Vincenzo Vinciguerra
email: vincenzo.vinciguerra07@gmail.com

Topological Quantum Interplay between Magnetic Flux and Mass in an Electron

Keywords: electron, topology, magnetic flux quantization, charge, mass, electroweak theory, Bohr's energy, W boson vector PACS numbers: Relativistic electron and positron, 41.75.Ht, Topology, 02.40.Pc, Electroweak interactions, 12.15.y, Atomic spectra, 32.30.-r, W bosons, 14.70.Fm

A topological quantum interplay between the magnetic flux Φ and the mass has been investigated, for the case of an electron, by evaluating a gauge-invariant phase factor (a Wilson loop) linked to the electromagnetic gauge field 𝐴 𝜇 of the particle. In particular, from this phase factor and the quantization of the magnetic flux variations, a relationship between the mass at rest of the electron 𝑚 𝑒 and its self-energy 𝛿𝑚, arising from radiative corrections, has been obtained also within a QED approach. Besides, a formulation of an energy scale comparable to the energy at rest of an electron-positron pair is proposed. Remarkably, a reckoning of the Bohr's energy of a 𝑊 + 𝑊 -pair is compatible with constants and parameters usually employed within the electroweak theory and comparable to the energy at rest of an 𝑒 -𝑒 + pair.

Introduction

Of all the massive elementary particles [START_REF] Nakamura | Review of Particle Physics[END_REF] investigated in the standard model (SM) [START_REF] Gaillard | The standard model of particle physics[END_REF], electrons have been definitely central in the development of the semiconductor industry [START_REF] Sze | Physics of Semiconductor Devices[END_REF], being their properties, either classical or quantal, widely exploited in the whole set of integrated circuits made of downscaled solid state devices. For technologists and experts involved in the investigation of matter, electrons are fundamental particles upon which devising explanations regarding atoms, molecules, condensed matter and general observed phenomena. Indeed, thanks also to the rise and reliability of computational methods, tested over the years and established on fundamental principles, scientists can implement today a set of theoretical frameworks to further understand the structure of matter. Examples are the ab-initio methods such as tight binding, Hartree Fock (HF) and self-consistent HF [START_REF] Cramer | Essentials of Computational Chemistry[END_REF][START_REF]Quantum materials: experiments and theory[END_REF]), as well as first principles methods, built on the theory of density functional (DFT) [START_REF] Kohn | Lecture: Electronic structure of matter -wave functions and density functionals[END_REF][START_REF] Jones | Density functional theory: Its origins, rise to prominence, and future[END_REF][START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF][START_REF] Kohn | Self-consistent equations including exchange and correlation effects[END_REF]. Their use has definitely had an impact on our understanding of quantum effects in solid state thin films (e.g. in the quantum integer and fractional Hall effect) [START_REF] Sasaki | Theory of the Integer & Fractional Quantum Hall Effects[END_REF] and in the development of new device technologies [START_REF] Hasnip | Density functional theory in the solid state[END_REF], too.

To complete the picture, the physical foundations of the methods illustrated above rely upon the theory of quantum electrodynamics (QED) [START_REF] Weinberg | The Quantum Theory of Fields[END_REF][START_REF] Greiner | Quantum Electrodynamics[END_REF] and its later developments within the electroweak theory; making QED the most successful theory in predicting and describing the interaction of electrons with light in every aspect. However, after more than a century from the discovery [START_REF] Thomson | Cathode rays[END_REF] of the electron as a fundamental building block of matter, the electron is still perceived mostly as an enigmatic particle in its essence. Indeed, the words of the scholar Wilczeck [START_REF] Wilczek | The enigmatic electron[END_REF] confirms such a picture: "An electron is a particle and a wave; it is ideally simple, and unimaginably complex; it is precisely understood, and utterly mysterious; it is rigid, and subject to creative disassembly. No single answer does justice to reality".

Recently topology [START_REF] Castelvecchi | The Strange topology that is reshaping physics[END_REF] is receiving considerable attention by condensed matter's physicists focusing on the properties of topological phases [START_REF] Haldane | Model for a Quantum Hall effect without Landau levels: condensedmatter realization of the "parity anomaly[END_REF][START_REF] Kosterlitz | metastability and phase transitions in twodimensional systems[END_REF]. Nonetheless, topological arguments have been already introduced in physics in the past [START_REF] Bhattacharjee | Use of Topology in physical problems in Topology and condensed matter physics[END_REF][START_REF] Nash | Topology and Physics a historical essay in History of topology[END_REF][START_REF] Morandi | The role of topology in classical and quantum physics in Lecture notes in physics[END_REF][START_REF] Nakahara | Geometry[END_REF][START_REF] Stanescu | Introduction to Topological Quantum Matter & Quantum Computation[END_REF]. Indeed, the charge quantization conditions determined by the existence of magnetic monopoles [START_REF] Dirac | Quantised singularities in the electromagnetic field[END_REF][START_REF] Shnir | Magnetic Monopoles[END_REF], the Arhanov-Bohm effect [START_REF] Aharonov | Significance of electromagnetic potentials in quantum theory[END_REF] and the Berry's phase [START_REF] Berry | Quantal phase factors accompanying adiabatic changes[END_REF] are just acknowledged examples of topological properties arising in specific physical systems.

In general, topology gives us insights on how different parts of a structure are linked together, or can be connected by paths. The Königsberg's bridge problem of Euler [28], the network of covalent bonds of atoms in a molecule [START_REF] Sauvage | From chemical topology to molecular machines[END_REF], the structure of a LAN or a WAN [START_REF] Chen | Fundamentals of Complex Networks: Models, Structures and Dynamics[END_REF] are also examples of topological structures that can occur in space configurations. More specifically, in geometry, surface topology investigates how paths (open or in loops) can be deformed equivalently and continuously without tears or cuts of the surface or, in a similar manner, how surfaces can be modified smoothly into one another (e.g. a cup into a donut!).

Topological aspects determining the quantization properties of the intrinsic magnetic flux of electrons and the significance of the magnetic flux quantization is acknowledged in the field of condensed matter [START_REF] Parks | Quantized magnetic flux in superconductors[END_REF][32][START_REF] Onsager | Proceedings of the International Conference on Theoretical Physics[END_REF] as determinant to understand the physics of superconducting state and proved experimentally by Deaver and Fairbank [START_REF] Deaver | Experimental evidence for quantized flux in superconducting cylinders[END_REF], as well as by Doll and Näbauer [START_REF] Doll | Experimental proof of magnetic flux quantization in a superconducting ring[END_REF]. Later the role played by the quantization of magnetic flux was extended to the Quantum Hall effect [START_REF] Laughlin | Quantized Hall conductivity in two dimensions[END_REF][START_REF] Avron | A topological look at the quantum Hall effect[END_REF].

In this work, we investigate the role that a quantization of the intrinsic magnetic flux in a particle plays in determining its properties. In particular, we discuss the topological interplay occurring between the intrinsic magnetic flux of a charged lepton, namely an electron, and its mass.

According to the literature, the topic has been investigated to some extent in the past and for a decade by Jehle [START_REF] Jehle | The relationship of flux quantization to charge quantization and the fine structure constant[END_REF][START_REF] Jehle | The relationship of flux quantization to charge quantization and the fine structure constant[END_REF][START_REF] Jehle | Relationship of flux quantization to charge quantization and the electromagnetic coupling constant[END_REF][START_REF] Jehle | Flux quantization and particle physics[END_REF][START_REF] Jehle | Flux quantization and fractional charges of quarks[END_REF][START_REF] Jehle | Topological characterization of leptons, quarks and hadrons[END_REF]. Indeed, in Jehle's works, it is reported about the possibility to formulate a theory of the electromagnetic field of charged leptons as resulting from quantized magnetic flux. Jehle pointed out that the magnetic flux is an option that electrodynamics provides to investigate the nature of charged leptons and proved that the Coulomb potential can be obtained as arising from the quantization of the magnetic flux [START_REF] Jehle | The relationship of flux quantization to charge quantization and the fine structure constant[END_REF]. Although the formulated model was phenomenological and in contradiction with the standard formulation of QED, because of the electromagnetic origin of the mass of the electron, his formulation was an embryonal case of a topological property associated with the magnetic flux of the electron.

More recently, Saglam [START_REF] Saglam | Spin-dependent quantized magnetic flux through the electronic orbits of Dirac hydrogen atom[END_REF] and Stein [START_REF] Stein | Quantized magnetic flux through the orbits of hydrogen-like atoms within the atomic model of Sommerfeld[END_REF] proved that the magnetic flux of an electron in a hydrogen atom is also quantized and that the electron spin contributes to the quantized flux according to -2𝜋𝑐 〈𝑠〉/𝑒, where 〈𝑠〉 is the expectation value of the spin of the particle, -𝑒 is its electric charge and 𝑐 the speed of light. At the best of the knowledge of the author, no further extensive investigations have been reported so far on the topic.

The paper is organized as follows. First, the topological quantum interplay between the magnetic flux Φ and the mass, for the case of an electron, is investigated by evaluating a gauge-invariant phase factor associated with the Wilson loop of the electromagnetic field 𝐴 𝜇 generated by the particle. In particular, from the phase factor and the quantization of the magnetic flux variations, a relationship between the mass at rest of the electron 𝑚 0 and its self-energy 𝛿𝑚 is obtained also within a QED approach. Hence, the formulation of an energy scale comparable with the energy at rest of an electron and compatible with constants and parameters usually employed within the electroweak theory is discussed. Finally the conclusions are reported.

Topological interplay between magnetic flux and mass

Whenever a particle of charge 𝑞 undergoes a space-time loop 𝐶, in the gauge field 𝐴 𝜇 generated by an electron, it is possible to evaluate a gauge-invariant quantity [START_REF] Wilson | Confinement of quarks[END_REF]

𝑒 𝑖𝑞 ℏ𝑐 ∮ A 𝜇 𝑑𝑥 𝜇 𝐶 |𝜓⟩ = 𝑒 𝑖𝑞𝛼 |𝜓⟩ (1)
that determines a phase shift 𝑞𝛼 of the state |𝜓⟩. From the derivative of eq. 1 with respect to 𝑞, it results that the quantity

On the other hand, the gauge-invariant quantity on the first member of eq. 2 can be evaluated independently. For example, for the case of an electron at rest with spin up, in a semi-classical approach, it reads as:

〈 ∮ A 𝜇 𝑑𝑥 𝜇 𝐶 〉 = 𝑐 -𝑒〈𝜑(𝑟 ⃗,𝑡)〉 -𝑒 𝑇 -Φ(↑) = 2𝑐 -𝑒 𝛿𝑚 𝑒𝑠 𝑐 2 𝑇 -Φ(↑) (3)
Where 〈𝜑(𝑟 ⃗, 𝑡)〉 is the expectation value of the time-component of the 4-vector potential 𝐴 𝜇 , 𝛿𝑚 𝑒𝑠 𝑐 2 works as an electrostatic energy and the period 𝑇 is a characteristic time of the electron, which hereafter will be considered equal to 𝑇 = 𝜋ℏ 𝑚 𝑒 𝑐 2 .

In order to calculate the magnetic flux of an electron we proceed by considering the line integral of the operator 𝐴 ⃗ on a closed circular loop 𝐶 𝑟 = {𝑟(𝑡)|𝑡 = 0 → 𝑇} in the space-coordinates and average it over all possible values:

Φ = 〈 ∮ 𝐴 ⃗ ⃗⃗ • 𝑑𝑟 ⃗ 𝐶 𝑟 〉 = ∫ 𝑑𝑉 ∮ 𝜓 † (𝑟, 𝑡)𝐴 ⃗ 𝜓(𝑟, 𝑡) • 𝑑𝑟 ⃗ = 𝐶 𝑟 ∫ 𝑑𝑉′ ∮ 𝜓 † (𝑟′, 𝑡′)𝐴 ⃗ 𝜓(𝑟′, 𝑡′) • 𝑑𝑟′ ⃗⃗⃗ 𝐶 𝑟′ (4)
In doing such a calculation, we can renominate 𝑟 in 𝑟 ′ , as we expect that the variables 𝑟 and 𝑟 ′ play a symmetric role.

In the semi-classical approach that we are following, according to eq. 4 the magnetic flux of an electron reads as:

Φ = 〈 ∮ 𝐴 ⃗ ⃗⃗ • 𝑑𝑟 ⃗ 𝐶 𝑟 〉 = 1 𝑐 〈 ∫ ∮ 𝑗 ⃗ •𝑑𝑟 ⃗ |𝑟 ⃗-𝑟 ′ ⃗⃗⃗ | 𝐶 𝑟 𝑑𝑉 ′ 〉. (5)
In particular, for the case of an electron at rest, the magnetic moment 𝑚 𝑧 of an electron depends only on the spin, and it's an observable that commutes with the Dirac's Hamiltonian. It can be shown that:

Φ = 1 𝑐 〈 ∫ ∮ 𝑗 ⃗ •𝑑𝑟 ⃗ |𝑟 ⃗-𝑟 ′ ⃗⃗⃗ | 𝐶 𝑟 𝑑𝑉 ′ 〉 = 4𝜋 𝑐 〈 ∫ 1 2 (𝑟 ⃗×𝑗 ⃗ ) 𝑧 |𝑟 ⃗-𝑟 ′ ⃗⃗⃗ | 𝑑𝑉 ′ 〉 = 4𝜋 𝑐 〈 ∫ 𝑚 𝑧 𝜓 † (𝑟′,𝑡′)𝜓(𝑟′,𝑡′) |𝑟 ⃗-𝑟 ′ ⃗⃗⃗ | 𝑑𝑉 ′ 〉 (6)
By reasoning on the quantized value that the magnetic moment 𝑚 𝑧 of an electron at rest assumes for a state with spin up, we can deduce that:

Φ(↑) = - 4𝜋ℏ𝑐 2𝑒𝑚 𝑒 𝑐 2 〈 ∫ 𝑒 2 𝜓 † (𝑟 ′ ,𝑡 ′ )𝜓(𝑟 ′ ,𝑡 ′ ) |𝑟 ⃗-𝑟 ′ ⃗⃗⃗ | 𝑑𝑉 ′ 〉. (7)
Hence the gauge-invariant quantity in eq. 2, can be evaluated as

〈 ∮ A 𝜇 𝑑𝑥 𝜇 𝐶 〉 = 𝛿𝑚 𝑒𝑠 𝑚 𝑒 2𝜋ℏ𝑐 𝑒 . ( 8
)
In the event, the phase in eq. 2 amounts to:

𝑞 ℏ𝑐 〈 ∮ A 𝜇 𝑑𝑥 𝜇 𝐶 〉 = 𝑞 ℏ𝑐 𝛿𝑚 𝑒𝑠 𝑚 𝑒 2𝜋ℏ𝑐 𝑒 = 𝛿𝑚 𝑒𝑠 𝑚 𝑒 2𝜋𝑞 𝑒 . (9)
Moreover, in the hypothesis that there's a certain arbitrariness in the choice of the charge 𝑞, we can choose it such that the gauge-invariant quantity in eq. 3 reads as:

𝑞 ℏ𝑐 〈 ∮ A 𝜇 𝑑𝑥 𝜇 𝐶 〉 = 2𝜋𝑛, with 𝑛 𝜖 ℤ (10)
If this holds, the ratio between 𝛿𝑚 𝑒𝑠 and 𝑚 𝑒 , must satisfy the equation:

𝛿𝑚 𝑒𝑠 𝑚 𝑒 = 𝑛 𝑞/𝑒 . ( 11
)
On the other hand, also the magnetic flux is quantized and equals

Φ(↑) = - 4𝜋ℏ𝑐 𝑒 𝛿𝑚 𝑒𝑠 𝑚 𝑒 = - 4𝜋ℏ𝑐 𝑒 𝑛 𝑞/𝑒 , (12)
for the case of an electron at rest with spin up, whereas, for an electron at rest with spin down the magnetic flux reads

Φ(↓) = 4𝜋ℏ𝑐 𝑒 𝛿𝑚 𝑒𝑠 𝑚 𝑒 = 4𝜋ℏ𝑐 𝑒 𝑛 𝑞/𝑒 . ( 13
)
By imposing that the variation of the magnetic flux from spin down to spin up is equal to the quantum of the magnetic flux:

Φ(↓) -Φ(↑) = 8𝜋ℏ𝑐 𝑒 𝑛 𝑞 𝑒 = 2𝜋ℏ𝑐 𝑒 , ( 14
)
we obtain that the charge 𝑞 must satisfy the condition 𝑞 = 4𝑛𝑒.

Moreover, by defining 𝛿𝑚 𝑒𝑚 𝑐 2 = 4𝛿𝑚 𝑒𝑠 𝑐 2 as an electromagnetic energy linked to the electron, the magnetic flux can be expressed as:

Φ(↑) = - 𝜋ℏ𝑐 𝑒 𝛿𝑚 𝑒𝑚 𝑚 𝑒 . ( 15
)
From classical arguments it's reasonable to expect such a dependence. In fact, the magnetic flux Φ(↑) can be expressed in terms of the magnetic moment of the electron -𝜇 𝐵 , where 𝜇 𝐵 is the Bohr magneton, as:

Φ(↑) = - 1 2 2𝜋ℏ𝑐 𝑒 𝛿𝑚 𝑒𝑚 𝑚 𝑒 = - 𝑒 𝑚 𝑒 𝑐 ℏ 2 2𝜋𝛿𝑚 𝑒𝑚 𝑐 2 𝑒 2 = -𝜇 𝐵 2𝜋𝛿𝑚 𝑒𝑚 𝑐 2 𝑒 2 . ( 16
)
Indeed, the magnetic dipole moment 𝜇 determined by a current 𝑖 flowing, e.g. in a circular coil of radius 𝑟, is 𝜇 = 𝑖𝜋𝑟 2 . The magnetic field generated at the center of the coil is 𝐵 = 2𝜋𝑖 𝑟 , and the magnetic flux is approximately Φ = 𝐵𝜋𝑟 2 . By combining previous equations, we can express the magnetic flux in terms of the magnetic moment 𝜇. In fact,

Φ = 𝐵𝜋𝑟 2 = 2𝜋𝑖 𝑟 𝜋𝑟 2 = 𝜇 2𝜋 𝑟 . A
comparison with equation ( 16) provides the value

1 𝑟 = 𝛿𝑚 𝑒𝑚 𝑐 2 𝑒 2
.

By considering also possible QED corrections for the magnetic moment and indicating with 𝛿𝑚 𝑒𝑠 𝑐 2 the contribution of the electrostatic energy, eq. 7 reads as:

Φ(↑) = - 𝑔 𝑠 2 8𝜋ℏ𝑐 2𝑒𝑚 𝑒 𝑐 2 𝛿𝑚 𝑒𝑠 𝑐 2 , ( 17
)
where 𝑔 𝑠 is the spin g-factor of the electron.

In conclusion, the gauge-invariant quantity appearing in eq. ( 2), can be expressed as:

𝑒 ℏ𝑐 〈 ∮ A 𝜇 𝑑𝑥 𝜇 𝐶 〉 = 2𝜋 𝑔 𝑠 2 𝛿𝑚 𝑒𝑚 𝑚 𝑒 = 2𝜋, (18)
and being quantized in units of 2𝜋, it implies that there must exist an interplay between the magnetic flux and the mass, such that:

𝑔 𝑠 2 𝛿𝑚 𝑒𝑚 𝑚 𝑒 = 1. (19)
Clearly such a conclusion has been achieved by pursuing a semi-classical approach. In fact, within the QED formulation it is not possible to evaluate a quantity such as the electrostatic selfenergy of the electron. However, from the gauge-invariant quantity appearing in eq. ( 2), we can consider that 𝑑𝑥 𝜇 = 𝛾 𝜇 𝑐𝑑𝜏, where 𝛾 𝜇 are the Dirac Matrices and 𝜏 is an intrinsic time, characteristic of the system. If this is the case, by evaluating the line integral as an integral over the time 𝜏, and the expectation value turns out the value of a quantity which has the dimension of an energy 𝑒〈A 𝜇 𝛾 𝜇 〉. On the other hand, within the QED formulation the self-interaction of the electron with its own electromagnetic field is determined as the self-energy 𝛿𝑚𝑐 2 of the electron. Consequently, the gauge-invariant quantity of eq (2) can be estimated as

𝑒 ℏ𝑐 〈 ∮ A 𝜇 𝑑𝑥 𝜇 𝐶 〉 = 𝑒 ℏ𝑐 〈 ∫ A 𝜇 𝑑𝑥 𝜇 𝑑𝑐𝜏 𝜏 0 𝑐𝑑𝜏〉 = 𝑒 ℏ𝑐 𝑐𝜏〈A 𝜇 𝛾 𝜇 〉 = 𝜏 ℏ 〈A 𝜇 𝑒𝛾 𝜇 〉 = 𝜏 ℏ δ𝑚𝑐 2 , ( 20
)
where the characteristic time 𝜏 is set as

𝜏 = 𝑔 𝑠 2 2𝜋ℏ
𝑚 𝑒 𝑐 2 , so that it can be related to the magnetic moment of the electron. In fact,

Φ(↑) = -𝜇 𝐵 2𝜋𝛿𝑚𝑐 2 𝑒 2 = - 𝑔 𝑠 2 𝑒 𝑚 𝑒 𝑐 ℏ 2 2𝜋𝛿𝑚𝑐 2 𝑒 2 = - 1 𝑚 𝑒 ℏ 2 2𝜋𝛿𝑚𝑐 𝑒 , (21)
and the variation of the magnetic flux provides the value

Φ(↓) -Φ(↑) =
) 22
In conclusion, the Wilson loop appearing in eq. ( 2) is equal to 1 and reads as

⟨𝜓|𝑒 𝑖𝑒 ℏ𝑐 ∮ A 𝜇 𝑑𝑥 𝜇 𝐶 |𝜓⟩ = 𝑒 𝑖 𝜏 ℏ δ𝑚𝑐 2 = 𝑒 𝑖2𝜋 𝑔 𝑠 2 δ𝑚 𝑚 𝑒 = 1, (23)
if the gauge invariant quantity appearing in eq. 1 is such that:

𝑔 𝑠 2 δ𝑚 𝑚 𝑒 = 1. (24)
In table I, we summarize such considerations by evaluating the normalized magnetic flux

Φ 𝑛𝑜𝑟𝑚 = -𝜇 𝐵 2𝜋𝛿𝑚 𝐸𝑀 𝑐 2 𝑒 2 2𝜋ℏ𝑐 𝑒 ⁄
as determined by the confinement size or scale length, for the cases of two representative spin ½ charged particles (antiparticles), namely the electron (positron) and the proton (antiproton). The case of the electron is reported for the Bohr radius, the classical radius of the electron, the Compton wavelength and is compared with the calculation obtained from the evaluation of the radiative corrections of the electron mass at first order as obtained from the QED theory. The case of the electron, considered as a point and structure-less particle, is compared with the case of the proton, which is different, since its size is determined by the confinement of its constituent quarks. for different scale lengths or confinement sizes of the spin 1/2 particle. Normalization has been done with respect to the quantum flux 2𝜋ℏ𝑐 𝑒 . For the case of an electron (positron) different scale lengths can be considered. By taking into account the electron spin g-factor, a variation of the normalized magnetic flux ∆Φ 𝑛𝑜𝑟𝑚 for an electron flipping from spin up to down is in module equal to ⁄ . Finally for the case of the proton, we consider that the minimum length scale is determined by the radius of the proton, which, being a composite particle, is determined by its constituent quarks.

In Figure 1, from a to d, a set of schematizations is also reported in order to show the evolution from classical models of the electromagnetic field of the electrons to QED model of the selfinteraction of the electron with its own electromagnetic field. In particular, in Fig. 1a we report a representation of the lines of force of the electric and magnetic fields that can be deduced from classical electrodynamics considerations [START_REF] Jackson | Classical Electrodynamics[END_REF]. In Fig. 1b we report a representation of the main features and properties of an electron at rest, according to quantum mechanics. Along with its charge -𝑒 and its mass at rest 𝑚 𝑒 , we have to consider the intrinsic angular momentum or spin ½ of the particle, up in this specific case, its magnetic moment -𝜇 𝐵 (where 𝜇 𝐵 is a Bohr magneton unit) and the associated flux of the magnetic field Φ, which is independent of the specific contour line 𝐶 𝑟 . In order to evidence the topology which is behind the quantum mechanical properties of an electron, we report an analogical model of the property of spin rotation which is well represented by the topological properties of a Möbius strip (Fig. 1c). Fig. 1d reports the one-loop Feynman diagram of the one-loop electron self-energy according to QED.

Fig. 1a Representation of the lines of force of the electric 𝐸 ⃗⃗ and magnetic fields 𝐵 ⃗⃗ as deduced from classical electrodynamics [START_REF] Pohl | The size of the proton[END_REF] for a moving charged particle, being 𝑣 ⃗ the vector velocity. . Fig. 1c A Möbius strip can work as an analogical model reproducing the topological properties of a spin ½ particle subject to a 2π rotation. In the specific case, a spin up will turn into a spin down particle, after following the one-sided surface of the strip and rotating of 2π.

Fig.1d

Feynman diagram of the one-loop electron selfenergy. Within the QED formulation the self-interaction of the electron with its own electromagnetic field is determined as the self-energy δm𝑐 2 of the electron. By evaluating the line integral in eq. 2 as an integral over the time 𝜏, the expectation value turns out an expectation value of a quantity which has the dimension of an energy 𝑒〈A 𝜇 𝛾 𝜇 〉.

Formulation of an energy scale comparable to the energy at rest of the electron

In the following an application of equation 19 is considered in order to provide a formulation of an energy scale which is comparable to the energy at rest of the electron. As a matter of fact, radiative corrections within the QED theory provide a contribution to the self-energy determination of the electron [START_REF] Weinberg | The Quantum Theory of Fields[END_REF][START_REF] Greiner | Quantum Electrodynamics[END_REF]. Indeed, by opportunely choosing the cut-off energy required in the QED formulation, a value of the self-energy which is comparable with the energy at rest of the electron can be determined. However, concerning the origin of masses, the diversification of lepton masses can be explained by a Yukawa mechanism of coupling of massless charged leptons with the Higgs field that spontaneously breaks the gauge symmetry [START_REF] Weinberg | The Quantum Theory of Fields[END_REF]. If this is the case we should be able to provide an evaluation of the electron mass within a minimal extension of a scheme which takes into account somehow of the electroweak forces.

In this respect, we may observe as in defining the classical radius 𝑟 𝑜 of the electron, the mass at rest of the electron is proportional to the fine structure constant 𝛼 , whereas the classical radius of the electron can be associated to the Compton wavelength of a more massive particle whose mass reads (𝑚 𝜇 -𝑚 𝑒 )𝑐 2 , we can observe as the mass associated to the classical radius of the electron is of the same order of magnitude of the mass of a muon. In such a way we can provide a value of the mass of the electron in terms of that of a muon. However, usually we do the opposite, namely we express the mass of heavier leptons in term of the lightest lepton mass, the electron.

On the other hand, we can conjecture that the dependence of the mass of the electron on the fine structure constant 𝛼 is reasonable but it may involve a higher power of 𝛼, such as 𝛼 2 . If this is the case, the 𝛼 2 quantity must be multiplied by the mass value of a heavier particle opportunely chosen. Moreover, we can also expect that the unknown equation expressing an energy, should be in a certain way related to standardized expressions of use within the quantum theory among which we can make a choice. In particular, a mass-energy formula that we can consider stems from the atomic theory. In fact, the equation that relates the Bohr energy of the hydrogen energy levels to the mass of the electron has the same dependence on 𝛼 2 . Moreover, if we want to pursue the determination of a formula for the mass of the electron within a minimal extension scheme which includes quantities appearing in the electroweak theory, we have to consider the possibility that such heavier mass might be related to the masses of the vector bosons mediating the weak force. Let us devise a thought experiment hence. Namely, let us consider a pair 𝑊 + 𝑊 -of charged 𝑊 vector bosons that undergoes a scattering process at very low energy. If this is the case, neglecting the very short lifetime of such particles, it results that the scattering amplitudes determined by the Coulomb scattering are influenced by the possible bound states determined by the 𝑊 + 𝑊 -system. The energy of such bound states is lowered with respect to 2𝑀 𝑊 𝑐 2 by the Bohr energy determined by the electromagnetic interaction of the 𝑊 + 𝑊 -pair, resulting, as a first approximation, in a binding energy , 𝐸 𝐵 = 1 2 𝛼 2 𝑀 𝑊 𝑐 2 2 , of 1.07 𝑀𝑒𝑉. This energy value is quite close to the energy of an 𝑒 + 𝑒 -pair, which amounts to 2𝑚 𝑒 𝑐 2 = 1.02𝑀𝑒𝑉 when both particles are at rest. And indeed, we can say that the energy scale of an electron at rest is comparable to half of the Bohr energy of a 𝑊 + 𝑊 -pair:

𝑚 𝑒 𝑐 2 ≈ 1 8 𝛼 2 𝑀 𝑊 𝑐 2 . ( 25
)
Moreover, the difference 2∆𝐸 between the Bohr energy 𝐸 𝐵 and the electron-positron pair mass, results of 48.15 keV:

2∆𝐸 = 1 4 𝛼 2 𝑀 𝑊 𝑐 2 -2𝑚 𝑒 𝑐 2 = 48.15 𝑘𝑒𝑉. (26)
Indicating as 𝛿𝑀 𝑊 = 2.085 ± 0.042 𝐺𝑒𝑉 [START_REF] Beringer | Review of particle physics[END_REF] the decay width of a charged vector boson, assuming that 2𝛿𝑀 𝑊 is the uncertainty of the mass of the 𝑊 + 𝑊 -pair, it results that:

In the hypothesis that such a difference turns out from an indetermination on the mass of the charged boson vector 𝑊, we can wright equation ( 26) as:

∆𝐸 = 1 8 𝛼 2 𝑀 𝑊 𝑐 2 -𝑚 𝑒 𝑐 2 ≈ 1 8 𝛼 2 2𝛿𝑀 𝑊 𝑐 2 . ( 28
)
Therefore, the error associated with the mismatch of the two energy scales, the mass of the 𝑚 𝑒 𝑐 2 and the corrected Bohr energy of a 𝑊 + 𝑊 -pair

1 8
𝛼 2 (𝑀 𝑊 -2𝛿𝑀 𝑊 )𝑐 2 is:

1 - 1 8 𝛼 2 𝑀 𝑊 𝑐 2 - 1 8 𝛼 2 2𝛿𝑀 𝑊 𝑐 2 𝑚 𝑒 𝑐 2 ≈ 0.7%, (29)
and differs only by 0.7 % from the mass at rest of an electron.

It is also possible to go further in our evaluation. Indeed, we can introduce an angle 𝜃 such that to take into account of the deviation of difference reported in eq. 26 with respect to the quantity calculated in eq. 27 :

and such a value differs only by 0.08% with respect to the mass of the electron. On the other hand, the Bohr energy is an electromagnetic form of energy, by considering the results reported in eq. ( 19), eq. ( 31) must be multiplied by half of the spin g-factor 𝑔 𝑠

2

. By doing such a correction, we can obtain a value of an energy scale which differs by only 0.04% with respect to the mass at rest of the electron: Is this what the mass of an electron should look like? Does eq. 32 determines the mass of the electron? Although the advantage of such a formulation is the use of parameters and quantities which are proper of the electroweak theory, in spite of the close value of the calculated energy scale in eq. 32 to the energy at rest of the electron, a skeptic reader might also consider that the previous heuristic reasoning leads only to a fortunate guess of an energy value which is very close to the mass of an electron. As a fact, we cannot exclude that such a conclusion is a pure coincidence arising from the calculation of the Bohr energy of a 𝑊 + 𝑊 -pair. Although of the many clues provided, such as the Bohr energy and its corrections which are very close to the energy at rest of an 𝑒 + 𝑒 -pair, the Bohr radius of the 𝑊 + 𝑊 -system which is comparable to the classical electron radius, a critical point to consider is the appropriate identification of the angle 𝜃. However, independently of the possible implications which are behind eq. 25 and 32, it is sure that eq. 25, a Bohr energy, is the value of an energy scale comparable to the energy scale of an electron at rest which has been formulated by means of the mass 𝑀 𝑊 of the 𝑊 particle.

𝑚 𝑒 𝑐

Conclusions

In conclusion, by evaluating a gauge-invariant phase factor (that is a Wilson loop) linked to the electromagnetic gauge field 𝐴 𝜇 of the particle, namely an electron, a topological quantum interplay between the magnetic flux Φ and the mass has been demonstrated, In particular, it has been shown that, from the phase factor and the quantization of the magnetic flux variations, a relationship between the mass at rest of the electron 𝑚 e and its self-energy 𝛿𝑚, arising from radiative corrections, can be obtained, also within a QED approach. Besides, a formulation of an energy scale comparable to the energy at rest of an electron-positron pair is proposed. It is remarkable that a reckoning of the Bohr energy of a 𝑊 + 𝑊 -pair, which is compatible with constants and parameters usually employed within the electroweak theory, is comparable to the energy at rest of an 𝑒 -𝑒 + pair. Moreover, it has been shown how corrections can be provided such that the calculated energy scale approaches better the value of an 𝑒 -𝑒 + pair. However at this stage we cannot exclude that such a conclusion is a pure coincidence arising from our speculation. Nevertheless, independently of such speculations, it is certain that the electron, the lightest lepton, will continue to be exploited in the entire spectrum of technological applications that will emerge. A better approximation can be obtained by introducing an angle 𝜃, such that a projection of the "segment"

Fig. 1b

 1b Fig.1bRepresentation of the main features of an electron at rest. An electron, even at rest, has a magnetic field which determines a magnetic moment of -𝜇 𝐵 (a Bohr magneton), for an electron with spin up. The associated magnetic flux determined by the electron is independent of the line path and is quantized in units of Φ(↑) = -2𝜋𝑐ℏ 2𝑒

. 3 2𝛼

 3 If we consider the Barut formula[START_REF] Barut | The mass of the muon[END_REF] 𝑚 𝜇 = 𝑚 𝑒 + 𝑚 𝑒

1 4𝛼 2 (

 12 2𝛿𝑀 𝑊 )𝑐 2 = 55.5 ± 1.1 𝑘𝑒𝑉.

Fig. 2a A

 2a Fig.2a A pair of charged 𝑊 vector bosons (𝑊 + 𝑊 -) is considered to survive long enough to arrange a hydrogen-like system. The Bohr's radius 𝑎 0𝑊 + 𝑊 -= 2ℏ 𝛼𝑀 𝑊 𝑐 of the system results comparable to the classical radius of the electron 𝑟 0 and to the Compton's wavelength of the muon. In particular, the radius of the n th Bohr's level 𝑟 𝑛𝑊 + 𝑊 -= 𝑛 2 2ℏ 𝛼𝑀 𝑊 𝑐

Fig. 2b . 1 2𝑛 2 𝛼 2 𝑀 𝑊 𝑐 2 2 25 • 1 .

 2b1222251 Fig.2b. Comparison of the Bohr's energy scale 𝐸 𝐵 of a 𝑊 + 𝑊 -pair, working as a hydrogen-like system, with the energy of an electron-positron pair 2𝑚 𝑒 𝑐 2 = 1.02𝑀𝑒𝑉. The energy associated to the n th Bohr's orbit with respect to 2𝑀 𝑊 𝑐 2 is lowered by the energy 1 2𝑛 2 𝛼 2 𝑀 𝑊 𝑐 2 2 and differs, with respect to the ground state, by an energy ∆𝐸 of ∆𝐸 = 1 2 (1 -1 𝑛 2 ) 𝛼 2 𝑀 𝑊 𝑐 2 2 . The energy ∆𝐸 associated to the ground state and n = 5 level is ∆𝐸 = 24 25 • 1.07𝑀𝑒𝑉 = 1.027 𝑀𝑒𝑉 .

1 4𝛼 2 (

 12 2𝛿𝑀 𝑊 )𝑐 2 , i.e.

1 4𝛼 2 ( 4 𝛼 2

 1242 2𝛿𝑀 𝑊 )𝑐 2 cos𝜃, on the vertical axis, equals the difference between ( 1 𝑀 𝑊 𝑐 2 -2𝑚 𝑒 𝑐 2 ). The evaluation of such an angle provides a value which is close to the Weinberg angle 𝜃 𝑊 . By adopting 𝜃 𝑊 = 𝜃, we can estimate a value of the electron mass which differs only by 0.08 % from the measured one. Multiplying such a value by

  |∆𝛷 𝑛𝑜𝑟𝑚 | =

	𝑔 𝑠 2	4𝛿𝑚 𝑒𝑠 𝑚 𝑒	=	𝑔 𝑠 2	𝛿𝑚 𝑒𝑚 𝑚 𝑒	=	𝑔 𝑠 2	𝛿𝑚 𝑚 𝑒	. Such a variation equals unity, if 𝛿𝑚 𝑒𝑚 = 𝑚 𝑒	𝑔 𝑠 2

  Such a value is comparable with the Weinberg angle 𝜃 𝑊 = 28.1°. In the hypothesis that the angle 𝜃 is the Weinberg angle 𝜃 𝑊 , we can gain a value for the energy scale that we are evaluating which is even closer to the energy at rest of the mass of the electron:

	1 4	𝛼 2 (2𝛿𝑀 𝑊 )𝑐 2 𝑐𝑜𝑠𝜃 =	1 4	𝛼 2 𝑀 𝑊 𝑐 2 -2𝑚 𝑒 𝑐 2 .	(30)
	It results that a reckoning of 𝑐𝑜𝑠𝜃 provides a value of 𝜃 which amounts to 𝜃 =
	𝑎𝑐𝑜𝑠 ( 55.5±1.1 𝑘𝑒𝑉 48.15 𝑘𝑒𝑉 ) = (29.8 ± 2)°. 𝑚 𝑒 𝑐 2 = 1 8 𝛼 2 𝑀 𝑊 𝑐 2 -	1 8	𝛼 2 (2𝛿𝑀 𝑊 )𝑐 2 𝑐𝑜𝑠𝜃 𝑊 = 0.5106(5) 𝑀𝑒𝑉 (0.08%),


Laurent Praly
email: laurent.praly@mines-paristech.fr

Observers to the aid of "strictification" of Lyapunov functions (Long version)

Keywords: Strict Lyapunov function, observer, auxiliary function

We present a procedure for modifying a weak Lyapunov function V into a strict one. For this we augment the given function with an auxiliary function V a obtained as a Lyapunov function associated with the error dynamics given by an observer designed for the system with the derivative of V as an output function.

Introduction

Problem statement

Let O be an open neighborhood of the origin in R n . We consider a system the dynamics of which on O are:

ẋ = f (x) , (1)
with x in O and f : O → R n a sufficiently many times differentiable function which is zero at the origin. We assume the knowledge of a C 1 positive definite 1 function V : O → R ≥0 and we denote L f V (x) its Lie derivative, evaluated at x, along the vector field f . When L f V is a negative definite function, V is called a strict Lyapunov function, it is a weak Lyapunov function if L f V is only non negative. It follows from LaSalle invariance theorem or Barbashin-Krasovskii theorem that the origin is asymptotically stable when V is a weak Lyapunov function and, for any strictly positive real number v, the largest invariant set contained in:

Z v = {x ∈ O : L f V (x) = 0 , V (x) = v} , (2)
is empty. In this case a strict Lyapunov function exists thanks to the Converse Lyapunov Theorem. We propose here a procedure which, starting from the knowledge of a weak Lyapunov function, builds a modification transforming it into a strict Lyapunov function with an as explicit as possible expression.

Our approach and related results

The problem of "strictifying" a Lyapunov function has received a lot of attention, the results of which are partly reported in the monograph [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF]. We address it by continuing with the idea, introduced in [START_REF] Angeli | Input-to-state stability of PD-controlled robotic systems[END_REF] and summarized in [1, §5.6], of exploiting output-to-state stability. More specifically, inspired by [START_REF] Matrosov | On the theory of stability of motion[END_REF], given the function V as above, we are interested in finding functions h : O → R p , V a : O → R ≥0 and W a : O → R ≥0 , a class K function α and a class K ∞ function γ satisfying:

L f V (x) ≤ -α(|h(x|)) , (3)
L f V a (x) ≤ -W a (x) + γ(|h(x)|) (4)
and such that the function x → W a (x) + |h(x)| is positive definite. The motivation is that, in this case,

V s (x) = µ(V (x)) + µ a (V a (x)) , (5)
where µ and µ a are appropriately chosen C 1 class K ∞ functions, is a good candidate for being a strict (global) Lyapunov function. Very advanced and sophisticated techniques for designing the functions µ and µ a are available. We refer to [START_REF] Sontag | Changing supply functions in input/state stable systems[END_REF][START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF][START_REF] Ito | Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions[END_REF] and many others for this and do not address this point in the paper besides what is in the appendix, extracted from these references, and formatted for our application. In this context, V a is called an auxiliary function. As noticed in [1, §5.6], the challenge is to find an expression for it. Below we propose a procedure to obtain it via the design of an observer.

According to [6, Section II], an observer is a dynamical system, with the given system output h(x) as input, and the output of which estimates a function τ of the given system state x. Hence the idea of choosing V a in (4) as a Lyapunov function quantifying the estimation error and the dynamic of which describes the observer convergence. This idea is not new. It has been exploited already for instance in [7, §V].

In section 2 we present our construction technique in the simplest possible way, concentrating on the idea of getting, via the introduction of an observer, an auxiliary function V a . In section 3, we demonstrate, by means of examples, the possible interest and the many flexibilities of the technique.

In a complementary section at the end of this report, we give more advanced results on Examples 2 and 4.

Notation:

In all what follows ω denotes a real number strictly larger than 1.

The observer design approach

The procedure

The starting point of the procedure is the function V with L f V non positive. We look for a function h : O → R p , called output function in our context, satisfying:

-L f V (x) ≥ α(|h(x)|) , (6)
where α is a class K function, and such that the system:

ẋ = f (x) , y = h(x)
is observable in such a way that we can design an observer.

What we mean by we can design an observer is that there exist continuous functions τ : O → R m and ϕ :

h(O) × R m → R m such that: 1. the function x → (h(x), τ (x)) is injective on O.
2. We have:

L f τ (x) = ϕ(h(x), τ (x)) , τ (0) = 0 . (7)
3. For the augmented system:

ẋ = f (x) , ż = ϕ(h(x), z) , (8)
we know a

C 1 function V : O ×R m → R ≥0 satisfying: V(x, τ (x)) = 0
and, for all (x, z) such that z = τ (x),

V(x, z) > 0 , ∂V ∂x (x, z)f (x) + ∂V ∂z (x, z)ϕ(h(x), z) = -W(x, z) < 0 . (9)
This does give an observer since, on one hand, when z is in the image τ (O), and the value h(x) is known, there is a unique x satisfying:

z = τ (x) ,
and, on the other hand, the solution (X(x, t), Z((x, z), t)) of ( 8), issued from (x, z) at time 0, is such that Z((x, z), t) converges to τ (X(x, t)) as t goes to infinity. See [8, Theorem 1] to find a more precise result.

By evaluating at (0, τ (x)) the functions involved in (9), we obtain:

∂V ∂z (0, τ (x))ϕ(0, τ (x)) = -W(0, τ (x)) < 0 ∀x ∈ O : τ (x) = 0 . (10)
This motivates us to define the auxiliary function V a as:

V a (x) = V(0, τ (x)) .
In view of [START_REF] Teel | Lyapunov-based sufficient conditions for exponential stability in hybrid systems[END_REF], we have the decomposition:

L f V a (x) = -W(0, τ (x)) + ∂V ∂z (0, τ (x)) [ϕ(h(x), τ (x)) -ϕ(0, τ (x))] .
Here, h(x) plays the role of a disturbance. For this to be useful in our context, we assume we are in an ISSlike context, i.e. there exist a class K ∞ function γ and a continuous positive definite function α a : τ (O) → R ≥0 satisfying:

∂V ∂z (0, τ (x)) ϕ(h(x), τ (x)) (11) ≤ -ω α a (V a (x)) + γ(|h(x)|) .
In this case, (4) holds and, with the help of Lemma 1 in appendix, [START_REF] Ito | Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions[END_REF] gives an expression of a strict Lyapunov function. Fortunately, [START_REF] Malisoff | Stabilization and robustness analysis for a chain of exponential integrators using strict Lyapunov functions[END_REF] holds always when |τ (x)| and |h(x)| are small enough, since the function z → W(0, z) is positive definite, and we have [START_REF] Faubourg | Control Lyapunov functions for homogeneous Jurdjevic-Quinn systems[END_REF] and continuity.

At the end, we keep only the functions τ and V from the procedure and we need explicit expressions for them. The other points are useful only to make sure that these functions do have the required properties.

In nonlinear system theory there are different approaches for observer design, hence as many variants for the construction of τ and V. In the next Section, we shall restrict our attention to high gain observers and nonlinear Luenberger observers. For the former, an expression of τ is given by successive differentiations of the output function and V is a quadratic form known up to a scalar constant gain . In the latter, V is a known quadratic form, but an expression has to be found for τ solving [START_REF] Teel | Lyapunov-based sufficient conditions for exponential stability in hybrid systems[END_REF].

For the time being, to help the reader in getting a better grip on the procedure, we propose the following example.

Example 1 ([1, Section 5.5.1]). Consider the system: Let V be the function defined as:

ẋ1 = -(x 1 + x 2 )(x 1 + a) , ẋ2 = bx 1 (x 2 + c) ,
V (x) = b x 1 -a ln x 1 + a a + x 2 -c ln x 2 + c c .
It is positive definite, proper and C 1 on O and satisfies:

˙ V (x) = - x 1 (x 2 + c) ẋ2 = -b x 2 1 .
This is (3), with:

h(x) = x 1 , α(s) = bs 2 .
It follows that the origin is stable and that x 1 converges to 0 along any solution. So, if by "measuring" (=knowing) that x 1 goes to 0, we can "estimate" (=deduce) that x 2 goes also to 0, we are done. For this "measure-estimate" process, we look for an observer for the system:

ẋ1 = -(x 1 + x 2 )(x 1 + a) , ẋ2 = bx 1 (x 2 + c) , y = x 1 .
We propose:

ż = -ab z -bx 1 (x 1 -c + (1 -b)a) .
We note that:

τ (x) = x 2 + bx 1
is a solution of (7) which reads here as:

- ∂τ ∂x 1 (x 1 , x 2 )(x 1 + x 2 )(x 1 + a) + ∂τ ∂x 2 (x 1 , x 2 )bx 1 (x 2 + c) = -ab τ (x) -bx 1 (x 1 -c + (1 -b)a) .
We define the function V as:

V(x, z) = (τ (x) -z) 2 2 = (x 2 + bx 1 -z) 2 2 .
It satisfies:

˙ V(x, z) = -ab (x 2 + bx 1 -z) 2 = -ab V(x, z) .
Hence we have a convergent observer providing an estimate of τ (x). Since the data of h(x) = x 1 and τ (x) = x 2 + bx 1 defines x = (x 1 , x 2 ) uniquely, we have actually an estimate of x.

From all this, we propose:

V a (x) = V(0, τ (x)) = (x 2 + bx 1 ) 2 2
as an auxiliary function. It satisfies:

˙ V a (x) = -ab x 2 + bx 1 2 -bx 1 x 2 + bx 1 x 1 -c + (1 -b)a , ≤ - ab 2 x 2 + bx 1 2 + b 2a x 2 1 x 1 -c + (1 -b)a 2 .
This is [START_REF] Malisoff | Stabilization and robustness analysis for a chain of exponential integrators using strict Lyapunov functions[END_REF] with:

γ(s) = b 2a s 2 s + |(1 -b) -c| 2 .
So, from Lemma 1 in appendix, with:

α a (s) = abs 2ω , κ(s) = a[1 + ln(3)] max s ab , s ab ,
we can get an expression of a strict Lyapunov function. More directly, since there exists 2 a continuous non decreasing function κ : R → R >0 satisfying:

κ x 1 -a ln x 1 + a a ≥ 1 + 1 a x 1 -c + (1 -b)a 2 ,
for all x 1 in the open interval (-a, +∞), the following function is a strict Lyapunov function:

V s (x) = V (x) 0 κ(s)ds + V a (x) .

Solution with a high gain observer

To ease the presentation in this paragraph, we restrict our attention to the case where the dimension p of h(x) is 1 and ( 6) reads:

-L f V (x) ≥ |h(x)| 2 ∀x ∈ O . ( 12
)
Assume the system (1) with h as an output function is strongly differentially observable of order m. This means that there exists an integer m such that the function τ defined as:

τ (x) =      h(x) L f h(x)
. . .

L m-1 f h(x)      (13)
is an injective immersion on O. In this case there exists a C 1 function ϕ m which is zero at the origin and satisfies:

ϕ m (τ (x)) = L m f h(x) ∀x ∈ O
and such that the components z i of z = τ (x) satisfy:

ż1 = z 2 , . . . , żm-1 = z m , żm = ϕ m (z) .
We rewrite these equations more compactly as:

ż = A z + B ϕ m (z) , z 1 = C z .
We choose a compact subset C of O which is forward invariant for (1), e.g. a bounded sublevel set of V contained in O. We consider the Kalman observer-like for τ (x):

ż = A z + B ϕ m (sat(z)) + K (h(x) -Cz) ,
where sat is a bounded function satisfying:

sat(τ (x)) = τ (x) ∀x ∈ C , 2 e.g. κ(s) = 1 + sup x>-a: x 1 -a ln(x 1 +a)≤s 2 a x 1 -c + (1 -b)a 2 .
and the vector K is chosen such that there exist positive definite matrices P and Q satisfying:

P (A -KC) + (A -KC) P ≤ -2 Q , 2 τ (x) P B ϕ m (τ (x)) ≤ τ (x) Q τ (x) ∀x ∈ C .
Since ϕ m (0) = 0, by invoking the high gain observer paradigm (see [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] for example), we are guaranteed that P and Q exist when K is in the form:

K =    k 1 . . . m k m    ,
where the gain is to be chosen large enough depending on the Lipschitz constant:

Lip = sup x∈C ∂ϕ m ∂z (τ (x)) .
Then, since we have:

˙ τ (x) = (A -KC) τ (x) + B ϕ m (τ (x)) + K h(x) , we obtain, for all x in C, ˙ τ (x) P τ (x) ≤ -τ (x) Q τ (x) + 2 τ (x) P Kh(x) , ≤ -1 -ρk 2 p 2 λmin(Q) τ (x) Q τ (x) + 1 ρ |h(x)| 2 ,
where k and p are norms of K and P , and ρ is a strictly positive real number. With this, Lemma 1 gives us again a strict Lyapunov function on C. A simpler expression is:

V s (x) = V (x) + µ a τ (x) P τ (x)
where µ a is a real number to be chosen. We get:

˙ V s (x) ≤ -1 - µ a ρ |h(x)| 2 -µ a 1 - ρk 2 p 2 λ min (Q) τ (x) Q τ (x) .
Since τ is injective on O and zero at the origin, the right hand side is negative definite when µ a and ρ satisfy:

µ a < ρ < λ min (Q) k 2 p 2 .
Remark 1. Our construction uses the function τ which comes directly from the successive Lie derivatives of a square root of -L f V . This is related, though simpler, to what is done in [START_REF] Faubourg | Control Lyapunov functions for homogeneous Jurdjevic-Quinn systems[END_REF] for a Jurdjevic-Quinn control design with the introduction of an auxiliary scalar field and used also in [1, Sections 4.4 and 5.2].

Our construction gives also a tool simpler than the one proposed in [1, Section 5.4] although with some resemblance. But, of course, here, we have the stronger assumption of strong differential observability.

Example 2. Consider the system:

ẋ1 = σ(1 -x 2 )x 1 , ẋ2 = 1 - x 2 x 1 x 2 , with (x 1 , x 2 ) in: O = (R >0 ) 2
and σ a constant strictly positive real number. This is the system [11, (6)] with:

M 1 = M 2 = 1 , D * = 1 σ , u = y -x , x 1 = exp(x) , x 2 = exp(y) .
In [START_REF] Malisoff | Stabilization and robustness analysis for a chain of exponential integrators using strict Lyapunov functions[END_REF] the authors exhibit a strict Lyapunov function allowing them to prove that (1, 1) is an asymptotically stable equilibrium with O as domain of attraction. By following the approach described above we can get another expression of a strict Lyapunov function valid, at least, on any compact subset of O.

Our starting point is to observe that:

(x 1 -1) x 2 dx 1 -σ(1 -x 2 )
x 2 1 dx 2 = 0 is an exact differential equation with:

V (x) = ln(x 1 ) + 1 x 1 -σ ln(x 2 ) + σ x 2
as potential. This function is defined, C 1 and proper on O with a unique stationary point (global minimum) at (1, 1). Moreover it satisfies:

˙ V (x) = -σ 1 x 1 (1 -x 2 ) 2 . ( 14
)
Hence V is a weak global Lyapunov function on O showing the stability of the point (1, 1). Let the compact set:

C v = {x ∈ O : V (x) ≤ v}
where v is any strictly positive real number. Let also the output function h : O → R be defined from ( 14) as:

h(x 1 , x 2 ) = x 2 -1 .
There exists a real number α satisfying:

- ˙ V (x) ≥ α h(x) 2 ∀x ∈ C v .
According to [START_REF] Outbib | On the stabilization of smooth nonlinear systems[END_REF], we define:

τ (x) = h(x) L f h(x) = x 2 -1 1 -x2 x1 x 2 .
The corresponding function is a diffeomorphism on O.

Then, by letting z = τ (x), we obtain:

ż1 = z 2 , ż2 = ϕ 2 (x) ,
where:

ϕ 2 (x) = 1 -2 x 2 x 1 1 - x 2 x 1 x 2 + σ x 2 2 x 1 (1 -x 2 ) .
It follows from what has been done above for the general case that, for each strictly positive real number v , for all µ a sufficiently small and sufficiently large 3 , the function:

ln(x 1 ) + 1 x 1 -σ ln(x 2 ) + σx 2 + µ a [x 2 -1] 2 - 1 [x 2 -1] 1 - x 2 x 1 x 2 + 1 2 1 - x 2 x 1 x 2 2
is a strict Lyapunov function on C v . Hence, we are left with tuning the two real numbers and µ a to get a strict Lyapunov function, whereas in [START_REF] Malisoff | Stabilization and robustness analysis for a chain of exponential integrators using strict Lyapunov functions[END_REF], it is a function which has to be tuned. However in the latter the obtained Lyapunov function is strict on O and not only on the compact subset C v as in our case.

Solution with a nonlinear Luenberger observer

Again to simplify the presentation, we assume (12), instead of ( 6), and we choose a compact subset C of O which is forward invariant for [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF]. Following [START_REF] Andrieu | On the existence of a Kazantzis-Kravaris/Luenberger observer[END_REF][START_REF] Luenberger | Observing the state of a linear system[END_REF][START_REF] Shoshitaishvili | Singularities for projections of integral manifolds with applications to control and observation problems[END_REF], a nonlinear Luenberger observer takes the form:

ż = F z + G h(x) ,
where z is in R m , and F is a Hurwitz matrix. Consider the partial differential equation:

L f τ * (x) = F τ (x) + G h(x) . (15)
According to [8, Theorem 2], a solution exists on C provided the real part of the eigen values of F are negative enough. It is C 1 and injective if:

• the system has no (backward) indistinguishable pair of states,

• the dimension of z is large enough ( m ≥ 2(n + 1) )

• and the eigen values of F are outside a zero Lebesgue measure set. See [START_REF] Andrieu | On the existence of a Kazantzis-Kravaris/Luenberger observer[END_REF]Theorem 3]. Let P be a positive definite symmetric matrix satisfying:

P F + F P = -Q < 0 . We get: ˙ V (x) + µ a τ (x) P τ (x) ≤ -|h(x)| 2 + 2µ a τ (x) P L f τ (x) , ≤ -|h(x)| 2 -µ a τ (x) Q τ (x) + 2µ a τ (x) P G h(x) , ≤ -1 2 |h(x)| 2 -µ a 1 -4µap 2 g 2 λmin(Q) τ (x) Q τ (x) .
Hence, by choosing µ a in the open interval 0, λmin(Q)

4p 2 g 2
, the following is a strict Lyapunov function:

V s (x) = V (x) + µ a τ (x) P τ (x) . 3 The observer gains are k 1 = and k 2 = 2 where is to be tuned depending on the Lipschitz constant on Cv of ϕ 2 with respect to z.

The interest of this approach is that distinguishability only is sufficient instead of strong differential observability for the high gain approach. But the drawback is the problem of finding an expression for a solution to the partial differential equation [START_REF] Anderson | Stability of adaptive systems: Passivity and averaging analysis, Signal Processing, Optimization, and Control[END_REF]. Fortunately approximations are allowed as claimed in [8, Theorem 5].

Example 3. Consider the system:

ẋ1 = x 2 , ẋ2 = -x 1 -x 2 exp - 1 x 2 2 , (16)
already studied in [13, Example 1] and [START_REF] Mazenc | Strong Lyapunov functions for systems satisfying the conditions of La Salle[END_REF]Example 3.4].

We have:

˙ V (x) = ˙ x 2 1 + x 2 2 = -2 x 2 2 exp - 1 x 2 2 .
We define the output function h as:

y = h(x) = x 2 exp - 1 2x 2 2
.

With such a function the system ( 16) is observable but not differentially observable. So we cannot proceed with a high gain observer. Unfortunately, we do not know either how to go with a nonlinear Luenberger observer since we have no expression for a solution to the partial differential equation corresponding to [START_REF] Anderson | Stability of adaptive systems: Passivity and averaging analysis, Signal Processing, Optimization, and Control[END_REF]. But, instead of the output y we can use as well a function of y, as long as we do not lose observability. Here we note that the function:

x 2 → y = x 2 exp - 1 2x 2 2
is strictly increasing and therefore invertible. Let h -1 denote its inverse. So, instead of y, we use x 2 as an output and actually even a function g of x 2 . In this case a Luenberger observer is for instance:

ż = -z + g(x 2 ) .
The partial differential equation corresponding to [START_REF] Anderson | Stability of adaptive systems: Passivity and averaging analysis, Signal Processing, Optimization, and Control[END_REF] is:

∂τ ∂x 1 (x 1 , x 2 ) x 2 - ∂τ ∂x 2 (x 1 , x 2 ) x 1 + x 2 exp - 1 x 2 2 = -τ (x 1 , x 2 ) + g(x 2 ) .
We find that by selecting g as:

g(x 2 ) = 2x 2 -x 2 exp - 1 x 2 2 = 2h -1 (y) 2 -y 2 h -1 (y) ,
an expression of a solution τ is simply:

τ * (x 1 , x 2 ) = x 1 + x 2 .
Since the function (

x 1 , x 2 ) → (h(x), τ (x)) = (x 1 , x 1 + x 2 )
is injective, the above observer is appropriate. It gives us the auxiliary function:

V a (x) = [x 1 + x 2 ] 2 .
It satisfies:

˙ V a (x) = -2 [x 1 + x 2 ] 2 -[x 1 + x 2 ]g(x 2 ) , ≤ -V a (x) + γ(|y|) ,
where γ is a class K ∞ function satisfying:

2h -1 (y) 2 -y 2 h -1 (y) 2 ≤ 4h -1 (|y|) 2 = γ(|y|) ≤ 4V (x) .
Hence we can get a strict Lyapunov function from Lemma 1 in appendix with:

α(s) = 2s 2 , α a (v) = v ω , γ -1 (s) = h √ s 2 , κ(v) = 4v .

Extensions via examples

The procedure described in Section 2 is only under its very elementary form. Many variations are possible. The following examples illustrate some degrees of freedom.

Example 4 ([15, Theorem 2.3]). Consider the linear time-varying system:

ẋ1 = A x 1 + B ψ(t) x 2 , ẋ2 = -ψ(t) B x 1 , (17)
with

A an n 1 × n 1 matrix, B a vector in R n1 , x 1 in R n1 , x 2 in R n2 .
The uniform complete observability of this system is studied in a complementary section of this report. In [15, Theorem 2.3 ii)], it is established that the origin is uniformly asymptotically stable if the function ψ : R → R n2 is bounded, locally integrable and there exist strictly positive real numbers θ, β and T satisfying: This result can be re-established as follows. The Lyapunov function:

A + A ≤ -β 2 I
V (x) = |x 1 | 2 + |x 2 | 2 satisfies: ˙ V (x) ≤ -β 2 |x 1 | 2 .
So the origin is uniformly stable. Let:

h(x) = β x 1 .
To make sure there exists a convergent observer, we check observability. The trick here is to consider the system: 4 If the derivative ψ exists almost everywhere and satisfies [START_REF] Mazenc | Stabilisation de trajectoires, ajout d'intégration, commandes saturées[END_REF] with u = x 1 as (known) input. In [15, pages 46-48], it is proved that, under [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF], this system ( 19) is uniformly completely observable. So, according to [16, Lemma 5 and Theorem 2], there exist strictly positive real numbers p and p and a solution t → P (t) to the following Riccati equation:

t+T t | ψs)| 2 ds ≤ Ψ then t+T t ψ(s)ψ(s) ds ≥ θ2 I implies (18). ẋ1 = Bψ(t) x 2 + Au , ẋ2 = -ψ(t)B u , y = x 1 ,
Ṗ (t) = 2Sym
0 Bψ(t) 0 0 P (t) -P (t) I 0 0 0 P (t) + I satisfying:

p ≤ P (t) ≤ p ∀t ≥ 0 .
With this, we have a Kalman filter for the system [START_REF] Mazenc | Stabilisation de trajectoires, ajout d'intégration, commandes saturées[END_REF].

There is no need to write it. It is sufficient to know that the error system can be analyzed with the quadratic form given by P (t) -1 . This leads to:

V s (x, t) = |x 1 | 2 + |x 2 | 2 + µ a x 1 x 2 P (t) -1 x 1 x 2
as a candidate strict Lyapunov function. It satisfies:

˙ V s (x, t) ≤ -[β 2 -µ a ]|x 1 | 2 -µ a P (t) -1 x 1 x 2 2 +2µ a x 1 x 2 P (t) -1 A -ψ(t)B x 1 , ≤ -β 2 -µ a - 2µ a [a 2 + ψ 2 b 2 ] p |x 1 | 2 - µ a 2 P (t) -1 x 1 x 2 2 ,
where a is a norm of A, b a norm for B and ψ a bound for |ψ|. So, for µ a strictly smaller than

β 2 p p+2[a 2 +ψ 2 b 2 ]
, V s is a strict Lyapunov function.

The same steps as the ones followed here for a linear adaptive system can be followed for a nonlinear one, as the system studied in [START_REF] Loria | Strict Lyapunov functions for model reference adaptive control: application to Lagrangian systems[END_REF].

Example 5 (Forwarding). In this example we construct a strict Lyapunov function for a system the feedback of which is designed by forwarding. We remain at the simplest level and refer to [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF] for more elaborate techniques and to [START_REF] Mazenc | Stabilisation de trajectoires, ajout d'intégration, commandes saturées[END_REF] for even more details.

With an appropriate choice of the coordinates5 for x 2 , the system is:

ẋ1 = a(x 1 ) + b(x 1 )u , ẋ2 = A x 2 + c(x 1 )u . ( 20
)
We assume: A1: we know a C 1 positive definite and proper function V 1 and a continuous positive definite function ᾱ satisfying:

L a V (x 1 ) ≤ -ᾱ(|x 1 |) -|L b V 1 (x 1 )| 2 .

A2:

A + A ≤ 0 . ( 21
)
A3: The pair (A, c(0) ) is detectable.

With letting:

V (x) = V 1 (x 1 ) + x 2 x 2 ,
we obtain:

˙ V (x) ≤ -ᾱ(|x 1 |) -|L b V 1 (x 1 )| 2 + L b V 1 (x 1 ) + 2x 2 c(x 1 ) u .
This motivates us for selecting:

u = -c(x 1 ) x 2 . ( 22
)
Indeed this gives:

˙ V (x) ≤ -ᾱ(|x 1 |) - 7 4 x 2 c(x 1 ) 2 .
With Assumption A3, we can conclude that the origin is asymptotically stable. But V is not a strict Lyapunov function. To go on following our procedure we pick the function h as:

h(x) = x 1 c(x 1 ) x 2 .
We note that inequality (21) gives:

A -c(0)c(0) + A -c(0)c(0) ≤ -2c(0)c(0) .
It follows from Assumption A3 that the matrix A-c(0)c(0) is strictly Hurwitz. So there exists a positive definite matrix P satisfying:

P A -c(0)c(0) + A -c(0)c(0) P = -I . ( 23
)
With this, the observer we consider is:

ż = A z + c(x 1 )u + c(0) c(x 1 ) x 2 -c(0) z ,
with u known, given by ( 22). The problem here, created by the second component of h, is the argument x 1 instead of 0 in the term between brackets. If we ignore it, a solution to the partial differential equation corresponding to [START_REF] Anderson | Stability of adaptive systems: Passivity and averaging analysis, Signal Processing, Optimization, and Control[END_REF] is simply:

τ (x) = x 2 .
The injectivity requirement is trivially satisfied. And, in view of (23), the function V is to be:

V(x, z) = [z -x 2 ] P [z -x 2 ] .
It corresponds the auxiliary function:

V a (x) = V(0, τ (x)) = x 2 P x 2 which satisfies: ˙ V a (x) = 2x 2 P A -c(x 1 )c(x 1 ) x 2 , = 2x 2 P A -c(0)c(0) x 2 -2x 2 P c(x 1 )c(x 1 ) x 2 + 2x 2 P c(0)c(0) x 2 , = -|x 2 | 2 + 2x 2 P c(0)c(0) -c(x 1 )c(x 1 ) x 2 .
We are in the case discussed in Remark 2 in appendix where:

γ(x) = 2x 2 P c(0)c(0) -c(x 1 )c(x 1 ) x 2
does not depend only on h(x). Fortunately, the continuity of c implies the existence of a strictly positive real number δ such that we have the property:

|x 1 | ≥ δ ∀x 1 : 1 ω ≤ 2 Sym P c(0)c(0) -c(x 1 )c(x 1
) .

On another hand, the function V being proper, there exists a class K function κ which satisfies, for all x : |x 1 | ≥ δ,

2 Sym P c(0)c(0) -c(x 1 )c(x 1 ) ≤ κ(V (x))ᾱ(|x 1 |) P .
This implies (see ( 26)):

1 1 + x 2 P x 2 γ(x) ≤ κ(V (x)) α(|x 1 |) ∀x : |x 1 | ≥ δ .
From all this and Remark 2, it follows that:

V s (x) = ω V (x) 0 κ(s)ds + x 2 P x2 0 1 1 + v dv is a strict Lyapunov function.
This function has some similarities with the one given in [20, Proposition 3.2] and obtained from the stabilizability of the pair (A, c(0)).

Conclusion

On one hand, auxiliary functions, as proposed by [START_REF] Matrosov | On the theory of stability of motion[END_REF], are shown in [START_REF] Malisoff | Constructions of strict Lyapunov functions[END_REF] to be efficient for strictifying a given Lyapunov function when combined with the output-to-state stability formalism advocated in [START_REF] Angeli | Input-to-state stability of PD-controlled robotic systems[END_REF].

On another hand zero-state detectability is often used complementary to LaSalle invariance principle to establish asymptotic stability.

The combination of these two facts leads to the procedure we have proposed for the strictification of a given Lyapunov function. Specifically, an observer is constructed based on the zero-state detectability. This gives us a Lyapunov function describing the convergence of the associated error system. This function is then used to construct an auxiliary function allowing us to propose a candidate strict Lyapunov function.

Acknowledgements

This report is based on a work done with Daniele Astolfi, from LAGEP, Université de Lyon, France.

Lemma 1. Let O be an open neighborhood of the origin in R n , V : O → R ≥0 be a C 1 positive definite function, h : O → R p be a C 0 function which is zero at the origin, V a : O → R ≥0 be a C 1 function, γ be a class K function, α a : R ≥0 → R ≥0 be a C 0 positive definite function and ω be a real number strictly larger than 1, such that the function x → α a (V a (x)) + |h(x)| is positive definite on O and we have, for all x in O,

L f V (x) ≤ -α(|h(x|)) , L f V a (x) ≤ -ω α a (V a (x)) + γ(|h(x)|) . (24)
Then, for any compact subset6 C in O, there exist a class K ∞ function µ and a class K function µ a such that the following is is a strict Lyapunov function on C :

V s (x) = µ(V (x)) + µ a (V a (x)) .
Proof. Because V is positive definite on O, γ is a class K ∞ function and h is continuous and zero at the origin, there exists a class K function κ satisfying:

γ(|h(x)|) ≤ κ(V (x)) ∀x ∈ C .
Let:

µ(v) = ω v 0 κ(s)ds , µ a (v) = v 0 κ a (s)ds .( 25
)
with the notation:

κ a (s) = α • γ -1 • α a (s) .
The function µ is of class K ∞ and the function µ a is of class K. They give:

L f V s (x) ≤ -ω κ(V (x)) α(|h(x)|) -κ a (V a (x)) ω α a (V a (x)) + κ a (V a (x)) γ(|h(x)|) .
If we have α a (V a (x)) ≥ γ(|h(x)|), this yields :

L f V s (x) ≤ -ω κ(V (x)) α(|h(x)|) -[ω -1] κ a (V a (x)) α a (V a (x)) ∀x ∈ C .
If instead we have γ(|h(x)|) > α a (V a (x)), this yields :

L f V s (x) ≤ -[ω -1] κ(V (x)) α(|h(x)|) -ω κ a (V a (x)) α a (V a (x)) -κ(V (x)) α(|h(x)|) -α • γ -1 • α a (V a (x)) , ≤ -[ω -1] α(|h(x)|) κ(V (x)) -ω κ a (V a (x)) α a (V a (x)) ∀x ∈ C .
For the right hand side to be zero we must have V (x) or (α a (V a (x)), h(x)) zero. But the functions V and x → α a (V a (x))+|h(x)| being positive definite, this implies L f V s is negative definite and therefore that V s is a strict Lyapunov function on C.

Remark 2. The proof above relies only on the property:

κ a (V a (x)) γ(|h(x)|) ≤ κ(V (x)) α(|h(x)|) . (26)
∀x ∈ C : α a (V a (x)) ≤ γ(x) .

In the case where, to satisfy (24), we must have γ depending on the whole x and not only on h(x), it is sufficient that, with κ a continuous function satisfying:

κ(v) ≥ sup x: V (x)≤v γ(x) ,
there exists a continuous positive definite function κ a satisfying:

κ a (v) ≤ inf x∈C:Va(x)=v, αa(v)≤γ(x) κ(V (x))α(|h(x)|) γ(x) .
Indeed (26) follows readily and we can pick (25) again.

Complements

Uniform complete observability of ( 17)

We reproduce here what can be found in [15, pages 46-48].

For the system: It gives:

ẋ1 = Bψ(t) x 2 , ẋ2 = 0 , y = x 1 , (27)
γ x (t, t+T ) = x Γ(t, t+T )x = t+T t |x 1 +BT(s, t) x 2 | 2 ds
where:

|x 1 | 2 -|x 1 + BT(s, t) x 2 | 2 = 2 s t x 2 ϕ(r)B T [x 1 + BT(r, t) x 2 ]dr ≤ 2|B| s t |x 2 | 2 |ϕ(r)| 2 dr s t |x 1 + BT(r, t) x 2 | 2 dr ≤ 2|B||x 2 | s t |ϕ(r)| 2 dr γ x (t, s)
This yields:

T |x 1 | 2 ≤ γ x (t, t + T ) + 2|B||x 2 | t+T t s t |ϕ(r)| 2 dr γ x (t, s)ds ≤ γ x (t, t + T ) + 2|B||x 2 |T t+T t |ϕ(r)| 2 dr γ x (t, t + T )
On another hand, we have:

t+T t |BT(r, t) x 2 | 2 dr = t+T t | -x 1 + [x 1 BT(r, t) x 2 ]| 2 dr ≤ 2T |x 1 | 2 + 2γ x (t, t + T )
With ( 18), we have finally obtained, for all x,

θ 2 |B| 2 |x 2 | 2 ≤ 4 x Γ(t, t + T )x × ×   x Γ(t, t + T )x + |B||x 2 |T t+T t |ϕ(r)| 2 dr  
The function ϕ being bounded, this implies uniform complete observability of the system (27). Note that, with the same kind of arguments as above, we can get:

t+T t |T(r, t) x 2 | 2 dr ≤ 2 t+T t s t x 2 ϕ(r)T(r, t) x 2 dr ds ≤ 2T t+T t |ϕ(r) x 2 | 2 dr t+T t |T(r, t) x 2 | 2 dr ≤ 4T 2 t+T t |ϕ(r) x 2 | 2 dr
Hence [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF] implies the same of type of bound for ϕ as for T.

A strict global Lyapunov for the system in Example 2

After a time change dt → x 1 dt and a coordinate change

x 1 → χ 1 = 1 x1 , the dynamics are: χ1 = -σ(1 -x 2 ) , ẋ2 = 1 χ 1 -x 2 x 2 .
Assuming x 2 is measured, a reduced order observer is:

ż = -σ(1 -x 2 ) -x 2 2 + x 2 z + 1 -x 2 .
It gives an estimate of χ 1 + x 2 and provides the auxiliary function:

V a (x) = 1 2 (χ 1 + x 2 -2) 2 = 1 2 1 x 1 + x 2 -2 2 .
We have:

˙ V a (x) = (χ 1 + x 2 -2) -σ(1 -x 2 ) -x 2 2 + x 2 χ 1 , = (χ 1 + x 2 -2) × × -σ(1 -x 2 ) + x 2 χ 1 (1 -x 2 ) 2 -x 2 (χ 1 + x 2 -2) , = - x 2 2 χ 1 (χ 1 + x 2 -2) 2 + (χ 1 + x 2 -2) (1 -x 2 ) -σ + x 2 χ 1 (1 -x 2 ) , ≤ - x 2 2 2χ 1 (χ 1 + x 2 -2) 2 + χ 1 2x 2 2 -σ + x 2 χ 1 (1 -x 2 ) 2 (1 -x 2 ) 2 , ≤ - x 2 2 2x 1 (1 + x 1 (x 2 -2)) 2 + 1 2x 1 x 2 2 (-σ + x 1 x 2 (1 -x 2 )) 2 (1 -x 2 ) 2 .
Going back to the initial time, we have:

˙ V a (x) ≤ - x 2 2 2x 2 1 (1 + x 1 (x 2 -2)) 2 + (-σ + x 1 x 2 (1 -x 2 )) 2 2σx 1 x 2 2 σ x 1 (1 -x 2 ) 2 .
It follows that a strict Lyapunov function is given by:

V s (x) = µ(V (x)) + V a (x) ,
with µ satisfying:

µ (V (x)) ≥ (-σ + x 1 x 2 (1 -x 2 )) 2 2σx 1 x 2 2 .
Such a function exists since V is proper on (R > 0) 2 . This construction is to be compared with what is in the first part of the proof of Theorem 2 in [START_REF] Malisoff | Stabilization and robustness analysis for a chain of exponential integrators using strict Lyapunov functions[END_REF].

Generalization of Example 4

In [START_REF] Loria | Strict Lyapunov functions for model reference adaptive control: application to Lagrangian systems[END_REF], the authors consider the following more general form of the system in Example 4. In that paper, the authors give an expression of a strict Lyapunov function for the system: ẋ1 = F (t, x 1 ) + Φ(t, x 1 )x 2 , ẋ2 = -Φ(t, x 1 )

∂V 1 ∂x 1 (t, x 1 ) ,

where the function x 1 → V 1 (t, x 1 ) is positive definite, radially unbounded, both uniformly in t, and satisfies:

∂V 1 ∂t (t, x 1 ) + ∂V 1 ∂x 1 (t, x 1 )F (t, x 1 ) ≤ -α(|x 1 |) .

To find an expression of a strict Lyapunov function for this system we start with picking:

V (t, x) = V 1 (t, x 1 ) + 1 2 |x 2 | 2 , W (t, x) = -α(|x 1 |) .
Then, as a preliminary task for designing an observer, we rewrite the system dynamics as:

ẋ1 = Φ o (t)x 2 + u 1 + δ 1 , ẋ2 = u 2 , y = x 1 ,
where u 1 and u 2 are known inputs, y is the measured output and δ 1 is a disturbance defined as: With ignoring the disturbance for the time being, we can go with a Kalman filter as an observer. This yields the auxiliary function:

u 1 = F (t,
V a (t, x) = x P (t) -1 x with P the solution of the Riccati equation coming with the Kalman filter. Because of the persistence of excitation and the boundedness of Φ o , there exist strictly positive real numbers p and p such that: p ≤ P (t) ≤ p ∀t .

Note that we have:

x P (t) -1 F (t, x 1 ) + K(t, x 1 )x 2 -Φ(t, x 1 ) ∂V1 ∂x1 (t, x 1 )

≤ [|x 1 | + |x 2 |] ρ(|x1|) p [2 + |x 2 |] , ≤ 1 p ρ(|x 1 |)|x 1 | + ρ(|x1|) 2 ε + ε|x 2 | 2 +ρ(|x 1 |)|x 1 | 1+|x2| 2 2 + ρ(|x 1 |)|x 2 | 2 , ≤ 1 (|x 1 |) + [ 2 (|x 1 |) + ε] |x 2 | 2 ,
where ε is a strictly positive real number, strictly smaller than 1 3p 2 , and the functions 1 and 2 are defined as:

1 (|x 1 |) = ρ(|x 1 |) p |x 1 | + ρ(|x 1 |) ε + 1 2 |x 1 | , 2 (|x 1 |) = ρ(|x 1 |) p 1 2 |x 1 | + 1 .
They are 0 at 0. This yields:

  where a, b and c are strictly positive real numbers and (x 1 , x 2 ) lives in the open rectangle: O = (-a, +∞) × (-c, +∞) .

  ds ≥ θ 2 I .[START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF]

  x 1 ) , u 2 = -Φ(t, x 1 ) ∂V 1 ∂x 1 (t, x 1 ) , δ 1 = K(t, x 1 )x 2 ,with the notation:K(t, x 1 ) = Φ o (t) -Φ(t, x 1 ) , Φ o (t) = Φ(t, 0) .Because of smoothness, uniform in t, and the fact that F (t, 0) and ∂V1 ∂x1 (t, 0) are zero, there exists a class K ∞ function, linearly bounded on a neighborhood of zero, satisfying :|F (t, x 1 )| + |K(t, x 1 )| + Φ(t, x 1 )∂V 1 ∂x 1 (t, x 1 ) ≤ ρ(|x 1 |) .

See [18, Section IV].

We can have C non compact and equal to O if V is proper on O.

Appendix

An existence result for µ and µ a From all the many published results concerning cascaded systems, small gain theorems, strictification of Lyapunov functions, ... we have extracted what follows without any novelty but formatted for our specific framework.

with the notation:

We let:

May be with increasing 1 , this is a continuous function according to Berge Maximum Theorem. Since

ξ 1 is positive definite and we have :

Let also κ be a continuous function satisfying:

We pick:

we obtain:

There is no problem when 1 3p V a (t, x) -γ(x) > 0. In the other case, we get:

V a (t, x) .

So we have an expression of a strict Lyapunov function up to solving the optimization problems (29) and (30).


B A Van Tiggelen
email: bart.van-tiggelen@grenoble.cnrs.fr

Stirring the quantum vacuum: Angular Casimir Momentum of a Landau Charge

Keywords:

We consider the angular momentum of a charge q rotating in a homogeneous magnetic field and study the role of the electromagnetic quantum vacuum. Its orbital angular momentum is caused by the recoil of energetic vacuum photons that grows as n 2 , i.e. faster than the kinetic angular momentum -2n of a Landau level.

I. INTRODUCTION

The radiation of the electromagnetic (EM) quantum vacuum is perfectly isotropic and possesses an energy density ωdω/π 3 c 3 0 in the frequency interval dω. This statement is Lorentz-invariant [START_REF] Milonni | The Quantum Vacuum[END_REF]. The mystery of its UV-divergence is still one of the major challenges in physics, but poses in general no problems to calculate the Casimir force between dielectric or metallic objects [START_REF] Milton | The Casimir Effect[END_REF]. Due to its perfect isotropy, energy current and momentum density of the quantum vacuum vanish. This is no longer true when the vacuum interacts with matter. In a pioneering work [START_REF] Feigel | [END_REF] Feigel predicted "Casimir" momentum in bi-anisotropic materials. Although the Poynting vector of the quantum vacuum still vanishes [4], a momentum density emerges that suffers from a divergence at high energies. Relativistic photons were already known to be relevant for the Lamb shift [START_REF] Milonni | The Quantum Vacuum[END_REF], in contrast to Casimir-Polder forces [START_REF] Milton | The Casimir Effect[END_REF], caused by lowenergy vacuum photons. In a microscopic bi-anisotropic quantum model, the divergency of EM momentum is removed by mass renormalization [5], resulting in a Casimir momentum of the order of α 2 times the classical Abraham momentum [6,7]. It was also demonstrated for a chiral quantum particle in a magnetic field [8,9].

EM angular momentum is of great recent interest [10] and in macroscopic media even controversial [11], since matter and radiation are hard to disentangle. In Ref. [12] Casimir-Polder torques were discussed to stir vortices in superfluids. How does "Casimir" angular momentum emerge on a microscopic scale? The simplest model for which this question can be answered is the well-known cyclotron problem of a non-relativistic charge q without spin rotating in a uniform magnetic field B 0 , in quantummechanics better known as the Landau problem. In the Coulomb gauge and in Gaussian units, the vector potential is A 0 (r, t) = B 0 (t)×r/2. With the quantum vacuum, the non-relativistic Hamiltonian reads,

H = 1 2µ p - q c 0 A 0 (r, t) - q c 0 A(r) 2 + kΠ ω k a † kΠ a kΠ (1)
Here, p the canonical momentum satisfying [r i , p j ] = i δ ij . It is customary to choose the z-axis along the direction of qB 0 and to introduce the cyclotron frequency ω c = qB 0 /µc 0 . It is convenient to express the vector potential of the quantum vacuum in terms of the spherical vector harmonics Φ Π ( k) defined on the unit sphere in reciprocal space [START_REF] Cohen-Tannoudji | Photons and Atoms[END_REF],

A(r) = 2π c 0 kΠ 1 k 3/2 a kΠ exp(-ik • r)Φ Π ( k) + a † kΠ exp(ik • r) ΦΠ ( k) (2)
The continuum limit has been taken,

k ≡ d 3 k/(2π) 3 and [a kΠ , a † k ′ Π ′ ] = δ kk ′ δ Π,Π ′ .
The vector index Π = {J, M, p} summarizes the 3 discrete quantum numbers of total EM angular momentum (J), its z-component (M ) and two transverse polarizations p, the longitudinal vector harmonic being excluded by the Coulomb gauge. Without the quantum vacuum, the kinetic momentum of the charge is p K = µdr/dt = µ[H 0 , r]/i = p -qA 0 (r, t)/c 0 . The canonical angular momentum l z = (r × p) z commutes with H 0 and has eigenvalues (|m|n) . The Hamiltonian H 0 has the eigenstates |n, m, k z of a 2D harmonic oscillator whose energy levels E n = (n + 1 2 ) ω c + 2 k 2 z /2µ are independent of m due to gauge invariance [START_REF] Goerbig | Quantum Hall Effects, in: Les Houches Summer School XCI, Ultracold Gases and Quantum Information[END_REF].

In general, total angular momentum L z is conserved, equal to the sum of the canonical angular momentum l z of the charge and the angular momentum of the transverse EM field J z . The angular kinetic momentum operator of the charge is not conserved and can be written as,

l K z = L z -J z - q c 0 (r × A) z -l L (3)
The "Lenz" term l L = µω c ρ 2 /2 is responsible for the electromotive force in classical electrodynamics, recently discussed quantum-mechanically [START_REF] Greenshields | [END_REF] as well as key element in the EM momentum controversy [16]. Without the quantum vacuum, the kinetic angular momentum of a Landau state is given by n, m|l K z |n, m = -(2n+1) = -2E n (rot)/ω c , again independent on m. When the quantum vacuum is included, l K z will decay radiatively, but slow enough for l K z to be well-defined and modified. We will use time-dependent perturbation theory to calculate the different contributions of the quantum vac-uum to l K z for an excited Landau state, and all proportional to qB 0 (t) for slow adiabatic changes. We identify the interaction W = -(q/µc 0 )p K • A(r) between rotating charge and quantum vacuum, and imagine to switch it on slowly like W (t) = W exp(ǫt/ ) at t 0 → -∞, when the total wave function is assumed to be in the pure state |N = |n, m, k z = 0 ⊗ |{0} [START_REF] Loudon | The Quantum Theory of Light[END_REF]. In this processus L z is conserved and equal to its initial value (|m|n) . The index N ′ refers to all possible product states {n, m, k z } ⊗ {n kΠ } of charge plus transverse photons. Explicit reference will be made nor to the highly degenerated levels m, neither to the momentum k z of the charge along the magnetic field. The degeneration of the m-levels is protected by gauge invariance, and the impact of photon recoil on the longitudinal displacement is negligible. Since W N N = 0, the wave function at t ≥ t 0 is perturbed as,

|Ψ N (t) = exp - i t t0 dt ′ [E N + ∆E N (t ′ )] |N + ′ N ′ |N ′ W N ′ N (t) E N -E N ′ + iǫ + • • • with ∆E N (t) = ′ N ′ W N N ′ (t)W N ′ N (t)/(E N -E N ′ + iǫ)
the second-order perturbation of the energy level E N , the sum ′ N ′ avoiding the initial level N [START_REF]We ignored the perturbation[END_REF]. For an excited state,

t t0 dt ′ ∆E N (t ′ ) = t t0 dt ′ (E L n + A n /2i) -1 2 i N n with N n
a time-independent normalization of the wave function. Lamb shift and spontaneous emission rate are

E L = ω c 2α 3π x log 2 x , A n = nω c 4α 3 x (4)
with x = ω c /µc 2 0 and α = q 2 / c 0 the fine structure constant; E L is equal for all Landau levels whereas radiative decay is proportional to rotational energy.

In reciprocal space, the transverse EM momentum J z in Eq. ( 3) is expressed as

(J z )(k) ij = -i δ ij (k × ∇ k ) z - i ǫ zij , i.e.
as a sum of orbital angular momentum and spin [START_REF] Cohen-Tannoudji | Photons and Atoms[END_REF]. Neither one of them behaves as a genuine angular momentum [START_REF] Van Enk | [END_REF] but this separation is physically useful. In Hilbert space J z reads,

J z = k α † i (k)J ij (k)α j (k) (5)
with the photon annihilation operator

α i (k) = k -1 Π a kΠ Φ Π,i ( k)
, and its associated creation operator α † i (k). The quantum expectation of J z is obtained by inserting the linearly perturbed eigenfunction (4) on both sides of the matrix element Ψ N (t)|J z |Ψ N (t) . This creates either a virtual or real photon with energy ω k and angular momentum Π ′′ out of the quantum vacuum at the cost of canonical angular momentum of the charge.

Working out the photon operators, leaves us with

J z = 2π q 2 µ 2 c 0 e 2ǫt k 1 k 2 k ′ Π ′ k ′′ Π ′′ δ kk ′ δ k ′′ k (k ′ ) 3/2 (k ′′ ) 3/2 × n|p K • Φ Π ′ ( k′ )e ik ′ r 1 E n -H ′ 0 -ω k -iǫ × ΦΠ ′ i ( k)J ij (k)Φ Π ′′ j ( k) × 1 E n -H ′ 0 -ω k + iǫ p K • ΦΠ ′′ ( k′′ )e -ik ′′ r |n
In principle is Π ′ = Π ′′ since the spherical harmonics are orthogonal eigenfunctions of J ij (k). The mathematics is easier by using their completeness, Π ΦΠ ( k)Φ Π ( k′ ) = δ kk′ ∆( k), with ∆( k) transverse to k imposed by the Coulomb gauge. The exponential exp(ik • r) can be moved by using the operator identity exp

(ik • r)f (p) = f (p -k) exp(ik • r)
, and which induces a photon recoil in H 0 (p). With

δ kk ′ = δ kk ′ δ kk ′ /k 2 , J z = 2π e 2ǫt/ q 2 µ 2 c 0 kk ′′ n|p K m 1 E n -H ′ 0 (p -k) -ω k -iǫ 1 k e ik ′ r ∆ mi ( k)J ij (k)∆ jl ( k′′ )e -ik ′′ r δ kk" 1 E n -H ′ 0 (p -k ′′ ) -ω k + iǫ p K l |n
From this expression the EM spin of the quantum vacuum can be identified as,

S z = 1 3 2π e 2ǫt/ q 2 µ 2 c 0 i ǫ zij × k 1 k n|p K i 1 |E n -H ′ 0 -E(k) + iǫ| 2 p K j |n (6)
We have performed the angular integral over k to eliminate ∆( k), neglected the photon recoil p K • k/µ irrelevant for spin, and defined the energy E(k) = ω k + 2 k 2 /2µ. The orbital angular momentum is associated with a differential operator acting on δ kk" , and an integration by parts is imposed to perform the integral over k ′′ . Since this operator acts only on angles, we find for the orbital angular momentum,

L z = 2π e 2ǫt/ q 2 µ 2 c 0 k n|p K m × i ǫ zst kt ∇ s ∆ mi ( k) 1 E n -H ′ 0 (p -k) -ω k -iǫ e ikr ∆ il ( k)e -ikr 1 E n -H ′ 0 (p -k) -ω k + iǫ p K l |n (7)
As k z = 0 the kinetic operator p K is located in the xy plane. It is customary to write p K x + (-)ip K y = (2µ ω c ) 1/2 c ( †) in terms of the raising and lowering operators of the Landau levels, in terms of which H 0 = ω c (c † c + 1 2 ). To evaluate the spin in Eq. ( 6) we use

ǫ zij p K i f (H 0 )p K j = -iµ ω c (c † f (E n-1 )c -cf (E n+1 )c † ).
The first term implies the release of a real photon with energy ω c . As ǫ ↓ 0, this part of S z is written as ( /2)

t t0 dt ′ exp(2ǫt ′ / )δ( ω c -ω k ), so that d dt S z = - 1 2 A n (8)
with A n defined in Eq. ( 4). The second term involves virtual photons and is finite as ǫ ↓ 0,

S z = (n + 1) q 2 3 ω c 3µc 0 ∞ 0 kdk ( ω c + E(k)) 2 = α 3π (n + 1) x log 2 x (9)
In expression (7) for L z the derivative ∇ k acts on 3 factors inside brackets. Its action on the factor in the middle, caused by the photon recoil, is a factor x smaller than the rest. The action on the first factor gives ǫ zst kt ∇ s (∆ mi )∆ il = -k -1 ǫ zlt km kt and produces an angular momentum L

(1) z = S z . Finally, the action of ∇ k on exp(ik • r) leads to the expression

L (2) z = 2π 2 e 2ǫt/ q 2 µ 2 c 0 × k n|p K m ∆ ml ( k) 1 H n -iǫ (r × k) z 1 H n + iǫ p K l |n
where

H n ≡ E n -H 0 -E(k) + p K • k/µ.
This time, the photon recoil cannot be ignored and we must expand either one of the denominators which produces an integral dk dk with integrand of the type

ǫ zst ∆ ml ( k)k u kt × p K m 1 H n -iǫ p K u 1 H n -iǫ r s 1 H n + iǫ p K l
Physically, this corresponds to the creation of a virtual photonic mode with finite orbital angular momentum.

In terms of the infinitely degenerated center (X, Y ) of the cyclotron orbit we associate x = Xp K y /µω c and y = Y + p K

x /µω c . The operators (X, Y ) drop out in the vacuum expectation value since they do not occur in pairs. Upon expressing (p K x , p K y ) in the operators c and c † , the integrand contains four transition operators. As was the case for S z , some contribute to spontaneous emission but are seen to be a factor x smaller than A n . We thus focus on terms where the limit ǫ ↓ 0 exists. For instance, the sequence

n|cH -1 n c † H -1 n cH -1 n c † |n = (n + 1)(n + 2) E(k) 3 leads to, L (2)
z

∼ (n + 1)(n + 2) 2π 2 q 2 µ 2 c 0 µ µ 2 ω c k k E(k) 3 ≈ 8π α(n + 1)(n + 2) x
Upon collecting all possibilities, performing the angular integral, and adding the complex conjugate, we obtain

J z = S z - 4 15π α(n + 1)(n + 4) x (10)
and d/dt J z = -A n /2. The last but one term in Eq. ( 3), the longitudinal EM angular momentum ∆L , is itself linear in the vacuum field. Its leading quantum expectation value is obtained using the linear perturbation of the eigenfunctions and the completeness of the spherical vector harmonics,

Ψ N |∆L |Ψ N = - q 2 µc 0 4π 3 k 1 k ǫ zij n|r i 1 H n + iǫ p K j |n
plus its c.c. Here, the k-integral diverges in the UV as dk/k. We recognize for large k the form (δµ/µ) n|l K z |n , with δµ the well-known Bethe-Kramers mass renormalization [START_REF] Milonni | The Quantum Vacuum[END_REF]. Upon adding it to the kinetic mass as l K = (µ + δµ)r × dr/dt, and upon subtracting it from the above equation, we obtain,

Ψ N |∆L |Ψ N = - q 2 µc 0 4π 3 k 1 k ′ n ′ i µ × ǫ ijz n|p K i |n ′ ∆E n ′ n + E(k) -iǫ n ′ |p K j |n E(k) + c.c.
where we have used the identity ∆E n ′ n n|r|n ′ = (i /µ) n|p K |n ′ . The operators p K x and p K y can be expressed in terms of c † , c, which results in,

∆L = 4α 3π x log 2 x (11)
which, like the Lamb shift in energy, is independent on n.

The last contribution of the quantum vacuum to the angular momentum is associated with the Lenz term l T in Eq. ( 3). The quantum vacuum comes in via N n and the second term in Eq. ( 4). With l T = (c † c + 1 + b † bib † c † /2 + icb/2) in terms of the raising (b † ) and lowering (b) operator of the degenerated m-levels [START_REF] Goerbig | Quantum Hall Effects, in: Les Houches Summer School XCI, Ultracold Gases and Quantum Information[END_REF], the mdependence is seen to cancel in the sum of both terms. The second term equals (n + 1) times 4π q 2 3µ 2 c 0

1 i t t0 dt ′ e 2ǫt ′ / k n|p K j 1 H n + iǫ p K j |n + c.c. = -A n t t0 dt ′ e 2ǫt ′ / - 2α 3π (n + 1) x log 2 x
and the first is times

4π q 2 3µ 2 c 0 e 2ǫt/ k 1 k n|p K j 1 H n + iǫ (c † c + 1) 1 H n -iǫ p K j |n = nA n t t0 dt ′ e 2ǫt ′ / + 2α 3π (n + 1)(n + 2)x log 2 x
In particular, n 2 terms also cancel and

δl T = 2 S z ; d dt δl T = -A n (12)
By adding up the four contributions S z , J z , J and δl T we find for the total angular momentum of the quantum vacuum

J QV z = 4α 3π (n + 2) x log 2 x - 4α 15π (n + 1)(n + 4)x (13)
and d J QV z /dt = -2 A n . We conclude that the quantum vacuum achieves an angular momentum that is, in units of , proportional to α × x = α × ω c /µc 2 0 ≈ 10 -12 /Tesla, which is time-dependent if B 0 is. In all momentum integrals the photon momentum k takes values up to µc 0 with nonetheless a significant weight of nonrelativistic momenta. A relativistic description of the rotating electron is thus relevant but should affect only numerical coefficients in Eq. [START_REF] Cohen-Tannoudji | Photons and Atoms[END_REF]. Even in a relativistic picture, Eq. ( 3) for the kinetic angular momentum is valid, and l K z remains quantized to -(2n + 1) [START_REF]The eigenfunctions are the same for the relativistic Hamiltonian without spin, and thus the Lenz moment l T remains unaltered[END_REF]. In the ground state, the existence of Casimir angular momentum makes the kinetic angular momentum slightly more negative than -, in states with large n it will be slightly less negative than -(2n + 1) , the correction growing like n 2 . Note that the gauge-invariant magnetic moment M z = (q/2µ)l K of the rotating charge is subject to the same correction. Due to the quantum vacuum, the kinetic angular -(2n + 1) decays to the Landau level n -1 with rate A n so that d l K /dt = +2 A n and l K z + J QV z = L z is conserved in the decay. The nonrelativistic analysis imposes that E n ≪ µc 2 0 , implying, for an electron in a field of 10 Tesla, that n ≪ 10 9 . Pushing our theory to this extreme synchrotron regime, the relative contribution of Casimir orbital angular momentum would be of order 10 -4 .

II. CONCLUSIONS

The main objective of this work is to establish the existence of angular momentum of the EM quantum vacuum, induced by the presence of a rotating charge in a magnetic field. It is instructive to look at the separate contributions of spin, orbital angular momentum and angular momentum directly associated with the gauge fields. All are oriented along the magnetic field and proportional to the product of fine structure constant and the small ratio of rotational energy to rest energy. Spin and orbital angular momentum decay in the sam way, their coupling being large, yet the orbital angular momentum of the quantum vacuum, induced by photon recoil, dominates angular momentum for highly energetic Landau levels. Casimir angular momentum is governed by virtual photons with energies up to the rest mass of the charge and, despite the UV renormalizability of the theory, would merit a relativistic treatment. A future challenge would be to study EM angular momentum in the fully relativistic synchrotron problem, or to investigate it for Rydberg orbits.