Maciej Piotr Cysewski
email: piotr.cysewski@cm.umk.pl

Maciej Przybyłek

Selection of effective cocrystals former for dissolution rate improvement of active pharmaceutical ingredients based on lipoaffinity index

Keywords: dissolution rate, pharmaceutical cocrystals, excipients screening, molecular descriptors, hydrophilicity Chemical compounds studied in this article Iloperidone (PubChem CID: 71360), Ritonavir (PubChem CID: 392622), Carbamazepine

Selection of effective cocrystals former for dissolution rate improvement of active pharmaceutical ingredients based on lipoaffinity index

Introduction

During last half century enormous in vitro and in vivo attempts have been made to model drug dissolution behavior [START_REF] Pestieau | In vitro biphasic dissolution tests and their suitability for establishing in vitro-in vivo correlations: A historical review[END_REF]. The dissolution profile measurement of active pharmaceutical ingredient (API) is the most fundamental method for assessment of drug absorption. Although the bioavailability is a complex issue and not straightforwardly correlated with in vitro results [START_REF] Lu | In vitro-in vivo correlation: perspectives on model development[END_REF] this approach is still valuable and effective at the pre-formulation stage of solid phase dosage form development. There are several basic techniques of dissolution profile determination but about 70% of results available on FDA (Food and Drug Administration) (FDA, 2017) dissolution rates database were obtained using paddle method [START_REF] Shohin | A brief review of the FDA dissolution methods database[END_REF]. Other approaches were applied to a lesser extent, including the basket method, flow-through cell measurements or reciprocating cylinder approaches [START_REF] Shohin | A brief review of the FDA dissolution methods database[END_REF]. Since tablet is the most frequently studied dosage form, many methods of crystal engineering have been applied for dissolution rate (DSR) enhancement rather than thermodynamic solubility. Improvement of dissolution profiles can be obtained by particle size reduction techniques [START_REF] Choi | Design of PVP/VA S-630 based tadalafil solid dispersion to enhance the dissolution rate[END_REF][START_REF] Dizaj | Nanosizing of drugs: Effect on dissolution rate[END_REF][START_REF] Khadka | Pharmaceutical particle technologies: An approach to improve drug solubility, dissolution and bioavailability[END_REF][START_REF] Sironi | Dynamic dissolution-/permeation-testing of nano-and microparticle formulations of fenofibrate[END_REF], crystal morphology/polymorphism modifications [START_REF] Amodwala | Statistically optimized fast dissolving microneedle transdermal patch of meloxicam: A patient friendly approach to manage arthritis[END_REF][START_REF] Jain | Effect of differential surface anisotropy on performance of two plate shaped crystals of aspirin form I[END_REF][START_REF] Keraliya | Effect of solvent on crystal habit and dissolution behavior of tolbutamide by initial solvent screening[END_REF][START_REF] Maghsoodi | Role of solvents in improvement of dissolution rate of drugs: crystal habit and crystal agglomeration[END_REF] or multicomponent

A C C E P T E D

M A N U S C R I P T 2 crystals synthesis [START_REF] Ross | Engineering and manufacturing of pharmaceutical cocrystals: a review of solvent-free manufacturing technologies[END_REF][START_REF] Schultheiss | Pharmaceutical cocrystals and their physicochemical properties[END_REF][START_REF] Yadav | Co-crystals: A novel approach to modify physicochemical properties of active pharmaceutical ingredients[END_REF]. The latter approach including bi-component cocrystallization deserves particular attention due to the variety of cocrystal preparation methods such as solvent evaporation techniques [START_REF] Bag | A kinetically controlled crystallization process for identifying new co-crystal forms: fast evaporation of solvent from solutions to dryness[END_REF][START_REF] Bag | Screening and selective preparation of polymorphs by fast evaporation method: A case study of aspirin, anthranilic acid, and niflumic acid[END_REF][START_REF] Cysewski | Exploring the cocrystallization potential of urea and benzamide[END_REF]Przybyłek et al., 2016aPrzybyłek et al., , 2016b)), antisolvent crystallization [START_REF] Chun | Combined anti-solvent and cooling method of manufacturing indomethacin-saccharin (IMC-SAC) co-crystal powders[END_REF][START_REF] Lee | Controlling the polymorphism of carbamazepine-saccharin cocrystals formed during antisolvent cocrystallization using kinetic parameters[END_REF][START_REF] Wang | Anti-solvent cocrystallization of carbamazepine and saccharin[END_REF], mechanochemical co-grinding [START_REF] Hasa | Cocrystal Formation through Mechanochemistry: from Neat and Liquid-Assisted Grinding to Polymer-Assisted Grinding[END_REF][START_REF] Karki | Screening for pharmaceutical cocrystal hydrates via neat and liquid-assisted grinding[END_REF][START_REF] Li | Mechanochemical synthesis of pharmaceutical cocrystal suspensions via hot melt extrusion: Feasibility studies and physicochemical characterization[END_REF][START_REF] Tröbs | Mechanochemical synthesis and characterisation of cocrystals and metal organic compounds[END_REF] or slurry methods [START_REF] Apshingekar | Synthesis of caffeine/maleic acid co-crystal by ultrasound-assisted slurry co-crystallization[END_REF][START_REF] Kojima | High-throughput cocrystal slurry screening by use of in situ Raman microscopy and multi-well plate[END_REF][START_REF] Takata | Cocrystal screening of stanolone and mestanolone using slurry crystallization[END_REF]. Furthermore, the variety of cocrystals formers applicable for pharmaceutical formulations offers fine tuning of physicochemical properties of new multicomponent solids ood and Rodr guez-Hornedo, 2009). Nutraceuticals [START_REF] Sekhon | Nutraceutical cocrystals: An overview[END_REF] is an exemplary class of non-toxic chemicals embracing among others phenolic acids, flavonoids, vitamins and saccharides [START_REF] Aitipamula | Pharmaceutical cocrystals of ethenzamide: structural, solubility and dissolution studies[END_REF][START_REF] Bethune | Improving the poor aqueous solubility of nutraceutical compound pterostilbene through cocrystal formation[END_REF][START_REF] Sekhon | Nutraceutical cocrystals: An overview[END_REF][START_REF] Sinha | Cocrystallization of nutraceuticals[END_REF].

There are also many potential candidates for cocrystallization with other chemicals found for example on EAFUS (Everything Added to Food in the United States) and GRAS (Generally Regarded as Safe) lists provided by FDA.

There are several interesting examples of improving rate of drug dissolution throughout cocrystal synthesis. For instance, according to [START_REF] Sanphui | Fast dissolving curcumin cocrystals[END_REF] dissolution rates of Curcumin cocrystals with Resorcinol and Pyrogallol are respectively 5 and 12 times higher than pure Curcumin. Another example is anticoagulant Apixaban, which exhibits poor aqueous solubility and bioavailability. [START_REF] Chen | Improving the solubility and bioavailability of apixaban via apixaban-oxalic acid cocrystal[END_REF] have found that this can be easily overcome via cocrystallization with oxalic acid. According to this study, the bioavailability of Apixaban, if dosed to beagle dogs, is about two times higher when the oxalic acid cocrystal was applied. Many more examples can be found in the literature, which were systematically collected for the purpose of this study. Extensive survey allowed for gathering of cases of pharmaceutical cocrystals, for which the dissolution rates have been studied experimentally. Unfortunately, these data are methodologically quite inconsistent what prevents from collecting of the actual values of DSR enhancements. However, available information allows for discrimination of excipients between DRS enhancers or reducers. Indeed, interesting examples of new forms of drugs with improved dissolution profile can be found in the case of such antibiotics as Clarithromycin and Prulifloxacin [START_REF] Nanjwade | Development and evaluation of prulifloxacin tablet[END_REF]Rajbhar et al., 2016aRajbhar et al., , 2016b)). Besides, such neuropsychiatric drugs as Carbamazepine, Agomelatine, Domperidone, Iloperidone and Paliperidone were also studied [START_REF] Hoffman | Co-crystals : Commercial Opportunities and Patent Considerations[END_REF][START_REF] Lahamage | Co-crystal approach for improving poorly soluble drug agomelatine by using acid coformer for tablets[END_REF][START_REF] Moradiya | Continuous cocrystallization for dissolution rate optimization of a poorly water-soluble drug[END_REF][START_REF] Mounika | Preparation and characterization of novel cocrystal forms of domperidone[END_REF][START_REF] Shayanfar | Solubility and dissolution rate of a carbamazepine-cinnamic acid cocrystal[END_REF][START_REF] Yamashita | Harvesting potential dissolution advantages of soluble cocrystals by depressing precipitation using the common coformer effect[END_REF][START_REF] Yan | Improving the solubility of agomelatine via cocrystals[END_REF][START_REF] Zhang | Dissolution and pharmacokinetic properties of two paliperidone cocrystals with 4-hydroxybenzoic and 4aminobenzoic acid[END_REF][START_REF] Zhang | Using dissolution and pharmacokinetics studies of crystal form to optimize the original iloperidone[END_REF]. Additionally, solid dispersions of analgesics and nonsteroidal anti-inflammatory drugs such as Acetylsalicylic acid, Aceclofenac, Acemetacin, Ethenzamide, Celecoxib, Etodolac, Indomethacin, Ketoprofen, Lornoxicam, Meloxicam, Piroxicam and Paracetamol were extensively studied [START_REF] Aitipamula | Pharmaceutical cocrystals of ethenzamide: structural, solubility and dissolution studies[END_REF][START_REF] Brown | Investigating pharmaceutical co-crystals as a means to improve the solubility of a drug[END_REF][START_REF] Cheney | Coformer selection in pharmaceutical cocrystal development: a case study of a meloxicam aspirin cocrystal that exhibits enhanced solubility and pharmacokinetics[END_REF][START_REF] Gadade | Cocrystallization of etodolac: prediction of cocrystallization, synthesis, solid state characterization and in vitro drug release[END_REF][START_REF] Haeria | Characterization and dissolution test of aspirin-nicotinamide cocrystal[END_REF][START_REF] Hiendrawan | Simultaneous cocrystallization and micronization of paracetamol-dipicolinic acid cocrystal by supercritical antisolvent (SAS)[END_REF][START_REF] Jung | Bioavailability of indomethacin-saccharin cocrystals[END_REF][START_REF] Moradiya | Continuous manufacturing of high quality pharmaceutical cocrystals integrated with process analytical tools for in-line process control[END_REF][START_REF] Patel | Preparation and characterization of lornoxicam Co-crystals[END_REF][START_REF] Remenar | Celecoxib:nicotinamide dissociation: Using excipients to capture the cocrystal's potential[END_REF][START_REF] Sanphui | Acemetacin cocrystals and salts: Structure solution from powder X-ray data and form selection of the piperazine salt[END_REF][START_REF] Siswandi | Virtual screening of co-formers for ketoprofen cocrystallization and the molecular properties of the co-crystal[END_REF][START_REF] Sohrab | Enhancement of dissolution rate of aceclofenac by formation of aceclofenac-nicotinic acid cocrystal using water soluble polymers[END_REF][START_REF] Zahid | Preparation of cocrystals of paracetamol with saccharin for obtaining improved dissolution profile[END_REF]. Many examples of cocrystals dissolution rates can be found also in case of antiinfective agents [START_REF] Chadha | Cocrystals of efavirenz with selected conformers: Preparation and characterization[END_REF][START_REF] Cherukuvada | Nitrofurantoin-p-aminobenzoic acid cocrystal: Hydration stability and dissolution rate studies[END_REF][START_REF] Childs | Metronidazole cocrystals and imipramine cocrystals U.S. Patent 20090258859 A1[END_REF][START_REF] Gao | Enhanced dissolution and stability of adefovir dipivoxil by cocrystal formation[END_REF][START_REF] Goud | Modulating the solubility of sulfacetamide by means of cocrystals[END_REF][START_REF] Gujar | Improvement in physicochemical properties of nevirapine[END_REF][START_REF] Imchalee | Gas anti-solvent processing of a new sulfamethoxazole-l-malic acid cocrystal[END_REF][START_REF] Jung | Liquid-assisted grinding to prepare a cocrystal of adefovir dipivoxil thermodynamically less stable than its neat phase[END_REF][START_REF] Kumar | Polymorphism and solubility of selected active pharmaceutical ingredients[END_REF][START_REF] Manin | Crystal structures, thermal analysis, and dissolution behavior of new solid forms of the antiviral drug arbidol with dicarboxylic acids[END_REF][START_REF] Nalte | Solubility enhancement of nevirapine by cocrystallisation technique[END_REF][START_REF] Paun | Improvement of physicochemical properties of tinidazole cocrystals: An influence of additives[END_REF][START_REF] Remenar | Crystal engineering of novel cocrystals of a triazole drug with 1,4-dicarboxylic acids[END_REF][START_REF] Sateesh Babu | Evaluation of physiochemical and anti-tubercular activity of co-crystal of isoniazid with methyl paraben[END_REF][START_REF] Sekhon | Nutraceutical cocrystals: An overview[END_REF][START_REF] Serrano | Modelling and shadowgraph imaging of cocrystal dissolution and assessment of in vitro antimicrobial activity for sulfadimidine/4-aminosalicylic acid cocrystals[END_REF][START_REF] Shah | Utilization of co-crystallization for solubility enhancement of a poorly soluble antiretroviral drug-ritonavir[END_REF]. This wide class of APIs includes such drugs as Adefovir, Dipivoxil, Arbidol, Efavirenz, Itraconazole, Nevirapine, Nitrofurantoin, Ritonavir, Sulfamethazine, Sulfamethoxazole, Tinidazole, Voriconazole, Isoniazid, Metronidazole and Sulfacetamide. Furthermore DSR enhancement was also studied for Exemestane and Megestrol acetate acting as antineoplastics [START_REF] Shiraki | Dissolution improvement and the mechanism of the improvement from cocrystallization of poorly water-soluble compounds[END_REF]. Such interest was also focused on some cardiovascular agents including Cilnidipine, Felodipine, Irbesartan, Telmisartan, Apixaban, Ezetimibe, Fenofibrate, Olanzapine, Sildenafil, Tadalafil [START_REF] Arafa | Xylitol as a potential co-crystal co-former for enhancing dissolution rate of felodipine: preparation and evaluation of sublingual tablets[END_REF][START_REF] Chadha | Cocrystals of telmisartan: characterization, structure elucidation, in vivo and toxicity studies[END_REF][START_REF] Chen | Improving the solubility and bioavailability of apixaban via apixaban-oxalic acid cocrystal[END_REF][START_REF] Info | ENHANCEMENT OF SOLUBILITY AND DISSOLUTION RATE OF TELMISARTAN BY[END_REF][START_REF] Khor | Comparison study of therapeutic properties of proteins and secondary metabolites from Carica papaya[END_REF][START_REF] Nijhawan | Cocrystals of irbesartan with hippuric acid[END_REF][START_REF] Punita | Co-Crystal Formation Of Cilnidipine With Urea And Benzoic Acid: An Efficient Approach To Enhance The Solubilty And Dissolution Rate[END_REF][START_REF] Renkoğlu | HPLC determination of olanzapine and carbamazepine in their nicotinamide cocrystals and investigation of the dissolution profiles of cocrystal tablet formulations[END_REF][START_REF] Shewale | Formulation and Solid State Characterization of Nicotinamide-based Co-crystals of Fenofibrate[END_REF][START_REF] Shiraki | Dissolution improvement and the mechanism of the improvement from cocrystallization of poorly water-soluble compounds[END_REF][START_REF] Sugandha | Solubility enhancement of ezetimibe by a cocrystal engineering technique[END_REF][START_REF] Vinesha | Enhancement of solubility of tadalafil by cocrystal approach[END_REF][START_REF] Žegarac | A sildenafil cocrystal based on acetylsalicylic acid exhibits an enhanced intrinsic dissolution rate[END_REF]. There are also single studies devoted to antiallergics as Fexofenadine [START_REF] Mounika | Preparation and characterization of novel co-crystal forms of fexofenadine[END_REF], antidiabetic agents as Gliclazide [START_REF] Prawiro | Enhancing dissolution rates of gliclazide via cocrystallization with nicotinamide[END_REF], AMG 517, Danazol, Famotidine [START_REF] Bak | The co-crystal approach to improve the exposure of a water-insoluble compound: AMC 517 sorbic acid co-crystal characterization and pharmacokinetics[END_REF][START_REF] Childs | Formulation of a danazol cocrystal with controlled supersaturation plays an essential role in improving bioavailability[END_REF][START_REF] Shi | Enhanced dissolution of famotidine by cocrystal formation with tartaric acid and maleic acid[END_REF], sodium channel blockers as 4-(4-

A C C E P T E D M A N U S C R I P T 3
Fluorophenoxy)benzaldehyde semicarbazone [START_REF] Mcnamara | Use of a glutaric acid cocrystal to improve oral bioavailability of a low solubility API[END_REF] and chelating agents as Deferiprone [START_REF] Zhang | Synthesis, characterization and dissolution of three pharmaceutical cocrystals based on deferiprone[END_REF]. In addition to the aforementioned typical APIs, several drug-like compounds and nutraceuticals such as benzamide [START_REF] Grossjohann | Characterisation, solubility and intrinsic dissolution behaviour of benzamide: dibenzyl sulfoxide cocrystal[END_REF], pterostilbene [START_REF] Bethune | Improving the poor aqueous solubility of nutraceutical compound pterostilbene through cocrystal formation[END_REF], curcumin [START_REF] Sanphui | Fast dissolving curcumin cocrystals[END_REF], baicalein [START_REF] Huang | Baicalein-nicotinamide cocrystal with enhanced solubility, dissolution, and oral bioavailability[END_REF], quercetin [START_REF] Smith | Cocrystals of quercetin with improved solubility and oral bioavailability[END_REF] and epigallocatechin-3-gallate [START_REF] Smith | Crystal engineering of green tea epigallocatechin-3-gallate (EGCg) cocrystals and pharmacokinetic modulation in rats[END_REF] were taken into account by some investigators.

As it was reported in our previous works [START_REF] Cysewski | In silico screening of dicarboxylic acids for cocrystallization with phenylpiperazine derivatives based on both cocrystallization propensity and solubility advantage[END_REF][START_REF] Przybyłek | Applicability of phenolic acids as effective enhancers of cocrystals solubility of methylxanthines[END_REF], in silico methods can be quite successful in cocrystal screening and predicting cocrystal solubility advantage. Particularly, it was shown [START_REF] Przybyłek | Applicability of phenolic acids as effective enhancers of cocrystals solubility of methylxanthines[END_REF] that compounds containing polar OH and COOH groups are excellent methylxanthines solubility enhancers. Solubility increase in the presence of hydrophilic compounds is consistent with many experimental data [START_REF] Brough | Use of polyvinyl alcohol as a solubility-enhancing polymer for poorly water soluble drug delivery (Part 1)[END_REF][START_REF] Savjani | Drug solubility: importance and enhancement techniques[END_REF]Thiry et al., 2017b). It is understandable that hydrophilic excipients will also improve the dissolution profile. However, there are no reports documenting rational criterions for selection of coformers suitable for efficient dissolution rate enhancements. This paper filling this gap offers new theoretical screening protocol relying on the two criterions narrowing the list of potential coformers, for which both probability of cocrystallization and DRS enhancement is high enough. This strategy can be helpful by suggesting appropriate compounds for the development of new solid forms of APIs with improved bioavailability.

Methods

Training set of API cocrystals

Based on literature survey the collection of active pharmaceutical ingredients cocrystalized with different excipients was prepared. Since dissolution rates have been experimentally studied using diverse measurements techniques, the quantified database cannot be built. Instead, a separation of cases into positive and negative sub-classes was conducted. The latter group comprised cases for which no sensible improvement of solubility kinetics was observed. On the contrary, the positive subset collected cocrystals, for which significant gain was observed in terms of dissolution rate improvements with respect of the pure drug. These data were used as training sets for performed chemometric analysis and comprised 102 positive and 17 negative systems. Detailed characteristics of all cocrystals used in this study can be found in supporting materials as collected in Table S1.

Cocrystallization probability assessments

The probability of formation of a homogeneous binary solid by two given components was estimated based on mixing paradigm [START_REF] Klamt | Solvent-screening and co-crystal screening for drug development with COSMO-RS[END_REF] relying on the assumption that intermolecular interactions directing formation of the intermolecular complexes in the solid state can be reliably estimated by components affinity expressed in terms of enthalpic contribution to mixing under hypothetical super-cooled conditions at ambient temperature. The formal definition is as follows

  2 2 2 1 1 1 2 12 2 1 12 1 H x H x H x H x H mix     
where x denotes molar fraction, superscripts represent solvent types and subscripts denote solutes. Thus, excess enthalpy can be obtained by subtracting enthalpies values characterizing pure components from the sum of molar enthalpies of components in the mixture. From the practical point of view three calculations are indispensable, namely two for each of pure components and one for mixture formed at given stoichiometry. This method requires the estimation of so called σprofiles defined in the framework of COSMO-RS (Conductor like Screening Model for Real Solvents) approach [START_REF] Klamt | Conductor-like Screening Model for Real Solvents: A New Approach to the Quantitative Calculation of Solvation Phenomena[END_REF] implemented in COSMOtherm software (COSMOlogic GmbH & Co. KG, 2016). For this purpose the geometries of the most significant conformations of any considered compound were fully optimized both in the gas and condensed phases using BP-RI/TZVP scheme and finally σprofiles generation on BP-RI/TZVPD level. All these calculations were performed with an aid of TURBOMOLE V7.0 (TURBOMOLE-GmbH, 2015) interfaced with TmoleX 4.2. Prior to this stage conformational analysis of all compounds was performed using OpenBable O' Boyle et al., 2011) capabilities. The whole procedure was applied both to drugs and potential excipients. The

A C C E P T E D M A N U S C R I P T

4 parametrization enabling for final H mix computations is defined in BP_TZVPD_FINE_C30_1601.ctd set (COSMOlogic GmbH & Co. KG, 2016) and stands for the current state of art of these kinds of computations.

Only such compounds were selected as potential coformers, which have the chance of being applied in real manufacturing of pharmaceutical cocrystals. Hence, the first choice of such coformers are collections termed EAFUS and GRAS published by FDA. Additionally, the set was also augmented by some neutracuetics including phenolic acids, their esters and some hydroxyl benzoic acids marked as non-toxic. The coformers list used for screening purposes comprised finally nine aliphatic dicarboxylic acids, 37 phenolic acids, 97 esters, 68 aldehydes, 89 ketones, 20 amines, 134 alcohols, 103 heterocyclic compounds, 56 sulfur containing species, 34 ethers and additionally 550 other compounds of great variety of structural diversities. In total 1194 compounds were used as potential cocrystal formers. In order to distinguish salts from neutral cocrystals, the criterion of probability of proton transfer can be utilized, which is typically indicated by differences in components pKa values. Hence, it is assumed that proton transfer between interacting conformers occurs if pKa = pKb-pKa>2.7. Since there are no available experimental data of pKa for all considered compounds, predicted values were used instead. For this purpose the free online tool provided by ChemAxon was utilized (http://epoch.uky.edu/ace/public/pKa.jsp) [START_REF] Szegezdi | A method for calculating the pKa values of small and large molecules[END_REF].

Molecular descriptors computation

Descriptors were computed using online facilities of ChemDes Version 1.0 [START_REF] Dong | ChemDes: an integrated web-based platform for molecular descriptor and fingerprint computation[END_REF]. The following five sets of molecular descriptors were included, namely Chemopy (Cao et al., 2013), Bluedesc, CDKit, andPaDel (Yap, 2011). All 2D-type molecular parameters were obtained based on information coded in smiles strings. Since there are some inconsistencies in smiles generation depending on the algorithms used, here for keeping coherency, one source of these data was used for all compounds. Features of the OpenBable O' Boyle et al., 2011) program were used for generation of smiles strings using sdf files with previously optimized geometries that were used during H mix computations. Also for computations of 3D descriptors the same geometry files were applied. Since the training set comprising experimental data was limited, the potential bias of cases collection was alleviated by randomization. Hence, the binary classification was performed 20 times for every molecule descriptors set after random selection of 50% of data. Then the collected false positive FP% and true positives TP% percentages were averaged. Also the associated standard deviation values were computed. It was assumed that this approach at least partly compensates shortage of experimental data used for finding molecular descriptors adequate for distinguishing positive and negative cases. The results of this screening was documented in supporting materials (see Table S2) by collecting molecular descriptors which include at least 50% of true positive cases assuming 95% confidence, what means inclusion of at most 5% false positive cases in prediction.

Results and Discussion

In the first stage of this study many various molecular descriptors were generated in terms of their applicability for dissolution rate improvement assessment. Then, the virtual cocrystal screening was performed for four drugs, namely Iloperidone, Ritonavir, Carbamazepine, Ethenzamide. All these compounds belonging to the II class of Biopharmaceutics Classification System (high permeability and low solubility) have been extensively studied in terms of bioavailability and dissolution profile improvement [START_REF] Aitipamula | Pharmaceutical cocrystals of ethenzamide: structural, solubility and dissolution studies[END_REF][START_REF] Danjo | Preparation and dissolution behavior of ethenzamide solid dispersion using various sugars as dispersion carriers[END_REF][START_REF] Foglio Bonda | Nanonized itraconazole powders for extemporary oral suspensions: Role of formulation components studied by a mixture design[END_REF][START_REF] Gadade | Cocrystallization of etodolac: prediction of cocrystallization, synthesis, solid state characterization and in vitro drug release[END_REF][START_REF] Ibrahim | Polymeric surfactant based etodolac chewable tablets: formulation and in vivo evaluation[END_REF][START_REF] Ige | Enhanced in vitro dissolution of Iloperidone using Caesalpinia Pulcherrima mucoadhesive microspheres[END_REF][START_REF] Jitkar | Spherical agglomeration of platy crystals: curious case of etodolac[END_REF][START_REF] Kojo | Improved oral absorption profile of itraconazole in hypochlorhydria by self-micellizing solid dispersion approach[END_REF][START_REF] Miwa | Lidocaine self-sacrificially improves the skin permeation of the acidic and poorly water-soluble drug etodolac via its transformation into an ionic liquid[END_REF][START_REF] Remenar | Crystal engineering of novel cocrystals of a triazole drug with 1,4-dicarboxylic acids[END_REF][START_REF] Six | Increased physical stability and improved dissolution properties of itraconazole, a class II drug, by solid dispersions that combine fast-and slow-dissolving polymers[END_REF]Thiry et al., 2017a[START_REF] Thiry | Investigation of a suitable in vitro dissolution test for itraconazole-based solid dispersions[END_REF]. It is understandable that not all pharmaceutically accepted excipients are miscible in the solid phase with a particular API in the form of a cocrystal. Therefore, an additional step characterizing probability of cocrystallization was performed.

Selection of appropriate molecular descriptor

The chemometric domain [START_REF] Xue | Molecular Descriptors in Chemoinformatics, Computational Combinatorial Chemistry, and Virtual Screening[END_REF] matured to offer a variety of molecular descriptors [START_REF] Mannhold | Molecular drug properties-measurement and prediction[END_REF][START_REF] Todeschini | Molecular Descriptors for Chemoinformatics[END_REF], which can be used for extracting

A C C E P T E D M A N U S C R I P T

precise information about miscellaneous physicochemical properties of chemical compounds. The vast number and a great diversity of molecular parameters enables for comprehensive characteristics of different contributions to the studied phenomena. Here, the differentiation between excipients being able to enhance the dissolution rates and those not possessing this property was the objective of the performed screening. Although the explicit target function was not defined, as it is the case for typical QSPR studies, it was still possible to achieve a quantification using a binary classification approach [START_REF] Powers | Evaluation: From Precision, Recall and F-Measure To Roc, Informedness, Markedness & Correlation[END_REF][START_REF] Vihinen | How to evaluate performance of prediction methods? Measures and their interpretation in variation effect analysis[END_REF]. This methods relies on the statistical analysis of the percentages of true positive (TP), false positive (FP), true negative (TN) and false negative (FN) cases varying as a function of a given threshold parameter. Here, such threshold value was addressed to all of considered molecular descriptor differences between API and coformer and was set as to not exceed 5% uncertainty (amount of FP%) as a typical and reasonable statistical uncertainty criterion. The final criterion applied for selection of the most useful molecular parameters was related the highest possible value of TP associated with an acceptable level of FP inaccuracy. The search for the highest percentage of positive cases included in the model with simultaneous preserving of at most 5% of FP was extended over all 3679 molecular descriptors used for classification. During this stage only the compounds included in the training set as positive and negative subsets were used. As a result, the final set of molecular descriptors was restricted to those offering efficiency higher than 65%. This was graphically documented in Fig. 1. Interestingly, the best descriptive potential was found to be addressed to lipoaffinity index (LA) (R Liu et al., 2001;Ruifeng Liu et al., 2001), for which the inclusion of as many as 70% true positive cases was possible. This descriptor quantifying lipoaffinity is not bundled with hydroaffinity since it represents summing over atomic E-state indices other than nitrogen and oxygen centers. Thus, the main contributions come from carbon atoms and atoms of low hydrogen bonding capacity such as halogen ones. Also those atoms, which are characterized by high polarizability, as for example P or S, have significant contributions to LA. The definition of lipoaffinity index uses weighting [START_REF] Gramatica | Modelling and prediction of soil sorption coefficients of non-ionic organic pesticides by molecular descriptors[END_REF][START_REF] Hall | Electrotopological State Indices for Atom Types: A Novel Combination of Electronic, Topological, and Valence State Information[END_REF]Ruifeng Liu et al., 2001) estimated based on linear regressions. The most important fact of LA is that it is defined on the basis of topological E-state indices, which do not require molecular 3-D information. Hence, no significant computations are really necessary for estimating values of this molecular descriptor. The second parameter of potential usefulness is MaxAasC. It is also related to atomic E-state characteristics but includes only the maximal value for carbon atom in each compound and so it encompasses similar information of molecules as LA. Finally, the third parameter of potential discriminating power is simply LogP in variant computations protocols. This is in accord with chemical intuition and pharmaceutical drug delivery strategies [START_REF] Brough | Use of polyvinyl alcohol as a solubility-enhancing polymer for poorly water soluble drug delivery (Part 1)[END_REF][START_REF] Ghanbarzadeh | Dramatic improvement in dissolution rate of albendazole by a simple, one-step, industrially scalable technique[END_REF][START_REF] Kalepu | Insoluble drug delivery strategies: review of recent advances and business prospects[END_REF][START_REF] Maggi | II. Technological approaches to improve the dissolution behavior of nateglinide, a lipophilic insoluble drug: Comilling[END_REF][START_REF] Sareen | Improvement in solubility of poor water-soluble drugs by solid dispersion[END_REF][START_REF] Savjani | Drug solubility: importance and enhancement techniques[END_REF][START_REF] Serajuddin | Solid dispersion of poorly water-soluble drugs: early promises, subsequent problems, and recent breakthroughs[END_REF][START_REF] Sruti | Improvement in the dissolution rate and tableting properties of cefuroxime axetil by melt-granulated dispersion and surface adsorption[END_REF] that differences in polarities between drug and excipient is a main driving force toward dissolution. Many examples of such cases can be found in the literature, including cocrystals but also other types of solid mixtures such as inclusion complexes with cyclodextrins [START_REF] Dua | Dissolution behavior of β-cyclodextrin molecular inclusion complexes of aceclofenac[END_REF][START_REF] Badr-Eldin | Aripiprazole-cyclodextrin binary systems for dissolution enhancement: effect of preparation technique, cyclodextrin type and molar ratio[END_REF]Thiry et al., 2017aThiry et al., , 2017b) ) and dispersions with soluble polymers [START_REF] Brough | Use of polyvinyl alcohol as a solubility-enhancing polymer for poorly water soluble drug delivery (Part 1)[END_REF][START_REF] Choi | Design of PVP/VA S-630 based tadalafil solid dispersion to enhance the dissolution rate[END_REF][START_REF] Ibrahim | Polymeric surfactant based etodolac chewable tablets: formulation and in vivo evaluation[END_REF][START_REF] Maggi | II. Technological approaches to improve the dissolution behavior of nateglinide, a lipophilic insoluble drug: Comilling[END_REF][START_REF] Six | Increased physical stability and improved dissolution properties of itraconazole, a class II drug, by solid dispersions that combine fast-and slow-dissolving polymers[END_REF]Thiry et al., 2017a). Noteworthy, [START_REF] Guha | Exploratory analysis of kinetic solubility measurements of a small molecule library[END_REF], found that APIs can be classified in terms of their kinetic solubility values using different hydrophilicity indices. Hence, all three molecular descriptors identified during screening procedure are related to differences in the polarity of cocrystals components. Indeed, the positive values of all three measures considered here suggest that the higher difference the higher chance of dissolution rate enhancement. For example, the threshold value of LA associated with 5% confidence is equal to 3.61. It is worth mentioning that despite similarities in both definition and meaning, the selected parameters are almost not correlated to each other. The values of determination coefficient, R 2 , between LA and MaxAasC or LogP are equal to 0.39 and 0.24, respectively. The correlation between last two descriptors is even lower (R 2 =0.10). Taking all these observations into account, it is rational to assume that for coformers screening the values of LA3.61 seems to be a sufficiently accurate criterion for classification of dissolution rate enhancement of cocrystals formed with active pharmaceutical ingredients. It is worth mentioning, that the value of LA maximizing TP% at 95% confidence corresponds also to the maximum of Matthews correlation coefficient distribution [START_REF] Matthews | Comparison of the predicted and observed secondary structure of T4 phage lysozyme[END_REF]. This index is a measure

A C C E P T E D M A N U S C R I P T

6 of correlation between the observed and predicted binary classifications and adopts values in the <-1,+1> range. It is worth mentioning that MCC is a contingency matrix method of calculating the Pearson product-moment correlation coefficient and has essentially the same interpretation [START_REF] Powers | Evaluation: From Precision, Recall and F-Measure To Roc, Informedness, Markedness & Correlation[END_REF]. Hence, it relies on ranking rather than mean values as in the case of ordinary correlation coefficient. The interpretation of MCC values is straightforward since +1 value represents a perfect match and -1 value denotes the total disagreement between prediction and observation. Finally, a zero denotes no better than random prediction. Hence, the maximization of MCC offering a quite useful criterion for quantification of theoretical approaches (Cysewski, 2016a) additionally confirms that the proposed method for threshold values is statistically meaningful. In the case analyzed here, the MCC reached a value of 0.54 suggesting a quite modest but acceptable performance of the statistical classification.

Exemplary applications

The above formulated criterions of coformers selection for increasing of cocrystals solubility rate do not include a very important question, whether the homogeneous bi-molecular complex is to be formed. It is well known that the outcome of solidification either as binary eutectics or cocrystal cannot be generally predicted from chemical formula (Cysewski, 2016a(Cysewski, , 2016b)). However, the probability of intermolecular complex formation in the solid state and quantification of the probability of cocrystal formation can be determined by computing the values of mixing enthalpy (Cysewski, 2016a). This indispensable additional criterion is a well-accepted measure of cocrystallization propensity. This kind of quantification of coformers affinity was successfully used for experimentally verified cocrystals screening [START_REF] Cysewski | In silico screening of dicarboxylic acids for cocrystallization with phenylpiperazine derivatives based on both cocrystallization propensity and solubility advantage[END_REF](Cysewski, , 2016a;;[START_REF] Cysewski | Exploring the cocrystallization potential of urea and benzamide[END_REF][START_REF] Klamt | Solvent-screening and co-crystal screening for drug development with COSMO-RS[END_REF][START_REF] Klamt | The COSMO and COSMO-RS solvation models[END_REF][START_REF] Klamt | Prediction of aqueous solubility of drugs and pesticides with COSMO-RS[END_REF][START_REF] Klamt | COSMO: a new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient[END_REF][START_REF] Loschen | Solubility prediction, solvate and cocrystal screening as tools for rational crystal engineering[END_REF]; Przybyłek et al., 2016b). In order to demonstrate the usefulness of the screening procedure proposed here, the search for DSR enhancers was extended over a quite comprehensive set of coformers defined in the methodology part. The potential pairs of drug and excipient were classified according to both high probability of cocrystal formation and a high chance for improvement of the dissolution rate by the following conjunction H mix <-0.18 and LA>3.61. This region can be identified by right-bottom quadrant on figures presenting the distributions of relative values of lipoaffinity index as a function of mixing enthalpy.

A C C E P T E D M A N U S C R I P T

Prediction of DSR enhancement of Iloperidone

Iloperidone is an atypical antipsychotic for the treatment of schizophrenia symptoms. The improvement in cognition of schizophrenic patients is attributed to its activity as dopamine D2 and 5-HT2A receptor antagonist. Although it is practically insoluble in water, the bioavailability of this medicament is as high as 96% due to good absorption from the GI tract. The maximal concentration is reached within 2-4 hours. This is achieved by tableting with very efficient dispersing agents. Alternatively, the increase of water solubility might be obtained via cocrystallization with an appropriate solubility enhancer. Indeed, such compounds as 3,5-dihydroxybenzoic acid, 3hydroxybenzoic acid or 2,3-dihydroxybenzoic acid were found [START_REF] Zhang | Using dissolution and pharmacokinetics studies of crystal form to optimize the original iloperidone[END_REF] to be well suited for this purpose. It is interesting to see if other compounds can also be used for improvement of dissolution rate of this drug. For this purpose the whole two-step screening procedure was applied and values of LA and H mix were estimated for all pairs of potential cocrystals formed with solids belonging to EAFUS/GRAS list. As a result, the distributions of these chemometric measures were obtained and plotted in Fig. 2. Inspection of these data leads to the conclusion that there is a broad range of compounds which might be considered as dissolution rate enhancers. Although many pairs comprising Iloperidone fulfil the criterion of lower lipoaffinity compared to Iloperidone, there are also numerous cases excluded due to low probability of cocrystallization. Thus, H mix criterion is quite restrictive in the case of Iloperidone pairs. There is a relatively low number of cases, which were additionally excluded due to too low values of LA. In total, as many as 141 potential coformers might be advised for experimental screening as indicated in Fig. 2 by green circles located in the rightbottom quadrant. The three cocrystals studied experimentally are also found in the target window. Ten new cocrystals of the highest relative values of lipoaffinity index were enumerated in Fig. 2. In this subset the first two compounds, Inositol and Sucralose, are rich in hydroxyl groups and the rest are carboxylic acids. The full list of dissolution rate enhancers of Iloperidone identified by the applied screening procedure is provided in supporting materials (see Table S3). On this list one can find not only other carboxylic acids but also some amines, heterocyclic compounds and other polar compounds.

Prediction of DSR enhancement of Ritonavir

Second example documents the results of classification of potential cocrystals of Ritonavir. This antiretroviral drug, sold under the trade name Norvir, is used to treat HIV/AIDS and takes active part in combination treatment known as highly active antiretroviral therapy (HAART). Ritonavir is a protease inhibitor with activity against Human Immunodeficiency Virus Type 1 (HIV-1). It can be administered either in the form of capsules or tablets but the latter are preferred from the bioavailability point of view. It is classified as practically non-soluble in water what might justify the cocrystals screening for solubility enhancement. Interestingly, many potential binary solids are worth studying experimentally as it was documented in Fig. 3. There are as many as 302 coformers resulting from the performed screening procedure. The top ten cases are enumerated in Fig. 3., while the rest are collected in supporting materials (see Table S4). Also in the case of Ritonavir, all three experimentally studied systems are found in the target quadrant in Fig. 3.

Excipient

Prediction of DSR enhancement of Carbamazepine

The third drug was selected for documenting some problematic cases. Carbamazepine sold under a variety of tradenames is a drug primarily used in the treatment of neuropathic pain and epilepsy. Its anticonvulsant activity is granted from structural similarities to tricyclic antidepressants and the capability to inhibit sustained repetitive firing by blocking use-dependent sodium channels. The synaptic transmission blockage in the trigeminal nucleus and seizure control is attributed to pain relief and reduction of post-tetanic potentiation of synaptic transmission in the spinal cord. Carbamazepine also possesses many other therapeutic effects including anticholinergic, central antidiuretic, antiarrhythmic, muscle relaxant, antidepressant, sedative, and neuromuscular-blocking properties. Carbamazepine is sparsely soluble in water (17.7 mg/L, https://www.drugbank.ca/) and in clinical studies both suspensions and conventional tablets are used with varying bioavailability, both in the terms of time to peak concentration and percentage of absorption. It seems then to be interesting to find excipients affecting the dissolution rate of Carbamazepine. In Fig. 4 there are presented the results of applied screening procedure. What is directly visible in the case of this drug, not all of experimentally studied systems were found within the target predictive region. This is the cost of binary classification, which excluded almost one fifth of systems despite the fact that they belong to true positive cases. This misfortunate situation is somewhat compensated with a relatively high number of other potential DSR enhancers. Indeed, as many as 114 cases were identified as potential coformers interesting from the perspective of dissolution rate improvement as enumerated in Table S5. The best ten were collected in Fig. 4. Although they all show quite modest affinities toward Carbamazepne, what is indicated by relatively high H mix values, they are still found within the target quadrant. It is worth noting that the majority of selected dissolution rate enhancers are aliphatic and aromatic acidic compounds including phenolic acids, aliphatic dicarboxylic and hydroxycarboxylic acids. Apart from dissolution rate, these promising classes of coformers have been extensively studied in the context of improving carbamazepine solubility [START_REF] Childs | Screening strategies based on solubility and solution composition generate pharmaceutically acceptable cocrystals of carbamazepine[END_REF][START_REF] Rodr Guez-Hornedo | Solubility Advantage of Pharmaceutical Cocrystals[END_REF][START_REF] Shayanfar | Solubility and dissolution rate of a carbamazepine-cinnamic acid cocrystal[END_REF][START_REF] Ullah | The development of carbamazepine-succinic acid cocrystal tablet formulations with improved in vitro and in vivo performance[END_REF]. Unfortunately only very few reports on the bioavailability of carbamazepine cocrystals can be found in the literature (Hickey et al., 2007;[START_REF] Ullah | The development of carbamazepine-succinic acid cocrystal tablet formulations with improved in vitro and in vivo performance[END_REF].

Prediction of DSR enhancement of Ethenzamide

For the above presented examples of cocrystals the application of screening procedure identified many candidates for dissolution rate enhancers. However, it is worth mentioning that not all drugs are so prone to cocrystallization with dissolution rate improvement. Ethenzamide seems to be such a case. This popular analgesic and antipyretic agent is a non-prescription drug belonging to non-steroidal anti-inflammatory agents with medicinal uses similar to those of aspirin. It is typically administered in combination with other components as Acetaminophen, Aspirin or Caffeine in the over-the-counter pain remedies. For this last example analyzed here only few new cocrystals of improved DSR can be suggested. Fortunately, all six experimentally studied cases were also positively identified by the applied procedure, but the list of new coformers is not very long. This indicates that it will be probably quite difficult to find effective enhancers of dissolution rate of Enthenzamide via cocrystallization. Apart from ten coformers listed in Fig. 5 there were found only thirteen additional ones, collected in supporting materials (see Table S6).

A C C E P T E D

M A N U S C R I P T

Conclusions

Recently there has been a growing interest in in-silico pharmaceutical excipients screening methods including bioavailability and solubility improvement agents such as soluble polymers, surfactants and nutraceutical cocrystals formers [START_REF] Bergström | Computational prediction of formulation strategies for beyond-rule-of-5 compounds[END_REF][START_REF] Christensen | Surface area, volume and shape descriptors as a novel tool for polymer lead design and discovery[END_REF][START_REF] Cysewski | In silico screening of dicarboxylic acids for cocrystallization with phenylpiperazine derivatives based on both cocrystallization propensity and solubility advantage[END_REF][START_REF] Przybyłek | Applicability of phenolic acids as effective enhancers of cocrystals solubility of methylxanthines[END_REF]. The major advantage of efficient theoretical approaches, important from the economic viewpoint, is reducing the time and reagents costs. The large number of experimental data available in the literature gives the opportunity to build models for predicting many pharmaceutically relevant features. However, the main problem with solubility and dissolution rate modeling are non-homogeneous experimental data sets. However, for the screening purposes, qualitative physicochemical properties classification methods can be very useful for providing an initial overview of the excipients applicability. The search for enhancers of drugs dissolutions rate is dictated by the importance of drugs bioavailability control. Although there are many ways of water dissolution improvement, the cocrystallization with appropriate excipient is one of the most common approaches. This paper documented a new way of classification of potential cocrystal formers into four classes depending on the values of mixing enthalpy and relative value of lipoaffinity index (expressed as the difference between descriptor calculated for API and excipient). The whole procedure relied on the training set comprising positive and negative cases of cocrystals found in the literature. The formers included 102 drug-excipient pairs for which experimentally proven increase of dissolution rate was observed with respect of pure drug. The latter set comprised 17 cocrystals for which such effect was not observed. Despite the fact that the only available data were of qualitative character, the performed binary classification statistical analysis allowed to formulate quantitative criterions. Among considered 3679 molecular descriptors the relative value of lipoaffinity index has been found as the most appropriate measure suited for discrimination of positive and negative cases. Assuming 5% precision the applied classification criterion led to inclusion of 70% positive cases in the final prediction. Since lipoaffinity index is a molecular descriptor computed using 2D information about a chemical structure, its estimation is straightforward and computationally inexpensive. The inclusion of additional criterion characterizing the cocrystallization probability completed the requirements for an effective and reliable theoretical screening procedure. The practical applications require only the computation of the relative value of lipoaffinity index of excipient with respect of considered drug and additionally their mutual affinity expressed in terms of H mix values. Hence, the conjunction criterion can be formulated as follows: From the graphical point of view this corresponds to the right-bottom quadrant of LA distributions plotted against H mix values. Hence, the practice of theoretical screening is very straightforward, provided that the collections of these two molecular descriptors are available. The values of LA can be computed virtually for any compound using freely available software [START_REF] Yap | PaDEL-descriptor: An open source software to calculate molecular descriptors and fingerprints[END_REF]. The computation of H mix values requires a commercial program, what is also not a restrictive requirement since it is considered as a standard of first principle thermodynamics computations [START_REF] Klamt | Solvent-screening and co-crystal screening for drug development with COSMO-RS[END_REF][START_REF] Klamt | The COSMO and COSMO-RS solvation models[END_REF][START_REF] Klamt | Prediction of aqueous solubility of drugs and pesticides with COSMO-RS[END_REF][START_REF] Klamt | COSMO: a new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient[END_REF]. What is worth mentioning, the sets of promising coformers can be identified for many drugs as detailed in provided supplementary materials. These lists offer experimentalist the first choice selection of compounds for screening of dissolution rate improvements and will be used also in our laboratory for future projects. Since the applied statistical procedure was formulated using 5% precision, it is also expected that the same accuracy is associated with provided sets of potential coformers. It is also important to notice that almost one third (30%) of actual true positive cases are excluded from the analysis due to the limited nature of discriminating power of lipoaffinity index itself. However, this molecular descriptor was found to be the best among many other indices. From the practical point of view finding all of DSR enhancers is not really necessary since any success in finding of at least some is sufficient for new forms of drugs developments. It should be however noted that there are not many negative examples available in the literature and therefore further studies should be carried out to find more precise cocrystal dissolution rate improvement criterions.

Fig. 1 .

 1 Fig.1. Threshold values of selected molecular descriptors affecting distributions of false positive (FP%) and true positive (TP%) percentages. The relative values representing differences between molecular descriptors values characterizing drugs with respect of excipients were added below plots ( 1 PaDEL, 2 Chemopy, 3 RDFKit). Error bars represent the standard deviation determined for 20 randomly selected subsets containing 50% of DSR dataset.

Fig. 2 .

 2 Fig.2. The distributions of relative values of lipoaffinity index (LA) as a function of mixing enthalpy (H mix ) characterizing pairs formed by Iloperidone and all solid coformers found on EAFUS/GRAS list. Potential cocrystals with enhanced dissolution rate are represented by green circles. Experimentally

Fig. 3 .

 3 Fig.3. The distributions of relative values of lipoaffinity index (LA) as a function of mixing enthalpy (H mix ) characterizing pairs formed by Ritonavir and all solid coformers found on EAFUS/GRAS list. Potential cocrystals with enhanced dissolution rate are represented by green circles. Experimentally confirmed positive examples are denoted by red circles. Black plus signs denote cases with high probability of salt formation (pKa>2.7).

Fig. 4 .

 4 Fig.4. The distributions of relative values of lipoaffinity index (LA) as a function of mixing enthalpy (H mix ) characterizing pairs formed by Carbamazepine and all solid coformers found on EAFUS/GRAS list. Potential cocrystals with enhanced dissolution rate are represented by green circles. Experimentally confirmed positive examples are denoted by red circles.

Fig. 5 .

 5 Fig.5. The distributions of relative values of lipoaffinity index (LA) as a function of mixing enthalpy (H mix ) characterizing pairs formed by enthenzamide and all considered coformers. Potential cocrystals with enhanced dissolution rate are represented by green circles. Experimentally confirmed positive examples are denoted by red circles.

Acknowledgements

Technical assistance of Anna Cieślińska MD) and Tomasz Miernik (MD) during computations and preparation of the manuscript is appreciated.

A C C E P T E D

M A N U S C R I P T


Philippe Martin

Lionel Rosier

Pierre Rouchon

Controllability of Parabolic Equations by the Flatness Approach

Keywords: 2000 Mathematics Subject Classification. -93B05,93C20 .degenerate parabolic equation, singular coefficient, null controllability, Gevrey functions, flatness

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The null controllability of parabolic equations has been extensively investigated since several decades. After the pioneering work in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF][START_REF] Jones | A fundamental solution for the heat equation which is supported in a strip[END_REF][START_REF] Luxemburg | Entire functions and Müntz-Szász type approximation[END_REF], mainly concerned with the one-dimensional case, there has been significant progress in the general N-dimensional case [START_REF] Fursikov | Controllability of evolution equations[END_REF][START_REF] Imanuvilov | Controllability of parabolic equations[END_REF][START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] by using Carleman estimates. The more recent developments of the theory were concerned with discontinuous coefficients [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF][START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF], degenerate coefficients [START_REF] Cannarsa | Persistent regional null controllability for a class of degenerate parabolic equations[END_REF][START_REF] Cannarsa | Carleman estimates for a class of degenerate parabolic operators[END_REF][START_REF] Cannarsa | Carleman estimates and null controllability for boundarydegenerate parabolic operators[END_REF][START_REF] Flores | Carleman estimates for degenerate parabolic equations with first order terms and applications[END_REF], or singular coefficients [START_REF] Cazacu | Controllability of the heat equation with an inverse-square potential localized on the boundary[END_REF][START_REF] Ervedoza | Control and stabilization properties for a singular heat equation with an inverse-square potential[END_REF].

In [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF], the authors derived the null controllability of a linear one-dimensional parabolic equation with (essentially bounded) measurable coefficients. The method of proof combined the Lebeau-Robbiano approach [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] with some complex analytic arguments.

Here, we are concerned with the null controllability of the system (a(x)u x ) x + b(x)u x + c(x)u -ρ(x)u t = 0, x ∈ (0, 1), t ∈ (0, T ), (1.1) α 0 u(0, t) + β 0 (au x )(0, t) = 0, t ∈ (0, T ), (1.2) α 1 u(1, t) + β 1 (au x )(1, t) = h(t), t ∈ (0, T ), (1.3) u(x, 0) = u 0 (x), x ∈ (0, 1), (1.4) where (α 0 , β 0 ), (α 1 , β 1 ) ∈ R 2 \ {(0, 0)} are given, u 0 ∈ L 2 (0, 1) is the initial state and h ∈ L 2 (0, T ) is the control input.

The given functions a, b, c, ρ will be assumed to fulfill the following conditions a(x) > 0 and ρ(x) > 0 for a.e. x ∈ (0, 1), (1.5)

( 1 a , b a , c, ρ) ∈ [L 1 (0, 1)] 4 , (1.6) ∃K ≥ 0, c (x) ρ(x)
≤ K for a.e. x ∈ (0, 1), (1.7) ∃p ∈ (1, ∞], a 1-1 p ρ ∈ L p (0, 1). (1.8) The assumptions (1.5)- (1.8) are clearly less restrictive than the assumptions from [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF]:

(1.9)

a, b, c, ρ ∈ L ∞ (0, 1) and a(x) > ε, ρ(x) > ε > 0 for a.e. x ∈ (0, 1)

for some ε > 0.

Let us introduce some notations. Let B be a Banach space with norm • B . For any t 1 < t 2 and s ≥ 0, we denote by G s ([t 1 , t 2 ], B) the class of (Gevrey) functions u ∈ C ∞ ([t 1 , t 2 ], B) for which there exist positive constants M, R such that (1.10)

u (p) (t) B ≤ M p! s R p ∀t ∈ [t 1 , t 2 ], ∀p ≥ 0. When (B, • B ) = (R, | • |), G s ([t 1 , t 2 ], B) is merely denoted G s ([t 1 , t 2 ]). Let L 1 ρ := {u : (0, 1) → R; ||u|| L 1 ρ := 1 0 |u(x)|ρ(x)dx < ∞}. Note that L 2 (0, 1) ⊂ L 1 ρ if ρ ∈ L 2 (0, 1)
. The main result in this note is the following

Theorem 1.1. -Let the functions a, b, c, ρ : (0, 1) → R satisfy (1.5)-(1.8) for some numbers K ≥ 0, p ∈ (1, ∞]. Let (α 0 , β 0 ), (α 1 , β 1 ) ∈ R 2 \ {(0, 0)} and T > 0.
Pick any u 0 ∈ L 1 ρ and any s ∈ (1, 2 -1/p). Then there exists a function h ∈ G s ([0, T ]), that may be given explicitly as a series, such that the solution u of (1.1)-(1.4) satisfies u(., T ) = 0. Moreover u, au x ∈ G s ([ε, T ], W 1,1 (0, 1)) for all ε ∈ (0, T ).

Clearly, Theorem 1.1 can be applied to parabolic equations with discontinuous coefficients that may be degenerate or singular at a point (or more generally at a sequence of points). The proof of it is not based on the classical duality approach, in the sense that it does not rely on the proof of some observability inequality for the adjoint equation. It follows the flatness approach developed in [START_REF] Laroche | Extension de la notion de platitude à des systèmes décrits par des équations aux dérivées partielles linéaires[END_REF][START_REF] Laroche | Motion planning for a 1-D diffusion equation using a Brunovsky-like decomposition[END_REF][START_REF] Laroche | Motion planning for the heat equation[END_REF][START_REF] Martin | Null controllability of the 1d heat equation using flatness[END_REF][START_REF] Martin | Null controllability of the 2d heat equation using flatness[END_REF][START_REF] Martin | Null controllability of the heat equation using flatness[END_REF][START_REF] Martin | Null controllability using flatness: a case study of 1-d heat equation with discontinuous coefficients[END_REF][START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF][START_REF] Martin | On the reachable states for the boundary control of the heat equation[END_REF][START_REF] Meurer | Control of Higher-Dimensional PDEs: Flatness and Backstepping Designs[END_REF]. This direct approach gives explicitly both the control and the trajectory as series, which leads to efficient numerical schemes by taking partial sums in the series [START_REF] Martin | Null controllability of the heat equation using flatness[END_REF]. The flatness approach was used in [START_REF] Martin | On the reachable states for the boundary control of the heat equation[END_REF] to improve the results in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF] concerning the reachable states for the heat equation. It was also successfully applied to Schrödinger equation in [START_REF] Martin | Controllability of the 1d schrödinger equation by the flatness approach[END_REF]. Let us describe its main steps. In the first step, following [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF], we show that after a series of changes of dependent/independent variables, system (1.1)-(1.4) can be put into the canonical form

u xx -ρ(x)u t = 0, x ∈ (0, 1), t ∈ (0, T ), (1.11) α 0 u(0, t) + β 0 u x (0, t) = 0, t ∈ (0, T ), (1.12) α 1 u(1, t) + β 1 u x (1, t) = h(t), t ∈ (0, T ), (1.13) u(x, 0) = u 0 (x), x ∈ (0, 1), (1.14)
where ρ(x) > 0 a.e. in (0, 1) and ρ ∈ L p (0, 1) with p ∈ (1, ∞]. In the second step, following [START_REF] Martin | Null controllability of the 1d heat equation using flatness[END_REF][START_REF] Martin | Null controllability of the heat equation using flatness[END_REF], we seek u in the form

u(x, t) = n≥0 e -λnt e n (x), x ∈ (0, 1), t ∈ [0, τ ], (1.15) u(x, t) = i≥0 y (i) (t)g i (x), x ∈ (0, 1), t ∈ [τ, T ], (1.16)
where τ ∈ (0, T ) is any intermediate time; e n : (0, 1) → R (resp. λ n ∈ R) denotes the n th eigenfunction (resp. eigenvalue) associated with (1.11)-(1.13) and satisfying [START_REF] Laroche | Extension de la notion de platitude à des systèmes décrits par des équations aux dérivées partielles linéaires[END_REF][START_REF] Laroche | Motion planning for a 1-D diffusion equation using a Brunovsky-like decomposition[END_REF] -e n = λ n ρ e n , x ∈ (0, 1) (1.17) α 0 e n (0) + β 0 e n (0) = 0, (1.18) α 1 e n (1) + β 1 e n (1) = 0, (1.19) while g i : (0, 1) → R is defined inductively as the solution to the Cauchy problem g 0 = 0, x ∈ (0, 1) (1.20) α 0 g 0 (0) + β 0 g 0 (0) = 0, (1.21) β 0 g 0 (0) -α 0 g 0 (0) = 1 (1.22) for i = 0, and to the Cauchy problem

g i = ρ g i-1 , x ∈ (0, 1) (1.23) g i (0) = 0, (1.24) g i (0) = 0 (1.25)
for i ≥ 1. Expanding u on generating functions as in (1.16) rather than on powers of x as in [START_REF] Laroche | Motion planning for the heat equation[END_REF][START_REF] Martin | Null controllability of the 1d heat equation using flatness[END_REF] was introduced in [START_REF] Laroche | Motion planning for a 1-D diffusion equation using a Brunovsky-like decomposition[END_REF] and studied in [START_REF] Laroche | Extension de la notion de platitude à des systèmes décrits par des équations aux dérivées partielles linéaires[END_REF].

The fact that the generating function g i is defined as the solution of a Cauchy problem, rather than the solution of a boundary-value problem, is crucial in the analysis developed here. First, it allows to prove that every initial state in the space L 1 ρ (and not only states in some restricted class of Gevrey functions) can be driven to 0 in time T . Secondly, from (1.23)-(1.25), we see by an easy induction on i that for ρ ∈ L ∞ (0, 1), the function g i is uniformly bounded by C/(2i)!, and hence the series in (1.16) is indeed convergent when y ∈ G s ([τ, T ]) with 1 < s < 2.

The corresponding control function h is given explicitly as

h(t) = 0 if 0 ≤ t ≤ τ, i≥0 y (i) (t)(α 1 g i (1) + β 1 g i (1)) if τ < t ≤ T.
It is easy to see that the function u(x, t) defined in (1.16) satisfies (formally) (1.11), and also the condition u(x, T ) = 0 if y (i) (T ) = 0 for all i ∈ N, so that the null controllability can be established for some initial states. The main issue is then to extend it to every initial state u 0 ∈ L 1 ρ . Following [START_REF] Martin | Null controllability of the 1d heat equation using flatness[END_REF][START_REF] Martin | Null controllability of the 2d heat equation using flatness[END_REF][START_REF] Martin | Null controllability of the heat equation using flatness[END_REF], we first use the strong smoothing effect of the heat equation to smooth out the state function in the time interval (0, τ ). Next, to ensure that the two expressions of u given in (1.15)-(1.16) coincide at t = τ , we have to relate the eigenfunctions e n to the generating functions g i .

It will be shown that any eigenfunction e n can be expanded in terms of the generating functions g i as

(1.26) e n (x) = ζ n i≥0 (-λ n ) i g i (x)
with ζ n ∈ R. Note that, for ρ ≡ 1 and (α 0 , β 0 , α 1 , β 1 ) = (0, 1, 0, 1), λ n = (nπ) 2 for all n ≥ 0, e 0 (x) = 1 and e n (x) = √ 2 cos(nπx) for n ≥ 0 while g i (x) = x 2i /(2i)!, so that (1.26) for n ≥ 1 is nothing but the classical Taylor series expansion of cos(nπx) at x = 0:

(1.27) cos(nπx) = i≥0 (-1) i (nπx) 2i (2i)! •
Thus (1.26) can be seen as a natural extension of (1.27), in which the generating functions g i , a priori not smoother than W 2,p (0, 1), replace the functions x 2i /(2i)!.

The condition (1.8) is used to prove the estimate

|g i (x)| ≤ C R 2i (i!) 2-1 p
needed to ensure the convergence of the series in (1.16) when y ∈ G s ([τ, T ]) with 1 < s < 2 -1/p. Theorem 1.1 applies in particular to any system (a(x)u x ) x -u t = 0, x ∈ (0, 1), t ∈ (0, T ), (1.28)

α 0 u(0, t) + β 0 (au x )(0, t) = 0, t ∈ (0, T ), (1.29) α 1 u(1, t) + β 1 (au x )(1, t) = h(t), t ∈ (0, T ), (1.30) u(x, 0) = u 0 (x), x ∈ (0, 1), (1.31)
where a(x) > 0 for a.e. x ∈ (0, 1) and a + 1/a ∈ L 1 (0, 1). This includes the case where a is measurable, positive and essentially bounded together with its inverse (but not necessarily piecewise continuous), and the case where a(x) = x r with -1 < r < 1. (Actually any r ≤ -1 is also admissible, by picking p > 1 sufficiently close to 1 in (1.8).) Note that our result applies as well to a(x) = (1 -x) r with 0 < r < 1, yielding a positive null controllability result when the control is applied at the point (x = 1) where the diffusion coefficient degenerates (see [START_REF] Cannarsa | Carleman estimates for a class of degenerate parabolic operators[END_REF]Section 2.7]). Note also that the coefficient a(x) is allowed to be degenerate/singular at a sequence of points: consider e.g. a(x)

:= | sin(x -1 )| r with -1 < r < 1. Then a + 1/a ∈ L 1 (0, 1).
The null controllability of (1.28)-(1.31) for a(x) = x r with 0 < r < 2 was established (in appropriate spaces) in [START_REF] Cannarsa | Carleman estimates for a class of degenerate parabolic operators[END_REF].

Another important family of heat equations with variable coefficients is those with inverse square potential localized at the boundary, namely

u xx + µ x 2 u -u t = 0, x ∈ (0, 1), t ∈ (0, T ), (1.32) u(0, t) = 0, t ∈ (0, T ), (1.33) α 1 u(1, t) + β 1 u x (1, t) = h(t), t ∈ (0, T ), (1.34) u(x, 0) = u 0 (x), x ∈ (0, 1), (1.35)
where µ ∈ R is a given number. Note that Theorem 1.1 cannot be applied to (1.32)-(1.35), for c(x) = µx -2 is not integrable on (0, 1). It was proved in [START_REF] Cazacu | Controllability of the heat equation with an inverse-square potential localized on the boundary[END_REF] that (1.32)-(1.35) is null controllable in L 2 (0, 1) when µ ≤ 1/4 by combining Carleman inequalities to Hardy inequalities. We note that this result can be retrieved by the flatness approach as well.

Theorem 1.2. -Let µ ∈ (0, 1/4], (α 1 , β 1 ) ∈ R 2 \{(0, 0)}, T > 0, and τ ∈ (0, T ). Pick any u 0 ∈ L 2 (0, 1) and any s ∈ (1, 2). Then there exists a function h ∈ G s ([0, T ]) with h(t) = 0 for 0 ≤ t ≤ τ and such that the solution u of (1.32)-(1.35) satisfies u(T, .

) = 0. Moreover, u ∈ G s ([ε, T ], W 1,1 (0, 1)) for all ε ∈ (0, T ). Finally, if 0 ≤ µ < 1/4 and r > (1 + √ 1 -4µ)/2, then x r u x ∈ G s ([ε, T ], W 1,1 (0, 1)) for all ε ∈ (0, T ).
The note is organized as follows. In Section 2 we give a sketch of the proof of Theorem 1.1. Section 3 is concerned with the numerical control of a heat equation with discontinuous coefficients, that may serve as a model for the heat conduction of a one-dimensional rod with constant thermals properties. Then B ∈ W 1,1 (0, 1), c ∈ L 1 (0, 1), and ã(x) > 0 and c(x) ≥ 0 for a.e. x ∈ (0, 1).

We introduce the solution v to the elliptic boundary value problem

-(ãv x ) x + cv = 0, x ∈ (0, 1), (2.1) v(0) = v(1) = 1, (2.2)
and set

u 1 (x, t) := e -Kt u(x, t), u 2 (x, t) := u 1 (x, t) v(x) • Finally, let (2.3) L := 1 0 (a(s)v 2 (s)e B(s) ) -1 ds, y(x) := 1 L x 0 (a(s)v 2 (s)e B(s) ) -1 ds and (2.4) û(y, t) := u 2 (x, t), ρ(y) := L 2 a(x)v 4 (x)e 2B(x) ρ(x)
for 0 < t < 1, y = y(x) with 0 < x < 1. Then the following result holds.

Proposition 2.1.

-(i) v ∈ W 1,1 (0, 1) and 0 < v(x) ≤ 1 for all x ∈ [0, 1]; (ii) y : [0, 1] → [0, 1
] is an increasing bijection with y, y -1 ∈ W 1,1 (0, 1); (iii) ρ(y) > 0 for a.e. y ∈ (0, 1), and ρ ∈ L p (0, 1); (iv) û solves the system ûyy -ρû t = 0, y ∈ (0, 1), t ∈ (0, T ), (2.5) α0 û(0, t) + β0 ûy (0, t) = 0, t ∈ (0, T ), (2.6) α1 û(1, t) + β1 ûy (1, t) = ĥ(t) := e -Kt h(t), t ∈ (0, T ), (2.7)

û(y(x), 0) = u 0 (x) v(x) , x ∈ (0, 1), (2.8)
for some (α 0 , β0 ), (α 1 , β1 ) ∈ R 2 \ {(0, 0)}.

Proof: (i) Let l := 1 0 ds/ã(s) and z(x) := l -1 x 0 ds/ã(s). Then, one can see that z is an increasing continuous bijection from [0, 1] to [0, 1], and (using a result due to Zareckii, see e.g. [START_REF] Bogachev | Measure Theory[END_REF]) that z -1 ∈ W 1,1 (0, 1). Introduce w(z) := v(x(z)), which solves the boundary value problem

- d 2 w dz 2 + (l 2 ãc)(x(z))w = 0, z ∈ (0, 1), w(0) = w(1) = 1.
It can be seen that 0 < w ≤ 1 on [0, 1]. The other properties (ii), (iii), and (iv) follow by direct calculations. See [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF] for more details.

L 2 ρ := f : (0, 1) → R; ||f || 2 L 2 ρ := 1 0 |f (x)| 2 ρ(x)dx < ∞ .
Then the following result holds.

Proposition 2.2. -Let p, ρ, α 0 , β 0 , α 1 , and β 1 be as above. Then there are a sequence (e n ) n≥0 in L 2 ρ and a sequence (λ n ) n≥0 in R such that (i) (e n ) n≥0 is an orthonormal basis in L 2 ρ ; (ii) For all n ≥ 0, e n ∈ W 2,p (0, 1) and e n solves -e n = λ n ρe n in (0, 1), (2.9) α 0 e n (0) + β 0 e n (0) = 0, (2.10)

α 1 e n (1) + β 1 e n (1) = 0. (2.11) (iii)
The sequence (λ n ) n≥0 is strictly increasing, and for some constant C > 0 (2.12)

λ n ≥ Cn for n 1.

Proof: Introduce for λ * 1 the boundary-value problem

-u + λ * ρu = ρf in (0, 1), (2.13) α 0 u(0) + β 0 u (0) = 0, (2.14) α 1 u(1) + β 1 u (1) = 0. (2.15)
Then, introducing a variational formulation and applying Lax-Milgram theorem, we obtain the existence and uniqueness of a solution u ∈ W 2,1 (0, 1) of (2.13)-(2.15) for any f ∈ L 2 ρ . The results in (i)-(ii) follow from an application of the spectral theorem. Finally, (iii) is established by using Prüfer substitution. We refer the reader to [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF] for more details.

We now turn our attention to the generating functions g i (i ≥ 0) defined along (1.20)-(1.25).

Proposition 2.3. - (i) g 0 (x) = (α 2 0 + β 2 0 ) -1 (β 0 -α 0 x)
(ii) There are some constants C, R > 0 such that

(2.16) ||g i || W 2,p (0,1) ≤ C R i (i!) 2-1 p ∀i ≥ 0•
Proof: (i) is obvious, and (ii) is obtained inductively from the formula

g i (x) = x 0 s 0 ρ(σ)g i-1 (σ)dσ ds.
See [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF] for more details.

The fact that we can expand the eigenfunctions in terms of the generating functions is detailed in the following Proposition 2.4. -There is some sequence (ζ n ) n≥0 of real numbers such that for all n ≥ 0 (2.17)

e n = ζ n i≥0 (-λ n ) i g i in W 2,p (0, 1).
Furthermore, for some constant C > 0, we have

(2.18) |ζ n | ≤ C(1 + |λ n | 3
2 ) ∀n ≥ 0.

Proof: Let ẽ := ζ n i≥0 (-λ n ) i g i , where ζ n ∈ R. Then we infer from (1.20)-(1.25) that ẽ = -λ n ρẽ and that α 0 ẽ(0) + β 0 ẽ (0) = 0. If we pick ζ n := β 0 e n (0) -α 0 e n (0), then we easily see that e n = ẽ. The estimate (2.18) is proved by direct calculations. We refer the reader to [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF] for more details.

Since p > 1, for any s ∈ (1, 2 -1 p ) and any 0 < τ < T , one may pick a function ϕ ∈ G s ([0, 2T ]) such that

ϕ(t) = 1 if t ≤ τ, 0 if t ≥ T.
We are in a position to derive the null controllability of (1.11)- (1.14). Let u 0 ∈ L 2 ρ . Since (e n ) n≥0 is an orthonormal basis in L 2 ρ , we can write

(2.19) u 0 = n≥0 c n e n in L 2 ρ with n≥0 |c n | 2 < ∞. Let (2.20) y(t) := ϕ(t) n≥0 c n ζ n e -λnt for t ∈ [τ, T ] and (2.21) u(x, t) = n≥0 c n e -λnt e n (x) if 0 ≤ t ≤ τ, i≥0 y (i) (t)g i (x) if τ < t ≤ T.
The main result in this section is the following Theorem 2.5. -Let p ∈ (1, ∞], ρ ∈ L p (0, 1) with ρ(x) > 0 for a.e. x ∈ (0, 1), T > 0, τ ∈ (0, T ), and (α 0 , β 0 ), (α 1 , β 1 ) ∈ R 2 \ {(0, 0)}. Let u 0 ∈ L 2 ρ be decomposed as in (2.19), and let y be as in (2.20). Then y ∈ G s ([τ, T ]), and the control

(2.22) h(t) = 0 if 0 ≤ t ≤ τ, i≥0 y (i) (t)(α 1 g i (1) + β 1 g i (1)) if τ < t ≤ T.
is such that the solution u of (1.11)-(1.14) satisfies u(., T ) = 0. Moreover u is given by

(2.21), h ∈ G s ([0, T ]), and u ∈ C([0, T ], L 2 ρ ) ∩ G s ([ε, T ], W 2,p (0, 1
)) for all 0 < ε ≤ T .

Proof: It is clear that the function u solves formally (1.11)-(1.14) on (0, τ ) and on (τ, T ), together with u(., T ) = 0. The main concern is thus the convergence of the two series in (2.21) and the fact that they coincide at t = τ . See [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF] for the details. As a possible application, we consider the boundary control by the flatness approach of radial solutions of the heat equation in the ball B(0, 1) ⊂ R N (2 ≤ N ≤ 3). Using the radial coordinate r = |x|, we thus consider the system

u rr + N -1 r u r -u t = 0, r ∈ (0, 1), t ∈ (0, T ), (2.23)
u r (0, t) = 0, t ∈ (0, T ), (2.24)

α 1 u(1, t) + β 1 u r (1, t) = h(t), t ∈ (0, T ) (2.25)
u(r, 0) = u 0 (r), r ∈ (0, 1). (2.26) Note that Theorem 1.1 cannot be applied directly to (2.23)-(2.26), for (1.7) fails. (Note that, in sharp contrast, the control on a ring-shaped domain {r 0 < |x| < r 1 } with r 1 > r 0 > 0 is fully covered by Theorem 1.1, the coefficients in (2.23) being then smooth and bounded.)

We use the following change of variables from [START_REF] Colton | Integral operators and reflection principles for parabolic equations in one space variable[END_REF] which allows to remove the term with the first order derivative in r in (2.23):

(2.27) u(r, t) = ũ(r, t) exp(- 1 2 r 0 N -1 s ds) = ũ(r, t) r N -1 2 • Then (2.23) becomes (2.28) ũrr + (N -1)(3 -N ) 4 ũ r 2 -ũt = 0.
This equation has to be supplemented with the boundary/initial conditions ũ(0, t) = 0, t ∈ (0, T ), (2.29)

(α 1 - N -1 2 β 1 )ũ(1, t) + β 1 ũr (1, t) = h(t), t ∈ (0, T ), (2.30) ũ(r, 0) = r N -1 2 u 0 (r), r ∈ (0, R). (2.31)
For N = 3, (2.28) reduces to the simple heat equation ũrr -ũt = 0 to which Theorem 1.1 can be applied.

For N = 2, (2.28)-(2.31) is of the form (1.32)-(1.35) with µ = 1/4. Therefore Theorem 1.2 can be applied.

Numerical control of the 1-D heat equation with discontinuous coefficients

We consider the heat conduction in a one-dimensional rod made of two sections with constant thermal properties. Without restriction, we can assume the rod has length 1, with one section of length X and the other of length 1 -X. The evolution of the temperature u is given by the heat equation

ρθ t (x, t) = (aθ x ) x (x, t).
a and ρ are the piecewise constant functions on (0, 1):

(a(x), ρ(x)) := (a 0 , ρ 0 ), 0 < x < X, (a 1 , ρ 1 ), X < x < 1
where a 0 , a 1 , ρ 0 and ρ 1 are strictly positive constants. At the 0-end, the rod is submitted to the constant ambient temperature θ 0 , and at the 1-end to a time-varying heat source (the control input) of temperature θ 1 (t). The heat flux -aθ x at the ends obeys the convection conditions

-(aθ x )(0, t) = h 0 (θ 0 -θ(0, t)), -(aθ x )(1, t) = h 1 (θ(1, t) -θ 1 (t))
with h 0 , h 1 some positive constants. Setting u(x, t) := θ(x, t) -θ 0 and taking as input h(t) := θ 1 (t) -θ 0 results in the boundary value problem (a(x)u x ) x -ρ(x)u t = 0, x ∈ (0, 1), t ∈ (0, T ), (3.1)

α 0 u(0, t) + β 0 (au x )(0, t) = 0, t ∈ (0, T ), (3.2)
α 1 u(1, t) + β 1 (au x )(1, t) = h(t), t ∈ (0, T ), (3.3) u(x, 0) = u 0 (x), x ∈ (0, 1), (3.4) where the constants α 0 , β 0 , α 1 , β 1 satisfy α 2 0 + β 2 0 > 0, α 2 1 + β 2 1 > 0, α 0 β 0 ≤ 0 and α 1 β 1 ≥ 0. Notice the two limiting cases: β i = 0 (Dirichlet conditions), obtained when taking h i → ∞; α i = 0 (Neumann conditions), obtained when taking as control input h(t) := h 1 (θ 1 (t) -θ(1, t)) and letting h 0 = 0.

Included in the formulation of the system is the fact that a solution u and its quasi-derivative au x are continuous on [0, 1], and in particular at x = X (whereas u x will in general be discontinuous at X). We could thus rewrite (3.1) more explicitly as the piecewise constant heat equation u t (x, t) = a0 ρ0 u xx (x, t), 0 < x < X, u t (x, t) = a1 ρ1 u xx (x, t), X < x < 1 together with the so-called interface conditions

u(X -, t) = u(X + , t), a 0 u x (X -, t) = a 1 u x (X + , t).
From the flatness approach developed in the previous sections, we know that the trajectory u of (3.1)-(3.4) is given explicitly in (1.15)-(1.16), the eigenfunctions e n 's being as in (1.17)- (1.19), and the generating functions g i 's being as in (1.20)- (1.25).

We refer the reader to [START_REF] Martin | Null controllability using flatness: a case study of 1-d heat equation with discontinuous coefficients[END_REF] for the details concerning the numerical computation of the e n 's and the g i 's.

The control input in the time interval (τ, T ) reads

(3.5) h(t) := i≥0 (α 1 g i (1) + β 1 g i (1))y (i) (t)
where

(3.6) y(t) := φ s ( t -τ T -τ ) n≥0 c n ζ n e -λnt
and the c n 's denote the Fourier coefficients of the initial state decomposed along the e n 's:

u 0 (x) := n≥0 c n e n (x).
The function φ s in (3.6) is the Gevrey "step function"

φ s (t) :=      1 if t ≤ 0, 0 if t ≥ 1, 1 - t 0 ϕs(z)dz 1 0 ϕs(z)dz if t ∈ (0, 1)
where the "bump function" ϕ s is defined as

ϕ s (t) = 0 if t ∈ (0, 1), exp( -1 M t k (1-t) k ) if t ∈ (0, 1)
with k = (s -1) -1 > 1 and M > 0 some constants. It is well known that both ϕ s and φ s are Gevrey of order s, with s ∈ (1, 2).

A practical problem when implementing the control (3.6) is to evaluate sufficiently many derivatives of φ s , i.e. of ϕ s . Numerical computations (with e.g. finite differences) or symbolic computations cannot be used in practice to evaluate more than 20 derivatives. Nevertheless, much more derivatives can be computed with accuracy by proceeding inductively. We first note that Derivating i times in (3.7) and using Leibniz' rule results in

p k+1 ϕ (i+1) s + i j=1 i j (p k+1 ) (j) ϕ (i+1-j) s = k( ṗϕ (i) s + ipϕ (i-1) s ) This formula gives ϕ (i+1) s in terms of ϕ (0) s , ..., ϕ (i)
s , and the derivatives of P := p k+1 that can be obtained similarly, by applying Leibniz' rule to both sides of p Ṗ = (k + 1) ṗP.

Note that, in order to avoid computing ratios of very large numbers, it is better in practice to use recursion formulas for φ(i)

s := ϕ (i)
s /(2i)! and for P (i) := P (i) /i! (see [START_REF] Martin | Null controllability using flatness: a case study of 1-d heat equation with discontinuous coefficients[END_REF].) Using this procedure, about 140 derivatives can be efficiently determined with Matlab double-precision arithmetics.

We conclude this note with some numerical simulation. We use as parameters: X = 1/2, a 0 = 10/19, ρ 0 = 15/8, a 1 = 10, ρ 1 = 1/8, α 0 = cos(π/3), β 0 = -sin(π/3), α 1 = cos(π/4), β 1 = sin(π/4). We take as initial condition u 0 (x) := 1 2 1 (1/2,1) (x) -1 2 1 (0,1/2) (x), and as control times: τ = 0.05 and T = 0.35. We pick also s = 1.65 and M = 2. The series for h and y in (3.5)-(3.6) were truncated at a "large enough" order for a good accuracy, namely ī = 130 and n = 60; a fairly large ī is needed here because (T -τ )a 0 /ρ 0 is rather small. Figure 1 shows the resulting temperature u; the discontinuity of u x at x = X is clearly visible. We refer the interested reader to [START_REF] Martin | Null controllability using flatness: a case study of 1-d heat equation with discontinuous coefficients[END_REF] for more details about the numerical investigation of the

Numerical control of a degenerate 1-D heat equation

We now apply the approach of the previous section to the degenerate heat equation θ t (x, t) -x γ θ x (x, t) x = 0, (x, t) ∈ (0, 1) × (0, T ) (4.1) α 0 θ(0, t) + β 0 x γ θ x (x, t) x=0 = u(t), (4.2) α 1 θ(1, t) + β 1 θ x (1, t) = 0, (4.3) with 0 ≤ γ < 1. The controllability properties of this system (with Dirichlet boundary conditions) are studied in [START_REF] Cannarsa | The cost of controlling degenerate parabolic equations by boundary controls[END_REF]; in particular, it is shown in [START_REF] Cannarsa | The cost of controlling degenerate parabolic equations by boundary controls[END_REF] that the cost in terms of the H 1 (0, T )-norm of the control achieving null controllability blows up as γ tends to 1 The assumptions (1.7)-(1.8) are clearly satisfied, with K = 0 and p = ∞. The system is put into its canonical form by setting x := x 1-γ and θ(x, t) := θ(x, t), which immediately yields (1 -γ) -2 x γ 1-γ θt (x, t) -θxx (x, t) = 0, (x, t) ∈ (0, 1) × (0, T ) α 0 θ(0, t) + (1 -γ)β 0 θx (0, t) = u(t), α 1 θ(1, t) + (1 -γ)β 1 θx (1, t) = 0.

For the actual scenario (γ = 0.5, T = 0.35, τ = T /10, s = 1.7), Neumann boundary conditions are used (α 0 = α 1 = 0, β 0 = β 1 = 1). The initial condition is the piecewise linear function θ 0 (x) := 1 -5x, x ∈ (0, 0.45), θ 0 (x) := 5.5, x ∈ (0.45, 0.55), and θ 0 (x) := 2x -0.5, x ∈ (0.55, 1).

We refer the interested reader to [START_REF] Martin | Flatness and null controllability of 1-d parabolic equations[END_REF] for the details about the numerical scheme.

2 .

 2 Sketch of the proof of Theorem 1.1 2.1. Reduction to the canonical form (1.11)-(1.14). -Let a, b, c, ρ, and p be as in (1.5)-(1.8). Set B(x) := x 0 b(s) a(s) ds, ã(x) := a(x)e B(x) c(x) := (Kρ(x) -c(x))e B(x) .

2. 2 .

 2 Null controllability of the control problem (1.11)-(1.14). -Assume given p ∈ (1, ∞], ρ ∈ L p (0, 1) with ρ(x) > 0 for a.e. x ∈ (0, 1), and (α 0 , β 0 ), (α 1 , β 1 ) ∈ R 2 \ {(0, 0)}. Let = d/dx, and let

Theorem 1 .

 1 1 follows from Theorem 2.5. Modifying slightly the first step in the proof of Theorem 1.1, we can reduce (1.32)-(1.35) to the canonical form (1.11)-(1.14), so that the conclusion of Theorem 1.2 follows from Theorem 2.5.

(3. 7 ) 1 k

 71 p k+1 φs = k ṗϕ s where p(t) := M t(1 -t) is a polynomial function of degree 2 (hence its derivatives of order > 2 are zero).

Figure 1 .

 1 Figure 1. Evolution of the temperature u(x, t)

Figure 2 .

 2 Figure 2. a) Evolution of θ (γ = 1/2); b) corresponding initial, intermediate, and final states.