Philippe Poizot

Lydia Laffont-Dantras

Jacques Simonet
email: jacques.simonet@univ-rennesl.fr

The one-electron cleavage and reductive homo-coupling of alkyl bromides at silver-palladium cathodes

Keywords:

a b s t r a c t

We have previously demonstrated that ''palladized" silver electrodes obtained by displacement reaction, thanks to an immersion in acidic Pd II -based solutions, display quite interesting capabilities in the cleavage of alkyl halides. Such chemically modified surfaces by an Ag-Pd alloy lead to large potential shifts compared to the use of glassy carbon or smooth silver cathodes as well. The present article now gives preliminary results focused on the particular electrochemical activity of primary and secondary alkyl bromides. Experimental data show that such electrodes may behave quite differently from smooth silver, especially, when acetonitrile and dimethylformamide are used as solvent. While smooth silver cathodes generally entail two-electron cleavage reactions, palladized silver interfaces always yield catalytic cleavages of C-Br bonds according to single one-electron reactions. The homo-coupling processes may generally occur with a rather high efficiency.

Introduction

The use of new metallic electrodes in the field of organic electrochemistry remains nowadays an important task to achieve selective reactions [START_REF] Dekker | Organic Electrochemistry[END_REF]. Since the use of mercury and lead is banned from laboratories for environmental reasons [START_REF] Brown | [END_REF]3], electrochemists generally choose as cathode material different carbon surfaces such as glassy carbon, which today seems to provide a sort of universal solid working electrode [4]. However, the reactivity of carbons toward ion insertion (e.g. lamellar compounds like graphite) or/and their tendency to graft some electro-generated species like free radicals, are often mentioned [5,6]. Alternatively, at least within conducting media presenting a certain level of moisture [7], platinum could be used although both cost and weak hydrogen over-voltage make its use beyond À1.5 V vs. Saturated Calomel Electrode (SCE) difficult [8]. The use of other metallic cathodes like copper [7], palladium [8], tin [9], nickel [10] or stainless steel [11] was also put forward during the last decade. However, some specific interfacial reactions, which constitute a major drawback, can occur. Lastly, a huge development of chemically modified interfaces, in particular solid-conducting surfaces, has been performed since the 1980s. For instance, carbons or metals like gold [14] or silicon [15] have been chemically modified either by grafting of groups or with functionalized conductive polymers. Such modified surfaces may have found some interest in interfacial synthesis (with electron transfer).

Very recently, the interest in using smooth silver [16], smooth palladium and palladized surfaces [17,18] was developed to improve the ease of carbon-halogen bond scission. In the same way, palladium associated with other transition metals in well-defined alloys has been successfully put forward specifically with copper and silver [19][20][21]. In particular, the Ag-Pd interface was successfully used to strongly speed up scission of carbon-iodine bonds [21]. In the case of carbon-heteroatom bonds, known to be more difficult to split under electron transfer, alkyl bromides are certainly interesting to study especially with the aim to improve the ease of the overall scission (destruction of wastes in industry) as well as form free radical transients useful in radical chemistry (addition to organic radicalophiles and synthesis of homo-dimers). The addition of such radicals may also present some interest in the chemical transformation of surfaces too.

Concerning the r bond of alkyl bromides, it has been reported that their reductive scission requires high activation energy at glassy carbon surfaces [22][23][24][25][26][START_REF] Savéant | Advances in Physical Organic Chemistry[END_REF]. Therefore, the cleavage occurs in most cases under a dissociative electron transfer at potential lower than À2 V vs. SCE. In order to efficiently speed up the cleavage of such carbon-halogen bonds at much higher potentials, palladized smooth silver cathodes have been tested toward the electrochemical activity of alkyl bromides. Here are gathered the first results on the cleavage of a large series of primary, secondary and tertiary alkyl bromides RBrs. Reactions are compared to those currently observed at glassy carbon electrodes and at smooth silver using several dipolar solvents like acetonitrile (AN), dimethylformamide (DMF), and propylene carbonate (PC). The advantages of these Ag-Pd surfaces in terms of catalytic character and chemical stability doi:10.1016/j.jelechem.2008.07.021 are developed for the first time. The first results of this investigation using palladized silver substrates are also compared to the recent data obtained on smooth silver electrodes mainly in AN [START_REF] Mussini | [END_REF][29][30][31][32][33]. Finally, the aim of this work is also to stress the conditions of obtaining one-electron cleavages with C-Br bonds and to check the formation of homo-dimers in high yields.

Experimental

Chemical doping of smooth silver by palladium and characterization

The general approach to modify silver at the nanometric scale using palladium has been previously detailed [8]. Briefly, the procedure is based on the contact of metallic silver substrates with a fresh aqueous solution of palladium sulphate in sulphuric acid. More specifically, one gram of Pd(SO 4 ) Á 2H 2 O (Alfa Aesar) dissolved in 100 mL of 0.1 N H 2 SO 4 solution is put in the presence of silver. After a few tens of seconds, a shiny layer progressively covers the silver surface. The quasi-instantaneous deposition process onto the smooth silver surface is provoked by the displacement of silver by palladium (E°(Pd 2+ /Pd) = 0.92 V/NHE and E°(Ag + /Ag) = 0.799 V/NHE) concomitantly with an unexpected alloying reaction leading to a specific Ag-Pd interface. Note that palladium nitrate and chloride could be used as well.

It also appeared of interest to test silver electrodes elaborated by silverization of solid surfaces. Such electrodes were obtained by galvanostatic deposits of silver from a solution of AgNO 3 (i.e. 10 g L À1 in a 0.1 N HNO 3 solution). These deposits, preliminarily elaborated onto gold, nickel or platinum were achieved, thanks to fixed current electrolyses of 0.5 mA mm À2 for a passed charge ranging from 10 À3 to 4 Â 10 À3 C mm À2 corresponding to average thickness of the order of 1-5 lm. Then, the formation of the Ag-Pd alloy layer was simply obtained by dipping the silvered metallic substrates into Pd 2+ -based acidic solutions as previously explained. It is expected that the thickness of the layer is not deeply altered by the displacement process.

Characterization of the as-produced layers was carried out by a combined SEM, HRTEM, EDS study. Morphology of the layer formed onto the electrode was first investigated by SEM using a FEI QUANTA F200P coupled with an Oxford Link ISIS instrument for Energy Dispersive Spectroscopy analysis (EDS). The TEM and HRTEM imaging were performed using a FEI TECNAI F20 S-TWIN, whereas the elemental composition was also determined by energy dispersive spectroscopy (EDS) at nanometer resolution. The layer thickness looks strongly dependent on the time of contact with the palladium solution, but cannot be precisely measured owing to the displacement reaction and the fact that silver is slowly attacked by acid. SEM characterizations of the new surface lead to assess that the average thickness of the layer (depending on the contact time) is generally of the order of 1 lm. Note that the layer is efficiently attached to silver and is not easily removed by sonication.

Electrochemical procedure, salts and solvents

The electrochemical instrumentation has been described in previous publications [17][18]. Typically, all potentials are given against SCE, whereas each electrode used in voltammetry had an apparent surface area of 0.78 mm 2 . The supporting salt concentration was fixed at 0.1 M using either tetra-n-butylammonium tetrafluoroborate (TBABF 4 ) or tetra-n-butylammonium hexafluorophosphate (TBAPF 6 ); these two salts being purchased from Aldrich. Used solvents were dimethylformamide, acetonitrile (quality for analyses) both purchased from SDS and propylene carbonate (qual-ity for syntheses) purchased from Merck. Experiments performed with platinum and palladium electrodes as substrate did not require especially dry solutions since the explored potential ranges were not lower than À2 V vs. SCE. Finally, each electrochemical experiment was performed under inert atmosphere using dry argon.

Silver, glassy carbon, pure palladium, gold, nickel, and platinum were carefully polished with silicon carbide paper (Struer) or with Norton polishing paper (type 02 and 03). Before use, the electrodes were twice rinsed with water then alcohol and finally, acetone. Lastly, each electrode was dried with a hot air flow for about 30 s. Under such conditions, electrodes could be re-used leading to coherent data. In the present study, electrolysis products were found to be long-chain dimers often insoluble leading to their deposition onto the electrode surfaces. So, a rinsing step with acetone or ether could be necessary to renew the interface efficiency.

Coulometry and electrolyses

Coulometric measurements and electrolyses of alkyl bromides were carried out using three-electrode system cells with a total catholyte volume of about 5-10 mL. The anodic compartment was separated by a fritted glass of weak porosity. The experiments were completed on small substrate amounts (typically 0.1 mM). In order to avoid the disturbance resulting from the possible presence of silver oxide, which could depend on the history of solid substrate, a pre-electrolysis of the solution was systematically achieved prior to adding the RX compound to the cell.

Macroelectrolysis products (monomeric and dimeric forms from RBrs) were analyzed by GC/MS after extraction by a large amount of ether. Analyses were made on the extracts after rinsing by water and drying on MgSO 4 .

Results and discussion

Primary alkyl bromides

Basically, it is known that almost all alkyl bromides give a twoelectron irreversible step at glassy carbon electrodes at quite reducing potentials in DMF containing tetraalkylammonium salts (i.e. E <À2.4 V vs. SCE) [34][35][36][37][START_REF] Peters | Organic Electrochemistry[END_REF]. Under the same conditions, the use of smooth silver as a cathode induces significant potential shifts toward less cathodic potentials. However, two steps are typically observed (denoted (I) and (II)) with the specificity that the first one, located at a rather less negative potential (i.e. E p/2 $À1.3 V), is usually small. The curve B in Fig. 1 shows this particular behaviour with 1-bromooctane in DMF/TBAPF 6 . Additionally, it is worth noting that the results are quite similar in AN compared to DMF. As shown in Table 1 when using DMF/TBABF 4 as electrolyte, the sum of currents of peaks (I) and (II) is roughly equal to the current of the two-electron step usually obtained at glassy carbon. Peak (I) is apparently consistent with an activation process by silver involving a preliminary reaction between Ag and the alkyl bromide similar to what was previously described with RI [8] compounds, but the reaction is much slower. The existence of a reaction at silver could also be checked by SEM analysis (see Fig. 2 with 1-bromooctane). Electrochemical reduction in DMF of the modified silver layer at step (I) level gave crystal deposits identified as those of the dimer in C 16 (image b).

Generally speaking, secondary bromides exhibit similar results. This electrochemical behaviour, frequently met in DMF with primary and secondary alkyl bromides, strongly suggests that RBr compounds are reduced according to two very different processes at the silver interface: (i) a reduction step of RBr limited by the kinetic of a surface reaction and (ii) a reduction step at much more negative potential involving no specific effect of the metal surface. For the latter, the electron uptake is much less dependent on the cathode material, hence, carbon and silver electrodes give comparable results. This proposal is supported by two experimental facts observed with all primary and secondary alkyl bromides. Thus, the first step may exhibit an almost pure kinetic character (i.e. no variation of current upon the scan rate in voltammetry for 10 6 v 6 1 V s À1 ). The variation of the limit current with changes in the RBr concentration is approximately zero beyond a threshold value. The second step progressively increases as the scan rate increases because it is under diffusion control. It is therefore quite possible to expect peak (II) to represent the RBr reduction at silver without any catalytic implication. Regarding the reduction of primary and secondary bromides at smooth silver in AN one notices also a complex reduction process involving again two different steps as shown in Fig. 3 (curves A1 and A2 with 1-bromooctane and 2-bromopentane, respectively).

If keeping in mind that the cathodic reduction of RBr compounds at solid interfaces (such as platinum or glassy carbon) is much less easy than the one observed with the corresponding iodides, the catalytic effect by silver depends on several parameters: the activation process specific to RBr compounds and its kinetics at the metal surface, the choice of the electrolyte anion, the adsorption of the electrolyte at silver as well as the treatment of the surface. There are some experimental evidences to support the reduction scheme below:

As a matter of fact, the reduction RBrs entails a two-electron process onto smooth silver at least in DMF (see Table 1). Rather negative potentials are necessary to get a reasonable current density (fixed E Red < À1.6 V) apparently because the activation by silver is too slow. Therefore, it appears difficult to discriminate between the two processes suggested in Scheme 1. Thus, free alkyl radicals are readily reduced taking into account that the standard potentials for the transition RÅ/R À given by Lund [START_REF] Occhialini | [END_REF] are generally higher than À1.4 V vs. SCE. Note that macro-electrolyses achieved at À1.6 V vs. SCE with silver cathodes are mainly oriented to the obtaining of two-electron processes with major formation of the corresponding alkane.

E p/2 (V) I lim (lA) E p/2 (V) 1st step Total reduction current (lA) a E p/2 (V) b I lim (lA) E(V) Q (F mol À1 ) E(V) Q (F mol À1 ) 1

Reduction of alkyl bromides at Ag-Pd surfaces

If now the silver surface is efficiently palladized by means of the displacement reaction, dramatic changes could be observed in voltammetries: appearance of a large step replacing step (I) observed at smooth silver (see Table 1, Fig. 1, curve C in DMF and Fig. 3, curves B1, B2 in AN) with a concomitant potential shift to less cathodic potentials with a magnitude ranging from 0.1 to 0.2 V. Interestingly, the alkyl bromides reduction then occurred following a one-electron step as demonstrated by coulometry measurements achieved in DMF (see Table 1). It has been shown that the current of this main peak at Ag-Pd interface may keep a certain kinetic character since a weak variation of the peak current upon scan rate is observed (especially with [RBr] > 10 mM and v > 100 mV s À1 ). It is expected that the doping of the silver surface by palladium specifically leads to faster catalytic process than pure silver interface, but in many cases, observed kinetics remain quite close that of diffusion. Additionally, it could be underlined that long immersion times of the silver substrate in a Pd II -based solution (efficient coverage of silver surface) led to the loss of the kinetic character of the main one-electron step. The thickness of the modified layer would then contribute to influence the global rate of the surface catalysis.

We also achieved some preliminary experiments with galvanostatic deposits of silver onto gold, platinum, palladium, and nickel. Such silvered interfaces give systematically very reproducible results, which are somewhat better in terms of catalysis than conventional smooth silver. Similarly, the activation of silver deposits by palladization was achieved. For example, the activation of a silver layer on platinum or gold was achieved and these electrodes showed comparable results to the ones described above with massive Ag-Pd electrodes. In other words, very thin galvanostatic deposits of silver (average thickness $1 lm) at many solid-conducting surfaces may transform solid-conducting interfaces into efficient and sometime cheap Ag-Pd supported electrodes. Many other solid substrates can also be considered. Lastly, let us stress that the reactivity of free radicals on solid surfaces is now considered as quite general and effective especially on transition metals and carbons [6]. Using the same experimental conditions as those linked to Fig. 1 (curves C, case of 1-bromooctane), palladized silver surfaces clearly exhibit a second step equal in current to the first one. In most cases (for the whole series of RBrs), the threshold potential of these peaks ranges between À1.5 and À1.8 V vs. SCE. It is likely that these second steps correspond to the reduction of free alkyl radicals: the reduction can probably be disturbed by the specific chemisorption of these radicals onto the Ag-Pd surface, which may lead to a slowing down of the second electron exchange. Anyhow, the use of Ag-Pd interface would enable to efficiently separate the first step involving the formation of the free alkyl radical from the second one corresponding to its cathodic reduction. This electrochemical behaviour makes the use of pure silver electrodes very interesting (see coulometric data in Table 1 with Ag and Ag-Pd working electrodes). Addition-ally, it is worth noting that separation of the two steps is quite unusual for alkyl halides: it was first mentioned with tert-butyl iodide and bromide in non-aqueous media by Savéant [22,25]. Previous data on the use of pure palladium showed that primary alkyl iodides were similarly reduced stepwise [10].

Use of propylene carbonate as solvent

Surprisingly, the use of propylene carbonate containing tetraalkylammonium salts enables to obtain with smooth silver electrodes very similar results to those observed at Ag-Pd electrodes with DMF and AN. Such results are shown with butylbromide in Fig. 4, curve B. In general, it was found with all the RBrs studied that electrochemical responses at silver and Ag-Pd electrodes are strictly identical when using PC as solvent. It can be suggested that the high dielectric constant of PC (e r = 65.1) activates the dissociation of the C-Br r bond and speeds up the reaction with silver. It is therefore remarkable that RBrs in PC exhibit a very close threshold potential to those of parent iodides in any solvent. This point is clearly evidenced in Fig. 4 where curve B (RBr) is quite similar to curve D (RI). This reinforced the proposal of a transient with RBrs of the form: R-Ag + ,A À (Scheme 1, reaction 2) in which the electrolyte anion A À plays the major role. Unfortunately, macro-electrolyses achieved with RBrs in PC at smooth silver turned to be fast self-inhibited by the adsorption of free alkyl radicals on silver or coverage of the surface by insoluble R-R dimers.

Voltammetries of tertiary alkyl bromides

Tert-Butyl bromide exhibits almost identical data at smooth silver and Ag-Pd electrode whatever the solvent. The unique observed step located at E p = À1.24 V at smooth silver and À1.22 V at Ag-Pd electrodes (both in 0.1 M TBABF 4 /PC) is always one-electron. Within a similar potential range, 1-bromoadamentane does not exhibit any visible reduction step at smooth silver in DMF. On the contrary, at an Ag-Pd electrode, it is curious to see the clear presence of an adsorption type reduction peak at À1.10 V. This could be explained by the compact coverage of the silver surface by adamently-Ag + ions slowly formed, and their further reduction into adamentyl radicals. However, 1-bromoadamentane recently reported needs further investigations. Very weak currents with RBrs enabled to repeat data on the homo-coupling of 1-iodoadamentyl recently reported at silver electrodes [32].

Reduction products at macro Ag-Pd cathodes

Macro-reductions of several primary alkyl bromides were achieved using large Ag-Pd electrodes (surface areas of about 3-4 cm 2 ). In particular, results in DMF containing TBABF 4 do confirm the one-electron character of the global cathodic process. Most reductions achieved at quite moderate potentials ranging from À1.2 V to À1.4 V vs. SCE are somewhat similar to the data already reported [10][11][12][13] at palladium cathodes (case of primary alkyl iodides) and at copper-palladium electrodes (case of primary alkyl bromides) [20]. They underline the peculiar role of palladium in the cleavage of carbon-halogen bonds. As seen in Table 2, preliminary macro-electrolyses achieved with RBrs in C 7 , C 8 , C 10 , and C 12 are in agreement with the transient formation of a free radical RÅ able to dimerize with a good efficiency. Additionally, a minor disproportionation of radicals leads to corresponding alkane and alkene. With short alkyl chains, the dimeric forms are formed almost quantitatively. Quite successfully, reduction of 12-bromododecanoic acid gave homo-dimer (tetracosanedioic acid) in high yield. On the contrary, long alkyl chains RBrs with R > C 12 displayed in the course of electrolyses a fast decay of currents after a few coulombs have passed through the cell. This behaviour can be explained by involving the precipitation of dimer crystals onto the surface, which could progressively poison its catalytic activity. This is displayed by the coverage of Ag crystals first eletrodeposited onto nickel (Fig. 5, image B) in the course of the reduction of 1-bromodecane in PC with the formation of fine dimers crystals on the nanometric scale. Those crystals are surprisingly equally distributed onto the silver surface. Under these conditions, the capability of Ag to catalyze in PC the one-electron process is clearly evidenced. On the other hand, surfaces covered by an Ag-Pd layer are displayed in Fig. 5, image C. After electrolysis of a RBr compound, the surface state is dramatically changed (image D) with formation of channels through the surface with apparently loss of palladium. This fact strongly supports the chemical implication of palladium in the one-electron reduction of RBrs. It is then believed that reaction of RBr with cathode material would lead to efficiently activate palladium Pd * by dissociation of Ag-Pd (initiation reaction (6), followed by a chain (reactions (7) and ( 8)) according to Scheme 2.

Conclusions

The aim of the present work was to describe a new cathodic material capable to achieve, at moderate potentials, one-electron reduction processes with primary alkyl bromides taken here as target substrates. Thus, for the first time, a Ag-Pd alloy considered as modifier of smooth silver surfaces was used to reduce non-activated C-Br bonds. The energy gain is large, often larger than 1 V, when referred to the data obtained at quite common electrode material such as glassy carbon. These preliminary results are extremely encouraging because this new cathodic material combines two important properties: (i) the specific catalysis of silver toward primary and secondary alkyl bromides and (ii) the role of palladium metal in directing such cleavages to one-electron reduction whatever the solvent. Therefore, these preliminary results have pointed out that the use of such Ag-Pd electrodes could be considered as a source of free alkyl radicals as shown by means of voltammetric analysis, coulometries and fixed potential electrolyses. The reduction step was found to be generally diffusion-controlled and electrolyses could be achieved stepwise. The stability of the surface appeared to be good and macro-electrolyses (current densities up to 50 mA cm À2 ) allowed several runs by using the same electrode. However, until now, the intrinsic reason to obtain electro-catalysis at such electrodes is not totally understood, and the unexpected synergy [Ag + Pd] needs additional studies even if the presence of Pd nanoparticles close to the cathode surface should play a crucial. A general scope with other kinds of organic substrates, which also possessing a chemical bond capable to cleave under electron transfer, will be undertaken.

RBr + Ag-Pd

RAg + , Br -+ Pd * Pd * + RBr R Pd Br

Fig. 1 .

 1 Fig. 1. Voltammetries obtained at different working electrodes using 1-bromooctane (concentration: 10 mM) in 0.1 M TBAPF 6 /DMF. Apparent surface areas, S = 0.78 mm 2 . Scan rate: 100 mV s À1 : (A) Glassy carbon, (B) Smooth silver with the separation of two steps: I and II, (C) Ag-Pd electrode.

Fig. 2 .

 2 Fig. 2. SEM images showing the reaction of the pristine silver with pure 1-bromooctane in 0.1 M TBAPF 6 /DMF: (a) After immersion during 30 min and a rinsing step with alcohol and then acetone. (b) The former surface (without rinsing and in contact with air) maintained at a potential of À2.0 V vs. SCE until nil current.

Fig. 3 . 4 ) 5 )

 345 Fig. 3. Voltammetric responses of bromo-alkanes at smooth silver and palladized silver in 0.1 M TBABF 4 /AN. Apparent surface areas, S = 0.78 mm 2 . Scan rate: 50 mV s -1 . (a) 1-Bromooctane (concentration: 9 mM) at A1, smooth silver and B1, Ag-Pd electrodes; (b) 2-bromopentane (concentration: 9.5 mM) at A2, smooth silver and B2, Ag-Pd electrodes.

Fig. 4 .

 4 Fig. 4. Comparison, under very similar experimental conditions, between voltammetric responses of 1-iodobutane (concentration: 9 mM) and 1-bromobutane (concentration: 11 mM) in 0.1 M TBAPF 6 /PC. Apparent surface areas, S = 0.78 mm 2 . Scan rate: 50 mV s À1 : (A) 1-iodobutane at a glassy carbon electrode, (B) 1-bromobutane at a smooth silver electrode, (C) Same than (B) using an Ag-Pd electrode, (D) 1-iodobutane at a smooth silver electrode.

Fig. 5 .

 5 Fig. 5. SEM images of silver deposits by galvanostatic means onto smooth nickel (electricity amount: 0.3 C cm À2 , image A) and glassy carbon (electricity amount: 0.6 C cm À2 , image C), where the silver deposit was transformed into an Ag-Pd layer. These two kinds of surface were used as catalysts for the reductive coupling of RBrs in different solvents containing 0.1 M TBABF 4 at À1.3 V vs. SCE. Reduction of 1-bromodecane in PC at Ni-Ag (image B) and 1-bromooctane in DMF at Ag-Pd supported by glassy carbon (image D). The electricity amounts were 0.6 C per cm 2 in both experiments.

R

  Scheme 2.

Table 1

 1 Voltammetric data for some bromo-alkanes in 0.1 M TBABF4 in DMF

	Entry RBr (concentration)	Glassy carbon el.	Smooth silver el.	Palladized silver el. Coulometric measurements b
				Ag-Pd el. c	Smooth Ag el.

Table 2

 2 Macro-electrolyses performed with a U-shaped cell with two compartments at Ag-Pd electrodes Separator: fritted glass. Working electrode surface area about 3.5 cm 2 . Solvents: DMF or PC containing 0.1 M TBABF 4 . Applied potential: E Red = À1.3 V vs. SCE.

	Substrate	Solvent	R(-H)%	R(H)%	R-R%	n (F mol À1 )
	1-Bromoheptane	DMF	-	-	>99	1.05
	1-Bromooctane	DMF	-	-	>99	1.1
	1-Bromodecane	DMF	5.9	4.4	90.7	1.1
	1-Bromododecane	DMF	55.3	27.7	17.1	1.2
	1-Bromododecane	PC	16.6	4.8	78.6	1.1
	1-Bromotetradecane	DMF	68	28	3	1.3
	1-Bromooctadecane	DMF	39.4	58.9	2.5	1.7
	12-Bromododecanoic acid	DMF	-	-	>99	1.1

Acknowledgements

The authors are deeply thankful to Professor V. Jouikov, Université de Rennes1, for his efficient collaboration (GC/MS analysis), as well as Michèle Nelson for helpful assistance.


Xin-Xin Han

Denis Efimov
email: denis.efimov@inria.fr

Andrey Polyakov
email: andrey.polyakov@inria.fr

Kai-Ning Wu

Input-to-state stability analysis of heat equation with boundary finite-time control

Keywords: heat equation, input-to-state stability, switching, boundary control, backstepping

, based on the backstepping approach, a switching boundary control law depending on the system state is designed. By estimating the upper bound of kernel functions, switching levels are determined and commutation law is further constructed. For the chosen switched control, the well-posedness property is verified. It is proved that the resulting system is input-to-state stable, and the solutions of the system will not exceed the highest admissible level dependent on the disturbance amplitude. Meanwhile, a stronger result is also obtained, that is the finite-time stability for the disturbance-free system. The numerical example is provided to support the derived result.

Introduction

Analysis and design of stable dynamical processes have always been in the focus of control theory research. In the uncertainty-free case the rates of convergence to the desired mode constitute one of the main characteristics of a system such as asymptotic stability [START_REF] Egidio | Global asymptotic stability of limit cycle and H 2 /H ∞ performance of discrete-time switched affine systems[END_REF], finite-time and fixed-time stability [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit lyapunov function approach[END_REF], etc. However, in real world, the system under consideration may be disturbed by various perturbations or noises. In this situation, it is of vital importance to proceed to the robustness analysis. The concept of input-to-state stability (ISS) introduced by E. Sontag in [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] plays a fundamental role in the robust stability analysis, which provides a framework to tackle the stability with respect to disturbance inputs. Conceptually, ISS is characterized by the dynamical be-havior of the system being bounded with respect to arbitrary bounded input perturbations, and also asymptotically stable in the absence of external inputs. Nowadays, fruitful results in ISS for finite-dimensional systems have been carried out, see for instance [START_REF] Dashkovskiy | Input-to-state stability of nonlinear impulsive systems[END_REF][START_REF] Teel | A unified framework for input-to-state stability in systems with two time scales[END_REF].

On the other hand, the problem of ISS for infinitedimensional systems is as important as that for finitedimensional systems. To name a few examples, reaction process [START_REF] Christofides | Nonlinear and robust control of PDE systems: Methods and applica-tions to transport-reaction processes[END_REF], robotic arms [START_REF] Géradin | Mechanical vibrations: theory and application to structural dynamics[END_REF], aircraft wings [START_REF] Robu | Simultaneous H ∞ vibration control of fluid/plate system via reduced-order controller[END_REF], etc., all require the robust control methods in practical applications. The investigation of ISS for infinite-dimensional systems has also received a considerable attention (see e.g., [START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF][START_REF] Mironchenko | Input-to-state stability of infinite-dimensional systems: recent results and open questions[END_REF]). In [START_REF] Jayawardhana | Infinite-dimensional feedback systems: the circle criterion and input-to-state stability[END_REF] the authors developed the ISS theory to the infinite-dimensional systems for the first time. Afterwards, some qualitative results of ISS were established [START_REF] Guiver | Infinite-dimensional Lur'e systems: Input-to-state stability and convergence properties[END_REF][START_REF] Mironchenko | Characterizations of input-to-state stability for infinite-dimensional systems[END_REF]. Aiming at the infinite-dimensional systems governed by PDE, plenty of works on the ISS analysis and control have focused on the distributed disturbances [START_REF] Wei | Input-to-state stability of impulsive reaction-diffusion neural networks with infinite distributed delays[END_REF][START_REF] Mazenc | Strict lyapunov functions for semilinear parabolic partial differential equations[END_REF]. Meanwhile for the situation involving boundary disturbances, various methodologies have been proposed more recently [START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF][START_REF] Zheng | ISS-like estimates for nonlinear parabolic PDEs with variable coefficients on higher dimensional domains[END_REF]. For instance, spectral decomposition and finite difference scheme [START_REF] Karafyllis | ISS with respect to boundary disturbances for 1-D parabolic PDEs[END_REF], monotonicity-based approach [START_REF] Mironchenko | Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances[END_REF], Lyapunov methods [START_REF] Zheng | Input-to-state stability with respect to boundary disturbances for a class of semilinear parabolic equations[END_REF], De Giorgi iteration [START_REF] Zheng | A De Giorgi iteration-based approach for the establishment of ISS properties for Burgers equation with boundary and in-domain disturbances[END_REF], etc. The aforementioned ISS results indicate that the trajectories of the system in the disturbance-free case are asymptotically stable.

In particular, many real-world application settings also require the disturbance-free system to guarantee finitetime stability [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF][START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF]. Compared with the asymptotic stability, it is recognized that finite-time stability plays a more important role following some specific control engineering demands, such as finite-time deployment for large networks of multi-agent systems [START_REF] Meurer | Finite-time multi-agent deployment: A nonlinear PDE motion planning approach[END_REF], finite-time operation duration of solid propellant rockets [START_REF] Smyshlyaev | Closed-form boundary state feedbacks for a class of 1-D partial integrodifferential equations[END_REF], etc. Nevertheless, the issue that whether ISS and finite-time stability can be satisfied (or be guaranteed by some control approaches) for the disturbed and disturbancefree PDE systems respectively has not been sufficiently explored, which is the motivation of this study.

Currently, control placements of PDE systems from the perspective of its spatial distribution are either distributed over the entire spatial domain (which we usually call the distributed control [START_REF] Wu | Design of distributed H ∞ fuzzy controllers with constraint for nonlinear hyperbolic PDE systems[END_REF]), or only placed at the boundary of the spatial domain (that is the boundary control [START_REF] Rathnayake | Observer-based event-triggered boundary control of a class of reaction-diffusion PDEs[END_REF]), or applied at some specific points in the spatial domain (which is commonly known as pointwise control [START_REF] Ammari | Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force[END_REF][START_REF] Badkoubeh | Flatnessbased deformation control of an Euler-Bernoulli beam with in-domain actuation[END_REF][START_REF] Rebarber | Exponential stability of coupled beams with dissipative joints: a frequency domain approach[END_REF] or interior point control [START_REF] Byrnes | Interior point control of a heat equation using zero dynamics design[END_REF]). Among them boundary control, as a novel approach to effectively save the implementation cost, has gradually come into researchers' mind [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF][START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reactiondiffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF][START_REF] Guo | Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation[END_REF][START_REF] Wu | Fuzzy boundary control design for a class of nonlinear parabolic distributed parameter systems[END_REF]. One can refer to the foundational work from Miroslav Krstic by backstepping method [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF][START_REF] Boskovic | Boundary control of an unstable heat equation via measurement of domain-averaged temperature[END_REF]. Some existing results on the ISS boundary stabilization can be found in [START_REF] Zhang | Exponential input-to-state stabilization of an ODE cascaded with a reaction-diffusion equation to disturbances[END_REF][START_REF] Zhang | Input-to-state stability of coupled hyperbolic PDE-ODE systems via boundary feedback control[END_REF][START_REF] Zheng | A De Giorgi iteration-based approach for the establishment of ISS properties for Burgers equation with boundary and in-domain disturbances[END_REF] and references therein. Nevertheless, regarding the dual requirements of ISS boundary stabilization for the resulting PDE system and finite-time boundary stabilization for the corresponding disturbance-free system, serious technical challenges arise in the boundary feedback controller design and such stability analysis comparing to the asymptotic ISS case. In recent years, some efforts have been made to stabilize the (disturbance-free) parabolic PDE systems in finite time via different boundary control design methods [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF][START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF][START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF]. However, when disturbances enter the system, ISS analysis under the boundary finite-time control methods is a problem that are worthy of probing. For example, the results of [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF] guarantee finite-time stability through boundary control and time-varying kernel functions, whose utilization in the backstepping transformations with the presence of disturbances leads to unbounded gains for perturbations in the target dynamics. The results of [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF][START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF] are based on switching strategy with piecewise constant kernel functions for different boundary actuations, and growing to infinity control gains and rate of commutation are serious obstacles for analysis of these systems in the presence of exogenous inputs (the definition and existence of solutions is an issue). For these reasons, IS-S analysis via boundary finite-time control strategy becomes complicated and furthermore, the achievement of the aforementioned dual requirements is still open. We are going to fill this gap in the present work. This paper extends the results in [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF] to the system with disturbance input, steering the solution of the closed-loop system to achieve ISS-like stabilization. Based on [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF] a piecewise linear boundary controller with a statedependent switching law is presented. By estimating the bound of kernel functions, switching levels are determined and switching law is constructed. The main result states that the solution of the system cannot exceed the highest level dependent on the disturbance amplitude based on the switching framework we deigned, which makes it possible to achieve ISS property. It is worth mentioning that a stronger ISS result is obtained in this paper. Specifically, in addition to establishing an ISS-like estimate in the presence of disturbance, finitetime stability is also guaranteed for the disturbance-free case. The strategy of hybrid boundary control and analysis methods of ISS may be applicable for some other types of PDE systems.

The paper is organized as follows. Section 2 provides the preliminaries. The model description is proposed in Section 3.1. With the switching law described in Section 3.2, the well-posedness analysis is further discussed in Section 3.3. In Section 4, the main result of ISS-like stabilization is derived via rigorous computations. The numerical simulation is presented in Section 5 to illustrate the theoretical result. Section 6 concludes this study.

Notation: R is the set of real numbers and R + = {x ∈ R : x ≥ 0}, |x| corresponds to absolute value of x ∈ R. Z, Z + and Z -represent the sets of integers, nonnegative integers and negative integers, respectively. Denote

f (x) = df (x)
dx for a differentiable function f : R → R. For any interval I ⊆ R + , let C k (I, X) with k ∈ Z + be the space consisting of all k-order continuously differentiable functions defined on I, which take values in the space X. Denote by L 2 (I, R) the Lebsgue space of square integrable functions f :

I → R equipped with the L 2 -norm f = I f 2 (x)dx. Let Q (a,b) = (a, b) × (0, 1) for (a, b) ⊂ R + (or Q t for (a, b) = (0, t) with t ∈ R + ∪{∞}), we use L ∞ (Q (a,b) ) := L ∞ (Q (a,b) , R)
to denote the set of essentially bounded functions d :

Q (a,b) → R with corresponding norm d L ∞ (Q (a,b) ) = ess sup (s,x)∈(a,b)×(0,1) |d(s, x)| < ∞. J m (p) and I m (p)
with m ∈ Z denote the Bessel and modified Bessel functions of the first kind, respectively, of the order m for complex argument p.

Preliminaries

We recall that a continuous function θ : R + → R + belongs to class-K if it satisfies θ(0) = 0 and it is strictly increasing; θ is a class-K ∞ function if it belongs to class-K and θ(s) → ∞ as s → ∞ also. A continuous function ψ : R + × R + → R + is a class-KL function if ψ(•, t) belongs to class-K for each fixed t ≥ 0, and it decreases to zero as t → ∞ for each fixed s ≥ 0.

∈ C 1 ([0, l], R) with z(0) = z(l) = 0: l 0 z 2 (ξ)dξ ≤ l 2 π 2 l 0 z 2 ξ (ξ)dξ.
3 Model description and well-posedness analysis

Consider the following scalar system described by heat equation:

   u t (t, x) = u xx (t, x) + d(t, x), t > 0, x ∈ (0, 1), u(t, 0) = 0, u(t, 1) = U (t), t > 0, u(0, x) = u 0 (x), x ∈ (0, 1), (1)
where u(t, x) ∈ R is the system state at time t ∈ R + and spatial position x ∈ (0, 1), d(t, x) ∈ R is the external input which belongs to L ∞ (Q ∞ ). The initial condition u 0 ∈ L 2 ((0, 1), R) and satisfies zero order compatibility condition u 0 (0) = 0 and u 0 (1) = U (0), where U (t) ∈ R is the boundary control input.

Next we will formulate the definition of ISS estimate in L 2 norm.

Definition 3.1 It is said that system (1) admits an IS-S estimate in L 2 norm with respect to the initial value

u 0 ∈ L 2 ((0, 1), R) and input d ∈ L ∞ (Q ∞ ), if there exist functions ν ∈ K and β ∈ KL satisfying u(t, •) ≤ β( u 0 , t) + ν( d L ∞ (Qt) )
for all t > 0.

In this work, the main goal is to design a novel hybrid boundary feedback control law, via backstepping method, that provides the system (1) with ISS. The hybrid boundary controller is given as

U (t) = 1 0 K(1, y, 2 σ(t) )u(t, y)dy (2)
and

U (t) = 0 if u(t, •) = 0, where K : D → R defined in D := {(x, y, i) ∈ [0, 1] 2 × Z + : y ≤ x} is the kernel of the backstepping transformation, σ : R + → Z is a function being constant for t ∈ [t k , t k+1
), and t k for k ∈ Z + with t 0 = 0 is the sequence of time instances for control commutation. The shape of the kernel K and the values of σ(t) and t k will be specified later.

Backstepping transformation and inverse transformation

Following the standard framework of backstepping approach [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF], we search for the invertible Volterra integral transformation

w(t, x) = u(t, x) - x 0 K(x, y, 2 σ(t) )u(t, y)dy (3)
that maps system (1) into the following target system

       w t (t, x) = w xx (t, x) -2 σ(t) w(t, x) + d(t, x) - x 0 K(x, y, 2 σ(t) )d(t, y)dy, w(t, 0) = 0, w(t, 1) = 0, (4)
with

w(t k , x) = u(t k , x) - x 0 K(x, y, 2 σ(t k ) )u(t k , y)dy and t ∈ [t k , t k+1 ) (σ(t) is a constant on this interval of time).
After utilizing integration by parts and taking into account boundary conditions, substitution of transformation (3) into target system (4) and original system (1) leads to the following PDE that the kernel K(x, y, 2 σ(t) ) satisfies

       Kxx(x, y, 2 σ(t) ) -Kyy(x, y, 2 σ(t) ) -2 σ(t) K(x, y, 2 σ(t) ) = 0, K(x, 0, 2 σ(t) ) = 0, d dx K(x, x, 2 σ(t) ) = -2 σ(t)-1 . (5)
Here and after, we use the notation

d dx K(x, x, 2 σ(t) ) := K x (x, y, 2 σ(t) )| y=x + K y (x, y, 2 σ(t) )| y=x .
Note that after a straightforward calculation [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF][START_REF] Smyshlyaev | Closed-form boundary state feedbacks for a class of 1-D partial integrodifferential equations[END_REF], the solution of PDE [START_REF] Christofides | Nonlinear and robust control of PDE systems: Methods and applica-tions to transport-reaction processes[END_REF], with the help of modified Bessel function I 1 (•), is given by

K(x, y, 2 σ(t) ) = -y2 σ(t) I 1 2 σ(t) (x 2 -y 2 ) 2 σ(t) (x 2 -y 2 ) . (6)
Remark 3.1 Based on some properties of modified Bessel function I 1 (•), the well-posedness of ( 5) is ensured by explicitly giving an analytical solution [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF]. Note that it holds only for constant σ(t) instead of a function σ(t) of t. In other words, K(x, y, 2 σ(t) ) is a solution of PDE (5) when σ(t) is a constant, i.e., t ∈ [t k , t k+1 ). Therefore, target system (4) holds for t ∈ [t k , t k+1 ), and the state function of the first equation of target system (4) will jump if σ(t) switches to next mode. It is worth remarking that for the original system (1), it remains a continuous solution in L 2 norm even though there exist some jumps in the right boundary of the spatial domain. This will be proven in the following subsection and also illustrated in the numerical simulation.

On the other hand, the form of inverse backstepping transformation is determined as

u(t, x) = w(t, x) + x 0 L(x, y, 2 σ(t) )w(t, y)dy (7)
for t ∈ [t k , t k+1 ), where the kernel L(x, y, 2 σ(t) ) of the inverse transformation satisfies the following PDE:

       Lxx(x, y, 2 σ(t) ) -Lyy(x, y, 2 σ(t) ) + 2 σ(t) L(x, y, 2 σ(t) ) = 0, L(x, 0, 2 σ(t) ) = 0, d dx L(x, x, 2 σ(t) ) = -2 σ(t)-1 . (8)
Similarly, it can be verified [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF][START_REF] Smyshlyaev | Closed-form boundary state feedbacks for a class of 1-D partial integrodifferential equations[END_REF] that PDE (8) has a well-posed solution of the form

L(x, y, 2 σ(t) ) = -y2 σ(t) J 1 2 σ(t) (x 2 -y 2 ) 2 σ(t) (x 2 -y 2 ) ,
where J 1 (•) is the Bessel function of the first kind of order 1.

For further analysis, we provide the following estimates of the backstepping transformation and inverse transformation.

Proposition 3.1 For the transformations ( 3) and ( 7), the following estimates are satisfied

w(t, •) ≤ Ω 1 (2 σ(t) ) u(t, •) ; u(t, •) ≤ Ω 2 (2 σ(t) ) w(t, •) (9)
for all t ∈ [t k , t k+1 ),
where

Ω 1 (s) = 1 + s 1 0 x 0 y 2 I 0 ( s(x 2 -y 2 )) -I 2 ( s(x 2 -y 2 )) 2 dydx 2 , Ω 2 (s) = 1 + s 1 0 x 0 y 2 J 0 ( s(x 2 -y 2 )) + J 2 ( s(x 2 -y 2 )) 2 dydx 2 .
Proof. Using the Cauchy-Schwarz inequality and [START_REF] Temme | Special functions: An introduction to the classical functions of mathematical physics[END_REF], we have

I k (p) = p 2k (I k-1 (p) -I k+1 (p))
w(t, •) ≤ u(t, •) + x 0 K(x, y, 2 σ(t) )u(t, y)dy ≤ 1 + 1 0 x 0 K 2 (x, y, 2 σ(t) )dydx × u(t, •) = 1 + 2 σ(t) 1 0 x 0 y 2 (I0(p) -I2(p)) 2 dydx 2 × u(t, •) =Ω1(2 σ(t) ) u(t, •) , where p = 2 σ(t) (x 2 -y 2 ).
Similarly, the estimate of u(t, •) in ( 9) can be obtained further by using J k (p) = p 2k (J k-1 (p) + J k+1 (p)) [START_REF] Temme | Special functions: An introduction to the classical functions of mathematical physics[END_REF].

Switching law design

Define switching level η i for i ∈ Z recursively as Ω(2 i )η i = η i-1 with η 0 = 1, where Ω(s) = Ω 1 (s)Ω 2 (s).

In order to ensure system (1) to be ISS under the hybrid boundary feedback control law (2), we design the switching signal σ(t) following the rule:

σ(t) =    i + 1, if σ(t) = i and u(t, •) ≤ η i+1 i, if σ(t) = i and η i+1 < u(t, •) < η i-1 i -1, if σ(t) = i and u(t, •) ≥ η i-1
(10) for t < t * and σ(t) = 0 for all t ≥ t * , where

t * = inf{t ≥ 0| u(t, •) = 0}. Here, switching level η σ(t) determines the switching point t k , k ∈ Z + satisfying t k+1 = inf{t ∈ (t k , t * )| u(t, •) = η σ(t)+1 or u(t, •) = η σ(t)-1 }, t 0 = 0. Remark 3.2 It is obvious from (10) that σ(t) is a function of u(t, •) .
In the sequel, we present the relation between these two functions (see Proposition 3.2), which will be helpful for the following robustness analysis. Furthermore, for the function Ω(s) that determines the switching level, we can see that Ω(s) (s > 0) is an increasing function and Ω(s) → +∞ as s → +∞. Indeed, a straightforward calculation yields

Ω 1 (s) = (1 + s h 1 (s)) = h 1 (s) + 1 2 √ h1(s) 1 0 x 0 y 2 I1( √ s(x 2 -y 2 ))I2( √ s(x 2 -y 2 )) √ s(x 2 -y 2 ) dydx utiliz- ing recurrence relation I k (p) = I k+1 (p) + k p I k (p) [37],
where h 1 (s) = π 0 e p cos θ cos(kθ)dθ [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF]), we derive that Ω 1 (s) ≥ 0 and thus Ω 1 (s) increases. In addition, it can be deduced by

J k (p) = -J k+1 (p) + k p J k (p) [37] that Ω 2 (s) = h 2 (s) - 1 2 √ h2(s) 1 0 x 0 y 2 J1( √ s(x 2 -y 2 ))J2( √ s(x 2 -y 2 )) √ s(x 2 -y 2 )
dydx, where

h 2 (s) = 1 0 x 0 y 2 J1( √ s(x 2 -y 2 )) √ s(x 2 -y 2 ) 2 dydx. Then we infer from J 1 (p) = -J 0 (p) [37] that Ω 2 (s) ≥ 0 is equivalent to h 3 (s) := 1 0 x 0 y 2 J0( √ s(x 2 -y 2 ))J1( √ s(x 2 -y 2 )) √ s(x 2 -y 2 ) dydx = 1 2s 1 2 - 1 0 x 0 J 2 0 ( s(x 2 -y 2 ))dydx ≥ 0. Notice that |J k (p)| ≤ 1 by the representation J k (p) =
1 π π 0 cos (kθ -p sin(θ)) dθ [START_REF] Temme | Special functions: An introduction to the classical functions of mathematical physics[END_REF], thus h 3 (s) ≥ 0 holds and Ω 2 (s) is also increasing. To sum up, the monotonicity of Ω(s) = Ω 1 (s)Ω 2 (s) follows immediately from the fact that Ω 1 (s) > 0 and Ω 2 (s) > 0. On the other hand, using the representation I k (p) = 1 π π 0 e p cos θ cos(kθ)dθ [START_REF] Watson | A treatise on the theory of Bessel functions[END_REF], we know that

I 0 (p) -I 2 (p) = 1 π π 0 e p cos θ (1 - cos(2θ))dθ ≥ 1 π π 2 0 (1 -cos(2θ))dθ = 1 2 for p ≥ 0. Since Ω 2 (s) ≥ 1, then Ω(s) ≥ Ω 1 (s) ≥ 1 + s 8 √ 3 → +∞ for s → +∞. Hence, Ω(s) → +∞ as s → +∞. Proposition 3.2 There exist η, η ∈ K ∞ such that η(2 -σ(t) ) ≤ η σ(t) ≤ η(2 -σ(t)
) for all t ∈ [0, t * ) provided that σ(t) is defined by [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF] and thus we have

1 2η -1 ( u(t, •) ) ≤ 2 σ(t) ≤ 2 η -1 ( u(t, •) ) ( 11
)
holds for all t ∈ [0, t * ).

Proof. According to the switching law [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF], we have for all t ∈ [0, t * ) that η σ(t)+1 ≤ u(t, •) ≤ η σ(t)-1 holds. On the other hand, based on the relation

Ω(2 σ(t) )η σ(t) = η σ(t)-1 , we can get η σ(t) = σ(t) k=1 1 Ω(2 k )
when σ(t) ∈ Z + and η σ(t) = 0 k=σ(t)+1 Ω(2 k ) when σ(t) ∈ Z -, and η σ(t) → 0 as σ(t) → +∞ or η σ(t) → +∞ as σ(t) → -∞ (it is a decreasing step function). Therefore, there exists a non-decreasing function η : R + → R + such that η σ(t) = η(2 -σ(t) ). Indeed, in order to find two class K ∞ functions η, η, which form lower and upper bounds for η, we observe that η( 2

-σ(t) ) = η σ(t) → 0 as 2 -σ(t) → 0 (σ(t) → +∞) and η(2 -σ(t) ) = η σ(t) → +∞ as 2 -σ(t) → +∞ (σ(t) → -∞). In this case, η(2 -σ(t)
) is an increasing step function with respect to argument 2 -σ(t) . Thus the relation η( 2

-σ(t) ) ≤ η σ(t) ≤ η(2 -σ(t) ) holds. Furthermore, it implies that η(2 -σ(t)-1 ) ≤ u(t, •) ≤ η(2 -σ(t)+1
) for all t ∈ [0, t * ). Consequently,

1 2η -1 ( u(t,•) ) ≤ 2 σ(t) ≤ 2 η -1 ( u(t,•) ) is valid for all t ∈ [0, t * ).

Well-posedness analysis

Before stating the main result, let us discuss the wellposedness of the closed-loop system (1) with ( 2) and [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF]. First we give the following definition about the solution of system (1) with ( 2) and [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF]. Definition 3.2 The solution of system (1) with ( 2) and ( 10) is a tuple (u, σ): u ∈ C 0 (R + , L 2 ((0, 1), R)) and σ : R + → Z satisfying the switching rule [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF]. Here, u is a solution to the system (1) understood in the weak sense, that is the following equality

- 1 0 ξ(0, x)u0(x)dx - T 0 1 0 ξt(t, x)u(t, x)dxdt + T 0 ξx(t, 1) 1 0 K(1, y, 2 σ(t) )u(t, y)dydt - T 0 1 0 ξxx(t, x)u(t, x)dxdt - T 0 1 0 d(t, x)ξ(t, x)dxdt = 0
holds for a.e. T ∈ R + and for each ξ ∈ C 2 ([0, T ) × [0, 1]) with compact support in [0, T ) × [0, 1] and ξ vanished on [0, T ) × {0, 1}. Theorem 3.1 Assume that u 0 ∈ L 2 ((0, 1), R) and d ∈ L ∞ (Q ∞ ). Then closed-loop system (1) with ( 2) and ( 10) has a unique solution (u, σ): u ∈ C 0 (R + , L 2 ((0, 1), R)) and σ : R + → Z.

Proof. For simplicity of notation, we denote U (t) = K σ(t) u(t, •), where K σ(t) is a linear operator of the form

K σ(t) v = 1 0 K(1, y, 2 σ(t) )v(y)dy for v ∈ L 2 ((0, 1), R). Since |I k (p)| ≤ e p ≤ e √ 2 σ(t) where p = 2 σ(t) (x 2 -y 2 ) (0 ≤ y ≤ x ≤ 1), we can get |K(x, y, 2 σ(t) )| = |y2 σ(t) I0(p)-I2(p) 2 | ≤ 2 σ(t) e √ 2 σ(t) and thus |K σ(t) u(t, •)| ≤ 2 σ(t) e √ 2 σ(t) u(t, •) := M σ(t) u(t, •) holds. In fact, σ(t)
is a function of u(t, •) and depends on the number of switching. Defining q as the number of commutations in the system (1), in order to construct the resulting switching signal σ(t), we introduce the notation σ q (t) to represent the corresponding switching signal (step) function when the number of commutation is q.

In what follows, we use the fixed point argument [START_REF] Kantorovich | Functional analysis[END_REF]. Set U q 0 (t, x) = 0 and let U q n (t, x)(n ≥ 1) be the unique solution of

     U q n,t (t, x) = U q n,xx (t, x) + d(t, x), U q n (t, 0) = 0, U q n (t, 1) = K σ q (t) U q n-1 (t, •), U q n (0, x) = u 0 (x).
Define E q n (t, x) = U q n+1 (t, x) -U q n (t, x), then we obtain

     E q n,t (t, x) = E q n,xx (t, x), E q n (t, 0) = 0, |E q n (t, 1)| ≤ M σ q (t) E q n-1 (t, •) , E q n (0, x) = 0.
Step 0 (q = 0): For the given initial value u 0 (x), two scenarios are possible: either u 0 = 0, then by rule [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF]

σ(t) = 0 for all t ≥ t 0 . Since d ∈ L ∞ (Q ∞ ),
the considered system has a unique solution u ∈ C 0 (R + , L 2 ((0, 1), R)) and thus the pair (u, σ) exists [27, Def. 2.3, Ch. 4, pp. 106]; or u 0 = 0, then σ 0 (t) is determined by u 0 and now σ 0 (t) (also for M σ 0 (t) ) is a constant for t ≥ 0. Using the proof of Lemma 6 in [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF], we conclude that sequence {U 0 n (t, •)} will converge to its limit U 0 ∞ (t, •) and system (1) has a unique solution U 0 ∞ ∈ C 0 (R + , L 2 ((0, 1), R)). In this case, we determine a pair (U 0 ∞ (t, •), σ 0 (t)) for t ≥ 0.

Step 1 (q = 1): Note that from rule [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF] the first switching will happen either when U 0 ∞ (t, •) reaches the switching level η σ 0 (t)+1 or η σ 0 (t)-1 or there exists the point

t * = inf{t ≥ t 0 | U 0 ∞ (t, •) = 0}. Define switching point t 1 = inf{t > t 0 | U 0 ∞ (t, •) = η σ 0 (t)+1 or U 0 ∞ (t, •) = η σ 0 (t)-1 }. If t * < t 1 ,
the first switching happens at t * and by ( 10) σ(t) = 0 for t ≥ t * (for t ∈ [t 0 , t * ), σ(t) = σ 0 (t) ), then the solution U 0 ∞ (t, •) will exit or stay at the origin after t * . To conclude, system (1) has a unique solution u ∈ C 0 (R + , L 2 ((0, 1), R)) and the pair (u, σ) is determined. If t * ≥ t 1 , the first switching happens at t 1 . For t ∈ [t 0 , t 1 ), we have σ 1 (t) = σ 0 (t); and for t ≥ t 1 , σ 1 (t) is determined by U 0 ∞ (t 1 , •) . To sum up, σ 1 (t), t ≥ 0 is defined by U 0 ∞ (t, •). It is obvious that σ 1 (t), t ≥ 0 is a piecewise constant and then we have M σ 1 (t) ≤ C for some constant C > 0. Following the similar line as that in [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF], one can also verify the existence of the unique solution

U 1 ∞ ∈ C 0 (R + , L 2 ((0, 1), R)) and a pair (U 1 ∞ (t, •), σ 1 (t)) for t ≥ 0 is then confirmed.
Iterating the aforementioned procedure, we can obtain some pairs (U q ∞ (t, •), σ q (t)) for each q and the unique solution (u(t, •), σ(t)) of system ( 1) with ( 2) and ( 10) can be found.

Main result

We are in a position to state the criterion of ISS via hybrid boundary control in term of the designed statedependent switching law. We will omit the arguments (t, x) wherever no confusion arises. Theorem 4.1 For any initial switching mode σ(0) ∈ Z with u 0 ∈ (η σ(0)+1 , η σ(0) ] or u 0 = 0, system (1) with the hybrid boundary control law (2) under the switching rule [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF] admits an ISS estimate in L 2 norm with respect to the initial value u 0 ∈ L 2 ((0, 1), R) and input d ∈ L ∞ (Q ∞ ), which is given by [START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF]. Moreover, the system in the case of d(t, x) = 0 is finite-time stable, and the settling-time function T satisfies T < 3

+∞ j=σ(0) qj 2 j - q σ(0)
2 σ(0)-1 < +∞ , where q j = ln(Ω(2 j )).

Proof. Let w n be the classical solutions of system (4) with disturbance functions

d n ∈ C ∞ (R + × (0, 1), R) sat- isfying d n (t, •) -d(t,
•) → 0 as n → +∞ uniformly on any compact interval of time, then w n (t, •)w(t, •) → 0 as n → +∞ uniformly on any compact interval of time [27, Th. 2.7, Ch. 4, pp. 108], where w is the unique mild solution of system (4), which, due to the transformation [START_REF] Dashkovskiy | Input-to-state stability of nonlinear impulsive systems[END_REF], corresponds to the unique weak solution of system [START_REF] Ammari | Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force[END_REF]. Take the Lyapunov functional

V (w n (t, •)) = 1 2 1 0 w 2 n (t, x)dx.
In what follows, our proof is carried out on two intervals [0, t * ) and [t * , +∞), where

t * = inf{t ≥ 0| u(t, •) = 0}.
For t ∈ [0, t * ) (in this case, u 0 = 0 by default, otherwise we have t * = 0), calculating the derivative of sys-tem for w n on each subinterval [t k , t k+1 ) gives

V (w n (t, •)) = 1 0 w n w n,xx -2 σ(t) w 2 n + w n d n -w n x 0 K(x, y, 2 σ(t) )d n (t, y)dy dx. (12)
Applying integration by parts and utilizing the boundary conditions to w n with d n , we obtain by virtue of Lemma 2.1 that

1 0 w n w n,xx dx =w n w n,x 1 0 - 1 0 w 2 n,x dx ≤ -π 2 1 0 w 2 n dx. (13)
Substituting ( 13) into (12) yields

V (w n (t, •)) ≤ -π 2 1 0 w 2 n dx -2 σ(t) 1 0 w 2 n dx + 1 0 w n d n dx - 1 0 w n x 0 K(x, y, 2 σ(t) )d n (t, y)dydx ≤ -(2 σ(t) + π 2 ) w n (t, •) 2 + w n (t, •) • d n (t, •) + w n (t, •) • x 0 K(x, y, 2 σ(t) )d n (t, y)dy ≤ -(2 σ(t) + π 2 ) w n (t, •) 2 + w n (t, •) • d n (t, •) × 1 + 2 σ(t) 1 0 x 0 y 2 (I 0 (p) -I 2 (p)) 2 dydx 2 ≤ -2π 2 V (w n (t, •)) + Ω 1 (2 σ(t) ) d n (t, •) × 2V (w n (t, •)) (14) for t ∈ [t k , t k+1 ), where p = 2 σ(t) (x 2 -y 2 ). Notice that the solution of ODE ς(t) = -aς(t) + b(t) ς(t), t > t k with the initial value ς(t k ) > 0 is ς(t) = (e -a 2 (t-t k ) [ ς(t k ) + 1 2 ( t t k e a 2 (s-t k ) b(s)ds)]) 2 . Setting a = 2π 2 and b(t) = √ 2Ω 1 (2 σ(t) ) d n (t,
•) and applying the comparison principle to [START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reactiondiffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF], we obtain

V (w n (t, •)) ≤e -π 2 (t-t k ) V (w n (t k , •)) + Ω 1 (2 σ(t) ) √ 2 t t k (e π 2 (s-t k ) × d n (s, •) )ds , t ∈ [t k , t k+1 ). (15)
Letting n → ∞ in (15), we derive

V (w(t, •)) ≤e -π 2 (t-t k ) V (w(t k , •)) + Ω 1 (2 σ(t) ) √ 2π 2 d L ∞ (Qt) × (1 -e -π 2 (t-t k ) ), t ∈ [t k , t k+1 ).
Then it follows from (9) that

1 Ω 2 (2 σ(t) ) u(t, •) ≤ w(t, •) ≤e -π 2 (t-t k ) Ω 1 (2 σ(t) ) u(t k , •) + Ω 1 (2 σ(t) ) π 2 d L ∞ (Qt) × (1 -e -π 2 (t-t k ) ), t ∈ [t k , t k+1 ), that is u(t, •) ≤Ω(2 σ(t) ) e -π 2 (t-t k ) u(t k , •) + 1 π 2 (1 -e -π 2 (t-t k ) ) × d L ∞ (Qt) , t ∈ [t k , t k+1 ). (16)
Recalling the relation [START_REF] Géradin | Mechanical vibrations: theory and application to structural dynamics[END_REF] and using the properties of Ω(s), it holds that

α( u(t, •) ) = u(t, •) Ω( 2 η -1 ( u(t,•) ) ) ≤ u(t, •) Ω(2 σ(t) ) ≤e -π 2 (t-t k ) u(t k , •) + 1 π 2 (1 -e -π 2 (t-t k ) ) d L ∞ (Qt) ≤e -π 2 (t-t k ) u(t k , •) + 1 π 2 d L ∞ (Qt) , t ∈ [t k , t k+1 ) (17) where α(s) = s Ω( 2 η -1 (s)
) and α ∈ K ∞ .

Therefore, an ISS-like estimate has been obtained for each interval of time [t k , t k+1 ), and now our goal is to derive a similar estimate for all t ∈ [0, t * ). To this end, the following analysis will be proceeded.

Due to [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF], u(t k , •) = η σ(t) for k > 0 and u(t 0 , •) = u 0 . Hence, for k > 0 from [START_REF] Jayawardhana | Infinite-dimensional feedback systems: the circle criterion and input-to-state stability[END_REF] we have the condition of boundedness for a trajectory:

e -π 2 (t-t k ) η σ(t) + 1 π 2 (1 -e -π 2 (t-t k ) ) d L ∞ (Qt) ≤ η σ(t) or equivalently, 1 π 2 d L ∞ (Qt) ≤ η σ(t) . Therefore, for d L ∞ (Qt) = 0, there exists i * ∈ Z such that η i * +1 ≤ 1 π 2 d L ∞ (Qt) ≤ η i *
for any input d with a finite norm d L ∞ (Qt) . Moreover,

µ(2 -i * ) ≤ π 2 η(2 -i * -1 ) ≤ π 2 η i * +1 ≤ d L ∞ (Qt)
for some µ ∈ K ∞ , then

η i * ≤ ( d L ∞ (Qt) )
for (s) = η • µ -1 (s), where the functions η and η from the class-K ∞ were defined in the proof of Proposition 3.2. Note that due to [START_REF] Jayawardhana | Infinite-dimensional feedback systems: the circle criterion and input-to-state stability[END_REF] the trajectory u(t, •) cannot exceed the level η i * -1 for the given input d provided that u 0 < η i * -1 .

Based on the above analysis, we have the estimate u(t k , •) ≤ max{ u 0 , η i * }. Then it follows from ( 17) that

α( u(t, •) ) ≤e -π 2 (t-t k ) max{ u 0 , η i * } + 1 π 2 d L ∞ (Qt) ≤e -π 2 (t-t k ) u 0 + κ( d L ∞ (Qt) ), t ∈ [t k , t k+1 )
where κ(s) = (s) + s π 2 and κ ∈ K ∞ .

Finally, we will demonstrate that there exists a finite upper bound T of t k . To this end let us first exploit a known conclusion for the convergence time of system (1) without external input. It is shown in [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF] that system (1) under the hybrid boundary control law [START_REF] Badkoubeh | Flatnessbased deformation control of an Euler-Bernoulli beam with in-domain actuation[END_REF] with the switching rule [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF] in the case of d(t, x) = 0 is finitetime stable, and the convergence time to the origin is T ≤ 3

+∞ j=σ(0) qj 2 j - q σ(0)
2 σ(0)-1 < +∞, where q j = ln(Ω(2 j )). Hence, there exists an upper estimate T ∈ K of the settling time, i.e., u(t, •) = 0 for t ≥ T ( u 0 ), and T = T ( u 0 ) depends on the initial value u 0 . It is clear that T < T ≤ T . Therefore, we derive

α( u(t, •) ) ≤e -π 2 (t-T ( u0 ) ) u 0 + κ( d L ∞ (Qt) ) for t ∈ [0, t * ), which implies that u(t, •) ≤α -1 (2e -π 2 (t-T ( u0 ) ) u 0 ) + α -1 (2κ( d L ∞ (Qt) )) =β( u 0 , t) + θ 1 ( d L ∞ (Qt) ) (18
) for t ∈ [0, t * ), where β(s, t) = α -1 (2se -π 2 (t-T (s) )) and

θ 1 (x) = α -1 (2κ(x)). Moreover, θ 1 ∈ K ∞ and β ∈ KL since α -1 ∈ K ∞ .
Hence, the ISS estimate [START_REF] Karafyllis | ISS with respect to boundary disturbances for 1-D parabolic PDEs[END_REF] of the solutions to system (1) with the hybrid boundary control law (2) under the switching rule [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF] is derived for t ∈ [0, t * ).

Next we establish an ISS estimate for t ∈ [t * , +∞). According to the switching rule [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF], we have σ(t) = 0 for all t ≥ t * . Proceeding in the similar way as ( 12)-( 16) and bearing u(t * , •) = 0 in mind, we get

u(t, •) ≤ Ω(1) 1 + π 2 (1 -e -(1+π 2 )(t-t * ) ) d L ∞ (Qt) ≤ Ω(1) d L ∞ (Qt) 1 + π 2 = θ 2 ( d L ∞ (Qt) ), t ≥ t * (19
) where θ 2 (x) = Ω (1) 1+π 2 x and θ 2 ∈ K ∞ .

Combining [START_REF] Karafyllis | ISS with respect to boundary disturbances for 1-D parabolic PDEs[END_REF] and ( 19) yields the following estimate

u(t, •) ≤β( u 0 , t) + max{θ 1 ( d L ∞ (Qt) ), θ 2 ( d L ∞ (Qt) )} =β( u 0 , t) + θ( d L ∞ (Qt) ), t ≥ 0 ( 20
) where θ(x) = max{θ 1 (x), θ 2 (x)} and θ ∈ K ∞ . This completes the proof. Remark 4.1 It is worth stressing that continuous boundary feedback control U (t) = 1 0 K(1, y, λ)u(t, y)dy with constant decay rate λ > 0 can only provide the exponential ISS property, which indicates the exponential stability for the disturbance-free system. In order to achieve finite-time stability for the disturbance-free system, the kernel gain in the boundary feedback control law is required to vary in time, that is in the form of K(1, y, g(t)) with a function g(•). This idea is verified by effectively treating the problem of finite-time stabilization for various reaction-diffusion systems, see [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF][START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF][START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF] for more details. Highly inspired by [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF], the present paper is the attempt to develop a framework of ISS control issue for PDE systems. The design of switching linear boundary feedback (2) that depends on system state leads to the boundedness of the system trajectory as proven in Theorem 4.1 (i.e., the state norm will not exceed the highest level), and finally the resulting system turns out to be ISS, as well as finite-time stable in the distrurbance-free case. Remark 4.2 In the process of proving Theorem 4.1, it has been shown that the solution of the close-loop system (1) will converge to zero if there is no disturbance input, which corresponds to Theorem 1 in [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF]. Consequently, this paper extends available results in [START_REF] Polyakov | On boundary finite-time feedback control for heat equation[END_REF] to the heat equation with external input or disturbance in the distributed form. It is well known that, some disturbances or uncertainties are expressed as boundary disturbances [START_REF] Pisano | On the ISS properties of a class of parabolic DPS' with discontinuous control using sampled-in-space sensing and actuation[END_REF], such as the presence of control actuator errors that can be modeled by a PDE system with Dirichlet boundary conditions u(t, 0) = 0, u(t, 1) = U (t) + v(t) (v(t) is the unknown boundary disturbance). At present, some methods such as the eigenfunction expansion [START_REF] Karafyllis | ISS with respect to boundary disturbances for 1-D parabolic PDEs[END_REF][START_REF] Karafyllis | ISS in different norms for 1-D parabolic PDEs with boundary disturbances[END_REF], finite-difference scheme [START_REF] Karafyllis | ISS in different norms for 1-D parabolic PDEs with boundary disturbances[END_REF], monotonicity method [START_REF] Mironchenko | Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances[END_REF], De Giorgi iteration [START_REF] Zheng | A De Giorgi iteration-based approach for the establishment of ISS properties for Burgers equation with boundary and in-domain disturbances[END_REF], non-coercive Lyapunov function method [START_REF] Jacob | Remarks on input-to-state stability and non-coercive lyapunov functions[END_REF], etc., are presented, which provides ideas to deal with the ISS analysis in the case of Dirichlet boundary inputs.

Numerical example

Consider the scalar system (1) with disturbance input d(t, x) = 2 cos(5x) sin(5t). The initial value is chosen to be u 0 (x) = 9x(1 -x), x ∈ [0, 1] which satisfies compatibility conditions. We simulate numerically the state of system (1) by using the implicit Euler scheme with the step size h = 0.02 of space variable and the step size τ = h 2 of time variable. First-order forward difference scheme and second-order centred difference scheme are applied in the time variable and space variable, respectively. Note that the solution of open-loop system (i.e., U (t) = 0) with d(t, x) = 0 decays exponentially to zero as t → ∞. To illustrate this fact, the L 2 norm u(t, •) for the open-loop system (1) with d(t, x) = 0 is shown in Fig. 3(b) (blue line) in a logarithmic scale. It can be seen that open-loop system (1) without disturbance input is exponentially stable. On the other side, in the presence of disturbance input d(t, x), the simulation result for the L 2 norm is plotted in Fig. 1, and thus the open-loop system ( 1) is asymptotically ISS. Now the hybrid boundary feedback control (2) with the corresponding switching law [START_REF] Espitia | Boundary time-varying feedbacks for fixedtime stabilization of constant-parameter reactiondiffusion systems[END_REF] are utilized to stabilize the system (1) by using the kernel function [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF]. The L 2 norm u 0 of the initial value is numerically calculated as 1.6432 (here, we adopt the trapezoidal rule for the numerical integration) and satisfies u 0 ∈ (η -3 , η It is observed that after 6 switches, the L 2 norm of the system gradually decreases at first and finally converges to the bounded neighborhood of the origin. Compared with Fig. 1, we also notice from Fig. 2(b) that the L 2 norm of closed-loop system (1) remain confined in a smaller vicinity of the origin, which is less influenced by disturbance input d(t, x). Moreover, Fig. 3 shows the state response of the system (1) with switching boundary control U (t) in the absence of disturbance input. Similarly, the spatio-temporal evolution of the state u(t, x) and the time evolution of L 2 norm u(t, •) with a logarithmic scale (red line) respectively, are plotted in Fig. 3(a) and Fig. 3(b). It is clearly seen that the closed-loop system (1) with d(t, x) = 0 is finite-time stable. Obviously, the comparison result shows that under switching boundary control U (t), the convergence rate for the system (1) without external input is faster than exponential convergence and the disturbance attenuation is improved.

Conclusion

In this paper, we dealt with the switching boundary feedback control towards ISS for heat equation. The hybrid linear boundary control law was developed based on the standard backstepping method. Under a constructive switching rule, we have shown that closed-loop system can successfully achieve ISS. Meanwhile, a stronger result has also been obtained, that is the finite-time stability can be guaranteed for the disturbance-free system. Besides linear switching boundary feedback control in this paper, some other novel and challenging control schemes can deal with the problem of ISS (such as design of a suitable characterization that g(t) satisfies for the kernel gain with the form K(1, y, g(t)) as aforementioned), which will be of interest for future researches.

Lemma 2 . 1 (

 21 [START_REF] Dym | Fourier series and integrals[END_REF] Ch. 1, Wirtinger's inequality) For any bounded interval [0, l] ⊂ R + and z

I 2 2 ≥ 0

 220 (p) ≥ 0 and I1(p) p = I0(p)-I2(p) for p ≥ 0 (it can be easily obtained by the representation I k (p) = 1 π

Fig. 1 .

 1 Fig. 1. Numerical solution of the open-loop system (1) with disturbance input d(t, x) = 2 cos(5x) sin(5t).

  -4 ] = (1.6163, 1.6752]. Set the initial mode σ(0) = -4. Thus, the closed-loop system (1) is ISS according to Theorem 4.1. Fig. 2 depicts the dynamical behavior of closed-loop system (1), in which Fig. 2(a) shows the state u(t, x) of the system and Fig. 2(b) shows the L 2 norm of the state.

Fig. 2 .

 2 Fig. 2. State response of the closed-loop system (1) with disturbance input d(t, x) = 2 cos(5x) sin(5t).

Fig. 3 .

 3 Fig. 3. State response of the closed-loop system (1) with disturbance input d(t, x) = 0.

This work was partially supported by the China Scholarship Council (CSC) Scholarship (CSC no. 202106120182) and by Natural Science Foundations of Shandong Province under Grant ZR2021MF007. This paper was not presented at any IFAC meeting.


Emilie Roger
email: emilie.roger@univ-angers.fr

BEHAVIOR OF LIPID NANOCAPSULES AFTER ORAL ADMINISTRATION

The oral route is the most common route for drug delivery. However, some drugs present a low oral bioavailability due to low drug solubility, drug degradation along the gastrointestinal tract, or low intestinal permeability. To improve oral drug delivery, drug encapsulation into nanocarriers, such as lipid nanocapsules (LNCs), is currently one of the most promising technologies. LNCs, developed by our group, have proven to be very interesting for oral administration. These nanoparticles have a size that ranges from 20 to 130 nm and are prepared by a well-known low-energy emulsification process: the phase-inversion temperature method. They have already shown promising properties for the oral delivery of paclitaxel, Sn38, fondaparinux, decitabine, and miltefosine. However, despite the promising benefits, these studies focus only on monitoring drugs encapsulated in nanocarrier but ignore the performance of the nanocarrier itself.

Currently, the only method that can characterize the particle integrity of organic nanoparticles is by Förster resonance energy transfer (FRET). It is a subset of the fluorescence technique often dubbed the 'nano-ruler' since the technique is very sensitive to the distance change in nanoscale and is thus proved to be useful for monitoring the association or the dissociation of organic nanoparticles. In this context, FRET-LNCs were developed and the behavior of LNC after oral administration was studied. This lecture will try to explain how LNCs allow for enhancing the oral bioavailability of drugs. Special attention will be paid to the ability of those LNCs to maintain their full integrity using in vitro and in vivo models.

  her PharmD in 2006 and his PhD in 2009 from the University of Angers, France. After a postdoctorat position at pharmaceutical department, University of Minnesota (Minneapolis, Minnesota, USA), she started an assistant professor position at the CAMB laboratory (UMR 7199 CNRS, Strabsourg, France) in 2010 and then she joined the MINT laboratory (UMR INSERM U1066 CNRS 6021, Angers, France) in 2013. Her current research is focused on oral administration and on the development of new drug delivery system, with special attention on nanotechnology, and their application to the pharmaceutical fields. She published 31 publications in international journals with reviewers, 2 patents and 3 book chapters (total citations in Scopus 1692; H-index 19),. Major publications during last 5 years Kaeokhamloed, N. , Legeay S. & Roger E.et al. FRET as the tool for in vivo nanomedicine tracking. J. Control. Release (in press) Deshayes, C., Arafath, M. N., Apaire-Marchais, V. & Roger, E. Drug Delivery Systems for the Oral Administration of Antimicrobial Peptides: Promising Tools to Treat Infectious Diseases. Front. Med. Technol. 3, 1-13 (2022). Bonnet, S., Elfatairi, R., Franconi, F., Roger, E. & Legeay, S. Organic nanoparticle tracking during pharmacokinetic studies. Nanomedicine 16, 2539-2563 (2021). Kaeokhamloed, N. et al. New in vitro coculture model for evaluating intestinal absorption of different lipid nanocapsules. Pharmaceutics 13, 1-16 (2021). Billat, P.-A. et al. Models for drug absorption from the small intestine : where are we and where are we going ? Drug Discov. Today 22, (2017). Roger, E., Gimel, J.-C., Bensley, C., Klymchenko, A. S. & Benoit, J.-P. Lipid nanocapsules maintain full integrity after crossing a human intestinal epithelium model. J. Control. Release 253, (2017).