Sylvain Ervedoza
email: sylvain.ervedoza@math.u-bordeaux.fr

Zhiqiang Wang

Jiacheng Zhang

Semi-global stabilization of a nonlinear transport equation with nonlocal velocity *

In this paper, we establish the semi-global feedback stabilization for a class of nonlinear transport equations with nonlocal velocity, which models a highly re-entrant system encountered in semi-conductor manufacturing. We design two new time-varying feedback laws: one yields a semi-global exponential stability of the corresponding closed-loop system, the other one a local exponential stability result for the corresponding closed-loop system. The crucial assumption on the velocity function λ(•) and the target equilibrium ρ is strongly reduced compared to the previous results. Numerical simulations are also provided as illustration of the theoretical results.

Introduction

In this paper, we study the stabilization of a nonlinear transport equation with nonlocal velocity

∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, ∞), (1.1)
in which W denotes the total mass of ρ:

W (t) = 1 0 ρ(x, t) dx, t ∈ (0, ∞), (1.2)
and λ : R → R * + is a smooth positive function. For problem (??)-(??) to be well-posed, since the velocity λ is positive, we further need to impose some boundary conditions at x = 0. Here we choose to impose the boundary condition through the influx: λ(W (t))ρ(0, t) = u(t), t ∈ (0, ∞),

(1.3) and u : R → R will be a control function. Furthermore, our goal is to design the control u from measurement made on the outflux y(t) = λ(W (t))ρ(1, t), t ∈ (0, ∞).

(1.4)

To be more precise, given ρ ∈ R, our goal is to design a feedback control u in terms of y, of the form

u(t) = F[y(•)], t ∈ (0, ∞), (1.5)
where F is a suitable function of the measurement y, such that the closed-loop system (??)-(??)-(??)-(??)-(??) is exponentially stable towards the state ρ.

The nonlinear transport equation model we study arose from the semiconductor manufacturing systems and was firstly introduced by Armbruster et al. in [?]. The performance of the wafer fabrication stage of integrated circuit (IC) production is closely related to the profits. These manufacture systems are characterized by their highly re-entrant feature with very high volume (number of parts manufactured per unit time) and very large number of consecutive production steps as well. The continuum model introduced in [?] has the advantage that it is more accurate with more parts, machines, and re-entrant such as Discrete Event Simulation method [?] and Queueing Networks method [?].

Control problems for conservation laws and general hyperbolic systems have been widely studied, see [?, ?] and the references therein. Regarding the controllability of nonlinear hyperbolic equations (or systems), one can refer to [?, ?, ?] for the context of classical solutions, and to [?, ?] for the context of entropy solutions. Asymptotic stability and stabilization of hyperbolic systems have also been studied in the literature, and three main strategies have been used. The first one relies on the analysis of the solution along the characteristic curves; see in particular [?, ?]. The second one is the Lyapunov function approach; see, in particular, [?, ?, ?, ?]. The third one consists in designing boundary feedback controls through the Backstepping method, which has been used for instance to stabilize exponentially the inhomogeneous quasilinear hyperbolic system in H 2 norm (see [?, ?]). One can also refer to [?] for many successful examples about feedback stabilization with this approach.

Concerning the manufacturing model (??) itself, the well-posedness of the open-loop system for (??)-(??) was firstly established in [?]. An optimal control problem motivated by [?, ?] and related to the Demand Tracking Problem is also considered in [?]. The objective of that optimal control problem is to minimize, by choosing the influx u, the L p -norm (p 1) of the difference between the actual outflux y and a given demand forecast y d over a fixed time period. Another related work [?] gave a necessary condition for the possible optimal controls. Based on the W 1,p -regularity results of solutions of the adjoint system, the article [?] exhibits the optimality conditions for a special type of cost functional, which is motivated by the L 2 tracking type cost functional from [?, ?, ?]. Let us also point out that these results have been generalized: the well-posedness of a coupled system of nonlocal conservation laws, modeling the highly re-entrant multi-commodity manufacturing network, for L p -, BV -and W 1,p -data and the existence of minimizers that solve the corresponding optimal control problem are proved in [?]. It is also worth mentioning that the exact controllability of the solution and the outflux for (??)-(??) were obtained in [?].

The output feedback stabilization (i.e. with a closed loop control) for the system (??) under a suitable feedback law is a natural and interesting problem. More precisely, the problem of asymptotical stabilization can be described as follows: For any given equilibrium ρ ∈ R and any initial data ρ 0 , can we find a feedback law (??) giving the control u in terms of y such that ρ is asymptotically stable for the corresponding closed-loop control system (??)-(??)-(??)-(??)-(??), namely, the weak solution ρ to the system (??)-(??)-(??)-(??)-(??) converges to ρ asymptotically when time t goes to +∞?

In order to achieve this goal, a natural way would be to propose a feedback law of the proportional form u(t) -ρλ(ρ) = k(y(t) -ρλ(ρ)), t ∈ (0, ∞), (1.6)

for some k ∈ (-1, 1). With this feedback law (??), the authors in [?] give a sufficient and necessary condition on the exponential stabilization for the corresponding linearized control system by spectral analysis. Then by a Lyapunov function approach and perturbation techniques, local exponential stabilization were also proved for the original closed-loop system in general nonlinear cases. In particular, the feedback law (??) with k ∈ (-1, 1) provides local exponential stability if and only if

d := ρλ (ρ) λ(ρ) satisfies d > -1. (1.7)
Note that this condition depends on the equilibrium ρ and the velocity function λ. It is also important to emphasize that if ρ > 0, the stabilization result (see [?], Theorem 4.2) for the nonlinear system is local in the sense that the initial data, and consequently the solution, belong to a small neighborhood of the equilibrium ρ > 0 . When ρ = 0, a global stabilization result in L 2 (0, 1) has been proved [?, Theorem 4.1] where there is no smallness limitation for the initial data and the solution.

Later on in [?], the stabilization results of [?] have been generalized upon using a Lyapunov function approach. Firstly if ρ = 0, the global stabilization result in L 2 (0, 1) for the closed-loop system (??)-(??)-(??)-(??)-(??) with a general velocity function λ(•) ∈ C 1 ([0, +∞); (0, +∞)) is generalized to L p (0, 1) (p 1) data. Secondly if ρ > 0, the global exponential stabilization result in L 2 (0, 1) for the closed-loop system (??)-(??)-(??)-(??)-(??) with a typical class of velocity functions

λ(s) = A B + s , s ∈ [0, ∞) A > 0, B > 0 (1.8)
is obtained. That is, the smallness restriction on the initial data in [?, Theorem 4.2] is removed by using the special feature of the velocity function (??). Additionally, stabilization results for the discrete counterpart of system (??)-(??) are obtained by the eigenvalue decomposition method and by a Lyapunov function method, and numerical examples are provided to show that if the velocity function λ(•) is not of the form (??), the solution of the closed-loop system might converge to a different equilibrium or even diverge as the time t goes to ∞. Let us also remark that when ρ = 0, the global stabilization result in L p (0, 1) (p 1, p = 2) remains an open problem even for systems satisfying (??). The situation for general systems with different velocity functions other than (??

) is widely open.

We also refer to [?] for a result on designing PI control to stabilize the linearized system of the nonlinear model (??) by spectral analysis as in [?], or to the more recent work [?] for the design of event-triggered boundary controls to stabilize this model. We also refer to [?] for a nonlocal model which has both local and nonlocal nature, and to [?] for a nonlocal traffic flow model, which also shares the non-local feature of the system (??) under consideration.

In this article, to design a suitable feedback law for (??)-(??)-(??)-(??), our key observations are the following:

• If we know the value of W for all time t > 0, the natural feedback law

u(t) -ρλ(W (t)) = k(y(t) -ρλ(W (t))), t ∈ (0, ∞), (1.9)
with k ∈ (-1, 1) will work, since it is equivalent to

ρ(0, t) -ρ = k(ρ(1, t) -ρ), k ∈ (-1, 1), (1.10)
which obviously stabilizes the system, as we will recall in Section ??.

• If we do not know the function W but only an approximation W on (0, ∞), the feedback control

u(t) -ρλ( W (t)) = k(y(t) -ρλ( W (t))), t ∈ (0, ∞), (1.11)
for k ∈ (-1, 1) should work as well, at least if W converges to W as t → ∞.

• The dynamics of the non-local term W is completely determined by the outflux and the influx, since

dW dt = u -y in D (0, ∞).
(1.12)

• If one wants to estimate W precisely, it is convenient to set u = y, so that the equation (??)-(??)-(??)-(??) can be seen as an equation on the torus with constant speed, which thus satisfies several nice properties, such as time periodicity for instance.

Based on these ideas and elements, we show a stabilization result by alternating two types of phases:

• Acquisition steps, in which we simply choose u = y and try to improve the approximation W of W ;

• Stabilization steps, in which we choose the feedback law (??) for k ∈ (-1, 1), with W solving

d W dt = u -y (1.13)
during these steps.

We therefore propose two new closed-loop systems with time varying feedback laws for which we can establish the local exponential stability of ρ under the weaker condition

g(s) := sλ(s) is a local C 1 diffeomorphism around ρ, (1.14)
which is in fact guaranteed if and only if g (ρ) = 0, i.e.

d = ρλ (ρ) λ(ρ) = -1. (1.15)
But as our first result is in fact a semi-global one, we will rather state in its more general version:

Theorem 1.1. Let ρ ∈ R, λ ∈ C 1 (R; R * + ), and k ∈ (-1, 1). Let R 0 > r 0 > 0 be such that the function g defined by g(s) = sλ(s) is a C 1 diffeomorphism of [ρ -R 0 , ρ + R 0 ] on its image. Assume that ρ 0 ∈ L p (0, 1) for some p ∈ (1, ∞) satisfies ρ 0 -ρ L p (0,1) r 0 .
(1.16)

Then there exists T 1 = T 1 (r 0 , R 0 ) and T 2 = T 2 (r 0 , R 0 ) such that, introducing the sequence of times

T 0 = 0 < T 1 < T 2 < • • • < T i < T i+1 < • • • -→ +∞, with T 2i+1 -T 2i = T 1 and T 2i+2 -T 2i+1 = T 2 , (1.17) the feedback given for all i ∈ N by u(t) = y(t) for t ∈ (T 2i , T 2i+1 ), (1.18)
and by

           W = (k -1)y(t) + (1 -k)ρλ( W (t)) for t ∈ (T 2i+1 , T 2i+2 ), W (T 2i+1 ) = g -1 1 T 2i+1 -T 2i T2i+1 T2i y(τ ) dτ , u(t) -ρλ( W (t)) = k(y(t) -ρλ( W (t)), for t ∈ (T 2i+1 , T 2i+2 ), (1.19)
is an exponentially stable feedback law around ρ for (??)-(??)-(??)-(??). More precisely, for p ∈ (1, ∞), there exist C p = C p (r 0 , R 0 ) > 0 and α = α(r 0 , R 0 ) > 0 such that, for any ρ 0 ∈ L p (0, 1) satisfying (??), the solution ρ of (??)-(??)-(??)-(??) with initial datum ρ 0 , with the feedback law given by (??) for t ∈ (T 2i , T 2i+1 ) and by (??) for t ∈ (T 2i+1 , T 2i+2 ) satisfies:

ρ(t) -ρ L p (0,1) C p exp(-αt) ρ 0 -ρ L p (0,1) , ∀t ∈ [0, ∞).
(1.20)

Remark 1.1. Strictly speaking, the feedback law proposed in Theorem ?? is not of the form (??), although the control u can be computed only from the knowledge of y.

First, we alternate between acquisition steps, corresponding to the time intervals (T 2i , T 2i+1 ), i ∈ N, and stabilization steps, corresponding to the intervals (T 2i+1 , T 2i+2 ), i ∈ N. Thus, our feedback law is timedependent, alternating between two different laws.

Second, it is important to notice that, for i ∈ N, W (T 2i+1 ) depends on y on (T 2i , T 2i+1 ). Therefore, to compute the control function u at time t = T + 2i+1 , one needs to know y at time t = T + 2i+1 and y| (T2i,T2i+1) . Third, let us remark that on time intervals of the form (T 2i+1 , T 2i+2 ), i ∈ N, the state of the system becomes augmented with the extra unknown W , which is introduced to track W on the time interval.

Remark 1.2. The stabilization result in Theorem ?? is semiglobal, since R 0 and r 0 are not supposed to be small.

For instance, the proof of Theorem ?? gives that, if λ(s) = A/(B + s) for some A and B positive constants, then for every ρ 0, for every non-negative initial state ρ 0 , the feedback law proposed in ?? applies and the corresponding solution ρ converges exponentially, as t → ∞, to ρ.

In fact, in addition to the proof of Theorem ??, one should only check that the function s → g(s) := sλ(s) is a global C 1 diffeomorphism of (0, +∞) on (0, A).

Remark 1.3. It is interesting to notice that, if the condition (??) holds, then Theorem ?? holds for R 0 and r 0 small enough. Accordingly, in this case, it provides a local stabilization result in the neighborhood of ρ.

Thus, the feedback law proposed here allows to consider situations in which the condition d > -1 (recall the definition of d in (??)) is not satisfied, which is the necessary and sufficient condition for the feedback law of the proportional form (??) to work see [?].

Also note that Theorem ?? generalizes the stabilization result of [?] in space L 2 (0, 1) to L p (0, 1) spaces

(p ∈ (1, ∞)).
One of the drawback of the stabilization law proposed in Theorem ?? is that T 1 and T 2 should be large enough and are not very explicit (see Remark ??). This might seem not very effective when considering the acquisition steps (the time intervals of the form [T 2i , T 2i+1 )), in which there is no stability process involved. Therefore, we propose below another approach for the acquisition step, in which the time for the acquisition step is reduced, but which is effective only locally.

Theorem 1.2. Let ρ ∈ R, λ ∈ C 1 (R; R * + ), p ∈ (1, ∞) and k ∈ (-1, 1). Set d = ρλ (ρ) λ(ρ) , (1.21)
and assume that d = -1.

Then there exists T 2 > 0 such that, introducing the sequence of times

T 0 = 0 < T 1 < T 2 < • • • < T i < T i+1 < • • • -→ +∞, with T 2i+2 -T 2i+1 = T 2 , (1.22)
the feedback given for all i ∈ N by u(t) = y(t) for t ∈ (T 2i , T 2i+1 ), (1.23)

with T 2i+1 = T 2i + 1 λ( W (T 2i )) , (1.24)
and by

           W = (k -1)y(t) + (1 -k)ρλ( W (t)) for t ∈ (T 2i+1 , T 2i+2 ), W (T 2i+1 ) = W (T 2i ) - 1 1 + d W (T 2i ) - T2i+1 T2i y(s)ds , u(t) -ρλ( W (t)) = k(y(t) -ρλ( W (t)), for t ∈ (T 2i+1 , T 2i+2 ), (1.25)
is a locally exponentially stable feedback law around ρ for (??)-(??)-(??)-(??). More precisely, for any T 2 such that T 2 > 1/λ(ρ), there exist r 0 > 0, C p > 0 and α > 0 such that for all ρ 0 ∈ L p (0, 1) and W 0 satisfying

ρ 0 -ρ L p (0,1) + | W 0 -W 0 | r 0 , (1.26)
the solution (ρ, W ) of (??)-(??)-(??)-(??) with initial datum (ρ 0 , W 0 ), with the feedback law given by (??) for t ∈ (T 2i , T 2i+1 ) and by (??

) for t ∈ (T 2i+1 , T 2i+2 ) satisfies 1 ρ(t) -ρ L p (0,1) + | W (t) -W (t)| C p exp(-αt) ρ 0 -ρ L p (0,1) + | W 0 -W 0 | , ∀t ∈ [0, ∞). (1.27)
Of course, in practice, one can alternate the acquisition steps proposed in Theorem ?? and Theorem ??, keeping in mind that the one proposed in Theorem ?? is in principle effective only locally around the stationary state ρ.

The organization of this paper is as follows: In Section ??, we prove the well-posedness of the closed-loop system. In Section ??, we prove a stability result under the additional condition that the total mass W 0 is known. Then in Section ??, we give the proof of Theorem ??. In Section ??, we focus on the proof of Theorem ??. Finally, some numerical simulations are shown in Section ??.

Wellposedness of the closed-loop system

In this section, we prove the well-posedness of the closed-loop system presented in Theorem ??.

The boundary condition y = u is classical and we refer to [?, Lemma 2.1] for the following well-posedness result:

Proposition 2.1. Let T > 0, ρ ∈ R. Then for all ρ 0 ∈ L 1 (0, 1), there exists a unique solution ρ ∈

C 0 ([0, T ]; L 1 (0, 1)) of    ∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, T ), u(t) = y(t), t ∈ (0, T ), ρ(x, 0) = ρ 0 (x),
x ∈ (0, 1),

(2.1)

1 Properly speaking, W is defined only on the time intervals [T 2i+1 , T 2i+2 ], and for convenience of notation, we simply set

W (t) = W (T 2i ) for t ∈ (T 2i , T 2i+1 ).
where

       W (t) = 1 0 ρ(x, t)dx, t ∈ (0, T ), u(t) = λ(W (t))ρ(0, t), t ∈ (0, T ), y(t) = λ(W (t))ρ(1, t), t ∈ (0, T ), (2.2) in the following sense: ρ ∈ C 0 ([0, T ]; L 1 (0, 1)), (W, u, y) ∈ W 1,1 (0, T ) × L 1 (0, T ) × L 1 (0, T )
, and for every s ∈ (0, T ], and every ), then the solution ρ of (??) belongs to C 0 ([0, T ]; L p (0, 1)), and

ϕ ∈ C 1 ([0, 1] × [0, s]) satisfying ϕ(x, s) = 0, for all x ∈ [0, 1], one has s 0 1 0 ρ(x, t)(∂ t ϕ(x, t) + λ(W (t))∂ x ϕ(x, t))dxdt - s 0 y(t)ϕ(1, t)dt + s 0 u(t)ϕ(0, t)dt + 1 0 ρ 0 (x)ϕ(x, 0)dx = 0. (2.3) Furthermore, if ρ 0 ∈ L p (0, 1) for some p ∈ [1, ∞
(W, u, y) ∈ W 1,p (0, T ) × L p (0, T ) × L p (0, T ).
We then focus on the feedback law (??) for the system (??)-(??)-(??)-(??), and check that this corresponds to a well-posed system. We first introduce the corresponding definition of weak solution.

Definition 2.1. Let T > 0, ρ ∈ R, k ∈ R, ρ 0 ∈ L 1 (0, 1) and w 0 ∈ R. A pair of functions (ρ, w) ∈ C 0 ([0, T ]; L 1 (0, 1)) × W 1,1 (0, T ) is a weak solution of the Cauchy problem            ∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, T ), u(t) = ky(t) + (1 -k)ρλ(w(t)), t ∈ (0, T ), w (t) = u(t) -y(t), t ∈ (0, T ), ρ(x, 0) = ρ 0 (x), x ∈ (0, 1), w(0) = w 0 , (2.4)
where (W, u, y) are as in (??), if the following properties hold:

(ρ, w) ∈ C 0 ([0, T ]; L 1 (0, 1)) × W 1,1 (0, T ), (W, u, y) ∈ W 1,1 (0, T ) × L 1 (0, T ) × L 1 (0, T )
, and for every s ∈ (0, T ], and every

ϕ ∈ C 1 ([0, 1] × [0, s]) satisfying ∀x ∈ [0, 1], ϕ(x, s) = 0, one has s 0 1 0 ρ(x, t)(∂ t ϕ(x, t) + λ(W (t))∂ x ϕ(x, t))dxdt - s 0 y(t)ϕ(1, t)dt + s 0 u(t)ϕ(0, t)dt + 1 0 ρ 0 (x)ϕ(x, 0)dx = 0 (2.5) and w = u -y in D (0, T ). (2.6)
Remark 2.1. In the previous work [?, Eq. (3.26)], it has been observed that, when ρ satisfies the transport equation (??) (1) the total mass W (t) of ρ, defined by (??) (1) , satisfies

W = u -y, in D (0, T ).
where (W, u, y) ∈ W 1,1 (0, T ) × L 1 (0, T ) × L 1 (0, T ) are defined in system (??). Accordingly, a solution (ρ, w) of (??) should satisfy that w -W is a constant function. This motivates the choice done on (??) (1) .

Remark 2.2. Note that the problem (??) is of course completely equivalent to the Cauchy problem

           ∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, T ), u(t) = ky(t) + (1 -k)ρλ(w(t)), t ∈ (0, T ), w (t) = (k -1)y(t) + (1 -k)ρλ(w(t)), t ∈ (0, T ), ρ(x, 0) = ρ 0 (x), x ∈ (0, 1), w(0) = w 0 , (2.7)
where (W, u, y) are as in (??), which is of course completely similar to the feedback law given in (??) for the system (??)-(??)-(??)-(??).

We now give a well-posedness result for the system (??).

Theorem 2.1. Let ρ ∈ R and k ∈ [-1, 1] be given. For any initial data (ρ 0 , w 0 ) ∈ L 1 (0, 1) × R, there exists a time T * > 0 such that the Cauchy problem (??) has a unique weak solution (ρ, w)

∈ C 0 ([0, T * ); L 1 (0, 1)) × W 1,1 loc ([0, T * )) with (W, u, y) ∈ W 1,1 loc ([0, T * )) × L 1 loc ([0, T * )) × L 1 loc ([0, T * ))
. Moreover, for p ∈ (1, ∞), and ρ 0 ∈ L p (0, 1), the solution (ρ, w) of the Cauchy problem (??) belongs to the set

C 0 ([0, T * ); L p (0, 1)) × W 1,p loc ([0, T * )) with (W, u, y) ∈ W 1,p loc ([0, T * )) × L p loc ([0, T * )) × L p loc ([0, T * )).
Besides, if we denote by T * the maximal time of existence in this class, T * is finite if and only if lim T →T * ρ(t) L p (0,1) = ∞.

Proof of Theorem ??. We first prove the local result of Theorem ??, and show that there exists δ > 0 depending on (ρ 0 , w 0 ), such that the Cauchy problem (??) has a unique weak solution (ρ, w) ∈ C 0 ([0,δ];L 1 (0,1)) × W 1,1 (0,δ) with (W,u,y) ∈ W 1,1 (0,δ) × L 1 (0,δ) × L 1 (0,δ). The proof of this local well-posedness result is based on fixed point arguments, similarly as in [?] for a similar open-loop system and [?] for a similar closed-loop system.

Clearly, since we are looking for W ∈ W 1,1 (0, δ) and since λ takes values in (0, ∞), the system (??) can be rewritten as

           ∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, δ), ρ(0, t) = kρ(1, t) + (1 -k)ρ λ(w(t)) λ(W (t)) , t ∈ (0, δ), w (t) = u(t) -y(t), t ∈ (0, δ), ρ(x, 0) = ρ 0 (x),
x ∈ (0, 1).

(2.8)

with

W (t) = 1 0 ρ(x, t)dx, t ∈ (0, δ).
(2.9)

As mentioned in Remark ??, this implies that

∀t ∈ (0, δ), w(t) = W (t) -W 0 + w 0 , with W 0 = 1 0 ρ 0 (x)dx.
(2.10) Therefore, for parameters δ > 0, and M > 0, that will be chosen later on, we define the following subspace of L 1 (0, δ):

Σ δ,M = {W ∈ L ∞ (0, δ) with W L ∞ (0,δ) M }.
(2.11)

We then introduce the mapping F δ : W ∈ Σ δ,M → W where W is obtained by solving the linear system

       ∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, δ), ρ(0, t) = k ρ(1, t) + (1 -k)ρ λ(W (t) -W 0 + w 0 ) λ(W (t)) , t ∈ (0, δ), ρ(x, 0) = ρ 0 (x),
x ∈ (0, 1),

(2.12) and setting

W (t) = 1 0 ρ(x, t)dx, t ∈ (0, δ).
(2.13) Note that the equation (??) is a classical Cauchy problem for a transport equation, and is thus easily seen to be well-posed.

Let us first prove that there exists M > 0 and δ > 0 such that the mapping

F δ maps Σ δ,M into itself. For M > 0, we introduce S(M ) = sup [-M,M ] λ and I(M ) = inf [-M,M ]
λ (which is positive by assumption). Now, for δ > 0 and M > 0 such that δS(M ) 1, which guarantees that no characteristic has the time to cross the whole domain during the time interval (0, δ). It is then easy to check that:

• ρ(1, •) ∈ L 1 (0, δ) and ρ(1, •) L 1 (0,δ) 1 I(M ) ρ 0 L 1 (1-S(M )δ,1) .
• ρ(0, •) thus belongs to L 1 (0, δ), and setting

g W (t) = (1 -k)ρ λ(W (t) -W 0 + w 0 ) λ(W (t)) , t ∈ (0, δ), using |k| 1, we have ρ(0, •) L 1 (0,δ) 1 I(M ) ρ 0 L 1 (1-S(M )δ,1) + g W L 1 (0,δ) .
• Accordingly ρ ∈ C 0 ([0, δ]; L 1 (0, 1)), and

ρ L ∞ (0,δ;L 1 (0,1)) ρ 0 L 1 (0,1) + S(M ) ρ(0, •) L 1 (0,δ) 1 + S(M ) I(M ) ρ 0 L 1 (0,1) + S(M ) g W L 1 (0,δ) . • Also W satisfies, in D (0, δ), W = λ(W (t))(ρ(1, t) -ρ(0, t)), so that W L 1 (0,δ) 2 I(M ) ρ 0 L 1 (1-S(M )δ,1) + g W L 1 (0,δ) .
Consequently,

W L ∞ (0,δ) |W 0 | + 2 I(M ) ρ 0 L 1 (1-S(M )δ,1) + g W L 1 (0,δ) . Since |k| 1, for W ∈ Σ δ,M , g W L 1 (0,δ) δ g W L ∞ (0,δ) 2δ|ρ| S(M + |W 0 | + |w 0 |) I(M ) , we deduce that if M > |W 0 |, there exists δ = δ(M ) > 0 such that for all δ ∈ (0, δ(M )], the mapping F δ maps Σ δ,M into itself.
We then prove that F δ is a contraction in Σ δ,M endowed with the topology of L 1 (0, δ) for δ small enough.

In order to do that, let us consider W 1 and W 2 in Σ δ,M and call ( ρ 1 , W 1 ), respectively ( ρ 2 , W 2 ), the corresponding solution of (??)-(??). We then compute:

d dt W 1 -W 2 = λ(W 1 (t))( ρ 1 (0, t) -ρ 1 (1, t)) -λ(W 2 (t))( ρ 2 (0, t) -ρ 2 (1, t)) = (k -1)(λ(W 1 (t)) ρ 1 (1, t) -λ(W 2 (t)) ρ 2 (1, t)) + λ(W 1 (t)) -λ(W 2 (t)) 2 (g W1 (t) + g W2 (t)) + λ(W 1 (t)) + λ(W 2 (t)) 2 (g W1 (t) -g W2 (t)) . (2.14)
Now, one one hand, there exists C M > 0 such that for all t ∈ (0, δ),

λ(W 1 (t)) -λ(W 2 (t)) 2 (g W1 (t) + g W2 (t)) + λ(W 1 (t)) + λ(W 2 (t)) 2 (g W1 (t) -g W2 (t)) C M |W 1 (t) -W 2 (t)|. (2.15)
On the other hand, using the characteristic formula, for i ∈ {1, 2}, for all τ ∈ (0, δ),

τ 0 λ(W i (t)) ρ i (1, t) dt = 1 1-ξi(τ ) ρ 0 (x) dx,
where

ξ i (τ ) = τ 0 λ(W i (t)) dt. Accordingly, for τ ∈ (0, δ), δ 0 τ 0 λ(W 1 (t)) ρ 1 (1, t) dt - τ 0 λ(W 2 (t)) ρ 2 (1, t) dt dτ δ 0 1-ξ2(τ ) 1-ξ1(τ ) ρ 0 (x) dx dτ 1 1-S(M )δ |ρ 0 (x)| δ 0 1 (x-ξ1(τ ))(x-ξ2(τ ))<0 dτ dx C M δ 0 |W 1 (t) -W 2 (t)| dt 1 1-S(M )δ |ρ 0 (x)| dx, (2.16)
C M being a constant depending on M only and not on δ, where we have used that for i ∈ {1, 2}, τ → ξ i (τ ) is strictly increasing with derivative bounded from below by I(M ), and that sup τ ∈(0,δ)

|ξ 1 (τ ) -ξ 2 (τ )| dτ sup [-M,M ] |λ | δ 0 |W 1 (t) -W 2 (t)| dt.
Accordingly we obtain from (??)-(??)-(??) that

δ 0 | W 1 (t) -W 2 (t)| dt C M δ 0 |W 1 (t) -W 2 (t)| dt 1 1-S(M )δ |ρ 0 (x)| dx + C M δ δ 0 |W 1 (t) -W 2 (t)| dt. (2.17)
It follows that for δ > 0 small enough, the mapping F δ is a contraction on Σ δ,M endowed with the topology of L 1 (0, δ). Thus there exists a unique fixed point W of F δ in Σ δ,M . By construction, this fixed point corresponds to a solution (ρ, w) of (??), and we easily check that (ρ, w) belongs to

C 0 ([0, δ); L 1 (0, 1)) × W 1,1 (0, δ) with (W, u, y) ∈ W 1,1 (0, δ) × L 1 (0, δ) × L 1 (0, δ).
Also note, for later use, that if ρ 0 ∈ L p (0, 1) for some p > 1, M can be chosen as M = 2 ρ 0 L 1 (0,1) , and the contractivity of F δ is guaranteed by taking δ > 0 such that

C M ρ 0 L p (0,1) (S(M )δ) 1-1/p + C M δ = 1 2 .
Accordingly, the time of local existence is bounded from below uniformly for ρ 0 in bounded balls of L p (0, 1). The fact that this solution is unique comes from the fact that M can be taken arbitrarily large in the above process.

When ρ 0 ∈ L p (0, 1) for some p > 1, of course the proof above applies and yields the existence and uniqueness of a solution (ρ, w) of (??

) in C 0 ([0, T * ); L 1 (0, 1)) × W 1,1 loc ([0, T * )) with (W, u, y) ∈ W 1,1 loc ([0, T * )) × L 1 loc ([0, T * )) × L 1 loc ([0, T * )) for some T * > 0.
It is easy to check in the above proof that ρ 0 ∈ L p (0, 1) implies that, for W ∈ Σ δ,M , the solution ( ρ, W ) of (??)-(??) satisfies:

ρ(1, •) ∈ L p (0, δ), ρ(0, •) ∈ L p (0, δ), ρ ∈ C 0 ([0, δ]; L p (0, 1
)), and W ∈ W 1,p (0, δ). Accordingly, the solution (ρ, W ) of (??) constructed above as the fixed point of

F δ belongs to C 0 ([0, δ]; L p (0, 1)) × W 1,p loc (0, δ) with (W, u, y) ∈ W 1,p (0, δ) × L p (0, δ) × L p (0, δ).
Finally, if we denote by T * the maximal time of existence of the solution (ρ, W ) of (??), if T * is finite, then necessarily lim T →T * ρ(T ) L p (0,1) = ∞. This is indeed a straightforward consequence of the fact, already proved, that the time of local existence is bounded from below uniformly for ρ 0 in bounded balls of L p (0, 1).

Remark 2.3. Note that in Theorem ??, we have stated that T * < ∞ if and only if lim T →T * ρ(T ) L p (0,1) = ∞ for some p ∈ (1, ∞).

The case L 1 is more delicate since the contractivity of the map F δ in the above proof is not quantified in terms of the L 1 norm of ρ 0 , recall (??).

However, in the specific case k = 1, which corresponds to u = y in (??), the situation is very degenerate, as W and w does not depend on time. In this case, it is easy to check that the problem (??) is globally wellposed in L 1 : the solution (ρ, w) belongs to

C 0 ([0, ∞); L 1 (0, 1)) × W 1,1 loc ([0, ∞)) with (W, u, y) ∈ W 1,1 loc ([0, ∞)) × L 1 loc ([0, ∞)) × L 1 loc ([0, ∞)).
For later use, we also give the following well-posedness result, which can be proved in a similar -and in fact, easier -way as the previous result, whose proof is thus left to the reader:

Theorem 2.2. Let k ∈ R. For ρ 0 ∈ L 1 (0, 1) and h ∈ L ∞ (0, ∞), there exists a time T * and a unique weak solution (ρ, W ) of    ∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, T ), u(t) = ky(t) + h(t)λ(W (t)), t ∈ (0, T ), ρ(x, 0) = ρ 0 (x),
x ∈ (0, 1),

(2.18)

where (W, u, y) are as in (??), with (ρ,

W ) ∈ C 0 ([0, T * ); L 1 (0, 1)) × W 1,1 loc ([0, T * )), and (u, y) ∈ L 1 loc ([0, T * )) × L 1 loc ([0, T * )).
Moreover, for p ∈ (1, ∞), and ρ 0 ∈ L p (0, 1), the solution (ρ, w) of the Cauchy problem (??) belongs to the set C 0 ([0, T * );

L p (0, 1)) × W 1,p loc ([0, T * )) with (W, u, y) ∈ W 1,p loc ([0, T * )) × L p loc ([0, T * )) × L p loc ([0, T * )). In this case, T * is finite if and only if lim T →T * ρ(t) L p (0,1) = ∞.
3 An exponentially stable feedback law when the initial mass is known

In this section, we assume that the initial total mass 1 0 ρ 0 (x)dx is known and we design a suitable feedback law in this case. This provides some inspiration about the design of the feedback law proposed in Theorems ?? and ??, which alternates between a natural feedback law and some acquisition step to get knowledge, or rather improved knowledge, on the total mass W .

Theorem 3.1. Let ρ ∈ R and k ∈ (-1, 1). If ρ 0 ∈ L 2 (0, 1) and

1 0 ρ 0 (x)dx is known, (3.1)
then the closed-loop system

                               ∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, ∞), u(t) = ky(t) + (1 -k)ρλ(w(t)), t ∈ (0, ∞), w (t) = (1 -k) (ρλ(w(t)) -y(t)) , t ∈ (0, ∞), ρ(x, 0) = ρ 0 (x),
x ∈ (0, 1),

w(0) = 1 0 ρ 0 (x)dx,
where

W (t) = 1 0 ρ(x, t)dx, t ∈ (0, ∞), u(t) = λ(W (t))ρ(0, t), t ∈ (0, ∞), y(t) = λ(W (t))ρ(1, t), t ∈ (0, ∞), (3.2)
admits a global solution (ρ, w), which converges exponentially to (ρ, ρ), i.e. there exist constants C > 0 and α > 0 depending on k, ρ and ρ 0 L 2 (0,1) , such that

|w(t) -ρ| + ρ(•, t) -ρ L 2 (0,1) Ce -αt ρ 0 (•) -ρ L 2 (0,1) , t ∈ [0, ∞). (3.3)
Furthermore, for all times t 0, w(t) = W (t).

For sake of simplicity, we assumed in Theorem ?? that the initial datum ρ 0 belongs to L 2 (0, 1), but it is easy to check that the proof of Theorem ?? can be adapted to deal with initial data in L p (0, 1) for p ∈ (1, ∞), in which case the exponential decay (??) of the L 2 norm ρ(•, t) -ρ L 2 (0,1) should be replaced by an exponential decay of the L p norm ρ(•, t) -ρ L p (0,1) .

Proof of Theorem ??. Let us first denote by T * the maximal time of existence of the solution (ρ, w) of (??).

Note that from Remark ?? W (t) = u(t) -y(t), t ∈ (0, T * ).

(3.4)

Using (??) (2,3) , we get that w = W in D (0, T * ). Since we also have w(0) = W (0) by (??) (5) , we get that for all t 0, w(t) = W (t). Consequently, the boundary condition (??) (2) simply reduces to

ρ(0, t) -ρ = k(ρ(1, t) -ρ), t ∈ (0, T * ). (3.5)
We then show that the solution of (??) is global, that is T * = ∞. In order to do that, let us remark that in D (0, T * ),

d dt 1 0 |ρ(x, t) -ρ| 2 dx = -λ(W (t)) 1 0 d dx |ρ(x, t) -ρ| 2 dx = λ(W (t))(|k(ρ(1, t) -ρ)| 2 -|ρ(1, t) -ρ| 2 ) 0, (3.6) since |k| 1. Thus t → 1 0 |ρ(x, t) -ρ| 2 dx is a non-increasing function and ∀t ∈ (0, T * ), 1 0 |ρ(x, t) -ρ| 2 dx 1 0 |ρ 0 (x) -ρ| 2 dx.
Consequently, we obtain, for all t ∈ (0, T * ),

|W (t)| 1 0 |ρ(x, t) -ρ|dx + ρ ρ 0 -ρ L 2 (0,1) + ρ ρ 0 L 2 (0,1) + 2ρ. (3.7)
These uniform bounds on W and the L 2 norm of ρ 0 -ρ implies that T * = ∞, i.e. that the solution of (??) exists for all t ∈ [0, +∞).

For later use, let us also remark that inf

t 0 {λ(W (t))} inf |s| ρ0 L 2 (0,1) +2ρ {λ(s)} := α, (3.8)
where α is a positive constant depending on ρ 0 L 2 (0,1) and ρ.

We now introduce the Lyapunov functional

V (t) = 1 0 e -βx (ρ(x, t) -ρ) 2 dx, t 0, (3.9)
where β > 0 is a constant to be determined. Clearly, the quantity

V (t) is equivalent to ρ(•, t) -ρ 2 L 2 (0,1) , uniformly with respect to t ∈ (0, ∞). Moreover, in D (0, ∞), dV (t) dt = -λ(W (t))e -βx (ρ(x, t) -ρ) 2 x=1 x=0 -βλ(W (t)) 1 0 e -βx (ρ(x, t) -ρ) 2 dx = λ(W (t))(k 2 -e -β )(ρ(1, t) -ρ) 2 -βλ(W (t))V (t).
(3.10)

Since k ∈ (-1, 1), we can take β > 0 such that e -β k 2 .

(3.11)

With this choice, that we keep up from now on, the above estimate yields

dV (t) dt -βλ(W (t))V (t), in D (0, ∞).
(3.12)

Combined with (??), this yields

dV (t) dt -2αV (t), in D (0, ∞), (3.13) with α = β α/2. Consequently, V (t) e -2αt V (0), ∀t ∈ [0, +∞), (3.14)
Using then that for all t 0,

e -2β ρ(•, t) -ρ 2 L 2 (0,1) V (t) ρ(•, t) -ρ 2 L 2 (0,1) we conclude that ρ(•, t) -ρ L 2 (0,1) e β-αt ρ 0 (•) -ρ L 2 (0,1) , ∀t ∈ [0, +∞). (3.15)
This finishes the proof of Theorem ??, since we immediately have by Cauchy-Schwarz inequality that, for all t 0,

|W (t) -ρ| = 1 0 (ρ(x, t) -ρ) dx ρ(•, t) -ρ L 2 (0,1) ,
and we have already seen that w(t) = W (t) for all t 0.

4 Proof of Theorem ??

The goal of this section is to prove Theorem ??.

As underlined in Section ??, the knowledge of W is critical to design a suitable exponentially stable feedback law, and this is why the algorithm we propose alternate between two phases, one corresponding to an acquisition step in which we try to estimate W at best, and another one corresponding to the natural feedback law that one would use if W were known.

To make the proof of Theorem ?? easier to follow, we split this section in several parts: Section ?? studies the acquisition step, Section ?? focuses on the stabilizing step, and Section ?? put together the results obtained in each section to give the complete proof of Theorem ??.

The acquisition step in Theorem ??

Here we focus on the acquisition steps in which the feedback law is given by u = y, recall (??). For convenience, we study this case in an infinite time horizon. We therefore focus on the equation

                   ∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, +∞), u(t) = y(t), t ∈ (0, +∞), ρ(x, 0) = ρ 0 (x),
x ∈ (0, 1),

where

W (t) = 1 0 ρ(x, t)dx, t ∈ (0, +∞), u(t) = λ(W (t))ρ(0, t), t ∈ (0, +∞), y(t) = λ(W (t))ρ(1, t), t ∈ (0, +∞). (4.1)
Note that the boundary condition (??) (2) also reads

ρ(0, t) = ρ(1, t), t ∈ (0, +∞). (4.2)
That way, we identify x = 0 and x = 1, and the equation (??) can now be thought as taking place in the torus T. We then easily get the following lemma, whose proof is left to the reader:

Lemma 4.1. The solution ρ of (??) with initial datum ρ 0 ∈ L 1 (0, 1) is global in time and satisfies:

• for all t 0,

W (t) = W 0 = 1 0 ρ 0 dx and λ(W (t)) = λ(W 0 ). (4.3)
• if ρ 0 ∈ L p (0, 1) for some p ∈ [1, ∞), ρ belongs to C 0 (R + ; L p (0, 1)) and for all constant ρ ∈ R, for all t 0, ρ(t) -ρ L p (0,1) = ρ 0 -ρ L p (0,1) . (4.4)

Once we have noticed that the velocity t → λ(W (t)) does not depend of time according to (??), the solution ρ of (??) with initial datum ρ 0 ∈ L 1 (0, 1) is periodic in time of period 1/λ(W 0 ), and y(t) = λ(W (t))ρ(1, t) has the same period. Our claim is that we can then recover λ(W 0 )W 0 in the following way: Proposition 4.1. Let ρ 0 ∈ L 1 (0, 1), and ρ the corresponding solution of (??). Then y is 1/λ(W 0 ) periodic and we have the following identity:

λ(W 0 ) 1/λ(W0) 0 y(τ ) dτ = λ(W 0 )W 0 .
(4.5)

Besides, setting, for all t > 0,

F (t) = 1 t t 0 y(τ ) dτ, (4.6)
we get the following estimate:

∀t > 0, |F (t) -λ(W 0 )W 0 | 1 t 1 0 |ρ 0 (x) -W 0 | dx. (4.7)
Before going into the proof of Proposition ??, let us point out that this simple result is in fact crucial to our approach, as it states that F in (??) is an estimator of g(W 0 ) (recall that g is defined by g(s) = sλ(s)). In particular, if g is injective, one can recover W 0 from F .

Proof. The proof of (??) follows from the fact that the velocity λ(W 0 ) does not depend on time, so that y(t) = λ(W 0 )ρ(t, 1) and, from the characteristic formula,

y(t) = λ(W 0 )ρ 0 (1 -tλ(W 0 )) for all t ∈ [0, 1/λ(W 0 )].
We therefore have

1/λ(W0) 0 y(τ ) dτ = λ(W 0 ) 1/λ(W0) 0 ρ 0 (1 -tλ(W 0 )) dt = 1 0 ρ 0 (x) dx = W 0 ,
which immediately implies (??). Now, with F as in (??), using (??) we have

|F (t) -λ(W 0 )W 0 | = 1 t t 0 (y(τ ) -λ(W 0 )W 0 ) dτ 1 t sup t∈[0,1/λ(W0)] t 0 (y(τ ) -λ(W 0 )W 0 ) dτ 1 t 1/λ(W0) 0 |y(τ ) -λ(W 0 )W 0 | dτ 1 t 1 0 |ρ 0 (x) -W 0 | dx,
where the last identity comes from the formula y(t) = λ(W 0 )ρ 0 (1 -tλ(W 0 )) and the change of variable t → 1 -tλ(W 0 ). This completes the proof of (??).

Remark 4.1. During the acquisition step, we get an estimator of g(W 0 ) (recall the definition of g in (??)) and thus g has to be at least injective to recover W 0 from the knowledge of g(W 0 ). This is in fact a necessary and sufficient condition to identify W 0 when considering the system equation (??)-(??). Indeed, if g is not injective, we can find distinct W 0,a , W 0,b such that g(W 0,a ) = g(W 0,b ), and one easily checks that the constant functions ρ a , ρ b defined by ρ a (x, t) = W 0,a and ρ b (x, t) = W 0,b for all x ∈ (0, 1) and t 0 are solutions of (??) for which the corresponding outputs y a (t) = λ(W 0,a )ρ a (1, t) = g(W a

The stabilizing step

In this subsection, we focus on the equations (??)-(??) completed with boundary conditions of the form ρ(0, t) = kρ(1, t) + h(t), t ∈ (0, +∞), (4.8)

for some k ∈ (-1, 1) and in which we assume h ∈ L ∞ (0, +∞). (4.9)

Clearly, (??) is equivalent to u(t) = ky(t) + h(t)λ(W (t)), t ∈ (0, +∞), (4.10) which thus corresponds to the Cauchy problem (??).

Our goal is to prove the following result:

Proposition 4.2. Let k ∈ (-1, 1), h ∈ L ∞ (0, +∞) and ρ 0 ∈ L p (0, 1) for some p ∈ (1, ∞).
Then the solution ρ of (??), with initial datum ρ 0 belongs to C 0 ([0, +∞); L p (0, 1)). Besides, taking β > 0, ε ∈ (0, 1) such that e -β > |k|(1 + ε), (4.11)

and introducing, for all t 0,

λ min (t) = inf τ ∈[0,t] {λ(W (τ ))}, λ max (t) = sup τ ∈[0,t]
{λ(W (τ ))}, (4.12)

we get the following estimates:

ρ(t) L p (0,1) e β(1-λmin(t)t) ρ 0 L p (0,1) + (C p (ε)) 1/p e β 1 + 1 β h L ∞ (0,t) , (4.13)
where C p (ε) > 0 is a constant depending on ε and p.

Proof. For p ∈ (1, ∞), let T * be the maximal time of existence of the solution of (??). We have, in D (0, T * ),

d dτ 1 0 e -pβx |ρ(x, τ )| p dx = p 1 0 e -pβx |ρ(x, τ )| p-2 ρ(x, τ )∂ τ ρ(τ, x) dx = - 1 0 e -pβx λ(W (τ ))∂ x (|ρ(x, τ )| p ) dx = -e -pβ λ(W (τ )) 1-p |y(τ )| p + λ(W (τ )) 1-p |u(τ )| p -pβλ(W (τ )) 1 0 e -pβx |ρ(τ, x)| p dx .
In particular, using (??), for all t ∈ (0, T * ) and τ ∈ (0, t), we obtain for some constant C p (ε) > 0 that

d dτ 1 0 e -pβx |ρ(x, τ )| p dx + pβλ(W (τ )) 1 0 e -pβx |ρ(x, τ )| p dx λ(W (τ )) 1-p -e -pβ |y(τ )| p + (1 + ε) p k p |y(τ )| p + C p (ε)λ(W (τ )) p |h(τ )| p .
Following, for all t ∈ (0, T * ), we have, in D (0, t),

d dτ 1 0 e -pβx |ρ(x, τ )| p dx + pβλ(W (τ )) 1 0 e -pβx |ρ(x, τ )| p dx λ(W (τ ))C p (ε) h p L ∞ (0,t) .
This inequality yields, for all t ∈ (0, T * ),

1 0 e -pβx |ρ(x, t)| p dx e -pβ t 0 λ(W (τ ))dτ 1 0 e -pβx |ρ 0 (x)| p dx + 1 pβ C p (ε) h p L ∞ (0,t) e -pβλmin(t) 1 0 e -pβx |ρ 0 (x)| p dx + 1 pβ C p (ε) h p L ∞ (0,t) ,
where we used

t 0 exp -pβ t τ λ(W (s)) ds λ(W (τ )) dτ 1 pβ .
This proves in particular that for all t ∈ (0, T * ), e -βx ρ(t) L p (0,1) e -βλmin(t)t e -βx ρ 0 L p (0,1)

+ (C p (ε)) 1/p 1 + 1 β h L ∞ (0,t) , (4.14)
where we used that, as p ∈ (1, ∞),

1 pβ 1/p 1 β 1/p 1 + 1 β .
According to Theorem ??, this implies T * = +∞. Besides, immediate bounds on e -βx in (0, 1) immediately yields (??). This concludes the proof Proposition ??.

Proof of Theorem ??

We assume the setting of Theorem ??, with ρ 0 ∈ L p (0, 1) for some p ∈ (1, ∞) and satisfying (??) for some r 0 .

For later use, we introduce the notations

λ min = inf s∈[ρ-R0,ρ+R0]
{λ(s)}, (4.15)

λ max = sup s∈[ρ-R0,ρ+R0]
{λ(s)}, (4.16)

γ g = min{|g(ρ + R 0 ) -g(ρ + r 0 )|, |g(ρ -R 0 ) -g(ρ -r 0 )|}, (4.17) λ ∞ = λ L ∞ ([ρ-R0,ρ+R0]) , (4.18) (g -1 ) ∞ = (g -1 ) L ∞ (g([ρ-R0,ρ+R0])) . (4.19)
We then set for all i ∈ N,

δ p (i) = ρ(T i ) -ρ L p (0,1) , (4.20)
and, for i ∈ N, we denote by e(2i + 1) the error between W (T 2i+1 ) and W (T 2i+1 ) given by

e(2i + 1) = W (T 2i+1 ) -g -1 1 T 2i+1 -T 2i T2i+1 T2i y(τ ) dτ = |W (T 2i+1 -W (T 2i+1 )| . (4.21)
We will construct the iteration such that for all i ∈ N,

δ p (2i) r 0 . (4.22)
(Recall the assumption (??), which can be rewritten under the form δ p (0) r 0 .) In order to show that our construction makes sense, we fix some i ∈ N such that our solution ρ has been constructed on [0, T 2i ] and δ p (2i) r 0 . (4.23) which in particular implies that |W (T 2i ) -ρ| r 0 , and we show that for a suitable choice of T 1 = T 2i+1 -T 2i and T 2 = T 2i+2 -T 2i+1 , we get δ p (2i + 2) r 0 .

Iteration: The acquisition step

According to Section ??, we have

ρ(t) -ρ L p (0,1) = δ p (2i) for all t ∈ [T 2i , T 2i+1 ], (4.24)
and in particular, δ p (2i + 1) = δ p (2i). (4.25)

Recall then that W (t) is constant on [T 2i , T 2i+1 ] from Lemma ??. We thus set

W 2i = W (T 2i ) and λ 2i = λ(W (T 2i )).
Besides, if for t > T 2i , we introduce

F 2i (t) = 1 t -T 2i t T2i y(τ ) dτ, Proposition ?? then implies that for t ∈ (T 2i , T 2i+1 ), |F 2i (t) -g(W 2i )| 1 t -T 2i 1 0 |ρ(x, T 2i ) -W 2i | dx 1 t -T 2i 1 0 |ρ(x, T 2i ) -ρ| dx + |ρ -W 2i | 2 t -T 2i δ p (2i).
Taking

T 2i+1 -T 2i 2r 0 γ g , (4.26)
where γ g is defined in (??), and noting (??), we get immediately that 2 T 2i+1 -T 2i δ p (2i) γ g , and from the above that (??) that F (T 2i+1 ) ∈ g([ρ -R 0 , ρ + R 0 ]), and thus

W (T 2i+1 ) = g -1 1 T 2i+1 -T 2i T2i+1 T2i y(τ ) dτ is well defined and belongs to [ρ -R 0 , ρ + R 0 ].
Moreover, using (??), we obtain

e(2i + 1) 2 T 2i+1 -T 2i (g -1 ) ∞ δ p (2i), (4.27)
where (g -1 ) ∞ is defined in (??).

In the following, we also assume that

2 T 2i+1 -T 2i (g -1 ) ∞ R 0 -r 0 4r 0 , (4.28)
so that we have

|W (T 2i+1 ) -ρ| δ p (2i) r 0 and |W (T 2i+1 ) -W (T 2i+1 )| R 0 -r 0 4 and | W (T 2i+1 ) -ρ| R 0 + 3r 0 4 . (4.29)
In the following, we always assume that T 2i+1 -T 2i = T 1 is chosen so that conditions (??) and (??) are satisfied, i.e.

T 1 max 2r 0 γ g , 8r 0 (g -1 ) ∞ R 0 -r 0 . (4.30)

Iteration: The stabilization step

Setting ρ(x, t) = ρ(x, t) -ρ, for all x ∈ (0, 1) and t ∈ (T 2i+1 , T 2i+2 ),

λ(s) = λ(s + ρ), for s ∈ R,
we easily check that ρ solves

∂ t ρ + λ( W )∂ x ρ = 0 for (x, t) ∈ (0, 1) × (T 2i+1 , T 2i+2 ), (4.31)
in which W denotes the total mass of ρ:

W (t) = 1 0 ρ(x, t) dx for t ∈ (T 2i+1 , T 2i+2 ), (4.32)
with boundary conditions

ρ(0, t) = k ρ(1, t) + h(t) for t ∈ (T 2i+1 , T 2i+2 ), (4.33)
where

h(t) = (1 -k)ρ λ( W (t)) λ(W (t)) -1 . (4.34)
and W (t) is given by (??). We then remark that, at least while the solution is well-defined, integrating the equation (??), we also have, for all t ∈ (T 2i+1 , T 2i+2 ),

W (t) = u(t) -y(t) = (k -1)y(t) + (1 -k)ρλ( W (t)) = ( W ) (t), i.e. ( W (t) -W (t)) = 0 for t ∈ (T 2i+1 , T 2i+2 ). Since | W (T 2i+1 ) -W (T 2i+1 )| = e(2i + 1), we therefore get ∀t ∈ (T 2i+1 , T 2i+2 ), | W (t) -W (t)| = e(2i + 1). (4.35)
• The solution is well-defined on (T 2i+1 , T 2i+2 ) and W and

W belong to [ρ-R 0 , ρ+R 0 ] on [T 2i+1 , T 2i+2 ].
Let us denote by T * 2i+2 the largest time smaller than T 2i+2 such that

| W (t) -ρ| R 0 and |W (t) -ρ| R 0 for t ∈ (T 2i+1 , T * 2i+2 ) and ρ(t) L p (0,1) < ∞. (4.36)
Our goal is to show that for suitable choices of T 1 and T 2 , T * 2i+2 = T 2i+2 . Then using (??) and (??), (??) and (??),

h L ∞ (T2i+1,T * 2i+2 ) (1 -k)ρ λ ∞ λ min e(2i + 1) (1 -k)ρ λ ∞ λ min 2 (g -1 ) ∞ T 2i+1 -T 2i δ p (2i), (4.37)
where we used the notations (??)-(??)-(??). For β > 0 and ε > 0 small enough to satisfy e -β > |k|(1 + ε), applying Proposition ?? for (??)-(??)-(??)-(??), we get for all t ∈ (T 2i+1 , T * 2i+2 ) that

ρ(t) L p (0,1) e β(1-λmin(t-T2i+1)) δ p (2i) + (C p (ε)) 1/p e β 1 + 1 β (1 -k)ρ λ ∞ λ min 2 (g -1 ) ∞ T 2i+1 -T 2i δ p (2i), that is ρ(t) -ρ L p (0,1) e β(1-λmin(t-T2i+1)) δ p (2i) + (C p (ε)) 1/p e β 1 + 1 β (1 -k)ρ λ ∞ λ min 2 (g -1 ) ∞ T 2i+1 -T 2i δ p (2i). (4.38)
Therefore for all t ∈ (T 2i+1 , T * 2i+2 ),

|W (t) -ρ| ρ(t) -ρ L p (0,1) e β δ p (2i) + M ε T 2i+1 -T 2i δ p (2i), (4.39)
where

M ε := (C p (ε)) 1/p e β 1 + 1 β (1 -k)ρ λ ∞ λ min 2 (g -1 ) ∞ . (4.40)
According to the estimates (??)-(??) and Theorem ??, the solution ρ is well-defined on the whole interval (T 2i+1 , T 2i+2 ), and T * 2i+2 necessarily equals to T 2i+2 if we manage to show that W and W cannot get outside [ρ -R 0 , ρ + R 0 ] on the time interval (T 2i+1 , T 2i+2 ). This is the goal of the analysis below. Now for fixed R 0 > r 0 > 0, we take β > 0 small enough such that

e β < min 1 + R 0 -r 0 3r 0 , 1 |k| , (4.41)
ε > 0 such that (??) holds, and, in accordance with (??), we choose

T 1 = T 2i+1 -T 2i such that T 1 max 2r 0 γ g , 8r 0 (g -1 ) ∞ R 0 -r 0 , 3M ε r 0 R 0 -r 0 , (4.42)
where M ε is given by (??).

Then it follows from (??) that for t ∈ (T 2i+1 , T * 2i+2 ),

|W (t) -ρ| δ p (2i) 1 + 2(R 0 -r 0 ) 3r 0 2R 0 + r 0 3 < R 0 for t ∈ [T 2i+1 , T * 2i+2 ].
On the other hand, we have from (??) that for all t ∈ (T 2i+1 , T * 2i+2 ),

| W (t) -ρ| | W (t) -W (t)| + |W (t) -ρ| | W (T 2i+1 ) -W (T 2i+1 )| + 2R 0 + r 0 3 R 0 -r 0 4 + 2R 0 + r 0 3 < R 0 .
Accordingly, T * 2i+2 = T 2i when β is chosen as in (??) and T 1 = T 2i+1 -T 2i is chosen as in (??). • Decay of the L p -norms.

By (??) and (??), we have, with M ε as in (??), that for all t ∈ [T 2i+1 , T 2i+2 ],

ρ(t) -ρ L p (0,1) = ρ(t) L p (0,1) e β(1-λmin(t-T2i+1)) δ p (2i) + M ε T 2i+1 -T 2i δ p (2i). (4.43)
In particular, for t = T 2i+2 in (??), we obtain by (??) that

δ p (2i + 2) e β(1-λmin(T2i+2-T2i+1)) δ p (2i) + M ε T 2i+1 -T 2i δ p (2i).
Therefore, if T 2 = T 2i+2 -T 2i+1 and T 1 = T 2i+1 -T 2i satisfy, in addition to (??),

e β(1-λminT2) 1 4 and M ε T 1 1 4 i.e., T 2 1 λ min 1 + log 4 β and T 1 4M ε , (4.44)
we get

δ p (2i + 2) 1 4 δ p (2i) + 1 4 δ p (2i) 1 2 δ p (2i). (4.45) 4.3.3 Complete Iteration: From T 2i to T 2i+2
For fixed R 0 > r 0 > 0 and k ∈ (-1, 1), we take β > 0 such that (??) holds, ε > 0 such that (??) holds, and, with M ε given by (??), we set

T 1 = max 2r 0 γ g , 8r 0 (g -1 ) ∞ R 0 -r 0 , 3M ε r 0 R 0 -r 0 , 4M ε , (4.46)
T 2 = 1 λ min 1 + log 4 β . (4.47)
By Section ?? and Section ??, we obtain that for all i ∈ N, the solution ρ of (??)-(??)-(??)-(??) with the feedback law given by (??) for t ∈ (T 2i , T 2i+1 ) and by (??) for t ∈ (T 2i+1 , T 2i+2 ) satisfies, if the initial condition ρ 0 is such that ρ 0 -ρ L p (0,1) r 0 , that for all i ∈ N,

δ p (2i + 1) = δ p (2i) and δ p (2i + 2) 1 2 δ p (2i).
In particular, we also get ∀i ∈ N, δ p (2i) 2 -i δ p (0). (4.48)

Let us now explain how to obtain the estimate (??) from there. Recall that T 0 = 0 and

T 2i+1 = (i + 1)T 1 + iT 2 and T 2i+2 = (i + 1)T 1 + (i + 1)T 2 for all i ∈ N, (4.49)
where T 1 , T 2 be given by (??) and (??). Then for any t ∈ [0, ∞), j = t/(T 1 + T 2 ) is the integer such that T 2j t < T 2j+2 . If t ∈ [T 2j , T 2j+1 ], we get from (??) and (??) that ρ(t) -ρ L p (0,1) = δ p (2j) 2 -j δ p (0) = 2 -j ρ 0 -ρ L p (0,1) 2e -log 2

T 1 +T 2 t ρ 0 -ρ L p (0,1) .

If t ∈ [T 2j+1 , T 2j+2 ], we get from (??), (??) and (??) that ρ(t) -ρ L p (0,1) e β ρ(t)e -βx L p (0,1)

e β δ p (2j) + 1 4 δ p (2j) e β + 1 4 2 -j δ p (0) 2 e β + 1 4 e -log 2
T 1 +T 2 t ρ 0 -ρ L p (0,1) .

These last two estimates conclude the proof of Theorem ??.

Remark 4.2. From the above proof, it is clear that Theorem ?? can be proven for any T 2 > 1/λ min , by taking, with the notations in the above proof,

T 1 = max 2r 0 γ g , 8r 0 (g -1 ) ∞ R 0 -r 0 , 3M ε r 0 R 0 -r 0 , 1 2(1 -e β(1-λminT2) ) M ε .
5 Proof of Theorem ??

Similarly as in Section ??, we will distinguish the study of the acquisition steps from the stabilizing steps.

Since only the acquisition step differs from the one in Theorem ??, we will focus on it before proving Theorem ??.

The acquisition step in Theorem ??

In this section, we still consider the Cauchy problem (??).

Let us recall that the problem is that we only have an approximation W 0 of W 0 , and our goal is to get a better approximation of W 0 based on the outflux y.

In particular, our goal is to explain why the value

W 1 = W 0 - 1 1 + ρλ ( W 0 )/λ( W 0 ) W 0 - T0 0 y(s) ds , (5.1)
or

W 1 = W 0 - 1 1 + ρλ (ρ)/λ(ρ) W 0 - T0 0 y(s) ds , (5.2) with T 0 = 1 λ( W 0 ) , (5.3)
is a better approximation of W 0 than W 0 , at least if the initial data are close to ρ.

Theorem 5.1. Assume that ρλ (ρ) + λ(ρ) = 0 (equivalently ρλ (ρ)/λ(ρ) = -1). Let ε 0 > 0 be such that for all s ∈ [ρ -ε 0 , ρ (5.4) setting ( W 1 , W 1 , T 0 ) as in (??)-(??), we have

+ ε 0 ], ρλ (s) + λ(s) = 0. Then, for p ∈ (1, ∞) there exists C > 0 such that if (ρ 0 , W 0 ) ∈ L p (0, 1) × R satisfies | W 0 -ρ| + |W 0 -ρ| ε 0 ,
| W 1 -W 0 | C |W 0 -W 0 | 2 + ρ 0 -ρ L p (0,1) |W 0 -W 0 | 1-1/p , (5.5) and | W 1 -W 0 | C |W 0 -W 0 | 2 + | W 0 -ρ| 2 + ρ 0 -ρ L p (0,1) |W 0 -W 0 | 1-1/p .
(5.6)

Before going into the proof of Theorem ??, let us mention that although the estimate (??) is valid under the only condition of | W 0 -W 0 | being small enough, it expresses that W 1 is closer to W 0 than W 0 only when ρ 0 -ρ L p (0,1) is at most of the order | W 0 -W 0 | 1/p . This means that the estimate (??) on W 1 can be used mainly when ρ 0 is close to ρ, in which case we can also use W 1 instead of W 1 , thus yielding to the estimate (??).

Proof of Theorem ??. In order to prove Theorem ??, it is convenient to rewrite the system (??) on R with a initial datum ρ 0 extended by 1-periodicity to R. Indeed, since u = y in (??), W is independent of time (recall Lemma ??), and the velocity λ(W (t)) is thus constant equal to λ(W 0 ), denoted by λ 0 up to the end. It is then clear that, given ρ 0 ∈ L 1 (0, 1), the solution ρ of

   ∂ t ρ + λ 0 ∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, +∞), ρ(0, t) = ρ(1, t), t ∈ (0, +∞), ρ(x, 0) = ρ 0 (x),
x ∈ (0, 1), coincides, on (0, 1) × (0, ∞), with the solution ρ of

∂ t ρ + λ 0 ∂ x ρ = 0, (x, t) ∈ R × (0, +∞), ρ(x, 0) = ρ 0 (x), x ∈ R, (5.7)
where, with a slight abuse of notation, we denote the same for ρ 0 and its 1-periodic extension on R.

We then work on the solutions ρ of (??). It is obvious that

ρ(x, t) = ρ 0 (x -tλ 0 ), (x, t) ∈ R × (0, ∞).
Accordingly,

T0 0 y(s) ds = T0 0 λ 0 ρ 0 (1 -tλ 0 ) ds = 1 1-λ0 T0 ρ 0 (x) dx = 1 0 ρ 0 (x) dx - 1-λ0 T0 0 ρ 0 (x) dx = W 0 -ρ(1 -λ 0 T 0 ) - 1-λ0 T0 0 (ρ 0 -ρ) dx = W 0 -ρ 1 - λ(W 0 ) λ( W 0 ) - 1-λ(W0)/λ( W0) 0 (ρ 0 -ρ) dx.
(5.8)

We then estimate the two last terms. On one hand, we get

1 - λ(W 0 ) λ( W 0 ) + λ ( W 0 ) λ( W 0 ) (W 0 -W 0 ) sup [ρ-ε0,ρ+ε0] λ inf [ρ-ε0,ρ+ε0] λ |W 0 -W 0 | 2 .
On the other hand, we have

1-λ(W0)/λ( W0) 0 (ρ 0 -ρ) dx ρ 0 -ρ L p (0,1) 1 - λ(W 0 ) λ( W 0 ) 1-1/p ρ 0 -ρ L p (0,1) |W 0 -W 0 | 1-1/p sup [ρ-ε0,ρ+ε0] λ inf [ρ-ε0,ρ+ε0] λ 1-1/p
.

By plugging these two estimates into the identity (??), we get

T0 0 y(s) ds + ρλ ( W 0 ) λ( W 0 ) W 0 -W 0 1 + ρλ ( W 0 ) λ( W 0 ) C |W 0 -W 0 | 2 + ρ 0 -ρ L p (0,1) |W 0 -W 0 | 1-1/p . (5.9)
We then immediately deduce the estimate (??).

To prove the estimate (??), we proceed similarly, the only difference being in the estimate of the second term in (??), which is now as follows:

1 - λ(W 0 ) λ( W 0 ) + λ (ρ) λ(ρ) (W 0 -W 0 ) sup [ρ-ε0,ρ+ε0] λ λ | W 0 -ρ|| W 0 -W 0 | + sup [ρ-ε0,ρ+ε0] λ inf [ρ-ε0,ρ+ε0] λ |W 0 -W 0 | 2 .
The end of the proof of estimate (??) then follows line to line the one of (??).

Remark 5.1. Considering the above proof, the estimate (??) could be generalized for arbitrary W 0 as follows: for all ε > 0, there exists a positive constant C = C(ε) such that for all W 0 satisfying

| W 0 -W 0 | + | W 0 -ρ| ε, we have | W 1 -W 0 | C(ε) | W 0 -W 0 | 2 + | W 0 -ρ| 2 + ρ 0 -ρ L p (0,1) |W 0 -W 0 | 1-1/p .
(5.10) However, since in the following, we will have to restrict ourselves to local stabilization results to get the exponential decay of the solutions of the system under consideration in Theorem ??, the restriction (??) is irrelevant in the proof of Theorem ??.

For later use, let us also remark that due to the estimate

∀a, b 0, ab (p + 1) 2p a 2p/(p+1) + (p -1) 2p b 2p/(p-1) ,
and |W 0 -ρ| ρ 0 -ρ L p (0,1) , under the setting of Theorem ??, we get the existence of a constant C depending on p such that

| W 1 -W 0 | C | W 0 -W 0 | 2 + ρ 0 -ρ 2p/(p+1) L p (0,1)
.

(5.11)

Proof of Theorem ??

We assume the setting of Theorem ??. We choose ε 0 as in Theorem ??, and we introduce

λ min = inf s∈[ρ-ε0,ρ+ε0]
{λ(s)}, (5.12)

λ ∞ = sup s∈[ρ-ε0,ρ+ε0]
{λ (s)}.

(5.13)

We assume that ρ 0 ∈ L p (0, 1) and W 0 ∈ R are such that

ρ(0) -ρ L p (0,1) + | W 0 -W 0 | r 0 ,
for a bound r 0 > 0 to be determined, and we let (ρ, W ) be the solution of (??)-(??)-(??)-(??) with the feedback law given by (??) for t ∈ (T 2i , T 2i+1 ) and by (??) for t ∈ (T 2i+1 , T 2i+2 ). We then set for all i ∈ N,

δ p (i) = ρ(T i ) -ρ L p (0,1)
, and e

(i) = | W (T i ) -W (T i )|.
(5.14)

We assume that r 0 ε 0 , with ε 0 as in Theorem ??, so that we get δ p (0) + e(0) r 0 ε 0 .

(5.15)

Let us take i ∈ N such that δ p (2i) + e(2i) r 0 (5.16) and let us show that, under suitable smallness assumption on r 0 , we can guarantee that δ p (2i + 2) + e(2i + 2) (δ p (2i) + e(2i))γ, (5.17)

for some γ ∈ (0, 1). During the acquisition step, since W is constant on (T 2i , T 2i+1 ) from Lemma ??, we get from (??) that e(2i + 1) C e(2i) 2 + δ p (2i) 2p/(p+1) , and δ p (2i + 1) = δ p (2i).

(5.18)

During the stabilizing step, as argued in the proof of Theorem ??, let us defined T * 2i+2 as the largest time larger than T 2i+1 and smaller than T 2i+2 such that

∀t ∈ (T 2i+1 , T * 2i+2 ), ρ(t) -ρ L p (0,1) + | W (t) -ρ| ε 0 . Then we have ∀t ∈ (T 2i+1 , T * 2i+2 ), | W (t) -W (t)| = e(2i + 1). (5.19)
Besides, taking β > 0 and ε > 0 such that (??) holds, applying Proposition ??, we get that for all t ∈ (T 2i+1 , T * 2i+2 ), (5.20) where the second estimate comes from (??), and where the used the notations (??)-(??). This guarantees that for all t ∈ (T 2i+1 , T * 2i+2 ),

ρ(t) -ρ L p (0,1) e β δ p (2i + 1) + (C p (ε)) 1/p e β 1 + 1 β h L ∞ (T2i+1,T * 2i+2 ) e β δ p (2i) + (C p (ε)) 1/p e β 1 + 1 β (1 -k)ρ λ ∞ λ min e(2i + 1),
|W (t) -ρ| e β δ p (2i) + C(C p (ε)) 1/p e β 1 + 1 β (1 -k)ρ λ ∞ λ min e(2i) 2 + δ p (2i) 2p/(p+1) , | W (t) -ρ| e β δ p (2i) + C 1 + (C p (ε)) 1/p e β 1 + 1 β (1 -k)ρ λ ∞ λ min e(2i) 2 + δ p (2i) 2p/(p+1) ,
where the second estimate is deduced from the first one, together with (??) and (??). Therefore, if r 0 > 0 is small enough (recall that r 0 quantifies the smallness condition in (??)), we can guarantee that for all t ∈ (T 2i+1 , T * 2i+2 ),

|W (t) -ρ| + | W (t) -ρ| ε 0 2 .
Combined with (??) and Theorem ??, we deduce that T * 2i+2 = T 2i+2 . We also have e(2i + 2) = e(2i + 1), and, with β > 0 and ε > 0 such that (??) holds, applying Proposition ??, we get that for all t ∈ (T 2i+1 , T 2i+2 ),

ρ(t) -ρ L p (0,1) e β(1-λmin(t-T2i+1)) δ p (2i) + (C p (ε)) 1/p e β 1 + 1 β (1 -k)ρ λ ∞ λ min e(2i + 1). (5.21) Taking t = T 2i+2 , we get e(2i + 2) + δ p (2i + 2) e β(1-λminT2) δ p (2i) + C 1 + (C p (ε)) 1/p e β 1 + 1 β (1 -k)ρ λ ∞ λ min e(2i) 2 + δ p (2i) 2p/(p+1) .
It is then clear that, for T 2 such that λ min T 2 > 1, there exists r 0 small enough and γ ∈ (0, 1) such that, if the condition (??) holds at time T 2i , we have (??), and in particular the condition (??) holds at time T 2(i+1) = T 2i+2 .

• T 2i+1 is computed when t n becomes larger than T 2i with the formula

T 2i+1 = T 2i + 1 λ( W n ) .
We also fix a stopping time T stop given by the stopping criterion ρ n -ρ e 2 N ≤ 10 -2 , where e = (1, • • • , 1) and the 2 norm of a = (a 0 , • • • , a N ) is defined as

a 2 N = ∆x N i=1
a 2 j .

Example 1

We choose the velocity function λ, the steady state ρ and the initial datum ρ 0 as follows:

λ(s) = 1 1 + s 2 , s ∈ R, ρ = 0.3, ρ 0 (x) = sin (πx) , x ∈ (0, 1).
With this example, we show that the feedback control (??)-(??) of Theorem ?? can stabilize the system under consideration (Note that in this case, the proportional control (??) in [?] also applies locally, since d = ρλ (ρ)/λ(ρ) = -18/109 > -1 in this case). Since ρ 0 -ρ L 1 (0,1) 2/π < 0.65, Theorem ?? applies, with r 0 = 0.65 and R 0 = 0.7, and we choose T 1 = 6 and T 2 = 5. Theorem ?? also applies, at least locally, with the same choice of T 2 since T 2 > 1/λ(0.3) = 1.09. Figure ?? left shows the convergence of ρ -0.3 e 2 N to 0 for various choices of the parameter k when using the algorithm of Theorem ??, and Figure ?? right plots the same numerical experiments but when using the algorithm of Theorem ??. pic/bar03smallloglh2-eps-converted-to.pdf pic/bar03smallloglh2second-eps-converted-to.pdf N for different values of k in Example 1 using, in the left, the algorithm of Theorem ??, and in the right, the algorithm of Theorem ??.

In Table ?? below, we also compare the different stopping time for the feedback proposed in Theorem ??, Theorem ?? and the simpler proportional feedback (??) in [?] with the same value k, and we do it for several choices of k. k = 0.3 k=0.5 k=0.7 k=0.9 T stop with the algorithm of Theorem ??

9 As expected since d = ρλ (ρ)/λ(ρ) = -18/109 > -1 in this case, the proportional feedback controller (??) stabilizes the system and provides a faster decay of the solutions of the system compared to the algorithms of Theorem ?? and ??, since these both algorithm use acquisition steps during which the stabilization process is momentarily stopped.

However, as underlined in Figures ??, the stopping times for the algorithm of Theorem ?? are significantly reduced compared to the ones of Theorem ??, in which the acquisition step is larger. Also note that in fact, the performance of Theorem ?? is rather close to that of the proportional feedback controller (??) in most cases.

Also note that, as expected, for the three algorithms presented in Table ??, the convergence of the solution is faster when taking |k| smaller.

Let us finally mention that the algorithm of Theorem ??, which theoretically should provide stabilization only locally, is working well in all the above mentioned numerical experiments, although the initial datum is not in a small neighborhood of ρ (recall ρ 0 -ρ L 1 (0,1) 2/π < 0.65).

Example 2

We choose the velocity function λ, the steady state ρ and the initial datum ρ 0 as follows: λ(s) = 1 1 + s 2 , s ∈ R, ρ = 5, ρ 0 (x) = 5 sin πx 2 , x ∈ (0, 1). ( 6.2)

In this case, explicit computations show that d = ρλ (ρ) λ(ρ) = -25 13 < -1, so that the proportional feedback controller (??) does not stabilize the system (??)-(??)-(??)-(??) (recall [?]). Despite this, the feedback control proposed in Theorem ?? and Theorem ?? still work. Note that ρ 0ρ L 1 (0,1) = 5 -10/π < 3, so we can choose r 0 = 3, R 0 = 4, and we choose T 1 = 45 and T 2 = 50 (T 2 > 1/λ(ρ) with this choice) in the algorithm of Theorem ??. Similarly, we choose T 2 = 50 in the algorithm of Theorem ??. Regarding the algorithm of Theorem ??, It might seem surprising that the algorithm of Theorem ?? performs worse than the algorithm of Theorem ?? on that example. In fact, to understand this, one should look at the way both algorithms approximate W , that is Figures ?? and ?? left. Since the initial datum is rather far from the equilibrium, the algorithm of Theorem ?? has difficulty to suitably estimate W at the beginning of the process, while W computed with the algorithm of Theorem ?? catches W rather fast.

Note that this example is the one corresponding to Example 2 in [?, Section 5.2], in which the proportional control (??) drives the solution ρ to the equilibrium ρ = 0.2 instead of the desired equilibrium ρ = 5. The reason lies in the fact that the function g : s → sλ(s) coincides at ρ and ρ, but the value of d = ρλ (ρ)/λ(ρ) at ρ is strictly less than -1, while its value at ρ is strictly larger than -1, which implies in particular that the proportional feedback controller given by (??) is not locally stabilizing around ρ.

Example 3

We choose the velocity function λ, the steady state ρ and the initial datum ρ 0 as follows: λ(s) = 1 1 + s 2 , s ∈ R, ρ = 5, ρ 0 (x) = 10 + sin πx 2 . (6.3)

Note that example (??) differs from example (??) from the choice of the initial datum. This example is the one given in [?, Section 5.2], where it is shown that the proportional control (??) does not work and the solution diverges.

However, the feedback control provided by Theorem ?? and Theorem ?? still work provided that T 2 is large enough, as it is shown in Figure ?? plotting ρ with k = 0.3. In both numerical tests, we choose T 2 = 130, so that it is larger than 1/λ(ρ) and 1/λ(W 0 ) (Since ρ 0 (x) 11 for all x ∈ (0, 1), 1/λ(W 0 ) 1 + 11 2 ). The stopping times are then 634.01 for the algorithm of Theorem ?? and 606.04 for the one of Theorem ??.

pic/bar5bigori3d-eps-converted-to.pdfpic/bar5bigori3dsecond-eps-converted-to.pdf

  ) and y b (t) = λ(W 0,b )ρ b (1, t) = g(W b ) are the same.

Figure 1 :

 1 Figure 1: Convergence as t → ∞ of log ρ -0.3 e 2N for different values of k in Example 1 using, in the left, the algorithm of Theorem ??, and in the right, the algorithm of Theorem ??.

  Figure ?? left shows how W approximates W and Figure ?? right shows the 3D plot of ρ; the stopping time is 169.51. Figure ?? shows the same plots for the algorithm of Theorem ??; the stopping time is then 345.3. pic/bar5smallWhatW-eps-converted-to.pdf pic/bar5small3d-eps-converted-to.pdf

Figure 2 :

 2 Figure 2: Example 2 with the algorithm of Theorem ?? and k = 0.25. Left, plot of W and its approximation W versus time. Right, the 3-d plot of the solution ρ.

Figure 4 :

 4 Figure 4: Example 3. Left, the 3-d plot of the solution ρ provided by the algorithm of Theorem ?? with k = 0.3, T 1 = 120, and T 2 = 130. Right, the 3-d plot of the solution ρ provided by the algorithm of Theorem ?? with k = 0.3, and T 2 = 130.

Table 1 :

 1 Stopping times T stop for k = 0.3, 0.5, 0.7, 0.9 in Example 1.

	.94	18.59	30.18	87.26

* S.E. is partially supported by the Agence Nationale de la Recherche, Project TRECOS, under grant ANR-20-CE40-0009. Z.W. is partially supported by the Natural Sciences Foundation of China, under grant 11971119.

Accordingly, by induction, we get that, if e(0) + δ p (0) r 0 , then, for all i ∈ N, e(2i) + δ p (2i) (e(0) + δ p (0))γ 2i .

Since the times T 2i+1 -T 2i are uniformly bounded from above by 1/λ min , we can conclude as in the proof of Theorem ?? to show the exponential stability (??) for solutions of (??)-(??)-(??)-(??) with the feedback law given by (??) for t ∈ (T 2i , T 2i+1 ) and by (??) for t ∈ (T 2i+1 , T 2i+2 ). Details are thus left to the reader.

Numerical simulations

In this section, we give some numerical simulations which illustrate our theoretical results established before.

Numerical implementations

The numerical scheme is implemented as follows.

For N ∈ N * , the space interval (0, 1) is divided into N points, corresponding to a mesh size ∆x = 1/N . The corresponding time discretization parameter is ∆t = ∆x/λ M , where λ M = max{λ(s), s ∈ R}.

The function ρ is then approximated by a sequence of vectors (

n∈N , in the sense that ρ n m denotes the discrete approximation of the value of ρ at (t n , x m ) where t n = n∆t and x m = m∆x. Similarly, (W n ) n∈N is a sequence of real number approximating W at time t n and given by

The sequence ( W n ) n∈N corresponds to the approximation of W at times (t n = n∆t) n∈N .

Corresponding to the algorithm presented in Theorem ??, we use the following explicit solver, given for n ∈ N by

Corresponding to the algorithm presented in Theorem ??, we do as in (??) when t n belongs to intervals of the form [T 2i , T 2i+1 ) and when t n belongs to intervals of the form [T 2i+1 , T 2i+2 ], except that:

• when t n < T 2i+1 t n+1 for some i, we set

where v i ∈ N is such that T 2i < t vi T 2i + ∆t.


Loris Pace
email: loris.pace@ec-lyon.fr

Matthieu Beley

Mohamed El Khattabi
email: mohamed.el-khattabi@cnrs.fr

Arnaud Bréard

Design and Modeling of a 100W 1MHz GaN-Based Single-Switch Resonant Converter for High Power Density Inherent PFC LED Driver

Keywords: "Resonant converter", "High frequency power converter", "Gallium Nitride (GaN)", "Lighting", "High power density systems"

This paper presents the design of a 100W 1MHz single-switch GaN-based resonant converter for high power density LED driver. The proposed prototype is based on the class E topology and performs Zero Voltage Switching (ZVS) as well as inherent Power Factor Correction (PFC) capability. The design procedure using analytical equations is detailed. A high frequency model of the converter including PCB parasitic is achieved and simulation results are confronted to measurements. The designed converter shows an efficiency of 81 % for an output power of 100W at 1MHz.

Introduction

In a today context of green energy usage, Light Emitting Diodes (LEDs) are gradually replacing incandescent light bulbs and fluorescent lamps in most of the applications. This is mainly due to their higher efficiency, robustness and their control capabilities [START_REF] Lamar | Latest Developments in LED Drivers[END_REF]. However, the forward voltage of LEDs can vary according to several parameters such as temperature, ageing… Therefore, the most efficient solution for lighting is to use a power converter, called LED driver, that is able to regulate the current through the LEDs.

Along with the increasing use of LEDs, there is an increasing demand for efficient, compact and high power density LED drivers. Increasing the switching frequency is the most viable way to reduce the size of passive components and to increase the power density of power converters [START_REF] Madsen | Very high frequency resonant DC/DC converters for LED lighting[END_REF]. This is now enabled by the emergence of Wide BandGap (WBG) power semiconductors such as Gallium Nitride (GaN) transistors and Silicon Carbide (SiC) diodes.

As Switching-Mode Power Supplies (SMPS) are nonlinear systems, current harmonics tend to be injected to the AC grid when rectifying the grid voltage for supplying the LEDs. The IEC 61000-3-2 limits the percentage of current harmonics for LED driver with a power greater than 25W. Therefore, a Power Factor Correction (PFC) circuit is required to ensure quasi sinusoidal current absorption. The conventional two-stage of a LED driver is then given in Fig. 1 [START_REF] Tung | Design and Implementation of 150 W AC/DC LED Driver with Unity Power Factor, Low THD, and Dimming Capability[END_REF]. It is constituted of a full bridge rectifier converting AC to DC, a PFC DC/DC converter and a step-down DC/DC converter to supply the LEDs and possibly regulate the output current. Most of the time the PFC converter is a current source input topology controlled to absorb a rectified sinusoidal current [START_REF] Tung | Design and Implementation of 150 W AC/DC LED Driver with Unity Power Factor, Low THD, and Dimming Capability[END_REF], which explains the need of an additional converter in the structure to regulate the LED current. Recently, authors in [START_REF] Mahdi | A Class-E-Based Resonant AC-DC Converter With Inherent PFC Capability[END_REF] have shown that a resonant class E-based converter is able to perform inherent PFC function without any specific control strategy. The tested prototype uses a 1200V SiC MOSFET with a switching frequency of 90kHz. The converter shows an efficiency of 88% for an output power of 211W. Some other related works have emerged these last years. A 10W class E/F prototype operating at 200kHz is presented in [START_REF] Ashique | A Class E/F 3 based 10W LED Driver with ZVS Capability[END_REF]. The converter achieves high power factor with an efficiency around 80%. A 50W 1MHz resonant converter based on the association of a charge pump circuit and a class DE inverter is proposed in [START_REF] Ammar | Analysis and Design of a Charge-Pump-Based Resonant AC-DC Converter With Inherent PFC Capability[END_REF]. The prototype achieves PFC functionality inherently with a maximal efficiency of 88%.

In this paper, it is proposed to design and model a 1MHz 100W single-stage LED driver based on a class E/D DC/DC converter as presented in Fig. 2. The several advantages of the proposed converter are the following:

• High switching frequency (1MHz) for higher power density capability This work also highlights the importance of high frequency modeling of the power converter taking into account the effects of the printed circuit board (PCB) [START_REF] Pace | Parasitic Loop Inductances Reduction in the PCB Layout in GaN-Based Power Converters Using S-Parameters and EM Simulations[END_REF]. The proposed modeling method enables to predict the performances of the converter before manufacturing and potentially to optimize the design. An original EM-Circuit model of the proposed converter is built in this work.

The following section focuses on the characterization of a commercial 100W LED driver in order to collect the specifications for the design of the proposed converter. The third section details the design procedure of the Class E/D topology. The fourth section presents the modeling method of the designed high frequency converter by taking into account the PCB layout effects. The fifth section gives a comparison between experimental and simulation results of the proposed Class E/D DC/DC converter. Then, the performances of the proposed resonant LED driver are presented. Finally, a conclusion discusses on the obtained results and gives prospects on future work.

Characterization of a Commercial LED Driver

In order to collect some specifications for the designed power converter in this work, it is proposed to characterize a commercial LED driver with a conventional 2-stage topology. The tested converter is the XC-100W3A-HTP shown in Fig. 3. It is a 100W LED driver with a fixed average output current of 3A and an output voltage between 29V and 35V. A power factor of 0.95 and an efficiency of 90% are specified by the manufacturer. The power density is estimated to be 350W/l. The experimental power factor is 0.94 and the measured efficiency is 90% which validates the specified values given by the manufacturer. The equivalent resistance of the LED, , is estimated to 11Ω. This value shall be used in the design of the proposed resonant converter.

Design of the Class E/D Converter

The electrical scheme of the proposed class E/D DC/DC converter is given in Fig. 5. It is constituted of a class E inverter and a class D rectifier linked through an L-C resonant circuit. First, the class D rectifier can be modeled by its equivalent input resistance expressed as in (1) [START_REF] Kkelis | Comparison of current driven Class-D and Class-E half-wave rectifiers for 6.78 MHz high power IPT applications[END_REF]. Then, the electrical scheme in Fig. 5 can be simplified as shown in Fig. 6.

1 - 2 + - = 2 (2) = + 2 -1 2 1 - + 2 (3)
Where is the desired transferred power and is the phase shift between and the switch turnon instant.

Knowing that the average voltage of ! " is , the capacitance # $ is calculated using equation ( 4). This demonstration is well detailed in [START_REF] Mahdi | A Class-E-Based Resonant AC-DC Converter With Inherent PFC Capability[END_REF].

# $ = ∆ & (4)
Where, & = 2 ' is the angular frequency at the desired switching frequency and ∆ is a parameter that only depends on the duty cycle . The expression of ∆ is given in [START_REF] Ashique | A Class E/F 3 based 10W LED Driver with ZVS Capability[END_REF].

∆ = 1 -( 1 - + ) + (5)
The components of the resonant tank * and # are sized for a given loaded quality factor + and using the First Harmonic Approximation (FHA) as represented by the equivalent circuit in Fig. 7. The FHA hypothesis is valid as long as the resonant current is sinusoidal, which means that the power is transmitted only at the switching frequency. Fig. 7: Equivalent circuit of the class E output circuit using FHA Thus, * and # are the solutions of the 2equations system given in [START_REF] Ammar | Analysis and Design of a Charge-Pump-Based Resonant AC-DC Converter With Inherent PFC Capability[END_REF]. Where " , is the RMS value of the first harmonic of ! " .

" , can be expressed by [START_REF] Pace | Parasitic Loop Inductances Reduction in the PCB Layout in GaN-Based Power Converters Using S-Parameters and EM Simulations[END_REF]. Whereis a parameter that depends only on the duty cycle . The expression ofis given through equations ( 8) to [START_REF] Wang | A Multiloop Method for Minimization of Parasitic Inductance in GaN-Based High-Frequency DC-DC Converter[END_REF].

⎩ ⎪ ⎨ ⎪ ⎧ + = 1 * # " , = 2* & - 1 # & 3 + (6) " , = - # $ & (7)
-= -4 + -5 2 (8)
-4 = 1 - 2 -2 + 6 2 + 7(2 1 -) 2 + 7( - 4 + ) 4 (9) -4 = 1 - 2 -2 + 6 2 + 7(2 1 -) 2 + 7( - 4 + ) 4 (10) 7 = 2 1 - 2 + - (11) 6
= 1 - + -1 + (12)
Finally, similarly to the conventional Boost converter, the input inductance * should be sized to limit the input current ripple ∆ as given in [START_REF] Galeano | Transmission Line Approach for the PCB Gate Interconnection Design in GaN-Based High-Frequency Power Converters[END_REF]. The output capacitor # 9 is sized using equation ( 14) to limit the output voltage ripple ∆ :;< and ensure a constant current in the LED during the switching period.

* = ∆ ' (13)
# 9 = ∆ :;< ' (14)
For the design of the proposed converter, the power is set to 100W, the switching frequency ' is 1MHz, the input voltage is 100V (average rectified value) and the loaded quality factor + is chosen to be equal to 7. The required passive components values and characteristics are listed in Table I.

Table I: Passive components values and requirements

Value

Requirements

# $ 10nF C0G, > 400V, ' $ ≫ 1>?@ # 15nF C0G, > 400V, ' $ ≫ 1>?@ * 3.9µH
Air core, > 7A, ' $ ≫ 1>?@ * 100µH

Ferrite core, > 2A,

' $ > 1>?@ # 9 1µF X7R, > 100V, ' $ > 1>?@
The maximal voltage " BCD applied to the switch depends on the duty cycle as given in (15). For this application the maximal voltage is evaluated to 250V using (15).

" BCD = E # $ & (15)
Where E is a parameter that only depends on the duty cycle . To obtain the E parameter, it is first necessary to calculate the angle F G4H to which the ! " voltage reaches its maximal value. The expressions of F G4H and E are given by equations ( 16) and (17) respectively.

F G4H = 2 -+ asin 2 1 7 3 (16) E = F G4H -2 + 7(cos F G4H + -cos 2 + ) (17)
The RMS current " in the transistor is calculated using (18) . It should be noted that this expression is only valid for low duty cycle values.

" = 3 - 2 3 sin 2 + (18)
According to these constraints, the selected GaN device is the GS66504B 650V 15A with a

The reverse voltage RRS applied to the rectifier diodes and the RMS diodes current can be calculated using ( 19) and (20). The selected SiC Schottky diodes for this work are the FFSD1065A 650V 10A.

RRS = T (19)

= 2 (20)

EM-Circuit Simulation Model of the Class E/D Converter

In this section an EM-Circuit simulation model of the resonant DC/DC converter that takes into account the influence of the PCB is proposed.

The followed procedure to build the simulation model is given in Fig. 8. First, the passive and active devices are characterized and their equivalent circuit models are determined. Then an EM model of the PCB is built in Advanced Design System (ADS®) software. All models are finally lumped in a transient simulation to analyze the electrical performances of the power converter.

Passive Devices Modeling

The input inductor, * , and the resonant inductor, * , are characterized with the impedance analyzer E4991B-100 within a frequency range of 1 MHz to 100 MHz. Both inductors are then modeled by the equivalent circuit given in Fig. 9(a) by fitting the experimental impedance curves versus frequency [START_REF] Zdanowski | Analytical and experimental determination of the parasitic parameters in high-frequency inductor[END_REF]. Table II summarizes the inductors model parameters.

The impedance curves of the multilayer ceramic capacitors # $ , # and # 9 are extracted from data given by the manufacturer. All capacitors are then modeled by the equivalent circuit given in Fig. 9(b). Table III summarizes the model parameters of the capacitors.

Active Devices Modeling

A complete methodology to build the high frequency models of packaged GaN transistors and SiC Schottky diodes using S-parameter and pulsed I(V) measurements has already been presented in previous work [START_REF] Pace | Extraction of Packaged GaN Power Transistors Parasitics Using S-Parameters[END_REF], [START_REF] Pace | Electrothermal Modeling of GaN Power Transistor for High Frequency Power Converter Design[END_REF]. This method is rigorously followed in this work to get the simulation model of the GaN transistor GS66504B and the SiC diodes FFSD1065A.

PCB EM Modeling

The realized prototype of the Class E/D DC/DC converter and its PCB layout are presented in Fig. 10. The power converter is designed on a 0.4 mm thick FR4 substrate with 35 µm top and bottom copper layers. The bottom layer is dedicated to a ground plane in order to minimize the parasitic inductances in the commutation loops [START_REF] Wang | A Multiloop Method for Minimization of Parasitic Inductance in GaN-Based High-Frequency DC-DC Converter[END_REF]. The procedure detailed in Fig. 8 is followed to build the EM model of the designed PCB using ADS Momentum. The main parameters for the EM simulation are given in Table IV.

Gate Resistors Selection

In this converter, the most critical commutation loop is the gate circuit. Therefore, the gate loop is minimized as shown in Fig. 11.

Fig. 11: Gate loop of the designed converter A S-parameter simulation is performed on the EM model of the PCB in order to determine the gate loop parasitic inductance X Y [START_REF] Pace | Parasitic Loop Inductances Reduction in the PCB Layout in GaN-Based Power Converters Using S-Parameters and EM Simulations[END_REF]. The parasitic inductance is estimated from the imaginary part of the gate loop impedance and its evolution over frequency is shown in Fig. 12. A parasitic gate loop inductance X Y = 2.4nH is found at 1MHz and it can be observed that this value is stable between 1MHz and 100MHz. Using the extracted X Y , the gate resistor Y is selected according to equation (21) in order to limit ! Y" overshoots without excessively slowing down the switching process [START_REF] Galeano | Transmission Line Approach for the PCB Gate Interconnection Design in GaN-Based High-Frequency Power Converters[END_REF].

Y = X Y # $$ (21)
Where # $$ is the input capacitance of the GaN transistor.

All the active and passive devices are modeled by equivalent circuits and the EM Model of the PCB is obtained. Then, a transient simulation can be performed on the complete EM-circuit model of the resonant converter. The next section highlights the main experimental and simulation results obtained in this work.

Experimental and Simulation Results

A comparison between experimental and simulated waveforms of ! " , and :;< is performed at the test conditions listed in Table V.

A DC resistive load of 11.3Ω is used to emulate the power LED. It is constituted of a seriesparallel connection of several power resistors TKH45P50R0FE and TKH45P2R00FE (Ohmite). The rated power for the DC load is 225W.

The results are presented in Fig. 13. It can be seen that the proposed simulation model permits to estimate the experimental waveforms with good accuracy. A non-perfect ZVS switching is obtained as shown in Fig. 13(a). This can be due to the shift on the * value and the presence of parasitic (devices, PCB…) that were not taken into account in the design process. Another observation is that the maximal voltage constraint on the GaN transistor is 2.5 times higher than the input voltage. This can be verified with equation (15).

Table V: Nominal test conditions

Input voltage 100V

Switching frequency 1MHz

Duty cycle 18%

DC load resistance 11.3Ω The obtained output power is 160W for an input voltage = 100 . While a theoretical power of 100W was used in the design procedure. The difference is explained by the design hypothesis that are not completely verified. Indeed, the current through the input inductance presents a ripple of 200mA while the resonant current presents a slight harmonic distortion. Furthermore, the presence of parasitic elements in the circuit and the tolerances, in particular on the resonant inductor value, impact the rated power of the converter. This last point highlights the fact that the design procedure must be improved in future work to take parasitic effects and devices imperfections into consideration.

The voltage waveform at the output of the class E inverter stage, ! ] , is given in Fig. 14. The simulation model permits to estimate the voltage rise time and ringing at 140MHz with good precision as shown in Fig. 14. . These results highlight the inherent PFC operation of the proposed power converter (resistive behavior). Using these results, the AC input current resulting from a rectified AC input voltage of 120V 60Hz can be estimated over an AC grid period as presented in Fig. 16(b). In future work, this waveforms shall be verified experimentally. Experimental results have shown an efficiency of 81% at 100W output power. Future work will focus on improving this efficiency. Using the developed HF simulation model, the air-core resonant inductor * could be designed on PCB in order to better manage the inductance value and the parasitic. An integration of * in a PCB assembly is considered as a perspective. Also, the diodes of the Class D rectifier shall be replaced by lower voltage rating Schottky diodes with better dynamics and lower losses. Finally, a major future prospect is to improve the design procedure by considering circuit and devices parasitic as parts of the converter topology.

Fig. 1 :

 1 Fig. 1: Conventional 2-stage LED driver

Fig. 2 :

 2 Fig. 2: Proposed LED driver structure based on a class E/D DC/DC converter

Fig. 3 :

 3 Fig. 3: Experimental test bench of the commercial LED driver

Fig. 4 :

 4 Fig. 4: Input and output current waveforms of the commercial LED driver

Fig. 5 :

 5 Fig. 5: Electrical scheme of the proposed class E/D DC/DC converter

Fig. 6 :

 6 Fig. 6: Simplified equivalent scheme of the resonant converter for the class E inverter design

2

 2

Fig. 8 :

 8 Fig. 8: EM-Circuit simulation procedure

Fig. 9 :

 9 Fig. 9: Equivalent circuit model of inductors (a) and capacitors (b)

Fig. 10 :

 10 Fig. 10: Prototype of the Class E/D DC/DC converter

Fig. 12 :

 12 Fig. 12: Parasitic gate loop inductance X Y versus frequency (EM Simulation)

Fig. 13 :

 13 Fig. 13: Experimental vs simulated waveforms: (a) ! " (b) (c) :;<

Fig. 14 :

 14 Fig. 14: Experimental vs simulated waveform of the voltage ! ]The efficiency of the designed class E/D DC/DC converter is estimated by measuring the DC input and output powers. Fig.15presents the evolution of the efficiency versus the output power level. An efficiency of 81% is reached for an output power of 100W ( = 80 ).

Fig. 15 :

 15 Fig. 15: Efficiency vs. output power

Fig. 16 :

 16 Fig. 16: (a) Input and output current vs. input voltage (b) AC input current vs. AC input voltage

Table II : Inductors model parameters

 II

	*	100	120	8.8	9.4
	*	3.6	343	41	3.8

* (µH) U $ (mΩ) U V (kΩ) V (pF)

Table III : Capacitors model parameters

 III

	# $	10	15	2
	#	15.5	10	2
	# 9	1000	5	0.65

# (nF) W U (mΩ) W X (nH)

Table IV : EM simulation model parameters

 IV

	EM	Mesh	Mesh
	simulation	frequency	density
	RF Mode	1GHz	350 cells/m

" PQ of 100mΩ and an output capacitance # :$$ of 80pF at 100V.