (d(s)x, d(s)q) = e 2βs d(s)f (x, q). To be a strictly monotone dilation in V the group d must have the parameters (see [START_REF] Engel | One-Parameter Semigroups for Linear Evolution Equations[END_REF]) satisfying the inequalities

rβ > 0 and (2α -nβ -rβ)/µ-α+nβ > nβ/2.
By Corollary 2, the considered system is ISS if 0 < µ < 2.

Case 2) Let n = 1, Ω = (0, 1). For d(s) = e s I, s ∈ R the closed densely defined operator ∆ is d-homogeneous of degree 0 as any linear operator.

Let V = L 2 (R, R) and f : L 2 ((0, 1), R) × V → L 2 ((0, 1), R) be defined as follows

[f (x, q)](z) = R K(x, z, τ )q(τ )dτ, z ∈ (0, 1)
where q ∈ V and K : L 2 ((0, 1), R) × (0, 1) × R → R is such that f satisfies the regularity assumption (2).

For q = 0 we have the well-known heat system, which is globally uniformly asymptotically stable.

Let the dilation d in V be defined as ( d(s)q)(τ ) = e αs q(e βs τ ), where s, τ ∈ R and α > β/2, α, β ∈ R, β = 0.

If for all x ∈ B, z ∈ (0, 1), τ, s ∈ R we have K(e s x, z, τ ) = e (1-α+β)s K(x, z, e βs τ ),

then f (d(s)x, d(s)q) = d(s)f (x, q)
and, by Corollary 2, the considered system is ISS. All requested conditions hold, for example, if

1/2 < α ≤ 2, β = 1 and K(x, z, τ ) = x 2-α ze -|τ | x .
The ISS analysis for systems with non-homogeneous boundary conditions can be done in the similar way using the so-called Dirichlet lifting [START_REF] Mironchenko | Input-to-state stability of infinite dimensional systems: Recent results and open questions[END_REF]. However, if the disturbances appear in boundary conditions, the use of Dirichlet lifting may involve the time derivative of the disturbances and, consequently, the obtained ISS property may not be derived in its conventional form (see, e.g. [START_REF] Karafyllis | ISS with respect to boundary disturbances for 1-D parabolic PDEs[END_REF] for more details). An exception is the case of monotone systems as shown in [START_REF] Mironchenko | Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances[END_REF]. The natural question in this context: may the homogeneity provide similar advantages for ISS analysis as the monotonicity? This is the interesting problem for future investigations.

Conclusion

The main contributions of the paper are Theorem 3 and Corollary 2. The first one provides a characterization (in terms of a locally Lipschitz continuous Lyapunov function) of the uniform asymptotic stability of homogeneous infinite dimensional systems of a certain class, while the second one proposes a possible way for their ISS analysis. Theorem 3 proves the existence of the so-called coercive Lyapunov function for any uniformly asymptotically stable homogeneous system. The latter allows us to prove that the asymptotic stability of the unperturbed system implies its ISS provided that the perturbations are incorporated to the system in a homogeneous way. Of course, the ISS analysis can always be done using the conventional approach based on ISS Lyapunov function [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF], [START_REF] Dashkovskiy | Local iss of reaction-diffusion systems[END_REF], [START_REF] Prieur | ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws[END_REF], [START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF], [START_REF] Mironchenko | Input-to-state stability of infinite dimensional systems: Recent results and open questions[END_REF]. The homogeneity may simplify the mentioned analysis, since only the unperturbed homogeneous system needs to be studied. The presented scheme is demonstrated on examples. An extension of the obtained results to other classes of (locally or globally) homogeneous infinite dimensional systems is the interesting problem for future investigations.

Appendix

Auxiliary results

Lemma 2 and Corollary 3 are consequences of the classical results about well-possedness of evolution systems with globally Lipschitz continuous nonlinearities (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], Theorems 1.2 and 1.4 in Chapter 6).

Lemma 2 (Existence of Solutions) Let A : D(A) ⊂ B → B be closed densely defined linear operator which generates a strongly continuous semigroup Φ of linear bounded operators on B and f : B × V → B satisfy (2). Let δ > 1 be an arbitrary real number and

f δ (x, q) = a δ ( x )f (x, q), x ∈ B, q ∈ V where a δ ∈ C ∞ c (R + , R
) such that a δ (s) = 0 for s / ∈ (1/(2δ), 2δ) and a δ (s) = 1 for s ∈ [1/δ, δ]. Then for any x 0 ∈ B and any q ∈ L ∞ ([t 0 , +∞), B) the initial value problem

ẋ = Ax + f δ (x, q), t ≥ t 0 x(t 0 ) = x 0 ( 16
)
has a unique mild solution x δ : [t 0 , +∞) → B, which, for x 0 ∈ K(δ), coincides with the mild solution x x0 of (1) as long as x x0 (t) ∈ K(δ). Moreover, for any x 0 ∈ B\{0} the system (1) has a mild solution x x0 , which is uniquely defined as long as 0 < x x0 (t) < +∞.

Corollary 3 (Continuous dependence of initial data)

If all conditions of Lemma 2 are fulfilled then for any T > 0 and any q > 0 there exists L δ,T,q > 0 such that for all x 1 , x 2 ∈ B and all q ∈ L ∞ ((t 0 , +∞), V ) : q ∞ ≤ q one has

x δ x1,q (t)-x δ x2,q (t) ≤ L δ,T,q x 1 -x 2 , ∀t ∈ [t 0 , t 0 +T ]
where x δ xi,q (t) is a mild solution of (16) with x(t 0 ) = x i .

Corollary 4 Let all conditions of Lemma 2 be fulfilled. If x 0 ∈ D(A) then the corresponding mild solution x δ x0,q of (16) is a locally Lipschitz continuous function of time.

Proof. See [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], page 189.

Corollary 5 Let all conditions of Lemma 2 be fulfilled. Let the origin of (1) with q = 0 be globally uniformly Lyapunov stable. Let r > 1 be an arbitrary real number and

δ > max ε(r), 1 ε -1 (1/r) ,
where ε ∈ K ∞ is given by (9) and ε -1 is the inverse function to ε. Then 1) for any x 0 ∈ B : 1/r ≤ x 0 ≤ r we have

x δ x0 (t) ≤ ε( x 0 ) ≤ δ, ∀t ≥ 0 2) if ∃t 1 ≥ 0 such that x δ (t 1 ) ≤ 1/δ then x δ x0 (t) ≤ 1/r, ∀t ≥ t 1 ,
where x δ x0 is the unique mild solution of (16) with q = 0 and t 0 = 0.

Proof. 1) The inequality (9) and x x0 (0) = x 0 yield ε(r) ≥ r and 1/δ < 1/r < r < δ. Since x δ x0 coincide with x x0 as long as x δ x0 (t) ∈ [1/δ, δ] then the inequality (9) implies that x δ x0 (t) ≤ ε( x 0 ) ≤ δ for all t ≥ 0 and any x 0 : x 0 ≤ [1/r, r]. 2) Suppose the contrary, i.e. ∃t 2 > t 1 such that x δ x0 (t 2 ) > 1/r and x δ x0 (t) > 1/δ for t ∈ (t 1 , t 2 ]. In this case, we have x δ x0 (t) = x x0 (t) for all t ∈ (t 1 , t 2 ] and using the inequality (9) we conclude

x δ x0 (t) ≤ ε( x δ x0 (t 1 ) ) = ε(1/δ) ≤ ε(ε -1 (1/r)) = 1/r
for all t ≥ t 1 .

The proof of Theorem 3

Sufficiency. Let δ > 1 be an arbitrary number. Then by Lemma 2 the solution x δ x0 of the system

ẋ = Ax + f δ (x), t > 0, x(0) = x 0 , (17)
with f δ (x) = a δ ( x )f (x). coincides with the solution x x0 of (8) as long as

x δ x0 (t) ∈ K(δ). If x 0 ∈ D(A) ∩ K(δ) then by Corollary 4 the function t → V (x δ x0 (t)
) is a locally Lipschitz continuous function of time and differentiable almost everywhere. In this case, using the inequality [START_REF] Deimling | Multivariable Differential Equations[END_REF] we conclude

V (x δ x0 (t))
a.e.

= D + V (x δ x0 (t))
a.e.

≤ -c x δ x0 (t) µ+m d , the function t → V (x δ x0 (t) is strictly decreasing and

V (x δ x0 (t)) ≤ V (x 0 ) -c t 0 x δ x0 (t) µ+m d ds ≤ V (x 0 ) - c k t 0 V µ+m (x δ x0 (t))ds as long as x δ x0 (t) ∈ K(δ).
Let x 0 ∈ K(δ). Since D(A) dense in B then there exists

x i ∈ D(A) ∩ K(δ) such that x i → x 0 as i → +∞.
Using the continuous dependence of solutions on initial conditions (see Corollary 3), we derive

sup t∈[0,T ] x δ x0 (t) -x δ xi (t) → 0 as i → +∞ for any T > 0 such that x δ x0 (t) ∈ K(δ), ∀t ∈ [0, T ].
Tending δ → +∞ we conclude that for any x 0 ∈ B\{0} the function t → V (x x0 (t)) is strictly decreasing and

V (x x0 (t)) ≤ V (x 0 ) - c k t 0 V µ+m (x x0 (t))ds
as long as x x0 (t) = 0. Hence, for any r > 0 we derive

V (x x0 (t)) ≤ r, ∀t ≥ max 0, k(V 1-µ-m (x0)-r 1-µ-m ) c .
The latter implies that mild solutions of ( 8) converges to zero uniformly on the initial data, i.e. for any R > r > 0 there exists T = T (R, r) such that x x0 (t) ≤ r for all x 0 ∈ B(r) and for all t ≥ T . Moreover, for all x 0 ∈ B\{0} we have

k x x0 (t) m d ≤ V (x x0 (t)) ≤ V (x 0 ) ≤ k x 0 m d and x x0 (t) ≤ ε( x 0 ) := σ -1 k k 1/m σ( x 0 ) ( 18
)
as long as x x0 (t) = 0, where σ -1 is the inverse function to σ (see Theorem 1). Since

σ, σ ∈ K ∞ then ε ∈ K ∞ .
Let us prove that if x x0 (t 0 ) = 0 then x x0 (t) = 0 for all t ≥ t 0 . Suppose the converse, i.e. ∃t 1 > t 0 : := x(t 1 ) > 0. Then due to continuity of t → x(t) there exists t 2 ∈ (t 0 , t 1 ) such that x(t 2 ) = δ := min{ /2, ε -1 ( /2)} and x(t) > δ for all t ∈ (t 2 , t 1 ], where ε -1 ∈ K ∞ is the inverse function to ε. In this case, from the inequality (18) we derive

x x0 (t) ≤ ε( x(t 2 ) ) = ε(δ) ≤ ε(ε -1 ( /2)) = /2
for all t ∈ (t 2 , t 1 ]. This contradicts to the supposition x x0 (t 1 ) = . Therefore, the inequality [START_REF] Hermes | Nilpotent approximations of control systems and distributions[END_REF] holds for all t ≥ 0 and any solution of (8) with x 0 ∈ B.

Necessity. We design the Lyapunov function in two steps: first, we construct a function V r : B → R + such that t → V r (x x0 (t)) is strictly decreasing as long as x x0 (t) ∈ K(r) for some r > 1; next, from V r we derive a d-homogeneous Lyapunov function V : B → R + by means of a special integral transformation inspired by [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF].

I. Let s 0 > 0, r 0 := sup s∈[-s0,s0] d(s)
and r > r 2 0 > 1. Let δ > r and a δ ∈ C ∞ c be defined as in Corollary 5. Then for any x 0 ∈ B the system (17) has the unique mild solution x δ x0 : [0, +∞) → B which, for x 0 ∈ K(δ), coincides with the unique solution x x0 of (8) as long as x δ x0 (t) ∈ K(δ). Moreover, by Corollary 3, for any T > 0 there exists L δ,T > 0 such that

x δ x1 (t) -x δ x2 (t) ≤ L δ,T x 1 -x 2 , for all t ∈ [0, T ].
Since the origin of ( 8) is globally uniformly asymptotically stable then there exists T r > 0 such that

x 0 < r ⇒ x x0 (t) ≤ 1/r, ∀t ≥ T r .
In the view of Lemma 2 and Corollary 5 we have x δ x0 (t) = x x0 (t) as long as x x0 (t) ≥ 1/δ provided that x 0 ∈ K(r). Indeed, the global uniform Lyapunov stability of ( 8) and the selection of the parameter δ > r imply that x δ x0 (t) ≤ δ for all t ≥ 0 and all x 0 ∈ K(r). Moreover, if there exists t 1 > 0 such that x δ x0 (t 1 ) ≤ 1/δ (resp. x x0 (t 1 ) ≤ 1/δ) then x δ x0 (t) ≤ 1/r (resp. x x0 (t) ≤ 1/r) for all t ≥ t 1 . In this case, for the functional V 0 : B → R + given by

V 0 (x 0 ) := sup t≥0 x x0 (t) we have V 0 (x 0 ) = sup t∈[0,Tr] x δ x0 (t)
for all x 0 ∈ K(r). Hence, using the triangle inequality we derive

|V 0 (x 1 ) -V 0 (x 2 )| ≤ sup t∈[0,Tr] x δ x1 (t) -x δ x2 (t) ≤ L δ,Tr ( x 1 -x 2 ), ∀x 1 , x 2 ∈ K(r).
The latter means that the functional V 0 satisfy the Lipschitz condition on K(r).

The global uniform Lyapunov stability of (8) implies that x ≤ V 0 (x) ≤ ε( x ), where ε ∈ K is given by [START_REF] Dashkovskiy | Local iss of reaction-diffusion systems[END_REF]. Moreover, the function t → V 0 (x x0 (t)) is decreasing as long as x x0 (t) ∈ K(r).

Let us consider the functional

V 1 (x 0 ) := +∞ 0 ã( x x0 (s) ) x x0 (s) ds, where ã ∈ C ∞ c is such that ã(ρ) = 0 for s / ∈ (1/r, r), 0 < ã(ρ) ≤ 1 for ρ ∈ (1/r, r) and ã(ρ) = 1 for ρ ∈ [1/r 0 , r 0 ]. Notice that ã( x x0 (s) ) x x0 (s) = 0 if x x0 (s) / ∈ K(r).
Repeating the above considerations we conclude that

V 1 (x 0 ) = Tr 0 ã( x δ x0 (s) ) x δ x0 (s) ds for all x 0 ∈ K(r). Since x → ã( x )
x satisfy the Lipschitz condition on B with some Lipschitz constant L > 0 then

|V 1 (x 1 )-V 1 (x 2 )| ≤ L Tr 0 x δ x1 (s) -x δ x2 (s) ds ≤T r L δ,Tr L x 1 -x 2 , ∀x 1 , x 2 ∈ K(r),
i.e., V 1 also satisfies the Lipschitz condition on K(r).

Moreover, if h > 0 and t ≥ 0 are such that

x x0 (t + θ) ∈ K(r), ∀θ ∈ [0, h] then x x0 (t + θ) = x δ xx 0 (t) (θ) for all θ ∈ [0, h] and V 1 (x x0 (t + h)) -V 1 (x x0 (t)) = Tr 0 ã( x δ x(t+h) (s) ) x δ x(t+h) (s) ds - Tr 0 ã( x δ x(t) (s) ) x δ x(t) (s) ds = Tr 0 ã( x δ x(t+h) (s) ) x δ x(t+h) (s) ds - h 0 ã( x δ x(t) (s) ) x δ x(t) (s) ds - Tr+h h ã( x δ x(t) (s) ) x δ x(t) (s) ds = Tr 0 ã( x δ x(t+h) (s) ) x δ x(t+h) (s) ds - h 0 ã( x δ x(t) (s) ) x δ x(t) (s) ds - Tr 0 ã( x δ x(t) (s + h) ) x δ x(t) (s + h) ds = - h 0 ã( x δ x(t) (s) ) x δ x(t) (s) ds = - h 0 ã( x x(t) (s) ) x x(t) (s) ds = -ã( x x0 (t + θ h ) ) x x0 (t + θ h ) h, θ h ∈ [0, h],
where the mean value theorem for integrals is utilized on the last step. The latter means that lim sup

h→0 + V1(xx 0 (t+h))-V1(xx 0 (t)) h ≤ -ã( x x0 (t) ) x x0 (t) as long as x(t) ∈ K(r). Notice that 0 ≤ V 1 (x 0 ) ≤ T r ε( x 0 ).
In this case, the functional V r : B → R + given by

V r (x) := V 0 (x) + V 1 (x)
satisfies the Lipschitz condition on K(r) and

x ≤ V r (x) ≤ (1 + T r )ε( x )
for all x ∈ B. Moreover, for any x 0 ∈ K(r) the function t → V r (x x0 (t)) is strictly decreasing and lim sup

h→0 + Vr(xx 0 (t+h))-Vr(xx 0 (t)) h ≤ -W r ( x x0 (t) )
as long as x x0 (t) ∈ K(r), where W r (ρ) := ã(ρ)ρ with ρ ∈ R + .

Due to the homogeneity of the system (8) the solutions x x0 are symmetric (see Theorem 2). Hence, if d(-s)x x0 (t + θ) ∈ K(r) for all θ ∈ [0, h] then using the identity d(-s)x x0 (t) = x d(-s)x0 (e µs t), ∀s ∈ R we derive

V r (d(-s)x x0 (t + h)) -V r (d(-s)x x0 (t)) =V r (x d(-s)x0 (e µs (t + h))) -V r (x d(-s)x0 (e µs t)) ≤-ã( x d(-s)x0 (e µs t + θ h ) ) x d(-s)x0 (e µs t+θ h ) e µs h,
where θ h ∈ [0, e µs h], and µ ∈ R is the homogeneity degree of the system (8), i.e. the function

t → V r (d(-s)x x0 (t)
) is strictly decreasing and lim sup

h→0 + V r (d(-s)x x0 (t + h)) -V r (d(-s)x x0 (t)) h ≤ -ã( d(-s)x x0 (t) ) d(-s)x x0 (t) e µs as long as d(-s)x x0 (t) ∈ K(r).
II. Inspired by [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] let us consider the functional V :

B → R + given by V (x) := +∞ -∞ e ms â(V r (d(-s)x))ds where â ∈ C ∞ is an increasing function such that â(ρ) = 1 for ρ ≥ (1+T r )ε(r), â(ρ) = 0 for ρ ≤ 1/r and â (ρ) > 0 for 1/r < ρ < (1 + T r )ε(r). By construction, the func- tional V : B → R + is d-homogeneous of degree m > 0. Moreover, V (x) ≤ +∞ -∞ e ms â((1 + T r )ε( d(-s)x) ))ds ≤ s(x) -∞ e ms â((1 + T r )ε( d(-s)x) ))ds ≤ e ms(x)
where s(x) := ln

x d /σ ε -1 1 r(1+Tr) and σ ∈ K ∞ is defined in Theorem 1. Indeed, if s ≥ s(x) then from Theorem 1 we derive σ ( d(-s)x ) ≤ e -s x d ≤ σ ε -1 1 r(1+Tr) and taking into account σ, ε ∈ K ∞ we conclude d(-s)x) ≤ ε -1 1 r(1+Tr)
or, equivalently, (1 + T r )ε( d(-s)x) ) ≤ 1/r for all s ≥ s(x). The latter means V r (d(-s)x) ≤ 1/r and â(V r (d(-s)x))) = 0 for all s ≥ s(x).

Similarly, we derive

V (x) ≥ +∞ -∞ e ms â( d(-s)x) )ds ≥ s(x)
-∞ e ms ds = e ms(x) , where s(x) := ln ( x d /σ ((1 + T r )ε(r))) . Indeed, if s ≤ s(x) then from Theorem 1 we have σ( d(-s)x ) ≥ e -s x d ≥ σ((1 + T r )ε(r)).

Since σ ∈ K ∞ then d(-s)x ≥ (1 + T r )ε(r) and â( d(-s)x )) = 1 for all s ≤ s(x). Therefore, we have proven that the inequality (10) holds for k := 1 σm ((1 + T r )ε(r))

, k := 1

σ m ε -1 1 r(1+Tr) . If x(t + θ) ∈ K(r 0 ) for all θ ∈ [0, h] then d(-s)x(t + θ) ∈ K(r) for all θ ∈ [0, h] and for all s ∈ [-s 0 , s 0 ]. Indeed, d(-s)x(t + θ) ≤ d(-s) • x(t + θ) ≤ r 2 0 = r and 1/r 0 ≤ x(t + θ) = d(s) • d(-s)x(t + θ) .
The latter means that

1/r ≤ V r (d(-s)x x0 (t + θ)) ≤ (1 + T r )ε(r) for all s ∈ [-s 0 , s 0 ] and all θ ∈ [0, h]. Using V r (d(-s)x x0 (t + h)) -V r (d(-s)x x0 (t)) ≤ -W r (x d(-s)x0 (e µs t + θ h
))e µs h = -W r (d(-s)x x0 (t + e -µs θ h ))e µs h we derive

V (x x0 (t + h)) -V (x x0 (t)) = +∞ -∞ e ms [â(V r (d(-s)x x0 (t+h)))-â(V r (d(-s)x x0 (t)))]ds = s0 -s0 e ms [â(V r (d(-s)x x0 (t+h)))-â(V r (d(-s)x x0 (t)))]ds = s0 -s0 e ms â (θ)[V r (d(-s)x x0 (t+h))-V r (d(-s)x x0 (t))]ds ≤ -h s0 -s0
e ms â (θ)W r (d(-s)x x0 (t + e -µs θ h ))e µs ds, where the mean value theorem is utilized with θ ∈ e (m+µ)s-q â (V r (d(-s)x)) d(-s)x ds ≥

[V r (d(-s)x x0 (t + h)), V r (d(-s)x x0 (t))]. Therefore, the function t → V (x x0 (t)) is strictly decreasing and lim sup h→0 + V (x x0 (t + h)) -V (x x0 (t)) h = D + V (x(t)) ≤ - +∞ -∞ e (m+µ)s â (V r (d(-s)x x0 (t)))W r ( d(-s)x x0 (t) )ds = -W (x x0 (t)),
1 r 0 e (m+µ)s * â (V r (d(-s * )x)),
where

s * ∈ [-s 0 , s 0 ]. Since d(-s * )x ∈ K(r 0 ) then 1 r < 1 r 0 ≤ V r (d(-s * )x) ≤ (1 + T r )ε(r 0 ) < (1 + T r )ε(r).
Hence, we derive inf x =1 W(x) ≥ c, where c = min{e (m+µ)s 0 ,e -(m+µ)s 0 } min We have proven that

D + V (x(t)) ≤ -W (x x0 (t))
as long as x x0 (t) ∈ K(r 0 ).

If x x0 (t) ∈ d(τ )K(r 0 ), where τ ∈ R, then d(-τ )x x0 (t) ∈ K(r 0 ). Using homogeneity of V we derive

V (x x0 (t)) = V (d(-τ )d(τ )x x0 (t)) = e -τ V (d(τ )x x0 (t)).
Since by Theorem 2 we have x d(τ )x0 (e -µτ t) = d(τ )x x0 (t) then the function t → V (x x0 (t)) is strictly decreasing as well and lim sup

h→0 + V (xx 0 (t+h))-V (xx 0 (t)) h = D + V (x(t)) =e -τ s lim sup h→0 + V (x d(τ )x 0 (e -µτ (t+h)))-V (x d(τ )x 0 (e -µτ t)) h ≤ -e -τ (1+µ) W (x d(τ )x0 (e -µτ t)) = -W (d(-τ )x d(τ )x0 (e -µτ t)) = -W (x x0 (t))
as long as x x0 (t) ∈ d(τ )K(r 0 ), where τ ∈ R is any real number. Taking into account τ ∈R d(τ )K(r 0 ) = B\{0} we complete the proof.

The proof of Corollary 1

Proof. Since B is a reflexive Banach space then by Theorem 1.6, Chapter 6, [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] any mild solution x of ( 8) with x 0 ∈ D(A) is a strong solution and by Theorem 2.9, Chapter 4, [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] we have x(t)

a.e ∈ D(A). In this case, using the representation

x(t+h) =Φ(t + h)x 0 + t 0 Φ(t -s)f (x(s))ds =x(t)+(Φ(h)-I)Φ(t)x 0 + t+h 0 Φ(t+h-s)f (x(s))ds - t 0 Φ(t -s)f (x(s))ds =(Φ(h) -I) Φ(t)x 0 + t 0 Φ(t-s)f (x(s))ds + t+h t Φ(t + h -s)f (x(s))ds =x(t)+(Φ(h)-I)x(t)+ h 0 Φ(s)f (x(t+h-s))ds we derive for x(t) ∈ D(A) x(t+h)-x(t)-h(Ax(t)+f (x(t)) h ≤ Ax(t) -(Φ(h)-I)x(t) h + f (x(t)) - 1 h h 0 Φ(s)f (x(t + h -s))ds ≤ Ax(t)-(Φ(h)-I)x(t) h + f (x(t))-1 h h 0 Φ(s)f (x(t))ds + 1 h h 0 Φ(s) f (x(t)) -f (x(t + h -s)) ds → 0 as h → 0,
where the properties of the semigroup Φ (see [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF], page 5), the local Lipschitz continuity of f and continuity of x are utilized on the last step. Since

V (x(t+h))-V (x(t))) h = V (x(t+h))-V (x(t)+h(Ax(t)+f (x(t))) h + V (x(t)+h(Ax(t)+f (x(t)))-V (x(t)) h
and V is locally Lipschitz continuous then, taking into account

|V (x(t+h))-V (x(t)+h(Ax(t)+f (x(t)))| h ≤ L x(t+h)-x(t)-h(Ax(t)+f (x(t)) h → 0 as h → 0, x(t) ∈ D(A), we derive D + V (x(t)) a.e. = D + V (x(t); Ax(t) +f (x(t))
if B is a reflexive Banach space and x 0 ∈ D(A)\{0}.

Sufficiency. The latter identity and the inequality [START_REF] Driver | Analysis Tools with Applications[END_REF] imply that the inequality (11) of Theorem 3 is fulfilled.

Necessity. In the proof of Theorem 3 we design a Lyapunov function satisfying 1), 2) and

D + V (x(t)) ≤ -W (x(t)),
as long as x(t) ∈ B\{0}.

Since B is a reflexive Banach space then for any x 0 ∈ D(A) the mild solution x is the strong one and x(t)

a.e ∈ D(A). Let t j → 0, t j > 0 as j → +∞ be an arbitrary sequence of time instances such that the latter inclusion holds. Since any mild solution is continuous function of time then for any h > 0 there exists i :

x(t i ) -x 0 < h 2 and x(t i + h) -x(h) < h 2 Obviously, i → +∞ as h → 0. Let us consider V (x0+h(Ax0+f (x0))-V (x0) h = V (x0+h(Ax0+f (x0))-V (x0+h(Ax(ti)+f (x(ti)) h + V (x0+h(Ax(ti)+f (x(ti)))-V (x(ti)+h(Ax(ti)+f (x(ti)) h + V (x(ti)+h(Ax(ti)+f (x(ti)))-V (x(ti+h)) h + V (x(ti+h))-V (x(h)) h + V (x(h))-V (x0) h .
The third term in the above sum tends to zero as h → 0 + (see above). Since the operator A is closed then x(t i ) → x 0 with x(t i ) ∈ D(A) yields Ax(t i ) → Ax 0 and

|V (x0+h(Ax0+f (x0))-V (x0+h(Ax(ti)+f (x(ti))| h ≤L Ax 0 -Ax(t i ) +L f (x 0 )-f (x(t i )) → 0 as h → 0.
Using the local Lipschitz continuity of V we derive

|V (x0+h(Ax(ti)+f (x(ti)))-V (x(ti)+h(Ax(ti)+f (x(ti))| h ≤ L x0-x(ti) h ≤ Lh → 0 as h → 0.
Similarly, we obtain

|V (x(ti+h))-V (x(h))| h ≤ Lx 0 x0-x(ti) h ≤ Lh → 0 as h → 0. Therefore, lim sup h→0 + V (x(h))-V (x0) h = lim sup h→0 + V (x0+h(Ax0+f (x0))-V (x0) h ≤ -W (x 0 ), ∀x 0 ∈ D(A)\{0}.
The proof is complete.

The proof of Theorem 4

Since the d-homogeneous operator A generates a strongly continuous semigroup Φ, then according Lemma 1 we have Φ(τ )d(s) = d(s)Φ(e µs τ ), ∀τ ≥ 0, ∀s ∈ R.

Recall (see e.g. [START_REF] Driver | Analysis Tools with Applications[END_REF], page 187) that K Let x x0,q be a solution of (1). Since f (x x0,q (•), q(•)) ∈ L 1 ((t 0 , t 0 + T ), B) then

d(s)x x0,q (t 0 + e µs (t -t 0 )) =d(s)Φ(e µs (t-t 0 ))x 0 + t0+e µs (t-t0) t0 d(s)Φ(t 0 +e µs (t-t 0 )-τ )• f (x x0,q (τ ), q(τ )) dτ =Φ(t -t 0 )d(s)x 0 + e µs t t0 d(s)Φ(e µs (t -σ))• f (x x0,q (t 0 + e µs (σ -t 0 )), q(t 0 + e µs (σ -t 0 ))) dσ =Φ(t -t 0 )d(s)x 0 + e µs t t0 Φ(t -σ)• d(s)f (x x0,q (t 0 +e µs (σ-t 0 )), q(t 0 +e µs (σ-t 0 ))) dσ =Φ(t)d(s)x 0 + t t0 Φ(t -σ)•
f (d(s)x x0,q (t 0 +e µs (σ-t 0 )), d(s)q(t 0 +e µs (σ-t 0 )))dσ, where the linearity of the operator Φ(t -σ) and the dhomogeneity of Φ, f are utilized on the last two steps. The proof is complete.

The proof of Corollary 2

I. Let q > 0 be an arbitrary positive number. Let r > 1 and δ > r be defined for f (•, 0) as in Corollary 5. Since for q = 0 the origin of ( 1) is uniformly asymptotically stable then exists T r > 0 such that

x x0,0 (t 0 ) ∈ K(r) ⇒ x x0,0 (t) ≤ 1/r, ∀t > t 0 + T r . and x x0,0 (t 0 ) ∈ K(r) ⇒ x x0,0 (t) ≤ δ, ∀t ≥ t 0 ,
Moreover, Corollary 5 yields x x0,0 (t) = x δ x0,0 (t), as long as x x0,0 (t) ∈ K(δ). For any x 0 ∈ B we have

x δ x0,q (t) = Φ(t)x 0 + t t0 Φ(t -s)f δ (x δ x0,q (τ ), q(τ ))dτ = x δ x0,0 (t)+ t t0 Φ(t-s)(f δ (x δ x0,q (τ ), q(τ ))-f δ (x δ x0,0 (τ ), 0))dτ.
Let M = sup s∈[0,Tr] Φ(s) and L δ,Tr,q > 0 be defined as in Corollary 3. Then for any q ∈ L ∞ ((0, T r ), V) : q ∞ ≤ q we have

x δ x0,q (t) -x δ x0,0 (t) ≤ M L δ,Tr,q t t0 x δ x0,q (τ ) -x δ x0,0 (τ ) + M t t0 a δ ( x δ x0,0 (τ ) ) f (x δ x0,0 (τ ), q(τ ))-f (x δ x0,0 (τ ), 0) dτ for all t ∈ [t 0 , t 0 + T r ] such that x x0,q (t) < +∞.
By assumption, there exist Lδ,q > 0 such that

f (x 1 , q 1 ) -f (x 2 , q 2 ) ≤ Lδ,q ( x 1 -x 2 + q 1 -q 2 γ )
for all x 1 , x 2 ∈ K(2δ) and for all q 1 , q 2 ∈ B(q). Since a δ ( x ) = 0 for x / ∈ K(2δ) then

x δ x0,q (t) -x δ x0,0 (t) ≤M L δ,Tr,q t t0 x δ x0,q (τ )-x δ x0,0 (τ )
+ M Lδ,q t t0 q(τ ) γ dτ and using Grönwall-Bellman inequality we derive

x δ x0,q (t) -x δ x0,0 (t) ≤ Lδ,q M e L δ,Tr , q M t t t0 q(τ ) γ dτ, for all t ∈ [0, T r ] and any q ∈ L ∞ ((0, T r ), V) : q ∞ ≤ q.

Using Theorem 3 (see also its proof) we conclude that there exists a d-homogeneous Lyapunov functional V : B → R of degree 1 such that for any x 0 ∈ B\{0} we have V (x x0,0 (h)) -V (x x0,0 (0)) ≤ - For any h > 0 : x x0,0 (θ), x x0,q (θ) ∈ K(δ), ∀θ ∈ [0, h] we have

x x0,0 (θ) = x δ x0,0 (θ), x x0,q (θ) = x δ x0,q (θ), θ ∈ [0, h].

Then taking into account the local Lipschitz continuity of V and x x0,q (0) = x x0,0 (0) = x 0 we derive V (x x0,q (h)) -V (x x0,q (0)) =V (x x0,0 (h))-V (x x0,0 (0))+V (x x0,q (h))-V (x x0,0 (h))

≤ - x x0,0 (τ ) d dτ +L V,δ Lδ,q M e L δ,Tr , q M h h 0 q(τ ) γ dτ, for any q ∈ L ∞ ((0, h), V) : q ∞ ≤ q. Therefore, one has lim sup h→0 + V (xx 0 ,q (h))-V (xx 0 ,q (0)) h ≤ -x 0 1+µ d + lim sup h→0 +

C(q)

h 0 q(τ ) γ dτ, h where C(q) := L V,δ Lδ,q M e L δ,Tr , q M Tr . The function q → C(q) is uniformly bounded on any compact from R + .

Since x x0,q (t + h) = x x * ,q * (h) with x * = x x0,q (t) and q * (h) := q(t + h) then the function t → V (x x0,q (t)) is strictly decreasing and

D + V (x x0,q (t)) ≤ -x x0,q (t) µ+1 d + C(q) lim sup h→0 + 1 h h 0 q(t + τ ) γ dτ
as long as x x0,q (t) ∈ K(δ) and

x x0,q (t) µ+1 d > C(q) lim sup

h→0 + 1 h h 0 q(t + τ ) γ dτ.
The latter conclusion holds for any q ∈ L ∞ (R, V) such that q L ∞ ((t0,t),V) ≤ q.

II. Let d(s)x x0,q (t) ∈ K(r) or, equivalently, x x0,q (t) ∈ d(-s)K(r) for all t ∈ [t 0 , t 0 + T ] and some T > 0. From Theorem 4 we derive x d(s)x0,q ( t) ≤ d(s)x x0,q (t 0 +e µs ( t-t 0 )), ∀ t ∈ t 0 , t 0 + T e µs , where t := t 0 + e -µs (t -t 0 ), x d(s)x0,q : [t 0 , t 0 + e -µs T ] → B is a solution of the system (1) for x 0 replaced with d(s)x 0 and q ∈ L ∞ ((t 0 , t 0 + T ), V) replaced with q ∈ L ∞ ((t 0 , t 0 + e -µs T ), V) given by q( t) = d(s)q(t 0 + e µs ( t -t 0 )), ∀ t ∈ t 0 , t 0 + T e µs .

Since x d(s)x0,q ( t) = d(s)x x0,q (t 0 +e µs ( t-t 0 )) ∈ K(r) then D + V (x d(s)x0,q ( t)) ≤ -x d(s)x0,q ( t) 1+µ d + C(q) lim sup h→0 + h 0 q( t+τ ) γ dτ h for all t ∈ [t 0 , t 0 + e -µs T ].

Taking into account homogeneity of V and • d we derive

V (x d(s)x0,q ( t)) =V (d(s)x x0,q (t 0 +e µs ( t -t 0 ))) =e s V (x x0,q (t 0 +e µs ( t -t 0 ))) and x d(s)x0,q ( t) µ+1 d = e (µ+1)s x x0,q (t 0 +e µs ( t -t 0 )) d .

The latter means that for any q ∈ L ∞ (R, V) : q L ∞ ((t0,t),V) ≤ q and any x 0 ∈ B\{0}we have D + V (x x0,q (t)) ≤ -x x0,q (t) 1+µ e -(µ+1)s C(q) h 0 d(s)q(t+e µs τ ) γ dτ h as long as x x0,q (t) ∈ d(-s)K(r).

Since the dilation d is strictly monotone then there exists β > 0 such that d(s) ≤ e βs , ∀s ≤ 0.

By Theorem 1 there exists σ ∈ K ∞ such that q d ≤ σ( q ) ≤ σ(q) for all q ∈ V : q ≤ q. Hence, using the group property of d, we derive d(s)q = d(s + ln σ(q)) d(ln q d -ln σ(q)) d(ln q d)q

but the strict monotonicity of d yields d(s)q ≤ d(s + ln σ(q)) ≤ e β(s+ln σ(q)) , ∀s ≤ -ln σ(q).

The latter means that D + V (x x0,q (t)) ≤ -x x0,q (t) 1+µ

d + e ( βγ-(µ+1))s C(q)σ(q)

as long as x x0,q (t) ∈ d(-s)K(r) and s ≤ -ln σ(q).

Selecting s = -ln x x0,q (t) d we derive d(ln x x0,q (t) d )x x0,q (t) = 1 and x x0,q (t) ∈ d(-s)K(r). For any q ∈ L ∞ (R, V) : q L ∞ ((t0,t),V) ≤ q and any x 0 ∈ B\{0} we have D + V (x x0,q (t)) ≤ -x x0,q (t) 1+µ d 1-C(q)σ γ (q)

x x0,q (t) βγ d as long as x x0,q (t) d ≥ σ(q). Since C : R + → R + is uniformly bounded on compacts from R + and σ ∈ K ∞ then then there exists ξ ∈ K ∞ such that ξ(q) ≥ max{σ(q), (2C(q)σ(q)) 1/( βγ) }.

Therefore, we have D + V (x x0,q (t)) ≤ -0.5 x x0,q (t) 1+µ d as long as x x0,q (t) d ≥ ξ(q), where x x0,q is any solution of (1) with an arbitrary q ∈ L ∞ (R, V) : q L ∞ ((t0,t),V) ≤ q. The latter means that V is the ISS Lyapunov function of the system. Repeating the conventional arguments (see e.g. [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] for more details) we complete the proof.

Definition 6 (

 6 [START_REF] Polyakov | On homogeneous finite-time control for linear evolution equation in hilbert space[END_REF]) The functional • d : B → R + given by 0 d = 0 andu d = e su , where s u ∈ R : d(-s u )u = 1, u = 0(4) is called the canonical homogeneous norm in the Banach space B, where d is a strictly monotone dilation in B. Obviously, d(s)u d = e s u d and u d = -u d for ∀u ∈ B and ∀s ∈ R. The homogeneous norm • d is called canonical since it is induced by the canonical norm • in B and u d = 1 ⇔ u = 1. Notice that • d = • provided that d is the standard dilation d(s) = e s I, s ∈ R.

Lemma 1 ([ 37 ], Lemma 8 . 1 )

 13781 Let a linear closed densely defined operator A : D(A) ⊂ B → B generate a strongly continuous semigroup Φ of linear bounded operators on B and d be a group of linear bounded invertible operators on B. If the operator A is d-homogeneous of degree µ ∈ R then Φ(t)d(s) = d(s)Φ(e µs t), ∀t ≥ 0, ∀s ∈ R. (6)

Theorem 2 ([ 37 ], Theorem 8 . 1 )

 23781 Let d be a group of linear bounded invertible operators on B. Let a linear closed densely defined operator A : D(A) ⊂ B → B generate a strongly continuous semigroup Φ of linear bounded operators on B. Let A and f be d-homogeneous operators of degree µ ∈ R. If x : [0, T ) → B is a mild solution of the evolution equation

  Example 1 it is already shown that the Laplace operator ∆ is d-homogeneous of degree 2β with the dilation (d(s)x)(z) = e αs x(e βs z), where α > nβ/2. Simple computations show d(s)x = R ne αs x(e βz ) 2 dz = e α-nβ/2 x .[START_REF] Engel | One-Parameter Semigroups for Linear Evolution Equations[END_REF]

Theorem 4

 4 Let d be a group of linear bounded invertible operators in B, the operator A : D(A) ⊂ B → B be d-homogeneous of degree µ, closed densely defined generator of a strongly continuous semigroup Φ of linear bounded operators on B, the function f : B × V → B satisfy (2) and

  as long as x x0 (t) ∈ K(r 0 ), whereW (x) := +∞ -∞ e (m+µ)s â (V r (d(-s)x))W r ( d(-s)x )ds.The functional W is, obviously, nonnegative and d-homogeneous of degree µ + 1. If x = 1 then d(-s)x ∈ K(r 0 ) and W r ( d(-s)x ) = d(-s)x for all s ∈ [-s 0 , s 0 ]. Using the mean value theorem for integrals we deriveW (x) ≥ s0 -s0

1/r 0

 0 ≤ρ≤(1+Tr )ε(r 0 ) â (ρ) r0 > 0 and using d-homogeneity of W we concludeW (x) =W (d(ln x d )d(-ln x d )x) = x m+µ d W (d(-ln x d )x) ≥ c x m+µ d .

a

  Kξ(s)ds for any bounded linear operator K : B → B and any Bochner integrable function ξ ∈ L 1 ((a, b), B).

h 0 x

 0 x0,0 (τ ) µ+1 d dτ as long as x x0,0 (h) = 0.

h 0 x

 0 x0,0 (τ ) µ+1 d dτ + L V,δ x x0,q (h) -x x0,0 (h) ≤ -h 0

  Theorem 1 ([37], Lemmas 7.1, 7.2) If d is a strongly continuous strictly monotone dilation then • d is singlevalued, positive definite, locally Lipschitz continuous on B\{0} and there exist σ

The system (1) is locally ISS if in Definition 8 x0 and q are restricted to some balls in B and V, respectively.