Paul-Etienne Vidal

Jean-Marc Diénot

Trajin Baptiste

• Champ magnétique.

• Activités de conception

• Caractérisation, intégration et validation initiale.

Contexte: mise en oeuvre de composants Grand Gap q ! efficacité ; ! contraintes environnementales et en fonctionnement;

q " temps de commutation à Plus fort gradient de température et points chauds locaux à Plus de EMI/EMC q Projet ANR Captif : intégration de plusieurs capteurs à Amélioration des modèles de connaissance comportementaux des assemblages (électro-thermomécanique)

à Estimer de l'état de santé et prédiction la durée de vie, q Modèle multiphysique à Déterminer les zones intéressantes pour l'implémentation des capteurs.

q Capteur de déformation et de temperature à base de technologie de type Nanoparticules ; capteurs électromagnétiques à validation et intégration.

q Procédé avancé pour le traitement des données à Méthode d'estimation et traitement du signal à Adapté à plusieurs capteurs (de type variés).

q Procédé compatible avec un procédé d'intégration d'assemblage en salle blanche q Grande capacité de surveillance;

q Modèles de comportement électro-thermomécanique des assemblages;

q Conception de plusieurs capteurs intégrés avec leurs algorithmes de traitement associés.

Conclusion

q Nouveaux modules avec capteurs intégrés q Nouveaux capteurs de déformation et de température pour l'En de P;

q EMI/EMC caractérisation.

q Estimation de la durée de vie.

q Meilleurs connaissance du comportement. q Modèle électro-thermomécanique q O : point d'estimation de la temperature;

q C : mesure du capteur q S : source de puissance q P : température locale ou puissance introduite q F1, F2, F3, F4 et F5 modèle de type "transfert" entre points.

  M. Dienot (1), F. Rotella (1), I. Sakhraoui (1), B. Trajin (1), G. Vine (1), I. Zambettakis (1), A. Renaud (2), E. Woirgard (2), G. Bayle (3), S. Behar (3), P. Lasserre (4), J. Favre (5) (1) : Laboratoire Génie de Production, Université de Toulouse, INPT -ENIT, 47 avenue d'Azereix 65000 TARBES, paul-etienne.vidal@enit.fr (2) : Intégration du Matériau au Système, CNRS UMR5218, (3) : Nanolike SAS, (4) : Primes Association, (5) : aPSI3D SAS

q

  Comparaison des simulations aux mesures q Définition d'un indicateur de vieillissement • Densité d'énergie de déformation• #$% = & ' ( ' ) *+,• Test de vieillissement.Traitement des signaux -estimationq Méthode adaptée et conception d'algorithmes pour le traitement des données en vue de la surveillance d'un module.


Gurvan Jodin
email: jodin@laplace.univ-tlse.fr

Johannes Scheller
email: scheller@mit.edu

Jean-Franc ¸ois Rouchon
email: rouchon@laplace.univ-tlse.fr

Marianna Braza
email: braza@imft.fr

On the multidisciplinary control and sensing of a smart hybrid morphing wing

Morphing wing technology is of great interest for improving the aerodynamic performance of future aircraft. A morphing wing prototype using both surface embedded Shape Memory Alloys (SMA) and piezoelectric macro fiber composite (MFC) actuators has been designed for wind tunnel experiments. This smart wing is a mechatronic system that contains embedded sensors to measure the surrounding flow and control the actuators. This article will focus on the control of the cambering system which is achieved using a group of nested control loops as well as on the perspective of a novel control strategy using in-situ temperature measurements. It will be shown that by exploiting the inherent hysteretic properties of the SMAs cambering a significant reduction in power consumption is possible by appropriately tailoring the control strategy. Furthermore, by comparing the post-processed pressure signals recorded during the wind tunnel experiments to the aerodynamic performance gains a perspective for a novel in-situ control will be shown.

I. INTRODUCTION

Nowadays the environmental compatibility of aircraft is an important issue. This leads to the design of more efficient aircraft. With respect to aerodynamic performance, current planes have their wings optimized for one cruise flight step. During flight, the altitude, the weight and the speed are changing so one fixed wing shape is sub-optimal. It has been shown that changing the shape of the wing can save several percents of fuel burn for a regional passenger aircraft [START_REF] Lyu | Aerodynamic shape optimization of an adaptive morphing trailing edge wing[END_REF]. The concept of real time wing shape adaptation is called morphing.

While the benefits of morphing wings are known, the ways to obtain these benefits are a subject of intense research. Actuation system weigh, safety and reliability are examples of common issues. Some research focuses on relatively high Technology Readiness Level (TRL) targeting current industrial airliners at true scale. The European research project SARISTU [START_REF] Dimino | Smart Intelligent Aircraft Structures (SARISTU): Proceedings of the Final Project Conference[END_REF] and the Adaptive Compliant Trailing Edge Flap [START_REF] Kota | Flight testing of the flexfloil tm adaptive compliant trailing edge[END_REF] concept from the USA both address morphing wings for industrial airliners. Flight tests are currently in progress, demonstrating the effectiveness and airworthiness of the solutions. However, these new adaptive structures are actuated through conventional actuators like servomotors.

Recent advances in the field of smart materials show the potential to overcome difficulties to make a wing both stiff enough to withstand the loads and flexible enough to be easily deformed [START_REF] Barbarino | A review of morphing aircraft[END_REF]. As complex multidisciplinary approaches are needed, such technologies are primarily limited to laboratories, with low TRL. Considering electroactive materials, Shape Memory Alloys (SMAs) and piezoelectric materials are commonly used. SMAs are characterized by thermomechanical behaviors, and most of applications use Joule heating to activate the SMAs. Diverse morphing concepts have been developed [START_REF] Barbarino | A review on shape memory alloys with applications to morphing aircraft[END_REF]. Typical applications are shape adaptation at low deformation speed. Some SMAs are able to carry more than 400 MPa and can recover strain of over 4%. Research shows the ability of SMAs to reach one million actuation cycles [START_REF] Jani | A review of shape memory alloy research, applications and opportunities[END_REF]. Piezoelectric materials' electromechanical behavior is activated via the electric field [START_REF] Barbarino | A review of morphing aircraft[END_REF]. The most used material is lead zirconate titanate (PZT). The developments of piezoelectric composites allows for a simple implementation; piezoelectric composite patches glued on the structure are often used as bending actuators that control the shape of a wing. Despite the brittle nature of the PZT ceramics they provide high frequency actuation with an excellent life cycle. The previously cited work generally achieves the desired functions using one specific smart material. A combination of both SMAs and PZT can enhance the global performance. The synergistic smart morphing aileron [START_REF] Pankonien | Synergistic smart morphing aileron: Experimental quasi-static performance characterization[END_REF] is the combination of a SMA actuated hinge followed by a flexible piezoelectric driven trailing edge.

The French laboratories LAPLACE and IMFT have been focused on studying electroactive morphing for more than ten years in various collaborative research projects. A SMA actuated plate [START_REF] Chinaud | Trailing-edge dynamics and morphing of a deformable flat plate at high reynolds number by time-resolved {PIV}[END_REF] and a NACA0012 wing with a vibrating trailing edge [START_REF] Scheller | Trailing-edge dynamics of a morphing {NACA0012} aileron at high reynolds number by high-speed {PIV}[END_REF] are the precursors of a NACA4412 hybrid morphing wing [START_REF] Scheller | A hybrid morphing naca4412 airfoil concept[END_REF]. This prototype both embeds SMAs and trailing edge piezoelectric fiber actuators [START_REF] Scheller | A combined smart-materials approach for next-generation airfoils[END_REF]. This allowed both large deformations (∼10% of the chord) at limited frequency (≤ 1 Hz) and small deformations (several mm) at higher frequencies (≤ 100 Hz). It has been shown that the trailing edge vibration interacts with the shear layer. The wing's wake energy has been reduced, leading to an improvement in aerodynamic performance [START_REF] Scheller | A hybrid morphing naca4412 airfoil concept[END_REF]. To the authors' knowledge, the originality of the present work is the study of the links between camber control, vibrating trailing edge and the airflow dynamics. Previous work has primarily targeted the Micro Air Vehicle (MAV) scale; now the purpose of the present work is the up-scaling towards commercial passenger airliners. In cooperation with Airbus, a first step is a reduced scale wing with an innovative electroactive hybrid morphing flap design. This paper firstly introduces the wing prototype. Secondly, a focus on the link between control law, power consumption and actuator design is provided. Thirdly, wind tunnel experimental results are presented; showing some answers on the use of embedded sensors to estimate morphing performances.

II. ELECTROACTIVE HYBRID MORPHING WING PROTOTYPE A. Hybrid morphing

The airflow around a wing can be decomposed in two spacetime domains: a global time averaged effect -influenced by large wing shape deformation at low frequencies -and small amplitude higher frequency dynamics -influenced by small amplitude but higher frequency actuation. These two different effects justify the use of two actuators based on two different electroactive materials. The here presented morphing wing embeds both camber control and Higher Frequency Vibrating Trailing Edge (HFVTE) actuators. Fig. 1 presents the hybrid morphing prototype. The chord length is 700 mm, the span is 590 mm.

B. Higher Frequency Vibrating Trailing Edge actuator

The HFVTE actuation system is based on piezoelectric Macro Fiber Composite (MFC) patches which expand in length when subject to an electric field. Piezoelectric patches are glued on both sides of a metallic substrates. By alternatingly applying a voltage to the patches, the patches generate a bending moment which results in actuator vibration. To respect the shape of the airfoil, the HFVTE actuators are placed at the trailing edge, covered by a specifically designed silicone shell. As presented in Fig. 2, vibrations of 0.7 mm can be reached at frequencies up to 100 Hz. (See [START_REF] Jodin | Dimensionnement d'une maquette pour l'investigation du morphing électroactif hybride en soufflerie subsonique[END_REF] and [START_REF] Jodin | Implementation of a hybrid electro-active actuated morphing wing in wind tunnel[END_REF] for more details on the design.)

C. Camber control actuator

The camber actuation system is designed to bend the trailing edge structure while withstanding aerodynamic loads. The ac- tuator is based on surface embedded SMAs. Three SMA wires are embedded under the upper skin and three other SMA wires are embedded on the lower skin, as shown in Fig. 3. SMAs are characterized by a strongly non-linear thermomechanical coupling, which has been modeled and understood during the last decades [START_REF] Lexcellent | Shape-memory alloys handbook[END_REF]. The actuator has been modeled in [START_REF] Jodin | Dimensionnement d'une maquette pour l'investigation du morphing électroactif hybride en soufflerie subsonique[END_REF] and [START_REF] Jodin | Implementation of a hybrid electro-active actuated morphing wing in wind tunnel[END_REF]; to sum up the principle, cold SMA are pseudo-plastic and can be easily stretched thanks to a low stress plateau. The SMAs are heated through Joule heating generating strong forces to recover their initial shapes. To easily design the mock-up and as cold SMA are very flexible, the SMA stiffness is neglected. This creates a sub-optimal design with a large amount of SMA, but the design solution is reliable and stiff regarding to aerodynamic loads. The maximum trailing edge deflection is about -25 mm to +20 mm. This corresponds to a controllable camber modification range of 22% (from -10% to +12%) of the actuated chord.

III. CAMBER CONTROL: CONTROL LOOP AND

PERFORMANCES

To precisely reach the desired deformations while controlling 6 independent SMA wires, temperature and deformation sensors are implemented. One thermocouple per SMA wire is used. One strain gauge bridge is implemented per up and down SMA pair, to reconstruct the trailing edge displacement. Two strain gauges are glued on the upper skin and the other two are on the lower skin. This atypical gauges distribution has been calibrated using an external displacement sensor that measured the trailing edge displacement. A polynomial interpolation ensures an accurate trailing edge displacement reconstruction from the bridge's signal. As shown in Fig. 4, the average absolute error is under 0.4 mm for trailing edge displacements up to 10 mm. Over 10 mm, hysteresis decreases the accuracy to 1 mm. These sensors are used in two nested closed loops to control the camber. The reconstruct displacement is compared to a camber reference. Then a Proportional Integral (PI) controller send a SMA temperature reference to the temperature controllers. These controllers use temperature sensor information to provide heating and cooling commands via PI controllers. The control architecture is summed up in Fig. 5. Temperature references are output from displacement controller, with limited slope, in order to ensure a uniform variation of every SMA actuator. The displacement controller has a custom antiwindup on the PI's integrator in order to avoid over integration during the temperature slopes.

The performance of the nested-loop control is visible Fig. 6 for two displacement controller settings. A displacement step is applied on the controller. Rising the trailing edge displacement from 5 mm to 10 mm needs an increase of heating power to rise SMA temperature. The control strategy with displacement overshoot seems to be more efficient in steady power consumption than the strategy without overshoot. Overshoots in displacement lead the controller to decrease the final SMA temperature reaching the reference displacement. The overshoot effect is often seen as a control drawback; but due to the SMA activation temperature hysteresis, the final required temperature to obtain trailing edge position when decreasing the displacement is lower than the temperature needed to obtain the same position when increasing the displacement. This effect is clearly visible on Fig. 7, where temperatures are drawn depending on displacement for the same experiment as presented in Fig. 6. The required SMA temperature difference between the two strategies is over 6 • C. Finally, a simple change in the control strategy leads to a reduction of more than 20% of required heating power to maintain the displacement.

IV. WIND TUNNEL EXPERIMENTS

A. Experimental set-up

The wing model is tested in a subsonic wind tunnel. The test section is 592 mm width per 712 mm high. The wing chord c is 700 mm as previously mentioned. The wing is mounted at a maximal incidence of 10 • . As a result a blockage ratio of 18% is obtained. The blockage ratio effects are found to be acceptable in the present experiments which focus on the morphing effects. This is done by comparing a morphing configuration to the non-actuated case, both of which are subject to the same blockage ratio. The turbulence intensity of the inlet section is about 0.1% of the free stream velocity. Measurements are performed at ambient temperature (22 • C). Fig. 8 presents the experimental wind tunnel set up. An aerodynamic balance based on strain gauges has been designed to measure lift and drag force on the wing. Precautions in design and mounting have been taken to ensure the quality of the measures despite the vibrations and the wires coming from the embedded sensors and actuators. Three dynamic pressure transducers (MEGGIT 8507C-1) are implemented on the suction surface (upper skin) of the prototype. They are lined up one behind the other. The respective locations of the sensor P1, P2 and P3 are closed to the trailing edge, i.e. respectively at 80%, 86% and 93% of the wing chord. An experimental campaign exploring the influence of free stream velocity, angle of attack, camber, HFVTE frequency and HFVTE amplitude is performed. Illustrations of the amplitude of deformations are given in Fig. 8. For every experimental morphing experiment, forces, pressures, wind conditions and actuator information are recorded for 20 s. This record time has been found long enough to ensure a good statistical convergence.

B. Morphing effects 1) Hybrid morphing -Camber and HFVTE coupled effects:

To investigate the impact of all variables, thousands of runs are performed. Fig. 9 presents the results of 476 experiments, at incidence of 10 • and Reynolds number Re = 10 6 (Re = U∞•c ν where U ∞ is the free stream velocity -21.5m/s here -and ν is the kinematic viscosity). Lift coefficients are computed from time averaged aerodynamic force data on acquisitions for camber control that displace the trailing edge from -10 to +10 mm (i.e. +/ -1.5%c) and trailing edge vibrations at frequencies from F a = 0 -250Hz (i.e. f * a =0 to 8.3), where F a is the HFVTE actuation frequency and the dimensionless actuation frequency f a is defined by f a = Fa•c U∞ . The lift coefficient surface is plotted as a function of camber and vibration frequency. Each experiment testing a HFVTE configuration is preceded by a reference measure with the same camber, the same velocity and the same incidence, but with no trailing edge vibration. This ensures the accuracy in capturing the higher frequency-low amplitude actuation effects. At a given camber, the additional modifications due to the HFVTE are represented by vertical bars. The colors of these bars indicate if the HFVTE actuation increases the lift (green bars) or decreases the lift (red bars) compared to the experiment at same camber without HFVTE actuation.

The first noticeable result is the well known effect of the camber on the lift. Positively increasing the camber raises the lift. Notice on one hand that up and down deformed shapes are different, leading to non-symmetrical lift effects for negative and positive cambers respectively. On the other hand, characterizing the effects of trailing edge vibrations is less evident. In fact, according to Fig. 9, a frequency range from 100 Hz to 250 Hz (f * a ∈ [3.3 • • • 8.3]) seems to be beneficial for the lift, especially for large cambers. On the other side, at low cambers the effect of the vibration on the lift is reduced and the frequency range is shifted. The strong coupling between the camber and the vibrating trailing edge is complex. As the effects of camber have already been largely assessed (see for instance Ref. [START_REF] Abbott | Theory of Wing sections[END_REF]), the following section focuses on the effect of the vibrating trailing edge on the baseline cambered airfoil.

2) HFVTE effects at initial cambered airfoil: Dedicated experiments focus on the influence of the trailing edge vibrations on the flow, at fixed camber. For instance, Fig. 10 presents the processed data from 500 experiments, each during 20 s. HFVTE frequency range is 12 Hz to 450 Hz (i.e. f * a ∈ [0.8 • • • 30]); 5 amplitude levels are tested from 20% to 100% of the maximum amplitude of the actuators ; Reynolds number here is Re = 10 5 . Particular attention has been payed to repeatability and accuracy of the measures. The order of the measures has been generated randomly, in order to prevent any memory or hysteresis effect from the sensors. Finally, one measure at a given HFVTE setting is compared to a previous and a following reference measure with no actuation. Fig. 10 shows the variation of the lift coefficient in relation with the actuation frequency. The different actuation amplitude are drawn in different colors. Pale areas described the variations of the measures between the two experiments whereas the darker lines are the average values. The experiments are repeatable with +/ -0.14% lift difference. The first observation is the effect of the amplitude. Generally, the larger the HFVTE amplitude the larger the gain. This clearly indicates that if the actuator could vibrates at larger amplitudes, more gains can be achieved. But increasing amplitude over a certain limit would decreases the gains, as reported in [START_REF] Collis | Issues in active flow control: theory, control, simulation, and experiment[END_REF] for the forced active control of airflows. Secondly, a frequency threshold appears close to 100 Hz, separating low HFVTE effects area from higher gain area. This threshold is linked to the first resonance of the HFVTE actuator which can be seen by the 110 Hz peak in Fig. 2. Complex structural vibrations are detected with accelerometer on the wing, but the understanding of the coupling between wing structural resonance modes and the effects on flow is the purpose of next work.

Finally, it is worth noting that for the two tested velocity -i.e. the two Reynolds numbers Re = 10 6 or 5 • 10 5experimental gains above 2% have been identified, thanks to small vibrations of the trailing edge.

C. Morphing and sensing

Experimental morphing performance enhancements are characterized using an aerodynamic balance. This instrumental device cannot easily be embedded on aircraft. Other embedded sensors are needed. The three dynamic pressure sensors P1, P2 and P3 are used to this purpose.

The recorded pressure signals are useful. For instance they contain vortex signature in their frequency spectra that could be analyzed. A simpler analyze that can be used as morphing signal processing to determine an accurate observer of the aerodynamic gains.

V. CONCLUSION

This article presented the multidisciplinary coupling between an innovative mechatronic hybrid morphing wing prototype and the aerodynamic forces. The different time scales of aerodynamic phenomena justify the use of two electroactive smart material based actuators. Integration of piezoelectric and shape memory alloy actuators in the prototype requires a multidisciplinary design approach to fulfill the design target during the construction and to achieve accurate control of the innovative prototype, before moving towards the dedicated wind tunnel experiments. It has been shown that the control law of the SMAs can decrease the steady power consumption by more than 21%. Also, the aerodynamic performance enhancements are closely linked to the coupled actuator-structure dynamic behavior.

The camber modifications is able to change the lift by 23% and the drag by 35%, leading a lift over drag ratio enhancement of 16%. Additionally the actuated small vibrations of the trailing edge can increase the performances by 2% in addition to the camber gains. Embedded pressure sensors show the ability to estimate the morphing gains without needing external sensors.

The next step focuses on both the sensors and actuators. A better understanding of the fluid mechanism is needed to estimate in real time the gains with embedded sensors. A full scale flap demonstrator equipped with hybrid morphing actuator is currently under construction.

Fig. 1 .

 1 Fig. 1. Picture of the hybrid wing model on its stand out of wind tunnel.

Fig. 2 .

 2 Fig.2. Measured amplitude displacement A of the trailing edge, depending on the actuation frequency Fa. This frequency response measurement of the actuator has been performed with 1 kV applied voltage on the MFC patches, using a high-speed camera.

  wire (intrados) Silicon encapsulated SMA wire (extrados) 200[mm]

Fig. 3 .

 3 Fig. 3. Trailing edge with surface embedded SMA actuators.

Fig. 4 .

 4 Fig. 4. Characterization of the reconstructed displacement evaluated from embedded strain gauges.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Nested control loop architecture of the camber control actuator.

Fig. 7 .

 7 Fig. 7. Control strategy influence of the camber control actuator. Temperature -displacement plane representation of the experiments.

Fig. 8 .

 8 Fig. 8. Scheme of the wind tunnel experiment.

Fig. 9 .

 9 Fig. 9. Modification of the measured lift coefficient C l as a function of the dimensionless camber and the dimensionless actuation frequency. The percentage lift modification is normalized to the measured lift coefficient without any morphing C l0 . Corresponding 2D plots of three selected cambers are presented on the right. The blue areas around the data correspond to the 95% confidence level. Camber range is -10 to +10mm and frequency range is 0 -250Hz. Reynolds number for this experiment is Re = 10 6 .

Fig. 10 .

 10 Fig. 10. Impact of the HFVTE frequency and amplitude on the lift coefficient C l . Reynolds number for this experiment is Re = 5 • 10 5 .


English translation of : « О решении уравнения Лиувилля (1850, 1853) и его влиянии » in Институт истории естествознания и техники им. С.И. Вавилова. Годичная научная конференция,2021. М.: ИИЕТ РАН, 2021, R.A. Fando [Р.А. Фандо], E.V. Minina [Е.В. Минина ] and V.M. Savenkova [В.М. Савенкова], eds.

(ISBN 978-5-6046393-4-4), pp. 81--86.

The solution of Liouville's equation (1850, 1853) and its impact1

E.M. Bogatov, S. Kichenassamy Е.М. Богатов, С. Кишнассами

Liouville's 1853 paper [START_REF] Liouville | Sur l'équation aux différences partielles 𝑑 2 log 𝜆 𝑑𝑢 𝑑𝑣 ± 𝜆 2𝑎 2 = 0[END_REF], in which he derived in closed form the general local solution of equation 𝑢 𝑧𝑧̅ = 𝑒 𝑢 , is one of the few papers from the 19 th century that 21 st century mathematicians routinely quote as motivation for their work. We try and understand the reasons for the enduring importance of this paper. Because it does not distinguish the real and complex domains, Liouville's paper simultaneously opened the way to a representation of the general solution of two equations, namely the elliptic equation Δ𝑢 = 𝐾𝑒 𝑎𝑢 (Le) and the hyperbolic equation

𝑢 𝑥𝑦 = 𝐾𝑒 𝑎𝑢 , (Lh)
where 𝐾 and 𝑎 are nonzero real constants. Generalizations of (Le) -(Lh) are too numerous to be mentioned here.

Liouville announced his result in his 1850 commentary on Monge's Applications de l'analyse à la géométrie [START_REF] Liouville | des deux rayons de courbure principaux en chaque point d'une surface[END_REF]. His original proof is lost: he only published a simpler one [START_REF] Liouville | Sur l'équation aux différences partielles 𝑑 2 log 𝜆 𝑑𝑢 𝑑𝑣 ± 𝜆 2𝑎 2 = 0[END_REF]. He obtained (Le) from the expression for the Gaussian curvature of a surface in isothermal coordinates (see Lützen [START_REF] Lützen | Master of pure and applied mathematics[END_REF]), and obtained his general solution of (Lh) depending on two functions of one variable; thus, for

𝑎 = 𝐾 = 1, 𝑢(𝑥, 𝑦) = ln 2𝑓 ′ (𝑥)𝑔 ′ (𝑦) (𝑓(𝑥)+𝑔(𝑦)) 2 .
The same formula may be interpreted as giving a real solution of (Le) depending on one analytic function. The regularity and determination of the arbitrary functions was clarified rather recently (see [5, §10.6] and [START_REF] Crowdy | General solutions to the 2D Liouville equation[END_REF]).

Darboux [7, p.170] called (Lh) Liouville's equation, and Picard [START_REF] Picard | Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives[END_REF] gave the same name to (Le). Picard's line of thought led to the boundary blow-up problem for (Le), in relation to conformal mapping [START_REF] Poincaré | Les fonctions fuchsiennes et l'équation Δu = e u[END_REF]- [START_REF] Bieberbach | Δ𝑢 = 𝑒 𝑢 und die automorphen Funktionen[END_REF]. Darboux inserted (Lh) in a general theory of the solution of second-order PDEs in two variables. Nonetheless, Liouville's solution was not forgotten. Thus, Bianchi obtained from Liouville's solution a majorant series for the solution for what we now call the sine-Gordon equation [12, cap. XX §298, p. 214-217; cap. XXIV, §370, p. 386]. These lines of thought were apparently cut short by WWI, to be taken up again only after the birth of the theory of solitons.

Equation (Le), and its counterpart in three dimensions also arose as models for isothermal gas spheres in astrophysics [13, p. 131-132], thermo-ionic emission [14, p. 50], as a special model of incompressible fluid flow as well as other applications [START_REF] Crowdy | General solutions to the 2D Liouville equation[END_REF][START_REF] Joseph | Quasilinear Dirichlet problems driven by positive sources // Archive for Rational Mechanics and Analysis[END_REF][START_REF] Bandle | Harmonic Radius and Concentration of Energy; Hyperbolic Radius and Liouville's Equations Δ𝑢 = 𝑒 𝑢 and Δ𝑢 = 𝑢 (𝑛+2)/(𝑛-2)[END_REF]]. Liouville's exact solution attracted considerable interest in the late 70s as an exactly solvable model of field theory [START_REF] Onofri | An identity for T-ordered exponentials, with applications to Quantum Mechanics[END_REF][START_REF] Witten | Some exact multipseudoparticle solutions of classical Yang-Mills theory[END_REF]. The contemporary emergence of the theory of solitons [START_REF] Kichenassamy | Nonlinear Wave Equations[END_REF]Ch. 4] triggered a renewed interest in Bäcklund transformations, particularly the transformation relating (Lh) to the wave equation 𝑣 𝑥𝑦 = 0 [START_REF] Lamb | Bäcklund transformations at the turn of the century[END_REF]. Liouville's solution was also applied to the motion of singularities [START_REF] Джорджадзе | М. К. Сингулярные решения уравнения □φ+(m 2 /2)exp φ=0 и динамика особенностей // Теоретическая и математическая физика[END_REF][START_REF] Pogrebkov | The Liouville and sinh-Gordon equations. Singular solutions, dynamics of singularities and the inverse problem method[END_REF], and was treated by inverse scattering [START_REF] Андреев | Применение метода обратной задачи рассеяния к уравнению 𝜎 𝑥𝑡 = 𝑒 𝜎 // Теоретическая и математическая физика[END_REF]. The modern relevance of Liouville's formula seems to be due to two facts: it is not adequately interpreted in terms of the Cauchy and Dirichlet problems; and it admits of a generalization to large classes of non-integrable equations in any number of space dimensions. On the one hand, the Cauchy problem for (Lh) gives a solution depending on three functions (two Cauchy data, and the equation of the curve on which they are prescribed) whereas Liouville's solution only contains two, that are essentially the two singularity data that determine a solution near blow-up [START_REF] Kichenassamy | Blow-up Surfaces for Nonlinear Wave Equations, I & II // Communications in Partial Differential Equations[END_REF], and this observation generalizes to non-integrable problems in any dimension; in fact, it was the starting point of the method of Fuchsian Reduction [START_REF] Kichenassamy | Fuchsian Reduction: Applications to Geometry[END_REF]. On the other hand, for (Le), the Dirichlet problem requires two data (the domain and the boundary condition), whereas the boundary blow-up problem requires only one [START_REF] Kichenassamy | Boundary Blow-up and Degenerate Equations[END_REF]- [START_REF] Kichenassamy | Régularité du rayon hyperbolique[END_REF].

It is instructive to give special attention to the reception of Liouville equations in Russia and the CCCP, that seems to show at work mechanisms that may be of general significance F. Minding, who worked since 1844 at the Dorpat University in Russia, made a great contribution to the development of the geometry of surfaces [START_REF] Галченкова | Фердинанд Миндинг[END_REF]. Minding was aware of all the achievements in this area but apparently, he did not know about Liouville equation, possibly because he was not interested in isothermal coordinates on surfaces. In his works, as well as in the articles of his Russian student K.M. Peterson (the founder of the Moscow geometric school), no references to the equation in either form could be found. Moscow mathematicians of that time apparently dispensed with the use of equation (Le) or (Lh) in their research. Even now, one could argue that special coordinates detract from the identification of intrinsic quantities such as are given by tensor calculus.

By contrast, representatives of the St. Petersburg Mathematical School related to the line of P.L. Chebyshev, mentioned Liouville's results. In particular, equation (Le) was present in D. Grave's article in 1927, published in the prestigious collection « In memoriam N. I. Lobatschevskii » in Kazan in 1927 [28, p. 26]. But it this paper the above equation was not associated with Liouville's name. It is not known how long it would have taken for this association to appear in Russian sources if the forced emigration of S.B. Bergman to Tomsk from Nazi Germany had not happened. Bergman organized a scientific seminar at the University of Tomsk on the theory of surfaces using the methods of the theory of analytic functions, which met from 1934 to 1936 [29, p. 50-51]. Apparently, it was under the influence of Bergman that B.A. Fuchs, who was one of the participants of this seminar, began to use the name "Liouville's equation" in his works using the equivalent of equation (Lh)

1 𝑇 𝜕 log 𝑇 𝜕𝑢𝜕𝑣 = 𝐾
(Llog) to describe a constant curvature surface's metric in 1937 [30, p. 584]. However, after 1938, Fuchs stopped referring to Liouville in connection with the use of equation (Llog).

It so happens that in spite of the interest of Soviet mathematicians to equations of the form (Llog) or (Le), associating them with Liouville name in the mass consciousness of Soviet scientists was lost before the end of the 1970s. As a result, equation (Le), obtained during the creating of the stationary theory of thermal explosion and based on the Arrhenius law, (D.I. Frank-Kamenetskii, 1939 [31, p. 740]) was also not accompanied by the references to Liouville.

It is important to note that the Dirichlet problem formed by adding to equation (Le), with 𝐾 < 0, the condition u = 0 on a surface  of the form 𝑟̅ = 𝜆𝑟 0 ̅ , (𝑚𝑎𝑥|𝑟 0 ̅ | = 1; 𝑟̅ is radius-vector of the point of the surface) was subjected to group-theoretic analysis by I.M. Gelfand in his famous article [START_REF] Гельфанд | Некоторые задачи теории квазилинейных уравнений // Успехи математических наук[END_REF] (1959). He showed that for 𝐾 = -2 and for symmetric domains , the solution to the Dirichlet problem for the equation (Le) has bifurcation points when λ reaches the threshold value 𝜆 0 , which depends on the form of  [32, p. 140-142].

In addition, the integral equation obtained from (Le) by using Green's function for the Laplace equation turned out to be a model example of an equation of which the solutions do not belong to the space L p . That is, the Liouville equation could well have been an additional stimulus for the creation of Orlicz spaces theory and its development by M.A. Krasnoselskii (for the history of this issue see [START_REF] Богатов | Oб истории развития нелинейных интегральных уравнений в СССР. Сильные нелинейности // Научные ведомости[END_REF]). Amazingly, there were no references to Liouville here either!

The situation changed only after the publication of the textbook "Modern Geometry" (1979) [START_REF] Дубровин | Современная геометрия: методы и приложения[END_REF]. In Chapter 2, §13 (Conformal form of metrics) of this textbook, finally, equations (Le) and (Llog) were given with an indication of Liouville. At about the same time, the Liouville equation began to appear in the USSR in the works of physicists. Thus, V.A. Andreev constructed an N-soliton solution to equation (Lh), describing one moving and N-1 stationary solitons [START_REF] Андреев | Применение метода обратной задачи рассеяния к уравнению 𝜎 𝑥𝑡 = 𝑒 𝜎 // Теоретическая и математическая физика[END_REF]. Russian and Belarusian scientists, in relation to the two-dimensional field theory, have studied singular and regular solutions of (Lh), and studied generalizations of (Le) and (Lh) etc. [START_REF] Джорджадзе | М. К. Сингулярные решения уравнения □φ+(m 2 /2)exp φ=0 и динамика особенностей // Теоретическая и математическая физика[END_REF]- [START_REF] Pogrebkov | The Liouville and sinh-Gordon equations. Singular solutions, dynamics of singularities and the inverse problem method[END_REF]. Soviet scientist A.M. Polyakov, known for a number of fundamental contributions to quantum field theory, found a new use for the equation (Le) in high energy physics, in the quantum geometry of bosonic strings [START_REF] Polyakov | Quantum geometry of bosonic strings[END_REF]: indeed, the equation of motion associated with the so-called "Liouville action" Thus, this study has shown that Liouville's paper has not been fully superseded by any of the later developments originating from it for four reasons:

𝑆[𝜑] = 𝐶 ∫ (
1. Mathematical traditions have focused on only some aspects of it, playing an essential role in the spread (or inhibition) of mathematical ideas. The study of the Russian situation revealed the existence of several mathematical communities that received Liouville's paper differently. 2. His paper was one of the motivations of several modern theories such as nonlinear partial differential equations and functional analysis; the theory of analytical functions and conformal mapping; the theory of automorphic functions; the theory of Orlicz spaces; group-theoretical analysis; the theory of positive operators; the theory of solitons. At the same time, no single later theory can reproduce all of the known results on (Le) -(Lh). Furthermore, because Liouville's solution has robust features that are present in non-integrable systems in any dimension, without symmetry, Liouville's paper stimulated new theories such as Fuchsian Reduction.

;

  𝐶 = 𝑐𝑜𝑛𝑠𝑡. reduces to the equation (Le).

This work is a continuation of research carried out in[START_REF] Богатов | Об истории уравнения u = ke u и вкладе отечественных математиков // Обозрение промышленной и прикладной математики[END_REF].