Jamal Alfaisal

Alice Machado

Mathilde Galais

Véronique Robert-Hebmann

Laetitia Arnauné- Pelloquin

Lucile Espert
email: lucileespert@irim.cnrs.fr

Martine Biard-Piechaczyk

HIV-1 Vpr inhibits autophagy during the early steps of infection of CD4 T cells Running title: Incorporated HIV-1 Vpr inbibits autophagy

Keywords: autophagy, HIV-1, Vpr, FOXO3a, LC3

Background information: Autophagy is induced during HIV-1 entry into CD4 T cells by the fusion of the membranes triggered by the gp41 envelope glycoprotein. This anti-HIV-1 mechanism is inhibited by the viral infectivity factor (Vif) neosynthesized after HIV-1 integration to allow viral replication. However, autophagy is very rapidly controlled after HIV-1 entry by a still unknown mechanism. As HIV-1 viral protein R (Vpr) is the only auxiliary protein found within the virion in substantial amount, we studied its capability to control the early steps of HIV-1 envelope-mediated autophagy.

Results: We demonstrated that ectopic Vpr inhibits autophagy in both the Jurkat CD4 T cell line and HEK.293T cells. Interestingly, Vpr coming from the virus also blocks autophagy in CD4 T cells, the main cell target of HIV-1. Furthermore, Vpr decreases the expression level of two essential autophagy proteins (ATG), LC3B and Beclin-1, and an important autophagy-related protein, BNIP3 as well as the level of their mRNA. We also demonstrated in HEK.293T cells that Vpr degrades the FOXO3a transcription factor through the ubiquitin proteasome system.

Conclusion:

Vpr, the only well expressed HIV-1 protein incorporated into viruses, is able to negatively control autophagy induced during HIV-1 entry into CD4 T cells.

Significance: We provide insights of how HIV-1 controls autophagy very early after its entry into CD4 T cells and discovered a new function of Vpr. These results open the route to a better understanding of the roles of Vpr during HIV-1 infection through FOXO3a degradation and could be important to consider additional therapies that counteract the role of Vpr on autophagy.

-FOXO3a: forkhead transcription factor 3a -HIV-1: human immunodeficiency virus type 1 -MAPLC3B: microtubule-associated protein light chain 3B, hereafter referred to as LC3 -PBS: phosphate buffer saline -ROS: reactive oxygen species -RT-PCR: reverse transcriptase polymerase chain reaction -To: torin 1 -ULK1 (Unc-51 like autophagy activating kinase 1) -UPS: ubiquitin proteasome system -Vpr: HIV-1 viral protein R

Introduction

Autophagy is a lysosomal degradation mechanism involving autophagy related proteins (ATGs).

Sequential ATG complexes, including the ULK1 complex, the Beclin-1/class III PI3K complex and the ATG5/ATG12 and ATG8/LC3 conjugation systems, orchestrate and mediate the formation of double membrane vesicles (autophagosomes) that encompass cytoplasmic material [START_REF] Wesselborg | Autophagy signal transduction by ATG proteins: from hierarchies to networks[END_REF]. The autophagosomes then fuse with lysosomes for cargo degradation. BNIP3 (BCL2/adenovirus E1B 19kDa protein-interacting protein 3), a member of the BH3-only subfamily of Bcl-2 family proteins, is a potent inducer of autophagy in different cell types [START_REF] Bellot | Hypoxia-induced autophagy is mediated through hypoxia-inducible factor induction of BNIP3 and BNIP3L via their BH3 domains[END_REF][START_REF] Hamacher-Brady | Response to myocardial ischemia/reperfusion injury involves Bnip3 and autophagy[END_REF][START_REF] Kanzawa | Arsenic trioxide induces autophagic cell death in malignant glioma cells by upregulation of mitochondrial cell death protein BNIP3[END_REF]. BNIP3 can thus be defined as an ATG by regulating autophagy induction [START_REF] Fullgrabe | Transcriptional regulation of mammalian autophagy at a glance[END_REF]. BNIP3 triggers also mitophagy [START_REF] Ney | Mitochondrial autophagy: Origins, significance, and role of BNIP3 and NIX[END_REF] and apoptosis [START_REF] Landes | The BH3-only Bnip3 binds to the dynamin Opa1 to promote mitochondrial fragmentation and apoptosis by distinct mechanisms[END_REF], and its precise functions are not yet clearly defined [START_REF] Burton | The role of Bcl-2 family member BNIP3 in cell death and disease: NIPping at the heels of cell death[END_REF].

Autophagy is regulated by several transcription factors, including the forkhead transcription factor FOXO3a [START_REF] Warr | FOXO3A directs a protective autophagy program in haematopoietic stem cells[END_REF]. Many autophagy genes are under the control of FOXO3, including MAPLC3B (microtubule-associated protein light chain 3B, hereafter referred to as LC3), ATG9A, ATG4B, ATG5, ATG10, Beclin 1, BNIP3 (BCL2/adenovirus E1B 19kDa protein-interacting protein 3) and ULK1 (Unc-51 like autophagy activating kinase 1), depending on the cell type studied [START_REF] Audesse | FOXO3 directly regulates an autophagy network to functionally regulate proteostasis in adult neural stem cells[END_REF][START_REF] Fullgrabe | Transcriptional regulation of mammalian autophagy at a glance[END_REF][START_REF] Li | Forkhead box O3 (FoxO3) regulates kidney tubular autophagy following urinary tract obstruction[END_REF][START_REF] Mammucari | FoxO3 controls autophagy in skeletal muscle in vivo[END_REF][START_REF] Sanchez | AMPK promotes skeletal muscle autophagy through activation of forkhead FoxO3a and interaction with Ulk1[END_REF] Besides its role in maintaining homeostasis, autophagy plays a crucial role in the immune responses against pathogens. Autophagy is induced into CD4 T cells by HIV-1 envelope (Env), via the fusogenic function of gp41 [START_REF] Denizot | HIV-1 gp41 fusogenic function triggers autophagy in uninfected cells[END_REF]. In bystander CD4 T cells, which are not productively infected, autophagy is uncontrolled and leads to their death by apoptosis [START_REF] Espert | Autophagy is involved in T cell death after binding of HIV-1 envelope proteins to CXCR4[END_REF]. This mechanism is thus responsible, at least in part, for HIV-1 pathogenesis. Autophagy is differentially controlled in HIV-1-infected cells [START_REF] Espert | Differential role of autophagy in CD4 T cells and macrophages during X4 and R5 HIV-1 infection[END_REF]. In macrophages, autophagy is required for HIV-1 replication, and IRGM (immunity-associated GTPase family M) plays an important role through its binding to Nef [START_REF] Gregoire | IRGM is a common target of RNA viruses that subvert the autophagy network[END_REF]. In THP-1 macrophages, transfected viral protein R (Vpr) can also trigger autophagy [START_REF] Zhou | The role of autophagy in THP-1 macrophages resistance to HIV-vpr-induced apoptosis[END_REF]. However, the autophagy flux can degrade HIV-1 and Nef acts as an antiautophagic maturation factor through interaction with beclin-1 [START_REF] Kyei | Autophagy pathway intersects with HIV-1 biosynthesis and regulates viral yields in macrophages[END_REF] and through transcription factor EB sequestration [START_REF] Campbell | Human Immunodeficiency Virus Type 1 Nef Inhibits Autophagy through Transcription Factor EB Sequestration[END_REF]. In CD4 T cells, autophagy is an anti-HIV-1 process by selectively degrading the viral transactivator Tat, an essential protein for viral transcription [START_REF] Sagnier | Autophagy restricts HIV-1 infection by selectively degrading Tat in CD4+ T lymphocytes[END_REF]. We and others have previously reported that autophagy is blocked in CD4 T cells that were productively infected by HIV-1 [START_REF] Espert | Differential role of autophagy in CD4 T cells and macrophages during X4 and R5 HIV-1 infection[END_REF][START_REF] Zhou | Human immunodeficiency virus type-1 infection inhibits autophagy[END_REF], thus allowing HIV-1 replication to occur. We already know that neo-synthesized Vif blocks autophagy in CD4 T cells infected by HIV-1 for at least 4 days [START_REF] Borel | HIV-1 viral infectivity factor interacts with microtubule-associated protein light chain 3 and inhibits autophagy[END_REF]. However, the Env-induced autophagy level is very rapidly controlled in infected CD4 T cells, before its complete inhibition in the late steps of infection [START_REF] Sagnier | Autophagy restricts HIV-1 infection by selectively degrading Tat in CD4+ T lymphocytes[END_REF]. This result suggests that another viral component, present in the incoming viral particles, acts very early after HIV-1 entry, before integration. Among the HIV-1 auxiliary proteins, Vpr is the only one packaged into virions in large quantity [START_REF] Cohen | Human immunodeficiency virus vpr product is a virion-associated regulatory protein[END_REF]. It is a multifunctional viral protein essential for viral replication in macrophages [START_REF] Heinzinger | The Vpr protein of human immunodeficiency virus type 1 influences nuclear localization of viral nucleic acids in nondividing host cells[END_REF], but it also facilitates viral replication in CD4 T cells [START_REF] Hohne | Virion encapsidated HIV-1 Vpr induces NFAT to prime non-activated T cells for productive infection[END_REF][START_REF] Rucker | Vpr and Vpu are important for efficient human immunodeficiency virus type 1 replication and CD4+ T-cell depletion in human lymphoid tissue ex vivo[END_REF][START_REF] Zhou | Evidence for Vpr-dependent HIV-1 replication in human CD4+ CEM.NKR T-cells[END_REF]. Vpr is known to play multiple roles at different stages of the HIV-1 viral life cycle such as arresting the cell cycle at the G2/M phase, increasing the activity of the HIV-1 long terminal repeat and regulating apoptosis [START_REF] Guenzel | HIV-1 Vpr-a still "enigmatic multitasker[END_REF] by hijacking E3 ubiquitin ligases. Indeed, Vpr binds to DCAF1 (DDB1 (DNA damage-binding protein 1) and CUL4-associated factor 1), also known as viral protein R binding protein (VprBD), which is part of two E3 ubiquitin ligases, DCAF1-DDB1-CUL4-ROC1 and DCAF1-DDB1-DYRK2-EDD. Thus, Vpr targets multiple cellular proteins to proteasomal degradation and in consequence causes global remodeling of the cellular proteome [START_REF] Greenwood | Promiscuous Targeting of Cellular Proteins by Vpr Drives Systems-Level Proteomic Remodeling in HIV-1 Infection[END_REF].

We demonstrate here that ectopic Vpr inhibits the basal and induced autophagy flux in both the Jurkat CD4 T cell line and HEK.293T cells. It is worth noting that HEK.293 cells expressing the receptors of HIV-1 also undergo Env-mediated autophagy [START_REF] Espert | Autophagy is involved in T cell death after binding of HIV-1 envelope proteins to CXCR4[END_REF]. In the context of HIV-1 infection of CD4 T cells, Vpr coming from the viruses decreases autophagy triggered during viral entry. Interestingly, Vpr decreases the expression of three ATGs, LC3, Beclin-1 and BNIP3, at their transcriptional level. Furthermore, Vpr induces the degradation of the transcription factor FOXO3a.

Results

Vpr inhibits the autophagy process. Basal autophagy and chemically-induced autophagy (Torin 1, To) were first analyzed by immunofluorescence in the Jurkat CD4 T cell line expressing, or not, HA-Vpr, by counting the number of LC3 dots per cell. The autophagy flux was also studied using chloroquine (CQ), which blocks the maturation step of the process. As expected, autophagy is induced by To when Vpr is not expressed (Fig. 1a, white bars). However, the number of LC3 puncta remains very low in cells expressing HA-Vpr (red cells) whatever the conditions used, i.e. cells treated by To, CQ, or To + CQ (Fig. 1a, black bars). This result was reproduced with the ectopically expressed DsRed-Vpr in Jurkat cells (Fig. 1b), indicating that the tag has no influence on the inhibitory effect of Vpr on autophagy. It is important to note that cell nucleofection with a plasmid expressing the tag alone does not affect the autophagy flux (data not shown). Ectopic expression of Vpr in HEK.293 cells also inhibits autophagy induced by To (Suppl. Fig. 1)

To confirm these results, we analyzed the effect of Vpr on autophagy by western blot by following the level of lipidated LC3 (LC3-II) in Jurkat cells expressing, or not, HA-Vpr. As shown in figure 1c, LC3-II expression increases in presence of To + CQ without HA-Vpr, whereas the level of LC3-II stays very low in the presence of HA-Vpr. Altogether, these results confirm that Vpr inhibits both basal and induced autophagy. Interestingly, we observed that Vpr also decreases the expression of LC3-I, the non-lipidated form of LC3 (Fig. 1c). In accordance, red cells (Vpr positive cells) have a low LC3-I labeling (diffuse green fluorescence) in the cell cytoplasm (Fig. 1a and Fig. 1b).

Vpr decreases the expression of several ATGs and acts at the transcriptional level. As we showed that Vpr decreases the LC3-I expression level, we then analyzed the expression of two others ATGs, i.e.

Beclin-1 and BNIP3. As presented in Figure 2, Vpr triggers a significant decrease in the expression of the three ATGs analyzed. It is worth noting that LC3 and HA-Vpr cannot be revealed on the same blot due to their very similar molecular weight. This result suggests that Vpr may either trigger degradation of the three proteins, or, more probably, regulate the expression of these ATGs upstream by acting at their transcriptional level.

To analyze whether Vpr could modulate the transcription of the LC3, Beclin-1 and BNIP3 genes, we studied the expression of their mRNA by RT-PCR in Jurkat cells expressing, or not, HA-Vpr. As shown in Figure 3, the mRNA level of these three genes is strongly decreased in presence of HA-Vpr. Expression of HA-Vpr was controlled by Western blot (data not shown).These results demonstrate that the Vprmediated decrease in the expression of the ATGs tested is transcriptionally regulated.

Vpr, coming from the virion, inhibits HIV-1-induced autophagy. As Vpr is incorporated into the viral particles, we next analyzed the capability of Vpr coming from virions to block Env-induced autophagy at the early steps of infection, before the integration of the provirus into the genome of the host cells [START_REF] Mohammadi | 24 hours in the life of HIV-1 in a T cell line[END_REF]. To this aim, we infected Jurkat cells with HIV-1 (NL4-3 or NL4-3∆Vpr strains) and analyzed autophagy after 8h of infection, in presence or absence of CQ to follow the autophagy flux. The quantity of viruses were adjusted to have the same percentage of infected Jurkat cells (40%) by flow cytometry at 24h of infection. Autophagy was analyzed in HIV-1-infected Jurkat cells by counting the number of endogenous LC3 dots in cells expressing the viral capsid (p24 positive cells).

As shown in Figure 4, the number of autophagosomes is significantly higher in HIV-1ΔVpr-infected cells compared to wt HIV-1-infected cells, in presence or absence of CQ. These results demonstrate that Vpr, coming from the virion, negatively controls autophagy at the early step of infection, before de novo expression of the viral proteins.

The FOXO3a transcription factor is a target of Vpr. Several transcription factors are known to trigger autophagy by regulating the expression of autophagy-related genes [START_REF] Lapierre | Transcriptional and epigenetic regulation of autophagy in aging[END_REF]. Among them, FOXO3a hold our attention because its activation increases the abundance of mRNA and protein levels of LC3 and BNIP3 in several cell types, including cardiomyocytes, mouse primary renal proximal tubular cells or skeletal muscle cells [START_REF] Kume | Calorie restriction enhances cell adaptation to hypoxia through Sirt1-dependent mitochondrial autophagy in mouse aged kidney[END_REF][START_REF] Mammucari | FoxO3 controls autophagy in skeletal muscle in vivo[END_REF][START_REF] Sengupta | FoxO transcription factors promote autophagy in cardiomyocytes[END_REF].

FOXO3a is also known to induce expression of Beclin-1 in muscle cells [START_REF] Mammucari | FoxO3 controls autophagy in skeletal muscle in vivo[END_REF][START_REF] Webb | FOXO transcription factors: key regulators of cellular quality control[END_REF] and kidney tubular cells [START_REF] Li | Forkhead box O3 (FoxO3) regulates kidney tubular autophagy following urinary tract obstruction[END_REF]. As Vpr is known to target several cellular proteins to degradation by the ubiquitin-proteasome system (UPS), we analyzed whether Vpr could also trigger FOXO3a degradation. We thus analyzed the expression level of Flag-FOXO3a in presence or absence of HA-Vpr and MG132, a UPS inhibitor. As shown in Figure 5a, the expression level of FOXO3a is strongly decreased in presence of HA-Vpr without MG132, and this expression is almost completely restored in presence of the UPS inhibitor. This result indicates that Vpr triggers the UPSmediated FOXO3a degradation.

FOXO3a is mainly regulated through its subcellular localization. Indeed, its transcriptionally active form is located in the nucleus, while its inactive form is sequestered in the cytoplasm. To analyze whether Vpr could influence the FOXO3a localization into cells, we analyzed its expression level after cell fractionation in presence or absence of HA-Vpr. The Figure 5b shows that Flag-FOXO3a and HA-Vpr are present in both fractions, but the latter is more expressed into the nucleus. In presence of HA-Vpr, expression of Flag-FOXO3a is dramatically decreased in the cytoplasm but is also strongly reduced in the nucleus, indicating that the overall pool of FOXO3a is subjected to Vpr-mediated degradation (Fig. 5b), and that Vpr does not trigger FOXO3a shuttling. Vpr-mediated FOXO3a degradation is UPSdependent since addition of MG132 restores the expression level of FOXO3a in both fractions (Fig. 5c).

Discussion

During HIV-1 entry into CD4 T cells, autophagy is induced through the fusogenic function of gp41 [START_REF] Denizot | HIV-1 gp41 fusogenic function triggers autophagy in uninfected cells[END_REF]. Our previous results have shown that Env-mediated autophagy is an anti-viral mechanism by degrading specifically Tat [START_REF] Sagnier | Autophagy restricts HIV-1 infection by selectively degrading Tat in CD4+ T lymphocytes[END_REF]. Thus, autophagy must be negatively regulated in infected CD4 T cells, the main target cells of HIV-1, for an efficient viral replication. After provirus integration, neosynthesized viral infectivity factor (Vif) blocks autophagy [START_REF] Borel | HIV-1 viral infectivity factor interacts with microtubule-associated protein light chain 3 and inhibits autophagy[END_REF], but a first negative control of autophagy occurs earlier after HIV-1 entry, before synthesis of new viral proteins [START_REF] Sagnier | Autophagy restricts HIV-1 infection by selectively degrading Tat in CD4+ T lymphocytes[END_REF]. Vpr was a good candidate for this action since it is the only auxiliary viral protein well incorporated into virions. Many functions of this viral protein are linked to its capability to bind to E3 ligases in order to degrade their substrates. In addition, several E3 ligases, and especially CUL-RING E3 ligases, can regulate the autophagy machinery and the upstream regulators of autophagy [START_REF] Cui | Cullin-RING ligases in regulation of autophagy[END_REF]. We demonstrate here that the expression level of the proteins BNIP3, LC3 and Beclin-1 dramatically decreases in presence of Vpr. We thus investigated their level of transcription in the presence of Vpr. Interestingly, these three ATG genes were negatively regulated by Vpr at the transcriptional level, as well as the ULK1 gene (data not shown), suggesting that Vpr acts upstream, on a transcriptional factor. Several positive transcriptional regulators of ATGs are known, including transcription factor EB (TFEB), FOXO3a and E2F1 [START_REF] Fullgrabe | Transcriptional regulation of mammalian autophagy at a glance[END_REF]. All these factors induce the expression of LC3 [START_REF] Polager | E2F1 regulates autophagy and the transcription of autophagy genes[END_REF][START_REF] Sanchez | AMPK promotes skeletal muscle autophagy through activation of forkhead FoxO3a and interaction with Ulk1[END_REF][START_REF] Settembre | TFEB links autophagy to lysosomal biogenesis[END_REF] whereas BNIP3 is overexpressed by both E2F1 and FOXO3a [START_REF] Mammucari | FoxO3 controls autophagy in skeletal muscle in vivo[END_REF][START_REF] Shaw | Molecular regulation of autophagy and apoptosis during ischemic and non-ischemic cardiomyopathy[END_REF][START_REF] Yurkova | The cell cycle factor E2F-1 activates Bnip3 and the intrinsic death pathway in ventricular myocytes[END_REF] and Beclin-1 only by FOXO3a [START_REF] Sanchez | AMPK promotes skeletal muscle autophagy through activation of forkhead FoxO3a and interaction with Ulk1[END_REF]. Very recent data also show that FOXO3 controls many autophagy genes in adult neural stem cells [START_REF] Audesse | FOXO3 directly regulates an autophagy network to functionally regulate proteostasis in adult neural stem cells[END_REF]. In addition, we already know that Env triggers the production of ROS [START_REF] Molina | Proteomic analysis of the cellular responses induced in uninfected immune cells by cell-expressed X4 HIV-1 envelope[END_REF] and numerous data demonstrate that FOXO3a is activated in response to oxidative stress [START_REF] Storz | Forkhead homeobox type O transcription factors in the responses to oxidative stress[END_REF]. For all these reasons, we focused our work on FOXO3a.. However, since Vpr drives DCAF1-dependent degradation of many cellular proteins [START_REF] Greenwood | Promiscuous Targeting of Cellular Proteins by Vpr Drives Systems-Level Proteomic Remodeling in HIV-1 Infection[END_REF], we cannot exclude a role of other transcriptional factors in Vpr-mediated inhibition of autophagy.

FOXO3a is regulated by various post-translational modifications including phosphorylation that dictates its subcellular localization. One study have indirectly linked Vpr to FOXO3a, through its interaction with 14-3-3 [START_REF] Kino | Vpr protein of human immunodeficiency virus type 1 binds to 14-3-3 proteins and facilitates complex formation with Cdc25C: implications for cell cycle arrest[END_REF]. 14-3-3 are a highly conserved family of molecules that inhibit FOXO3a by inducing its accumulation in the cytoplasm [START_REF] Brunet | Akt promotes cell survival by phosphorylating and inhibiting a Forkhead transcription factor[END_REF][START_REF] Brunet | 14-3-3 transits to the nucleus and participates in dynamic nucleocytoplasmic transport[END_REF][START_REF] Dobson | Bimodal regulation of FoxO3 by AKT and 14-3-3[END_REF][START_REF] Lehtinen | A conserved MST-FOXO signaling pathway mediates oxidative-stress responses and extends life span[END_REF][START_REF] Tzivion | FoxO transcription factors; Regulation by AKT and 14-3-3 proteins[END_REF]. In a context of insulin resistance/lipodystrophy syndrome, Vpr binding to 14-3-3 inhibits insulin/PI3K/Akt-1 signaling pathway, triggering FOXO3a translocation to the nucleus (Kino et al., 2005a). We demonstrated here another function of Vpr, where Vpr triggers the degradation of FOXO3a by the proteasome both in the cytoplasm and the nucleus of cells. Further studies are needed to know the E3 ligase involved in this process. However, as FOXO3a is involved in many essential cellular processes, including the control of the cell cycle, the protection from oxidative stress, apoptosis and longevity, these results are of major interest to understand the actions of Vpr at different steps of the HIV-1 life cycle. It would also be interesting to analyze the role of Vpr on FOXO3a in macrophages where ectopic Vpr was shown to activate autophagy [START_REF] Zhou | The role of autophagy in THP-1 macrophages resistance to HIV-vpr-induced apoptosis[END_REF]. This also underlines the complexity of HIV-1 infection depending on the cell type where autophagy is repressed in CD4 T cells but induced and regulated in macrophages [START_REF] Espert | Differential role of autophagy in CD4 T cells and macrophages during X4 and R5 HIV-1 infection[END_REF].

In conclusion, these data demonstrate that during HIV-1 infection of CD4 T cells, incoming Vpr acts to inhibit the autophagy process activated during viral entry to "prepare" the cells for an efficient viral replication. These results also open the route to a better understanding of the roles of Vpr during HIV-1 infection through FOXO3a degradation.

Materials and methods

Cell lines, viruses and infections. The Jurkat CD4 T cell line and the adherent HEK.293T cell line were cultured in RPMI and DMEM, respectively, supplemented with 1% penicillin/streptomycin and 10% fetal calf serum. HIV-1 infection of Jurkat cells was performed with normalized amounts of supernatant of HEK.293T cells that have been transfected with X4 viruses (NL4-3 or NL4-3∆Vpr strains) for two days.

Follow-up of Jurkat cell infection was done by analysing the percentage of infected cells after 24h of infection by flow cytometry using the HIV KC57 FITC kit according to the manufacturer's instructions (Beckman Coulter). Briefly, cells were fixed and permeabilized, and the anti-p24 antibody was added to the cells. After staining for 30min, cells were washed with phosphate-buffered saline (PBS), and fluorescence intensity was measured on a Coulter Epics XL Flow Cytometer (Beckman Coulter).

Reagents, plasmids and antibodies. Torin 1 (To) was purchased from Tocris, chloroquine (CQ) and MG132 from Sigma-Aldrich. The pNL4-3 molecular clone and pNL4-3 deleted of Vpr were obtained from the NIH AIDS Reagent Program. Plasmids that express HA-Vpr and DsRed-Vpr were a gift from S.

Benichou (Institut Cochin, Paris), and plasmid expressing Flag-FOXO3a was purchased from Addgene.

Anti-LC3, anti-GAPDH, anti-Flag, anti-BNIP3 and anti-HA antibodies were obtained from Sigma-Aldrich; anti-Beclin-1 antibody was purchased from Santa-Cruz Biotechnologies. The monoclonal anti-p24 antibody was from the NIH AIDS Reagent program.

Western blotting. Cells were washed twice in PBS and the cell pellet was resuspended in Laemmli buffer. Cell lysates were electrophoresed in 12% Prosieve TM 50 gel (Lonza) and blotted to PVDF membranes. After a blocking step for 1 hour at room temperature in PBS containing 0.5% casein, blots were incubated with the primary antibody in the blocking buffer. After 3 washes with PBS and 0.05% Tween, the blots were incubated with peroxidase-coupled antiserum diluted in the blocking buffer.

After further washes, the immune complexes were revealed by ECL (Biorad) and autoradiographed. Quantification of protein expression was performed by densitometry analysis using the ImageJ software and statistical analysis was performed using ANOVA.

Analysis of autophagy.

The autophagy flux was analyzed by western blot by following the expression level of LC3-II and by immunofluorescence after labeling of endogenous LC3-II using in both cases the rabbit anti-LC3 antibody L7543 (1µg/ml, Sigma Aldrich). To analyze the flux, Torin 1 (To, 2µM, Sigma Aldrich) was added for 3h, in presence or absence of the autophagic flux blocker chloroquine (CQ, 50µM, Sigma Aldrich). Basal autophagy was also analyzed in parallel.

Immunofluorescence studies. Cells were fixed in 1% paraformaldehyde for 10min and then in 100% methanol for 5min at room temperature. After three washes in PBS, the first antibody was incubated for 1h at room temperature. After further washes, the fluorochrome-conjugated secondary antibody (Life Technologies) is then incubated for 30min, the nuclei were labeled with Dapi, and the cells were examined by epifluorescence using a Leica microscope. To analyze autophagy in HIV-1-infected CD4 T cells, cells were incubated with the rabbit anti-LC3 antibody L7543 (1µg/ml, Sigma Aldrich) and the mouse monoclonal anti-p24 antibody for 1h. After further washes, the cells were incubated with Alexa-488 and -568-conjugated secondary antibodies (Life Technologies) for 30min. The number of LC3 dots (green) per infected CD4 T cell (red) was counted. More than 100 CD4 T cells were analyzed for each culture condition by 2 independent investigators. Statistical analysis was performed using ANOVA.

Transfection. Jurkat cells were nucleofected using the Amaxa technology and HEK.293T cell transfection was performed using the Turbofect transfection reagent (Thermo Scientific) according to the manufacturer's instructions.

Reverse transcription PCR (RT-PCR)

. RNA was purified from 10 6 cells using the Nucleospin RNA plus (Macherey-Nagel), and RT-PCR was performed using the QIAGEN OneStep RT-PCR kit according to the Cellular fractionation. Cells were lysed in hypotonic buffer (10mM Tris-HCl, pH 7.65, 1.5mM MgCl2, 1mM DTT, 20mM N-ethylmaleimide, and protease inhibitors) and centrifuged at 500 × g for 15min.

The supernatant constituted the cytoplasmic fraction and the pellet resuspended in Laemmli buffer constituted the nuclear fraction. with or without Vpr were quantified by densitometry and normalized with GAPDH expression levels.

Author contribution

Results are from at least three independent experiments.

  manufacturer's instructions. The oligonucleotides used for LC3B, Beclin-1, BNIP3 and GAPDH mRNA are: LC3B FOR: GATGTCCGACTTATTCGAGAGC; LC3B BACK: TTGAGCTGTAAGCGCCTTCTA; Beclin-1 FOR: BACK: AGTGTTTGCCGCTGTCAGTAC. GAPDH FOR: CCCATCACCATCTTCCAG; GAPDH BACK: CCTGCTTCACCACCTTCT. A control RT-PCR was done without RNA.

  J.A. and A.M. performed experiments and analyzed data; M.G. and V.R-H. carried out experiments; L.A-P. and L.E. contributed to the interpretation of the results. M.B-P. supervised the project and wrote the manuscript with input from all authors. All authors discussed the results and contributed to the final manuscript.

Figure Legends Figure 1 .

 Legends1 Figure Legends

Figure 2 .

 2 Figure 2. Vpr decreases the expression level of LC3, Beclin-1 and BNIP3. Jurkat cells were transfected

Figure 3 .

 3 Figure 3. Vpr acts at the transcriptional level. Jurkat cells were transfected with a plasmid expressing

Figure 4 .

 4 Figure 4. Vpr coming from the viruses inhibits autophagy. Jurkat cells were infected by wt HIV-1 or HIV-

Figure 5 .

 5 Figure 5. Vpr triggers UPS-dependent degradation of FOXO3a. (a) HEK293T cells were transfected with

Acknowledgments

We thank S. Benichou for providing plasmids expressing Vpr and M. Mougel for technical advises and very helpful scientific discussions. The authors have no financial interests to declare.

Funding

This work was supported by institutional funds from the Centre National de la Recherche Scientifique (CNRS) and the Montpellier University and grants from the Agence Nationale de Recherches sur le SIDA et les Hépatites Virales (ANRS) and Sidaction. J.A. was benefiting from the Dunia Beam Erasmus Mundus scholarship, funded with the support of the European Commission.


Arnaud Münch
email: arnaud.munch@uca.fr.

Emmanuel Trélat
email: emmanuel.trelat@sorbonne-universite.fr.

Constructive exact control of semilinear 1D wave equations by a least-squares approach

Keywords: AMS Classifications: 35Q30, 93E24 Semilinear wave equation, exact controllability, least-squares approach

It has been proved by Zuazua in the nineties that the internally controlled semilinear 1D wave equation ∂tty -∂xxy + g(y) = f 1ω, with Dirichlet boundary conditions, is exactly controllable in H 1 0 (0, 1) ∩ L 2 (0, 1) with controls f ∈ L 2 ((0, 1) × (0, T )), for any T > 0 and any nonempty open subset ω of (0, 1), assuming that g ∈ C 1 (R) does not grow faster than β|x| ln 2 |x| at infinity for some β > 0 small enough. The proof, based on the Leray-Schauder fixed point theorem, is however not constructive. In this article, we design a constructive proof and algorithm for the exact controllability of semilinear 1D wave equations. Assuming that g does not grow faster than β ln 2 |x| at infinity for some β > 0 small enough and that g is uniformly Hölder continuous on R with exponent s ∈ [0, 1], we design a least-squares algorithm yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order 1 + s after a finite number of iterations.

Introduction

Let Ω := (0, 1), let ω := ( 1 , 2 ) with 0 1 < 2 1 and let T > 0. We set Q T := Ω × (0, T ), q T := ω × (0, T ) and Σ T := ∂Ω × (0, T ). We consider the semilinear 1D wave equation

     ∂ tt y -∂ xx y + g(y) = f 1 ω in Q T , y = 0 on Σ T , (y(•, 0), ∂ t y(•, 0)) = (u 0 , u 1 ) in Ω, (1)
where (u 0 , u 1 ) ∈ V := H 1 0 (Ω) × L 2 (Ω) is the initial state of y and f ∈ L 2 (q T ) is a control function. Here and throughout the paper, g : R → R is a function of class C 1 such that |g(x)| C(1 + |x|) ln 2 (2 + |x|) for every x ∈ R, for some C > 0. Then, (1) has a unique global (weak) solution in C 0 ([0, T ]; H 1 0 (Ω)) ∩ C 1 ([0, T ]; L 2 (Ω)) (see [START_REF] Cazenave | Équations d'évolution avec non linéarité logarithmique[END_REF]).

We say that (1) is exactly controllable in time T if, for any (u 0 , u 1 ) ∈ V and (z 0 , z 1 ) ∈ V , there exists a control function f ∈ L 2 (q T ) such that the solution of (1) satisfies (y(•, T ), ∂ t y(•, T )) = (z 0 , z 1 ). The exact controllability problem for [START_REF] Cannarsa | One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms[END_REF] has been addressed in [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF]. Theorem 1. [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF] Assume that T > 2 max( 1 , 1 -2 ). There exists β > 0 (only depending on Ω and T ) such that, if

lim sup |x|→+∞ |g(x)| |x| ln 2 |x| < β (2)
then (1) is exactly controllable in time T .

Moreover, it is proved in [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF] that, if g behaves like -s ln p (|s|) with p > 2 as |s| → +∞, then the system is not exactly controllable in any time T > 0, due to an uncontrollable blow-up phenomenon. Theorem 1 has been improved in [START_REF] Cannarsa | One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms[END_REF], weakening the condition (2) into lim sup |x|→+∞ x 0 g(r) dr |x| +∞ k=1 ln [k] (e k + x 2 ) -2 < +∞ where ln [k] denotes the k th iterate of ln and e k > 0 is such that ln [k] (e k ) = 1. This growth condition is essentially optimal since the solution of (1) may blow up whenever g grows faster at infinity and has the bad sign. The multi-dimensional case in which Ω is a bounded domain of R d , d > 1, with a C 1,1 boundary has been addressed in [11]. Assuming that the support ω of the control function is a neighborhood of ∂Ω and that T > diam(Ω\ω), the exact controllability of (1) is proved under the growth condition lim sup |x|→+∞ |g(x)| |x| ln 1/2 |x| < +∞. For control domains ω satisfying the classical multiplier assumption (see [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]), exact controllability has been proved in [START_REF] Zhang | Explicit observability estimate for the wave equation with potential and its application[END_REF] assuming that g is globally Lipschitz continuous. We also mention [START_REF] Coron | Global steady-state stabilization and controllability of 1D semilinear wave equations[END_REF] where a positive boundary controllability result is proved for steady-state initial and final data and for T large enough by a quasi-static deformation approach.

The proof given in [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF] is based on a Leray Schauder fixed point argument introduced in [START_REF] Zuazua | Exact controllability for the semilinear wave equation[END_REF][START_REF] Zuazua | Exact boundary controllability for the semilinear wave equation[END_REF] that reduces the exact controllability problem to obtaining suitable a priori estimates for a linearized wave equation with a potential. More precisely, it is shown that the operator K : L ∞ (Q T ) → L ∞ (Q T ), where y ξ := K(ξ) is a controlled solution with the control function f ξ of the linear boundary value problem

     ∂ tt y ξ -∂ xx y ξ + y ξ g(ξ) = -g(0) + f ξ 1 ω in Q T , y ξ = 0 on Σ T , (y ξ (•, 0), ∂ t y ξ (•, 0)) = (u 0 , u 1 ) in Ω, g(x) :=    g(x) -g(0) x if x = 0 g (0) if x = 0 (3)
satisfying (y ξ (•, T ), ∂ t y ξ (•, T )) = (z 0 , z 1 ) has a fixed point. The control f ξ in [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF] is the one of minimal L 2 (q T ) norm. The existence of a fixed point for the operator K is proved by applying the Leray-Schauder degree theorem: it is shown that if β is small enough, then there exists M = M ( (u 0 , u 1 ) V , (z 0 , z 1 ) V ) > 0 such that K maps the ball B ∞ (0, M ) to itself. The objective of this article is to design an algorithm providing an explicit sequence (f k ) k∈N that converges strongly to an exact control for [START_REF] Cannarsa | One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms[END_REF]. A first idea that comes to mind is to consider the Picard iterations (y k ) k∈N associated with the operator K defined by y k+1 = K(y k ), k 0 initialized with any element y 0 ∈ L ∞ (Q T ). The resulting sequence of controls (f k ) k∈N is then so that f k+1 ∈ L 2 (q T ) is the control of minimal L 2 (q T ) norm for y k+1 solution of

     ∂ tt y k+1 -∂ xx y k+1 + y k+1 g(y k ) = -g(0) + f k+1 1 ω in Q T , y k+1 = 0 on Σ T , (y k+1 (•, 0), ∂ t y k+1 (•, 0)) = (u 0 , u 1 ) in Ω. (4)
Such a strategy usually fails since the operator K is in general not contracting, even if g is globally Lipschitz. We refer to [START_REF] Fernández-Cara | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF] for numerical simulations providing evidence of the lack of convergence in parabolic cases (see also Remark 11 in Appendix A).

A second idea is to use a Newton type method in order to find a zero of the C 1 mapping F : Y → W defined by

F (y, f ) := ∂ tt y -∂ xx y + g(y) -f 1 ω , y(• , 0) -u 0 , ∂ t y(• , 0) -u 1 , y(• , T ) -z 0 , ∂ t y(• , T ) -z 1 (5)
for some appropriate Hilbert spaces Y and W (see further): given (y

0 , f 0 ) in Y , the sequence (y k , f k ) k∈N is defined iteratively by (y k+1 , f k+1 ) = (y k , f k ) -(Y k , F k ) where F k is a control for Y k solution of      ∂ tt Y k -∂ xx Y k + g (y k ) Y k = F k 1 ω + ∂ tt y k -∂ xx y k + g(y k ) -f k 1 ω , in Q T , Y k = 0, on Σ T , Y k (•, 0) = u 0 -y k (•, 0), ∂ t Y k (•, 0) = u 1 -∂ t y k (•, 0) in Ω, (6)
such that Y k (•, T ) = -y k (•, T ) and ∂ t Y k (•, T ) = -∂ t y k (•, T ) in Ω.
This linearization makes appear an operator K N such that y k+1 = K N (y k ) involving the first derivative of g. However, as it is well known, such a sequence may fail to converge if the initial guess (y 0 , f 0 ) is not close enough to a zero of F (see [START_REF] Fernández-Cara | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF] where divergence of the sequence is shown for large data). The controllability of nonlinear partial differential equations has attracted a large number of works in the last decades (see [START_REF] Coron | Control and nonlinearity[END_REF] and references therein). However, as far as we know, few are concerned with the approximation of exact controls for nonlinear partial differential equations, and the construction of convergent control approximations for controllable nonlinear equations remains a challenge.

In this article, given any initial data (u 0 , u 1 ) ∈ V , we design an algorithm providing a sequence (f k ) k∈N converging to a controlled solution for (1), under assumptions on g that are slightly stronger than the one done in Theorem 1. Moreover, after a finite number of iterations, the convergence is superlinear. This is done by introducing a quadratic functional measuring how much a pair (y, f ) ∈ Y is close to a controlled solution for (1) and then by determining a particular minimizing sequence enjoying the announced property. A natural example of an error (or least-squares) functional is given by E(y, f

) := 1 2 F (y, f ) 2
W to be minimized over Y . Exact controllability for (1) is reflected by the fact that the global minimum of the nonnegative functional E is zero, over all pairs (y, f ) ∈ Y solutions of (1). In the line of recent works on the Navier-Stokes system (see [START_REF] Lemoine | Analysis of continuous H -1 -least-squares approaches for the steady Navier-Stokes system[END_REF]), we determine, using an appropriate descent direction, a minimizing sequence (y k , f k ) k 0 converging to a zero of the quadratic functional.

The paper is organized as follows. In Section 2, we define the (nonconvex) least-squares functional E and the corresponding (nonconvex) optimization problem [START_REF] Lemoine | Approximation of nulls controls for semilinear heat equations using a least-squares approach[END_REF]. We show that E is Gateaux-differentiable on A and that any critical point (y, f ) for E such that g (y) ∈ L ∞ (Q T ) is also a zero of E. This is done by introducing an adequate descent direction (Y 1 , F 1 ) for E at any (y, f ) for which E (y, f ) • (Y 1 , F 1 ) is proportional to E(y, f ). This instrumental fact compensates the failure of convexity of E and is at the base of the global convergence properties of our least-squares algorithm. The design of this algorithm is done by determining a minimizing sequence based on (Y 1 , F 1 ), which is proved to converge to a controlled pair for the semilinear wave equation [START_REF] Cannarsa | One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms[END_REF], in our main result (Theorem 2), under appropriate assumptions on g. Moreover, we prove that, after a finite number of iterations, the convergence is superlinear. Theorem 2 is proved in Section 3. We show in Section 4 that our least-squares approach coincides with the classical damped Newton method applied to a mapping similar to F , and we give a number of other comments. In Appendix A, we state some a priori estimates for the linearized wave equation with potential in L ∞ (Q T ) and source term in L 2 (Q T ) and we show that the operator K is contracting if ĝ L ∞ loc (R) is small enough. As far as we know, the method introduced and analyzed in this work is the first one providing an explicit, algorithmic construction of exact controls for semilinear wave equations.

Notations. Throughout, we denote by • ∞ the usual norm in L ∞ (R), by (•, •) X the scalar product of X (if X is a Hilbert space) and by •, • X,Y the duality product between X and Y . The notation • 2,q T stands for • L 2 (q T ) and • p for • L p (Q T ) , mainly for p = 2 and p = +∞.

Given any s ∈ (0, 1], we denote by C 1,s (R) the set of all functions g ∈ C 1 (R) such that g is uniformly Hölder continous with exponent s, meaning that

[g ] s := sup a,b∈R a =b |g (a) -g (b)| |a -b| s < +∞.
For s = 0, by extension, we set [g ] 0 := 2 g ∞ . In particular, g ∈ C 1,0 (R) if and only if g ∈ C 1 (R) and g ∈ L ∞ (R), and g ∈ C 1,1 (R) if and only if g is Lipschitz continuous (in this case, g is almost everywhere differentiable and g ∈ L ∞ (R), and we have [g ] s g ∞ ).

2 Least-squares algorithm and main result

Least-squares functional and minimization problem

Least-squares functional. We consider the Hilbert space

H := (y, f ) ∈ L 2 (Q T ) × L 2 (q T ) | ∂ tt y -∂ xx y ∈ L 2 (Q T ), y = 0 on Σ T , (y(•, 0), ∂ t y(•, 0)) ∈ V , (y(•, T ), ∂ t y(•, T )) ∈ V
endowed with the scalar product

((y 1 , f 1 ), (y 2 , f 2 )) H := (y 1 , y 2 ) 2 + (y 1 (•, 0), ∂ t y 1 (•, 0)), (y 2 (•, 0), ∂ t y 2 (•, 0)) V + ∂ tt y 1 -∂ xx y 1 , ∂ tt y 2 -∂ xx y 2 2 + (f 1 , f 2 ) 2,q T
and the norm (y, f

) H := ((y, f ), (y, f )) H .
In what follows, we fix some arbitrary (u 0 , u 1 ) ∈ V and (z 0 , z 1 ) ∈ V . The subspaces of H defined by

A := (y, f ) ∈ H | (y(•, 0), ∂ t y(•, 0)) = (u 0 , u 1 ), (y(•, T ), ∂ t y(•, T )) = (z 0 , z 1 ) in Ω , A 0 := (y, f ) ∈ H | (y(•, 0), ∂ t y(•, 0)) = (0, 0), (y(•, T ), ∂ t y(•, T )) = (0, 0) in Ω , Note that A = (y, f ) + A 0 for any (y, f ) ∈ A.
Given any (y, f ) ∈ A, it follows from the a priori estimate for the linear 1D wave equation that there exists C > 0, only depending on Ω and T , such that

(y, ∂ t y) 2 L ∞ (0,T ;V ) C ∂ tt y -∂ xx y 2 L 2 (Q T ) + (u 0 , u 1 ) 2 V y ∞ C (y, f ) H (7)
in particular y ∈ L ∞ (Q T ). Since g is of class C 1 , we have g(y) ∈ L 2 (Q T ) and g (y) ∈ L ∞ (Q T ). We define the least-squares functional E : A → R by

E(y, f ) := 1 2 ∂ tt y -∂ xx y + g(y) -f 1 ω 2 L 2 (Q T )
for every (y, f ) ∈ A.

Least-squares minimization problem. For any fixed (y, f ) ∈ A, we consider the (nonconvex) minimization problem inf

(y,f )∈A0 E(y + y, f + f ) (8)
In the framework of Theorem 1, the infimum of the functional of E is zero and is reached by at least one pair (y, f ) ∈ A, solution of (1) and satisfying (y(•, T ), ∂ t y(•, T )) = (z 0 , z 1 ). Conversely, any pair (y, f ) ∈ A such that E(y, f ) = 0 is a solution of (1). In this sense, the functional E is an error functional which measures the deviation of (y, f ) from being a solution of the underlying nonlinear equation.

A classical algorithmic way for computing the minimum consists in following descent directions, along the gradient of the functional. In descent algorithms, local minima are a usual issue to face with, unless the functional E is convex. Since (1) is nonlinear, here E fails to be convex in general. In spite of that, we are going to construct a minimizing sequence which always converges to a zero of E. Definition 2.1. Let T > 2 max( 1 , 1-2 ) be arbitrary. Given any (y, f ) ∈ A, over all pairs (Y 1 , F 1 ) ∈ A 0 solutions (the next result shows that there do exist some solutions) of

     ∂ tt Y 1 -∂ xx Y 1 + g (y) • Y 1 = F 1 1 ω + ∂ tt y -∂ xx y + g(y) -f 1 ω in Q T , Y 1 = 0 on Σ T , (Y 1 (•, 0), ∂ t Y 1 (•, 0)) = (0, 0) in Ω, (9)
we select the (unique) pair (Y 1 , F 1 ) ∈ A 0 such that the control F 1 , which is a null control for Y 1 , has a minimal L 2 (q T ) norm. In what follows, it is called the solution (Y 1 , F 1 ) ∈ A 0 of (9) of minimal control norm.

In the result hereafter, given any (y, f ) ∈ A, we establish some properties of the pair ((y, f ), (Y 1 , F 1 )) ∈ A × A 0 , where (Y 1 , F 1 ) is the solution of (9) of minimal control norm, which are at the base of the leastsquares algorithm that we propose in Section 2.2, and are useful in view of proving its convergence (see Theorem 2).

Proposition 1. Assume that T > 2 max( 1 , 1 -2 ).
There exists a positive constant C, only depending on Ω and T , such that, given any (y, f ) ∈ A:

(i) There exist solutions of (9). Moreover, the solution (Y 1 , F 1 ) ∈ A 0 of (9) of minimal control norm is unique and satisfies

(Y 1 , ∂ t Y 1 ) L ∞ (0,T ;V ) + F 1 2,q T Ce C √ g (y) ∞ E(y, f ) (10)
and

(Y 1 , F 1 ) H Ce C √ g (y) ∞ E(y, f ). (11)
In particular, Y 1

L ∞ (Q T ) Ce C √ g (y) ∞ E(y, f ). (ii) The derivative of E at (y, f ) ∈ A along the direction (Y 1 , F 1 ) satisfies E (y, f ) • (Y 1 , F 1 ) := lim λ→0 λ =0 E((y, f ) + λ(Y 1 , F 1 )) -E(y, f ) λ = 2E(y, f ). ( 12
)
(iii) Noting that the derivative E (y, f ) does not depend on (Y, F ) and defining the norm E (y, f

) A 0 := sup (Y,F )∈A0\{0} E (y, f ) • (Y, F ) (Y, F ) H
, where A 0 is the topological dual of A 0 , we have

1 √ 2 max 1, g (y) ∞ E (y, f ) A 0 E(y, f ) 1 √ 2 Ce C √ g (y) ∞ E (y, f ) A 0 . ( 13
) (iv) Assume that g ∈ C 1,s (R) for some s ∈ [0, 1]. Then E (y, f ) -λ(Y 1 , F 1 ) |1 -λ| + λ 1+s K(y)E(y, f ) s 2 2 E(y, f ) ∀λ ∈ R (14)
where

K(y) := C [g ] s Ce C √ g (y) ∞ 1+s . ( 15
)
Proof. Let us establish (i). The first estimate is a consequence of Lemma 1 in Appendix A, using the equality

∂ tt y -∆y + g(y) -f 1 ω 2 = 2E(y, f ). The second one follows from (Y 1 , F 1 ) H ∂ tt Y 1 -∂ xx Y 1 2 + Y 1 2 + F 1 2,q T + Y 1 (•, 0), ∂ t Y 1 (•, 0) V (1 + g (y) ∞ ) Y 1 2 + 2 F 1 2,q T + E(y, f ) C(1 + g (y) ∞ )e C √ g (y) ∞ E(y, f ) Ce (2+C) √ g (y) ∞ E(y, f ) using that (1 + s) e 2
√ s for every s 0. To prove (ii), we first check that, for every (Y, F ) ∈ A 0 , the functional E is differentiable at (y, f ) ∈ A along the direction (Y, F ) ∈ A 0 . For any λ ∈ R, simple computations lead to

E(y + λY, f + λF ) = E(y, f ) + λE (y, f ) • (Y, F ) + h((y, f ), λ(Y, F )) with E (y, f ) • (Y, F ) := ∂ tt y -∂ xx y + g(y) -f 1 ω , ∂ tt Y -∂ xx Y + g (y)Y -F 1 ω 2 (16)
and

h((y, f ), λ(Y, F )) := λ 2 2 ∂ tt Y -∂ xx Y + g (y)Y -F 1 ω , ∂ tt Y -∂ xx Y + g (y)Y -F 1 ω 2 + λ ∂ tt Y -∂ xx Y + g (y)Y -F 1 ω , (y, λY ) 2 + ∂ tt y -∂ xx y + g(y) -f 1 ω , (y, λY ) 2 + 1 2 ( (y, λY ), (y, λY ))
where (y, λY

) := g(y + λY ) -g(y) -λg (y)Y. ( 17
)
The mapping (Y,

F ) → E (y, f ) • (Y, F ) is linear continuous from A 0 to R since |E (y, f ) • (Y, F )| ∂ tt y -∂ xx y + g(y) -f 1 ω 2 ∂ tt Y -∂ xx Y + g (y)Y -F 1 ω 2 2E(y, f ) (∂ tt Y -∂ xx Y ) 2 + g (y) ∞ Y 2 + F 2,q T 2E(y, f ) max 1, g (y) ∞ (Y, F ) H . (18)
Similarly, for every λ ∈ R \ {0},

1 λ h((y, f ), λ(Y, F )) |λ| 2 ∂ tt Y -∂ xx Y + g (y)Y -F 1 ω 2 2 + |λ| ∂ tt Y -∂ xx Y + g (y)Y -F 1 ω 2 + 2E(y, f ) + 1 2 (y, λY ) 2 1 |λ| (y, λY ) 2 . Since g ∈ L ∞ loc (R) and y ∈ L ∞ (Q T ), we have 1 λ (y, λY ) = g(y + λY ) -g(y) λ -g (y)Y sup θ∈(0,1) g (y + θY ) ∞ + g (y) ∞ |Y | a.e. in Q T , and 1 λ (y, λY ) = g(y+λY )-g(y) λ -g (y)Y → 0 as λ → 0 a.e. in Q T . By the Lebesgue domi- nated convergence theorem, it follows that | 1 λ | (y, λY ) 2 → 0 as λ → 0 and then that |h((y, f ), λ(Y, F ))| = o(λ).
We deduce that the functional E is differentiable at the point (y, f ) ∈ A along the direction (Y, F ) ∈ A 0 . Finally, [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF] follows from the definition of (Y 1 , F 1 ) given in [START_REF] Lemoine | A fully space-time least-squares method for the unsteady Navier-Stokes system[END_REF].

Let us establish (iii). Note that, by ( 16), the derivative E (y, f ) does not depend on (Y, F ). Now, [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF] gives

E(y, f ) = 1 2 E (y, f ) • (Y 1 , F 1 )
where (Y 1 , F 1 ) ∈ A 0 is solution of (9) and, using (11),

E(y, f ) 1 2 E (y, f ) A 0 (Y 1 , F 1 ) A0 1 2 Ce C √ g (y) ∞ E (y, f ) A 0 E(y, f ).
Besides, for all (Y, F ) ∈ A 0 , the inequality [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF] leads to the left inequality in [START_REF] Münch | Numerical null controllability of the heat equation through a least squares and variational approach[END_REF].

|E (y, f ) • (Y, F )| 2E(y, f ) max(1, g (y) ∞ ) (Y, F ) H coming from
Let us finally establish (iv). We start by observing that, since g(y -λY 1 ) = (y, -λY 1 ) + g(y)λg (y)Y 1 , and since (Y 1 , F 1 ) ∈ A 0 is solution of (9), we have

E (y, f ) -λ(Y 1 , F 1 ) = 1 2 ∂ tt y -∂ xx y + g(y) -f 1 ω -λ ∂ tt Y 1 -∂ xx Y 1 + g (y)Y 1 -F 1 1 ω + (y, -λY 1 ) 2 2 = 1 2 (1 -λ) ∂ tt y -∂ xx y + g(y) -f 1 ω + (y, -λY 1 ) 2 2 . (19)
Now, for any (u, v) ∈ R 2 and any λ 0, writing g(u

+ λv) -g(u) = v λ 0 g (u + ξv) dξ, we have |g(u + λv) -g(u) -λg (u)v| λ 0 |v||g (u + ξv) -g (u)| dξ [g ] s |v| 1+s λ 1+s . It follows that | (y, -λY 1 )| = |g(y -λY 1 ) -g(y) + λg (y) Y 1 | [g ] s λ 1+s |Y 1 | 1+s
and thus, using [START_REF] Lemoine | Analysis of continuous H -1 -least-squares approaches for the steady Navier-Stokes system[END_REF],

(y, -λY 1 ) 2 [g ] s λ 1+s |Y 1 | 1+s L 2 (0,T ;L 2 (Ω)) [g ] s λ 1+s √ 2C Y 1 1+s L ∞ (Q T ) [g ] s λ 1+s √ 2C Ce C √ g (y) ∞ 1+s E(y, f ) 1+s 2 (20)
for some positive constant C only depending on Ω and T . Hence, using (19), we get

2E (y, f ) -λ(Y 1 , F 1 ) (1 -λ) ∂ tt y -∂ xx y + g(y) -f 1 ω 2 + (y, -λY 1 ) 2 |1 -λ| 2E(y, f ) + [g ] s λ 1+s |Y 1 | 1+s L 2 (0,T ;L 2 (Ω))
and, using (20), the estimate ( 14) follows.

Consequence. An important consequence of Proposition 1 and in particular of ( 13) is that any critical point (y, f ) ∈ A of E (i.e., E (y, f ) = 0) is a zero of E, and thus is a pair solution of the controllability problem. Moreover:

given any sequence (y k , f k ) k∈N in A such that E (y k , f k ) A 0 -→ k→+∞ 0 and such that g (y k ) ∞ is uniformly bounded, we have E(y k , f k ) -→ k→+∞ 0.
This is thanks to this instrumental property that a minimizing sequence for E cannot be stuck in a local minimum, and this, even though E fails to be convex (it has multiple zeros). Our least-squares algorithm, designed in the next section, and our main result, Theorem 2, are based on that property. Note that the left inequality in [START_REF] Münch | Numerical null controllability of the heat equation through a least squares and variational approach[END_REF] indicates the functional E is flat around its zero set. As a consequence, gradient-based minimizing sequences may have a low speed of convergence (see [START_REF] Lemoine | Analysis of continuous H -1 -least-squares approaches for the steady Navier-Stokes system[END_REF][START_REF] Münch | Numerical null controllability of the heat equation through a least squares and variational approach[END_REF] for such issues for the Navier-Stokes equation).

Least-squares algorithm

Assume that T > 2 max( 1 , 1 -2 ). By [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF] in Proposition 1, the vector -(Y 1 , F 1 ), solution of minimal control norm of (9), is a descent direction for E. This leads us to define, for any fixed m 1, the sequence (y k , f k ) k∈N in A defined by

         (y 0 , f 0 ) ∈ A (y k+1 , f k+1 ) = (y k , f k ) -λ k (Y 1 k , F 1 k ) ∀k ∈ N λ k = argmin λ∈[0,m] E (y k , f k ) -λ(Y 1 k , F 1 k ) (21)
where

(Y 1 k , F 1 k ) ∈ A 0 is the solution of minimal control norm of      ∂ tt Y 1 k -∂ xx Y 1 k + g (y k ) • Y 1 k = F 1 k 1 ω + (∂ tt y k -∂ xx y k + g(y k ) -f k 1 ω ) in Q T , Y 1 k = 0 on Σ T , (Y 1 k (•, 0), ∂ t Y 1 k (•, 0)) = (0, 0) in Ω. ( 22
)
The real number m 1 is arbitrarily fixed. It is used in the proof of convergence to bound the sequence of optimal descent steps λ k .

Main result

Given any s ∈ [0, 1], we set

β 0 (s) := s 2 C 2 (2s + 1) 2 (23)
where C > 0, only depending on Ω and T , is given by Proposition 1. Note that (2

+ 1 s )C β 0 (s) = 1.
Theorem 2. We assume that T > 2 max( 1 , 1 -2 ), that g ∈ C 1,s (R) for some s ∈ [0, 1], and that there exist α 0 and β ∈ [0, β 0 (s)) (with the agreement that β = 0 if s = 0), such that

|g (x)| α + β ln 2 (1 + |x|) ∀x ∈ R. ( 24
)
In the case where s = 0 (i.e., g ∈ L ∞ (R)) but g / ∈ C 1,s (R) for any s ∈ (0, 1], we assume moreover that

2 g ∞ C 2 e C √ g ∞ < 1.
Then:

• The sequence (y k , f k ) k∈N in A defined by (21), initialized at any (y 0 , f 0 ) ∈ A, converges to (y, f ) ∈ A, where (y, f ) is a solution of (1) such that (y(•, T ), ∂ t y(•, T )) = (z 0 , z 1 ).
• The sequence (λ k ) k∈N consists of positive real numbers and converges to 1.

• The decreasing sequence (E(y k , f k )) k∈N converges to 0.

Moreover, the convergence of all these sequences is at least linear, and is at least of order 1 + s after a finite number of iterations. 1

Remark 1. The limit (y, f ) ∈ A of the sequence (y k , f k ) k∈N , given by

(y, f ) = (y 0 , f 0 ) - +∞ k=0 λ k (Y 1 k , F 1 k ),
depends on the choice of the initialization (y 0 , f 0 ) ∈ A (see also Remark 10 further). It also depends on the selection criterion that we have chosen: in (22), F 1 k is the control of minimal norm.

Remark 2. In this remark, we assume that g ∈ L ∞ (R). When g is not uniformly Hölder continuous, a smallness condition on g ∞ is required in order to obtain the convergence. This condition is not required anymore as soon as g ∈ C 1,s (R) for some s ∈ (0, 1]: indeed, then, g satisfies the growth condition (24) with α = g ∞ and β = 0, and Theorem 2 can be applied.

Remark 3. In Theorem 2, we have assumed that the nonnegative coefficient β appearing in the growth condition (24) is lower than β 0 (s), i.e., lim sup

|x|→+∞ |g (x)| ln 2 |x| < s 2 (2s + 1) 2 C 2 (with the agreement that lim sup |x|→+∞ |g (x)| ln 2 |x| = 0 if s = 0), which, of course, implies that lim sup |x|→+∞ |g(x)| |x| ln 2 |x| < s 2 (2s + 1) 2 C 2 .
The threshold β 0 (s) is maximal when s = 1, i.e., when g is Lipschitz continuous, and we have β 0 (1) = 1 9C 2 . In comparison, the threshold β in Theorem 1 satisfies β < 1/(1 + C) 2 where C is another constant (only depending on Ω and T ), appearing in the a priori estimate (45) of Lemma 1 in Appendix A. 1 We recall that a sequence (u k ) k∈N of real numbers converges to 0 with order α 1 if there exists M > 0 such that |u k+1 | M |u k | α for every k ∈ N. A sequence (v k ) k∈N of real numbers converges to 0 at least with order α 1 if there exists a sequence (u k ) k∈N of nonnegative real numbers converging to 0 with order α 1 such that |v k | u k for every k ∈ N.

There exist cases covered by Theorem 1 (or, by the extension established in [START_REF] Cannarsa | One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms[END_REF]), in which exact controllability of (1) is true, but that are not covered by Theorem 2. Note however that the example g(x) = a + bx + 1 9C 2 +ε x ln 2 (1 + |x|) for any ε > 0 and any a, b ∈ R (which is somehow the limit case in Theorem 1) satisfies g ∈ C 1,1 (R) as well as (24).

While Theorem 1 was established in [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF] by a nonconstructive Leray-Schauder fixed point argument, we obtain here, in turn, a new proof of the exact controllability of semilinear 1D wave equations, which is moreover constructive, with an algorithm that converges unconditionally, at least with order 1 + s.

Remark 4. The convergence in Theorem 2 is unconditional. Anyway, a natural example of an initialization (y 0 , f 0 ) ∈ A is to take (y 0 , f 0 ) = (y , f ), the unique solution of minimal control norm of (1) with g = 0 (i.e., in the linear case).

Remark 5. As stated in Theorem 2, the convergence is at least of order 1 + s after a number k 0 of iterations. In this remark, we give the precise expression for k 0 in function of the various parameters. Given any s ∈ [0, 1], any α 0, any β ∈ [0, β 0 (s)) and any M 0, we set

c := [g ] s C 2+s e (1+s)C √ α (1 + M ) (1+s)C √ β (25)
with the agreement that, when s = 0, we take α = g ∞ and β = 0, so that c = 2

g ∞ C 2 e C √ g ∞
(because, by convention, [g ] 0 := 2 g ∞ ), which is the quantity required to be less than 1 in Theorem 2 when s = 0. With this convention, c is a continuous function of s on [0, 1].

If (1 + s) c E(y 0 , f 0 ) s 2 < 1 (this includes the case s = 0) then k 0 = 0, and otherwise,

k 0 = (1 + s) 1+ 1 s s c 1 s E 0 -1 + 1, ( 26
)
where • is the integer part, and where M > 0 is the minimal possible real positive number such that

1 (1 + s)[g ] s C 2+s e (1+s)C √ α (1 + M ) (1+s)C √ β E s 2 0 , C (y 0 , f 0 ) H + m s (1 + s) 1+ 1 s [g ] 1 s s C 3+ 2 s e (2+ 1
s )C √ α (1 + M ) (2+ 1 s )C √ β E 0 M.
The real number M is defined in an implicit way. More details are given in Section 3, at Step 7 of the proof, where we give in particular an explicit expression for M when s 0.

Remark 6. As a continuation to Remark 2, it is interesting to note that, assuming that g ∈ L ∞ (R):

• As stated in Theorem 2, if g / ∈ C 1,s (R) for any s ∈ (0, 1], to obtain convergence it is required to assume that 2 g ∞ C 2 e C √ g ∞ < 1, i.e., that c |s=0 < 1 with the notations of Remark 5, and we

have k 0 = 0. • If 2 g ∞ C 2 e C √ g ∞
1 and if g ∈ C 1,s (R) for some s ∈ (0, 1], then Theorem 2 applies and k 0 is given by (26). Moreover, k 0 is larger as s > 0 is smaller: more precisely, we have

k 0 ∼ e s 2 g ∞ C 2 e C √ g ∞ 1 s as s → 0.
Remark 7. Using (24), we have, for every (y 0 , f 0 ) ∈ A,

E(y 0 , f 0 ) ∂ tt y 0 -∂ xx y 0 2 + f 0 1 ω 2 + g(y 0 ) 2 (y 0 , f 0 ) H + T |g(0)| + T (α + β ln 2 (1 + y 0 ∞ )) y 0 ∞ (y 0 , f 0 ) H + T |g(0)| + T (α + β ln 2 (1 + (y 0 , f 0 ) H )) (y 0 , f 0 ) H . (27)
Remark 8. If s = 0 or if β = 0 in (24) then g ∈ L ∞ (R). In this case the proof of Theorem 2 is simpler. When s > 0 and β > 0, as alluded at the end of Section 2.1, in the proof of Theorem 2, the main difficulty is to prove that the sequence ( y k ∞ ) k∈N (defined in (21)) remains uniformly bounded, in particular in order to keep a uniform bound on the sequence of observability constants Ce C √ g (y k ) ∞ appearing in the estimates of Proposition 1. In the proof, done in Section 3, this difficulty is handled by an a priori assumption, which we prove to be satisfied a posteriori thanks to fine estimates.

Proof of Theorem 2

This section is devoted to proving Theorem 2. We assume that g ∈ C 1,s (R) for some s ∈ [0, 1].

Preliminary remark. Let (y 0 , f 0 ) ∈ A be arbitrarily fixed. In the sequel, we denote by

E k := E(y k , f k ) ∀k ∈ N.
By the minimization property in the definition (21) of the algorithm, we have

E k+1 = E (y k , f k ) - λ k (Y 1 k , F 1 k ) E (y k , f k ) -λ(Y 1 k , F 1 k ) for every λ ∈ [0, m].
Applying the estimate ( 14) of Proposition 1, Item (iv), to (y k , f k ), we infer that

E k+1 min λ∈[0,m] |1 -λ| + λ 1+s K(y k )E s 2 k 2 E k ∀k ∈ N (28)
where we recall that

K(y k ) = [g ] s C 2+s e (1+s)C √ g (y k ) ∞ .
The estimate (28) is instrumental in the proof of Theorem 2.

Having in mind Remark 8, we fix a constant M > 0, large enough, to be chosen later. In what follows, we make the a priori assumption

y k ∞ M ∀k ∈ N. (29)
We are going to see a posteriori that, if M is adequately chosen large enough, then ( 29) is indeed satisfied. The proof goes in several steps.

Step 1. There exists k 0 ∈ N (given by (26)) such that the sequence (E k ) k k0 decays to 0 with order greater than or equal to 1 + s.

Using the growth condition (24) and using the a priori assumption (29), we have

g (y k ) ∞ α + β ln 2 (1 + M ) ∀k ∈ N.
Here and in the sequel, we adopt the convention that, when s = 0, we take α = g ∞ and β = 0. Using the inequality √ a + b √ a + √ b for all a, b 0, we get

Ce C √ g (y k ) ∞ Ce C √ α+β ln 2 (1+M ) Ce C √ α (1 + M ) C √ β (30)
and thus K(y k ) c where K(y k ) is defined by [START_REF] Zhang | Explicit observability estimate for the wave equation with potential and its application[END_REF] and c is defined by (25) (including the case s = 0). By (28), we have

E k+1 min λ∈[0,m] e k (λ) E k with e k (λ) := |1 -λ| + λ 1+s c E s 2 k . (31)
Let λ k ∈ [0, m] be the minimizer of e k (λ) over [0, m] (not to be confused with λ k defined in ( 21)).

Let us first treat the case where s ∈ (0, 1]. Assuming that E k > 0 (otherwise there is nothing to do), we have

λ k = 1 and e k ( λ k ) = c E s 2 k if (1 + s) 1 s c 1 s √ E k < 1, λ k = 1 (1 + s) 1 s c 1 s √ E k and e k ( λ k ) = 1 - s (1 + s) 1+ 1 s c 1 s √ E k if (1 + s) 1 s c 1 s √ E k 1, (32)
and therefore, by (31),

c 1 s E k+1 c 1 s √ E k 1+s if (1 + s) 1 s c 1 s √ E k < 1, c 1 s √ E k - s (1+s) 1+ 1 s if (1 + s) 1 s c 1 s √ E k 1. (33)
• As a first case, let us assume that 0 < (1 + s) , the sequence (E k ) k∈N is decreasing and converges to 0 with order greater than or equal to 1 + s.

1 s c 1 s √ E 0 < 1. Then c 1 s √ E 0 <
• As a second case, let us assume that (1 + s)

1 s c 1 s √ E 0 1.
It follows from (31) that, as long as (1 + s)

1 s c 1 s √ E k 1, we have c 1 s √ E k c 1 s √ E 0 -k s (1+s) 1+ 1 s
. Hence there exists k 0 ∈ N such that

(1 + s) 1 s c 1 s √ E k < 1 for every k k 0 .
This means that, after a finite number of iterations, we turn back to the first case. The minimal number of iterations is given by the formula (26).

Finally, let us treat the case where s = 0. The function e k (λ) is piecewise linear, and is increasing whenever c > 1: this is why we need the smallness condition

c = 2 g ∞ C 2 e C √ g ∞ c < 1. Thanks to
this assumption, the minimizer of e k (λ) is λ k = 1 and thus e k ( λ k ) = c. Hence E k+1 cE k (which is also what we obtain by taking the limit s → 0 + in the first case of ( 33)) and thus (E k ) k∈N is decreasing and converges to 0 at least linearly. In this case we have k 0 = 0.

Remark 9. For every k k 0 , we have (assuming that E k > 0)

E k+1 E k c 2 E s k .
Since E k → 0 at least with order 1 + s, it follows that E k+1 E k → 0 as k → +∞, at least with order 1 + s. Note also that λ k > 0 for every k ∈ N because the sequence (E k ) k∈N is decreasing.

Step 2. The sequence (λ k ) k k0 defined in (21) converges to 1 as k → +∞ at least with order 1 + s.

Applying (19) to (y, f ) = (y k , f k ), (Y 1 , F 1 ) = (Y 1 k , F 1 k ) and λ = λ k , we have, since λ k m (and assuming that E k > 0), (1 -λ k ) 2 = E k+1 E k -(1 -λ k ) ∂ tt y k -∂ xx y k + g(y k ) -f k 1 ω , (y k , -λ k Y 1 k ) 2 E k - (y k , -λ k Y 1 k ) 2 2 2E k E k+1 E k -(1 -λ k ) ∂ tt y k -∂ xx y k + g(y k ) -f k 1 ω , (y k , -λ k Y 1 k ) 2 E k E k+1 E k + m √ 2 (y k , λ k Y 1 k ) 2 √ E k . By (20), we have (y k , λ k Y 1 k ) 2 λ 1+s k √ 2K(y k )E 1+s 2 k m 1+s √ 2 c E 1+s 2 k
, and thus

(1 -λ k ) 2 E k+1 E k + 2m 2+s c E s 2 k ∀k ∈ N.
Since E k → 0 at least with order 1 + s by Step 1 and E k+1 E k → 0 at least with order 1 + s by Remark 9, it follows that λ k → 1 at least with order 1 + s.

Step 3. We have e 0 ( λ 0 ) < 1, and the sequence (e k ( λ k )) k∈N decays to 0. Indeed, since e 0 (0) = 1 and e 0 (0) < 0, we have e 0 ( λ 0 ) = min λ∈[0,m] e 0 (λ) < 1 (also in the case where s = 0 thanks to the smallness condition). The rest of the statement follows from (32).

Step 4. The series k 0 √ E k converges, and

+∞ k=p E k 1
1 -e 0 ( λ 0 ) E p for every p ∈ N.

The fact that the series k 0 √ E k converges already follows from Remark 9 since E k+1 E k → 0. We will however use the rough estimate stated here. To prove it, we observe that, since the sequence (e k ( λ k )) k∈N is decreasing, we have e k ( λ k ) e p ( λ p ) e 0 ( λ 0 ) < 1 for all k, p ∈ N such that k p, and we infer from (31) that E k (e 0 ( λ 0 )) k-p E p ∀k, p ∈ N, k p and the result follows.

Step 5. The series k 0 λ k (Y 1 k , F 1 k ) converges in A 0 , and

q k=p λ k (Y 1 k , F 1 k ) H mCe C √ α (1 + M ) C √ β 1 1 -e 0 ( λ 0 ) E p ∀p, q ∈ N, q p.
Since λ k m, it follows from (11) in Proposition 1 and from (30) that

λ k (Y 1 k , F 1 k ) H mCe C √ α (1 + M ) C √ β E k ∀k ∈ N
and the result follows, using Step 4.

Step 6. The sequence (y k , f k ) k∈N defined by (21) converges to the element (y, f ) ∈ A given by

(y, f ) = (y 0 , f 0 ) - +∞ k=0 λ k (Y 1 k , F 1 k )
and the convergence is at least of order 1 + s after k 0 iterations (where k 0 is given by Step 1). Moreover, (y, f ) is a solution of (1) such that (y(•, T ), ∂ t y(•, T )) = (z 0 , z 1 ).

Indeed, by (21), we have (

y n , f n ) = (y 0 , f 0 ) - n-1 k=0 λ k (Y 1 k , F 1 k )
, hence (y k , f k ) converges to (y, f ) defined above. Let us prove that f is a null control for ȳ solution of [START_REF] Cannarsa | One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms[END_REF]. Using that (Y 1 k , F 1 k ) ∈ A 0 converges to zero as k → +∞, passing to the limit in ( 22), we infer that (y, f ) ∈ A solves

     ∂ tt y -∂ xx y + g(y) = f 1 ω in Q T , y = 0 on Σ T , (y(•, 0), ∂ t y(•, 0)) = (y 0 , y 1 ) in Ω. ( 34
)
Since (y, f ) ∈ A, we have (y(•, T ), ∂ t y(•, T )) = (z 0 , z 1 ) in Ω, i.e., f is a control for ȳ solution of (1). Now, for every k ∈ N, we have

(y, f ) -(y n , f n ) A0 = +∞ k=n λ k (Y 1 k , F 1 k ) H mC 1 -e 0 ( λ 0 ) e C √ α (1 + M ) C √ β E n (35)
The convergence to 0 with order greater than or equal to 1 + s after a finite number of iterations follows from Lemma 1.

Remark 10. The estimate ( 35) is a kind of coercivity property for the functional E. We emphasize, in view of the non-uniqueness of the zeros of E, that an estimate (similar to (35)) of the form (y, f ) -(y, f ) H C E(y, f ) does not hold for every (y, f ) ∈ A. We also insist on the fact that the sequence (y k , f k ) k∈N and its limit (y, f ) are uniquely determined by the initialization (y 0 , f 0 ) and by our selection criterion for the control F 1 .

Step 7.

If (2 + 1 s )C √ β < 1 whenever s ∈ (0, 1], and if 2 g ∞ C 2 e C
√ g ∞ < 1 whenever s = 0, then there exists M > 0 sufficiently large (depending on the initialization (y 0 , f 0 ), on α and on β) such that the a priori assumption (29) is indeed satisfied.

Let us summarize what we have done, under the growth condition (24) and under the a priori assumption (29). By (21), we have (y k , f k ) = (y 0 , f 0 ) -k-1 j=0 λ j (Y 1 j , F 1 j ), and then, using [START_REF] Fernández-Cara | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF] and Step 5, we get the a posteriori estimate

y k ∞ C (y k , f k ) H C (y 0 , f 0 ) H + mC 2 e C √ α (1 + M ) C √ β √ E 0 1 -e 0 ( λ 0 ) (36)
with the agreement that α = g ∞ and β = 0 if s = 0. Hence, to prove that the a priori assumption (29) is satisfied, it suffices to choose M > 0 large enough so that the right-hand side of (36) is less than or equal to M .

Recalling that c = [g ] s C 2+s e (1+s)C √ α (1 + M ) (1+s)C √ β for s ∈ (0, 1] and c = 2 g ∞ C 2 e C
√ g ∞ for s = 0, we infer from the proof of Step 1 (in particular, from (32)) that:

• For any s ∈ [0, 1], if (1 + s) c E s 2 0 < 1 (37) then e 0 ( λ 0 ) = cE s 2
0 and thus, by (36),

y k ∞ C (y 0 , f 0 ) H + C s 1+s m [g ] 1 1+s s c 1 1+s √ E 0 1 -cE s 2 0 . (38)
These estimates include the case s = 0 (with

[g ] 0 = 2 g ∞ ), in which (37) is exactly the smallness condition 2 g ∞ C 2 e C √ g ∞ < 1.
Here, we choose the minimal real number M > 0 such that

C (y 0 , f 0 ) H + C s 1+s m [g ] 1 1+s s c 1 1+s √ E 0 1 -cE s 2 0 M. (39)
This is possible by assuming that E 0 is sufficiently small, because then, there exist real numbers M (which cannot be arbitrarily large) satisfying both (37) and (39). This observation follows by inspecting both inequalities, either with M large, or with E 0 small. The above choice of M is implicit and unfortunately cannot be made explicit for any s. We can however give explicit expressions when s → 0, as follows. For s = 0, (37) is written as

c = 2 g ∞ C 2 e C √ g ∞ < 1
(smallness condition) and (39) gives

M |s=0 = C (y 0 , f 0 ) H + m C 2 e C √ g ∞ 1 -2 g ∞ C 2 e C √ g ∞ E 0 .
Now, when s → 0, s > 0, we must have β → 0, and then, taking equivalents, (37) gives

[g ] s C 2 e C √ α E s 2 0 < 1 while (39) gives M s 0 = C (y 0 , f 0 ) H + m C 2 e C √ α 1 -[g ] s C 2 e C √ α E 0
which is in accordance with the case s = 0. With this choice, (37) gives a smallness condition on the initialization when s > 0.

• For any s ∈ (0, 1], if (1 + s) c E s 2 0 1 (40)
(now s = 0 is excluded) then

1 1-e0( λ0) = (1+s) 1+ 1 s s c 1 s

√

E 0 and thus, by (36),

y k ∞ C (y 0 , f 0 ) H + C s 1+s m(1 + s) 1+ 1 s s[g ] 1 1+s s c 1+2s s(1+s) E 0 . (41)
Here, we choose M > 0 large enough such that

C (y 0 , f 0 ) H + C s 1+s m(1 + s) 1+ 1 s s[g ] 1 1+s s c 1+2s s(1+s) E 0 M. (42)
This is possible because there exist (large) real numbers M satisfying (42). Indeed, taking M large, (42) is of the kind Cst

M (2+ 1 s )C √ β E 0 M
which has solutions because, by assumption, (2

+ 1 s )C √ β < 1.
More precisely, here, we choose the minimal real number M > 0 such that

1 (1 + s)[g ] s C 2+s e (1+s)C √ α (1 + M ) (1+s)C √ β E s 2 0 , C (y 0 , f 0 ) H + m s (1 + s) 1+ 1 s [g ] 1 s s C 3+ 2 s e (2+ 1
s )C √ α (1 + M ) (2+ 1 s )C √ β E 0 M.
As before, the above choice of M is implicit. We can anyway give explicit formulas when s → 0. Indeed, when s 0, s > 0, we have then

β → 0, C √ β s < 1, and (40) gives [g ] s C 2 e C √ α M C √ β E s 2 0 1 while (42) gives M 1-C √ β s C (y 0 , f 0 ) H + me s [g ] 1 s s C 2 s e 1 s C √ α E 0
and therefore

M |s 0 = max    1 [g ] s C 2 e C √ α E s 2 0 1/C √ β , C (y 0 , f 0 ) H + me s [g ] 1 s s C 2 s e 1 s C √ α E 0 s/(s-C √ β)    .

Conclusion and further comments

Exact controllability of (1) has been established in [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF], under a growth condition on g, by means of a Leray-Schauder fixed point argument that is not constructive. In this paper, under a slightly stronger growth condition and under the additional assumption that g is uniformly Hölder continuous with exponent s ∈ [0, 1], we have designed an explicit algorithm and proved its convergence of a controlled solution of (1). Moreover, the convergence is super-linear of order greater than or equal to 1 + s after a finite number of iterations. In turn, our approach gives a new and constructive proof of the exact controllability of (1).

Several comments are in order.

Minimization functional. Among all possible admissible controlled pair (y, v) ∈ A 0 , we have selected the solution (Y 1 , F 1 ) of ( 9) that minimizes the functional J(v) = v 2 2,q T . This choice has led to the estimate [START_REF] Lemoine | Analysis of continuous H -1 -least-squares approaches for the steady Navier-Stokes system[END_REF] which is one of the key points of the convergence analysis. The analysis remains true when one considers the quadratic functional J(y, v) = w 1 v 2 2,q T + w 2 y 2 2 for some positive weight functions w 1 and w 2 (see for instance [START_REF] Cîndea | Numerical controllability of the wave equation through primal methods and Carleman estimates[END_REF]).

Newton method. Defining

F : A → L 2 (Q T ) by F (y, f ) := (∂ tt y -∂ xx y + g(y) -f 1 ω ), we have E(y, f ) = 1 2 F (y, f ) 2 L 2 (Q T )
and we observe that, for λ k = 1, the algorithm (21)-( 22) coincides with the Newton algorithm applied to F (see 6). This explains the super-linear convergence property obtained in Theorem 2, in particular the quadratic convergence when s = 1. Optimizing the parameter λ k gives a global convergence property of the algorithm and leads to the so-called damped Newton method applied to F . For this method, global convergence is usually achieved with linear order under general assumptions (see [START_REF] Deuflhard | Newton methods for nonlinear problems[END_REF]Theorem 8.7]). As far as we know, damped type Newton methods have been little applied to partial differential equations in the literature. We mention [START_REF] Lemoine | A fully space-time least-squares method for the unsteady Navier-Stokes system[END_REF][START_REF] Saramito | A damped Newton algorithm for computing viscoplastic fluid flows[END_REF] in the context of fluid mechanics.

Another variant. To simplify, let us take λ k = 1, as in the standard Newton method. Then, for each k ∈ N, the optimal pair (Y 1 k , F 1 k ) ∈ A 0 is such that the element (y k+1 , f k+1 ) minimizes over A the functional (z, v) → J(z -y k , v -f k ) with J(z, v) := v 2,q T (control of minimal L 2 (q T ) norm). Alternatively, we may select the pair (Y 1 k , F 1 k ) so that the element (y k+1 , f k+1 ) minimizes the functional (z, v) → J(z, v). This leads to the sequence (y k , f k ) k∈N defined by

     ∂ tt y k+1 -∂ xx y k+1 + g (y k )y k+1 = f k+1 1 ω + g (y k )y k -g(y k ) in Q T , y k = 0, on Σ T , (y k+1 (•, 0), ∂ t y k+1 (•, 0)) = (u 0 , u 1 ) in Ω. (43)
In this case, for every k ∈ N, (y k , f k ) is a controlled pair for a linearized wave equation, while, in the case of the algorithm (21)-( 22), (y k , f k ) is a sum of controlled pairs (Y 1 j , F 1 j ) for 0 j k. This formulation used in [START_REF] Fernández-Cara | Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods[END_REF] is different and the convergence analysis (at least in the least-squares setting) does not seem to be straightforward because the term g (y k )y k -g(y k ) is not easily bounded in terms of E(y k , f k ).

Local controllability when removing the growth condition (24). Let us remove the growth condition (24) on g . We have the following convergence result, under the assumption that E(y 0 , f 0 ) is small enough. Proposition 2. Assume that g ∈ C 1,s (R) for some s ∈ (0, 1]. There exists C([g ] s ) > 0 such that, if E(y 0 , f 0 )

C([g ] s ), then the sequence (y k , f k ) k∈N in A defined in (21) converges to (y, f ) ∈ A, where f is a null control for y solution of (1). Moreover, there exists k 0 ∈ N such that the sequence ( (y, f ) -(y k , f k ) H ) k k0 is decreasing and converges to 0 at least with order 1 + s.

The proof is a variant of the arguments given in this paper. We do not provide any details. In the case g(0) = 0, the smallness assumption on E(y 0 , f 0 ) is satisfied as soon as (u 0 , u 1 ) V is small. Therefore, the convergence result stated in Proposition 2 is equivalent to the local controllability property for (1). Proposition 2 can actually be seen as a consequence of the usual convergence of the Newton method: when E(y 0 , f 0 ) is small enough, i.e., when the initialization is close enough to the solution, then λ k = 1 for every k ∈ N and we recover the standard Newton method.

Multi-dimensional case. Let Ω is a bounded subset of R d , 1 d 3, and let ω be a nonempty open subset of Ω. Assume that the triple (Ω, ω, T ) satisfies the multiplier condition introduced in [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]. Then, we conjecture that Theorem 2 remains true in this context, strengthening the growth condition on g into |g (x)| α + β ln 1/2 (1 + |x|), for every x ∈ R and some β > 0 small enough. Establishing the result should require to use estimates of [11,[START_REF] Zhang | Explicit observability estimate for the wave equation with potential and its application[END_REF] (see also [START_REF] Lemoine | Approximation of nulls controls for semilinear heat equations using a least-squares approach[END_REF] for the case of a semilinear heat equations).

Boundary control. In this paper we have taken internal controls. Our approach may also be extended, with few modifications, to boundary controls considered in particular in [START_REF] Zuazua | Exact boundary controllability for the semilinear wave equation[END_REF]. We leave this issue open.

A Appendix: controllability results for the linearized wave equation

We recall in this appendix some a priori estimates for the linearized wave equation with potential in L ∞ (Q T ) and source term in L 2 (Q T ).

Lemma 1. Let A ∈ L ∞ (Q T ), let B ∈ L 2 (Q T ) and let (z 0 , z 1 ) ∈ V . Assume that T > 2 max( 1 , 1 -2 ).
There exists u ∈ L 2 (q T ) such that the solution of

     ∂ tt z -∂ xx z + Az = u1 ω + B in Q T , z = 0 on Σ T , (z(•, 0), ∂ t z(•, 0)) = (z 0 , z 1 ) in Ω, (44)
satisfies (z(•, T ), ∂ t z(•, T )) = (0, 0) in Ω. Moreover, the unique control u minimizing the L 2 (q T ) norm and its corresponding solution z satisfy

u 2,q T + (z, ∂ t z) L ∞ (0,T ;V ) C B 2 e (1+C) √ A ∞ + z 0 , z 1 V e C √ A ∞ (45)
for some constant C > 0 only depending on Ω and T .

Proof. The proof is based on estimates obtained in [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF]. The control of minimal L 2 (q T ) norm is given by u = ϕ1 ω where ϕ solves the adjoint equation

     ∂ tt ϕ -∂ xx ϕ + Aϕ = 0 in Q T , ϕ = 0 on Σ T , (ϕ(•, 0), ∂ t ϕ(•, 0)) = (ϕ 0 , ϕ 1 ) in Ω, (46)
where (ϕ 0 , ϕ 1 ) ∈ H := L 2 (Ω) × H -1 (Ω) is the unique minimizer of

J(ϕ 0 , ϕ 1 ) := 1 2 q T ϕ 2 + Q T Bϕ -(z 0 , z 1 ), (ϕ 0 , ϕ 1 ) V ,H with (z 0 , z 1 ), (ϕ 0 , ϕ 1 ) V ,H := z 0 , ϕ 1 H 1 0 (Ω),H -1 (Ω) -(z 1 , ϕ 0 ) L 2 (Ω),L 2 (Ω) .
In particular, the control v satisfies the optimality condition

q T ϕ ϕ + Q T Bϕ -(z 0 , z 1 ), (ϕ 0 , ϕ 1 ) V ,H = 0 ∀(ϕ 0 , ϕ 1 ) ∈ H from which we deduce that u 2 2,q T B 2 ϕ 2 + (z 0 , z 1 ) V (ϕ 0 , ϕ 1 ) H . From [18, Lemma 2], we get (ϕ, ∂ t ϕ) 2 L ∞ (0,T ;H) B 1 (ϕ 0 , ϕ 1 ) 2 H (1 + A 2 ∞ )e B2 √ A ∞
for some constants B 1 , B 2 > 0, and it follows that ϕ [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF]Theorem 4], there exists C > 0 such that (ϕ 0 , ϕ 1 ) 2

2 2 T B 1 (ϕ 0 , ϕ 1 ) 2 H (1 + A 2 ∞ )e B2 √ A ∞ . More- over, from

H

Ce C √ A ∞ ϕ 2 2,q T . Combining these inequalities, we get

u L 2 (q T ) B 2 √ T B 1 (1 + A 2 ∞ ) 1/2 e B 2 2
√ A ∞ + (z 0 , z 1 ) V √ Ce C 2 √ A ∞ .
Using the inequality (1 + s 2 ) 1/2 e √ s for every s 0, we get the result. Then, from [18, Lemma 1], we have (z,

∂ t z) 2 L ∞ (0,T ;V ) D 1 (z 0 , z 1 ) 2 H (1 + A ∞ ) + u1 ω + B 2 2 e D2 √ A ∞
for some constants D 1 , D 2 > 0, and we infer that

(z, ∂ t z) 2 L ∞ (0,T ;V ) D 1 (z 0 , z 1 ) 2 H (2 + A ∞ ) + 2 B 2 2 1 + T B 1 1 + A 2 ∞ e B2 √ A ∞ e D2 √ A ∞ .
Using that (1 + s) 1/2 e √ s and (1 + s 2 ) e 2 √ s for every s 0, we get the estimate.

We next discuss some properties of the operator K :

L ∞ (Q T ) → L ∞ (Q T ) defined by K(ξ) = y ξ , a null controlled solution of the linear boundary value problem (3) with the control f ξ of minimal L 2 (q T ) norm. Lemma 1 with B = -g(0) gives (y ξ , ∂ t y ξ ) L ∞ (0,T ;V ) C u 0 , u 1 V + g(0) 2 e (1+C) √ g(ξ) ∞ e C √ g(ξ) ∞ . (47)
As in [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF], the growth condition (24) implies that there exists d > 0 such that g(y

) ∞ d + β ln 2 (1 + y ∞ ) for every y ∈ L ∞ (Q T ), and it follows that e C √ ĝ(ξ) ∞ e C √ d (1 + ξ ∞ ) C √ β . Using (47), we infer that y ξ ∞ C u 0 , u 1 V + g(0) 2 e (1+2C) √ d (1 + ξ ∞ ) (1+2C) √ β .
Taking β small enough so that (1+2C) √ β < 1, we conclude that there exists M > 0 such that ξ ∞ M implies K(ξ) ∞ M . This is the argument of [START_REF]Exact controllability for semilinear wave equations in one space dimension[END_REF]. Note that, in contrast to β, M depends on u 0 , u 1 V (and increases with u 0 , u 1 V ).

The following result gives an estimate of the difference of two controlled solutions.

Lemma 2. Let a, A ∈ L ∞ (Q T ) and let B ∈ L 2 (Q T ). Let u and v be the null controls of minimal L 2 (q T ) norm for y and z respectively solutions of (49)

Then,

y -z L ∞ (Q T ) C 4 a ∞ e C √ A+a ∞ e (2+3C) √ A ∞ B 2 e (1+C) √ A ∞ + u 0 , u 1 V
for some constant C > 0 only depending on Ω and T .

Proof. The controls of minimal L 2 norm for y and z are given by u = ϕ1 ω and v = ϕ a 1 ω , where ϕ and ϕ a respectively solve the adjoint equations Lemma 1 implies that Ψ 2,q T + (Z, ∂ t Z) L ∞ (0,T ;V ) C ψ1 ω -ay 2 e (1+2C)

√ A ∞ .
Moreover, energy estimates applied to ψ give ψ L 2 (q T ) C a ∞ ϕ 2 e C √ A+a ∞ and

ϕ 2 C ϕ 0 , ϕ 1 H e (1+C) √ A ∞ Ce (1+C) √ A ∞ 2
u 2,q T using that ϕ 0 , ϕ 1 H Ce C √ A ∞ u 2,q T so that

ψ L 2 (q T ) C a ∞ e C √ A+a ∞ Ce (1+C) √ A ∞ 2
u 2,q T from which we deduce that

Z L ∞ (Q T ) C ψ1 ω + a L ∞ (Q T ) y 2 e (1+2C) √ A ∞ C a ∞ e C √ A+a ∞ Ce (1+C) √ A ∞ 2 u L 2 (q T ) + y L 2 (Q T ) C a ∞ e C √ A+a ∞ Ce (1+C) √ A ∞ 2 + 1 + C B 2 e (1+C) √ A ∞ + u 0 , u 1 V e C √ A ∞
leading to the result.

This result allows to establish the following property for the operator K.

Lemma 3. Under the assumptions done in Theorem 1, let M = M ( u 0 , u 1 V , β) be such that K maps B ∞ (0, M ) into itself and assume that ĝ ∈ L ∞ (0, M ). For any ξ i ∈ B ∞ (0, M ), i = 1, 2, there exists c(M ) > 0 such that K(ξ 2 ) -K(ξ 1 ) ∞ c(M ) ĝ L ∞ (0,M ) ξ 2 -ξ 1 ∞ .

Proof. For any ξ i ∈ B ∞ (0, M ), i = 1, 2, let y ξ i = K(ξ i ) be the null controlled solution of

    
∂ tt y ξ i -∂ xx y ξ i + y ξ i g(ξ i ) = -g(0) + f ξ i 1 ω in Q T , y i ξ = 0 on Σ T , (y ξ i (•, 0), ∂ t y ξ i (•, 0)) = (u 0 , u 1 ) in Ω, with the control f ξ i 1 ω of minimal L 2 (q T ) norm. We observe that y ξ 2 is solution of

    
∂ tt y ξ 2 -∂ xx y ξ 2 + y ξ 2 g(ξ 1 ) + y ξ 2 ( g(ξ 2 ) -g(ξ 1 )) = -g(0) + f ξ 2 1 ω in Q T , y ξ 2 = 0 on Σ T , (y ξ 2 (•, 0), ∂ t y ξ 2 (•, 0)) = (u 0 , u 1 ) in Ω.

It follows from Lemma 2 applied with B = -g(0), A = ĝ(ξ 1 ), a = ĝ(ξ 2 ) -ĝ(ξ 1 ), that

y ξ 2 -y ξ 1 ∞ A(ξ 1 , ξ 2 ) g(ξ 2 ) -g(ξ 1 ) ∞ ( 51
)
where the positive constant

A(ξ 1 , ξ 2 ) := C 2 e C √ ĝ(ξ 2 ) ∞ Ce (1+C) √ ĝ(ξ 1 ) ∞ 2 g(0) 2 e (1+C) √ ĝ(ξ 1 ) ∞ + u 0 , u 1 V e C √ ĝ(ξ 1 ) ∞
is bounded by some c(M ) > 0 for every ξ i ∈ B ∞ (0, M ). The result follows from (51).

∂∂

  tt y -∂ xx y + Ay = u1 ω + B in Q T , y = 0 on Σ T , (y(•, 0), ∂ t y(•, 0)) = (u 0 , u 1 ) tt z -∂ xx z + (A + a)z = v1 ω + B in Q T , z = 0 on Σ T , (z(•, 0), ∂ t z(•, 0)) = (u 0 , u 1 )in Ω.

∂∂∂∂∂∂∂

  tt ϕ -∂ xx ϕ + Aϕ = 0 in Q T , ϕ = 0 on Σ T , (ϕ(•, 0), ∂ t ϕ(•, 0)) = (ϕ 0 , ϕ 1 ) tt ϕ a -∂ xx ϕ a + (A + a)ϕ a = 0 in Q T , ϕ = 0 on Σ T , (ϕ(•, 0), ∂ t ϕ(•, 0)) = (ϕ a,0 , ϕ a,1 )in Ω, for some appropriate (ϕ 0 , ϕ 1 ), (ϕ a,0 , ϕ a,1 ) ∈ H.Hence Z := z -y solves tt Z -∂ xx Z + (A + a)Z = Φ1 ω -ay in Q T , Z = 0 on Σ T , (Z(•, 0), ∂ t z(•, 0)) = (0tt Φ -∂ xx Φ + (A + a)Φ = -aϕ in Q T , Φ = 0 on Σ T , (Φ(•, 0), ∂ t Φ(•, 0)) = (ϕ a,0 -ϕ 0 , ϕ a,1 -ϕ 1 )in Ω.We decompose Φ = Ψ + ψ where Ψ and ψ solve respectively tt Ψ -∂ xx Ψ + (A + a)Ψ = 0 in Q T , Ψ = 0 on Σ T , (Ψ(•, 0), ∂ t Ψ(•, 0)) = (ϕ a,0 -ϕ 0 , ϕ a,1 -ϕ 1 ) tt ψ -∂ xx ψ + (A + a)ψ = -aϕ in Q T , ψ = 0 on Σ T , ψ(•, 0), ∂ t ψ(•, 0)) = (0, 0) in Ω,and we deduce that Ψ1 ω is the control of minimal L 2 norm for Z solution of tt Z -∂ xx Z + (A + a)Z = Ψ1 ω + ψ1 ω -ay in Q T , Z = 0 on Σ T , (Z(•, 0), ∂ t Z(•, 0)) = (0, 0) in Ω.

  < 1, i.e., we remain in this first case, and since E k+1 c 2 E 1+s k

	iteration, c k ∈ N, we have (1 + s) 1 s √ E k < 1 for every k ∈ N and the sequence (c 1 s c 1 s √ E k	1 s	√	1 and, using (33), by E k ) k∈N is decreasing. Hence, for every

Acknowlegment. The first author warmly thanks Jérome Lemoine (Laboratoire Mathématique Blaise Pascal, Clermont Auvergne University) for fruitful discussions.