Maxime Gauthier

Jérémy Defrance

Catherine Jumarie

Emmanuelle Vulliet

Jeanne Garric

Monique Boily

Olivier Geffard

Disruption of oogenesis and molting by methoprene and glyphosate in Gammarus fossarum: involvement of retinoic acid?

Keywords: Reprotoxicity, Endocrine disruption, Amphipod, Vitamin A, Juvenile hormone agonists, Insecticide

In the last decade, the freshwater amphipod Gammarus fossarum proved to be a promising sentinel species in active biomonitoring programs to assess the effects of environmental contamination on non-target organisms. Given that the highly conserved retinoid (RETs) metabolism supports many biological functions and is perturbed by xenobiotics and used as biomarker for vertebrates, we explored the RETs functions in the crustacean model Gammarus fossarum. More specifically, we studied the implication of all-trans retinoic acid (atRA) in the reproduction (embryo, oocyte, and juvenile production) and development (success and delay of molting) by exposing G. fossarum females to atRA and citral (CIT), a known inhibitor of RA synthesis. In parallel, we exposed gammarids to methoprene (MET) and glyphosate (GLY), two pesticides suspected to interfere with atRA metabolism and signaling and frequently found in water systems. After 14 days of exposure, atRA, CIT, and MET reduced the number of oocytes, whereas only MET caused a reduced number of embryos. After 44 days, MET and GLY showed a tendency to decrease juvenile production. The duration of the molting cycle increased following the exposures to atRA and MET, while the treatment with CIT caused a typical endocrine disruptive inverted U-shaped curve. The exposure to GLY led to increased duration of the molting cycle at the lowest concentrations and lowered molting success at the highest concentration tested. This study highlights for the first time the implication of RA in the oogenesis and molting of G. fossarum and suggests that it may be a potential mediator of METinduced effects on these processes. This study adds to the comprehension of the reproductive and developmental control in G. fossarum and opens new research avenues to study the effects of xenobiotics on the RET system in this sentinel species. Ultimately, our study will drive the development of RET-based biomarkers for non-target aquatic invertebrates exposed to xenobiotics.

Introduction

Retinoids (RETs) are small molecules structurally similar to retinol (ROL, vitamin A) and are obtained from diet-derived carotenoids (CARs) or RET esters. Retinol is metabolized to retinoic acid (RA), the biologically active form of RETs. Retinoic acid plays crucial roles in cell growth, differentiation, and apoptosis, and is therefore implicated in embryonic development, growth, reproduction, and immunity through the activation of the Retinoic Acid Receptor (RAR) and the Retinoid X Receptor (RXR, [START_REF] Kam | Retinoic acid synthesis and functions in early embryonic development[END_REF]. For example, RA determines the onset of meiosis during gonadal development in mammals, birds, fish, and amphibians [START_REF] Smith | Onset of meiosis in the chicken embryo; evidence of a role for retinoic acid[END_REF][START_REF] Wallacides | Evidence for a conserved role of retinoic acid in urodele amphibian meiosis onset[END_REF][START_REF] Clagett-Dame | Vitamin A in reproduction and development[END_REF][START_REF] Peng | Retinoic acid and androgen influence germ cells development and meiotic initiation in juvenile orangespotted grouper, Epinephelus coioides[END_REF]. Because CARs are essential precursors of RETs, and knowing that imbalances of RA tissular concentrations have crucial implications for vertebrates, and their metabolism. CARs are involved in (1) protection against UV radiation and oxidative damages, (2) pigmentation, and (3) supply for bioconversion to RETs. In crustaceans, CARs are also well known for their essential role in reproduction [START_REF] Wade | A review of carotenoid utilisation and function in crustacean aquaculture[END_REF]. During early ovarian maturation, free or esterified forms of CARs are accumulated in the hepatopancreas, transported to the ovaries, and then incorporated into the developing oocytes in association with the vitellogenin [START_REF] Wang | Reproductive mechanisms in Crustacea focusing on selected prawn species: vitellogenin structure, processing and synthetic control[END_REF]. However, to this date, it is still unknown if these functions are assigned to CARs or if they are carried out by RETs. This exemplifies the remarkable lack of understanding relative to the RET functions in crustaceans. More recently, several endogenous RET isomers were studied in crustaceans tissues. More precisely, 13-cis/alltrans retinaldehyde (RALD) and 13-cis ROL were measured in the developing ovaries and all-trans ROL was measured in the eyestalks of Litopenaeus vanamei [START_REF] Linan-Cabello | Induction factors derived from carotenoids and vitamin A during the ovarian maturation of Litopenaeus vannamei[END_REF]. The 9-cis and all-trans isomers of RA and the isomers 13-cis and all-trans of RALD were measured in the regenerating blastemas of Uca pugilator (Hopkins et al. 2008) and the isomers 9-cis and all-trans of RA were measured in the mature ovaries of Scylla serrata [START_REF] Venkaiah | Detection and mode of action of retinoids on ovarian development in the mud crab Scylla Serrata[END_REF]. Isomers of RA (13-cis and alltrans), RA metabolites (13-cis 4-oxo and all-trans 4-oxo RA) and isomers of RALD (11-cis, 13-cis and all-trans RALD) were also found in whole-tissue homogenates of G. fossarum [START_REF] Gauthier | The retinoid metabolism of Gammarus fossarum is disrupted by exogenous all-trans retinoic acid, citral, and methoprene but not by the technical formulation of glyphosate[END_REF]. In this study, it was also showed that RA metabolites and the sum of two RALD isomers (11-cis and all-trans) were modulated throughout the reproductive cycle [START_REF] Gauthier | The retinoid metabolism of Gammarus fossarum is disrupted by exogenous all-trans retinoic acid, citral, and methoprene but not by the technical formulation of glyphosate[END_REF]. RXR is expressed in the eyestalks of Callinectes sapidus, Oziotelphusa senex senex, and Macrobrachium nipponence [START_REF] Techa | Ecdysone and retinoid-X receptors of the blue crab, Callinectes sapidus: cloning and their expression patterns in eyestalks and Y-organs during the molt cycle[END_REF][START_REF] Li | Identification of spliced mRNA isoforms of retinoid X receptor (RXR) in the Oriental freshwater prawn Macrobrachium nipponense[END_REF][START_REF] Girish | Expression of RXR, EcR, E75 and VtG mRNA levels in the hepatopancreas and ovary of the freshwater edible crab, Oziothelphusa senex senex (Fabricius, 1798) during different vitellogenic stages[END_REF], in the ovaries of U. pugilator, Gecarcinus lateralis, Marsupenaeus japonicus, Fenneropenaeus chinensis, Carcinus maenas, S. serrata, and Daphnia magna [START_REF] Durica | Characterization of crab EcR and RXR homologs and expression during limb regeneration and oocyte maturation[END_REF][START_REF] Kim | Ecdysteroid-responsive genes, RXR and E75, in the tropical land crab, Gecarcinus lateralis: differential tissue expression of multiple RXR isoforms generated at three alternative splicing sites in the hinge and ligand-binding domains[END_REF][START_REF] Asazuma | Molecular cloning and expression analysis of ecdysone receptor and retinoid X receptor from the kuruma prawn, Marsupenaeus japonicus[END_REF][START_REF] Wang | Reproductive mechanisms in Crustacea focusing on selected prawn species: vitellogenin structure, processing and synthetic control[END_REF][START_REF] Priya | Molecular characterization and effect of RNA interference of retinoid X receptor (RXR) on E75 and chitinase gene expression in Chinese shrimp Fenneropenaeus chinensis[END_REF][START_REF] Nagaraju | Molecular cloning and sequence of retinoid X receptor in the green crab Carcinus maenas: a possible role in female reproduction[END_REF][START_REF] Girish | Expression of RXR, EcR, E75 and VtG mRNA levels in the hepatopancreas and ovary of the freshwater edible crab, Oziothelphusa senex senex (Fabricius, 1798) during different vitellogenic stages[END_REF] and in homogenates of whole Gammarus fossarum [START_REF] Gouveia | Identification, expression, and endocrine-disruption of three ecdysoneresponsive genes in the sentinel species Gammarus fossarum[END_REF]. To ensure adequate RA tissular concentrations, the RET metabolism is regulated by a cascade of oxidase/reductase and isomerase as well as many intra-and extracellular binding proteins. Interestingly, citral (CIT), a compound known to inhibit the endogenous RA synthesis, induces limb deformities in developing chick embryos [START_REF] Tanaka | Citral, an inhibitor of retinoic acid synthesis, modifies chick limb development[END_REF] and has similar effects in the regenerating limb bud of the crustacean U. pugilator [START_REF] Hopkins | Limb regeneration in the Fiddler Crab, Uca pugilator: hormonal and growth factor control[END_REF]. However, studies reporting exposure of crustaceans to CIT are limited, and to our knowledge, none evaluated the effects of CIT on crustacean reproduction. On the other hand, exogenous exposures to RA in crustaceans were shown to influence circulating levels of glucose and hormones as well as reproduction and regeneration [START_REF] Hopkins | Limb regeneration in the Fiddler Crab, Uca pugilator: hormonal and growth factor control[END_REF][START_REF] Reddy | Effect of retinoic acid on hemolymph glucose regulation in the fresh water edible crab Oziotelphusa senex senex[END_REF][START_REF] Sreenivasula | 13-Cis-retinoic acid-induced hyperglycemia in the fresh water edible crab, Oziothelphusa senex senex is mediated by triggering release of hyperglycemic hormone from eyestalks[END_REF][START_REF] Girish | Evidence for retinoic acid involvement in the regulation of vitellogenesis in the fresh water edible crab, Oziotelphusa senex senex[END_REF][START_REF] Venkaiah | Detection and mode of action of retinoids on ovarian development in the mud crab Scylla Serrata[END_REF]. Thus, the detection and variations of several endogenous isomers of RETs, the expression of RXR, and the responsiveness of crucial physiological processes to RA in crustaceans point toward important functions of RA in their physiology and deserve further research. The massive usage of numerous pesticides for public health purposes (e.g., control of the mosquito populations) or in agriculture raises concern for aquatic non-target organisms [START_REF] Lawler | Environmental safety review of methoprene and bacterially-derived pesticides commonly used for sustained mosquito control[END_REF][START_REF] Zaller | Editorial: non-target effects of pesticides on organisms inhabiting agroecosystems[END_REF]. Among these, methoprene (MET) more specifically its enantiomer S-MET, a juvenile hormone analog (JHA), is used to prevent the metamorphosis of mosquitoes in many countries. For vertebrates, both MET and its metabolite methoprenic acid are strong ligands for RXR [START_REF] Harmon | Activation of mammalian retinoid X receptors by the insect growth regulator methoprene[END_REF]. Because xenobiotics can interfere with the RET system, RETs have served as the basis for developing biomarkers of environmental contamination in several species of mammals, birds, fish, and amphibians [START_REF] Boily | Retinoides : biomarqueurs et base moleculaire d'effets de substances toxiques. Ecotoxicologie Moleculaire : Principes fondamentaux et perspectives de developpements[END_REF]. A recent study observed perturbed RET concentrations in G. fossarum following exposure to the insecticide MET [START_REF] Gauthier | The retinoid metabolism of Gammarus fossarum is disrupted by exogenous all-trans retinoic acid, citral, and methoprene but not by the technical formulation of glyphosate[END_REF]. Other studies included the insect Apis mellifera in this list. Honey bees exposed to thiamethoxam, glyphosate (GLY), or atrazine had altered levels of some CARs and RETs [START_REF] Helmer | Effects of realistic doses of atrazine, metolachlor, and glyphosate on lipid peroxidation and dietderived antioxidants in caged honey bees (Apis mellifera)[END_REF][START_REF] Jumarie | Mixtures of herbicides and metals affect the redox system of honey bees[END_REF][START_REF] Gauthier | Chronic exposure to imidacloprid or thiamethoxam neonicotinoid causes oxidative damages and alters carotenoid-retinoid levels in caged honey bees (Apis mellifera)[END_REF]. Such investigations on the perturbation of the RET metabolism by pesticides were never undertaken in crustaceans and would provide avenues to suggest new RET-based biomarkers in crustaceans. Reproduction is a meaningful and critical endpoint from an ecological perspective as it allows the maintenance of a population. For this reason and because it was shown to be useful to study the effects of environmental contaminants, in vitro tests assessing the reproduction of non-target aquatic species gained much scientific interest. The reproductive toxicity bioassay developed by [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: application for reproductive toxicity assessment[END_REF] for the non-target crustacean amphipod G. fossarum is particularly interesting for several reasons. G. fossarum plays a significant role in wild ecosystems: it is a scavenger, thus participates in the degradation of organic matter, and is a feeding source for many organisms. It is widely distributed in Europe (and, to some extent, in America), and easy to collect and maintain under laboratory conditions. Furthermore, gammarids are non-target organisms and are sensitive to chemicals and xenobiotics, such as pesticides [START_REF] Bundschuh | Effects of peak exposure scenarios on Gammarus fossarum using field relevant pesticide mixtures[END_REF]. The bioassay allows the identification of reproduction and molting defects in G. fossarum. In healthy amphipods, ovarian maturation is synchronized with the molting cycle and happens concurrently. In this context, the synchronization of ovarian maturation and molting offers a biological framework and is exploited to allow not only the observation the identification of endocrine disruptive compounds (EDC). Besides, the assay offers an interesting experimental context to acquire new knowledge on the molecular mediators of these hormone-regulated processes. For vertebrates, the RET signaling pathway is currently being examined and discussed as a potential candidate to evaluate EDC (Organisation for Economic Cooperation and Development 2021). Considering the evolutionary conservation of RET signaling throughout many invertebrate phyla, the study of RET functions in crustaceans may prove helpful in identifying new mechanisms of EDC, which pose a contemporary challenge for non-target aquatic crustaceans. The objective of this study was to assess the potential use of RETs as biomarkers in the sentinel freshwater amphipod model G. fossarum. Thus, for the first time, the implication of RA in the reproduction and development of this model was assessed using the in vitro bioassay developed by [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: application for reproductive toxicity assessment[END_REF]. In the first experiment, we investigated its physiological involvement by monitoring parameters of reproduction (embryo, oocyte, and juvenile production) and development (success and delay of molting) following the exposure to atRA and citral (CIT), a known RA synthesis inhibitor. In the second experiment, to probe the RET metabolism for biomarker development, we studied the toxicological effects of the two following model pesticides: methoprene (MET), a typical EDC in crustaceans, and glyphosate (GLY). Both are known or suspected to intervene with the RA metabolism or signaling pathway [START_REF] Harmon | Activation of mammalian retinoid X receptors by the insect growth regulator methoprene[END_REF][START_REF] Paganelli | Glyphosate-based herbicides produce teratogenic effects on vertebrates by impairing retinoic acid signaling[END_REF][START_REF] Helmer | Effects of realistic doses of atrazine, metolachlor, and glyphosate on lipid peroxidation and dietderived antioxidants in caged honey bees (Apis mellifera)[END_REF][START_REF] Gauthier | Chronic exposure to imidacloprid or thiamethoxam neonicotinoid causes oxidative damages and alters carotenoid-retinoid levels in caged honey bees (Apis mellifera)[END_REF]) and are frequently found in water systems [START_REF] Lawler | Environmental safety review of methoprene and bacterially-derived pesticides commonly used for sustained mosquito control[END_REF][START_REF] Matozzo | The effects of glyphosate and its commercial formulations to marine invertebrates: A review[END_REF].

Materials and methods

Chemicals and exposure solutions

Acetone, analytical grade all-trans RA, CIT, GLY (n-(phosphomethyl)glycine), and MET (PESTANAL®) were purchased from Sigma Aldrich (St. Quentin Fallavier, France). All-trans RA, CIT, and MET were dissolved in acetone, and GLY was dissolved in groundwater. We used acetone which is the solvent routinely used in our lab for similar experiences because it was previously demonstrated as having no effects on reproductive parameters in G. fossarum [START_REF] Gouveia | Identification, expression, and endocrine-disruption of three ecdysoneresponsive genes in the sentinel species Gammarus fossarum[END_REF]. Glyphosate readily dissolved in groundwater and, therefore, no vehicle was needed. For both experiments, atRA, CIT, and MET solutions were further diluted with acetone to constitute the working solutions (WS) while GLY solution was diluted serially with groundwater to obtain the WS. WSs for atRA, CIT, MET, and GLY were kept in a -80 °C freezer and used daily to expose gammarids. Nominal concentrations were used in the text; however, concentrations of the first dilution for each compound were confirmed by chemical analysis and were all within 10% of the theoretical values for both experiments (Table S1). Analyses for atRA, CIT, and MET were conducted by the TRACES research team of the Institut des Sciences Analytiques de Lyon UMR5280 (Rhone, France) whereas GLY analyses were performed by La Drome Laboratoire (Drome, France).

Gammarid collection and acclimatization

Gammarids used in experiments 1 and 2 were collected in February and in May 2020, respectively, in La Cressonniere du Bugey, a watercress bed (Ain, France) known for its very low anthropogenic pressure and used as a control site. Gammarids were selected by sieving (2-2.5 mm) were brought back to the laboratory for a 10-day acclimation period (aerated 30 L aquariums, 16 °C ± 0.5 °C, 16/8 h light/dark cycles), fed ad libitum with Alnus spp. leaves and provided with continuous fresh groundwater. Groundwater was obtained from a forage well connected to the water table in Villeurbanne (Rhone, France), filtered through sand and a 10 μm filter with activated charcoal. In amphipods, the reproductive cycle of the female is synchronized with molting. After molting, eggs are laid in the marsupium and fertilized by a male later during the reproductive amplexus. The embryos will continue to develop, and the hatched eggs will be released from the marsupium before the upcoming molt [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: application for reproductive toxicity assessment[END_REF]. For both experiments, to ensure that females used were all in the reproductive stage A and within 24 h of each other, females were selected as follows(see experiment 2 for stage description). A large number of couples (± 1200 for experiment 1, and ± 1600 for experiment 2) were isolated. All females that have laid their eggs were discarded after 48 h. The following day, the same process was carried out. This time, freshly molted females in the reproductive cycle A were pooled and distributed randomly in each exposure condition (see the "Experiments" section).

Experiments

We conducted two experiments based on the bioassay developed by [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: application for reproductive toxicity assessment[END_REF]. For both experiments, we monitored survival daily (dead individuals were discarded) to ensure the observation of sublethal effects and confirmed that no significant differences were noted between CTRL and SOLV groups (Figs. S1 andS2). The first experiment lasted 14 days and was realized to observe the effects of atRA, CIT, and MET on the reproduction parameters of female G. fossarum (production of embryos and oocytes). These two parameters were observed for each females. For this reason, we tested a range of concentrations of atRA and For both pesticides, we also wanted to encompass concentrations that are found in the environment (up to 45 μg/L for MET and up to 105 mg/L for GLY, [START_REF] Brovini | Glyphosate concentrations in global freshwaters: are aquatic organisms at risk?[END_REF][START_REF] Hershey | Effects of methoprene and Bti (Bacillus thuringiensis var. israelensis) on non-target insects[END_REF] and selected only the active ingredients alone given the annual shifts in the commercial formulations employed. In our design, MET was also selected as a reference chemical to illustrate the effects of a typical EDC. In the second experiment, we wished to evaluate the effects of atRA, CIT, MET, and GLY on juvenile production and molting. Therefore, gammarids were exposed for 44 days, which allowed the observation of a whole reproduction cycle. That allowed us to monitor the juvenile production, success of molting, and duration of the molting cycle.

Experiment 1 Previously selected and synchronized females were randomly allocated to beakers (500 mL) containing groundwater in triplicates to three concentrations of atRA (7, 70, and 210 μg/L), CIT (11, 110, and 330 μg/L), and MET (6, 60, and 600 μg/L). Controls (CTRL) and control for acetone (SOLV) were also made in triplicates. In this experiment, we used 7 synchronized females per beaker. To avoid atRA photodegradation, exposures to this compound were made in beakers wrapped in aluminum foil. To ensure constant exposure throughout both experiments, we renewed the media daily using groundwater (aerated for 48 h) and 25 μL of freshly thawed WS for each compound. According to [START_REF] Gouveia | Identification, expression, and endocrine-disruption of three ecdysoneresponsive genes in the sentinel species Gammarus fossarum[END_REF], the acetone concentration we used (0.005%v/v) has no effects on gammarid reproduction. A square of Nitex nylon mesh fabric (5 . 10 cm) was placed in each beaker for enrichment and to allow movement and dispersion of gammarids. After 14 days, females typically in the C2-D1 stage readily allow the observation of the production of embryos (number of embryos in the marsupium) and oocytes (number of oocytes located dorsally). Hence, using a binocular, embryos were emptied from the marsupium and they were counted for each female. Oocytes were counted with a microscope. Given the direct influence of female size on the production of embryos and oocytes, both parameters were normalized by their size (mm) which were measured on photographs (. 10) by image analysis using ImageJ (v1.53a). Because only part of the primary oocytes (stage B and C1) goes through the secondary vitellogenesis (stage C2) as shown by [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: application for reproductive toxicity assessment[END_REF], the count of oocytes in the present experiment was performed on females that showed no delay in their development (C2 and D1). We assessed molting stages of females (A, B, C1, C2, D1, and D2) based on the observation of the integument changes on the dactylopodite and protopodite of individuals as described by [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: application for reproductive toxicity assessment[END_REF]. The postmolt stage A is characterized by a fine and soft cuticle that hardens during stage B, easily observable on the dactylopodite setae. During the intermolt stage (C), there is progressive epidermal retraction. In stage C1, the setae retract along the protopodite, while in stage C2, the new dactylopodite is completely retracted. Premolt stage (D) is characterized by the synthesis of a new cuticle visible on the new dactylopodite (D1) and the protopodite setae (D2). Given the short duration of stages A and B, they were merged as stage AB.

Experiment 2 To further investigate the sub-lethal effects of atRA, CIT, and MET on ecdysis and reproduction over a long period, gammarids were exposed for 44 days to the same compounds following similar exposure conditions. In addition to survival, temperature, conductivity, and dissolved oxygen concentration (mg/L) were monitored daily (Table S2). According to the results from the first experiment and to optimize survival for a longer exposure period, concentrations of the compounds for the second experiment were adjusted as follows: atRA (0.11, 1.1, and 11 μg/L), CIT (6, 60, and 600 μg/L), and MET (0.25, 2.5, and 25 μg/L). We also investigated the effects of GLY at the following concentrations: 0.45, 4.5, and 45 mg/L. In this experiment, we used 10 synchronized females per beaker (triplicates). When females reached the C2 stage (after 17 days), one ready-tomate male (in copulating amplexus) per female was added to each beaker and left for copulation. They were removed when no more dyads were found which corresponds, in our study, to 32 days. The reproductive stage of every female was registered daily to estimate the molting cycle duration. We measured the duration (days) of the female reproductive cycle from the beginning of the exposure (day 1) until the female was freed from copulating amplexus and had fertilized eggs in their marsupium (AB stage). During this period, the biological endpoints observed were (1) the number of produced juveniles per female, (2) the number of females succeeding to complete molting and egg-laying, and (3) the duration of the molting cycle. For the juvenile production, the newly hatched juveniles were counted every day and standardized by the number of females in that beaker. Up to 10 juveniles per beaker were photographed daily using a binocular. Juveniles were also measured by image analysis using ImageJ (v1.53a). On day 32, females who had not started mating or molting were registered and removed from experimental units.

Statistical analyses

Survival was tested with Log-rank (Mantel-Cox) tests comparing curves from each condition to their control group, respectively. Results for gammarids exposed to atRA, CIT, and MET were compared with the SOLV group (to assess the effect of solvent), whereas the ones exposed to GLY were compared with the CTRL group. Embryo and oocyte count and duration of the reproductive cycle were tested using the GLM (General Linear Model) one-way analyses of variance. We verified the normality of data distribution and the homogeneity of variance with Shapiro-Wilk and Levene tests respectively. Outliers were identified and discarded following a Grubb's test (< 2% of data for all parametric tests). Significant models (p < 0.05) were followed by a Dunnett "t" test against the respective control group. For mean juvenile production (data per beaker) and when normality and homogeneity of variance were not met, the non-parametric Kruskal-Wallis (KW) test was used, followed by Dunn's test (pairwise comparison with the respective control group). The non-parametric Jonckheere-Terpstra (JT) test, based on the medians ordered in a particular direction, was performed to detect a possible trend related to a concentration-response relationship when neither parametric nor KW detected significant effects. When applicable, unilateral statistical tests were employed. Mouting success was analyzed with Fisher's exact test by comparing predicted and observed events of molting. The p-values were considered significant when they were lower than 0.5 and marginally significant when between 0.5 and 1. All statistical analyses were performed with RStudio (version 1.4.1717) except for survival analyses that were performed with GraphPad Prism (version 9.3.1).

Results

Experiment 1

In the first experiment, we did not find significant differences in survival among groups (Fig. S1). Embryo count of atRAexposed females showed a significant dose-dependent diminution. The mean diminution of embryo count was equivalent to -35% and -25% at the two highest concentrations of atRA, respectively (70 μg/L and 210 μg/L, Fig. 1). Embryo count following CIT exposure did not show any significant differences. On contrary, MET caused a clear diminution of embryo count for all concentrations tested (6 μg/L, 60 μg/L, and 600 μg/L) and represented -53%, -50%, and 64.5% of the embryo count in unexposed females. AtRA, CIT, and MET all caused clear oocyte count diminutions at every concentration tested.

Experiment 2

In this experiment, no statistical differences were found between the CTRL and the SOLV group for survival, respectively (Fig. S2). Similarly, for gammarids exposed to atRA, CIT, and GLY, no statistically significant differences were found between the respective control and the compounds. However, it was observed a marginally significant model for gammarids exposed to MET (p = 0.053). This result was associated with significantly higher survival of gammarids exposed to the highest concentration of MET (25 μg/L). The juvenile hatching kinetic showed very similar rates among all conditions (Fig. 3A-D). It was also reported that newly hatched juveniles were significantly smaller following exposure to atRA, CIT, MET, and GLY (Fig. 3E-H). Although statistically significant in some cases, the effects of atRA, CIT, MET, and GLY on juvenile size were minute (< 3%) and will not be further discussed. In the same experiment, the majority of females molted successfully (Table S3), except for females exposed to GLY at the highest concentration (45 mg/L). At this concentration, a significant reduction in molting success was observed where 58.1% of females molted successfully compared to 91.7% in the CTRL group (odds ratio = 7.85, n = 45, p < 0.05). Mean juvenile production from females exposed to atRA and CIT was not influenced by the exposure, but it showed a tendency to be decreased dose-dependently following exposure to MET and GLY (Fig. 2A-D). These diminutions for MET (-24.8%, 0.25 μg/L; -13.8%, 2.5 μg/L; and = -29.7%, 25 μg/L) and GLY (-18.6%, 0.45 mg/L; -21.6%, 4.5 mg/L; and -17.4%, 45 mg/L) were substantial compared to their respective control. For gammarids exposed to CTRL and SOLV, the mean duration of their reproductive cycles was 26.3 days ± 1.8 and 25.1 days ± 1.4, respectively, and did not display significant differences when compared to one another (Fig. 2E-H). Among the different tested compounds, different patterns of effects were observed. AtRA increased the duration of the molting cycle specifically at 0.11 μg/L and 11 μg/L, but not at the intermediate concentration (1.1 μg/L). Similarly, females exposed to MET displayed longer cycles at the lowest (0.25 μg/L) and the highest concentration (25 μg/L), whereas they were only marginally significant at the intermediate concentration (2.5 μg/L, p = 0.076). CIT and GLY had significant effects at the lowest concentrations whereas CIT showed a U-shape response. CIT increased very clearly the duration of cycles for females exposed to the two lowest concentrations (6 μg/L and 600 μg/L) but had no effect at the highest concentration (600 μg/L). Similarly, GLY showed a significant increase in the duration of the cycle, but only at the lowest concentration (0.45 mg/L). E-H) following exposures to increasing concentrations of atRA, CIT, MET, and GLY. The hatched embryos that had been released for the marsupial pouch of females were counted daily and reported in daily percentages of the total cumulated juvenile production. Δ Significant JT trend test, *p < 0.05, **p < 0.01, ***p < 0.001

Discussion

In this study, we investigated the implication of RA on the reproductive and developmental physiology of G. fossarum. We exposed gravid females to atRA and CIT to simulate an excess or a deficiency of RA. We also monitored the toxicological effects of two pesticides commonly used and frequently found in surface water, MET and GLY that were either known or suspected to affect the RA metabolism and signaling. In a previous study by [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: application for reproductive toxicity assessment[END_REF], the oocyte and embryo count per female for unexposed gammarids ranged from 12.4 ± 2.8 to 21.5 ± 3.4 and from 10.9 ± 1.5 to 12.9 ± 2.0, respectively, throughout the reproductive cycle. In our first experiment, the reported mean oocyte counts per female for CTRL and SOLV were 14.4 ± 2.5 and 21.8 ± 4.5, respectively. When oocyte count was standardized by the size of females, our values also resembled the ones obtained by [START_REF] Cribiu | A " population dynamics " perspective on the delayed life-history effects of environmental contaminations : an illustration with a preliminary study of cadmium transgenerational effects over three generations in the crustacean Gammarus[END_REF] for unexposed gammarids. For embryo count per female, we reported values of 7.6 ± 9.2 and 14.0 ± 7.9 for CTRL and SOLV, respectively, which roughly compares to reference values from [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: application for reproductive toxicity assessment[END_REF]. In the first experiment, the CTRL group had a high proportion of females who were unexplainably not bearing embryos and caused very high standard deviations. For this reason, we excluded the CTRL group for the first experiment of our study. Results for gammarids exposed to atRA, CIT, and MET were then compared to SOLV, a condition in which > 80% of females were bearing embryos and showed similar values to what was obtained by [START_REF] Cribiu | A " population dynamics " perspective on the delayed life-history effects of environmental contaminations : an illustration with a preliminary study of cadmium transgenerational effects over three generations in the crustacean Gammarus[END_REF] once corrected by female sizes. In the second experiment, females in the CTRL and SOLV group generated 11.8 ± 2.0 and 11.6 ± 1.8 juveniles, respectively. According to the observations of [START_REF] Geffard | Ovarian cycle and embryonic development in Gammarus fossarum: application for reproductive toxicity assessment[END_REF], these numbers are in the same range as the embryo count of healthy females in C2 and D1 in the first experiment (results not shown). In the CTRL and SOLV groups, the newly hatched juveniles had mean sizes of 3.47 mm ± 0.17 and 3.45 mm ± 0.23, respectively, and hatched in a very synchronized manner for all conditions. At the end of the second experiment, we estimated the duration of the female molting cycles for the CTRL and SOLV groups to be around 26.3 days ± 1.8 and 25.1 days ± 0.9, respectively, which were expected according to the predicted development time modeled by [START_REF] Chaumot | In situ reproductive bioassay with caged Gammarus fossarum (Crustacea): part 1-gauging the confounding influence of temperature and water hardness[END_REF] for G. fossarum at 16 °C.

Implication of RA in reproduction and development

According to our results, atRA reduced oocyte production in gravid G. fossarum females and increased the duration of the molting cycle. As stated before, in a wide variety of vertebrates, the onset of the gonadic germ cells meiosis is dictated by RA levels through the activation of RXR, which could also be the case in crustaceans. Impaired differentiation of blastemal cells following exposure to emulsions of atRA was reported in the crustacean U. pugilator in conjunction with an increased UpRXR expression [START_REF] Hopkins | Effects of all-trans retinoic acid on regenerating limbs of the fiddler crab, Uca pugilator[END_REF][START_REF] Chung | Cloning of crustacean ecdysteroid receptor and retinoid-X receptor gene homologs and elevation of retinoid-X receptor mRNA by retinoic acid[END_REF]. As RXR is also measured in whole-tissue homogenates of G. fossarum throughout the reproductive cycle [START_REF] Gouveia | Identification, expression, and endocrine-disruption of three ecdysoneresponsive genes in the sentinel species Gammarus fossarum[END_REF], it is plausible that exposure to atRA perturbed the oogonial differentiation as a result of disrupted RA tissular concentrations (not observed in our study). Recent studies demonstrated the modulatory action of 13-cis and 9-cis RA on the production and release of hormones in secretory organs together with enhanced ovarian maturation in the crustacean O. senex senex and S. serrata [START_REF] Girish | Evidence for retinoic acid involvement in the regulation of vitellogenesis in the fresh water edible crab, Oziotelphusa senex senex[END_REF][START_REF] Venkaiah | Detection and mode of action of retinoids on ovarian development in the mud crab Scylla Serrata[END_REF]. These studies highlighted the endocrine control of some RA isomers on ovarian cell processes notably vitellogenesis [START_REF] Girish | Evidence for retinoic acid involvement in the regulation of vitellogenesis in the fresh water edible crab, Oziotelphusa senex senex[END_REF][START_REF] Venkaiah | Detection and mode of action of retinoids on ovarian development in the mud crab Scylla Serrata[END_REF]). However, atRA did not induce ovarian maturation in L. vannamei (Linan-Cabello and Paniagua-Michel 2004). Having evidence of the complex crosstalk regulation network implicating RAs and RXRs and the differential affinities of -cis and -trans isomers of RA to RXR for vertebrates, the nature of the RA isomer could influence the outcome in crustaceans and should thus be considered carefully. The apparent crucial importance of the molecular configuration of RA draws attention to the several isomerization processes that occur in crustacean secretory and target tissues, such as in the ovaries where RA isomerization from -trans to -cis forms was suggested to occur (Linan-Cabello and Paniagua-Michel 2004). To gain mechanistic insights into this complex biological network, effective extraction procedures coupled with discriminant analytical methods for the separation and quantification of a wide variety of RA isomers in crustacean matrices will be necessary. We could, in a future study, analyze the tissue contents of RETs, in particular, the isomers of RA in gammarids. In our study, CIT reduced the oocyte production of gravid G. fossarum females. CIT also delayed molting, however in a very distinct manner compared to atRA. Citral displayed a typical U-shaped non-monotonic dose-response curve (NMDRCs) which are typical for EDC in many animal models (see review by [START_REF] Vandenberg | Hormones and endocrine-disrupting chemicals: low-dose effects and nonmonotonic dose responses[END_REF]. To this date, there is limited knowledge on the mechanism of action and the various effects of CIT in the animal kingdom. Yet, its RA synthesis-inhibiting properties have been confirmed in many vertebrates and linked to perturbed oogonial meiosis or limb bud cell differentiation in the fish Oryzias latipes and the crustacean U. pugilator, respectively [START_REF] Hopkins | Limb regeneration in the Fiddler Crab, Uca pugilator: hormonal and growth factor control[END_REF][START_REF] Adolfi | Retinoic acid and meiosis induction in adult versus embryonic gonads of medaka[END_REF]. In G. fossarum, we previously reported that CIT reduces the tissular RA concentrations and successfully perturb the ratios between RALD and RA [START_REF] Gauthier | The retinoid metabolism of Gammarus fossarum is disrupted by exogenous all-trans retinoic acid, citral, and methoprene but not by the technical formulation of glyphosate[END_REF]. If CIT inhibits the RA biosynthesis in the target and the secretory organs of G. fossarum, deficiency in RA could result in RA signaling disorders, which, in turn, could have detrimental effects on oogenesis either directly or via the endocrine system. In 2007, Abramson et al. reported that molting of the insect Rhodnius prolixus was delayed following exposure to CIT vapors. Interestingly, NMDRCs are more likely to occur with chemicals that act through more than one mechanism of action, which, by exerting an effect on RA tissular levels, may be the case for CIT. Retinoic acid is already known to induce certain effects when deficient or in excess such as teratogenicity for vertebrates. In our study, it is important to acknowledge that neither CIT nor atRA reduced molting success. Nevertheless, atRA delayed molting without displaying a clear dose-response relation. Taken together, the similar effects of atRA and CIT on oocyte production support the hypothesis that RA plays a non-negligible role in crustacean oogenesis, potentially through an effect on the oogonial differentiation. It also reinforces the hypothesis that RA has a certain regulatory role in the crustacean molting process. As the exposure to RA synthesis inhibitor CIT elicited a more pronounced effect than the exposure to atRA, we hypothesized that gammarids managed RA excess better than a RA deficiency.

MET-and GLY-induced reproductive and developmental toxicity

Because MET is known to alter embryonic development and metamorphosis of insects, we expected a strong effect on the functions under the endocrine control, specifically the reproduction and development of G. fossarum. Unsurprisingly, MET reduced the production of embryos, oocytes, and juveniles and increased the duration of the molting cycle. Embryotoxicity was observed by [START_REF] Kim | Endocrine disruptors in developing embryo on Daphnia magna[END_REF] following 3-week exposure to approximately 15.5 μg/L of MET. In parallel, repeated injections of sublethal MET concentrations (15 μg/L, one-tenth of the 48 h-LD50) in the freshwater crab Travancoriana schirnerae also induced lowered diameters of the oogonia and the oocytes in all phases of the vitellogenesis and important histopathological anomalies in the ovaries [START_REF] Padmanabhan | Impact of methoprene on oogenesis of the freshwater crab Travancoriana schirnerae Bott, 1969 (Decapoda: Gecarcinucidae)[END_REF]. In another study, juveniles of Neocaridina davidi molted 21.1% and 25.3% less frequently following 56-day exposure to 100 and 200 μg/L of MET, respectively, resulting in smaller individuals compared to the controls [START_REF] Hu | Effects of two juvenile hormone analogue insecticides, fenoxycarb and methoprene, on Neocaridina davidi[END_REF]. To this date, the precise molecular targets explaining the MET-induced physiological perturbations in crustaceans are in isomers of RA in favor of a more transcriptionally active isomer all-trans retinoic acid [START_REF] Gauthier | The retinoid metabolism of Gammarus fossarum is disrupted by exogenous all-trans retinoic acid, citral, and methoprene but not by the technical formulation of glyphosate[END_REF]. Also, the fact that RA, MET, and its metabolite methoprenic acid are potent ligands for RXR in vertebrates [START_REF] Harmon | Activation of mammalian retinoid X receptors by the insect growth regulator methoprene[END_REF] raises interrogation relative to the interaction of MET and RXR in crustaceans. This interaction was suggested as an explanation for developmental abnormalities observed in the regenerating limb of U. pugilator [START_REF] Stueckle | Limb regeneration and molting processes under chronic methoprene exposure in the mud fiddler crab, Uca pugnax[END_REF]). Given our results, although RA, CIT, and MET all showed clear effects on the oocyte production and, to some extent on molting, they had different effects on embryo production. Our results support a potential implication of RA in the MET-induced effect on oogenesis and molting. However, it is not the case for embryotoxicity. Although both MET and RA may share RXR as a ligand in some secretory or target tissues, it is not known whether the interaction of MET and RXR influences the affinity of RXR for RA isomers, or even for the ecdysone receptor (EcR), and if such mechanism could lead to RA or ecdysteroid signaling disorders in crustaceans. As free and conjugated ecdysteroids are known to orchestrate vitellogenesis and ultimately carry the oocyte throughout various developmental checkpoints, disruption of the hormonal system could lead to the oogenesis and molting defects observed in our study. In our study, we showed that GLY affects molting in female G. fossarum. Whereas the lowest concentration caused significant molting delays, the highest concentrations completely inhibited the molting process, and consequently, the molting success. GLY also caused a trend of decreased juvenile production by females. Similar effects of GLY and GLY-based herbicides (GBH) on reproduction and development have also been shown in other crustaceans [START_REF] Mensah | Using growth measures in the freshwater shrimp Caridina nilotica as biomarkers of Roundup® pollution of South African freshwater systems[END_REF][START_REF] Cuhra | Clone-and age-dependent toxicity of a glyphosate commercial formulation and its active ingredient in Daphnia magna[END_REF]. Disruption of gene expression implicated in the endocrine system in the crustacean Macrobrachium potiuna has been observed and may explain some effects on crustacean reproduction and development [START_REF] De Melo | Effects of low glyphosate-based herbicide concentrations on endocrine-related gene expression in the decapoda Macrobrachium potiuna[END_REF]. However, the toxicodynamics of GLY in crustaceans is not well understood. As exposures to atRA and CIT were unsuccessful to reduce molting success, reduced molting success at high concentrations of GLY is not likely to be mediated by RA. However, the distinct effect of GLY on molting recorded at the lowest concentration could be the result of perturbed RA tissular levels. Glyphosate has already been associated to perturbation of RET concentrations in invertebrates. In the honeybee Apis mellifera, a 10-day exposure to GLY caused a diminution of a RA precursor, all-trans ROL (Hedrei [START_REF] Helmer | Effects of realistic doses of atrazine, metolachlor, and glyphosate on lipid peroxidation and dietderived antioxidants in caged honey bees (Apis mellifera)[END_REF]. At this point, our results are not refuting a possible mediation of RA in the GLY-induced molting delays observed at the lowest concentration tested. However, additional research will be needed to further study the effects of GLY on tissular RET levels in crustaceans. Following a 3-month trial study where MET was applied for mosquito control in lakes following the recommended application rates, MET was detected in 50% of water surface samples with a mean concentration of 26 μg/L and ranging at up to 45 μg/L [START_REF] Hershey | Effects of methoprene and Bti (Bacillus thuringiensis var. israelensis) on non-target insects[END_REF]. Because we observed reproductive and developmental defects in G. fossarum females following exposure to low concentrations of MET (respectively 0,6 μg/L and 0.25 μg/L), MET may pose risks to non-target invertebrates. GLY is also frequently found in surface waters draining farming areas. For example, in Canada and Brazil, GLY was found at concentrations ranging up to 140 μg/L and 2.02 mg/L, respectively [START_REF] Freire | Monitoring of toxic chemical in the basin of Maringa stream[END_REF]; the latter being close to the concentrations investigated in our study and susceptible to induce molting delays. If the extent of the molting delays (from 30 to 55 h) and the reduced oocyte count (from 14.6 to 64.5% of reduction) were to be observed in situ, a lowered reproductive success in the following cycles would be expected and could severely hinder the population dynamics. The results obtained in our study relative to the environmental risks posed by pesticides are conservative as pesticides are rarely used alone, and most of the time associated with surfactants that enhance their toxicity in live organisms. It is important to highlight that the observed molting delays following exposure to all compounds occurred in conditions that did not influence the juvenile hatching kinetics (Fig. 3) which attest to the synchronization of reproductive cycles in exposed females. This also further confirms the effects occurring specifically on reproduction or molting.

Conclusion

In this study, we investigated the influence of four compounds of physiological and toxicological interest (atRA, CIT, MET, and GLY) on reproductive and developmental markers in gravid G. fossarum females. First, our results showed the implication of RA on two endocrine-controlled processes, notably oogenesis and molting, with CIT eliciting a typical EDC NMDRC on molting. Secondly, we confirmed that MET, known for its ED properties in crustaceans, has important negative effects on the production of oocytes, embryos, juveniles, and molting in G. fossarum. In parallel, we showed that GLY delays or completely stops molting at low and high concentrations, respectively. That led us to hypothesize that RA could be a mediator of MET-induced effects on oogenesis and molting whereas it is unlikely to be implicated in the effects on embryos. In addition, we also hypothesize that RA may be implicated in the GLY-induced inhibition found at high concentration. RETs are currently investigated as potential EDC targets for vertebrates. This raises interrogation relatively to the modulation of RETs in crustacean's neuroendocrine system and may be an avenue to decipher this complex regulatory network. Also, these results suggest that the RET metabolism leading to the formation of RA is an interesting candidate for biomarker development in G. fossarum and merits further research. A better understanding of the implication of RETs in the reproductive and developmental physiology of G. fossarum paves the way to the finding of new biomarkers in this model, which ultimately will profit biomonitoring programs.

Fig. 1

 1 Fig. 1 Production of embryos (A-C) and oocytes (D-F) for females G. fossarum exposed for 14 days to increasing concentrations of atRA, CIT, and MET. Production of embryos and oocytes was calculated by normalizing the number of embryos or oocytes per female by the dorsal length of each female. For oocyte count, only females that were non-delayed in their development were selected (C2 and D1). Δ Significant JT trend test, *p < 0.05, **p < 0.01, ***p < 0.001

Fig. 2

 2 Fig. 2 Production of juvenile per female (A-D) and duration of molting cycle (E-H) for gammarids exposed for 44 days to increasing concentrations of atRA, CIT, MET, and GLY. Concentrations are displayed in μg/L, except for GLY (mg/L). See the "Materials and methods" section for estimations of juvenile production and reproductive cycles. Arrows represent dose-dependent relationship, # 0.05 > p < 0.1, *p < 0.05, **p < 0.01, ***p < 0.001

Acknowledgements The authors would like to thank Nicolas Delorme, Laura Garnero, and Julien Labrie (INRAE, gammarid collection, maintenance, and sorting); Herve Queau (reproductive and developmental endpoints); and Gaelle Daniele, Barbara Giroud, and Florent Lafay (ISA, chemical analyses).

Author contribution M.G. codesigned the experiments, carried out the laboratory work and statistical analyses, interpreted the results, built the corresponding tables and figures, and wrote the manuscript. J.D. participated to the laboratory work and preliminary statistical analyses. E.V. supervised the chemical analyses and approved the final version of the manuscript. O.G. codesigned the experiments supervised the project and revised the manuscript. M.B., C.J., and J.G. revised and approved the final version of the manuscript.

Funding This study has been funded by the Agence Nationale de la Recherche under a grant awarded to Olivier Geffard (ANR-18-CE34-0013-01) and by the Natural Sciences and Engineering Research Council of Canada (NSERC) under a grant given to Monique Boily (#RGPIN-2016-05142) and Catherine Jumarie (#RGPIN-2017-05106). Maxime Gauthier received financial support from the Fondation UQAM (2017-2018) and the Fonds de recherche du Quebec -Nature et technologies (FRQNT, 2018(FRQNT, -2021)).

Data availability Data will be made available upon reasonable request.

Supplementary information

The online version contains supplementary material available at https:// doi. org/ 10. 1007/ s11356-023-28327-w.


Michael Malisoff

Frédéric Mazenc

Corina Barbalata

Event-Triggered Control under Unknown Input and Unknown Measurement Delays using Interval Observers

Keywords:

We provide a new input-to-state stabilizing eventtriggered feedback design for linear systems with unknown input delays, unknown measurement delays, and unknown additive disturbances. Our trigger times are computed using only the matrices defining the system and time-lagged sampled state values. We use the theory of positive systems, interval observers, and a vector version of Halanay's inequality. We illustrate our method using a marine robotic model.

I. INTRODUCTION

This paper continues our development (begun in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], [START_REF] Mazenc | Event-triggered predictionbased delay compensation approach[END_REF], [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF], [START_REF] Mazenc | Event-triggered control for linear time-varying systems using a positive systems approach[END_REF]) of event-triggered control methods for continuous time systems using interval observers (as defined in [START_REF] Gouze | Interval observers for uncertain biological systems[END_REF], [START_REF] Raissi | Interval state estimation for a class of nonlinear systems[END_REF]), positive systems, and the use of matrices of absolute values instead of Euclidean norms. Although our prior eventtriggered works allowed input delays (as well as time-varying coefficients in linear systems and additive uncertainty [START_REF] Mazenc | Event-triggered control for linear time-varying systems using a positive systems approach[END_REF]), they assume that the input delays are known in advance.

Here we study more realistic systems having unknown delays in the input. We allow time-varying delays, and unknown disturbances. Although neither this work nor our prior eventtriggered controlled works assume that our systems to be controlled are positive, our analysis uses new synergies of the theory of positive systems, which has been shown in our prior works to lead to less triggering than traditional approaches, and the vector Halanay's inequality approach that was presented in [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF] in the absence of event triggering.

We prove input-to-state stability (or ISS), and we rule out Zeno's phenomenon. Allowing unknown input delays (as compared with our prior chain predictor based work [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] that required exact knowledge of the input delays to design the control and constantness of the delays), and trigger rules that only require sampled delayed state measurements with unknown measurement delays, make this work novel and significant. These innovations are made possible by our new event trigger rule and new Hurwitzness condition which have never appeared before, and which (as we illustrate below) let us achieve ISS under realistic bounds on the unknown time-varying input and measurement delays. This also contrasts with other notable event-triggered works for nonlinear systems, such as [START_REF] Borri | Event-triggered control of nonlinear systems with time-varying state delays[END_REF], [START_REF] Li | Observer-based event-triggered control for nonlinear systems with mixed delays and disturbances: The input-to-state stability[END_REF], [START_REF] Nozari | Event-triggered stabilization of nonlinear systems with time-varying sensing and actuation delay[END_REF] (where the delays are M. Malisoff is with Department of Mathematics, Louisiana State University (LSU), Baton Rouge, LA 70803, USA (e-mail: malisoff@lsu.edu).

F. Mazenc is with Inria EPI DISCO, L2S-CNRS-CentraleSupélec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France (e-mail: frederic.mazenc@l2s.centralesupelec.fr).

C. Barbalata is with Department of Mechanical and Industrial Engineering, LSU, Baton Rouge, LA 70803, USA (email: cbarbalata@lsu.edu).

Supported by US National Science Foundation Grant 2009659.

known) and [START_REF] Zhang | Event-triggered control for nonlinear time-delay systems[END_REF], [START_REF] Zhang | Event-triggered stabilization of a class of nonlinear time-delay systems[END_REF] (where the measurement delays were only allowed to capture the effects of sampling). Since eventtriggered controls only change values when it is essential to change them [START_REF] Heemels | An introduction to eventtriggered and self-triggered control[END_REF], they are useful for underwater robotics, owing to constraints in underwater communication. Hence, another contribution of this work is our application to a model of the BlueROV2 underwater vehicle, which is used to study corals, under uncertain input and measurement delays.

We use standard notation. The dimensions of our Euclidean spaces are arbitrary, unless we indicate otherwise. We set

Z 0 = {0, 1, 2, ...}. For a matrix G = [g ij ] ∈ R r×s , we set |G| = [|g ij |]
, so the entries of |G| are the absolute values of the corresponding entries g ij of G. By G + , we denote the matrix whose entries are max{0, g ij } in the ith row and jth column for each i and j, and

G -= G + -G. Note for later use that |G| = G + + G -. For matrices D = [d ij ] and E = [e ij ] of the same size, we write D < E (resp., D ≤ E) provided d ij < e ij (resp., d ij ≤ e ij )
for all i and j, 0 denotes the zero matrix, and I denotes the identity matrix. A matrix M ∈ R n×n is called Metzler provided all of its off-diagonal elements are nonnegative. Given M ∈ R n×n , let D M denote the diagonal matrix such that all main diagonal entries of M -D M are equal to zero, R M = D M + (M -D M ) + , and N M = (M -D M ) -. For matrix valued functions M (r) = [m ij (r)], the supremum is entrywise, i.e., sup r∈S M (r) has the (i, j) entry sup r∈S m ij (r) for each subset S of the domain of M and all i and j when these suprema are finite. We also use the standard definitions of ISS (which we also use to denote input-to-state stable) and comparison functions KL and K ∞ , e.g., from [START_REF] Sontag | Input-to-state stability: basic concepts and results[END_REF]. We use ||•|| to denote the usual Euclidean norm and corresponding matrix norm, and || • || J denotes the supremum in this norm over any interval J.

II. STUDIED SYSTEM

We first consider the general class of linear systems

ẋ(t) = Ax(t) + Bu(t) + δ(t) (1)
with x valued in R n and the input u (which will contain an unknown nonzero time-varying delay ∆ in the sequel) valued in R m , and where A ∈ R n×n and B ∈ R n×m are known matrices, and δ is an unknown locally bounded piecewise continuous function (but see Section IV for local analogs for nonlinear systems). Our control values will have the form u(t) = Kx(t i -∆(t i )) for all t ∈ [t i , t i+1 ) and i ∈ Z 0 for constant matrices K ∈ R m×n (which arises from zero-order hold and input delays on the controller side) where the t i 's admit positive constants T and T such that T ≤ t i+1 -t i ≤ T for all i ∈ Z 0 where t 0 = 0. The t i 's will be event trigger times, and we will add an assumption on K. We assume that the initial functions for (1) are constant. The t i 's will be found using sampled values of the time-lagged measurements y(t) = x(t -λ(t)) of the state of (1) at triggering times t i , whose delay λ is also unknown. Our first assumption is: Assumption 1: There are known constants ∆ ≥ 0 and λ ≥ 0 and a known function δ : R → [0, +∞) n such that

0 ≤ ∆(t) ≤ ∆, 0 ≤ λ(t) ≤ λ, and |δ(t)| ≤ δ(t) (2)
hold for all t ≥ 0.

Fixing A, B, and K as above, we use the function

Ω(r) = e Ar + r 0 e Aℓ dℓBK. (3)
Since Ω(0) = I, there is a constant ν > 0 such that the matrix Ω(r) is nonsingular for each r ∈ [0, ν]. Fixing ∆, λ, δ, and ν satisfying the preceding requirements and setting

Γ 1 = sup s∈[0,ν] I -Ω(s) -1 , Γ 2 = (Γ 1 + I) sup L∈[0,T ] L 0 e Aℓ BKdℓ , Γ 3 = sup ℓ∈[0,T ] |e Aℓ A|, and Γ 4 = sup ℓ∈[0,T ] |e Aℓ BK| (4)
for any constant T ≥ ν + ∆, we also assume:

Assumption 2: With the above notation and the choice H = A+BK, the matrix K ∈ R m×n is such that the matrix

M * = R H + N H + |BK|Γ 1 + max ∆, λ |BKA| +|(BK) 2 | + |BK|Γ 2 |A| + |BK| +λ|BK| (I + Γ 1 )(Γ 3 + Γ 4 ) + |A| + |BK| (5)
is Hurwitz.

Our novel Hurwitzness condition from Assumption 2 can always be satisfied after a change of coordinates if (A, B) is controllable and if ∆, λ, and ν are small enough. This is done by choosing K so that A + BK has distinct negative eigenvalues, then using a diagonalizing change of coordinates as in [START_REF] Mazenc | Event-triggered control for linear time-varying systems using a positive systems approach[END_REF], so A, B, and K and x are replaced by P -1 AP , P -1 B, KP , and P -1 x respectively for an invertible matrix P . After this coordinate change, H = R H + N H is diagonal and Hurwitz. Then, by continuity of eigenvalues as functions of the matrix entries, Assumption 2 holds when ∆, λ, and ν are small enough; see Section V for an illustration of how Assumption 2 can allow realistic delay values.

III. MAIN RESULT A. Control and Statement of Result

In terms of the control u that is defined by

u(t) = Kx(t i -∆(t i )) for all t ∈ [t i , t i+1 ) and i ∈ Z 0 (6)
and the notation from the previous section, the closed-loop system and triggering times t i we consider are defined by:

ẋ(t) = Ax(t) + BKx(t i -∆(t i )) + δ(t) for all t ∈ [t i , t i+1 ) and i ∈ Z 0 , ḣi (t) = Ah i (t) + BKx(t i -λ(t i ))
for all t ≥ t i and i ∈ Z 0 , h i (t i ) = x(t i -λ(t i )) for all i ∈ Z 0 , and

t i+1 = sup {b ∈ [t i , t i + T ] : |h i (b) -h i (t i )| ≤ Γ 1 |h i (b)|} for all i ∈ Z 0 , (7)
where t 0 = 0. Then t 1 is the supremum of all b ≥ t 0 = 0 such that the inequality in the supremum in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] holds with i = 0 or t 1 = t 0 +T = T (whichever is smaller), and then the other trigger times t i for i ≥ 2 are inductively defined in the same way. Our supremum in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] ensures that t i+1 -t i ≤ T for all i ≥ 0, so there are infinitely many times t i . Also, only the discrete state measurements h i (t i ) are needed to find the t i 's, which are found using solutions h i (t) of the initial value problem for the h i dynamics from the second and third equalities in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], and T can be arbitrarily large. Hence, the triggering rule does not need to continuously monitor x(t) to determine future triggering times, which is a significantly novel feature. We prove the following, whose last conclusion rules out Zeno's phenomenon: Theorem 1: Let the system (1) satisfy Assumptions 1-2. Then (1) in closed loop with the control from ( 6)-( 7) is ISS to the origin on R n . Also, t i+1 -t i ≥ ν for all i ≥ 0.

Remark 1: Our theorem is new, even in the special case where there are no measurement delays, in which case we can choose λ = 0 and then Assumption 2 is the requirement that

R H + N H + |BK|Γ 1 + ∆ |BK|Γ 2 |A| + |BK| + |BKA| + |(BK) 2 |
is Hurwitz. The proof of Theorem 1 builds comparison functions in the final ISS estimate, namely, β ∈ KL and γ ∈ K ∞ such that all solutions x(t) of ( 7) for all initial states x(0) ∈ R n and δ's satisfy

||x(t)|| ≤ β(||x(0)||, t) + γ ||δ|| [0,t] (8)
for all t ≥ 0, with positive constants c a and c b such that these requirements hold with β(r, t) = c a re -c b t ; see also Remark 2 below. Hence, all solutions x(t) converge to 0 ∈ R n exponentially at rate c b when δ is zero, with an overshoot (given by the second right side term in ( 8)) for nonzero δ.

B. Proof of Theorem 1

1) Ruling Out Zeno's phenomenon: We prove that t i+1t i ≥ ν for all i ∈ Z 0 . We proceed by contradiction.

Suppose that there were an i ∈ Z 0 such that t i+1 -t i < ν. By applying variation of parameters to the h i dynamics from [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], we deduce from our choice of

Ω in (3) that h i (t) = Ω(t -t i )h i (t i ) for all t ≥ t i . Since Ω(r) is invertible for each r ∈ [0, ν] and t i+1 -t i < ν, this provides a constant ε ∈ (0, ν + t i -t i+1 ) such that h i (t i ) = Ω(t -t i ) -1 h i (t) for all t ∈ [t i , t i+1 + ε), because 0 ≤ t i+1 + ε -t i < ν. Then |h i (t i ) -h i (t)| ≤ |I -Ω(t -t i ) -1 ||h i (t)| ≤ Γ 1 |h i (t)| for all t ∈ [t i , t i+1 + ε).
This contradicts the definition of t i+1 . Hence, t i+1 -t i ≥ ν holds for all i ∈ Z 0 .

2) Stability Analysis: The analysis has three parts. First, we derive a vector valued bound for |x(t) -x(t i )| at times t ∈ [t i , t i+1 ) for all i ∈ Z 0 . In the second part, we use the bound on |x(t) -x(t i )| from the first part to find a vector valued upper bound on ṡ(t) where s(t) = x(t) -x(t) is the difference between upper and lower vectors in an interval observer (x, x) for x. In the final part, we use the bound on ṡ(t) from the second part and a variant of the ISS vector Halanay's inequality from [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF] to obtain our ISS estimate [START_REF] Mazenc | Event-triggered predictionbased delay compensation approach[END_REF].

First Part. Defining the function σ by σ(t) = t i when t ∈ [t i , t i+1 ) and i ≥ 0, and setting z(t) = Ax(t)+BKx(σ(t)-∆(t i )), we observe that our choice

H = A + BK gives ẋ(t) = Hx(t) + BK[x(t i ) -x(t)] +BK[x(t i -∆(t i )) -x(t i )] + δ(t) = Hx(t) + BK[x(t i ) -x(t)] -BK ti (ti-∆(ti)) + (z(ℓ) + δ(ℓ))dℓ + δ(t) (9)
for all t ∈ [t i , t i+1 ) and i ∈ Z 0 , where we used the constantness of the initial functions for the x dynamics. Substituting our formula for z into [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF] gives 2 and R 3 = -BK. Next note that our choice of t i+1 from [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] gives

ẋ(t) = Hx(t) + BK[x(σ(t)) -x(t)] +R 1 σ(t) (σ(t)-∆(σ(t))) + x(ℓ)dℓ + R 2 σ(t) (σ(t)-∆(σ(t))) + x(σ(ℓ) -∆(σ(ℓ)))dℓ +R 3 σ(t) (σ(t)-∆(σ(t))) + δ(ℓ)dℓ + δ(t) (10) for all t ≥ 0 with R 1 = -BKA, R 2 = -(BK)
|h i (t) -h i (t i )| ≤ Γ 1 |h i (t)| (11)
for all t ∈ [t i , t i+1 ) and all i ∈ Z 0 . On the other hand, [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] and the Fundamental Theorem of Calculus give

ẋ(t)-ḣi (t) = A(x(t) -h i (t)) +BK[x(t i -∆(t i )) -x(t i -λ(t i ))] +δ(t) for all t ∈ [t i , t i+1 ) x(t i )-h i (t i ) = ti (ti-λ(ti)) + [Ax(ℓ) + δ(ℓ)] dℓ +BK ti (ti-λ(ti)) + x(σ(ℓ)-∆(σ(ℓ)))dℓ (12) for all i ∈ Z 0 . Since |z(t)| ≤ |Ax(t)| + |BKx(σ(t) - ∆(σ(t)
))|, we can use the second equality of (12) to get

|x(t i -∆(t i )) -x(t i -λ(t i ))| ≤ ti (ti-∆ ♯ ) + [|A||x(ℓ)| + |δ(ℓ)|] dℓ +|BK| ti (ti-∆ ♯ ) + |x(σ ♭ (ℓ))|dℓ, (13)
|e (t-ti)A (x(t i ) -h i (t i ))| ≤ Γ 3 ti (ti-λ) + |x(ℓ)|dℓ +Γ 4 ti (ti-λ) + |x(σ ♭ (ℓ))|dℓ + Γ 5 ti (ti-λ) + |δ(ℓ)|dℓ, (14)
and |x(t i ) -h i (t i )| ≤ ti (ti-λ) + [|A||x(ℓ)| + |δ(ℓ)|] dℓ +|BK| ti (ti-λ) + |x(σ ♭ (ℓ)))|dℓ (15) for all t ∈ [t i , t i+1 ) and i ∈ Z 0 , where the function σ ♭ is defined by σ ♭ (ℓ) = σ(ℓ) -∆(σ(ℓ)) for all ℓ ≥ 0, ∆ ♯ = max ∆, λ , Γ 3
and Γ 4 are defined in (4), and

Γ 5 = sup ℓ∈[0,T ] |e Aℓ |.
Letting G j (t i ) for j = 1, 2 denote the right sides of ( 13) and ( 14) respectively, and applying the method of variation of parameters to the dynamics for x -h i from ( 12) on [t i , t], it now follows from ( 12), [START_REF] Sontag | Input-to-state stability: basic concepts and results[END_REF], and (14) that

|x(t) -h i (t)| ≤ t ti e A(t-ℓ) δ(ℓ)dℓ + t ti e A(t-ℓ) BKdℓ G 1 (t i ) + G 2 (t i ), (16)
and so also 11) and ( 16) and then (17), and letting G 3 (t i ) denote the right side of (15), we can bound each of the three right side terms in |x(t) -

|h i (t)| ≤ |x(t)| + t ti e A(t-ℓ) δ(ℓ)dℓ + t ti e A(t-ℓ) BKdℓ G 1 (t i ) + G 2 (t i ) (17) for all t ∈ [t i , t i+1 ) and all i ∈ Z 0 , since |h i (t)| -|x(t)| ≤ |x(t) -h i (t)|. Using (
x(t i )| ≤ |x(t) - h i (t)| + |h i (t) -h i (t i )| + |h i (t i ) -x(t i )| (
which is obtained by applying the triangle inequality twice) separately to get

|x(t) -x(t i )| ≤ Γ 1 |h i (t)| + t ti e |A|(t-ℓ) δ(ℓ)dℓ + t ti e A(t-ℓ) BKdℓ G 1 (t i ) +G 2 (t i ) + G 3 (t i ) ≤ Γ 1 |x(t)| + Γ 1 t ti e |A|(t-ℓ) δ(ℓ)dℓ +Γ 2 G 1 (t i ) + t ti e |A|(t-ℓ) δ(ℓ)dℓ + (I + Γ 1 )G 2 (t i ) + G 3 (t i ) (18)
for all t ∈ [t i , t i+1 ) and i ∈ Z 0 , so ( 14)- [START_REF] Zhang | Event-triggered stabilization of a class of nonlinear time-delay systems[END_REF] give

|x(t)-x(t i )| ≤ Γ 1 |x(t)| + Γ 2 |A| ti (ti-∆ ♯ ) + |x(ℓ)|dℓ +Γ 2 |BK| ti (ti-∆ ♯ ) + |x(σ ♭ (ℓ))|dℓ +Γ 2 t (ti-∆ ♯ ) + δ(ℓ)dℓ +(Γ 1 + I) t ti e |A|(t-ℓ) δ(ℓ)dℓ +(Γ ♯ 3 + |A|) ti (ti-λ) + |x(ℓ)|dℓ +(Γ ♯ 4 +|BK|) ti (ti-λ) + |x(σ ♭ (ℓ))|dℓ +(I + Γ ♯ 5 ) t (ti-λ) + δ(ℓ)dℓ, (19)
where Γ ♯ j = (I +Γ 1 )Γ j for j = 3, 4, 5. Setting Γ 6 = Γ 2 |BK| and Γ 7 = (Γ 1 + I)e T |A| + Γ 2 , this provides the bounds

|x(t) -x(t i )| ≤ Γ 1 |x(t)| + Γ 2 |A| ti (ti-∆ ♯ ) + |x(ℓ)|dℓ +Γ 6 ti (ti-∆ ♯ ) + |x(σ ♭ (ℓ))|dℓ +Γ 7 t (ti-∆ ♯ ) + δ(ℓ)dℓ +(Γ ♯ 3 + |A|) ti (ti-λ) + |x(ℓ)|dℓ +(Γ ♯ 4 +|BK|) ti (ti-λ) + |x(σ ♭ (ℓ))|dℓ +(I + Γ ♯ 5 ) t (ti-λ) + δ(ℓ)dℓ. ( 20
)
Second Part. We next introduce the interval observer

                                         ẋ(t) = R H x(t) -N H x(t) +(BK[x(σ(t)) -x(t)]) + + σ(t) (σ ♭ (t)) + (R 1 x(ℓ)) + dℓ + σ(t) (σ ♭ (t)) + (R 2 x(σ ♭ (ℓ))) + dℓ + σ(t) (σ ♭ (t)) + (R 3 δ(ℓ)) + dℓ + δ(t) + ẋ(t) = R H x(t) -N H x(t) -(BK[x(σ(t)) -x(t)]) - - σ(t) (σ ♭ (t)) + (R 1 x(ℓ)) -dℓ - σ(t) (σ ♭ (t)) + (R 2 x(σ ♭ (ℓ))) -dℓ - σ(t) (σ ♭ (t)) + (R 3 δ(ℓ)) -dℓ -δ(t) - (21)
which is defined for all t ≥ 0. Since the matrix

R H N H N H R H ∈ R 2n×2n (22)
is Metzler, formula [START_REF] Mazenc | Event-triggered control for linear time-varying systems using a positive systems approach[END_REF] for the x dynamics, (21), and the formula Hx(t) = R H x(t) -N H x(t) imply that if

x(0) ≤ x(0) ≤ x(0), x(0) ≤ 0, and 0 ≤ x(0), (23) then x(t) ≤ x(t) ≤ x(t), x(t) ≤ 0, and 0 ≤ x(t)

hold for all t ≥ 0, e.g., by applying [5, Lemma 1] to the dynamics for (x, -x), and the dynamics for (x -x, x -x). Now, let us consider initial conditions such that (23) are satisfied and let s(t) = x(t) -x(t). Then (21) gives

ṡ(t) = (R H + N H )s(t) + |BK[x(σ(t)) -x(t)]| + σ(t) (σ ♭ (t)) + |R 1 x(ℓ))|dℓ + σ(t) (σ ♭ (t)) + |R 2 x(σ ♭ (ℓ))|dℓ + σ(t) (σ ♭ (t)) + |R 3 δ(ℓ)|dℓ + |δ(t)| ≤ (R H + N H )s(t) + |BK[x(σ(t)) -x(t)]| +|R 1 | σ(t) (σ(t)-∆) + |x(ℓ)|dℓ +|R 2 | σ(t) (σ(t)-∆) + |x(σ ♭ (ℓ))|dℓ + σ(t) (σ(t)-∆) + |R 3 δ(ℓ)|dℓ + |δ(t)| (25)
for all t ≥ 0. By using (20) to bound the second term in the upper bounds in (25), we can use (25) to obtain

ṡ(t) ≤ (R H + N H )s(t) + |BK|Γ 1 |x(t)| +|BK|Γ 2 |A| σ(t) (σ(t)-∆ ♯ ) + |x(ℓ)|dℓ +|BK|Γ 6 σ(t) (σ(t)-∆ ♯ ) + |x(σ ♭ (ℓ))|dℓ + σ(t) (σ(t)-∆) + |R 3 δ(ℓ)|dℓ + |δ(t)| +|BK|Γ 7 t (σ(t)-∆ ♯ ) + δ(ℓ)dℓ +|R 1 | σ(t) (σ(t)-∆) + |x(ℓ)|dℓ +|R 2 | σ(t) (σ(t)-∆) + |x(σ ♭ (ℓ))|dℓ +|BK|(Γ ♯ 3 + |A|) σ(t) (σ(t)-λ) + |x(ℓ)|dℓ +Γ * σ(t) (σ(t)-λ) + |x(σ ♭ (ℓ))|dℓ +|BK|(I + Γ ♯ 5 ) t (σ(t)-λ) + δ(ℓ)dℓ, (26)
where Γ * = |BK|(Γ ♯ 4 + |BK|). Since (24) gives x(t) -x(t) ≤ x(t) ≤ x(t) -x(t) and so also |x(t)| ≤ s(t) for all t ≥ 0, we can upper bound each right side term in (26) that involves x using s to get

ṡ(t) ≤ (R H + N H + |BK|Γ 1 )s(t) +|BK|Γ 2 |A| σ(t) (σ(t)-∆ ♯ ) + s(ℓ)dℓ +|BK|Γ 6 σ(t) (σ(t)-∆ ♯ ) + s(σ ♭ (ℓ))dℓ + δ(t) + |R 1 | σ(t) (σ(t)-∆) + s(ℓ)dℓ +|R 2 | σ(t) (σ(t)-∆ ♯ ) + s(σ ♭ (ℓ))dℓ + (|BK|Γ 7 + |R 3 |) t (σ(t)-∆ ♯ ) + δ(ℓ)dℓ +|BK|(Γ ♯ 3 +|A|) σ(t) (σ(t)-λ) + |s(ℓ)|dℓ + Γ * σ(t) (σ(t)-λ) + |s(σ ♭ (ℓ))|dℓ + |BK|(I + Γ ♯ 5 ) t (σ(t)-λ) + δ(ℓ)dℓ (27)
for all t ≥ 0, by also assuming that the s dynamics has constant initial functions. Collecting terms in (27) gives

ṡ(t) ≤ (R H +N H +|BK|Γ 1 )s(t) + Γ 8 sup ℓ∈[(t-2T -2∆ ♯ ) + ,t] s(ℓ) + (|BK|(Γ ♯ 5 +Γ 7 +I)+|R 3 |) t (σ(t)-∆ ♯ ) + δ(ℓ)dℓ + δ(t) (28)
for all t ≥ 0, with

Γ 8 = ∆ ♯ |BK|[Γ 2 |A| + Γ 6 ] + ∆ ♯ (|R 1 | + |R 2 |) + λ|BK|(Γ ♯ 3 + Γ ♯ 4 + |A| + |BK|). Third Part.
We apply [6, Theorem 1] to the function w(t) = s(t) and τ = 2(T + ∆ ♯ ), after extending s to the domain [-τ, +∞) by defining s to be constant on [-τ, 0], so we use a piecewise C 1 function w instead of assuming that w is C 1 as was assumed in [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF]; this is valid because the proof of [6, Theorem 1] remains valid if the C 1 requirement on w in [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF] is relaxed to only requiring it to be piecewise C 1 . This ensures that s satisfies an ISS property when

R H + N H + |BK|Γ 1 + Γ 8 (
β 0 ∈ KL and γ 0 ∈ K ∞ such that ||s(t)|| ≤ β 0 (||s|| [0,h] , t) + γ 0 (||δ|| [0,t] ) (30)
for all t ≥ 0 for all solutions of the s(t) dynamics from (25).

The remaining part of the proof converts the estimate (30) to an ISS estimate for the state variable x(t), as follows.

Note that [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF] provides the constant c1 = max{1,

||A|| + ||BK||} such that || ẋ(t)|| ≤ c1 (||x|| [t-T -∆,t] + ||δ|| [0,t]
) for all t ≥ 0. Hence, the fundamental theorem of calculus gives

||x(t)|| ≤ ||x(0)||+c 1 t 0 ||x|| [ℓ-T -∆,ℓ] dℓ+c 1 t||δ|| [0,t] (31)
for all t ≥ 0. Recalling our assumption that the initial functions are constant, it follows from (31) that the function

S(t) = ||x|| [t-T -∆,t] satisfies S(t) ≤ ||x(0)|| + c1 t 0 S(ℓ)dℓ + c1 t||δ|| [0,t] (32
||s|| [0,h] ≤ c3 (3 + 2h)||x(0)||+||δ|| [0,h] . (33)
Since |x(t)| ≤ s(t) holds for all t ≥ 0, and since the values x(t) do not depend on δ(ℓ) values for times ℓ ≥ t, we can substitute (33) into (30) and then use the fact that β 0 (a + b, t) ≤ β 0 (2a, t) + β 0 (2b, 0) holds for all a ≥ 0, b ≥ 0 and t ≥ 0, to satisfy the ISS requirement with β(r, t) = β 0 (2(3 + 2h)c 3 r, t) and γ(s) = γ 0 (s) + β 0 (2c 3 s, 0). Remark 2: In terms of diagonal matrices D whose main diagonal entries are all positive, matrices P > 0, constants h > 0 and τ > 0, and a constant c 0 ∈ (0, 1) and U 0 ∈ [1, +∞) n such that M 0 U 0 = c 0 U 0 where M 0 = e -Dh + 0 -h e Dℓ dℓP , the β 0 from [6, Theorem 1] and [6, Remark 1] is β 0 (r, t) = n||U 0 ||e (t-h) ln(c 0 )/(τ +h) r. The existence of c 0 and U 0 follows because M 0 > 0 is Schur stable [START_REF] Mazenc | ISS inequalities for vector versions of Halanay's inequality and of the trajectory-based approach[END_REF]. This and the last part of the proof of Theorem 1 ensure the exponential stability condition from Remark 1. We next use the preceding β 0 to obtain local analogs for nonlinear systems.

IV. LOCAL RESULT FOR NONLINEAR SYSTEMS

We provide local exponential stability analogs for

ẋ(t) = Ax(t) + Bu(t) + Φ(x(t)) (34)
where Φ satisfies the following (but similar arguments allow additive uncertainty δ(t) in (34) with suitable bounds on δ):

Assumption 3: Each component Φ i of Φ is continuous and satisfies lim z→0 Φ i (z)/||z|| = 0 for i = 1, . . . , n.
The preceding assumption is very general, because we can rewrite any

C 1 nonlinear function G : R n → R n that satisfies G(0) = 0 in the form G(x) = G 0 x + Φ(x)
, where Φ is the remainder term in the first order Maclaurin approximation of G (which will be of second order at the origin and so satisfy Assumption 3), and then combine the constant matrices A and G 0 to get one constant coefficient of x(t) when the given system has the form ẋ(t) = Ax(t) + Bu(t) + G(x(t)) for such a function G. Assumption 3 covers cases where each component Φ i grows quadratically near the origin, and we assume that (34) satisfies the standard uniqueness of solutions property, as in Section II. We continue our other notation from the preceding sections, where ∆ ♯ = max ∆, λ and Γ 5 = sup ℓ∈[0,T ] |e Aℓ |, and we use the following matrix:

B * = I + T +∆ ♯ |BK| (1+Γ 1 )(Γ 5 +e T |A| )+Γ 2 +2I (35)
Also, like in Theorem 1, our closed loop system will use ( 6), whose trigger times t i are defined by

t 0 = 0, ḣi (t) = Ah i (t) + BKx(t i -λ(t i )) for all t ≥ t i and i ∈ Z 0 , h i (t i ) = x(t i -λ(t i )) for all i ∈ Z 0 , and t i+1 = sup {b ∈ [t i , t i + T ] : |h i (b) -h i (t i )| ≤ Γ 1 |h i (b)|} for all i ∈ Z 0 , (36)
where T and ν satisfy the requirements from Section II. With H = A + BK where K satisfies Assumption 2, and letting D c denote the diagonal matrix whose main diagonal entries are all c and B c denote the closed 2-norm ball of radius c centered at 0 for any constant c > 0, and 1 ∈ R n×n be the matrix of all 1's, we prove the following, where the existence of the ϵ such that M * +ϵ1 is Hurwitz follows from continuity of eigenvalues of a matrix as functions of the matrix entries:

d 3 = d 2 2||BK|| + ∆(||BKA|| + ||(BK) 2 || +d 0 ||BK||) + d 0 max{1, h}e ||R H +N H ||h . (37)
Then, we can find positive constants c * and r * such that for each initial state x(0) in

S 0 = x 0 ∈ B x : max d 2 ||x 0 ||, β 0 [3 + 2h]d 3 ||x 0 ||, h < x ,
the solution x : [0, +∞) → R n of (34), in closed loop with ( 6) and (36), satisfies ||x(t)|| ≤ c * e -r * t ||x(0)|| for all t ≥ 0. Proof: The proof has two parts. In the first part, we find constants c * > 0 and r * > 0 such that the local exponential stability estimate holds for all t ≥ 0 and all solutions x : [0, +∞) → R n of the closed loop system such that

||x|| [0,+∞) ≤ x (38)
holds. In the second part, we show that all solutions x(t) of the closed loop system such that x(0) ∈ S 0 satisfy (38). Second Part. We show that (38) holds along all solutions x(t) of the closed loop system such that x(0) ∈ S 0 . First, we show that ||x|| [0,h] ≤ x. To this end, note that the first part of the proof gives ||x|| [0,t] ≤ d 2 ||x(0)|| for all initial states x(0) and t ∈ [0, h] for which the corresponding solution x(t) of the closed loop system satisfies ||x|| [0,t] ≤ x. Now we argue by contradiction. If x(0) ∈ S 0 were such that x(t) satisfied ||x|| [0,h] > x, then choose the largest t * ∈ [0, h) such that ||x|| [0,t * ] ≤ x. By the continuity of ||x|| [0,t] as a function of t, this gives ||x|| [0,t * ] = x. On the other hand, the preceding argument and our choice of S 0 give ||x|| [0,t * ] ≤ d 2 ||x(0)|| < x, which is a contradiction. Hence, ||x|| [0,h] ≤ x.

Therefore, we can apply the first part of the proof of this theorem to get ||x(t)|| ≤ β(||x(0)||, t) for all initial states x(0) ∈ B x and all t ≥ 0 for which ||x|| [0,t] ≤ x, where β(r, t) = β 0 ([3 + 2h]d 3 r, t) is from the proof of Theorem 1 except with the c i 's replaced by the d i 's and where we can use 3 + 2h instead of 2 [3 + 2h] in the β formula because δ = 0. Also, our choices of β and S 0 give β(||x 0 ||, h) = β 0 [3 + 2h]d 3 ||x 0 ||, h < x for all x 0 ∈ S 0 .

We next use the preceding observations to check that ||x|| [h,+∞) ≤ x. We again argue by contradiction. If x(0) ∈ S 0 were such that the corresponding solution x(t) satisfied ||x|| [h,+∞) > x, then choose the largest t * * ≥ h such that ||x|| [h,t * * ] ≤ x. Again using the continuity of ||x|| [0,t] , this gives ||x|| [h,t * * ] = x. On the other hand, the preceding paragraph implies that ||x|| [h,t * * ] ≤ β(||x(0)||, h) < x, which is a contradiction. Hence, (38) holds for all solutions of the closed loop system having initial states x(0) ∈ S 0 .

V. ILLUSTRATION Consider a dynamics for the depth and pitch degreesof-freedom of an autonomous underwater vehicle, e.g., the BlueROV2 which is used for environmental surveys such as the study of corals. The vehicle has a Doppler Velocity Logger (or DVL) that estimates its velocity. When close to the sea floor, the DVL experiences bottom lock, which makes it impractical to continuously change the control values. Hence, we use Theorem 1 to design a control system for the depth plane, under input and measurement delays, where unlike in [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], the delays are time-varying and unknown. This is an innovative significant departure from event-triggered works like [START_REF] Mazenc | Event-triggered control for continuous-time linear systems with a delay in the input[END_REF], which did not allow measurement delays, and where the chain predictors required exact knowledge of a constant input delay. Following [START_REF] Mazenc | Event-triggered control using a positive systems approach[END_REF], we linearize the dynamics to get (1) with the diagonal matrix A = diag{-0.387, -1.8} and B = [0.038, 1.5] ⊤ . To choose K, we searched for a K that yielded the largest allowable ∆ and λ subject to the constraint that Assumption 2 holds with T = 1.

This produced K = [-0.0085, -0.054], which enabled us to satisfy Assumptions 1-2 with the lower bound ν = 0.16 on the intersample times, the bound λ = 2.86 on λ, and the bound ∆ = 0.64 on ∆. To validate the efficacy of our method, we used MATLAB to simulate the closed loop system from Theorem 1 with the preceding parameter values, and with random delays that took their values on the allowable intervals λ(t) ∈ [0, 1] and ∆(t) ∈ [0, 0.64]. We report the results in Fig. 1, where (w, q) represent the depth and pitch components of the state respectively, which have subscripts 1 and 2 to indicate two initial state vectors (0.2, 0.6) and (-0.15, -0.35) in our two simulations in each panel. In Fig. 1(b), the uncertainty had the constant components δ i (t) = 0.1 for i = 1, 2. Since our simulations exhibit ISS under realistic delays, they illustrate our result.

VI. CONCLUSION

Our new global input-to-state stability theorem for eventtriggered linear systems allowed uncertain input delays and uncertain measurement delays, and used new synergies of interval observers, positive systems, and an ISS vector Halanay's inequality approach. Our trigger times are computed from time-lagged sampled state measurements with unknown measurement delays. We also provided a basin of attraction estimate for a local analog that allows nonlinearities. We aim to develop methods to tune the control parameters to maximize the convergence rate, minimize overshoots, and maximize the basin of attraction in the nonlinear case. We also aim to use marine robotic experiments in our LSU lab to further compare the performance of our event-triggered controls with other possible solutions, such as adjustable time-delay processes based on time-delay tools that solve the event-triggered control problem for systems with delays.

Theorem 2 :

 2 Let H, K, M * , and Φ satisfy Assumptions 2-3, where ∆ and λ are constants such that 0 ≤ ∆(t) ≤ ∆ and 0 ≤ λ(t) ≤ λ hold for all t ≥ 0, and let d max be the largest main diagonal element of H. Let d 0 , ϵ, and x be positive values such that ||Φ(x)|| ≤ d 0 ||x|| for all x ∈ B x , d 0 B * 1 ≤ ϵ1, and M * + ϵ1 is Hurwitz. Choose any constants h > 0 and d > max{0, d max }, and then choose the function β 0 from Remark 2 with P = M * + ϵ1 + D d , D = D d , and τ = 2(T + ∆ ♯ ). Choose the constants d 1 = ||A|| + ||BK|| + d 0 , d 2 = e d1h , and

First Part.

  We indicate the changes that are needed in the proof of Theorem 1. The argument up through (26) remains the same, except with δ(t) and δ(t) replaced by Φ(x(t)) and |Φ(x(t))|, respectively. With these replacements, we collect the terms involving Φ and use the bounds B * |Φ(x)| ≤ d 0 B * 1|x| ≤ ϵ1|x| (which follow from our assumptions, since ||x|| ≤ |x 1 |+. . .+|x n |), and then the bound |x(t)| ≤ s(t) in the result to replace the terms involving |x|'s by terms that use s, to get (28) except with Γ 8 replaced by Γ 8 + ϵ1 and δ replaced by 0, if (38) holds. Since ||Φ(x)|| ≤ d 0 ||x|| for all x ∈ B x , we can then use the reasoning from the third part of the proof of Theorem 1 except with c i replaced by d i for i = 1, 2, 3 and δ = 0, to get the local exponential stability estimate for those solutions x(t) such that (38) holds.

Fig. 1 :

 1 Fig. 1: Simulations (a) without and (b) with uncertainty

  ) for all t ≥ 0. Hence, we can apply Gronwall's inequality to S to check that the constant c2 = max{1, c1h}e c1h is such that ||x(t)|| ≤ S(t) ≤ c2 (||x(0)||+||δ|| [0,t] ) for all t ∈ [0, h].This allows us to use the s dynamics from the equality in (25) and to then apply Gronwall's inequality to the function ||s(t)|| to get the constant c3 = max{1, h}(2c2 ||BK|| + c 2 (||R 1 || + ||R 2 ||)∆ + ∆||R 3 || + 1)e ||R H +N H ||h such that

	||s(t)|| ≤ c3 (||s(0)|| + ||x(0)|| + ||δ|| [0,t] ) for all t ∈ [0, h].
	Since we can assume that x(0) ≤ (1 + h)|x(0)| and x(0) ≥

-(1 + h)|x(0)|, it follows that we can then use the formula s = x -x to get ||s(0)|| ≤ 2(1 + h)||x(0)||, and so also ||s(t)|| ≤ c3 (3 + 2h)||x(0)|| + ||δ|| [0,t] for all t ∈ [0, h]. Hence,