A Roux

J.-P Martischang

M Baudoin

Self radiation force on a moving monopolar source

Keywords:

The radiation force exerted on an object by an acoustic wave is a widely studied phenomenon since the early work of Rayleigh, Langevin and Brillouin and has led in the last decade to tremendous developments for acoustic micromanipulation. Despite extensive work on this phenomenon, the expressions of the acoustic radiation force applied on a particle have so far been derived only for a steady particle, hence neglecting the effect of its displacement on the radiated wave. In this work we study the acoustic radiation force exerted on a monopolar source translating at a constant velocity small compared to the sound speed. We demonstrate that the asymmetry of the emitted field resulting from Doppler effect induces a radiation force on the source opposite to its motion.

Introduction

Since the seminal work by ?? and Langevin (work reported later by ??), much effort has been devoted to the derivation of theoretical expressions of the acoustic radiation force exerted by an acoustic wave on a particle. ?? was the first to recognize the tensorial nature of the acoustic radiation force, which is not necessarily orthogonal to the insonified interface. Later on, ? derived an expression of the axial acoustic radiation force exerted on a rigid spherical particle by a plane wave. This expression was extended to the case of a compressible fluid sphere and of an elastic particle by ? and ? respectively. The case of an incident focused wave was treated by ? for a rigid particle and ? for an elastic particle. Later on, the more general case of the axial force exerted by a Bessel beam was addressed by ??.

In parallel, a general expression of the acoustic radiation force exerted by an arbitrary wavefield on a spherical particle in the Long Wavelength Regime (LWR i.e. when ka 1 with k the wavenumber and a the particle radius) was obtained by ?. It was shown up to 3rd order (in ka) that the radiation force is proportional to the gradient of an acoustic potential, which is proportional to the difference between the time-averaged potential and kinetic acoustic energy weighed respectively by the monopole and dipole scattering coefficients. This expression was extended to the 6th order by ?, which is necessary when the average potential and kinetic energy are uniform in space (e.g. for plane propagating waves). This expression was also extended by ??? to consider the effect of the viscous and thermal boundary layers in some asymptotic limits (of the boundary layer size compared to the particle size) and in the general case by ? and ?. The case of nonspherical particles such as disks and spheroids was treated by ? and ? respectively, while the case of transient acoustic fields was recently addressed by ?. We can also note that some expression for the secondary radiation force (inter-particle force) have been derived by ?. The specific case of the radiation force exerted on a vibrating bubble known as the primary Bjerkness force was treated separately by ?, ?, ? and ?. Indeed, bubbles have some specificity: owing to their strong compressibility compared to the surrounding liquid, their monopolar resonance appears in the LWR. Hence bubbles can be attracted to the nodes or anti-nodes of a standing wave depending whether they are forced below or above their monopolar resonance frequency (see ?).

Recently, there have been some renewed interest in the calculation of the acoustic radiation force with the development of selective acoustical tweezers (see ? for a review of the subject). Acoustical tweezers rely on the acoustic radiation force to move objects. To reach selectivity, i.e. the ability to manipulate a single object independently of other neighboring objects, it is necessary to localize the acoustic energy close to the target particle to only affect it (see e.g. ?, ?,?). Hence, such selectivity cannot be reached in the LWR. In addition, to calculate the restoring force (i.e. the force which brings back the particle toward the trap center), it is necessary to compute the radiation force when the particle is out-centered from the trap position. Yet, all the aforementioned expressions were either limited to the calculation of the radiation force for an axisymmetric configuration or to the LWR. To cope with this issue, general expressions of the radiation force exerted by an arbitrary incident field on a spherical particle, without restriction on the particle size compared to the wavelength, were obtained by ? with an angular spectrum based method and by ?, and ? with a multipole expansion method. The equivalence between the formula obtained with the different approaches was demonstrated by ?. Note also that some general expressions have been proposed to compute the acoustic radiation torque exerted by an arbitrary acoustic field on a particle of arbitrary size by ? and ?.

Yet, in all the theoretical developments mentioned so far, the calculation of the radiation force is made for a steady particle, hence neglecting the effect of its motion. Some account of the interaction between the oscillatory and translational motion of bubbles can be found in the literature. Following ?, ? showed that even in an inviscid fluid, self-propulsion of a bubble can be achieved by nonlinear interactions between adjacent surface deformation modes. In their work however, the surface modes deformation are supposed to be known a priori. This work was extended later on by ? to account for the parametric excitation of surface modes by the isotropic volume mode and ? who considered the direct coupling between translational motion, volume and shape modes. Finally, ? obtained an expression of this coupling whatever the shape modes, their natural frequency, and the type of excitation (parametric forcing by the volume mode or direct excitation through externally induced pressure gradients at the surface of the bubble). In parallel it was also shown by ? and ? that even if the bubble oscillations remain spherical, some complex coupling between volumetric oscillations and translational motion leading to erratic motion of the bubble can still occur when the bubble is excited by acoustic standing waves of high intensity. Finally, we can mention the work of ? who computed how the viscous drag applied on a translating bubble is modified by its oscillation.

But none of these works considered the effect of the asymmetry of the acoustic wave radiated by a translating source on the acoustic radiation force. In this paper, we consider a monopolar source translating in a quiescent inviscid fluid at a constant velocity U along a fixed axis and demonstrate that the asymmetry of the acoustic field due to Doppler effect (Figure ??) induces a self-induced radiation force on the source resisting its motion.

Figure 1: Sketch illustrating the asymmetry of the acoustic field synthesized by a translating monopolar source. The normalized field is calculated with eq. (??) and for the sake of illustration, the asymmetry is magnified by choosing a Mach number M = 0.5. This result is obtained by inserting the well known solution of the wavefield radiated by a moving monopolar source into a far-field integral expression of the radiation force exerted on a moving source, and finally computing this integral within the approximation of slow translating speed compared to the sound speed.

Wavefield radiated by a translating monopolar source

The first step to compute the self-induced radiation force exerted on a moving monopolar source is to compute the wavefield radiated by this source in a fixed reference frame. This classic calculation can be found in the acoustics textbook of ?. In this first section, we recall the main steps of the derivation. Here we suppose the fluid to be inviscid.

Wave equation for a translating monopolar source

In acoustics, a monopolar source can be seen as a source of mass, whose strength is specified by the instantaneous mass flow rate q(t) created by this source. In the following, the source is supposed to be periodic of period T. For a punctual source translating at a velocity U = U x = M c o x along a fixed axis x, the mass and momentum conservation equations become:

∂ρ ∂t + ∇ • (ρv) = q(t)δ(x -M c 0 t)δ(y)δ(z), (2.1) ∂ρv ∂t + ∇ (ρv ⊗ v) = ∇ σ, (2.2)
with R = (O, (x, y, z), t) a Galilean reference frame, M the mach number, c o the sound speed, ρ the density, v the fluid velocity, σ the stress tensor equal to -p Ī for an inviscid fluid, Ī the identity tensor and p the pressure. If we (i) make the classic asymptotic development of equations (??) and (??) up to first order:

     ρ = ρ 0 + ρ 1 (2.3a) p = p 0 + p 1 (2.3b) v = v o + v 1 (2.3c)
with 1, and obtain the linearized mass and momentum balance:

∂ρ 1 ∂t + ρ 0 ∇ • v 1 = q(t)δ(x -M c 0 t)δ(y)δ(z), (2.4) ρ 0 ∂v 1 ∂t = -∇p 1 , (2.5)
(ii) introduce the sound speed:

c 2 0 = ∂p ∂ρ s = p 1 ρ 1 , (2.6)
where s is the entropy and (iii) combine the time derivative of equation (??) with the divergence of equation (??), we obtain the wave equation:

∆p 1 - 1 c 2 0 ∂ 2 p 1 ∂t 2 = - ∂ ∂t [q(t)δ(x -M c 0 t)δ(y)δ(z)] .
(2.7)

Note that to ease the resolution of this problem, it is convenient to introduce the velocity potential ψ 1 defined by v 1 = -1 ρ0 ∇ψ 1 such that p 1 = ∂ψ1 ∂t , which enables to suppress the time derivative in the rhs of equation (??):

∆ψ 1 - 1 c 2 0 ∂ 2 ψ 1 ∂t 2 = -q(t)δ(x -M c 0 t)δ(y)δ(z), (2.8)
and will make the future change of variables easier.

Resolution of the wave equation and Lorentz transformation

The solution of the wave equation (??) is well-known for a fixed monopolar source (M = 0):

ψ 1 (r, t) = q(t ± r/c 0 ) 4πr , (2.9)
with r = x 2 + y 2 + z 2 the radial distance. To solve the problem for the moving source, the idea is to rewrite equation (??) in a reference frame wherein the source is fixed and the wave equation remains unchanged. This can be achieved by using the invariance of the wave equation by the Lorentz transformation, which is at the core of special relativity:

         x = γ (x -M c 0 t) (2.10a) y = y (2.10b) z = z (2.10c) c 0 t = γ (c 0 t -M x) , (2.10d)
where γ defined by γ -1 = √ 1 -M 2 is the "Lorentz acoustic boost". With this transformation, the wave equation (??) becomes:

∆ ψ 1 - 1 c 2 0 ∂ 2 ψ 1 ∂t 2 = -q (γ(t + M x /c)) δ(x /γ)δ(y )δ(z ).
(2.11)

Since the rhs of equation (??) is null when x = 0 for all t and using δ(x /γ) = γδ(x ), it comes:

∆ ψ 1 - 1 c 2 0 ∂ 2 ψ 1 ∂t 2 = -γq(γt )δ(x )δ(y )δ(z ).
(2.12)

If we now introduce a second set of variables:

         x = γx (2.13a) y = γy (2.13b) z = γz (2.13c) c 0 t = γc 0 t , (2.13d)
the wave equation becomes:

∆ ψ 1 - 1 c 2 0 ∂ 2 ψ 1 ∂t 2 = -γ 2 q(t )δ(x )δ(y )δ(z ), (2.14)
which now resembles the static monopolar source problem and whose solution is:

ψ 1 (r , t ) = γ 2 q(t ± r /c 0 ) 4πr .
(2.15)

If we now perform the inverse transformations to obtain the potential as a function of (x, y, z, t), we obtain:

ψ 1 (r , t ) = γ 2 q(γ(t ± r /c 0 )) 4πγr , ( 2.16)
with:

γ(t ± r c 0 ) = γ c 0 γ(c 0 t -M x) ± [γ(x -M c 0 t)] 2 + y 2 + z 2 (2.17) = t - M (x -M c 0 t) ± (x -M c 0 t) 2 + (y 2 + z 2 )(1 -M 2 ) c 0 (1 -M 2 ) , (2.18)
If we introduce the distance R ± between the emission and the observation points:

R ± = M (x -M c 0 t) ± R 1 1 -M 2 , (2.19) with R 1 = (x -M c 0 t) 2 + (y 2 + z 2 )(1 -M 2 ), (2.20)
then the solution becomes:

ψ 1 (r, t) = q(t -R/c 0 ) 4πR 1 , (2.21)
with R = R + in the subsonic case (M < 1).

Integral expression of the radiation stress in the far field

The next step is to derive a far-field integral expression of the radiation stress exerted on a moving source. Indeed in acoustics, a monopolar point source constitutes a far-field approximation of a real source of finite extent and hence the above expressions are only valid in the far field.

Far field expression of the radiation force for a moving source

The acoustic radiation stress exerted on an object of surface S(t) is by definition the time average of the surface integral of the stress exerted by the acoustic wave on its surface:

F rad = S(t) σndS , ( 3.22)
where f = 1 T t+T t f (t)dt is the time average of the function f and T the period of the function f (t). In general, there are two difficulties when computing this integral: (i) the surface of the object is vibrating and hence depends on time (S = S(t)) and (ii) an expression of the wave scattered by the object in the near field must be known. Hence generally this integral is converted into an integral over a closed surface at rest surrounding the object in the far field by using the divergence theorem and Reynolds transport theorem (see e.g. the review by ? for details of this process). Here an additional difficulty comes from the fact that the particle, in addition to its vibration, is translating at a constant velocity U . To solve this issue, we will transpose our integral of the stress on the surface of the object into an integral over a spherical surface S ∞ of radius r ∞ λ, centered on the source, and hence translating at the velocity U in R (see Figure ??), with λ = c 0 /f the wavelength and f the frequency. The volume between S and S ∞ is named V. The integral of equation (??) over V gives:

V ∂ρv ∂t + ∇ ρv ⊗ v -σ dV = 0. (3.23)
Using the divergence theorem, this volume integral turns into:

V ∂ρv ∂t dV - S ρv ⊗ v -σ ndS + S∞ ρv ⊗ v -σ n ∞ dS = 0, (3.24)
with n and n ∞ the outgoing normal vectors to the surface S and S ∞ respectively (see Figure ??). Another equation can be obtained by applying the Reynolds transport theorem to the momentum density ρv:

d dt V ρvdV = V ∂ρv ∂t dV - S (v • n)ρvdS + S∞ (U • n ∞ )ρvdS, (3.25)
since the surface S follows the object surface displacement (equal to the fluid displacement at the interface due to the continuity condition) and the surface S ∞ is translating at a constant velocity v. If we study the steady regime, then:

d dt V ρvdV = 0. (3.26)
Finally, if we substract the time average of equation (??) to the time average of equation (??) and take into account (??), we obtain:

F rad = S∞ σ -ρv ⊗ v n ∞ dS + S∞ (U • n ∞ )ρvdS .
(3.27)

Expression as a function of the first order acoustic field

In order to compute the previous integrals and since the time average of first order terms are equal to 0 for a periodic signal, we need to express the terms appearing in equation (??) up to second order. Here we suppose again that the fluid is inviscid and hence σ = -p Ī. If we push the asymptotic development up to second order:

     ρ = ρ 0 + ρ 1 + 2 ρ 2 (3.28a) p = p 0 + p 1 + 2 p 2 (3.28b) v = v 1 + 2 v 2 , ( 3.28c)
the momentum balance (??) at second order in the volume V becomes:

ρ 0 ∂v 2 ∂t + ρ 1 ∂v 1 ∂t + ρ 0 (v 1 ∇v 1 ) = -∇p 2 . (3.29)
If we take the time-average of this equation, we obtain:

-∇p 2 = ρ 1 ∂v 1 ∂t + ρ 0 (v 1 ∇v 1 ) , (3.30) since ∂v2 ∂t = 0 in the steady regime. From the first order momentum conservation (??), we have ∂v1 ∂t = -1 ρ0 ∇p 1 and from the state equation (??) we have ρ 1 = p 1 /c 2 0 leading to:

ρ 1 ∂v 1 ∂t = - 1 2ρ 0 c 2 0 ∇p 2 1 . (3.31)
And since the acoustic field is by definition irrotationnal:

ρ 0 (v 1 ∇v 1 ) = ρ 0 2 ∇v 1 2 . (3.32)
Finally, by replacing (??) and (??) in (??), we obtain:

-∇p 2 = - 1 2ρ 0 c 2 0 ∇p 2 1 + ρ 0 2 ∇v 1 2 . (3.33)
From this equation, we obtain the second order averaged stress tensor:

σ2 = -p 2 Ī = ρ 0 v 2 1 2 - 1 ρ 0 c 2 0 p 2 1 2 Ī . (3.34)
Then at second order, we have:

ρv ⊗ v = ρ 0 v 1 ⊗ v 1 (3.35) ρv = ρ 1 v 1 , (3.36)
so that if we replace equations (??), (??) and (??) in (??), we obtain the final expression of the radiation force as a function of the first order acoustic fields:

F rad = S∞ ρ 0 v 2 1 2 - 1 ρ 0 c 2 0 p 2 1 2 Ī -ρ 0 v 1 ⊗ v 1 ndS + S∞ (U • n ∞ )ρ 1 v 1 dS .
(3.37)

Expression of the radiation force for a slowly translating source

The final step is to compute this integral from the first order acoustic field.

Pressure, density and velocity fields at first order

The first step to compute equation (??) is to determine the pressure, density and velocity fields at first order from the expression of the velocity potential previously (equation (??)). First,

p 1 = ∂ψ 1 ∂t = (1 -1 c0 ∂R ∂t )q (t -R/c 0 ) 4πR 1 - q(t -R/c 0 ) ∂R1 ∂t 4πR 2 1 . (4.38)
Moreover, the coordinate of the velocity v 1i along the direction x i = (x, y, z) is:

v 1xi = - 1 ρ 0 ∂ψ 1 ∂x i = - 1 ρ 0 -1 c0 ∂R ∂xi q (t -R/c 0 ) 4πR 1 - q(t -R/c 0 ) ∂R1 ∂xi 4πR 2 1 . (4.39)
Of course, we have:

ρ 1 = p 1 /c 2 0 .
Here R and R 1 are time and space dependent functions. To simplify the calculation of the derivatives and the force integral, we perform the change of variable corresponding to the Galilean transformation from R to R * :

         x * = x -M c 0 t (4.40a) y * = y (4.40b) z * = z (4.40c) t * = t (4.40d)
R 1 and R become:

R 1 = x * 2 + (y * 2 + z * 2 )(1 -M 2 ) and R = M x * + R 1 1 -M 2 . (4.41)
The Jacobian matrix J * of this transformation is:

J * =     1 0 0 -M c 0 0 1 0 0 0 0 1 0 0 0 0 1     with det J * = 1. (4.42)
We use the spherical coordinates (r * , θ * , ϕ * ) to compute the integral (see Figure ?? for the notations):

     x * = r * cos θ * (4.43a) y * = r * sin θ * cos ϕ * (4.43b) z * = r * sin θ * sin ϕ * . (4.43c)
The key point in the following derivation of the self radiation force is that we consider M 1, which means that the source translates at low velocity compared to the sound speed. This enables us to get simpler expressions of the pressure and velocity fields. First, we derive the expressions of R 1 and R at the first order in M :

R 1 = r * 2 -M 2 (y * 2 + z * 2 ) = r * 2 -M 2 r * 2 sin 2 θ * r * 1 - M 2 2 sin 2 θ * r * + O(M ), (4.44) and R M x * + r * 1 -M 2 2 sin 2 θ * 1 -M 2 r * + M x * . (4.45)
In order to get the velocity and the pressure fields at the first order in M , we compute at first the different time derivatives of R 1 and R: (4.46) with:

∂R ∂t = ∂x * ∂t ∂R ∂x * + ∂t * ∂t ∂R ∂t * = -M c 0 ∂r * ∂x * + M ,
∂r * ∂x * = x * r * = cos θ * . (4.47)
Hence we obtain:

∂R ∂t = -M c 0 (cos θ * + M ) . (4.48) Then: ∂R 1 ∂t = ∂x * ∂t ∂R 1 ∂x * + ∂t * ∂t ∂R 1 ∂t * = -M c 0 cos θ * . (4.49)
Finally, the acoustic pressure is:

p 1 = [1 + M (cos θ * + M )] q (t -R/c 0 ) 4πr * + q(t -R/c 0 )(M c 0 cos θ * ) 4πr * 2 . (4.50)
We choose a sphere S ∞ with a radius r ∞ huge compared to all the other lengths of the problem. We then write the acoustic pressure in the far field approximation. At first order in M we obtain:

p 1 = 1 + M cos θ * 4πr * q (t -R/c 0 ). (4.51)
In order to derive the velocity field we compute the different space derivatives of R 1 and R:

               ∂R ∂x = ∂x * ∂x ∂R ∂x * = cos θ * + M (4.52a) ∂R ∂y = ∂y * ∂y ∂R ∂y * = sin θ * cos ϕ * (4.52b) ∂R ∂z = ∂z * ∂z ∂R ∂z * = sin θ * sin ϕ * , ( 4.52c)
and

∂R 1 ∂xi = x * i r * . (4.53)
We obtain:

                 v 1x * = 1 ρ 0 c 0 (cos θ * + M )q (t -R/c 0 ) 4πr * + c 0 q(t -R/c 0 ) 4πr * 2 x * r * (4.54a) v 1y * = 1 ρ 0 c 0 sin θ * cos ϕ * q (t -R/c 0 ) 4πr * + c 0 q(t -R/c 0 ) 4πr * 2 y * r * (4.54b) v 1z * = 1 ρ 0 c 0 sin θ * sin ϕ * q (t -R/c 0 ) 4πr * + c 0 q(t -R/c) 4πr * 2 z * r * . (4.54c)
We finally write the velocity field in the far field approximation:

v 1x * = 1 ρ 0 c 0 q (t -R/c 0 ) 4πr * (cos θ * + M ) v 1y * = 1 ρ 0 c 0 q (t -R/c 0 ) 4πr * sin θ * cos ϕ * v 1z * = 1 ρ 0 c 0 q (t -R/c 0 ) 4πr * sin θ * sin ϕ * . (4.55) (4.56) (4.57)

Calculation of the integrals

With the change of variables (??) and since the Jacobian of this change of variable is equal to 1, the integral expression of the radiation force (??) becomes:

F rad = S∞ * ρ 0 v 2 1 2 - 1 ρ 0 c 2 0 p 2 1 2 Ī -ρ 0 v 1 ⊗ v 1 n * dS * + S∞ * U • n ∞ * )ρ 1 v 1 dS * , (4.58) with S ∞ * the surface defined by r * = r * ∞ a constant radius, the infinitesimal element of surface dS * = r * ∞ 2 sin θ * dθ * dϕ * , θ * ∈ [0, π] and ϕ * ∈ [0, 2π]. Finally, n * = n ∞ * = e * r , with (e *
r , e * θ , e * ϕ ) the spherical coordinates unit vector of R * . Since we have already expressed the pressure, velocity and density fields as a function of the spherical coordinates (r * , θ * , ϕ * ), we can now compute integral (??). In the following paragraphs, we compute separately each term of this integral.

Potential energy term

S∞ * 1 ρ 0 c 2 0 p 2 1 2 n * dS * = 1 2ρ 0 c 2 0 2π ϕ * =0 π θ * =0 1 + M cos θ * 4πr * ∞ q (t -R/c 0 ) 2 e * r r * ∞ 2 sin θ * dθ * dϕ * (4.59)
The integration over ϕ * along the y * and z * axis is null. Hence only the term along x * remains:

S∞ * 1 ρ 0 c 2 0 p 2 1 2 n * dS * = π ρ 0 c 2 0 π 0 1 + M cos θ * 4πr * ∞ q (t -R/c 0 ) 2 cos θ * sin θ * dθ * x .
(4.60) If we swap the time and space integrals and since the function q(t) is periodic, we obtain:

S∞ * 1 ρ 0 c 2 0 p 2 1 2 n * dS * = q (t) 2 16πρ 0 c 2 0 π 0 (1 + M cos θ * ) 2 cos θ * sin θ * dθ * x. (4.61)
Since the last integral is equal to 4M/3, we obtain:

S∞ * 1 ρ 0 c 2 0 p 2 1 2 n * dS * = q 2 M 12πρ 0 c 2 0 x. (4.62) 4.2.2. Kinetic energy term S∞ * ρ 0 v 2 1 2 n * dS * = 1 2ρ 0 c 2 0 2π ϕ * =0 π θ * =0 q (t -R/c 0 ) 4πr * ∞ 2 (cos θ * + M ) 2 + sin 2 θ * e * r r * ∞ 2 sin θ * dθ * dϕ * (4.63)
Using the same arguments than previously we have:

S∞ * ρ 0 v 2 1 2 n * dS * = q 2 16πρ 0 c 2 0 π 0 1 + 2M cos θ * + M 2 cos θ * sin θ * dθ * x (4.64)
Since the last integral is equal to 4M/3, we obtain:

S∞ * ρ 0 v 2 1 2 n * dS * = q 2 M 12πρ 0 c 2 0 x (4.65)

Convective term

Due to the symmetry of the problem (invariance by rotation over ϕ * angle), no force can exist along y and z directions. Hence, we only need to compute the following components of v 1 ⊗ v 1 :

(v 2 1x * )x ⊗ x + (v 1x * v 1y * )x ⊗ y + (v 1x * v 1z * )x ⊗ z (4.66)
Then due to the dependence of these terms over ϕ * given in equations (??) to (??) and since e * r = cos θ * x + sin θ * cos ϕ * y + sin θ * sin ϕ * y, we obtain:

S∞ * [ρ 0 v 1 ⊗ v 1 ]n * dS * = q 2 8πρ 0 c 2 0 π 0 (cos θ * + M ) 2 cos θ * + (cos θ * + M ) sin θ * 2 sin θ * dθ * x (4.67)
The integral term is equal to 8M/3, so that:

S∞ * [ρ 0 v 1 ⊗ v 1 ]n * dS * = q 2 M 3πρ 0 c 2 0 x (4.68)

Source translation term

The term due the translation of the sphere S ∞ is:

S∞ * (U • n * ∞ )ρ 1 v 1 dS * = M q 2 8πρ 0 c 2 0 π 0 (1 + M cos θ * )(cos θ * + M ) cos θ * sin θ * dθ * x
Since at leading order, the last integral is equal to 2/3, we finally obtain:

S∞ * (U • n ∞ )ρ 1 v 1 dS * = q 2 M 12πρ 0 c 2 x (4.69)

Final expression of the self radiation force and discussion

Final expression of the self-radiation force

If we now replace (??), (??), (??) and (??) in (??), we obtain the final expression of the self acoustic radiation force exerted on a monopolar source:

F rad = - q 2 M 4πρ 0 c 2 0 x.
(5.70)

There are many interesting things to notice in the above calculations and final expression. First, we see that the potential and kinetic energy terms cancel at leading order in M, so that the self radiation force is solely due to the convective and translation terms. Second, the radiation force is proportional to the radiated intensity, and inversely proportional to the sound speed square, which is classical in radiation force calculations. In addition, here the self-radiation force is proportional to the hydrodynamic Mach number M , which is expected since the force results from the asymmetry of the radiated field due to the translation of the source. Finally and most importantly F rad • U is always negative, which means that this force always slows down the movement of the bubble.

An example of a monopolar oscillator: a vibrating bubble

In the following section, we estimate this force for a translating and oscillating bubble in a liquid, which constitutes one example of acoustic monopolar source. Indeed, bubbles are exceptional resonators, which exhibit strong monopolar resonances in the long wavelength regime. Let's consider a spherical bubble of mean radius r b in a liquid of density ρ 0 and sound speed c 0 vibrating periodically at its resonance frequency, called the Minnaert frequency:

ω M = 1 r b 3γp 0 ρ 0 , (5.71)
with γ the heat capacity ratio of the gas in the bubble and p 0 the pressure of the surrounding fluid. At resonance, λ/r b = 2πc0 ω M r b = 2πc 0 ρ0 3γp0 1 (of the order of 5 × 10 2 for an air bubble in water). Since the bubble is very small compared to the wavelength in this case, it can be considered as a point source a few wavelengths away from the bubble surface. The oscillation of this bubble creates a periodic mass flux q(t) = Q cos(ω M t), whose magnitude Q is basically equal to the surface of the bubble 4πr 2 b , times the surrounding liquid mass density ρ 0 , times the amplitude of the oscillations αr b (where α designates a dimensionless parameter fixing the magnitude of the bubble oscillation), times the pulsation ω b :

Q ∼ 4πr 2 b ρ 0 αr b 1 r b 3γp 0 ρ 0 = 4πr 2 b α 3γp 0 ρ 0 .
(5.72)

Consequently, we have: (5.73) and:

q 2 ∼ 1 2 ω 2 M Q 2 = 72(πγp 0 αr b ) 2 ,
| F rad | = q 2 4πρ 0 c 3 0 U ∼ 18πγ 2 p 2 0 α 2 r 2 b ρ 0 c 3 0 U.
(5.74)

For small bubbles, it is interesting to compare this radiation force to the Stokes drag: (5.76)

|F Sto | = Cπµr b U, ( 5
For a bubble rising in water at ambient temperature and pressure, we have ρ 0 ∼1 × 10 3 kg m -3 , µ ∼1 × 10 -3 Pa s, c 0 ∼1.5 × 10 3 m s -1 , γ ∼ 1.4, p 0 =1 × 10 5 Pa, so that with α ∼ 0.5, we obtain:

| F rad | |F Sto | ∼ 7r b .
(5.77)

Hence, the self radiation force would be small compared to the Stokes drag for a millimetric bubble. However the self radiation force could become significant in cryogenic liquids such as liquid nitrogen or superfluid helium. In liquid nitrogen at T 77 K, ρ 0 =8 × 10 2 kg m -3 , µ =2 × 10 -4 Pa s, c 0 =8 × 10 2 m s -1 , γ ∼ 1, p 0 =1 × 10 5 Pa, so that for α ∼ 0.5 we obtain: (5.78) which means that for a bubble of a few millimeters in radius, the two phenomena would be of the same orders of magnitude. Note first that this calculation constitutes a rough comparison of the radiation force and Stokes drag since (i) for large bubble oscillations, the bubble dynamics becomes nonlinear, and (ii) the Stokes drag is also modified by the bubble oscillations as demonstrated by ?. Note also that the case of bubbles moving in a liquid constitutes just one possibility over the many configurations covered by equation (??) since the above theory applies for an arbitrary monopolar source moving in an arbitrary fluid as soon as (i) the monopolar source emits a signal in the long wavelength regime and (ii) that the source is moving at low speed compared to the sound speed.

| F rad | |F Sto | ∼ 140r b ,

Conclusion

In this paper, we calculated the theoretical expression of the radiation force exerted on a translating monopolar source by its own acoustic field. We showed that the asymmetry of the radiated field due to the motion of the source creates a self-induced radiation force opposite to its motion. This first calculation opens many perspectives to calculate this effect for different types of sources, beyond the long wavelength regime and potentially unveil situations wherein this force is strong and could be in the opposite direction leading to a propulsive force, and hence with a scatterer surfing on its own acoustic field.

Figure 2 :

 2 Figure2: S represents the source surface, varying over time. The surface S ∞ is centered on the source and moves with it at the velocity U in R. R * is the frame of the source.

Figure 3 :

 3 Figure 3: We make the change of variables corresponding to the Galilean transformation from R to R * and then use the local spherical coordinates (r * , θ * , ϕ * ).

  .75) with C = 4 for a bubble in a pure liquid moving at low Reynolds number, C = 6 if the surface is polluted so that the slip boundary condition is turned into a no slip boundary condition (see e.g. the book ?) and C = 12 for an undeformed bubble at large Reynolds numbers (see ?). If we compare equation (??) to (??), we obtain:

† Email address for correspondence: michael.baudoin@univ-lille.fr

Acknowledgements

We acknowledge support from ISITE ERC Generator program and stimulating discussions with Pr. J. Bush, which motivated us to perform this work.

Declaration of Interests

The authors report no conflict of interest.


Tarek Ahmed-Ali
email: tarek.ahmed-ali@ensicaen.fr

Iasson Karafyllis

Miroslav Krstic
email: krstic@ucsd.edu

Francoise Lamnabhi-Lagarrigue

Robust Stabilization of Nonlinear Globally Lipschitz Delay Systems

This paper studies the application of a recently proposed control scheme to globally Lipschitz nonlinear systems for which the input is delayed and applied with zero order hold, the measurements are sampled and delayed, and only an output is measured (i.e., the state vector is not available). The control scheme consists of an observer for the delayed state vector, an inter-sample predictor for the output signal, an approximate predictor for the future value of the state vector, and the nominal feedback law applied with zero order hold and computed for the predicted value of the future state vector. The resulting closed-loop system is robust with respect to modeling and measurement errors and robust to perturbations of the sampling schedule.

Introduction

Predictor feedback is used frequently in the literature for systems with large input delays. The literature on predictor feedback under non-constant input delays is reviewed in [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF][START_REF] Bekiaris-Liberis | Nonlinear Control Under Nonconstant Delays[END_REF] (where time-varying delays and state-dependent input delays are studied). For nonlinear systems with constant input delays, various forms of predictors have been used:

• Exact predictors based on the knowledge of the solution mapping [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF][START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF][START_REF] Krstic | Input delay compensation for forward complete and strict-feedforward nonlinear systems[END_REF].

• Approximate predictors based on successive approximations [START_REF] Karafyllis | Stabilization by means of approximate predictors for systems with delayed input[END_REF][START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF].

• Approximate predictors based on numerical schemes [START_REF] Karafyllis | Numerical schemes for nonlinear predictor feedback[END_REF][START_REF] Karafyllis | Predictor-based tracking for neuromuscular electrical stimulation[END_REF].

• Approximate predictors for which the prediction is provided by the output of a properly constructed control system [START_REF] Ahmed-Ali | Global exponential sampled-data observers for nonlinear systems with delayed measurements[END_REF][START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF][START_REF] Karafyllis | Global stabilization of nonlinear delay systems with a compact absorbing set[END_REF].

Moreover, the literature on predictor feedback for nonlinear systems with constant input delays has considered cases where the input is applied continuously (as in [1-3, 6, 8, 12], or [START_REF] Karafyllis | Predictor-based tracking for neuromuscular electrical stimulation[END_REF]) or is applied with zero order hold or ZOH (as in [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF] and [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF]). There is also a wide literature of predictor feedback design and implementation for linear systems with constant input delays; see the references in [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF] and [START_REF] Krstic | Input delay compensation for forward complete and strict-feedforward nonlinear systems[END_REF].

This chapter considers the application of a recently proposed control scheme to globally Lipschitz nonlinear systems for which the input is applied with ZOH, the measurements are sampled and delayed, and only an output is measured (since the state vector is not available). Moreover, we also consider the effect of possible modeling errors and measurement noise. The control scheme consists of an observer for the delayed state vector, an inter-sample predictor for the output signal, an approximate predictor for the future value of the state vector, and the nominal feedback law applied with ZOH and computed for the predicted value of the future state vector. The control scheme has been applied to globally Lipschitz nonlinear systems previously (as in [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF]) but in this work we have used a different prediction action, namely, we are using approximate predictors for which the prediction is provided by the output of a properly constructed control system (namely, dynamic approximate predictors) instead of predictors that are based on successive approximations. The chapter generalizes the results provided in [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF] to various directions:

• We show that the convergence is independent of the lower diameter of the sampling schedule (in contrast with [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF], where the estimates depended on the lower diameter of the sampling schedule). • We provide assumptions which can be applied to general nonlinear globally Lipschitz systems (in contrast with [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF], where only triangular single input systems were considered). • We provide explicit formulae for the asymptotic gains of various inputs (in contrast with [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF], where only qualitative estimates were provided). • We provide explicit inequalities for the upper diameter of the sampling partition and the holding period, which can be used in straightforward way by the potential control practitioner.

The application of the proposed control scheme guarantees robustness with respect to modeling errors, measurement noise and perturbations of the sampling schedule.

Prior to the submission of the present chapter, we were informed of the work [START_REF] Sun | Predictive control of nonlinear continuous networked control systems with large time-varying transmission delays and transmission protocols[END_REF]. The results in [START_REF] Sun | Predictive control of nonlinear continuous networked control systems with large time-varying transmission delays and transmission protocols[END_REF] also deal with globally Lipschitz systems using the control scheme proposed in [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF]. The results [START_REF] Sun | Predictive control of nonlinear continuous networked control systems with large time-varying transmission delays and transmission protocols[END_REF] cover various cases of transmission protocols and generalized the results of [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF] to non-triangular globally Lipschitz systems.

The results of the present paper are less conservative than the results in [START_REF] Sun | Predictive control of nonlinear continuous networked control systems with large time-varying transmission delays and transmission protocols[END_REF] and the main difference between the present work and the results of [START_REF] Sun | Predictive control of nonlinear continuous networked control systems with large time-varying transmission delays and transmission protocols[END_REF]i st h eu s eo f dynamic approximate predictors instead of predictors which are based on successive approximations.

Notation

Throughout this chapter, we adopt the following notation:

• For a vector x ∈ R n we denote by |x| its usual Euclidean norm, by x ′ its transpose. For a real matrix A ∈ R n×m , A ′ ∈ R m×n denotes its transpose and |A| = sup {|Ax| ; x ∈ R n , |x| = 1 } is its induced norm. I ∈ R n×n denotes the identity matrix. • R + denotes the set of non-negative real numbers. For every t ≥ 0, [t] denotes the integer part of t ≥ 0, i.e., the largest integer being less or equal to t ≥ 0. A partition

π = {T i } ∞ i=0 of R + is an increasing sequence with T 0 = 0 and T i →+∞. • Let x :[a -r, b) → R n with b > a ≥ 0 and r ≥ 0. By x t we denote the "history" of x from t -r to t, i.e., (x t ) (θ ) = x(t + θ) ; θ ∈[-r, 0],fort ∈[a, b).By xt we denote the "open history" of x from t -r to t, i.e., ( xt
) (θ ) = x(t +θ) ; θ ∈[-r, 0), for t ∈[a, b). • Let I ⊆ R + be an interval. By L ∞ (I ; U ) (L ∞ loc (I ; U ))
we denote the space of measurable and (locally) bounded functions u( • ) defined on I and taking values in U ⊆ R m . Notice that we do not identify functions in L ∞ (I;U) which differ on a measure zero set. For

L ∞ ([-r, 0]; R n ) or x ∈ L ∞ ([-r, 0); R n ) we define x = sup θ ∈[-r,0] |x(θ )| or x = sup θ ∈[-r,0) |x(θ )|. Notice that sup θ ∈[-r,0] |x(θ )|
is not the essential supremum but the actual supremum. By PC(I ; R m ) we denote the space of piecewise continuous functions u( • ) defined on I and taking values in R m .

• By C 0 (A; Ω), where A ⊆ R n and Ω ⊆ R m , we denote the class of continuous functions taking values in Ω ⊆ R m . A continuous mapping F :

C 0 ([-r, 0]; R l ) × R m → R n is
→ R + such that |F(x, u) -F(y, u)| ≤ Q ( x + y + |u|) x -y for all x, y ∈ C 0 ([-r, 0]; R l ) and for all u ∈ R m .

Statement of Main Results

We consider a time-invariant control system of the form

ẋ(t) = f (x(t), u(t -τ)) + Gv(t) x ∈ R n , u ∈ R m , v ∈ R q , t ≥ 0( 1 )
where f : R n × R m → R n is a continuous vector field with f (0, 0) = 0, τ>0isa constant, and G ∈ R n×q is a real constant matrix. The input v(t) ∈ R q quantifies the effect of possible modeling errors. We assume that the following assumptions hold for system (1).

(H1) There exist a continuous mapping k : R n → R m with k(0) = 0, a constant µ>0 and a symmetric, positive definite matrix P ∈ R n×n such that the following inequality holds for all x ∈ R n :

x ′ Pf(x, k(x)) ≤-4µ |x| 2 (2)
(H2) There exist constants L 1 , L 2 , K ≥ 0 such that the following inequalities hold for all x, y ∈ R n , u, v ∈ R m :

|k(x) -k(y)| ≤ K |x -y| (3) | f (x, u) -f (y, u)| ≤ L 1 |x -y| (4) | f (x, u) -f (x, v)| ≤ L 2 |u -v| (5)
(H3) There exist matrices L ∈ R n× p , H ∈ R p×n , a constant ω>0 and a symmetric, positive definite matrix Q ∈ R n×n such that the following inequality holds for all x, e ∈ R n , u ∈ R m :

e ′ Q ( f (x + e, u) -f (x, u) + LHe) ≤-2ω |e| 2 (6)
Discussion of the assumptions: Assumption (H1) guarantees that the "continuously applied" feedback law u(t) = k(x(t)) would globally exponentially stabilize the equilibrium point 0 ∈ R n of system (1) if the input delay τ were absent, i.e., if τ = 0. Assumption (H2) guarantees that both the "nominal" feedback law k : R n → R m and the mapping f : R n × R m → R n are globally Lipschitz mappings. Assumption (H3) guarantees that the system ż(t) = f (z(t), u(tτ)) + L (Hz(t)y(t)) would be a global exponential observer for system [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF] provided that the output y(t) = Hx(t) were available for all t ≥ 0 and that no modeling errors were present. System (1) under assumptions (H1), (H2), and (H3) would be globally exponentially stabilized by the dynamic output feedback law ż(t) = f (z(t), u(t)) + L (Hz(t)y(t)) with u(t) = k(z(t)) if (a) the input delay τ were absent, (b) the input u(t) were allowed to be continuously adjusted, (c) no modeling errors were present, and (d) the output y(t) = Hx(t) were available for all t ≥ 0. In this work, we will assume that none of the previous requirements hold. More specifically, we assume that:

• The output measurement is sampled, corrupted and delayed, i.e., there is a partition

{τ i } ∞ i=0 of R + with sup i≥0 (τ i+1 -τ i ) ≤ T s where T s > 0 is a constant, an input ξ ∈ L ∞ loc (R + ; R p ) and a constant r ≥ 0 so that y(t) = Hx(τ i ) + ξ(τ i ), for all t ∈[ τ i ,τ i+1
), i = 0, 1, 2, .... The number T s > 0 is called the upper diameter of the partition and is known, while the sampling times {τ i } ∞ i=0 are not known. • The input delay τ is present and modeling errors are present as well.

• The input cannot be continuously adjusted and can only be applied with ZOH, i.e., there exists a constant T H > 0 (the holding period) such that u(t) = u l ∈ R m ,for all t ∈[lT H ,(l + 1)T H ), l = 0, 1, 2, ....

One (or all) of the above complications are present when the control system is networked and there are communication or computation delays along the operation of the network; see the discussion in [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF]. Furthermore, the existence of sampled, corrupted and delayed measurements is common for (bio)chemical processes and the inability of continuous adjustment of the input is also common for many systems. Our main result is given next.

Theorem 1 Consider system (1) under assumptions (H1), (H2), and (H3). Let

T s > 0,T H > 0, and r ≥ 0 be real constants and N > 0 be an integer that satisfy the inequalities

exp(L 1 T H )(L 1 +L 2 K )T H 1-exp(L 1 T H )(L 1 +L 2 K )T H |P| L 2 K ≤ 2µ and K 4 2K 3 |QL||H | L 1 T s <ω,L 1 (r + τ) < N (7)
where K 3 and K 4 > 0 are constants that satisfy K 3 |x| 2 ≤ x ′ Qx ≤ K 4 |x| 2 for all x ∈ R n . Then for every c > 0, there exist constants θ>0 and Θ>0 such that for every partition

{τ i } ∞ i=0 of R + satisfying sup i≥0 (τ i+1 -τ i ) ≤ T s (8)
and for all choices of

x 0 ∈ C 0 ([-r, 0]; R n ),z j,0 ∈ C 0 ([-δ, 0]; R n ) (j = 0, ..., N), ȗ0 ∈ L ∞ ([-r -τ, 0); R m ),v∈ L ∞ loc (R + ; R q ), and ξ ∈ L ∞ loc (R + ; R p ), the solution (x(t), z 0 (t), ..., z N (t), u(t), w(t)) ∈ R n × R Nn × R m × R p (9)
of the system (1) with the choices

ż0 (t) = f (z 0 (t), u(t -τ -r )) + L (Hz 0 (t) -w(t)) , t ≥ 0( 1 0 ) ż j (t) =ż j-1 (t) + f (z j (t), u(t + jδ -r -τ)) -f (z j (t -δ), u(t + ( j -1)δ -r -τ)) -c z j (t) -z j-1 (t) - t t-δ f (z j (s), u(s + jδ -r -τ))ds , t ≥ 0, j = 1, ..., N (11) ẇ(t) = Hf(z(t), u(t -τ -r )), t ∈[τ i ,τ i+1 ), i = 0, 1, 2, ... ( 12
)
w(τ i ) = Hx(τ i -r ) + ξ(τ i ), i = 0, 1, 2, ... (13)
u(t) = k (z N (lT H )) , t ∈[lT H ,(l + 1)T H ), l = 0, 1, 2, ... ( 14
)
and δ = (r + τ)/N , and with the initial conditions x(s)

= (x 0 )(s) for s ∈[ -r, 0], z j (s) = (z j,0 )(s) for s ∈[ -δ, 0], and u(s) = ( ȗ0 )(s) for s ∈[ -r -τ, 0) corre- sponding to any inputs v ∈ L ∞ loc (R + ; R q ) and ξ ∈ L ∞ loc (R + ; R p )
, is unique, exists for all t ≥ 0 and satisfies the following estimates:

|x(t)| ≤ K 2 K 1 exp (-θ(t -τ)) sup 0≤s≤τ (|x(s)|) +Ξ 1 Ξ 2 ΩΛ N |QL| exp(θ (τ +T H )) 1-g sup 0≤s≤t (|ξ (s)|) +Ξ 1 |PG| + |G||P| C + Ω exp (θ T H ) Λ N +1 -Λ Λ-1 |G| exp (θτ) sup 0≤s≤t (|v(s)|) +Ξ 1 Ξ 2 ΩΛ N exp(θ (τ +T H )) 1-g (|QG| × exp(-θ r ) + |QL||HG| T s ) sup 0≤s≤t (|v (s)|) +Ξ 1 Ω exp (-θ(t -τ -T H )) Λ N +1 -Λ Λ-1 L 2 δ ȗ0 + 3m a x l=1,...,N z l,0 +Ξ 1 Ω exp (-θ(t -r -τ -T H )) sup 0≤s≤r (|x (s + τ ) -z N (s)|) +Ξ 1 Ω exp (-θ(t -r -τ -T s -T H )) Λ N 1-g K 4 K 3 sup r ≤s≤r +T s (|x (s -r ) -z 0 (s)|) +Ξ 1 Ω exp (-θ(t -r -τ -T H )) Λ N -1 Λ-1 × max j=1,...,N sup -δ≤s≤r
x (sr + jδ)z j (s) (15) and

x t + N j=0 z j,t + ȗt ≤ Θ exp (-θ t) x 0 + N j=0 z j,0 + ȗ0 + sup 0≤s≤t (|v(s)|) + sup 0≤s≤t (|ξ (s)|) , (16) where
Ω = |P| L 2 K (1 + C), Ξ 1 = K 2 2µK 1 (µ-θ K 2 ) , Ξ 2 = K 4 2ωK 3 (ω-θ K 4 ) ,Λ= θ θ -L 1 (exp(θ δ)-1) , g = K 4 2ωK 3 (ω-θ K 4 ) |QL||H | L 1 exp(θ T s )-1 θ
, and

C = exp(L 1 T H )L 2 KT H 1-exp(L 1 T H )(L 1 +L 2 K )T H . (17)
Inequality ( 16) guarantees the input-to-state stability property (as defined in [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF]) for the closed-loop system given by ( 1), ( 10)-( 14) with respect to modeling errors and measurement noise. More specifically, estimate (16) shows that the gain function for the external inputs v and ξ is linear. On the other hand, inequality (15)allowusto estimate explicitly the asymptotic gains of the external inputs v and ξ to the output Y (t) = x(t): the asymptotic gain of the modeling error v is guaranteed to be any number greater than

Ξ 1 |PG| + |G||P| C + Ω Λ N +1 -Λ Λ -1 |G| + Ξ 2 ΩΛ N 1 -g |QG| + Ξ 2 ΩΛ N 1 -g |QL||HG| T s
and the asymptotic gain of the measurement error ξ is guaranteed to be any number greater than

Ξ 1 Ξ 2 ΩΛ N |QL| 1 -g ,
where

Ω = |P| L 2 K (1 + C), Ξ 1 = 1 µ K 2 2K 1 ,Ξ 2 = 1 ω K 4 2K 3 ,Λ= 1 1-L 1 δ , g = Ξ 2 |QL||H | L 1 T s , and C = exp(L 1 T H )L 2 KT H 1-exp(L 1 T H )(L 1 +L 2 K )T H . ( 18
)
Robustness to perturbations of the sampling schedule is also guaranteed.

Estimates [START_REF] Heemels | Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance[END_REF] and ( 16) are independent of the lower diameter of the sampling partition (i.e., of inf i≥0 (τ i+1τ i ). This feature is in sharp contrast with the result in [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF]. This difference is explained by a different methodology in the proof; if the same methodology were followed in [START_REF] Karafyllis | Stabilization of nonlinear delay systems using approximate predictors and high-gain observers[END_REF] then a similar result would be proved. A few words are needed for the explanation of the hybrid dynamic feedback given by ( 10)-( 14).

• (10) is an observer for the delayed state vector x(tr ). However, [START_REF] Ahmed-Ali | Global exponential sampled-data observers for nonlinear systems with delayed measurements[END_REF] does not use the continuous signal y(t) = Hx(tr ), which is not available. The signal w(t) replaces the output signal y(t) = Hx(tr ). • (12), ( 13) is an inter-sample predictor for the non-available output signal; it uses the output values at the sampling times and "tries" to predict the output signal between two consecutive sampling times. • System ( 11) is an approximate predictor of the future value of the state vector x(t + τ). The approximate predictor uses the estimated value z 0 (t) of the delayed state vector x(t -r ), which is provided by the observer, and provides z N (t), which is an approximation of x(t + τ). • Finally, ( 14) is the "nominal" feedback law computed at the predicted value of the future state vector x(t + τ) applied with ZOH (emulation).

Remarks:

• Contrary to the approach in [START_REF] Heemels | Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance[END_REF], no tradeoff between the delays (in the input and the output) and the upper diameter of the sampling partition and the holding period is present for our control scheme. • From (7), we see that for long delays, we can use sufficiently many predictors that ensure the robustness properties [START_REF] Heemels | Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance[END_REF] and ( 16). The counterpart of this is that the gains corresponding to measurement and modeling errors will increase.

The proof of Theorem 1 is in the following section and is demanding because even the existence/uniqueness of the solution of the overall closed-loop system is not trivial, since the closed-loop system is a hybrid system with delays. For the proof of Theorem 1, we need the following two lemmas, which are stated below. Their proofs are simple and are omitted.

Lemma 1 Let r ≥ 0 be a constant and F : C 0 ([-r, 0]; R n ) × R m → R n be a continuous mapping that is Lipschitz on bounded sets and satisfies the inequal- for t ∈[ t 0 , b) a.e., where t 0 ≥ 0,b∈ (t 0 , +∞],c> 0, and a j ≥ 0 (j = 1, ..., m) are constants, p j ≥ 1 are integers and v j ∈ L ∞ loc (R + ; R p j ) (j = 1, ..., m) are measurable and locally essentially bounded functions. Then for every µ ∈ (0, c/K 2 ) and for every t ∈[t 0 , b), the estimate

ity |F(x, u)| ≤ L x + M |u| for all (x, u) ∈ C 0 ([-r, 0]; R n ) × R m for cer- tain constants L > 0 and M ≥ 0. Then for every t 0 ≥ 0,b ∈ (t 0 , +∞), x 0 ∈ C 0 ([-r, 0]; R n ), and u ∈ L ∞ ([t 0 -r, b); R m ),
sup t 0 ≤s≤t (|x(s)| exp (µs)) ≤ K 2 K 1 |x(t 0 )| exp (µt 0 ) + K 2 K 1 (c-µK 2 ) m j=1 √ a j sup t 0 ≤s≤t v j (s) exp (µs) (21)
holds.

Key Lemmas 5 Proof of Theorem 1

We divide the proof into the following three parts:

Part I: Existence and Uniqueness of Solutions

In this part of the proof, we show that for every

x 0 ∈ C 0 ([-r, 0]; R n ), z j,0 ∈ C 0 ([-δ, 0]; R n ) ( j = 0, ..., N ), ȗ0 ∈ L ∞ ([-r -τ, 0); R m ), v ∈ L ∞ loc (R + ; R q
), and ξ ∈ L ∞ loc (R + ; R p ), the closed-loop system given by ( 1), ( 10)-( 14) has a unique solution with initial condition x(s) = (x 0 )(s) for s ∈[r, 0], z j (s) = (z j,0 )(s) for s ∈[δ, 0], and u(s) = ( ȗ0

)(s) for s ∈[ -r -τ, 0) corresponding to inputs v ∈ L ∞ loc (R + ; R q ) and ξ ∈ L ∞ loc (R + ; R p ),
and defined for all t ≥ 0. Part II: Proof of [START_REF] Heemels | Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance[END_REF] In this part of the proof, we show that estimate [START_REF] Heemels | Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance[END_REF] holds for an appropriate constant θ>0.

Part III: Proof of (16)

In this part of the proof, we show that estimate (16) holds.

Part I: Existence and Uniqueness of Solutions

First we prove that for every

x 0 ∈ C 0 ([-r, 0]; R n ), z j,0 ∈ C 0 ([-δ, 0]; R n ) ( j = 0, ..., N ), ȗ0 ∈ L ∞ ([-r -τ, 0); R m ), v ∈ L ∞ loc (R + ; R q ),
and ξ ∈ L ∞ loc (R + ; R p ), the closed-loop system given by ( 1)-( 14) has a unique solution with initial condition

x(s) = (x 0 )(s) for s ∈[ -r, 0], z j (s) = (z j,0 )(s) for s ∈[ -δ, 0], and u(s) = ( ȗ0 )(s) for s ∈[ -r -τ, 0) corresponding to inputs v ∈ L ∞ loc (R + ; R q ) and ξ ∈ L ∞ loc (R + ; R p ).
The solution is defined for all t ≥ 0 and is constructed step-by-step using the following claim: Claim 1 Assume that there exist an integer a ≥ 0 and x ∈ C 0 ([-r, aT H ]; R n ) and z j ∈ C 0 ([-δ, aT H ]; R n ) for j = 0, ..., N that are absolutely continuous on [0, aT H ], and w ∈ PC([0, aT H ]; R p ) and u ∈ L ∞ ([-rτ,aT H ); R m ) that satisfy x(s) = (x 0 )(s) for s ∈[r, 0] and z j (s) = (z j,0 )(s) for all s ∈[δ, 0] and u(s) = ( ȗ0 )(s) for all s ∈[rτ, 0),aswellasEqs.( 1), ( 10), [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF], and (12)for t ∈[ 0, aT H ) a.e., Eq. ( 13) for all integers i ≥ 0 with τ i ≤ aT H , and Eq. ( 14)f o r l = 0, ..., a -1 (only when a > 0). Then there exist x ∈ C 0 ([-r,(a 1), ( 10)-( 12)fort ∈[ 0,(a + 1)T H ) a.e., Eq. ( 13) for all integers i ≥ 0 with τ i ≤ (a + 1)T H , and Eq. ( 14)forl = 0, ..., a.

+ 1)T H ]; R n ) and z j ∈ C 0 ([-δ, (a + 1)T H ]; R n ) ( j = 0, ..., N ) that are absolutely continuous on [0,(a+1)T H ], w ∈ PC([0,(a+1)T H ]; R p ), and u ∈ L ∞ ([-r -τ, (a+1)T H ); R m ) that satisfy x(s) = (x 0 )(s) for s ∈[ -r, 0], z j (s) = (z j,0 )(s) for s ∈[ -δ, 0], u(s) = ( ȗ0 )(s) for s ∈[ -r -τ, 0),Eqs.(
Proof (Claim 1) Using ( 14)f o rl = a, we can (uniquely) define u on [aT H , (a

+ 1)T H ). Since u is constant on [aT H ,(a + 1)T H ), we know that u ∈ L ∞ ([-r -τ, (a + 1)T H ); R m ).
Since the right-hand side of (1) satisfies a linear growth condition and since u is defined on [-rτ, (a + 1)T H ), it follows from Lemma 1 that we can uniquely define x on [aT H ,(a + 1)T H ]. The extended mapping x :[r,(a + 1)T H ]→R n satisfies x ∈ C 0 ([-r,(a + 1)T H ]; R n ), is absolutely continuous on [0,(a + 1)T H ], and satisfies (1)fort ∈[0,(a + 1)T H ) a.e.

Since lim i→∞ τ i =+∞, there are only a finite number of sampling times τ i in the interval [aT H ,(a + 1)T H ] (and possibly none). The right-hand sides of ( 10) and ( 12) satisfy a linear growth condition and since u is defined on [-rτ, (a + 1)T H ) and x is defined on [-r,(a + 1)T H ], it follows from Lemma 1 that we can uniquely define (z 0 , w) on [aT H ,(a + 1)T H ]. The extended mapping z 0 :[δ, (a + 1)T H ]→R n satisfies z 0 ∈ C 0 ([-δ, (a + 1)T H ]; R n ), is absolutely continuous on [0,(a + 1)T H ] and satisfies [START_REF] Ahmed-Ali | Global exponential sampled-data observers for nonlinear systems with delayed measurements[END_REF] and ( 12)f o rt ∈[ 0,(a + 1)T H ) a.e.. Moreover, the extended mapping w :[0,(a + 1)T H ]→R p satisfies w ∈ PC([0,(a + 1)T H ]; R p ).

Finally, using [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF] and Lemma 1, we can define z 1 and next z 2 , ..., z N on [aT H ,(a + 1)T H ]. The extended mappings z j :[δ, (a + 1)T H ]→R n (for j = 1, ..., N ) satisfy z j ∈ C 0 ([-δ, (a + 1)T H ]; R n ), are absolutely continuous on [0,(a + 1)T H ] and satisfy [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF]fort ∈[0,(a + 1)T H ) a.e. Therefore, the claim holds. ⊳ Part II: Proof of [START_REF] Heemels | Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance[END_REF] We next present three inequalities, which are direct consequences of ( 1), ( 2), ( 6) and ( 10):

(z 0 (t) -x(t -r )) ′ Q d dt (z 0 (t) -x(t -r )) ≤-2ω |z 0 (t) -x(t -r )| 2 -(z 0 (t) -x(t -r )) ′ QGv(t -r ) -(z 0 (t) -x(t -r )) ′ QL (w(t) -Hx(t -r )) for a.e. t ≥ T, ( 22
) x ′ (t)P ẋ(t) ≤ -4µ |x(t)| 2 + x ′ (t)PGv(t) -x ′ (t)P ( f (x(t), k(x(t))) -f (x(t), u(t -τ))
) , for a.e. t ≥ 0, and

z ′ 0 (t)Q ż0 (t) ≤-2ω |z 0 (t)| 2 + z ′ 0 (t)Qf(0, u(t -r -τ)) -z ′ 0 (t)QLw(t), for a.e. t ≥ 0, (23)
where T = min {τ i : τ i ≥ r , i = 1, 2, ...} is the smallest sampling time for which τ i ≥ r holds.

The following equations hold for all j = 1, ..., N , t ≥ 0 and are direct consequences of (11):

z j (t) = z j-1 (t) + t t-δ f (z j (s), u(s + jδ -r -τ))ds+ exp (-ct) z j (0) -z j-1 (0) - 0 -δ f (z j (s), u(s + jδ -r -τ))ds for all t ≥ 0 (25) x(t -r + jδ) = x(t -r + ( j -1)δ) + t t-δ f (x(s -r + jδ), u(s + jδ -r -τ))ds + t-r + jδ t-r +( j-1)δ Gv(s)ds for all t ≥ r (26)
Completing the squares in ( 22) and (24), and using (23) and ( 5), we get:

(z 0 (t) -x(t -r )) ′ Q d dt (z 0 (t) -x(t -r )) ≤ -ω |z 0 (t) -x(t -r )| 2 + 1 2ω |QG| 2 |v(t -r )| 2 + 1 2ω |QL| 2 |w(t) -Hx(t -r )| 2 for a.e. t ≥ T (27) x ′ (t + τ)P ẋ(t + τ) ≤ -4µ |x(t + τ)| 2 + |PG||v(t + τ)||x(t + τ)| + |P| L 2 |x(t + τ)||k(x(t + τ)) -u(t)| for a.e. t ≥ 0. ( 28
)
z ′ 0 (t)Q ż0 (t) ≤-ω |z 0 (t)| 2 + 1 2ω |Q| 2 L 2 2 |u(t -r -τ)| 2 + 1 2ω |QL| 2 |w(t)| 2 , for a.e. t ≥ 0. (29)
For all t ≥ 0, we can use [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF] to get

u(t) = k (z N (lT H )), where l =[ t/T H ].I t follows from Eq. (1) that |x(t + τ) -x(lT H + τ)| ≤ t+τ lT H +τ | f (x(s), u(s -τ))| ds + |G| t+τ lT H +τ |v(s)| ds. (30)
Therefore, we get the following for all t ≥ 0, where l =[t/T H ]:

|x(t + τ) -x(lT H + τ)| ≤ L 1 t lT H |x(s + τ) -x(lT H + τ)| ds + (L 1 + L 2 K ) T H |x(lT H + τ)| +L 2 KT H |z N (lT H ) -x(lT H + τ)| + |G| T H sup lT H ≤s≤t (|v(s + τ)|)
(using the triangle inequality, (4) and ( 5) and the fact that t ∈[lT H ,(l + 1)T H )) and

|x(t + τ) -x(lT H + τ)| ≤ exp (L 1 T H )(L 1 + L 2 K ) T H |x(lT H + τ)| + exp (L 1 T H ) L 2 KT H |z N (lT H ) -x(lT H + τ)| + exp (L 1 T H ) |G| T H sup lT H ≤s≤t (|v(s + τ)|)
(using the Gronwall-Bellman lemma and the fact that t ∈[lT H ,(l + 1)T H )), and

|x(t + τ) -x(lT H + τ)| ≤ exp(L 1 T H )(L 1 +L 2 K )T H 1-exp(L 1 T H )(L 1 +L 2 K )T H |x(t + τ)| + exp(L 1 T H )L 2 KT H 1-exp(L 1 T H )(L 1 +L 2 K )T H |z N (lT H ) -x(lT H + τ)| + exp(L 1 T H )|G|T H 1-exp(L 1 T H )(L 1 +L 2 K )T H sup lT H ≤s≤t (|v(s + τ)|)
(using the triangle inequality and the fact that exp

(L 1 T H )(L 1 + L 2 K ) T H < 1).
The above inequality in conjunction with the triangle inequality, (3), ( 28) and ( 14) (which implies that u(t) = k (z N (lT H )), where l =[t/T H ]) and the inequality

exp(L 1 T H )(L 1 +L 2 K )T H 1-exp(L 1 T H )(L 1 +L 2 K )T H |P| L 2 K ≤ 2µ (31) give x ′ (t + τ)P ẋ(t + τ) ≤-2µ |x(t + τ)| 2 + |P|L 2 K (1-L 1 T H exp(L 1 T H )) 1-exp(L 1 T H )(L 1 +L 2 K )T H |x(t + τ)||x(lT H + τ) -z N (lT H )| + exp(L 1 T H )|G||P|L 2 KT H 1-exp(L 1 T H )(L 1 +L 2 K )T H + |PG| |x(t + τ)| sup lT H ≤s≤t (|v(s + τ)|) (32)
for t ≥ 0 a.e. and l =[t/T H ]. Completing the squares in (32), we get

x ′ (t + τ)P ẋ(t + τ) ≤ -µ |x(t + τ)| 2 + 1 2µ |P|L 2 K (1-L 1 T H exp(L 1 T H )) 1-exp(L 1 T H )(L 1 +L 2 K )T H 2 |x(lT H + τ) -z N (lT H )| 2 =+ 1 2µ exp(L 1 T H )|G||P|L 2 KT H 1-exp(L 1 T H )(L 1 +L 2 K )T H + |PG| 2 sup lT H ≤s≤t |v(s + τ)| 2 (33)
for t ≥ 0 a.e. and l =[t/T H ]. Combining ( 25) and ( 26), we obtain the following for all j = 1, ..., N and t ≥ r :

z j (t) -x(t -r + jδ) ≤ |G| sup t-r +( j-1)δ≤s≤t-r + jδ (|v(s)|) + z j-1 (t) -x(t -r + ( j -1)δ) + L 1 t t-δ z j (s) -x(s -r + jδ) ds + exp (-ct) z j (0) -z j-1 (0) - 0 -δ f (z j (s), u(s + jδ -r -τ))ds (34)
It follows from (34) that the following inequalities hold for all θ ∈ (0, c], j = 1, ..., N , and t ≥ r :

sup r ≤s≤t z j (s) -x(s -r + jδ) exp(θ s) ≤ sup r ≤s≤t z j-1 (s) -x(s -r + ( j -1)δ) exp(θ s) +L 1 exp(θ δ)-1 θ sup r -δ≤s≤t z j (s) -x(s -r + jδ) exp(θ s) + z j (0) -z j-1 (0) - 0 -δ f (z j (s), u(s + jδ -r -τ))ds + |G| exp(θ t)sup 0≤s≤t-r + jδ (|v(s)|) (35)
Since L 1 δ<1, there is a small enough θ ∈ (0, c] such that L 1 (exp(θ δ) -1) <θ. It follows from (35) that the following inequalities hold for all θ ∈ (0, c] sufficiently small, j = 1, ..., N and t ≥ r : sup r ≤s≤t z j (s)x(sr + jδ) exp(θ s) ≤ Λsup r ≤s≤t z j-1 (s)x(sr + ( j -1)δ) exp(θ s) +sup r -δ≤s≤r z j (s)x(sr + jδ) exp(θ s)

+Λ |G| exp(θ t)sup 0≤s≤t-r + jδ (|v(s)|) +Λ z j (0) -z j-1 (0) - 0 -δ f (z j (s), u(s + jδ -r -τ))ds (36)
where

Λ = θ θ -L 1 (exp(θ δ) -1) . ( 37
)
Using ( 4), ( 5), (36), the fact that L 1 δ<1, and induction, we conclude that the following inequalities hold for all sufficiently small θ ∈ (0, c] and all j = 1, ..., N and t ≥ r :

sup r ≤s≤t z j (s) -x(s -r + jδ) exp (θ s) ≤ Λ j sup r ≤s≤t (|z 0 (s) -x(s -r )| exp (θ s)) + Λ j -1 Λ-1 exp (θ r ) max l=1,..., j sup r -δ≤s≤r (|z l (s) -x(s -r + lδ)|) +Λ Λ j -1 Λ-1 L 2 δ u 0 + 3m a x l=1,..., j z l,0 +Λ Λ j -1 Λ-1 |G| exp(θ t)sup 0≤s≤t-r + jδ (|v(s)|) (38)
Using Lemma 2 and inequalities ( 27) and (33) we obtain:

sup 0≤s≤t (|x(s + τ)| exp (θ s)) ≤ K 2 K 1 |x(τ )| + K 2 2µK 1 (µ-θ K 2 ) |PG| + exp(L 1 T H )|G||P|L 2 KT H 1-exp(L 1 T H )(L 1 +L 2 K )T H exp (θ t) sup τ ≤s≤t+τ (|v(s)|) + K 2 2µK 1 (µ-θ K 2 ) |P|L 2 K (1-L 1 T H exp(L 1 T H )) 1-exp(L 1 T H )(L 1 +L 2 K )T H × exp (θ T H ) sup 0≤s≤t (|x (s + τ ) -z N (s)| exp (θ s)) (39)
for all t ≥ 0 and θ ∈ (0,µ/K 2 ), where K 1 and K 2 > 0 are constants such that the inequality

K 1 |x| 2 ≤ x ′ Px ≤ K 2 |x| 2 holds for all x ∈ R n and sup T ≤s≤t (|z 0 (s) -x(s -r )| exp (θ s)) ≤ K 4 K 3 |z 0 (T ) -x(T -r )| exp (θ T ) + K 4 2ωK 3 (ω-θ K 4 ) |QG| sup 0≤s≤t-r (|v(s)| exp (θ s)) + K 4 2ωK 3 (ω-θ K 4 ) |QL| sup T ≤s≤t (|w(s) -Hx(s -r )| exp (θ s)) (40)
for all t ≥ T and θ ∈ (0,ω/K 4 ), where T = min {τ i : τ i ≥ r , i = 1, 2, ...}. Combining (39) and (38)f o r j = N , we obtain the following for all t ≥ 0 and sufficiently small θ>0:

sup 0≤s≤t (|x(s + τ)| exp (θ s)) ≤ K 2 K 1 |x(τ )| + K 2 2µK 1 (µ-θ K 2 ) |PG| + exp(L 1 T H )|G||P|L 2 KT H 1-exp(L 1 T H )(L 1 +L 2 K )T H × exp (θ t) sup τ ≤s≤t+τ (|v(s)|) + K 2 2µK 1 (µ-θ K 2 ) |P|L 2 K (1-L 1 T H exp(L 1 T H )) 1-exp(L 1 T H )(L 1 +L 2 K )T H × exp (θ(r + T H )) sup 0≤s≤r (|x (s + τ ) -z N (s)|) + K 2 2µK 1 (µ-θ K 2 ) |P|L 2 K (1-L 1 T H exp(L 1 T H )) 1-exp(L 1 T H )(L 1 +L 2 K )T H exp (θ T H ) ×Λ Λ N -1 Λ-1 L 2 δ u 0 + 3m a x l=1,...,N z l,0 + K 2 2µK 1 (µ-θ K 2 ) |P|L 2 K (1-L 1 T H exp(L 1 T H )) 1-exp(L 1 T H )(L 1 +L 2 K )T H exp (θ T H ) ×Λ N sup r ≤s≤t (|x (s -r ) -z 0 (s)| exp (θ s)) + K 2 2µK 1 (µ-θ K 2 ) |P|L 2 K (1-L 1 T H exp(L 1 T H )) 1-exp(L 1 T H )(L 1 +L 2 K )T H exp (θ(r + T H )) × Λ N -1 Λ-1 max j=1,...,N sup -δ≤s≤r x (s -r + jδ) -z j (s) + K 2 2µK 1 (µ-θ K 2 ) |P|L 2 K (1-L 1 T H exp(L 1 T H )) 1-exp(L 1 T H )(L 1 +L 2 K )T H exp (θ(t + T H )) ×Λ Λ N -1 Λ-1 |G| sup 0≤s≤t+τ (|v (s)|) (41)
Combining ( 1) and ( 12), we obtain the following for all t ∈[τ i ,τ i+1 ) with τ i ≥ T :

|w(t) -Hx(t -r )| ≤ |w(τ i ) -Hx(τ i -r )| + |H | L 1 t τ i |z 0 (s) -x(s -r )| ds + |HG| t τ i |v(s -r )| ds (42)
Using ( 13), (42) and the facts that t ∈[τ i ,τ i+1 ), τ i ≥ T , and sup i≥0 (τ i+1τ i ) ≤ T s , we get the following for all θ>0: 41) and (46) and using the fact that sup i≥0 (τ i+1τ i ) ≤ T s (which implies that T = min {τ i : τ i ≥ r , i = 1, 2, ...} satisfies T ≤ r + T s ), we obtain estimate [START_REF] Heemels | Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance[END_REF] for all t ≥ 0 and sufficiently small θ>0.

Part III: Proof of (16) Our strategy for the proof of estimate ( 16) is described next.

Using (38), ( 46) and ( 15) in conjunction with a standard causality argument, we conclude that there exists a constant κ>0 such that the following inequality holds for all t ≥ 0:

x t + N j=0 z t ≤ κ exp (-θ t) sup 0≤s≤τ +r +T s x s + ȗs + N j=0 z j,s +κ sup 0≤s≤t (|ξ (s)|) + sup 0≤s≤t (|v (s)|)

(47) Using (3), ( 14) and the fact that k(0) = 0, we obtain the following for all t ≥ 0:

ȗt ≤ ȗ0 exp (-θ(trτ)) +K exp (-θ(trτ -T H )) sup 0≤s≤t (|z N (s)| exp(θs))

Therefore, inequalities (47) and (48) allow us to conclude the existence of a constant κ>0 such that the following inequality holds for all t ≥ 0:

x t + ȗt + N j=0 z t ≤ κ exp (-θ t) sup 0≤s≤τ +r +T s x s + ȗs + N j=0 z j,s +κ sup 0≤s≤t (|ξ (s)|) + sup 0≤s≤t (|v (s)|) (49) respect to measurements and modelling errors. We also provided explicit estimates of the asymptotic gains of the external inputs v and ξ . This can be extended easily o networked control systems with uniformly globally exponentially stable scheduling protocols.

K 4 2ωK 3 1 exp 4 K 3 4 2ωK 3 ( 4 2ωK 3

 431434343 |w(t) -Hx(tr )| exp (θ t) ≤ |ξ(τ i )| exp (θ t) + |HG| T s exp (θ t) sup τ i ≤s≤t (|v(sr )|) + |H | L 1 sup τ i ≤s≤t (|z 0 (s)x(sr )| exp (θ s)) exp(θ T s )-1 θ (43)Estimate (43) implies the following estimate for all t ≥ T and θ>0:sup T ≤s≤t (|w(s) -Hx(sr )| exp (θ s)) ≤ exp (θ t) sup T ≤s≤t (|ξ(s)|) + |H | L 1 sup T ≤s≤t (|z 0 (s)x(sr )| exp (θ s)) exp(θ T s )-1 θ + |HG| T s exp (θ t) sup 0≤s≤t-r (|v(s)|)(44)Since K 4 2K 3 |QL||H | L 1 T s <ω, it follows that there exists θ>0 such that g = (ω-θ K 4 ) |QL||H | L and (44), we get this for all t ≥ T and for all sufficiently small θ>0:sup T ≤s≤t (|z 0 (s)x(sr )| exp (θ s)) |z 0 (T )x(Tr )| exp (θ T ) + exp(θ t) (ω-θ K 4 ) (|QG| exp (-θ r ) + |QL||HG| T s ) sup 0≤s≤t (ω-θ K 4 ) |QL| exp (θ t) sup T ≤s≤t (|ξ(s)|)(46)Combining (

  said to be Lipschitz on bounded sets if there exists a non-decreasing function Q : R +

To show (16), we use inequality (49) and the following claim: Claim 2 For every integer i ≥ 0 there exists a constant M i > 0 such that x t + ȗt + N j=0 z j,t ≤ M i x 0 + N j=0 z j,0 + ȗ0 + sup 0≤s≤t (|v(s)|) + sup 0≤s≤t (|ξ(s)|) for all t ∈[0, iT H ] (50) Indeed, if Claim 2 holds, then inequality ( 16) is a direct consequence of (49). Therefore, the rest of the proof is devoted to the proof of Claim 2.

Using Lemma 2 and inequality (29) we obtain for all t ≥ 0:

Using ( 4), ( 5) and ( 12) we obtain the following for all t ∈[ τ i ,τ i+1 ) and i = 0, 1, 2, ...:

Combining the estimate (52) with ( 13) and using the facts that t ∈[ τ i ,τ i+1 ) and sup i≥0 (τ i+1τ i ) ≤ T s , we obtain:

for all t ≥ 0. Using our bound (45), the combination of (51) and (53)i m p l i e st h e existence of constants Θ i > 0(i = 1, 2, 3, 4) such that the following estimate holds for all t ≥ 0:

Combining ( 53) and (54), we obtain:

for all t ≥ 0, for appropriate constants Θi > 0(i = 1, 2, 3, 4).

Inequality (4) implies the following estimates for all j = 1, ..., N and t ≥ 0:

Using (56) and the Gronwall lemma, we get this for all j = 1, ..., N and t ≥ 0:

Using ( 55) and ( 57) repeatedly, we obtain the following for all t ≥ 0:

for appropriate constants Θ>0 and σ>0. We are now ready to prove Claim 2.

Proof (Claim 2)

We use induction. The claim holds automatically for i = 0. Using (3), [START_REF] Karafyllis | Stability and Stabilization of Nonlinear Systems[END_REF], and (50) for a certain integer i ≥ 0 and the fact that k(0) = 0, we get:

for all t ∈[0,(i + 1)T H ].Using(4), (5), Lemma 1 and (59), we obtain

for all t ∈[ 0,(i + 1)T H ] for an appropriate constant R i > 0. Inequality (50)f o r i + 1 and an appropriate constant M i+1 > 0 is a direct consequence of (58), (59), and (60). The proof of Claim 2 is complete, so the proof of Theorem 1 is complete.

Concluding Remarks

In this chapter, we proposed a novel control scheme for nonlinear globally Lipschitz systems for which the input is delayed and applied with zero order hold, the measurements are sampled and delayed, and only an output is measured. The novelty of our work is in the use of a chain of approximate (dynamic) predictors to handle long delays in both input and output. Using small gain arguments, sufficient conditions on both the upper diameter of the sampling partition and the holding period, and a sufficient number of predictors in the chain, we proved that the closed loop is robust with


T Ahmed-Ali

F Giri

M Krstic

F Lamnabhi-Lagarrigue

L Burlion

Adaptive Observer for a Class of Parabolic PDEs

Keywords: Nonlinear system, observer design, PDE systems

The problem of state observation, based on spatiallysampled output measurements, is addressed for a class of infinite dimensional systems, modelled by a semi-linear heat equation augmented with a structured uncertain part involving a set of unknown parameters. An adaptive observer is designed that provides online estimates of the system (spatially distributed) state and unknown parameters based on sampled data (in space). Sufficient conditions for the observer to be exponentially convergent are established. These include an ad-hoc persistent excitation condition as well as a condition on how the observer gain must be selected in relation with the space sampling interval.

I. INTRODUCTION

Adaptive state observers are resorted to deal with online state and parameter estimation. The first adaptive observers have been developed for finite-dimensional continuous-time linear systems and an extensive survey can be found in [START_REF] Narendra | Stable Adaptive Systems[END_REF] and [START_REF] Ioannou | Robust Adaptive Control[END_REF]. Then, research efforts have been devoted to designing nonlinear adaptive observers for (finite-dimensional) nonlinear systems, e.g., [START_REF] Bastin | Stable adaptive observers for nonlinear time-varying systems[END_REF]- [START_REF] Besançon | On adaptive observers for state affine systems[END_REF]. More recently, sampled-data (in time) observers have been developed for (finitedimensional) nonlinear systems where output measurements are only available at sampling instants, e.g., [START_REF] Ahmed-Ali | Continuousdiscrete adaptive observers for state affine systems[END_REF]- [START_REF] Folin | Sampled-data adaptive observer for a class of state-affine nonlinear systems with output injection[END_REF].

The problem of observer design for infinite dimensional systems (IDSs) has also been given a great deal of interest, especially since the eighties. Several observer design techniques have been developed including the infinite dimensional Luenberger observer for linear IDSs (e.g., [START_REF] Curtain | An Introduction to Infinite Dimensional Linear Systems Theory[END_REF], [START_REF] Lasiecka | Control Theory for Partial Differential Equations: Continuous and Approximation Theories[END_REF]), the boundary observer design of bilinear IDSs (e.g., [START_REF] Amann | Feedback stabilization of linear and semilinear parabolic systems[END_REF]- [START_REF] Vries | A Luenberger observer for infinite dimensional bilinear system: A UV desinfection example[END_REF], [START_REF] Guo | Parameter estimation and non-collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance[END_REF]), backstepping-based boundary observers for parabolic partial integro-differential IDSs [START_REF] Smyshlyaev | Backstepping observers for a class of parabolic PDEs[END_REF], initial state recovery algorithms for various linear and nonlinear IDSs [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]- [START_REF] Fridman | Observers and initial state recovering for a class of hyperbolic systems via Lyapunov method[END_REF], sampled-data (in time and space) observer for semilinear diffusion IDSs [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF].

In the last few years, much interest has been paid to simultaneous parameter and state estimation for IDSs, within various contexts. In [START_REF] Smyshlyaev | Output-feedback adaptive control for parabolic PDEs with spatially varying coefficients[END_REF], simultaneous state and parameter estimation has been introduced to deal with output-feedback adaptive control design for parabolic PDEs. In this context, the unknown parameters are tuned by gradienttype laws while the (spatially distributed) state is estimated using openloop filters. The convergence of the estimates to their true values is not established. But, this is not required for the achievement of control objectives. In [START_REF] Smyshlyaev | Adaptive identification of two unstable PDEs with boundary sensing and actuation[END_REF], simultaneous state and parameter estimation has been performed to solve a parameter identification problem for reactionadvection type systems involving a single unknown parameter. Open-and closed-loop adaptive identifiers have been proposed where the unknown parameter is estimated using gradient-type estimators, while the (spatially distributed) state is estimated using open-loop filters. It is shown that the parameter estimate converges to its true values, by just using constant exciting inputs. In [START_REF] Guo | Parameter estimation and non-collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance[END_REF], simultaneous parameter and state estimation has been considered within the context of adaptive stabilization for a wave equation subject to a boundary harmonic disturbance linearly parameterized along a known set of functions. An adaptive observer estimating the system state and the (disturbance) unknown parameters is proposed and the estimation error system is shown to be asymptotically stable.

In this technical note, the problem of parameter and state estimation is addressed for IDSs that are described by a semilinear heat equation, based on sampled (in space) state measurements. The system includes a structured uncertain part, involving linearly a set of unknown parameters. Note that, in the absence of parameter uncertainty, the observer proposed in [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF] applies to the present system. On the other hand, the class of systems considered here can be viewed as a generalization of the system considered in [START_REF] Smyshlyaev | Adaptive identification of two unstable PDEs with boundary sensing and actuation[END_REF] as this involves a single unknown parameters. The corresponding observation problem is dealt with using an adaptive observer providing online estimates of the system (spatially distributed) state and unknown parameters. The observer design involves a backstepping state transformation, inspired from adaptive observers of nonlinear ODE systems [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], [START_REF] Smyshlyaev | Adaptive Control of Parabolic PDEs[END_REF]. The observer state and parameter estimators are derived so that the transformed system coincides with an exponentially stable target system. The observer derivation entails an ad-hoc persistent excitation condition, under which all state and parameter estimation errors are guaranteed to be exponentially vanishing. A second sufficient condition is established showing that the observer gain depends on the space sampling period. The technical note is organized as follows: the observation problem statement, including the class of IDSs, is described in Section II; the adaptive observer design and analysis are dealt with in Section III; a conclusion and a reference list end the technical note.

II. OBSERVATION PROBLEM STATEMENT

The system under study is described by a parabolic type PDE of the form

u t (x, t)=u xx (x, t)+θ T φ (y(t),t) , 0 <x<1,t>0 (1a)
with the boundary condition

u x (0,t)=0,t≥ 0 (1b)
and boundary input actuation:

u(1,t)=U (t),t ≥ 0, (control input) (1c)
where φ : R p × R → R n is a known C 1 function; θ ∈ R n is a fixed vector of unknown components but its dimension n is known.

The quantity θ T φ(y(t),t) might represent a possible structured modelling error. The system is observed via an output vector y(t) ∈ R p including all measurements acquired on the system at time t. Specifically, the spatial domain 0 ≤ x ≤ 1 is divided in p known subintervals [x j ,x j+1 ], with p ≥ 1 and

x 0 =0≤ x j ≤ x j+1 ≤ x p =1 (j =0•••p -1)
. A sensor is placed at the middle of each subinterval providing online state measurements at those positions y j (t)=u(x j ,t); x j = x j + x j+1 2 (j =0,...,p-1). (2a) Then, the system output vector y(t) is defined as follows:

y(t)=[y 0 (t),...,y p-1 (t)] T . ( 2b
)
Note that the case of single sensor p =1, or equivalently ∆=1, is not ruled out, where ∆ denotes the maximum space sampling interval, i.e., ∆= max j=0,...,p-1

(x j+1 -x j ). ( 2c
)
The goal is to generate accurate online estimates û(x, t) and θ, respectively of the system state u(x, t) ( 0 ≤ x ≤ 1; t ≥ 0)a n dt h e parameter vector θ, based on the output measurements y(t). To achieve this objective, the following assumption is considered: Assumption 1: u(x, t) is bounded and the function φ(., t) is uniformly bounded with respect to the argument t.

Remark 1:

a) Equation (1a) may capture several heat phenomena. For instance, the simpler case θ ∈ R boils down to the heat system (1) in [START_REF] Smyshlyaev | Adaptive identification of two unstable PDEs with boundary sensing and actuation[END_REF] (with the sensor being placed at the boundary x = 0). In the general case θ ∈ R n , (1a) may be viewed as an approximation of a parabolic type equation of the form

u t (x, t)=u xx (x, t)+θ T 1 0 ψ (u(x, t),t) dx.
Then, the approximation of the integral term by the rectangle method leads to an equation similar to (1a). b) The above boundedness assumption entails the existence of (not necessarily known) scalars

-∞ <u m <u M < ∞ and -∞ < φ m <φ M < ∞ such that, ∀ x ∈ [0, 1], ∀ y ∈ [u m ,u M ] p , ∀t ≥ 0 u m ≤ u(x, t) ≤ u M ,φ m ≤ φ(y, t) ≤φ M .
This assumption will prove to be crucial for the varying gain matrix λ(x, t) of the observer (defined in the next section) to be bounded.

III. OBSERVER DESIGN AND ANALYSIS

A. Observer Design

The system model (1a)-(1c) suggests the following observer structure:

ût (x, t)=û xx (x, t)+φ T (y(t),t) θ(t)-K (û(x j ,t) -y j (t)) + v(x, t),x j ≤ x<x j+1 , (j =0,...,p-1) (3a) ûx (0,t)=0 (3b) û(1,t)=U (t) (3c)
where θ(t) is a parameter vector estimate, v(x, t) is an additional correction term, and K ≥ 0 is the observer gain. Suitable choices of these quantities will be made based on the subsequent analysis. First, introduce the following errors:

u(x, t)=û(x, t) -u(x, t)(state estimation error) (4a) θ(t)= θ(t) -θ (parameter estimation error). ( 4b
)
Subtracting (1a) to (3a), it follows using (1a) and (4a), (4b) that u(x, t) undergoes the following equation:

u t (x, t)= u xx (x, t)+φ T (y(t),t) θ(t) -K u(x j ,t)+v(x, t), x j ≤ x<x j+1 , (j =0,...,p-1) (5a)
with the following boundary conditions:

u x (0,t)= u(1,t)=0(using (1b-c) and (4b-c)). (5b)
Now, introduce the backstepping transformation, inspired by [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] z

(x, t)= u(x, t) -λ(x, t) θ(t) (6)
where λ(x, t) ∈ R 1×n is an auxiliary vector function to be defined later. It is worth noting that, the above transformation has originally been introduced in [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] for finite dimensional systems described by nonlinear ODEs. It has proved in many places to be useful for the design of adaptive observers [START_REF] Besançon | On adaptive observers for state affine systems[END_REF]- [START_REF] Folin | Sampled-data adaptive observer for a class of state-affine nonlinear systems with output injection[END_REF].

It follows from ( 6) that z(x, t) undergoes the following equation:

z t (x, t)= u xx (x, t)+φ T (y(t),t) θ(t) -K u(x j ,t)+v(x, t) -λ t (x, t) θ(t) -λ(x, t) ˙ θ(t) (7)
for all t ≥ 0, x j ≤ x<x j+1 and j =0,...,p-1. Equation ( 7) suggests the following choice of v(x, t):

v(x, t)=λ(x, t) ˙ θ(t) (8)
Doing so, (7) simplifies to

z t (x, t)= u xx (x, t)+φ T (y(t),t) θ(t)-K u(x j ,t)-λ t (x, t) θ(t) (9)
for all t ≥ 0, x j ≤ x<x j+1 and j =0,...,p-1.I nv i e wo f( 6 ) , z(x, t)+λ(x, t) θ(t) can be substituted to u(x, t) on the right side of [START_REF] Hann | Continuous adaptive observer for state affine sampled-data systems[END_REF]. Doing so, one gets

z t (x, t)=z xx (x, t)+λ xx (x, t) θ(t)+φ T (y(t),t) θ(t) -K z(x j ,t)+λ(x j ,t) θ(t) -λ t (x, t) θ(t) = z xx (x, t) -Kz(x j ,t) + λ xx (x, t)+φ T (y(t),t) -Kλ(x j ,t) -λ t (x, t) θ(t) (10)
for all t ≥ 0, x j ≤ x<x j+1 and j =0,...,p-1. Equation ( 10) suggests the following trajectory for the auxiliary state vector λ(x, t):

λ t (x, t)=λ xx (x, t) -Kλ(x j ,t)+φ T (y(t),t); t ≥ 0,x j ≤ x<x j+1 ,j=0,...,p-1 (11a)
with the following boundary and initial conditions:

λ x (0,t)=λ(1,t)=0, and λ(x, 0) = 0. (11b)
Doing so, (10) boils down to

z t (x, t)=z xx (x, t)-Kz(x j ,t ); t ≥ 0,x j ≤ x<x j+1 , (j=0,...,p-1). (12a
) In view of (11b) and (5b), one gets from (6) the following boundary conditions:

z x (0,t)=z(1,t)=0 (12b)
Owing to the unknown parameter vector, the following adaptive law is used:

θ(t)= -R(t)Λ T (t) y(t) and θ(t) ∈ R n (13a) Ṙ(t)=R(t) -R(t)Λ T (t)Λ(t)R(t) with R(t) ∈ R n×n (13b) Λ(t)= ⎡ ⎢ ⎣ λ(x 0 ,t) . . . λ(x p-1 ,t) ⎤ ⎥ ⎦ ∈ R p×n (13c) y(t)=[û(x 0 ,t) -y 0 (t),...,û(x p-1 ,t) -y p-1 (t)] T (13d)
where the initial conditions θ(0) = θ 0 and R(0) = R 0 are arbitrarily chosen but R 0 = R T 0 > 0. The structure of the parameter adaptive law (13a)-(13d) is similar to that used in (finite-dimensional) system observers, e.g., [START_REF] Besançon | Remarks on nonlinear adaptive observer design[END_REF]- [START_REF] Besançon | On adaptive observers for state affine systems[END_REF]. The main difference is that the adaptive gain Λ(t) is presently generated by a PDE equation [namely, (11a), (11b)], while such a gain was generated by an ODE equation in the case of finite-dimension system observers. Except for this difference, the adaptive law (13a), (13b) is expected to perform as in the finitedimensional case. In particular, the matrix R(t) [generated by (13b)] will be required to stay all the time symmetric and positive definite. The observer thus designed is constituted of (3a)-(3c), (11a), (11b) and (13a)-(13c). For convenience, these equations are rewritten together as follows:

ût (x, t)=û xx (x, t)+φ T (y(t),t) θ(t)-K (û(x j ,t)-y j (t)) + λ(x, t) θ(t), for x j ≤ x<x j+1 (j =0,...,p-1) (14a) ûx (0,t)=0, û(1,t)=U (t) (14b) λ t (x, t)=λ xx (x, t) -Kλ(x j ,t)+φ T (y(t),t); t ≥ 0,x j ≤ x<x j+1 ,j =0,...,p-1 (14c) λ x (0,t)=λ(1,t)=0,λ (x, 0) = 0 (by definition) ((4d) θ(t)= -R(t)Λ T (t) y(t) (14e)
Ṙ(t)=R(t) -R(t)Λ T (t)Λ(t)R(t). (14f)

B. Observer Analysis

The observer analysis amounts to analyzing the stability of the following error system:

z t (x, t)=z xx (x, t) -Kz(x j ,t);
for all t ≥ 0 and x j ≤ x<x j+1 (j =0,...,p-1)

(15a) ˙ θ(t)= -R(t)Λ T (t) Z(t)+Λ(t) θ(t) with θ = θ -θ (15b) Z(t)=[z(x 0 ,t),...,z(x p-1 ,t)] T ∈ R p (15c) z x (0,t)=z(1,t)=0 (15d)
where (15a) and (15c) are, respectively, copies of (12a) and (14c), while (15b) is obtained from (14e) replacing there y(t) by Z(t)+ Λ(t) θ(t), due to (2a), (2b), [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], and (15d). The error equations (15a), (15b) are completed by the λ-equation (11a) and the associated boundary and initial conditions (11b).

It is worth noting that the solution (z =0,λ =0, θ =0)is an equilibrium of the system (15a), (15b). Indeed, it is readily checked that, if (z(x, t 0 )=0,λ(x, t 0 )=0, θ(t 0 )=0),forall0 <x<1 and some t 0 ≥ 0, then one has (z(x, t)=0,λ(x, t)=0, θ(t)=0),forallt ≥ t 0 .

For space limitation, the well posedness of the problem at hand is concisely discussed in the following remark.

Remark 2:

a) It is readily checked that, the system of (1a), (15a), and (14c) are simpler forms of equation ( 21) in [START_REF] Fridman | Robust sampled-data control of a class of semilinear parabolic systems[END_REF]. Following mutatis mutandis the well-posedness analysis developed there, one concludes that a strong solution (u(x, t),z(x, t),λ(x, t)) exists in the Hilbert space

H 1 2 = D (-A) 1 2 = w ∈ H 1 (0, 1) : w x (0) = w(1) = 0
where A = ∂ 2 /∂x 2 , (-A) 1/2 the square root of (-A),a n d H 1 (0, 1) is the Sobolev space of absolutely continuous scalar functions w :

[0, 1] → R or R n with dw/dx ∈ L 2 [0, 1].
Then, the existence and uniqueness of R(t) and θ(t) is immediately obtained from (14f) and (15b), applying the usual existence theorem of ODEs. Then, the existence and uniqueness of u(x, t)= z(x, t)+λ(x, t) θ(t) follows from [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF]. This also implies the existence of û(x, t)= u(x, t)+u(x, t). b) Also, the following Wirtinger's inequalities will be repeatedly used in the forthcoming analysis [START_REF] Hardy | Inequalities[END_REF] b a

w 2 (x, t)dx ≤ 4(b -a) 2 π 2 b a w 2 x (x, t)dx (16a) max a≤x≤b w 2 (x, t) ≤ b a w 2 x (x, t)dx (16b) whatever the function w ∈ H 1 (a, b) such that w(a)=0 or w(b)=0.
The next result is on the boundedness of the auxiliary vector λ(x, t). Proposition 1: The auxiliary state vector λ(x, t), generated by (14c)-(14d), is uniformly bounded, provided that the gain observer K in (14a) and the space sampling interval ∆ satisfy the condition:

0 ≤ K∆ 2 < 4π 2 .
Proof: See Appendix A. Using Proposition 1, it is readily follows from (13c) that:

Λ(t) ≤Λ m with Λ m = λ m √ p and λ m =s u p 0≤x≤1,t≥0 λ(x, t) . (17)
To ensure the exponential stability of the system (15a)-(15d), it is required that the time-varying matrix R(t) [solution of (14f)] exists is symmetric positive definite. Now, it is shown in many places (e.g., [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], [START_REF] Besançon | On adaptive observers for state affine systems[END_REF]) that R(t) enjoys the required properties if the following persistent excitation (PE) condition holds:

∃δ, ε 0 ,ε 1 > 0, ∀ t>0:ε 0 I (m+p)×(m+p) < t+δ t Λ T (s)Λ(s)ds < ε 1 I (m+p)×(m+p) . ( 18
)
Similar PE conditions are needed in system identification and adaptive observation. Note that, the right inequality simply means that Λ(s) is bounded which actually is the case due to [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]. The left inequality means that the column vectors of Λ(s) span the vector space R n+p on any finite time interval [t, t + δ],forallt. Under condition [START_REF] Ramdani | Recovering the initial state of an infinite-dimensional system using observers[END_REF], it turns out that the inverse R -1 is also bounded and symmetric positive definite, i.e., there are positive scalars (r, r), such that

rI (m+p)×(m+p) ≤ R -1 (t)= R -1 (t)
T ≤ rI (m+p)×(m+p) , ∀t ≥ 0.

(19) In the sequel, condition ( 18) is supposed to be true, so that one can make use of [START_REF] Fridman | Observers and initial state recovering for a class of hyperbolic systems via Lyapunov method[END_REF]. Then, one has the following main result:

Theorem 1: Let the adaptive observer described by (4a)-(4c), ( 9), (12a), (12b), and (14a), (14b) be applied to the system (1a), (1b), (2a), (2b) subject to Assumption 1 and the PE assumption [START_REF] Ramdani | Recovering the initial state of an infinite-dimensional system using observers[END_REF]. Let the observer gain K and the space sampling interval be selected such that K∆ < 2π 2 . Then, the estimation errors θ(t) and the quantity max 0≤x≤1 | u(x, t)| are globally exponentially vanishing.

Proof: Consider the following Lyapunov function candidate:

V (z, θ)=V 0 ( θ)+ μ 0 2 1 0 z 2 (x, t)dx + 1 2 1 0 z 2 x (x, t)dx (20a)
with V 0 ( θ)= θ T R -1 θ and μ 0 any positive scalar such that

μ 0 >p 1 - K∆ 2π 2 -1 . (20b)
Note that μ 0 exists because K∆ < 2π 2 by assumption (Theorem 1). From (20a), one gets the following time-derivative:

V (z, θ)= V0 ( θ)+μ 0 1 0 z(x, t)z t (x, t)dx+ 1 0 z x z xt (x, t)dx. (21)
Using (15b) and (14f), one immediately gets

V0 ( θ)= θ T (t) Ṙ-1 (t) θ(t)+2 θ T (t)R -1 (t) ˙ θ(t) = θ T (t) -R -1 +Λ T (t)Λ(t) θ(t) -2 θ T (t) Λ T (t)Λ(t) θ(t)+Λ T (t)Z(t) ≤-θ T (t)R -1 (t) θ(t)+ Z(t) 2 (using Young's inequality) ≤-θ T R -1 θ + p max 0<x<1 |z(x, t)| 2 (using (15c)) ≤-V 0 ( θ)+p 1 0 z 2 x (x, t)dx (22)
using Wirtinger's inequality (16b). On the other hand, using (15a), the third term on the right side of ( 21) develops as follows:

1 0 z(x, t)z t (x, t)dx = p-1 j=0 x j+1 x j z(x, t)(z xx (x, t) -Kz(x j ,t)) dx = p-1 j=0 x j+1 x j z(x, t)z xx (x, t)dx - p-1 j=0 x j+1 x j Kz(x j ,t)z(x, t)dx = 1 0 z(x, t)z xx (x, t)dx- p-1 j=0 x j+1
x j K (z(x j ,t)-z(x, t)) z(x, t)dx

- p-1 j=0 x j+1 x j Kz 2 (x, t)dx = - 1 0 z 2 x (x, t)dx-K ⎛ ⎜ ⎝ p-1 j=0 x j+1
x j (z(x j ,t)-z(x, t)) z(x, t)dx

- 1 0 z 2 (x, t)dx ⎞ ⎠ (23)
where the last equality is obtained using an integration by parts. By Young's inequality, one has

- p-1 j=0 x j+1 x j K (z(x j ,t)-z(x, t)) z(x, t)dx ≤ Kξ 2 1 0 z 2 (x, t)dx + 1 2ξ p-1 j=0 x j+1 x j K (z(x j ,t) -z(x, t)) 2 dx (24)
whatever ξ>0. Also, the application of (16a) gives

K 2ξ p-1 j=0 x j+1 x j (z(x j ,t)-z(x, t)) 2 dx = K 2ξ p-1 j=0 x j x j (z(x j ,t)-z(x, t)) 2 dx + K 2ξ p-1 j=0 x j+1 x j (z(x j ,t) -z(x, t)) 2 dx ≤ K∆ 2 2ξπ 2 1 0 z 2 x (x, t)dx.
This together with ( 23) and ( 24) yields

1 0 z(x, t)z t (x, t)dx ≤- 1 0 z 2 x (x, t)dx + Kξ 2 1 0 z 2 (x, t)dx + K∆ 2 2ξπ 2 1 0 z 2 x (x, t)dx -K 1 0 z 2 (x, t)dx ≤-1 - K∆ 2 2ξπ 2 1 0 z 2 x (x, t)dx -K 1 - ξ 2 1 0 z 2 (x, t)dx. (25)
Using ( 22) and ( 25), one gets

θ T (t) Ṙ-1 (t) θ(t)+2 θ T (t)R -1 (t) ˙ θ(t)+μ 0 1 0 z(x, t)z t (x, t)dx ≤-θ T R -1 θ + p 1 0 z 2 x (x, t)dx -μ 0 ⎛ ⎝ 1- K∆ 2 2ξπ 2 1 0 z 2 x (x, t)dx+K 1- ξ 2 1 0 z 2 (x, t)dx ⎞ ⎠ ≤-θ T R -1 θ -μ 0 1 - K∆ 2 2ξπ 2 - p μ 0 1 0 z 2 x (x, t)dx -Kμ 0 1 - ξ 2 1 0 z 2 (x, t)dx. ( 26
)
Focusing on the last term on the right side of ( 21), one has

1 0 z x (x, t)z xt (x, t)dx = 1 0 z x (x, t)z tx (x, t)dx (27)
using a similar argument as Remark A1 in Appendix A of [START_REF] Fridman | Sampled-data distributed Hinfinity control of transport reaction systems[END_REF].

Then, one has, using successively an integration by part, (12a) and the boundary conditions z x (0,t)=z(1,t)=0

1 0 z x (x, t)z xt (x, t)dx = - 1 0 z xx (x, t)z t (x, t)dx = - 1 0 z 2 xx (x, t)dx + p-1 j=1 x j+1 x j Kz xx (x, t)z(x j ,t)dx = - 1 0 z 2 xx (x, t)dx + p-1 j=1 x j+1 x j Kz xx (x, t)z(x, t)dx + p-1 j=1 x j+1 x j Kz xx (x, t)(z(x j ,t) -z(x, t)) dx ≤- 1 0 z 2 xx (x, t)dx + 1 0 Kz xx (x, t)z(x, t)dx + p-1 j=1 x j+1 x j K η 2 z 2 xx (x, t)+K 1 2η (z(x j ,t) -z(x, t)) 2 dx ( 28
)
where Young's inequality has been used in the last inequality and η> 0 is arbitrary. Integrating by part the second integral on the right side of ( 28), this yields

1 0 z x (x, t)z xt (x, t)dx ≤- 1 0 z 2 xx (x, t)dx + K (z x (1,t)z(1,t) -z x (0,t)z(0,t)) -K 1 0 z 2 x (x, t)dx + Kη 2 1 0 z 2 xx (x, t)dx + K 2η p-1 j=1 x j+1 x j (z(x j ,t) -z(x, t)) 2 dx ≤ Kη 2 -1 1 0 z 2 xx (x, t)dx -K 1 0 z 2 x (x, t)dx + K 2η p-1 j=1 x j x j (z(x j ,t) -z(x, t)) 2 dx + K 2η p-1 j=1 x j+1 x j (z(x j ,t) -z(x, t)) 2 dx ( 29
)
where the conditions z x (0,t)=z(1,t)=0 have again been used. Applying (16a) to the last two terms on the right side of (29), one gets

1 0 z x (x, t)z xt (x, t)dx ≤-1 - Kη 2 1 0 z 2 xx (x, t)dx -K 1 0 z 2 x (x, t)dx + ∆ 2 2ηπ 2 1 0 z 2 x (x, t)dx ≤-1 - Kη 2 1 0 z 2 xx (x, t)dx -K 1 - ∆ 2 2ηπ 2 1 0 z 2 x (x, t)dx). (30)
Let the free parameters η in (30) and ξ in (26) be such that:

ξ =∆ and η = ∆ π 2 . ( 31
)
This ensures that

1 - ∆ 2 2ηπ 2 =1- ∆ 2 > 0 and 1 - ξ 2 =1- ∆ 2 > 0 (32) because 0 < ∆ ≤ 1.
Then, the first term on the right side of (30) entails the following condition on K:

Kη 2 < 1 or, equivalently,K ∆ < 2π 2 (33)
which is nothing else than the condition in Theorem 1. Combining ( 21), [START_REF] Smyshlyaev | Adaptive Control of Parabolic PDEs[END_REF], and (30) gives, using (31)-(33)

V ≤-θ T R -1 θ -Kμ 0 1 - ∆ 2 1 0 z 2 (x, t)dx -μ 0 1 - K∆ 2π 2 - p μ 0 1 0 z 2 x (x, t)dx -K 1 - ∆ 2 1 0 z 2 x (x, t)dx. (34)
Note that by (20b), one has

1 - K∆ 2π 2 - p μ 0 > 0. (35)
Then, (34) yields

V ≤-V 0 ( θ)-μ 0 K 1- ∆ 2 + π 2 8 1- K∆ 2π 2 - p μ 0 × 1 0 z 2 (x, t)dx -K 1 - ∆ 2 + μ 0 2 1 - K∆ 2π 2 - p μ 0 × 1 0 z 2 x (x, t)dx ≤-V 0 ( θ)-σ 0 μ 0 2 1 0 z 2 (x, t)dx- σ 1 2 1 0 z 2 x (x, t)dx (36)
≤-σV (z, θ)

with σ =min(1,σ 0 ,σ 1 ),where

σ 0 =2K 1 - ∆ 2 + π 2 4 1 - K∆ 2π 2 - p μ 0 > 0 (38)
σ 1 = K 1 - ∆ 2 + μ 0 2 1 - K∆ 2π 2 - p μ 0 > 0. (39)
Clearly, inequality (37) implies that V is exponentially vanishing (as t →∞) and from (20a) so are θ, 1 0 z 2 (x, t)dx and 1 0 z 2 x (x, t)dx. Then, using Wirtinger's inequality (16b), one has

max 0≤x≤1 z 2 (x, t) ≤ 1 0 z 2 x (x, t)dx. (40)
This holds because z(1,t)=0 (due to (15d)). Then, in turn max 0≤x≤1 |z(x, t)| is also exponentially vanishing. On the other hand, one has from (6) that u(x, t)=z(x, t)+λ(x, t) θ(t).Asλ(x, t) is bounded (by Proposition 1), it follows that max 0≤x≤1 | u(x, t)| is also exponentially vanishing. This completes the proof of Theorem 1.

Remark 3:

a) As 0 < ∆ ≤ 1, the condition K∆/2π 2 < 1, introduced in Theorem 1, is more restrictive than the condition K∆ 2 /4π 2 < 1 required in Proposition 1. Therefore, only the former is retained. Under that condition, the gain K ≥ 0 can be arbitrarily chosen, but the spatial sampling interval ∆ must be selected accordingly. The larger K, the smaller ∆. b) Using (35), it follows from (38), (39) that σ 0 and σ 1 are increasing functions of the gain K. Then, it follows from (36) that the convergence rate of 1 0 z 2 (x, t)dx and 1 0 z 2 x (x, t)dx can be made arbitrarily high by letting K be sufficiently large.

Then, it follows from [START_REF] Guo | Parameter estimation and non-collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance[END_REF] that the convergence rate of V 0 ( θ) can be made arbitrarily close to that of V 0 ( θ(0))e -t by letting K be sufficiently large. Consequently, it follows from ( 6) and (40) that, the convergence rate of max 0≤x≤1 | u(x, t)| is also made higher with larger values of K. On the other hand, it has been pointed out in Part a that, large values of K entail small values of the sampling interval ∆ or, equivalently, large number p of required sensors.

IV. CONCLUSION

We have addressed the problem of estimating the state and parameters of the class of IDSs described by the model (1a)-(1c), (2a), (2b). The latter is basically a parabolic PDE augmented by the structured quantity θ T φ(y(t),t). The adaptive observer (14a)-( 14f) is designed and shown to enjoy exponential convergence, under the persistent excitation property [START_REF] Ramdani | Recovering the initial state of an infinite-dimensional system using observers[END_REF] and the condition (20b). The latter shows that the observer gain K must be selected taking into account the space sampling interval ∆. The smaller ∆ the larger may be the gain K.

APPENDIX A PROOF OF PROPOSITION 1

Consider the Lyapunov functional candidate:

W (t)= 1 2 1 0 λ(x, t)λ T (x, t)dx + 1 2 1 0 λ x (x, t)λ T x (x, t)dx (A1)
Using (14c), it follows from (A1):

Ẇ (t)= 1 0 λ t (x, t)λ T (x, t)dx + 1 0 λ x (x, t)λ T xt (x, t)dx. (A2)
Following the same approach as in the proof of Theorem 1, the two terms on the right side of (A2) will be successively upper bounded. The first term develops as follows:

1 0 λ t (x, t)λ T (x, t)dx = 1 0 λ xx (x, t)λ T (x, t)dx- p-1 j=0 x j+1
x j Kλ(x j ,t)λ T (x, t)dx

+ 1 0 φ T (y(t),t) λ T (x, t)dx = - 1 0 λ x (x, t) 2 dx - p-1 j=0 x j+1
x j Kλ(x j ,t)λ T (x, t)dx

+ 1 0 λ(x, t)φ (y(t),t) dx = - 1 0 λ x (x, t) 2 dx - p-1 j=0 x j+1
x j K (λ(x j ,t) -λ(x, t)) λ T (x, t)dx

+ 1 0 λ(x, t)φ (y(t),t) dx - 1 0 K λ(x, t) 2 dx (A3)
where the penultimate last equality is obtained using an integration by parts of the first term on the right side and the boundary conditions in (14d). Using Young inequality, one has

1 0 λ(x, t)φ (y(t),t) dx ≤ φ (y(t),t) 2 2ζ + ζ 2 1 0 λ(x, t) 2 dx (A4) - p-1 j=0 x j+1 x j K (λ(x j ,t) -λ(x, t)) λ T (x, t)dx ≤ 1 2ω p-1 j=0 x j+1 x j K λ(x j ,t) -λ(x, t) 2 dx + Kω 2 1 0 λ(x, t) 2 dx& (A5)
whatever ζ>0 and ω>0. Using Wirtinger's inequality (16a), the first term on the right side of (A5) is bounded as follows:

1 2ω p-1 j=0 x j+1 x j K λ(x j ,t) -λ(x, t) 2 dx ≤ 1 2ω p-1 j=0 x j x j K λ(x j ,t) -λ(x, t) 2 dx + 1 2ω p-1 j=0 x j+1 x j K λ(x j ,t) -λ(x, t) 2 dx ≤ K 2ω ∆ 2 π 2 p-1 j=0 x j+1 x j λ x (x, t) 2 dx (A6)
Using (A4)-(A6), it follows from (A3) that:

1 0 λ t (x, t)λ T (x, t)dx ≤-1 - K 2ω ∆ 2 π 2 1 0 λ x (x, t) 2 dx -K - Kω 2 - ζ 2 1 0 λ(x, t) 2 dx + φ (y(t),t) 2 2ζ . (A7)
Letting ω = K∆ 2 /2ϑπ 2 for some 0 <ϑ<1, inequality (A7) yields

1 0 λ t (x, t)λ T (x, t)dx ≤-(1 -ϑ) 1 0 λ x (x, t) 2 dx -K - K 2 ∆ 2 4ϑπ 2 - ζ 2 1 0 λ(x, t) 2 dx + φ (y(t),t) 2 2ζ . (A8)
Now, let us focus on the second term on the right side of (A2). First, notice that equality [START_REF] Fridman | Sampled-data distributed Hinfinity control of transport reaction systems[END_REF] still holds replacing there z(x, t) by λ(x, t). Then, one immediately has

1 0 λ x (x, t)λ T xt (x, t)dx = 1 0 λ x (x, t)λ T tx (x, t)dx = - 1 0 λ xx (x, t)λ T t (x, t)dx (A9)
using an integration by part and the boundary conditions (14d). The above equality develops further as follows, using (14c):

1 0 λ x (x, t)λ T xt (x, t)dx = - p-1 p=0 x j+1 x j λ xx (x, t)λ T t (x, t)dx = - p-1 p=0 x j+1 x j λ xx (x, t) 2 dx + K p-1 p=0 x j+1 x j λ xx (x, t)λ T (x j ,t)dx - 1 0 λ xx (x, t)φ (y(t),t)) = - 1 0 λ 2 xx (x, t)dx + K p-1 p=0 x j+1 x j λ xx (x, t)λ T (x, t)dx + K p-1 p=0 x j+1 x j λ xx (x, t) λ T (x j ,t) -λ T (x, t) dx - 1 0 λ xx (x, t)φ T (y(t),t)) dx = - 1 0 λ 2 xx (x, t)dx + K 1 0 λ xx (x, t)λ T (x, t)dx - 1 0 λ xx (x, t)φ T (y(t),t)) dx + K p-1 p=0 x j+1 x j λ xx (x, t) λ T (x j ,t) -λ T (x, t) dx. (A10)
Applying Young's inequality to the third and fourth quantities on the right side of (A10) and integrating by part the second integral, equality (A10) yields, using the boundary conditions (14d) The above inequality shows that, one first proceeds with the selection of the design parameters K, ∆ according to the following condition:

1 0 λ x (x, t)λ T xt (x, t)dx ≤- 1 0 λ xx (x, t) 2 dx -K 1 0 λ x (x, t) 2 dx + K p-1 p=0 x j+1 x j υ 2 λ xx (x, t) 2 + 1 2υ λ(x j ,t) -λ(x, t) 2 dx + ̟ 2
K∆ 2 4π 2 < 1 (A14)
This is nothing other than the assumption (made on K and ∆)i n Proposition 1. Then, let the free parameters (υ, ϑ, ̟) be such that

K∆ 2 4π 2 <ϑ<1, 0 < ζ 2 ≤ K - K 2 ∆ 2 4ϑπ 2 ∆ 2 2π 2 <υ< 2 K , ̟ 2 < 1 - Kυ 2 . (A15)
It is readily checked that, this choice ensures that

K - K 2 ∆ 2 4ϑπ 2 - ζ 2 ≥ 0, 1 -ϑ>0 1 - ∆ 2 2υπ 2 > 0, 1 - Kυ 2 - ̟ 2 > 0.
Then, applying Wirtinger's inequality (16a) to the second and fourth terms on the right side of inequality (A13), this develops as follows:

Ẇ (t) ≤- K - K 2 ∆ 2 4ϑπ 2 - ζ 2 + π 2 (1 -ϑ) 4
× 1 0 λ(x, t) 2 dx -K 1 - ∆ 2 2υπ 2 + π 2 4 1 - Kυ 2 - ̟ 2
× 1 0 λ x (x,
γ =2min K - K 2 ∆ 2 4ϑπ 2 - ζ 2 + π 2 (1 -ϑ) 4 , K 1 - ∆ 2 2υπ 2 + π 2 4 1 - Kυ 2 - ̟ 2 .
As φ(y(t),t) is bounded (by Assumption 1), it follows from (A16) that so is W (t). Then, it follows from (A1) that 1 0 λ x (x, t) 2 dx is bounded. Applying the inequality (16b), one gets that max 0≤x≤1 λ(x, t) 2 is bounded. Proposition 1 is proved.

  Applying the Wirtinger's inequality (16a) to the two terms in the penultimate line in (A11), one gets

		+		K 2υ	p-1 p=0	x j x j	λ(x j ,t) -λ(x, t) 2 dx
		+		K 2υ	p-1 p=0	x j+1 x j	λ(x j ,t) -λ(x, t) 2 dx
		+		̟ 2	1	λ 2 xx (x, t)dx +	1 2̟	φ (y(t),t)) 2	(A11)
							0
	whatever the scalars υ>0, ̟>0. 1
	λ x (x, t)λ T xt (x, t)dx
	0
		1								1
	≤-	0	λ xx (x, t) 2 dx -K	0	λ x (x, t) 2 dx
	+	Kυ 2	0	1	λ xx (x, t) 2 dx+	K∆ 2 2υπ 2	p-1 p=0	x j+1 x j	λ x (x, t) 2 dx
	+	̟ 2	1		λ xx (x, t) 2 dx +	1 2̟	φ (y(t),t)) 2
		0
	≤-1 -	Kυ 2	-	̟ 2	0	1	λ xx (x, t) 2 dx
	-K 1-	∆ 2 2υπ 2	0	1	λ x (x, t) 2 dx +	1 2̟	φ (y(t),t)) 2 .
										(A12)
	Combining (A12) and (A8), one obtains
		Ẇ (t) ≤-K -	K 2 ∆ 2 4ϑπ 2 -	ζ 2	0	1	λ(x, t) 2 dx
										1
							-(1 -ϑ)	0	λ x (x, t) 2 dx
							-K 1 -	∆ 2 2υπ 2	0	1	λ x (x, t) 2 dx
							-1 -	Kυ 2	-	̟ 2	0	1	λ xx (x, t) 2 dx
							+			1 2ζ	+	1 2̟	φ (y(t),t) 2 .	(A13)

1 0 λ xx (x, t) 2 dx + 1 2̟ φ (y(t),t)) 2 ≤-1 0 λ xx (x, t) 2 dx -K 1 0 λ x (x, t) 2 dx + Kυ 2 1 0 λ xx (x, t) 2 dx

  t) 2 dx

	+	1 2ζ	+	1 2̟		φ (y(t),t) 2
	≤-γW(t)+	1 2ζ	+	1 2̟	φ (y(t),t) 2	(A16)
	with