Hisashi Oishi

Sayaka Mashima

Yutaka Kuwahara

Makoto Takafuji

Kyohei Yoshida

Reiko Oda

Hongdeng Qiu

Hirotaka Ihara
email: ihara@kumamoto-u.ac.jp

Polymer encapsulation and stabilization of molecular gel-based chiroptical information for strong, tunable circularly polarized luminescence film †

We demonstrate a strategic approach for a completely organic, chiroptical polymer film, exhibiting strong (g lum = 0.01-0.05) and broad-band circularly polarized luminescence that can be realized by the stabilization of self-assembly-driven secondary chirality through polymer encapsulation.

Circularly polarized luminescence (CPL) has attracted attention as a next-generation light signal because it carries more information due to its chirality and no angular effect compared to normal and linearly polarized lights. A simple example can be seen in its use as a communication tool by some animals. 1 One technical approach for producing CPL is by converting polarized light from normal to linear using a quarter-wave plate. However, this approach can suffer from serious reductions in light energy with removal of light. The other solution for CPL generation is the use of chiral luminescent compounds that are totally organic. [2][3][4][5][6][7][START_REF] Bosson | [END_REF] Studies using this approach have realized stronger CPL by precise molecular design, skilled organic synthesis, and careful purification. Therefore, gram-scale synthesis has not yet been achieved without few exceptions. [START_REF] Bosson | [END_REF]9 In this work, we describe the fabrication of chiroptical polymer films that exhibit strong CPL and anticipate their possible applications in bio-sensing, chiroptical memory, detector system for security devices, and storage units. 4,10,11 For such purposes, gram-scale synthesis becomes increasingly more important.

We selected a binary system for CPL generation, 6,7 which has a distinct advantage on larger scale. Instead of using specially designed chiral luminescent materials, commercially available, non-chiral luminescent dyes can be used and chirality can be induced by chiral templates. Therefore, the essential problem becomes, rather than the scalability of synthesis and fabrication, the retention of chirality induction in the polymer system. In this study, we demonstrate that strong CPL-exhibiting polymer films can be fabricated by incorporating into the polymer a chirally self-assembled CPL-inducing system consisting of a nonchiral luminescent dye and associated chiral template.

We applied several cyanine dyes including anionic NK-2012 and cationic NK-77 as non-chiral luminescent species for CPL generation. Because these cyanine dyes are very sensitive to the chiral microenvironment, they function as good indicators for chirality evaluation. However, cyanine dyes are often unstable, especially in oxygen atmosphere under sunlight, and therefore, these systems are not suitable for our present purpose. Fig. 1 presents several commercially available fluorescent dyes as new chirality-induction candidates. To induce chirality in these non-chiral dyes, G-ca, having a carboxy head group, was chosen as a self-assembling organogelator (Fig. 1). The advantages of G-ca can be summarized in terms of (1) its good solubility in various organic solvents, which is very important because most of a bulk polymer system is hydrophobic and used in organic media; (2) the enhanced chirality due to the formation of secondary chirality through chiral arrangement;

(3) its ionizability to promote electrostatic interactions with the cationic dyes shown in Fig. 1; and ( 4) its possible gram-scale synthesis via a one-step coupling reaction between L-glutamide 12 and glutalic anhydride.

Mixed systems of G-ca and dyes were prepared by the following procedure: a 0.5 mM solution of G-ca in toluene was prepared by heating at 85 1C; cooling to 25 1C produced a clear solution. If the concentration was higher than a few millimolar in toluene, the solution became a clear gel. This phenomenon has been recognized as molecular gelation. 7,13,14 Fig. 2a shows the TEM image of the 0.5 mM G-ca solution after staining with uranyl acetate. G-ca forms nano-fibrillar aggregates whose smallest diameter is within 50 nm in the photo. This indicates that the gelation behavior is the result of entanglement among the well-developed nano-fibrillar aggregates of G-ca. Next, a methanolic solution of the dye was mixed in a given molar ratio with the G-ca solution. No significant change is observed by TEM, but confocal microscopy provides fibrillar fluorescence images, as shown in Fig. 2b. Since G-ca does not itself fluoresce, the obtained confocal image indicates that the dye is combined with G-ca and the fibrillar aggregation of G-ca is maintained after mixing with the dye.

The chirality of the G-ca/dye mixed systems was investigated by CD spectroscopy. As shown in Fig. 3a, the G-ca/NK-77 system exhibits a new exciton coupling ([y] 573 = 5 Â 10 4 deg cm dmol À1 ) near the absorption band (l max = 570 nm) of NK-77. Since NK-77 is non-chiral, the observed CD can be attributed to induced chirality from the G-ca aggregates. Although this induced CD intensity is not so small, while we have confirmed much larger values (41 Â 10 6 deg cm dmol À1 ) 15 in the case of an anionic cyanine dye, NK-2012, on poly(L-lysine). Therefore, we examined the effect of additives on the G-ca/dye system to promote their interaction.

When an organic amine such as pyridine or triethylamine (TEA) is added to the G-ca/dye system, a distinct increase in CD intensity is observed. Fig. 3b shows the effect of TEA on the CD spectrum. The peak intensity is amplified ca.20 times with the addition of ten equimolar of G-ca. This is probably due to the promotion of electrostatic interactions between the chiral G-ca and non-chiral NK-77 by neutralization of the iodide counter ion of NK-77. Therefore, in this study, TEA was applied in the promotion of induced CD for the other non-chiral dyes.

Polystyrene (PS) was selected as the bulk polymer for film fabrication. The typical fabrication procedure is as follows: a 5 wt% solution of PS was combined with G-ca in hot toluene, and then the methanolic dye solution was added to the mixture. Part of the mixture (400 mL) was drop-cast onto a glass plate (12 mm  25 mm) and dried in air. The thickness of the resultant polymer film was adjusted to be approximately 60 mm. The main composition rate was adjusted to be 2.0 wt% in the G-ca-in-polymer, and to be 1 : 20 in the molar ratio of dye and G-ca. Fig. 2d shows the confocal microscopy images of the PS film containing NK-77 with G-ca. The fibrillar fluorescence image indicates that not only does G-ca self-assemble in the polymer, but also that NK-77 is associated with the G-ca fibrils. Similar fluorescence images (Fig. S1,ESI †) were observed with the other dyes, including acriflavine (AC, l max = 530 nm) and Basic Red-5 (BR-5, l max = 591 nm). Notably, all the PS films are transparent, which means that the diameters of the aggregates are sufficiently small to escape the problem of undesirable light scattering. This is an essential and important requirement for optical applications, and therefore, we conclude that our strategic approach is unquestionably suitable for our research purposes.

Fig. 4 shows the CD spectra of the PS films containing the G-ca/dye systems. Large CD intensities are observed near the absorption bands of the dyes in all films. Since no CDs are observed in the absence of G-ca and the CD intensity is Our CPL generation system has multiple advantages. First, mixed-dye systems can be employed to provide broad-band CPL over a wide range of emission bands. For example, when AC and BR-5 are mixed, the resultant PS film provides two CD signals (Fig. S3, ESI †). Second, broad-band CPL can also be generated by adjusting the chiral microenvironment. For example, when the mixing ratio of AC to G-ca is adjusted, the luminescent peak width increases as shown in Fig. 5: that is, the Stokes shift can be made tunable by promoting the heterogeneity of chiral perturbation. Finally, by encapsulating G-ca in the polymer as well as selecting the proper dye, the CPL phenomenon can be remarkably stabilized. For example, we confirmed that the G-ca/AC polymer system maintained strong CPL over at least a month.

Conclusions

In conclusion, we have established a strong CPL (g lum E 0.05) generation system in a polymer thin film. Our system is advantageous because no chirality is needed in the fluorescent component, meaning that it is free from precise molecular design and complicated synthetic limitations, and therefore possible for industrial-scale application. In addition, the emission bands can be readily tuned by adjusting the molecular orientation of the chiral template system as well as the dye selection. These advantages could expand possible system applications, such as in optical uses (e.g., chiroptical detector for bio-signal sensing), in agriculture, or in solar cells as a reduced-angular wavelength converter.

Fig. 1

 1 Fig. 1 Chemical structures of non-chiral fluorescent compounds and organogelator G-ca used in this study.

Fig. 2

 2 Fig. 2 TEM and confocal micrographs of G-ca aggregates. (a) Initial concentration of G-ca: 0.1 mM in toluene. The dried sample was obtained by drop-casting on Cu mesh, stained with uranyl acetate and dried at room temperature. The confocal microscopy for the solution state (b) and the film (d) were observed with excitation wavelength at 488 nm. (b) G-ca: 0.5 mM and NK-77: 0.01 mM in toluene. (c) and (d) G-ca: 2 wt% in polystyrene. G-ca : NK-77 = 20 : 1.

Fig. 3

 3 Fig. 3 (a) CD spectra of the G-ca with NK-77 in toluene. [G-ca] = 0.5 mM; [NK-77] = 0.01 mM; [TEA] or [pyridine] = 5 mM. (b) Effect of TEA additive in the G-ca/dye system on the CD intensity.

Acknowledgements

This work was supported by the French-Japanese International Associated Laboratory, Chiral Nanostructures for Photonic Applications (LIA-CNPA), and a Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research and the Bilateral Joint Research project.

Conflicts of interest

There are no conflicts to declare.

Notes and references


Thibault Liard
email: thibault.liard@ls2n.fr

Ismaïla Balogoun
email: ismaila.balogoun@ls2n.fr

Swann Marx
email: swann.marx@ls2n.fr

Franck Plestan
email: franck.plestan@ec-nantes.fr

Boundary sliding mode control of a system of linear hyperbolic equations: a Lyapunov approach

Keywords: Sliding mode control, hyperbolic equations, robust control, stabilization of PDEs, nonlinear systems

This paper introduces a new sliding mode methodology for a system of two hyperbolic equations. The main difference with the existing literature is the definition of the sliding variable, given here by the gradient of a Lyapunov functional. We state and prove an existence theorem and a global asymptotic stability result. The efficiency of our feedback-law is illustrated by some numerical simulations relying on implicit schemes.

Introduction

This paper is concerned with the generalization of the sliding mode control (for short SMC) strategy for a class of partial differential equations (for short PDEs) subject to an unknown boundary disturbance. To be more precise, our focus will be on a system of two hyperbolic systems controlled from the boundary (see [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF] for a review on this class of system).

Stabilization of systems of linear transport equations of one dimension have been considered for many years. This is surely because this kind of systems models many physical phenomena. If [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF] provides a good overview of the actual research lines concerning this topic of stabilization, it is worth mentioning some important articles in that direction. In [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF], the local stabilization of a 2 × 2 nonlinear hyperbolic system is considered, where some distributed couplings are allowed. The feedback-law is built thanks to the backstepping method (see e.g., [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF] for an overview on this topic). In [START_REF] Bribiesca-Argomedo | Backsteppingforwarding control and observation for hyperbolic PDEs with Fredholm integrals[END_REF], hyperbolic systems with some non-local terms are studied. More precisely, the design method relies on a Fredholm transform instead of a more usual Volterra one. Generalizations to the case of systems of n linear hyperbolic equations have been considered in [START_REF] Di Meglio | Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input[END_REF] or [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF], to cite only few of the numerous papers dealing with such a topic. Let us also mention some recent papers about optimal time of stabilization [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic PDEs[END_REF][START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF] or about delay-robust stabilization [START_REF] Auriol | Delay-robust control design for two heterodirectional linear coupled hyperbolic PDEs[END_REF][START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF].

For many years, the SMC strategy has been proved to be efficient for the robust control of nonlinear systems described with ordinary differential equations (for short ODEs). Most of the classical techniques for ODEs are exposed in [START_REF] Shtessel | Sliding mode control and observation[END_REF][START_REF] Utkin | Sliding modes in control and optimization[END_REF]. Roughly speaking, this technique is decomposed into two steps: firstly, one selects a sliding variable such that the trajectories reach a sliding surface, where the global asymptotic stability is ensured; secondly, one designs a state feedback-law so that the sliding variable reaches the surface in finite-time on which the disturbance is rejected.

To the best of our knowledge, robust control for PDEs has been investigated for only few years. Most of the existing papers (see e.g., [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of pde systems[END_REF] or [START_REF] Lhachemi | Pi regulation of a reaction-diffusion equation with delayed boundary control[END_REF]) use PI controllers. In contrast with the sliding mode strategy, this imposes to consider disturbances which do not depend on time. In [START_REF] Deutscher | Finite-time output regulation for linear 2× 2 hyperbolic systems using backstepping[END_REF][START_REF] Deutscher | Output regulation for general linear heterodirectional hyperbolic systems with spatially-varying coefficients[END_REF], a feedback-law constructed using the backstepping method is proposed to reject periodic disturbances. This design, based on the internal model approach, allows to reject disturbances which might not match with the control, at the price of firstly assuming the disturbances to be periodic, and of secondly knowing the related frequencies. It is in contrast with the sliding mode approach, which only requires to know the bounds of the perturbations. However, the sliding mode approach needs the disturbances to match with the control.

The generalization of the SMC procedure to the PDE case is not new. However, many questions remain open. We believe that the main challenge is the choice of the sliding variable. Indeed, since one faces with an infinitedimensional system, there exist many ways to select it. In [START_REF] Guo | Sliding mode and active disturbance rejection control to stabilization of one-dimensional antistable wave equations subject to disturbance in boundary input[END_REF], [START_REF] Tang | Sliding mode control to the stabilization of a linear 2× 2 hyperbolic system with boundary input disturbance[END_REF] [START_REF] Pisano | Combined backstepping/second-order sliding-mode boundary stabilization of an unstable reaction-diffusion process[END_REF], a backstepping strategy is used in order to design a sliding mode controller. Note that an active disturbance rejection control 1 (ADRC for short) strategy is also followed in [START_REF] Guo | Sliding mode and active disturbance rejection control to stabilization of one-dimensional antistable wave equations subject to disturbance in boundary input[END_REF] and [START_REF] Feng | A new active disturbance rejection control to output feedback stabilization for a onedimensional anti-stable wave equation with disturbance[END_REF], which is a different approach from the SMC one. In [START_REF] Tang | Sliding mode control to the stabilization of a linear 2× 2 hyperbolic system with boundary input disturbance[END_REF], the use of the backstepping method imposes to measure the full-state, and therefore the boundaries, where are located the control and the disturbance. Let us also mention [START_REF] Baccoli | Boundary control of coupled reaction-diffusion processes with constant parameters[END_REF][START_REF] Barbu | Sliding mode control for a nonlinear phase-field system[END_REF][START_REF] Orlov | Exponential stabilization of the uncertain wave equation via distributed dynamic input extension[END_REF], where a distributed SMC is considered. In the latter case, the sliding variable is the full state itself.

In this paper, a different strategy is followed. The sliding variable is derived from the gradient of some well-known Lyapunov functional in the hyperbolic context [9, Section 2.1.2]. We thus assume that we measure this quantity, which does not imply the knowledge of the boundaries where are located the control and the disturbance. Moreover, in contrast with the usual backstepping strategy, the bound on the derivative of the disturbance does not need to be known. Furthermore, the global asymptotic analysis is provided in some L p space, which is not common in the literature. Indeed, most of the existing works give their stability results in the Hilbertian space L 2 . Let us mention however some recent works in that direction: [START_REF] Chitour | L p -asymptotic stability analysis of a 1d wave equation with a nonlinear damping[END_REF], [START_REF] Amadori | Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain[END_REF] or [START_REF] Chitour | One-dimensional wave equation with set-valued boundary damping: well-posedness, asymptotic stability, and decay rates[END_REF]. We also emphasize on the fact that our approach is adjustable to more general SMC methods such as adaptive approaches. Adaptive techniques [START_REF] Plestan | New methodologies for adaptive sliding mode control[END_REF][START_REF] Zhang | Adaptive robust control of floating offshore wind turbine based on sliding mode[END_REF] are interesting, due to the fact that reduced knowledge of the model is required (bounds of perturbations are not required). To that end, we illustrate numerically the efficiency of the adaptive SMC taken from [START_REF] Plestan | New methodologies for adaptive sliding mode control[END_REF] in our context.

It is also important to note that our approach could be extended to more general systems. Indeed, as soon as one has a Lyapunov function, the sliding variable and its related sliding control can be derived from it in a systematic way. As a consequence, one may consider the case of n × n linear hyperbolic equations with space varying velocities [9, Section 2.3] or distributed coupling terms as in [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF]. In the latter ones, the length of the domain has to be sufficiently small as explained in [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]Section 5.6]. Systems of nonlinear hyperbolic equations may be also considered, but the results would become local. However, in 1 ADRC method proposes to build an observer allowing the estimation of the disturbance. This estimation is then added in the feedback-law and allows to reject the perturbation. An interesting aspect of this strategy relies on the fact that the disturbance does not need to be matched with the control, i.e. it does not need to be located at the same place than the control.

order to avoid any further technicalities, and in order to ease the reading of our paper, we chose to focus on the case of a system of 2 × 2 linear hyperbolic equations. This paper is organized as follows. Section 2 presents the system of linear hyperbolic equations, the sliding mode method and introduces the notion of solutions that will be used all along the paper. Section 3 gathers the main results of the paper, namely an existence theorem and a global asymptotic result. Section 4 is devoted to the proof of the main theorems. Section 5 illustrates via numerical simulations the efficiency of our sliding mode control. Finally, Section 6 collects some remarks and introduces some future research lines to be followed.

Notation. The set of non-negative real numbers is denoted in this paper by R + . When a function f only depends on the time variable t (resp. on the space variable x), its derivative is denoted by ḟ (resp. f ). Given any subset of R denoted by Ω (R + or an interval, for instance), L p (Ω; R n ) denotes the set of (Lebesgue) measurable functions f 1 , . . ., f n such that, for i = {1, . . . , n}, Ω |f i (x)| p dx < +∞ when p = +∞ and such that sup ess x∈Ω |f i (x)| < +∞ when p = +∞. The associated norms are, for

p = +∞, (f 1 , . . . , f n ) p L p (Ω;R n ) := Ω |f 1 (x)| p dx + . . . + Ω |f n (x)| p dx and, for p = +∞, f 1 , . . . f n L ∞ (Ω;R n ) := sup ess x∈[0,L] |f 1 (x)| + . . . + |f n (x)|. For any p ∈ [1, ∞], the Sobolev space W 1,p (Ω; R n ) is defined by the set {(f 1 , . . . , f n ) ∈ L p (Ω; R n ) | (f 1 , . . . , f n ) ∈ L p (Ω; R n )}.
When n = 1, we simplify the notation and use L p (Ω) or W 1,p (Ω). Given an interval Ω of R, the set C ∞ c (Ω) denotes the set of infinitely differentiable functions with compact support. All along the paper, we will say a.e. to denote properties satisfied almost everywhere, i.e. properties satisfied everywhere except in a set of Lebesgue measure equal to 0.

Acknowledgement: We would like to thank Vincent Andrieu, Yacine Chitour and Vincent Perrollaz for fruitful discussions, which improve significantly the results of the paper.

Problem statement

Boundary linear Hyperbolic System

Let λ 1 > 0, λ 2 > 0 and k 2 ∈ IR. We consider the following linear hyperbolic system

           ∂ t R 1 (t, x) + λ 1 ∂ x R 1 (t, x) = 0, ∂ t R 2 (t, x) -λ 2 ∂ x R 2 (t, x) = 0, R 1 (t, 0) = u(t) + d(t), R 2 (t, L) = k 2 R 1 (t, L), R 1 (0, x) = R 0 1 (x), R 2 (0, x) = R 0 2 (x), (1)
where d(•) ∈ L ∞ (IR) is an unknown disturbance and u is a control function. When the system (1) is undisturbed (d = 0), it is nowadays well-known that the control function u(t) := k 1 R 2 (t, 0), allows to stabilize the system when |k 1 k 2 | < 1, see [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]Theorem 2.4.]. The proof, for the L 2 -topology, relies on the Lyapunov functional

V 2 (t) = L 0 p 1 λ 1 R 2 1 (t, x)e -νx λ 1 + p 2 λ 2 R 2 2 (t, x)e νx λ 2
dx.

(2) As classical in the SMC literature, our goal is to find a state feedback control u which allows to reject the disturbance and to globally asymptotically stabilize the system around the equilibrium point (0, 0) in the functional space L p (0, L; R 2 ). As mentioned in the introduction, one seeks a sliding surface (i.e., a subspace of the state space) where some desired global asymptotic stability properties are satisfied.

In our case, our aim is to find a sliding surface on which (1) becomes

       ∂ t R 1 (t, x) + λ 1 ∂ x R 1 (t, x) = 0, ∂ t R 2 (t, x) -λ 2 ∂ x R 2 (t, x) = 0, R 1 (t, 0) = k 1 R 2 (t, 0), R 2 (t, L) = k 2 R 1 (t, L), (3)
in finite time t 0 > 0 with k 1 chosen such that |k 1 k 2 | < 1. We know from [9, Theorem 2.1 and 2.2] that (0, 0) is exponentially stable for (3). The next section will provide a definition of this sliding surface (and its related sliding variable) together with the sliding mode controller.

Remark 1 (Generalization to n × n systems) Our approach could be extended to systems in the form

∂ t R(t, x) + Λ∂ x R(t, x) = 0, (t, x) ∈ R + × [0, L], (4)
where R(t, x) ∈ R n and Λ is a diagonal matrix defined by Λ := diag(λ 1 , . . . , λ m , -λ m+1 , . . . , -λ n ) with λ i > 0 for any i ∈ {1, . . . , n}. Associated to these velocities, one defines

R + (t, x) :=      R 1 (t, x) . . . R m (t, x)      , R -(t, x) :=      R m+1 (t, x) . . . R n (t, x)      (5)
In this case, one may consider

R + (t, 0) = B(u(t)+d(t)), R -(t, L) = K 2 R + (t, L) (6)
with u and d scalar valued functions, and B ∈ R m×1 and K 2 ∈ R (n-m)×m . As in (1), u denotes the control and d the disturbance.

On the sliding surface that one aims at reaching, the system becomes

         ∂ t R(t, x) + Λ∂ x R(t, x) = 0, (t, x) ∈ R + × [0, L], R + (t, 0) = BK 1 R -(t, 0), t ∈ R + , R -(t, L) = K 2 R + (t, L), t ∈ R + , R(0, x) = R 0 (x), (7)
with K 1 ∈ IR 1×(n-m) such that the pair (BK 1 , K 2 ) satisfies the condition given in [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]Theorem 3.2.]. This imposes to assume some stabilizability condition on the pair (K 1 , B).

Sliding surface

Let p ∈ [1, ∞].
We introduce the following sliding surface Σ defined as follows. For any t > 0

Σ := (R 1 (t, •), R 2 (t, •)) ∈ L p (0, L) 2 | S(t) = 0 , with S : IR + → IR defined by S(t) := L 0 p 1 λ 1 R 1 (t, x)e -νx λ 1 + p 2 λ 2 R 2 (t, x)e νx λ 2
dx.

(8) We will sometimes write S(t) by S(R 1 (t, •), R 2 (t, •)) to lighten the statements. This function represents the sliding variable mentioned earlier. As explained in [START_REF] Orlov | Robust control of infinite-dimensional systems via sliding modes[END_REF], this sliding variable represents the gradient of the Lyapunov function [START_REF] Acary | Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF]. We consider the set-valued sliding mode controller u defined by, for a.e t 0,

u(t) = k 1 R 2 (t, 0) -Ksign(S(t)), (9)
where k 1 is defined later, S is introduced in (8), the gain K > d L ∞ (R+) and the set-valued function sign is defined by

sign(z) =        -1 if z < 0, [-1, 1] if z = 0, 1 if z > 0.
From (9), the system (1) is rewritten as

           ∂ t R 1 (t, x) + λ 1 ∂ x R 1 (t, x) = 0, ∂ t R 2 (t, x) -λ 2 ∂ x R 2 (t, x) = 0, R 1 (t, 0) = k 1 R 2 (t, 0) -Ksign(S(t)) + d(t), R 2 (t, L) = k 2 R 1 (t, L), R 1 (0, x) = R 0 1 (x), R 2 (0, x) = R 0 2 (x). ( 10
)
For a.e t 0, we can deduce, after some formal integration by parts, that

Ṡ(t) = -νS(t) -[p 1 e -νL λ 1 -p 2 k 2 e νL λ 2 ]R 1 (t, L). +[p 1 k 1 -p 2 ]R 2 (t, 0) -Kp 1 sign(S(t)) + p 1 d(t). One picks p 2 = k 1 = e -ν L λ 1 + L λ 2 k2
and p 1 = 1. Then, we deduce that

Ṡ(t) = -νS(t) -Ksign(S(t)) + d(t).
Hence, since K > d L ∞ (R+) , for all t sufficiently large, the solution (R 1 , R 2 ) belongs to the sliding surface Σ and S(t) = Ṡ(t) = 0. This implies that the latter equation reduces to be Ksign(S(t))

+ d(t) = 0. Going back to R 1 (t, 0) in (1), one therefore obtains that R 1 (t, 0) = u(t) + d(t) = k 1 R 2 (t, 0)
. This shows (formally) that, on the sliding surface Σ, ( 1) is equal to (3).

Remark 2 (Space-varying velocities) When the velocities λ 1 and λ 2 are space-varying, we may consider the following sliding variable defined by

S(t) := L 0 (q 1 (x)R 1 (t, x) + q 2 (x)R 2 (t, x)) dx, with q 1 (x) = p 1 λ 1 (x) e - x 0 µ λ 1 (y) dy , q 2 (x) = p 2 λ 2 (x) e x 0 µ λ 2 (y) dy .
The positive constants p 1 and p 2 are chosen in an appropriate way.

Remark 3 (SMC for n × n systems (4)) At the light of Remark 1, the sliding variable would become for the system (4):

S(t) := m i=1 p i λ i L 0 e -µ λ i x R i (t, x)dx + n i=m+1 p i λ i L 0 e µ λ i R i (t, x)dx,
with the positive constants p i and µ chosen in an appropriate way.

Weak solution to the Cauchy problem

This subsection is devoted to the introduction of some notions of solutions. As usual in the PDE context, one needs to consider a weak notion of solutions, i.e. solutions admitting discontinuities in their domain. One can expect such phenomena since the initial data and the control under consideration in this paper are naturally discontinuous.

Definition 1 Let p ∈ [1, ∞] and R 0 1 , R 0 2 ∈ L p (0, L). We say that a map (R 1 , R 2 ) : [0, ∞) × (0, L) → R 2 is a solution of the Cauchy problem (10) if there exists u ∈ L p (IR) such that (1) If p = ∞, (R 1 , R 2 ) ∈ C 0 ([0, ∞); L p ((0, L); R 2 )). If p = ∞, (R 1 , R 2 ) ∈ C 0 ([0, ∞); L 1 ((0, L); R 2 )). (2) For every T > 0, for every test functions (ϕ 1 , ϕ 2 ) ∈ C 1 ([0, T ] × [0, L]; R 2 ) λ 1 ϕ 1 (t, L) -k 2 λ 2 ϕ 2 (t, L) = 0, (11)
we have

L 0 (ϕ 1 (T, x)R 1 (T, x) + ϕ 2 (T, x)R 2 (T, x)) dx - L 0 ϕ 1 (0, x)R 0 1 (x) + ϕ 2 (0, x)R 0 2 (x) dx - T 0 λ 1 ϕ 1 (t, 0)(u(t) + d(t)) -λ 2 ϕ 2 (t, 0)R 2 (t, 0)dt - L 0 T 0 (∂ t ϕ 1 (t, x) + λ 1 ∂ x ϕ 1 (t, x))R 1 (t, x) +(∂ t ϕ 2 (t, x) -λ 2 ∂ x ϕ 2 (t, x))R 2 (t, x) dtdx = 0. (12) (3) For a.e t 0, R 1 (t, 0) ∈ k 1 R 2 (t, 0) -Ksign(S(t)) + d(t),
with S given in [START_REF] Barbu | Sliding mode control for a nonlinear phase-field system[END_REF].

Remark 4 They may exist multiple measurable selection functions u (see e.g., [START_REF] Wagner | Survey of measurable selection theorems[END_REF] for a survey on such a notion) such that (R 1 , R 2 ) satisfies (1), ( 2) and (3) of Definition 1. This leads to the possible existence of multiple weak solutions of (10).

Main results

We are now in position to state our two main results, namely: an existence theorem and a robust stabilization theorem.

Theorem 1 (Existence) Let p ∈ [1, ∞] and R 0 1 , R 0 2 ∈ L p (0, L). The system (10) admits at least one weak so- lution (R 1 , R 2 ). Moreover, if p = ∞ then R 1 , R 2 ∈ L ∞ ((0, ∞) × (0, L)).
Remark 5 (Uniqueness) We may prove the uniqueness of the solution in this case. However, when considering for instance an adaptive controller, uniqueness fails to be true. For the sake of generality, we prove the robust stabilization for any possible weak solutions.

Theorem 2 ( Stabilization) Let p ∈ [1, ∞] and R 0 1 , R 0 2 ∈ L p (0, L) and K -d L ∞ (R+) > 0. There exist positive constants C, µ such that, for any t ∈ [0, ∞), (R 1 (t, •), R 2 (t, •)) T L p ((0,L);R 2 ) Ce -µt (R 0 1 (•), R 0 2 (•)) T L p ((0,L);R 2 ) ,
for any weak solution (R 1 , R 2 ) of (10).

Remark 6

We want to emphasize on the fact that our framework is quite general. We believe that it allows to use more sophisticated tools from the sliding mode literature. One may for instance consider adaptive SMC as in [START_REF] Plestan | New methodologies for adaptive sliding mode control[END_REF] or super-twisting controller as in [START_REF] Laghrouche | A Lyapunov approach to barrier-function based time-varying gains higher order sliding mode controllers[END_REF]. However, to make easier the reading of the paper, we do not want to enter in such details. In Section 5, we illustrate the efficiency of the algorithm proposed in [START_REF] Plestan | New methodologies for adaptive sliding mode control[END_REF] on the hyperbolic case.

4 Proof of Theorems 1 and 2

Existence of weak solutions

We consider the following ODE

γ(t) ∈ -νγ(t) -Ksign(γ(t)) + d(t), t ∈ IR + , γ(0) = γ 0 , (13)
where

ν > 0, d ∈ L ∞ (IR + ), K > d L ∞ (R+) and γ 0 ∈ IR.
The ODE is understood in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous righthand sides: control systems[END_REF]. A Filippov solution of ( 13) is an absolutely continuous map that satisfies [START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF] for almost all t 0. In [1, Example 2.19], the existence of Filippov solutions for (13) is proved.

Lemma 1 The ODE (13) admits a Filippov solution. Moreover, there exists t 0 > 0 such that, for any Filippov solution γ of (13),

γ ∈ W 1,∞ (IR) and γ(t) = 0, ∀ t t 0 . Remark 7 When K > d L ∞ (R+)
, there exists a unique Filippov solution and t 0 <

|γ(0)| K-d L ∞ (R + ) .
Let γ be a Filippov solution of ( 13) with initial condition γ(0) = S(R 0 1 , R 0 2 ). We consider the following PDE

           ∂ t Y 1 (t, x) + λ 1 ∂ x Y 1 (t, x) = 0, ∂ t Y 2 (t, x) -λ 2 ∂ x Y 2 (t, x) = 0, Y 1 (t, 0) = k 1 Y 2 (t, 0) + γ(t) + νγ(t), Y 2 (t, L) = k 2 Y 1 (t, L), Y 1 (0, x) = R 0 1 (x), Y 2 (0, x) = R 0 2 (x). ( 14
)
From Appendix A, the PDE ( 14) admits a unique weak solutions (Y 1 , Y 2 ) constructed using the method of characteristics, see [ We now prove that the weak solution (Y 1 , Y 2 ) of ( 14) satisfies Definition 1 (3). To that end, we introduce the function

σ(t) = S(Y 1 (t, •), Y 2 (t, •)),
where S is defined in ( 8) and (Y 1 , Y 2 ) the weak solution of [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF]. From Appendix B, for any T > 0, σ is a Carathéodory solution of

σ(t) = -νσ(t) + γ(t) + νγ(t), t ∈ [0, T ], σ(0) = S(R 0 1 , R 0 2 ). (15)
A Carathéodory solution of ( 15) is an absolutely continuous map that satisfies [START_REF] Deutscher | Finite-time output regulation for linear 2× 2 hyperbolic systems using backstepping[END_REF] for almost every t. Moreover, from [START_REF] Deutscher | Finite-time output regulation for linear 2× 2 hyperbolic systems using backstepping[END_REF] and Lemma 1, σ ∈ W 1,∞ (IR). We introduce

g ∈ W 1,1 (IR) ∩ W 1,∞ (IR) defined by g(t) = σ(t) -γ(t).
From ( 13) and ( 15) with γ(0

) = S(R 0 1 , R 0 2 ), g is solution of ġ(t) = -νg(t) g(0) = 0 (16)
Thus, for any t ∈ IR, g(t) = 0. By definition of g, we deduce that for any t ∈ IR, σ(t) = γ(t). Therefore, we have

Y 1 (t, 0) = k 1 Y 2 (t, 0) + γ(t) + νγ(t) ∈ k 1 Y 2 (t, 0) -Ksign(σ(t)) + d(t)
Thus, (Y 1 , Y 2 ) satisfies the point (3) of Definition 1. We conclude that, for any Filippov solution γ of (13) with initial condition γ(0) = S(R 0 1 , R 0 2 ), the associated weak solution (Y 1 (•), Y 2 (•)) of ( 14) is a weak solution of [START_REF] Bribiesca-Argomedo | Backsteppingforwarding control and observation for hyperbolic PDEs with Fredholm integrals[END_REF].

Remark 8 If the ODE (13) admits multiple Filippov solutions, then the PDE (10) has multiple weak solutions constructed from the coupled PDE-ODE (14)-(13).

Robust stabilization

Let (R 1 , R 2 ) a weak solution of [START_REF] Bribiesca-Argomedo | Backsteppingforwarding control and observation for hyperbolic PDEs with Fredholm integrals[END_REF]. From Definition 1, there exists u

∈ L p (IR) such that R 1 (t, 0) = u(t)+d(t) ∈ k 1 R 2 (t, 0) -Ksign(S(t)) + d(t), whence the existence of v ∈ L p (IR) such that u(t) = k 1 R 2 (t, 0) + v(t),
with v(t) ∈ -Ksign(S(t)). From Appendix B which holds replacing (B.1) by f (t) = v(t)+d(t), for any T > 0, S defined in ( 8) is a Carathéodory solution of

Ṡ(t) = -νS(t) + v(t) + d(t), t ∈ [0, T ], S(0) = S(R 0 1 , R 0 2 ). (17)
Since v(•) ∈ -Ksign(S(t)), then S is a Filippov solution of ( 13) with initial condition S(R 0 1 , R 0 2 ). From Lemma 1, S ∈ W 1,∞ (IR) and there exists a finite time t 0 (independent of v) such that Ṡ(t) + νS(t) = 0 for any t > t 0 .

Thus, from [START_REF] Di Meglio | Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input[END_REF], for any t > t 0 , v(t) + d(t) = 0.

We conclude that there exists a finite time t 0 such that for any weak solution (R 1 , R 2 ) of ( 10), for any t > t 0 , (R 1 , R 2 ) is a weak solution of

       ∂ t R 1 (t, x) + λ 1 ∂ x R 1 (t, x) = 0, ∂ t R 2 (t, x) -λ 2 ∂ x R 2 (t, x) = 0, R 1 (t, 0) = k 1 R 2 (t, 0), R 2 (t, L) = k 2 R 1 (t, L), (18)
From Appendix C, the system ( 18) is exponentially stable in L p -norm, that is to say there exist C, µ > 0 such that

(R 1 (t, •), R 2 (t, •)) T L p ((0,L);R 2 ) Ce -µ(t-t0) (R 1 (t 0 , •), R 2 (t 0 , •)) T L p ((0,L),IR 2 ) . (19)
Moreover, according to Theorem 1 and Appendix C, there exists C > 0 such that ∀t ∈ [0,

t 0 ], (R 1 (t, •), R 2 (t, •)) T L p ((0,L),I R 2 ) C (R 0 1 (•), R 0 2 (•)) T L p ((0,L),IR 2 ) , (20)
which concludes the proof of Theorem 2

Remark 9 Since S is a Filippov solution of (13) with initial condition γ(0) = S(R 0 1 , R 0 2 ), a weak solution of (10) is a weak solution of [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF]. As a consequence, the set of weak solutions of (10) is the set

{(Y γ 1 , Y γ 2 ) | γ Filippov solution of (13)} where (Y γ 1 , Y γ 2 )
is the unique weak solution of (14) associated to γ.

Numerical illustration

Conventional SMC

Numerical scheme: the approximate solution (R n 1 , R n 2 ) of ( 10) at time t n = n∆t is constructed using an upwind scheme as described in [START_REF] Leveque | Numerical methods for conservation laws[END_REF]Section 10.7]. The ghost-cell boundary conditions [START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF]Section 7] are computed as follows

R n 1 (t, 0) = k 1 Rn 2 (t, 0) -Ksign v (S( Rn 1 (t), Rn 2 (t))) + d(t), R n 2 (t, L) = k 2 Rn 1 (t, L),
where S is defined in ( 8), ( Rn 1 , Rn 2 ) are constructed using an upwind scheme from (R n-1

1 , R n-1 2
) and sign v is the Yoshida approximation of sign defined as

sign v (x) = sign(x) if |x| > v, x v if |x| v.
Note that the construction of the sliding mode control is inspired from [2, Section 5.1], where the sign function is approximated with the projection of this function on the interval [-K, K], which corresponds indeed to a saturation. As a consequence, the chattering phenomenon is reduced, see Figure 1. 10) at time t = 0 and at t = 50 are plotted with respect to x ∈ [0, L], see Figure 1, 2, 3, 4.

Simulations: let L = 3, λ 1 = 2, λ 2 = -1, k 2 = -1, ν = 0.1, K = 2 and v = 0.
(t, •), R n 2 (t, •)) of (
Figure 1 shows that the sliding surface is reached in finite time. In Figures 2 and3, the robust stabilization of R n 1 and R n 2 , respectively, is illustrated. Figure 4 shows the behavior of the controller. After a certain amount of time, one can note that the controller is almost equal to the disturbance -d, which is a classical behavior in the SMC literature. This shows that one can reconstruct the perturbation from a certain time thanks to the feedback control law defined in (9).

Adaptive SMC

If d L ∞ (R+) is an unknown parameter, we use the first adaptive sliding mode control law in [START_REF] Plestan | New methodologies for adaptive sliding mode control[END_REF]. As a consequence, the sliding mode control u in ( 9) is replaced by

u(t) = k 1 R 2 (t, 0) -K(t)sign(S(t)) (21)
where the adaptive gain K(•) is defined as follows. Let t 0 and ∆t the time step variation, if with K0 , K1 two positive constants and S defined in [START_REF] Barbu | Sliding mode control for a nonlinear phase-field system[END_REF]. If |S(t)| 4K(t)∆t then

|S(t)| > 4K(t)∆t then K(t) = K1 |S(t)|, K(0) = K0 ,
K(t) = K2 (t)|η| + K3 . Above, K2 (•) is a piecewise constant function with dis- continuous points (t i ) i defined by S(t i -∆t) > 4K(t i - ∆t)∆t and S(t i ) 4K(t i )∆t. For any t ∈ [t i , t i+1 ), K2 (t) = K(t i ). Note that t i is on-line updated. The function η is solution of τ η(t) + η(t) = sign(S(t)), η(0) = 0,
with τ > 0 and K3 is a positive constant. Figure 5 shows that the sliding surface is reached in finite-time. Figures 6 and7 show that R n 1 and R n 2 are stabilized around the equilibrium point (0, 0), meaning that our method is quite efficient. In Figure 8, the behavior of the adaptive gain K is given. One can see that it behaves almost like the function t → sin(t), which is the disturbance to be rejected. Finally, in Figure 9, the behavior of the controller u given in ( 21) is illustrated. Note that the amplitude of the adaptive control in Figure 9 is larger than the one in Figure 4 since the bound of the disturbance is unknown. From a certain time, it behaves like in the case where the control is not timevarying.

Conclusion

In this paper, we proposed a new approach for the sliding mode control applied to a specific class of PDEs, namely a system of two transport equations. It is a Lyapunov approach, since the sliding variable is based on the gradient of the classical Lyapunov function given in [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]. We have proved existence of the closed-loop system and have shown that the robust stabilization holds.

Many further research lines remain to be followed. It is important to note that our approach is a state-feedback approach, and one may instead consider an outputfeedback approach, which would require to introduce a notion of sliding mode observer. It might be also interesting to investigate the case of systems described with operators, as it has been done in [START_REF] Marx | Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques[END_REF], in the case of saturated feedback-laws. Finally, we may also consider sliding mode of higher order, as it has been done in [START_REF] Laghrouche | A Lyapunov approach to barrier-function based time-varying gains higher order sliding mode controllers[END_REF].

A Notion of solutions

Let p ∈ [1, ∞], R 0 1 , R 0 2 ∈ L p (0, L)
and γ be a Filippov solution of ( 13) with initial condition γ(0) = S(R 0 1 , R 0 2 ). We consider the linear hyperbolic system [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF].

Definition 2 (Y 1 , Y 2 ) is a weak solution of (14) if the points (1) and (2) of Definition 1 is satisfied replacing u(•) by -d(•) + γ(•) + νγ(•).
Lemma 2 There exists a unique weak solution Using the method of characteristics (see [9, Section 2.1]), we give an explicit weak solution (Y 1 , Y 2 ) of ( 14). We introduce the function f defined by

(Y 1 , Y 2 ) of (14). Moreover, if p = ∞ then Y 1 , Y 2 ∈ L ∞ ((0, ∞) × (0, L)).
f (t) = γ(t) + νγ(t). Let m 1 ∈ Z such that m 1 L λ 1 + L λ 2 t - x λ 1 < (m 1 + 1) L λ 1 + L λ 2 . (A.1) We define Y 1 for a.e (t, x) as follows. If m 1 < 0 then Y 1 (t, x) = R 0 1 (-λ 1 t + x)
We now assume that m 1 0 and we define

t i := t -i L λ 1 + L λ 2 - x λ 1 , i ∈ IN (A.2) If m 1 L λ1 + L λ2 t -x λ1 < m 1 L λ1 + L λ2 + L λ2 then Y 1 (t, x) = (k 1 k 2 ) m1 k 1 R 0 2 (λ 2 t m1 ) + m1 i=0 (k 1 k 2 ) i f (t i ). (A.3) If m 1 L λ1 + L λ2 + L λ2 t -x λ1 < (m 1 + 1) L λ1 + L λ2 then Y 1 (t, x) = (k 1 k 2 ) m1+1 R 0 1 -λ 1 t m1 -L λ2 + L + m1 i=0 (k 1 k 2 ) i f (t i ). (A.4) Let m 2 ∈ Z such that m 2 L λ 1 + L λ 2 t - L -x λ 2 < (m 2 + 1) L λ 1 + L λ 2 .
We define Y 2 for a.e (t, x) as follows. If m 2 < 0 then

Y 2 (t, x) = R 0 2 (λ 2 t + x).
We now assume that m 2 0 and we define ti :=

t -i L λ1 + L λ2 -L-x λ2 . If m 2 L λ1 + L λ2 t -L-x λ2 < m 2 L λ1 + L λ2 + L λ1 then two different cases occur. If m 2 = 0, Y 2 (t, x) = k 2 R 0 1 (-λ 1 t0 + L). Otherwise, Y 2 (t, x) = (k 1 k 2 ) m2 k 2 R 0 1 (-λ 1 tm2 + L) + m2-1 i=0 (k 1 k 2 ) i k 2 f ( ti - L λ 1 ). (A.5) If m 2 L λ1 + L λ2 + L λ1 t -L-x λ2 < (m 2 + 1) L λ1 + L λ2 then Y 2 (t, x) = (k 1 k 2 ) m2+1 R 0 2 λ 2 tm2 -L λ1 + m2 i=0 (k 1 k 2 ) i k 2 f ( ti -L λ1 ). (A.6)
An illustration of the construction of Y 1 with m 1 = 0 and Let us now prove the uniqueness.

L λ2 t -x λ1 < L λ1 + L λ2 in Figure A.1. x t t 0 L (t, x) × Y1(t, x) = Y1(t0, 0) Y1(t0, 0) = k1Y2(t0, 0) +f (t0) Y2(t0, 0) = Y2(t0 -L λ2 , L) Y2(t0 -L λ2 , L) = k2Y1(t0 -L λ2 , L) Y1(t1, L) = R 1 0 (-λ1(t0 -L λ2 ) + L)
Let (Y 1 1 , Y 1 2 ) and (Y 2 1 , Y 2
2 ) two weak solutions of [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF]. Then, the couple

(Y 1 , Y 2 ) defined by Y 1 = Y 2 1 -Y 1 1 and Y 2 = Y 2 2 -Y 1 2 is a solution of (14) with (R 0 1 , R 0 2 ) = (0, 0) and γ + νγ = 0. Let 1 p < ∞ and (ψ T 1 , ψ T 2 ) ∈ W 1,p ((0, L); IR 2 ) such that k 1 λ 1 ψ T 1 (0) -λ 2 ψ T 2 (0) = 0, λ 1 ψ T 1 (L) -k 2 λ 2 ψ T 2 (L) = 0. (A.7)
Using the method of characteristics as in the proof of Lemma 2, we state the existence of

ψ 1 , ψ 2 ∈ C 1 ([0, T ]; L p (0, L)) ∩ C 0 ([0, T ]; W 1,p (0, L)), (A.8) satisfying                  ∂ t ψ 1 (t, x) + λ 1 ∂ x ψ 1 (t, x) = 0, ∂ t ψ 2 (t, x) -λ 2 ∂ x ψ 2 (t, x) = 0, k 1 λ 1 ψ 1 (t, 0) -λ 2 ψ 2 (t, 0) = 0, λ 1 ψ 1 (t, L) -k 2 λ 2 ψ 2 (t, L) = 0, ψ 1 (T, x) = ψ T 1 (x) ψ 2 (T, x) = ψ T 2 (x
L 0 ψ T 1 (x)Y 1 (T, x) + ψ T 2 (x)Y 2 (T, x) = 0. (A.10)
Since the set of (ψ T 1 , ψ T 2 ) ∈ W 1,p ((0, L); R 2 ) satisfying (A.7) is dense in L p ((0, L); R 2 ) for any p = ∞, it follows from (A.10) that (Y 1 (T, x), Y 2 (T, x)) = (0, 0) for all T > 0 and for all x ∈ [0, L]. This achieves the proof of uniqueness of solution when 1 p < ∞. If p = ∞, the proof is similar as before choosing ψ T 1 , ψ T 2 ∈ W 1,1 (0, L).

B Conservation of mass

Let σ : t → S(Y 1 (t, •), Y 2 (t, •)) with S defined in ( 8) and (Y 1 , Y 2 ) a weak solution of [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF]. We consider γ a Filippov solution of ( 13) with initial condition γ(0) = S(R 0 1 , R 0 2 ) and we introduce the function f defined by

f (t) = γ(t) + νγ(t) (B.1) Lemma 3 For any T > 0, σ is a Carathéodry solution of σ(t) = -νσ(t) + f (t), t ∈ [0, T ], σ(0) = S(R 0 1 , R 0 2 ).
Proof 1 Using the change of variable

z 1 (t, x) = 1 λ 1 e -νx λ 1 Y 1 (t, x), z 2 (t, x) = k 1 λ 2 e νx λ 2 Y 2 (t, x), (B.2) (z 1 , z 2 ) is a weak solution of                                ∂ t z 1 + λ 1 ∂ x z 1 + νz 1 = 0, (t, x) ∈ [0, T ] × [0, L], ∂ t z 2 -λ 2 ∂ x z 2 + νz 2 = 0, (t, x) ∈ [0, T ] × [0, L], z 1 (t, 0) = 1 λ 1 (λ 2 z 2 (t, 0) + f (t)) t ∈ [0, T ], z 2 (t, L) = λ 1 λ 2 z 1 (t, L) t ∈ [0, T ], z 1 (0, x) = 1 λ 1 e -νx λ 1 R 0 1 (x), x ∈ [0, L], z 2 (0, x) = k 1 λ 2 e νx λ 2 R 0 2 (x), x ∈ [0, L],
(B.3) That is to say, (z 1 , z 2 ) satisfies the point (1) of Definition 1. Moreover, the point (2) of Definition 1 is replaced by, for any test function

ϕ 1 , ϕ 2 ∈ C 1 c (]0, T [×]0, L[), we have (0,T )×(0,L) z 1 (ϕ 1 t + λ 1 ϕ 1 x + νϕ 1 )dxdt + (0,T )×(0,L) z 2 (ϕ 2 t -λ 2 ϕ 2
x + νϕ 2 )dxdt = 0 (B.4) Moreover, the boundary conditions of (B.3) are satisfies almost everywhere. A standard density argument shows that the equation (B.4) is still admissible if ϕ 1 , ϕ 2 are just a Lipschitz functions. Let t > 0 and θ > 0, we define

ϕ i θ (t, x) := ψ i θ (t)φ i θ (x), with i ∈ {1, 2} and
ψ i θ (t) =        t θ if t θ, 1 if θ t t -θ, t-t θ if t -θ t t, (B.5)
and

φ i θ (x) =        x θ if x θ, 1 if θ x L -θ, L-x θ if L -θ x L, (B.6) When θ → 0 in (B.4), we have L 0 z 1 ( t, x)dx - L 0 z 1 (0, x)dx + λ 1 t 0 z 1 (t, L)dt -λ 1 t 0 z 1 (t, 0)dt + ν (0, t)×(0,L) z 1 (t, x)dxdt + L 0 z 2 ( t, x)dx - L 0 z 2 (0, x)dx -λ 2 t 0 z 2 (t, L)dt +λ 2 t 0 z 2 (t, 0)dt + ν (0, t)×(0,L) z 2 (t, x)dxdt = 0.
Using (B.2) , (B.3) and (8) we have for any t ∈ (0, T ),

S( t) = S(0) -ν t 0 S(u)du + t 0 f (u)du (B.7)
which conclude the proof of Lemma 3.

C Exponential stability

The proof of exponential stability is divided into two parts: first we prove the Lyapunov stability of the system (14) over the time interval [0, t 0 ] and then we prove the exponential stability of the system [START_REF] Feng | A new active disturbance rejection control to output feedback stabilization for a onedimensional anti-stable wave equation with disturbance[END_REF]. Let us start by proving the first part.

From Appendix A, ( 14) admits a unique weak solution constructed using the method of characteristics and given by (A.3),(A.4), (A.5) and (A.6). Assuming that when p = ∞. From (A.3) and (A.4) and k 1 k 2 < 1, we deduce that there exists a positive constant C such that for all t ∈ [0, t 0 ] we have

Y 1 (t, •) p L p ((0,L),I R) C R 0 1 (•), R 0 2 (•) p L p ((0,L),I R 2 ) + C L 0 m1(t,x) i=0 f (t i (t, x)) p dx (C.1)
where m 1 (t, x) and t i (t, x) are defined in Appendix A.

For the sake of clarity, the dependence of m 1 and t i with respect to t and x is emphasized in (C.1). From (A.1), for any t 0, the map x → m 1 (t, x) is a decreasing function and 0 m 1 (t, 0)m 1 (t, L) 1. Thus, two different cases occur. In the first case, there exists m 0 ∈ IN such that for every x ∈ [0, L], m 1 (t, x) = m 0 then using the change of variable s = t i with t i defined in Appendix A and t t 0 , there exists C > 0 such that

Y 1 (t, •) p L p ((0,L),I R) C R 0 1 (•), R 0 2 (•) p L p ((0,L),I R 2 ) + C f p L p ((0,t0),I R) (C.2)
In the second case, there exist m 0 ∈ IN and x 0 ∈ [0, L] such that m 1 (t, x) = m 0 for any 0 x x 0 and m 1 (t, x) = m 0 -1 for any x 0 < x L. Then,

L 0 m1(t,x) i=0 f (t i (t, x)) p dx = x0 0 m0 i=0 f (t i (t, x)) p dx + L x0 m0-1 i=0 f (t i (t, x)) p dx (C.3)
Using the change of variable s = t i with t i defined in Appendix A for each right-hand term of (C.3) and t t 0 , we deduce that there exists C > 0 such that (C.2) holds.

On the other hand, we have

f p L ∞ (0,t0) = Ṡ(t) + νS(t) p L ∞ (0,t0) = v + d p L ∞ (0,t0) . (C.4) This implies that f p L ∞ (0,t0) (2K) p . Now, us- ing K > d L ∞ (R+) and Remark 7, we know that t 0 < |S(0)| K-d L ∞ (R + ) . Therefore, there exists C > 0 such that f p L p ((0,t0),IR) C|S(0)| p . (C.5)
By continuous injection of L p ((0, L) in L 1 ((0, L), there exists C > 0 such that

|S(0)| R 0 1 (•), R 0 2 (•) L 1 ((0,L),IR 2 ) C R 0 1 (•), R 0 2 (•) L p ((0,L),IR 2 ) (C.6)
From (C.2), (C.5) and (C.6), we can conclude that, when 1 p < ∞, there exists a positive constant C such that for all t ∈ [0, t 0 ] we have Y 1 (t, •) L p ((0,L),IR) C R 0 1 (•), R 0 2 (•) L p ((0,L),IR 2 ) .

(C.7) Similarly, we prove that there exists a positive constant C such that for all t ∈ [0, t 0 ] we have Now, we provide a global asymptotic stability result related to the linear system [START_REF] Feng | A new active disturbance rejection control to output feedback stabilization for a onedimensional anti-stable wave equation with disturbance[END_REF]. From Appendix A with f (•) := γ(•) + νγ(•) = 0, (18) admits a unique weak solution constructed using the method of characteristics. In particular, the weak solution R 1 , R 2 of ( 18) is written as (A.3), (A.5), (A.4) and (A.6) replacing f by 0 and (R 0 1 , R 0 2 ) by (R 1 (t 0 , •), R 2 (t 0 , •). Thus, the inequality [START_REF] Filippov | Differential equations with discontinuous righthand sides: control systems[END_REF]

follows from k 1 k 2 = e -ν L λ 1 + L λ 2
< 1 (see also [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]Theorem 2.1]).

For the sake of completeness, we give the proof when p ∈ [2, ∞) using the following Lyapunov function. (C.9) with a > 0 and b > 0. We will consider classical solutions and deduce the result in the space L p (0, L) by a standard density argument.

V (t) = L 0 a λ 1 e -νx
The time derivative of V along the classical solutions to [START_REF] Feng | A new active disturbance rejection control to output feedback stabilization for a onedimensional anti-stable wave equation with disturbance[END_REF] along the trajectories of the system (18) which are of class C 1 . By density, this inequality also holds in the sense of distribution, i.e. for every solution to [START_REF] Feng | A new active disturbance rejection control to output feedback stabilization for a onedimensional anti-stable wave equation with disturbance[END_REF] in C 0 ([0, ∞); L p (0, L)).

On the other hand,there exists γ > 0 such that

Fig. 1 .Fig. 2 .

 12 Fig. 1. Plotting of the approximate sliding surfaceS(R n 1 (t, •), R n 2 (t, •)) with respect to the time t where (R n 1 (t, •), R n 2 (t, •)) is an approximate solution of (10) at time t and d(t) = sin(t)

xFig. 3 .Fig. 4 .

 34 Fig. 3. Plotting of R n 2 (t, •) at time t = 0 (-) and t = 50 (--) with respect to x ∈ [0, L] with d(t) = sin(t). The equilibrium point 0 is plotted in red straight line.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Plotting of the approximate sliding surface S(R n 1 (t, •), R n 2 (t, •)) (--) and t → 4K(t)∆t (-) with respect to the time t where (R n 1(t, •), R n 2 (t, •)) is an approximate solution of (1) with u defined in[START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF] and d(t) = sin(t)

xFig. 7 .Fig. 8 .

 78 Fig. 7. Plotting of R n 2 (t, •) at time t = 0 (-) and t = 50 (--) with respect to the time t with u defined in (21) and d(t) = sin(t). The equilibrium point 0 is plotted in red straight line.

tFig. 9 .

 9 Fig. 9. Plotting of the adaptive sliding mode control u defined in (21) with respect to the time t.

Fig. A. 1 . 2 t -x λ 1 .

 121 Fig. A.1. Construction of Y1(t, x) using the method of characteristics with m1 = 0 and L λ 2 t -x λ 1 .

Y 2

 2 (t, •) L p ((0,L),IR) C R 0 1 (•), R 0 2 (•) L p ((0,L),IR 2 ) .(C.8) Now, when p = ∞, according to (A.3), (A.4), (A.5), (A.6), (C.4) and (C.6) we obtain also (C.7) and (C.8), which conclude the proof of the first part.

λ 1

 1 |R 1 (t, x)| p + b λ 2 e νx λ |R 2 (t, x)| p dx

+ b e νL λ 2 k p 2 1 + L λ 2 ) k p 1 k p 2 = 1 + L λ 2 ) < 1 ,k 2 +

 21221212 |R 1 (t, L)| p -|R 2 (t, 0)| p = -νV (t) -|R 2 (t, 0)| p (-ak p 1 + b) -|R 1 (t, L)| p (-be νL λ 2 k p 2 + ae -νL λ 1 ) (C.10) Since p ∈ [2, ∞), then e ν( L λ e (1-p)ν( L λthus, we can select a and b such that e -ν( ae -νL λ 1 > 0 andak 1 + b > 0.Hence, we see that V (t) -νV (t) ∀t t 0 .(C.11)

  9, Section 2.1]. In particular, (Y 1 , Y 2 ) satisfies the points (1) and (2) of Definition 1 with u(t) = -d(t) + k 1 Y 2 (t, 0) + γ(t) + νγ(t).

  ∂ t R 1 (t, x)e -νx λ 1 R 1 (t, x)|R 1 (t, x)| p-2 dx R 2 (t, x)|R 2 (t, x)| p-2 dx ∂ x R 1 (t, x)e -νx λ 1 R 1 (t, x)|R 1 (t, x)| p-2 dx |R 1 (t, L)| p -|R 1 (t, 0)| p ∂ x R 1 (t, x), e -νx λ 1 R 1 (t, x)|R 1 (t, x)| p-2 |R 2 (t, L)| p -|R 2 (t, 0)| p )

	is
	V (t) =	ap λ 1	0	L
	+ λ 2 = -ap bp λ 2 L 0 νx ∂ t R 2 (t, x)e L 0
		+ bp	L	∂ x R 2 (t, x)e	νx λ 2 R 2 (t, x)|R 2 (t, x)| p-2 dx
	=	0 e -νL λ 1 --ap 2 apν 2λ 1 L 0	e	-νx λ 1 |R 1 (t, x)| p dx
		+	ap(p -2) 2	0	L
		+ λ 2 -bp 2 νL (e bpν 2λ 2 L 0	e	νx λ 2 |R 2 (t, x)| p dx
		-	bp(p -2) 2

L 0 ∂ x R 2 (t, x), e νx λ 2 R 2 (t, x)|R 2 (t, x)| p-2 = -νV (t)a(e -νL λ 1 |R 1 (t, L)| p -|R 1 (t, 0)| p ) + b(e νL λ 2 |R 2 (t, L)| p -|R 2 (t, 0)| p )

Using the boundary condition of (

18

) we have, for all t t 0

V (t) = -νV (t)a e -νL λ 1 |R 1 (t, L)| pk p 1 |R 2 (t, 0)| p

  (t, •), R 2 (t, •)) L p ((0,L);R 2 ) γe -ν t p (R 1 (t 0 , •), R 2 (t 0 , •)) L p ((0,L);R 2 ) ∀t t 0 . (C.13)

	1 γ V (t) γ (R 1 (t, •), R 2 (t, •)) p L p ((0,L);R 2 ) (R 1 (t, •), R 2 (t, •)) p L p ((0,L);R 2 ) .	(C.12)
	Then, we have
	(R 1

This research was conducted in the framework of the regional programme "Atlanstic 2020, Research, Education and Innovation in Pays de la Loire", supported by the French Region Pays de la Loire and the European Regional Development Fund.


Roghayeh Heiran

Saghi Sepehri

Aliasghar Jarrahpour
email: jarahpor@shirazu.ac.ir

Carole Digiorgio

Hana Douafer

Jean Michel Brunel

Ahmad Gholami

Elham Riazimontazer

Edward Turos

Val352

Hem

Synthesis, docking and evaluation of in vitro anti-inflammatory activity of novel morpholine capped β-lactam derivatives

Keywords: 2-azetidinone, Staudinger reactions, morpholine, anti-inflammatory, NOS, In silico Cys110, Trp461, Cys115, Ile462, Ser118, Gly117, Asp460, Asn482 Asp460, Asn482, Gly117, Ile462, Cys115, Ser118, Cys110, Met480, Phe476, Glu479, Trp461, Gln478 Trp461 Zn 3j -7.49 -Trp461, Ile462, Met480, Asn482, Cys115, Cys110, Gly117 -Zn Met355, Val352, Asn370, Gln263, Ala351, Glu377, Pro350, Tyr373, Asp382 Glu377, Asp382, Arg381, Gln263, Val352, Ala351, Pro350 Arg381, Asp382, Glu377, Gln263, Pro350, Ala351, Val352, Gly371, Asn370, Phe369 Gly371, Asn370, Ala351, Tyr373, Pro350, Val352, Glu377, Tyr347, --3o -8.25 -Ile462, Met480, Trp461, Asn482, Ala459, Asp460, Gly117, Ser118, Cys115 Phe369, Asn354, Ala262, Ala351, Val352, Gln263, Pro350, Glu377, Tyr373, Asp382, Arg388, Arg381 --5b -10.62 -Arg388, Asp382, Arg381, Ala351, Phe369, Glu377, Tyr373, Gln263, Ala262, Val352, Pro350 Gly371, Asn370, Pro350, Val352, Tyr373, Ala351, Ala262, Arg388, Asn354, Glu377, Arg381, Asp382 --5d -8.04 Tyr373 Asn370, Ala351, Phe369, Pro350, Val352, Gly371, Met355, Ala262, Glu377, Hem -Gln263, Asp382, Arg381, Arg388 5e -9.55 Gln263 Asn370, Gly371, Phe369, Tyr373, Ala351, Pro350, Asp382, Arg388, Glu377, Arg381 --5f -10.58 Gln263 Arg388, Arg381, Asp382, Pro350, Glu377, Tyr373, Met355, Phe369, Val352, Asn354, Tyr491, Ala262, Asn370 --5g -8.44 Gln263 Arg388, Glu377, Asp382, Arg381, Ala262, Ala351, Val352, Phe369 --5h -8.23 -Gly117, Cys110, Cys115, Trp461, Met480 -Zn 6a -8.51 Asp382, Asn354 Tyr373, Pro350, Gly371, Trp372, Glu377, Asp382, Arg381, Glu377, Arg388, Tyr373, Asn354, Tyr491, Val352, Ala351, Pro350 Ala351, Gln263, Pro350, Val352, Asp382, Glu377, Arg381, Trp463 --6d -8.38 Tyr373 Asn370, Gly371, Pro350, Ala351, Val352, Glu377, Asp382, Arg381 --6e -8.38 -Pro350, Ala351, Asn370, Gln263, Val352, Gly371, Tyr347, Tyr373, Glu377, Asp382, Arg381, Arg388 --6f -9.25 Tyr373, Asn354 Asn370, Pro350, Gly371, Phe369, Val352, Glu377, Asp382, Gln263, Trp346, Tyr347, Tyr491, Ala262 --Dex -8.01 Tyr373, Arg388, Gln263, Glu377 Asp382, Val352, Phe369, Asn370, Pro350, Gly371 --

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Introduction:

Inflammation is one of the most important defenses of the body against tissue damage or infection. However, chronic inflammation may lead to disease, such as arthritis, atherosclerosis, psoriasis, cancer, infections, asthma, etc. [1,2] Therefore, treatment of inflammation is a high priority topic for researchers over the past decades. [3] Inflammation as a biological signal of the immune system can be caused by different factors such as pathogens, damaged cells and toxic compounds. [4] Abnormal activation of certain enzymes such as inducible nitric oxide synthase (iNOS) [5] is one reason for inflammation-related disease and inhibitation of iNOS may exert anti-inflammatory effects. [START_REF] Seo | Synthesis and biological evaluation of piperlongumine derivatives as potent anti-inflammatory agents[END_REF] Therefore, development of anti-inflammatory chemicals is still of great importance.

β-Lactams are among the most important classes of organic compounds in the last century due to their diverse biological activities, [7,[START_REF] Decuyper | Antibacterial and β-Lactamase Inhibitory Activity of Monocyclic β-Lactams[END_REF] including their well-known antibacterial properties, as well as cholesterol absorption inhibition, [START_REF] Wang | 2-Azetidinone derivatives: Design, synthesis and evaluation of cholesterol absorption inhibitors[END_REF][START_REF] Dražić | Synthesis and evaluation of novel amide amino-β-lactam derivatives as cholesterol absorption inhibitors[END_REF] cysteine protease inhibition, [START_REF] Setti | 3,4-Disubstituted azetidinones as selective inhibitors of the cysteine protease cathepsin K. Exploring P3 elements for potency and selectivity[END_REF] tryptase and chymase inhibition, [START_REF] Aoyama | Design, synthesis and pharmacological evaluation of 3-benzylazetidine-2one-based human chymase inhibitors[END_REF] and anticancer, [13,[START_REF] Mohammadi | Diethyl [(3-phenoxy-2-oxo-4-phenyl-azetidin-1-yl)-phenyl-methyl]-phosphonate as a potent anticancer agent in chemo-differentiation therapy of acute promyelocytic leukemia[END_REF] antifungal, [START_REF] O'driscoll | Studies on the antifungal properties of N-thiolated β-lactams[END_REF][START_REF] Patel | Synthesis and evaluation of antibacterial and antifungal activities of 4-thiazolidinones and 2-azetidinones derivatives from chalcone[END_REF] antimalarials, [START_REF] Alborz | Synthesis and biological evaluation of some novel diastereoselective benzothiazole β-lactam conjugates[END_REF][START_REF] Jarrahpour | Synthesis of New β-Lactams Bearing the Biologically Important Morpholine Ring and POM Analyses of Their Antimicrobial and Antimalarial Activities[END_REF] anti-HIV, [START_REF] Sperka | Beta-lactam compounds as apparently uncompetitive inhibitors of HIV-1 protease[END_REF] antidiabetic, [START_REF] Wang | Discovery of azetidinone acids as conformationally-constrained dual PPARα/γ agonists[END_REF] antioxidant, [START_REF] Mermer | Synthesis of novel Azol-β-lactam derivatives starting from phenyl piperazine and investigation of their antiurease activity and antioxidant capacity comparing with their molecular docking studies[END_REF] and anti-inflammatory activities. [START_REF] Kumar | Synthesis of 3-[4′-(p-chlorophenyl)-thiazol-2′-yl]-2-[(substituted azetidinone/thiazolidinone)-aminomethyl]-6-bromoquinazolin-4-ones as antiinflammatory agent[END_REF][START_REF] Borazjani | Three-component synthesis of chromeno β-lactam hybrids for inflammation and cancer screening[END_REF] Kumar et.al. synthesized a series of quinazolinone β-lactam conjugates and investigated their antiinflammatory activities. [START_REF] Kumar | Synthesis of 3-[4′-(p-chlorophenyl)-thiazol-2′-yl]-2-[(substituted azetidinone/thiazolidinone)-aminomethyl]-6-bromoquinazolin-4-ones as antiinflammatory agent[END_REF] Recently, our laboratories have prepared chromeno β-lactam hybrids and have examined their anti-inflammatory properties. [START_REF] Borazjani | Three-component synthesis of chromeno β-lactam hybrids for inflammation and cancer screening[END_REF] Another class of nitrogen ring compounds are the morpholine derivatives, often used as the building blocks, [START_REF] Nelson | Syntheses of morpholine-2,3-diones and 2-hydroxymorpholin-3-ones: intermediates in the synthesis of aprepitant[END_REF] selective norepinephrine reuptake inhibitors [START_REF] Xu | Design and synthesis of reboxetine analogs morpholine derivatives as selective norepinephrine reuptake inhibitors[END_REF] and dopamine D4 receptor ligands. [START_REF] Audouze | New series of morpholine and 1,4-oxazepane derivatives as dopamine D4 receptor ligands: synthesis and 3D-QSAR model[END_REF] Moreover, they have been shown anti-inflammatory, [START_REF] Khanum | Synthesis and Evaluation of Benzophenone-N-ethyl Morpholine Ethers as Anti-inflammatory Agents[END_REF][START_REF] Dravyakar | Design and Development of Novel 2-(Morpholinyl)-N-substituted Phenylquinazolin-4-amines as Selective COX-II Inhibitor, Antiinflamm[END_REF] antiparasitic, [START_REF] Kuettel | Synthesis and Evaluation of Antiparasitic Activities of New 4-[5-(4-Phenoxyphenyl)-2 H -pyrazol-3yl]morpholine Derivatives[END_REF] antimicrobial, [START_REF] Yancheva | Synthesis, structure and antimicrobial activity of 6-(propan-2-yl)-3-methylmorpholine-2,5-dione[END_REF][START_REF] Santos | Synthesis and Evaluation of the in vitro Antimicrobial Activity of Triazoles, Morpholines and Thiosemicarbazones[END_REF] antitumor, [START_REF] Li | Design, synthesis and evaluation of novel diaryl-1,5diazoles derivatives bearing morpholine as potent dual COX-2/5-LOX inhibitors and antitumor agents[END_REF] and antioxidant activities. [START_REF] Chrysselis | Hypocholesterolemic and Hypolipidemic Activity of Some Novel Morpholine Derivatives with Antioxidant Activity[END_REF][START_REF] Ünver | Synthesis of new 1,2,4-triazole compounds containing Schiff and Mannich bases (morpholine) with antioxidant and antimicrobial activities[END_REF] Linezolid is an example of a commercially important antibacterial agent possessing a morpholine ring (Scheme 1). [START_REF] Panneerselvam | Synthesis of Schiff bases of 4-(4-aminophenyl)-morpholine as potential antimicrobial agents[END_REF] Recently, Cebeci et.al. have reported β-lactam derivatives containing a morpholine moiety (Scheme 1) that possess antimicrobial, antioxidant, antitubercular, and antiurease activities, as well as acetylcholinesterase inhibitory properties. [START_REF] Cebeci | Synthesis of novel Schiff bases and azol-β-lactam derivatives starting from morpholine and thiomorpholine and investigation of their antitubercular, antiurease activity, acethylcolinesterase inhibition effect and antioxidant capacity[END_REF] Our interest in new structural variants of biologically active β-lactams has led us to study hybridized systems, wherein the β-lactam core is combined with other relevant heterocycles. In this current study, we detail our studies on the synthesis and anti-inflammatory properties of βlactam-morpholine hybrids. the H-3 and H-4 protons of the β-lactam ring (J = 4.6-5.1 Hz) in the 1 H-NMR spectra. Spiro-βlactam 3o showed a single peak at 5.38 Hz for the sole β-lactam ring hydrogen. X-ray crystallographic analysis of 3o [START_REF] Atioglu | 1-(Morpholin-4-yl)-4-(2nitrophenyl)-spiro[azetidine-3,90-xanthen]-2-one[END_REF] revealed that the β-lactam ring is nearly planar and the morpholine ring adopts a chair conformation (Fig. 1). In an earlier paper of our research group morpholine β-lactams conjugates linked via propyl reported as antifungal agents. [START_REF] Jarrahpour | Synthesis of Some New Monocyclic β-Lactams Bearing a Morpholine Moiety at their N1 Positions as Antifungal Agents[END_REF] Therefore, to investigate the influence of propyl linkage between β-lactam and morpholine rings, β-lactams 5a-h were synthesized via Staudinger reactions of the corresponding imines 4 with different substituted acetic acids 2 or 2' in the presence of ptoluenesulfonyl chloride and trimethylamine. Isolated product yields were moderate to good (Scheme 3; Table 2). All the products were characterized by spectral and elemental analyses. The cis stereochemistry of monocyclic β-lactams 5a-f was deduced from the analysis of their 1 H-NMR spectra. The spiro-β-lactams 5g and 5h showed a single peak for H-4 at 4.73 and 4.72 Hz, respectively. The X-ray crystallographic structures of 5f [START_REF] Atioǧlu | [3-(Morpholin-4yl)propyl]-3-[(naphthalen-2-yl)oxy]-4-(3-nitrophenyl) azetidin-2-one[END_REF] and 5h [START_REF] Çelik | 1-[3-(Morpholin-4-yl)propyl]-4-(3-nitrophenyl)spiro[azetidine-3, 9′-xanthen]-2-one[END_REF] revealed a chair conformation for the morpholine ring and a nearly planar β-lactam ring in 5h (Fig. 2). The aryl nitro substituent in compounds 5a-f could be selectively reduced with Raney nickel and aqueous hydrazine to give aminoaryl β-lactams 6a-f (Table 2). The cis ring stereochemistry of 6a-f was confirmed by 1 H-NMR spectroscopy.

Scheme 3. General synthetic pathway for the synthesis of β-lactams 5a-h and 6a-f.

Anti-Inflammatory Assay

Our interest in these morpholine-β-lactam hybrids was with respect to possible antiinflammatory activity, a program our laboratories initiated recently. Anti-inflammatory activity was evaluated through the inhibition of nitric oxide production by LPS-stimulated mouse macrophages. Nitric oxide (NO) is a short-lived molecule, which is released from a variety of cells in response to different pathologic stimuli. NO exerts modulatory activity under a variety of inflammatory conditions, and may act as a vasodilator and a platelet inhibitor, and prevent neutrophil adhesion. As such, NO plays a protective role during infections and other stresses to the immune system.

In the anti-inflammatory assay, inflammation reactions are induced by a simulated microbial infection in macrophage cultures. In response to the microbial infection, a strong production of NO is achieved by macrophages in the culture medium. The anti-inflammatory activity is based on the capacity of compounds to inhibit inflammation reactions, and then NO release in the culture medium. The anti-inflammatory potential in evaluated by calculating, for each compound, the ratio between the anti-inflammatory activity and the cytotoxicity.

Results of the anti-inflammatory assays are reported in Table 3. Nine compounds, namely 3c, 3h, 3k, 3o, 5c, 5d, 5f, 6d, and 6f, presented excellent IC50 values for NO release, similar or better than the dexamethasone corticosteroid medication. The IC50 values for cell viability were also measured, in order to determine anti-inflammatory ratios. From this, we observed that morpholine-β-lactams 3e, 3h, 3k, 5c, 5f, 6c, 6d, and 6f afford improved anti-inflammatory ratios to that of dexamethasone. Inspection of compounds 5c, 5d, 5f, 6c, 6d, and 6f indicates that the phenyl ring substituent (NO2 versus NH2) plays an important role on anti-inflammatory activities.

For instance, 6f bearing an NH2 at the meta position provides a better anti-inflammatory ratio (99%) compared to its para isomer 6d (55%).

Molecular Docking of the β-Lactams to the Human Nitric Oxide Synthase

Molecular docking studies were next performed to try to understand and further validate the anti-inflammatory activities, and to explore plausible binding interactions with the enzymatic pocket of the human inducible nitric oxide synthase protein. [START_REF] Mughal | Flavonols and 4-thioflavonols as potential acetylcholinesterase and butyrylcholinesterase inhibitors: Synthesis, structure-activity relationship and molecular docking studies[END_REF] The X-ray crystallographic structure of human inducible nitric oxide synthase (PDB ID: 4NOS) was selected as the docking template. The procedure was carried out by eliminating the structure of the co-crystallized ligand and redocking individual β-lactams (3a-o, 5a-h or 6a-f) into this region. Root-mean-square deviation (RMSD) of the Cartesian coordinates of the atoms of the ligand in the docked and crystallographic conformations is the principle of the docking validation (RMSD < 2 Å).

Validation of molecular docking showed RMSD value was for PDB ID: 4NOS = 0.4 Å. [START_REF] Borazjani | Three-component synthesis of chromeno β-lactam hybrids for inflammation and cancer screening[END_REF] The scores of docking results based on the free binding energies and the hydrophobic and hydrogen bond interactions involved in the binding mode are presented in Table 4. Hem -

3b -9.74 - Gly371, Val352, Ala262, Pro350, Ala351, Asn370, Glu377, Gln263, Tyr373, Asp382 - - 3c -10.20 - Asp382, Tyr373, Glu377, Val352, Gln263, Met355, Ala262, Pro350, Ala351, Phe369, Asn354, Tyr491 - - 3d -8.77 Met480 Trp461, Glu479, Ile462, Cys115, Gly117, Cys110 - Zn 3e -9.34 Met480 Ser118, Asn482, Ile462, Cys110, Cys115, Gly117, Trp461, Glu479 - Zn 3f -9.04 Met480 Trp461, Glu479, Ile462, Asp460, Asn482, Cys115, Gly117, Cys110 - Zn
and in the superimposition of 5c and co-crystallized ligand (Fig. 3), 5c computationally docks almost at the same position as the original co-crystallized ligand. Looking next at the 2D image of the modeled 5c-enzyme modeled (shown on the rightside of Fig. 3), the lactam carbonyl oxygen serves as a hydrogen bond acceptor with the Gln263 (highlighted in the yellow ball) while the other substituents of 5c form hydrophobic interactions with residues Gly371, Asn370, Pro350, Val352, Tyr373, Ala351, Ala262, Arg388, Asn354, Glu377, Arg381 and Asp382 residues. The presence of an aromatic ring at the C3 position of the β-lactam ring showed more potent anti-inflammatory activity than an aliphatic moiety at the same position. For example, compounds 3d, 3f, 3j and 3n having a methoxy group exhibited poor anti-inflammatory activity compared to 3a, 3e, 3g and 3k with a phenoxy group. Furthermore, replacing the phenyl ring (3a)

with the p-chlorophenyl ring (3b) decreased both the anti-inflammatory activity and the nitric oxide synthase docking score. A comparison of the computed binding mode of 3a and 3b in the active site of the nitric oxide synthase is shown in Fig. 4. These images indicate that the active site binding modes of 3a and 3b are slightly different, such that a non-optimal orientation of 3b might be the reason for the decrease in both the anti-inflammatory activity and the nitric oxide synthase docking score of 3b compared to 3a. Specifically, 3a is computed to form a π-π stacking interaction with Hem (Figure 4A), while 3b lacks this contact (Figure 4B). Similarly, nitrophenyl-substituted β-lactam analogues 3d, 3f and 3j were computationally docked into the nitric oxide synthase binding site. Fig. 5 shows that the binding modes of 3d and 3f are similar but that of 3j is different. The oxygen atom of the morpholine ring on the lactam nitrogen of 3d and 3f forms a tight hydrogen bond with the NH of Met480, but this stabilizing interaction is not observed for 3j. We attribute this difference to be a plausible explanation for the diminished anti-inflammatory activity of 3j to 3d or 3f. We next examined the difference between having the morpholine ring directly attached to the nitrogen of the β-lactam ring (analogue 3c), versus having it tethered to the nitrogen via a threecarbon alkyl chain (analogue 5d). N-Morpholine β-lactam 3c has weaker in vitro antiinflammatory activity than propylmorpholine β-lactam 5d (Table 3). Computational modeling suggests that increased alkyl chain length has a negative effect on the active site docking ability (Fig. 7). The greater distance and conformational flexibility of the alkyl tether between the morpholine moiety and the β-lactam ring is predicted to reduce the effectiveness of reinforcing active site interactions. This should lower the NO inhibitory activity of 5d versus 3c. However, the opposite is observed. We believe that the sizable difference in lipophilic character of the two compounds may override the potential for stronger active site binding of 3c over 5d. The calculated CLogP of 3c (CLogP = 2.32) is considerably lower than that of 5d (CLogP = 4.67), which predicts a much reduced capability of 3c to penetrate the cell membrane, and this could be a reason for 3c having lower anti-inflammatory potency than 5d. 3 and4). This suggests that the NO2 substituent may exert more favorable active site binding, as illustrated in the more fitted, flattened orientation of 5c versus 6c in the human nitric acid synthase active site (Fig. 8). further enhanced the in vitro anti-inflammatory activity and in silico enzyme binding affinity (Tables 3 and4). This suggests that the presence of two chlorine atoms enhances opportunities for favorable hydrophobic interactions with active site residues, versus the mono-chlorophenyl substituent (ClogP = 3.91 for 6c vs. ClogP = 3.38 for 6b). In summary, β-lactams 6a-e exhibited the weakest affinity to the substrate binding region of the NO synthase enzyme, compared to analogues 3a-o and 5a-h. Our molecular docking results suggest that compounds 3a, 3i, 3k, 3o, 5b, 5d, 6a, and 6b all form a favorable π-π stacking interaction with Hem or Trp461 in the binding pocket (Fig. 9).

1-Morpholino-4-(4-nitrophenyl)-3-phenoxyazetidin-2-one (3a)

Cream-colored solid (yield 32 %). mp 164-166 °C. IR (KBr, cm -1 ) 1774 (CO, β-lactam), 1342, 1520 (NO2).

3-(4-Chlorophenoxy)-1-morpholino-4-(4-nitrophenyl) azetidin-2-one (3b)

White crystals (yield 35 %). mp 216-218 °C. IR (KBr, cm -1 ) 1786 (CO, β-lactam), 1346, 1520 (NO2).

3-Methoxy-1-morpholino-4-(4-nitrophenyl) azetidin-2-one (3d)

White crystals (yield 45 %). mp 172-174 °C. IR (KBr, cm -1 ) 1782 (CO, β-lactam), 1350, 1520 (NO2).

1-Morpholino-4-(3-nitrophenyl)-3-phenoxyazetidin-2-one (3e)

Light brown solid (yield 25 %). mp 134-136 °C. IR (KBr, cm -1 ) 1770 (CO, β-lactam), 1350, 1527 (NO2).

3-Methoxy-1-morpholino-4-(3-nitrophenyl) azetidin-2-one (3f)

White crystals (yield 43%

3-(4-Chlorophenoxy)-1-morpholino-4-(2-nitrophenyl) azetidin-2-one (3h)

Cream-colored crystals (yield 23 %). mp 152-154 °C. IR (KBr, cm -1 ) 1782 (CO, β-lactam), 1342, 1527 (NO2).

1-Morpholino-3-(naphthalen-2-yloxy)-4-(2-nitrophenyl) azetidin-2-one (3i)

Cream-colored solid (yield 38 %); mp 202-204 °C; IR (KBr, cm -1 ) 1767 (CO, β-lactam), 1342, 1527 (NO2).

3-Methoxy-1-morpholino-4-(2-nitrophenyl) azetidin-2-one (3j)

White solid (yield 64 %). mp 180-182 °C; IR (KBr, cm -1 ) 1767 (CO, β-lactam), 1342, 1520

(NO2). 1 H-NMR (CDCl3) δ 3.08 (CH2-N, m, 4H), 3.21 (OMe, s, 3H), 3.61 (CH2-O, m, 4H), 4.68 (H

4-(1-Morpholino-4-oxo-3-phenoxyazetidin-2-yl)benzonitrile (3k)

White crystals (yield 35 %)

4-(3-(4-Chlorophenoxy)-1-morpholino-4-oxoazetidin-2-yl)benzo nitrile (3l)

White crystals (yield 29 %). mp 162-168 °C. IR (KBr, cm -1 ) 1789 (CO, β-lactam), 2229 (CN).

General Procedure for the Synthesis of Monocyclic β-Lactams 6a-f:

β-Lactams 5a-f (1 mmol) were dissolved in 80 mL of EtOH: H2O (9:1) by heating to reflux.

Then the temperature was reduced to 60 °C and an 80% aqueous solution of hydrazine hydrate (0.15 mL, 2.3 mmol) and Raney-Ni (1 g as aqueous slurry) were added. The mixture was refluxed for an appropriate time for complete conversion by TLC monitoring. The mixture was cooled to room temperature and then filtered through a pad of Celite. The solvent was evaporated under reduced pressure and purified by silica gel chromatography (eluent 5:1 EtOAc/EtOH) to yield pure β-lactams 6a-f.

4-(4-Aminophenyl)-1-(3-morpholinopropyl)-3-phenoxyazetidin-2-one (6a)

White solid (yield 96 % Anti-inflammatory ratio = IC50-cell viability / IC50-NO release

Molecular docking studies

Computer-simulated docking studies were accomplished by the AutoDock 4.2 software. Protein Data bank (www. rcsb.org). All bound water and ligands were eliminated from the protein, and polar hydrogen atom were added to the protein as it was required for the electrostatics interactions, and then non-polar hydrogen atoms were merged together. In all dockings, a grid map with 60 grid points in the X, Y, and Z directions was built. Among the three different search algorithms offered by AutoDock 4.2, the Lamarckian genetic algorithm (LGA) approach was applied. For all docking procedures, 150 independent runs with the step sizes of 0.2 Å for translations and 5 ° for orientations and torsions were considered. For the Lamarckian GA method, a maximum number of 25 × 10 5 energy evaluations; 27,000 maximum generations; a gene mutation rate of 0.02; and a cross-over rate of 0.8 were used. At the end of docking, the structures were ranked by energy. Ligand-receptor interactions were all visualized on the basis of docking results using Discovery Studio Visualizer 4.0 and Ligplus2012.

Conclusions

In this study, some new monocyclic β-lactam derivatives bearing a morpholine ring substituent on the nitrogen have been synthesized and evaluated for anti-inflammatory activities.

Fig. 1 .

 1 Fig. 1. X-Ray structure analysis of compound 3o.

Fig. 2 .

 2 Fig. 2. X-ray crystal structures of compounds 5f and 5h.

Table 4 .

 4 Results of computational docking of morpholine-substituted β-lactams 3a-o, 5a-h and 6a-f into the active site of human inducible nitric oxide synthase (PDB ID: 4NOS)

Fig. 3 .

 3 Fig. 3. Compound 5c in the human inducible nitric oxide synthase active site (left) and superimposition of 5c (brown) and co-crystallized ligand (purple) within binding site.

Fig. 4 .

 4 Fig. 4. Computational binding modes of compound 3a (A) and 3b (B) in the active site of human inducible nitric oxide synthase.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Superimposition of 3d (A), 3f (B) and 3j (C) within the nitric oxide synthase binding site.

Fig. 7 .

 7 Fig. 7. Computational docking of 3c (A) and 5d (B) into the active site of human inducible nitric oxide synthase

Fig. 8 .

 8 Fig. 8. Interactions of 5c (A) and 6c (B) in the active site of human inducible nitric oxide synthase

Fig. 9 .

 9 Fig. 9. Close-up view of the computational docking of 3k within the active site of human inducible nitric oxide synthase.

  1 H-NMR (CDCl3) δ 2.91 (CH2-N, m, 2H), 3.07 (CH2-N, m, 2H), 3.65 (CH2-O, m, 4H), 5.24 (H-4, d, J = 4.9 Hz, 1H), 5.28 (H-3, d, J = 4.9 Hz, 1H), 6.65 (ArH, d, J = 9.0 Hz, 2H), 7.07 (ArH, d, J = 9.0 Hz, 2H), 7.62 (ArH, d, J = 8.8 Hz, 2H), 8.18 (ArH, d, J = 8.8 Hz, 2H). 13 C-NMR (CDCl3) δ 53.5 (CH2-N), 62.

  1 H-NMR (CDCl3) δ 3.21 (CH2-N, m, 4H), 3.72 (CH2-O, m, 4H), 5.36 (H-4, d, J = 5.0 Hz, 1H), 5.81 (H-3, d, J = 5.0 Hz, 1H), 6.75 (ArH, d, J = 8.6 Hz, 2H), 7.09 (ArH, d, J = 8.6 Hz, 2H), 7.49 (ArH, t, J = 7.5 Hz, 1H), 7.68 (ArH, t, J = 7.5 Hz, 1H), 7.80 (ArH, d, J = 7.6 Hz, 1H), 8.00 (ArH, d, J = 8.1 Hz, 1H). 13 C-NMR (CDCl3) δ 53.2 (CH2-N), 59.6 (C-4), 66.5 (CH2-O), 80.4 (C-3), 117.3, 125.

  1 H-NMR (CDCl3) δ 3.16 (CH2-N, m, 4H), 3.72 (CH2-O, m, 4H,), 5.57 (H-4, d, J = 5.1 Hz, 1H), 5.92 (H-3, d, J = 5.1 Hz, 1H), 6.87 (ArH, d, J = 8.9 Hz, 1H), 7.26 (ArH, s, 1H), 7.31-7.45 (ArH, m, 3H), 7.57 (ArH, d, J = 8.9 Hz, 2H), 7.66 (ArH, d, J = 9.1 Hz, 2H), 7.85 (ArH, d, J = 7.9 Hz, 1H), 7.96 (ArH, d, J = 8.2 Hz, 1H). 13 C-NMR (CDCl3) δ 53.3 (CH2-N), 59.3 (C-4), 66.5 (CH2-O), 80.

  -4, d, J = 5.0 Hz, 1H),5.51 (H-3, d, J = 5.0 Hz, 1H), 7.47 (ArH, t, J = 8.7 Hz, 1H), 7.65 (ArH, t, J = 7.9 Hz, 1H), 7.72 (ArH, d, J = 7.9 Hz, 1H), 8.03 (ArH, d, J = 8.1 Hz, 1H).13 C-NMR (CDCl3) δ 53.2 (CH2-N), 59.0 (OMe), 59.8 (C-4), 66.5 (CH2-O), 83.6 (C-3), 124.9, 128.9, 129.9, 130.9, 133.2, 148.8 (aromatic carbons), 166.9 (CO, β -lactam). MS m/z = 307 [M + ]; Anal. Calcd. for C14H17N3O5: C, 54.72; H, 5.58; N, 13.67%. Found: C, 54.91; H, 5.57; N, 13.79 %.

  1 H-NMR (CDCl3) δ 2.90 (CH2-N, m, 2H), 3.06 (CH2-N, m, 2H), 3.62 (CH2-O, m, 4H), 5.20 (H-4, d, J = 4.8 Hz, 1H), 5.28 (H-3, d, J = 4.8 Hz, 1H), 6.66 (ArH, d, J = 9.0 Hz, 2H), 7.07 (ArH, d, J = 9.0 Hz, 2H), 7.55 (ArH, d, J = 8.6 Hz, 2H), 7.61 (ArH, d, J = 8.6 Hz, 2H). 13 C-NMR (CDCl3) δ 53.5 (CH2-N), 62.4 (C-4), 66.2 (CH2-O), 79.

[ 43 ]

 43 Lamarckian Genetic Algorithm of the AutoDock 4.2 program was used as the search algorithm. The Graphical User Interface program AutoDock Tools 1.5.6 (ADT) were used to prepare, run, and analyze the docking simulations. Molecular docking of compounds was performed with the published crystal structure (PDB ID: 4NOS (human inducible nitric oxide synthase)) by the Auto-Dock Tool 1.5.6. All two dimensional (2D) structures of the compounds were built using the ChemDraw program (ChemDraw Ultra 10.0, Cambridge soft.), and then moved into the Hyperchem 8.0 software (HyperChem, Release 8.0 for Windows, Molecular Modeling System: HyperCube, 2007). Molecules were subjected to energy minimization with MM + force field and then PM3 semi-empirical technique. Then the partial charges of atoms were calculated by the GasteigereMarsili procedure implemented in the AutoDock Tools package. The non-polar hydrogens of compounds were merged. The crystal structure of protein was taken from

Table 1 .

 1 Isolated yield of monocyclic β-lactams 3a-o.

	Entry	Ar	R	yield (%) Entry	Ar	R	yield (%)
	3a	4-NO2C6H4	C6H5	32	3i	2-NO2C6H4 Naphthyl	38
	3b	4-NO2C6H4 4-ClC6H4	35	3j	2-NO2C6H4	CH3	64
	3c	4-NO2C6H4 Naphthyl	38	3k	4-CNC6H4	C6H5	35
	3d	4-NO2C6H4	CH3	45	3l	4-CNC6H4 4-ClC6H4	29
	3e	3-NO2C6H4	C6H5	25	3m	4-CNC6H4 Naphthyl	49
	3f	3-NO2C6H4	CH3	43	3n	4-CNC6H4	CH3	36
	3g	2-NO2C6H4	C6H5	36	3o	2-NO2C6H4	-	41
	3h	2-NO2C6H4 4-ClC6H4	23

Table 2 .

 2 Isolated

	Entry	Ar	R	yield (%) Entry	Ar	R	yield (%)
	5a a	4-NO2C6H4	C6H5	76	5h	3-NO2C6H4	-	59
	5b	4-NO2C6H4	4-ClC6H4	74	6a	4-NH2C6H4	C6H5	96
	5c	4-NO2C6H4 2,4-Cl2C6H3	63	6b	4-NH2C6H4	4-ClC6H4	92
	5d	4-NO2C6H4	Naphthyl	79	6c	4-NH2C6H4 2,4-Cl2C6H3	67
	5e a	3-NO2C6H4	C6H5	63	6d	4-NH2C6H4	Naphthyl	79
	5f	3-NO2C6H4	Naphthyl	63	6e	3-NH2C6H4	C6H5	68
	5g	4-NO2C6H4	-	71	6f	3-NH2C6H4	Naphthyl	59
	a previously reported synthesis [38]

yield of monocyclic β-lactams 5a-h and 6a-f.

Table 3 .

 3 Anti-inflammatory activity of derivatives

3a-o, 5a-h, and 6a-f. Entry IC50-NO release (µM) IC50-cell viability anti

				-
			(µM)	inflammatory ratio
	3a	35.41 + 2.04	256.83 + 12.14	7
	3b	10.67 + 1.22	267.48 + 11.65	26
	3c	9.90 + 2.18	147.25 + 1.65	14
	3d	45.51 + 3.13	275.28 + 13.24	6
	3e	13.04 + 2.47	>500	>38
	3f	30.87 + 1.99	193.17 + 9.41	6
	3g	14.76 + 1.52	293.84 + 8.47	21
	3h	7.92 + 1.07	>500	>62
	3i	21.07 + 1.95	263.34 + 10.59	12
	3j	57.45+ 2.34	114.15 + 1.66	2
	3k	4.57 + 0.81	204.11 + 12.65	51
	3l	ND	ND	ND
	3m	20.88 + 3.21	>500	>25
	3n	91.88 + 2.64	264.17 + 8.74	3
	3o	2.93 + 0.66	14.19 + 0.88	5
	5a	27.63 + 2.37	>500	>18
	5b	17.18 + 1.56	20.41 + 1.56	1
	5c	4.14 + 0.63	290.16 + 12.47	72
	5d	5.06 + 0.94	69.88 + 4.68	14
	5e	28.73 + 1.89	157.08+ 5.64	5
	5f	5.13 + 0.95	257.21 + 9.51	51
	5g	60.41 + 8.34	323.39 + 12.14	5
	5h	12.37 + 2.08	148.24 + 1.11	12
	6a	70.44 + 5.49	129.90 + 10.23	2
	6b	17.33 + 2.91	33.68 + 1.26	2
	6c	14.46 + 1.34	>500	>35
	6d	6.53 + 1.24	328.61 + 15.63	55
	6e	24.73 + 1.26	272.92 + 8.29	11
	6f	5.73 + 0.56	>500	>99
	Dexamethasone	5.02 + 1.34	159.2 + 26.35	32

ND: Not determined due to insolubility in the media

  ). mp 156-158 °C. IR (KBr, cm -1 ) 1782 (CO, β-lactam), 1342, 1535

	(NO2). 1 H-NMR (CDCl3) δ 2.86 (CH2-N, m, 4H), 3.14 (OMe, s, 3H), 3.63 (CH2-O, m, 4H), 4.57
	(H-4, d, J = 4.8 Hz, 1H), 5.06 (H-3, d, J = 4.8 Hz, 1H), 7.55 (ArH, t, J = 7.9 Hz, 1H), 7.80 (ArH,
	d, J = 7.9 Hz, 1H), 8.22 (ArH, d, J = 8.1 Hz, 1H), 8.30 (ArH, s, 1H). 13 C-NMR (CDCl3) δ 53.5
	(CH2-N), 58.6 (OMe), 61.9 (C-4), 66.2 (CH2-O), 82.8 (C-3), 123.8, 129.3, 134.7, 136.8, 148.2
	(aromatic carbons), 165.3 (CO, β -lactam). MS m/z = 307 [M + ]. Anal. Calcd. for C14H17N3O5: C,
	54.72; H, 5.58; N, 13.67 %. Found: C, 54.67; H, 5.65; N, 13.71 %.
	1-Morpholino-4-(2-nitrophenyl)-3-phenoxyazetidin-2-one (3g)
	Cream-colored crystals (yield 36 %). mp 184-186 °C. IR (KBr, cm -1 ) 1767 (CO, β-lactam),
	1354, 1527 (NO2). 1 H-NMR (CDCl3) δ 3.16 (CH2-N, m, 4H), 3.66 (CH2-O, m, 4H), 5.42 (H-4, d,

J = 5.1 Hz, 1H), 5.84 (H-3, d, J = 5.1 Hz, 1H), 6.78 (ArH, d, J = 8.5 Hz, 2H), 6.89 (ArH, t, J = 7.5 Hz, 1H), 7.10 (ArH, t, J = 8.5 Hz, 2H), 7.47 (ArH, t, J = 8.6 Hz, 1H), 7.68 (ArH, t, J = 8.2 Hz, 1H), 7.81 (ArH, d, J = 8.6 Hz, 1H), 8.01 (ArH, d, J = 8.2 Hz, 1H). 13 C-NMR (CDCl3) δ 53.2 (CH2-N), 59.5 (C-4), 66.5 (CH2-O), 80.2 (C-3), 115.8, 122.5, 124.9, 129.2, 129.4, 129.9, 130.5, 133.2, 148.8, 156.8 (aromatic carbons), 165.8 (CO, β -lactam). MS m/z = 369 [M + ]. Anal. Calcd. for C19H19N3O5: C, 61.78; H, 5.18; N, 11.38 %. Found: C, 61.85; H, 5.37; N, 11.42 %.

  6H), 4.81 (H-4, d, J = 4.3 Hz, 1H), 5.34 (H-3, d, J = 4.3 Hz, 1H), 6.55 (ArH, d, J = 8.4 Hz, 2H), 6.72 (ArH, d, J = 8.4 Hz, 2H), 6.83 (ArH, t, J = 7.3 Hz, 1H), 7.08 (ArH, d, J = 8.3Hz, 2H), 7.14 (ArH, t, J = 8.6 Hz, 2H).13 C-NMR (CDCl3) δ 24.4 (aliphatic chain), 38.4 (aliphatic chain), 53.6 (morpholine ring), 56.2 (aliphatic chain), 62.2 (C-4), 66.9 (morpholine ring), 81.9 (C-3), 114.7, 115.6, 121.8, 122.2, 129.1, 129.8, 146.9, 157.0 (aromatic carbons), 166.1 (CO, β -

	). mp 136-138 °C. IR (KBr, cm -1 ) 1751 (CO, β-lactam), 3356, 3456
	(NH2). 1 H-NMR (CDCl3) δ 1.62 (aliphatic chain, m, 2H), 2.29 (aliphatic chain and morpholine
	ring, m, 6H), 2.95 (aliphatic chain, m, 1H), 3.43(aliphatic chain, m, 1H), 3.65 (morpholine ring
	and NH2, m,

lactam). MS m/z = 382 [M + ]. Anal. Calcd. for C22H27N3O3: C, 69.27; H, 7.13; N, 11.02 %. Found: C, 69.35; H, 7.15; N, 10.95 %.

Acknowledgments

The authors thank the Shiraz University Research Council for financial support (Grant No. 98-GR-SC-23).

In vitro anti-inflammatory activity [START_REF] Hwang | Anti-inflammatory effects of talosin A via inhibition of NF-kappaB activation in lipopolysaccharide-stimulated RAW264.7 cells[END_REF]

Principle of the assay

The in vitro anti-inflammatory assay is based on the ability of macrophages to generate a strong inflammatory response when stimulated with antigens. Mouse immortalized macrophages (RAW 264.7 cell line) are stimulated by E. coli LPS and exposed to the test material for 24 hours. At the end of the incubation period, NO production is evaluated indirectly by measuring the accumulation of nitrite/nitrate, the stable end-products of NO oxidation, in the culture medium using a spectrophotometric method based on the Griess reaction. [START_REF] Hwang | Anti-inflammatory effects of talosin A via inhibition of NF-kappaB activation in lipopolysaccharide-stimulated RAW264.7 cells[END_REF]

Cell line

Mouse macrophages (RAW 264.7, Sigma-Aldrich, N° P6110401, Lot. 09I006), low passage number (<50).

Culture medium

Complete medium: DΜΕΜ with L-glutamine (Dulbecco's Minimum Essential Medium, Pan Biotech Lot 974251) supplemented with penicillin-streptomycin (10,000 U/mL penicillin, 10 mg/mL streptomycin, Pan Biotech, Lot 20145123), and 10% inactivated calf serum (Pan Biotech, Lot P440008), pΗ 7.2, freshly prepared, stored no longer than 3 weeks.

Dilutions of the test material

The test materials were diluted into dimethylsulfoxide (DMSO, Sigma-Aldrich) at the final concentration of 10 mM.

Controls

Negative control: DMSO Positive control: Dexamethasone at 1, 5, 10, 50, and 100 µM in DMSO

Test procedure

Cells were seeded into 48-well tissue culture plates at the concentration of 1x10 5 cells/mL (200 µl/well) for 24 hours at 37°C (5% CΟ2). At the end of the incubation period the culture medium was replaced by 200 µL of medium containing the appropriate concentrations of the test materials, and cells were incubated at 37°C (5% CO2) for one hour. At the end of the incubation period, proinflammatory LPS from E. coli was added to the cell cultures (1 µg/mL). The cells were then incubated at 37°C (5% CO2) for 24 hours.

Assessment of NO release

NO release was measured in the culture supernatant by the Griess reaction. 100 µl of the supernatants were transferred into the wells of a 96-well tissue culture plate, and 100 µl of the Griess modified reagent (Sigma-Aldrich) were added in each well. After a 15 min period at room temperature, the Optical Density (OD) of each well was read at 540 nm by a fluorescenceluminescence reader Infinite M200 Pro (Tecan). The results obtained for wells treated with the test material were compared to those of untreated control wells (DMSO, 100% viability) and converted to percentage values.

Assessment of cell viability

In parallel to the assessment of NO release, viability was measured to validate the assay.

The WST-1 vital dye reagent was used to measure cell mitochondrial respiration. For this purpose, the culture medium was decanted and 100µl of WST-1 reagent (1/10 dilution) were added in each well. After a 30-min incubation period at 37°C (5% CO2), the Optical Density (OD) of each well was read at 450 nm by a fluorescence-luminescence reader Infinite M200 Pro (TECAN). The results obtained for wells treated with the test material were compared to those of untreated control wells (DMSO, 100% viability) and converted to percentage values.

Calculation of the IC50

Inhibition of NO release and inhibition of cell viability were expressed as percentages as compared to the negative controls: The concentrations of the test material causing respectively a 50% decrease of NO release (IC50-NO release) and a 50% decrease of cell viability (IC50-cell viability) were calculated using software Tablecurve Version 2.0. The anti-inflammatory ratio corresponded to the ratio between the antiinflammatory activity and the toxicity, and expressed as follows:

Some of the morpholino-β-lactam hybrids showed higher anti-inflammatory responses than the corticosteroid dexamethasone. The anti-inflammatory activity was observed to be dependent on the substituents at the C3 and C4 centers of the lactam ring. Members of the 3a-o and 5a-h series exhibit higher ΔGbinding than the 6a-e series. The higher lipophilicity and electron-withdrawing characters had positive effect for anti-inflammatory activity and score docking. These results suggest the possibility of carrying out further studies on the mechanism of action of our compounds.

Conflict of interest

The authors declare that there is no conflict of interests.