Ana Djurdjevac
email: adjurdjevac@zedat.fu-berlin.de.

Armen Shirikyan
email: armen.shirikyan@cyu.fr.

Stabilisation of a viscous conservation law by a one-dimensional external force

Keywords: AMS subject classifications: 35L65, 35Q93, 35R60, 93C20 viscous conservation law, exponential stabilisation, approximate controllability, one-dimensional control

published or not. The documents may come

Stabilisation of a viscous conservation law by a one-dimensional external force

Ana Djurdjevac, Armen Shirikyan

Introduction

This paper deals with the problem of stabilisation of a damped-driven conservation law in arbitrary space dimension. To simplify the presentation, we confine ourselves in the introduction to the one-dimensional case. We thus fix an interval I := [a, b] and consider Burgers' equation

∂ t u + ∂ x A(u) -ν∂ 2
x u = h(t, x), x ∈ I, (0.1) supplemented with Dirichlet's boundary condition

u(t, a) = u(t, b) = 0 (0.2)
and the initial condition u(0, x) = u 0 (x). (0.3)

Here ν > 0 is a parameter, A : R → R is a given continuously differentiable function, and h is an external force. It will be convenient to study (0.1)-(0.3) in Hölder's classes of functions, so we fix a number γ ∈ (0, 1) and denote by C γ (I) the space of continuous functions g : I → R such that

g C γ := sup x∈I |g(x)| + sup x1,x2∈I |g(x 1 ) -g(x 2 )| |x 1 -x 2 | γ < ∞.
It is well known that, for any continuous function h : R + × I → R such that

|||h||| γ := sup t≥0, x∈I |h(t, x)| + sup t 1 ,t 2 ≥0 x 1 ,x 2 ∈I |h(t 1 , x 2 ) -h(t 2 , x 2 )| |t 1 -t 2 | γ/2 + |x 1 -x 2 | γ < ∞,
and any initial state u 0 ∈ C γ (I) satisfying an appropriate compatibility condition (see (1.4)), problem (0.1)-(0.3) has a unique solution u : R + ×I → R, continuously differentiable in t and twice continuously differentiable in x. Moreover, u belongs to the space X γ of continuous functions v : R + × I → R whose derivatives ∂ t v, ∂ x v, ∂ 2

x v exist and have finite ||| • ||| γ norms. We now fix an arbitrary non-negative smooth function ϕ ∈ C ∞ 0 (I) that does not vanish identically and has compact support, and consider the controlled Burgers equation

∂ t u + ∂ x A(u) -ν∂ 2
x u = h(t, x) + ξ(t)ϕ(x), x ∈ I, where ξ is a control function. Given an arbitrary solution û ∈ X γ of (0.1), (0.2) and an initial condition u 0 ∈ C γ (I), we wish to find a piecewise-constant function ξ : R + → R such that the solution u ∈ X γ of (0.1)-(0.3) satisfies the inequality

u(t) -û(t) C 2 ≤ Ce -αt ,
where C and α are some positive numbers, and the C 2 norm is defined as the maximum of the sum of the space derivatives up to the second order. If this property holds for any reference solution û, then we shall say that Eq. (0.1) is exponentially stabilisable in C 2 by a one-dimensional external force (proportional to ϕ). The following theorem is a particular case of the main result of this paper formulated in Theorem 2.1.

Main Theorem. In addition to the above hypotheses, suppose that the function A ∈ C 2 (R) satisfies the inequality

A (u)u ≥ c u 2 -C, u ∈ R,
where C and c are some positive numbers. Then Burgers' equation (0.1) is exponentially stabilisable in the space C 2 .

It should be noted that the problem of control and stabilisation of Burgers' equation and more general viscous conservation laws attracted a lot of attention in the last thirty years. In particular, Fursikov and Imanuvilov [START_REF] Fursikov | On controllability of certain systems simulating a fluid flow[END_REF] studied the 1D Burgers equation (0.1) with quadratic flux A(u) = 1 2 u 2 and proved the local exact controllability and the absence of global approximate controllability. The latter property was refined by Diaz [START_REF] Diaz | Obstruction and some approximate controllability results for the Burgers equation and related problems, Control of partial differential equations and applications[END_REF], and further non-controllability results were obtained by Guererro-Imanuvilov [START_REF] Guerrero | Remarks on global controllability for the Burgers equation with two control forces[END_REF]. Glass and Guererro [START_REF] Glass | On the uniform controllability of the Burgers equation[END_REF], Léautaud [START_REF] Léautaud | Uniform controllability of scalar conservation laws in the vanishing viscosity limit[END_REF], Coron [START_REF] Coron | Some open problems on the control of nonlinear partial differential equations[END_REF], and Fernández-Cara-Guererro [START_REF] Fernández-Cara | Null controllability of the Burgers system with distributed controls[END_REF] proved global exact boundary controllability to constant states for Burgers' equation and established some estimates for the time and cost of control. Kröner-Rodrigues [START_REF] Kröner | Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations[END_REF] studied the stabilisation to a nonstationary solution with a control localised in the Fourier space, and the second author [START_REF] Shirikyan | Global exponential stabilisation for the Burgers equation with localised control[END_REF] proved the stabilisation of non-stationary trajectories by a control with support in an interval. In the two-dimensional case, Thevenet-Buchot-Raymond [START_REF] Thevenet | Nonlinear feedback stabilization of a two-dimensional Burgers equation[END_REF] established the exponential stabilisation of stationary solutions by a feedback control. In conclusion, let us mention that various results on controllability and stabilisation were established for other equations of fluid mechanics, such as the Euler, Navier-Stokes, and Boussinesq systems (see the books [START_REF]Controllability of Evolution Equations[END_REF][START_REF] Fursikov | Optimal Control of Distributed Systems[END_REF][START_REF] Coron | Control and Nonlinearity[END_REF] and the references therein). However, we do not mention them here, since they are not directly related to the subject of this paper, whose methods are based on the theory of positivity-preserving operators and are not likely to be applicable in the case of the above-mentioned PDEs.

In the context of damped conservation laws, many questions remain open. We formulate only two of them, closely related to the main result of this paper.

Open problem 1. Theorem 2.1 shows that, given a trajectory û(t, x) of a viscous conservation law and an arbitrary initial condition, one can find a one-dimensional localised control bringing the corresponding solution u(t, x) arbitrarily close to û at some finite time. In the case of Burgers' equation, the Fursikov-Imanuvilov local controllability result [START_REF]Controllability of Evolution Equations[END_REF] implies that the two trajectories can be made to be equal with the help of a localised control. Imanuvilov and Yamamoto [START_REF] Yu | Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations[END_REF] established a similar local exact controllability result in the multidimensional case for a nonlinear parabolic equation, assuming that the nonlinearity has at most linear growth at infinity. It is a challenging open problem to prove the property of exact controllability (in the multidimensional case) with no growth conditions on the flux A.

Open problem 2. The local-in-time L 1 norm of the one-dimensional stabilising control obtained in Theorem 2.1 goes to zero exponentially fast as t → ∞. On the other hand, it follows from (2.3) that the size of control is of order 1 on some intervals. From the point of view of applications, it would be important to prove that the control can be chosen in a feedback form, so that its size decays to zero as t → ∞. This type of (local) results are well known in the case of stationary reference trajectories; see [START_REF]Stabilization for the 3D Navier-Stokes system by feedback boundary control[END_REF][START_REF] Raymond | Feedback boundary stabilization of the threedimensional incompressible Navier-Stokes equations[END_REF].

The paper is organised as follows. In Section 1, we discuss the well-posedness of the damped-driven conservation law and establish some estimates for solutions. In Section 2, we formulate and prove the main result of this paper. The appendix gathers some known results used in the main text.

Acknowledgements

The research of the first author was funded by Deutsche Forschungsgemeinschaft (DFG) through the grant CRC 1114: Scaling Cascades in Complex Systems, Project Number 235221301. The research of the second author was carried out within the MME-DII Center of Excellence (ANR-11-LABX-0023-01) and supported by the Agence Nationale de la Recherche through the grant NONSTOPS (ANR-17-CE40-0006) and by the CY Initiative of Excellence through the grant Investissements d'Avenir ANR-16-IDEX-0008.

Notation

We denote by D a connected bounded domain in R d with a C 3 boundary ∂D and by J T the time interval [0, T ] and write Q T = J T × D. Given a number γ ∈ (0, 1), we denote by C γ (D) the space of continuous functions g : D → R such that

g C γ := sup x∈D |g(x)| + sup x1,x2∈D |g(x 1 ) -g(x 2 )| |x 1 -x 2 | γ < ∞.
Similarly, if k ≥ 1 is an integer, then C k+γ (D) stands for the space of functions g ∈ C γ (D) whose derivatives up to the order k belong to C γ (D). The corresponding norm is defined by the formula

g C k+γ := |α|≤k ∂ α x g C γ .
For a cylinder Q = J × D (where J ⊂ R is an interval), we write C 1,2 (Q) for the space of continuous functions h : Q → R that are continuously differentiable in t and twice continuously differentiable in x. For γ ∈ (0, 1), we denote by C γ 2 ,γ (Q) the space of continuous functions h such that

|||h||| γ := sup (t,x)∈Q |h(t, x)| + sup t 1 ,t 2 ∈J x 1 ,x 2 ∈D |h(t 1 , x 2 ) -h(t 2 , x 2 )| |t 1 -t 2 | γ/2 + |x 1 -x 2 | γ < ∞.
We write X γ (Q) for the space of functions in C 1,2 (Q) whose first-order derivative in t and second-order derivatives in x belong to C γ 2 ,γ (Q). In the case when J is replaced by R + , the spaces mentioned above will be denoted by C 1,2 , C γ 2 ,γ , and X γ , respectively. Finally, we denote by C b the space of bounded continuous functions on the cylinder R + × D.

1 Preliminaries on multidimensional viscous conservation laws

Existence and uniqueness of solution

Let us consider the following initial-boundary value problem in a bounded domain D ⊂ R d with a boundary ∂D ∈ C 3 :

∂ t u + div x A(u) -ν∆u = h(t, x), (1.1)
u ∂D = 0, (1.2) u(0, x) = u 0 (x). (1.3)
Here ν > 0 is a parameter, h and u 0 are continuous functions in R + × D and D, respectively, and A : R → R d is a twice continuously differentiable vector function such that A(0) = 0. We are interested in the existence and uniqueness of solutions for problem (1.1)-(1.3) in Hölder spaces.

Theorem 1.1. Let γ ∈ (0, 1), T > 0, and ν > 0 be any numbers. Then, for any data h ∈ C γ 2 ,γ (Q T ) and u 0 ∈ C 2+γ (D) satisfying the compatibility conditions

u 0 = 0, div x A(u 0 ) -ν∆u 0 -h(0) = 0 on ∂D, (1.4) problem (1.1)-(1.
3) possesses a unique solution u ∈ X γ (Q T ). Moreover, there is a number M > 0 depending continuously on T , |||h||| γ , and u 0 C 2+γ such that

u X γ (Q T ) ≤ M. (1.5)
Proof. We claim that Theorem 6.1 in [LSU68, Chapter V] on quasilinear parabolic PDEs of divergence form is applicable in our situation. Indeed, in view of that result, setting

a(t, x, u, p) = d j=1 A j (u)p j -h(t, x), we only need to check that |a(t, x, u, 0)| ≤ C for all (t, x, u) ∈ Q T × R.
This inequality is obvious, and the existence and uniqueness of solution in the space X γ (Q T ) follows. Inequality (1.5) is established in the proof of Theorem 6.1 in [LSU68, Chapter V].

Remark 1.2. The above result on the existence and uniqueness of solution remains valid if the right-hand side h of (1.1) is a function with the following property: there is partition 0

= t 0 < t 1 < • • • < t n = T of the interval J T such that, for any k ∈ [[1, n]], the restriction of h to Q k := (t k-1 , t k ) × D belongs to C γ 2 ,γ (Q k ).
Indeed, it suffices to apply Theorem 1.1 to each of the cylinders Q k and to consider the concatenation of the resulting solutions. The trajectory obtained by that procedure will not be an element of X γ (Q T ), however, it will be a continuous curve on J T with range in

C 2+γ (D).
In what follows, we shall need various bounds on solutions for (1.1) expressed in terms of some norms of the right-hand side h. To this end, we first recall a comparison principle for nonlinear parabolic PDEs and then apply it to derive required a priori estimates for solutions.

Comparison principle

Let us consider the nonlinear differential operator

F (u) := ∂ t u + d j=1 ∂ j A j (t, x, u) -ν∆u,
where A = (A 1 , . . . , A d ) is a C 1 vector function, and the partial derivative ∂ j under the sum applies to both the x-variable and the x-argument of u. We shall say that a pair of functions (u

+ , u -) belonging to the space C 1,2 (Q T ) is admissible for F (u) if F (u + ) ≥ F (u -) in Q T . (1.6)
Recall that the parabolic boundary of Q T is defined by

∂ p Q T = {(t, x) ∈ R d+1 : t = 0, x ∈ D or t ∈ J T , x ∈ ∂D}.
The following result is established in [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF] for weak solutions under more restrictive hypotheses. Even though its proof is based on well known ideas, we briefly sketch it for the reader's convenience.

Theorem 1.3. Let A ∈ C 1 (R d+2 , R d ) and let (u + , u -) be an admissible pair for F (u) such that u + ≥ u -on ∂ p Q T . Then u + ≥ u -in Q T .
Proof. The function u = u + -u -is non-negative on the parabolic boundary boundary ∂ p Q T and satisfies the differential inequality

∂ t u -ν∆u + div(b(t, x)u) ≥ 0, (1.7)
where we used (1.6) and set

b(t, x) = 1 0 (∂ u A) t, x, u -(t, x) + θu(t, x) dθ.
We need to prove that u(s, x) ≥ 0 for any (s, x) ∈ Q T . To this end, it suffices to show that, for any non-negative function

ϕ ∈ C ∞ 0 (D), D u(s, x)ϕ(x) dx ≥ 0. (1.8)
Together with (1.7), let us consider the dual problem

∂ t ψ + ν∆ψ + b(t, x), ∇ ψ = 0, (1.9) ψ(s, x) = ϕ(x), ψ ∂D = 0 (1.10)
in the domain (0, s) × D. Since ϕ ≥ 0, by the maximum principle (see Section 3.2 in [START_REF] Landis | Second Order Equations of Elliptic and Parabolic Type[END_REF]), we have ψ ≥ 0, so that ∂ n ψ ≤ 0 on ∂D, where n stands for the outward unit normal vector to ∂D. It follows from (1.7) and (1.9) that

d dt (u, ψ) = (∂ t u, ψ) + (u, ∂ t ψ) = - ∂D u∂ n ψ dσ ≥ 0.
Integrating in t ∈ [0, s], using (1.10), and recalling that ϕ ≥ 0 and u(0) ≥ 0, we obtain the required inequality (1.8).

A priori estimates

Let us consider the PDE (1.1), in which h(t, x) is a given bounded continuous function, and the vector function A ∈ C 2 (R, R d ) (which now depends only on u) satisfies the following sign condition.

(S) There are positive numbers c and C such that, for all u ∈ R,

d j=1 A j (u) sgn(u) ≥ c|u| -C. (1.11)
The following result is inspired by Lemma 9 in [Cor07b, Section 2.1], which deals with the 1D Burgers equation.

Proposition 1.4. Let A ∈ C 2 (R, R d ) be a vector function satisfying (S) and let h ∈ C b . Then, for any solution u ∈ C 1,2 of (1.1), (1.2) and any T > 0, we have

u(T ) L ∞ ≤ M, (1.12)
where M > 0 is a number depending only on T , C, c, and h L ∞ .

Proof. Given ε > 0, we define

u + ε (t, x) = B ε B ε + d j=1 x j -dα -+ Lε t + ε , where L = u(0) L ∞ , α -= min{x j : x ∈ D, 1 ≤ j ≤ d}, and B ε > 0 is a continuous increasing function of ε ∈ [0, 1].
We claim that, for an appropriate choice of B ε , the pair (u + ε , u) is admissible for F and satisfies the inequality

u + ε ≥ u on ∂ p Q T .
Indeed, the function u + ε is positive and satisfies the inequality u + ε (0) ≥ u 0 L ∞ . Furthermore, a simple calculation based on (1.11) shows that

F (u + ε )(t, x) ≥ 1 t + ε (B ε c -1)u + ε (t, x) -B ε C .
We wish to prove that the right-hand side of this inequality is greater than h L ∞ for all (t, x) ∈ Q T . It is not difficult to check that this is certainly the case if

B ε = max 2c -1 , 2C(T + ε), C -1 (T + ε) h L ∞ .
Thus, the conclusion of Theorem 1.3 is applicable to the pair (u + ε , u) for any ε > 0, whence it follows that

max x∈D u(T, x) ≤ lim ε→0 + max x∈D u + ε (T, x) ≤ M := T -1 B 0 (B 0 + d(α + -α -)), (1.13)
where

α + = max{x j : x ∈ D, 1 ≤ j ≤ d}.
To derive a lower bound, it suffices to apply a similar argument to the pair (u, u - ε ), where

u - ε (t, x) = - B ε B ε - d j=1 x j + dα + + Lε t + ε .
This results in the inequality

min x∈D u(T, x) ≥ lim ε→0 + min x∈D u - ε (T, x) ≥ -M,
where M is the same constant as in (1.13). The proof of the proposition is complete.

In what follows, we always assume that Condition (S) is fulfilled and do not follow the dependence of various quantities on the numbers c and C entering (1.11).

Corollary 1.5. Let A ∈ C 2 (R, R d ) be a vector function satisfying Condition (S), and let h ∈ C γ 2 ,γ and u 0 ∈ C 2+γ (D) be such that the compatibility conditions (1.4) hold. Then there is K > 0 depending continuously on |||h||| γ and u 0 C 2+γ such that the solution u ∈ X γ of problem (1.1)-(1.3) satisfies the inequality u X γ ≤ K.

(1.14)

Moreover, for any τ > 0 there is K τ > 0 depending only on |||h||| γ such that

u(τ + •) X γ ≤ K τ . (1.15)
Proof. This result is a consequence of the uniform bound (1.12), the a priori estimate (1.5), and the parabolic regularisation. Namely, suppose for any s > 0 we found a function C s : R 2 + → R + increasing in both arguments such that any solution u ∈ X γ satisfies the inequality

u(s) C 2+γ ≤ C s |||h||| γ , u 0 L ∞ .
(1.16)

Using this inequality with t = τ /2 and inequality (1.12) with T ≥ τ /2, we see that u(t) C 2+γ ≤ C τ /2 (|||h||| γ , M ) for any t ≥ τ . The a priori bound (1.5) now implies that (1.15) holds with some K τ > 0. Inequality (1.14) follows immediately from (1.15) and (1.5).

To complete the proof, it remains to establish inequality (1.16). Its proof is based on parabolic regularisation and a standard bootstrap argument, and we only outline it. We fix any s > 0 and use Theorem 10.1 in [LSU68, Chapter III] to find σ ∈ (0, γ) such that u ∈ C σ 2 ,σ (Ω s ) and

u C σ 2 ,σ (Ωs) ≤ M 1 |||h||| γ , u 0 L ∞ ,
where Ω s = (s/3, s) × D, and M i stand for some increasing functions of their arguments. Rewriting (1.1) as the linear parabolic equation

∂ t u -ν∆u + d j=1 a j (t, x)∂ j u = h(t, x),
(1.17)

where

a j = A j • u ∈ C σ 2 ,σ (Ω s )
, we can use inequalities (3.2) and (3.3) for Green's function to conclude that u ∈ X σ (Ω s ), where Ω s = (s/2, s) × D, and the norm u X σ (Ω s ) is bounded by M 2 (|||h||| γ , u 0 L ∞ ). It follows that the coefficients a j in (1.17) belong to C γ 2 ,γ (Ω s ), and their norms are bounded by M 3 (|||h||| γ , u 0 L ∞ ). Applying the above argument with σ replaced by γ, we arrive at the required bound (1.16).

2 Stabilisation by a one-dimensional control

Formulation of the main result

Let D ⊂ R d be a bounded domain with a boundary ∂D ∈ C 3 . We consider the controlled PDE

∂ t u + div x A(u) -ν∆u = h(t, x) + η(t, x), (2.1)
supplemented with the initial and boundary conditions (1.2), (1.3). Here ν > 0 is a fixed parameter,

A ∈ C 2 (R, R d ) is a function satisfying (1.11), h : R + × D → R is a given function in C γ 2 ,γ
, and η is a control supported in a fixed domain Π such that Π ⊂ D.

Theorem 2.1. Let the above hypotheses be satisfied, let û be a solution of (1.1), (1.2) that belongs to the space X γ , let ϕ ∈ C ∞ 0 (Π) be a non-negative function that is not identically zero, and let σ ∈ (0, γ). Then there are positive numbers α, β < 1, and C, not depending on û, such that for any u 0 ∈ C 2+γ (D) satisfying the compatibility condition (1.4) one can find a control η(t, x) = ξ(t)ϕ(x), with a piece-wise constant ξ, for which the solution u(t, x) of problem (2.1), (1.2), (1.3) satisfies the inequality

u(t) -û(t) C 2+σ ≤ Ce -αt min u 0 -û(0) β L 1 , 1 , t ≥ 1. (2.2)
Moreover, there are positive numbers K, κ and sequences ξ k ∈ R, τ k ∈ (0, 1/4) such that

|ξ k | = K, τ k = κ u(k -1) -û(k -1) L 1 for any k ≥ 1, (2.3) ξ(t) = 0 for t ∈ k -1, k -1 2 ∪ k -1 2 + τ k , k , ξ k for t ∈ k -1 2 , k -1 2 + τ k .
(2.4)

A straightforward consequence of the above theorem is the following approximate controllability result.

Corollary 2.2. Under the hypotheses of Theorem 2.1, for any ε > 0, γ ∈ (0, 1), and σ ∈ (0, γ) there is T > 0 such that, given initial states u 0 , û0 ∈ C 2+γ (D) satisfying the compatibility condition (1.4), one can find a piecewise constant function ξ :

J T → R such that u(t) -û(t) C 2+σ ≤ ε, 0 ≤ t ≤ T,
where û ∈ X γ (Q T ) denotes the solution of problem (1.1)-(1.3) with u 0 = û0 , and u : J T → C 2+γ (D) stands for the solution of the control system (2.1), (1.2), (1.3) with η(t, x) = ξ(t)ϕ(x).

Let us sketch the proof of Theorem 2.1, postponing the details to the next two subsections. Note that, by the Gagliardo-Nirenberg-Sobolev interpolation inequality (see [START_REF] Brezis | Where Sobolev interacts with Gagliardo-Nirenberg[END_REF]), for any 0 < σ < γ < 1 and f ∈ C 2+γ (D) we have

f C 2+σ ≤ C f θ L 1 f 1-θ C 2+γ , (2.5)
where θ = γ-σ d+γ+2 . Therefore, the exponential stabilisation (2.2) will be established if we find a control η(t) such that the L 1 -norm of the difference u(t) -û(t) goes to zero exponentially as t → ∞, while the C 2+γ -norm remains bounded. We confine ourselves to the main part of the proof dealing with the exponential decay of the L 1 -norm.

Let us consider the difference w = v -û, where v is another solution of Eq. (1.1). Then w must be a solution of the linear problem

∂ t w -ν∆w + div a(t, x)w = 0, w ∂D = 0, (2.6)
where we set

a(t, x) = 1 0 A û(t, x) + sw(t, x) ds.
(2.7)

A key property that will enable us to prove the theorem is that either the L 1 -norm of w squeezes by the time t = 1 2 , or |w( 1 2 , x)| is minorised on Π by a positive constant. Namely, we have the following result.

Proposition 2.3. Let T > 0 and ρ > 0 be some numbers and let a ∈ C(Q T , R d ) be a function continuously differentiable in x such that sup

(t,x)∈Q T |a(t, x)| + |∇ x a(t, x)| ≤ ρ.
(2.8)

Then there are numbers q < 1 and δ > 0 depending only on T , ρ, ν and Π such that any solution w ∈ C 1,2 (Q T ) of (2.6) satisfies one of the following inequalities:

w(T ) L 1 ≤ q w(0) L 1 , (2.9) inf x∈Π |w(T, x)| ≥ δ w(0) L 1 .
(2.10)

In view of Corollary 1.5, inequality (2.8) holds for the vector function a(t, x) defined by (2.7). Applying Proposition 2.3 with T = 1 2 to the function w = v -û, we distinguish between two cases. If (2.9) holds, then we choose ξ ≡ 0, so that the solution u of the controlled problem (2.1), (1.3) coincides with v and, hence, in view of the contraction of the L 1 -norm, satisfies the inequality

u(1) -û(1) L 1 ≤ u( 1 2 ) -û( 1 2 ) L 1 ≤ q u 0 -û(0) L 1 .
In the opposite case, w satisfies (2.10) with T = 1 2 . Let us fix a parameter τ ∈ (0, 1 2 ), set T τ = 1 2 + τ , and define the function η(t, x) = ξ(t)ϕ(x), where ξ is zero on the set [0, 1 2 ) ∪ [T τ , 1) and is equal to a contant ξ on [ 1 2 , T τ ). The difference z = u -û is a solution of the problem

∂ t z -ν∆z + div A(û + z) -A(û) = ξ ϕ(x), z ∂D = 0, z( 1 2 ) = z (2.11) on the interval [ 1 2 , T τ ]
, where z = w( 1 2 ). Note that, in view of the L 1 -contraction for (2.6) and inequality (2.10) with T = 1 2 , we have z

L 1 ≤ d, inf x∈Π |z(x)| ≥ δd, (2.12)
where d = u 0 -û(0) L 1 . A key observation is that the solution of (2.11) is close to that of the ODE

∂ t Z = ξ ϕ(x), Z( 1 2 ) = z.
(2.13) Namely, we have the inequality

z(T τ ) -Z(T τ ) L 1 ≤ C 1 τ, (2.14)
where C 1 > 0 does not depend on u 0 , û(0), and τ ; see Lemma 2.4 below. In view of the second inequality in (2.12) and the continuity of z, we can assume, for instance, that z(x) ≥ δd for x ∈ Π. In this case, we set ξ = -(τ M ) -1 θδd, where M = ϕ L ∞ and θ ∈ (0, 1) is a small parameter to be fixed later. It follows from (2.12)-(2.14) that

u(T τ ) -û(T τ ) L 1 = z(T τ ) L 1 ≤ z(T τ ) -Z(T τ ) L 1 + Z(T τ ) L 1 ≤ C 1 τ + z -M -1 θδd ϕ L 1 ≤ C 1 τ + d 1 -M -1 θδ ϕ L 1 .
(2.15)

Choosing τ = εd with a small ε > 0 and using again the L 1 -contraction for (2.6), we obtain

u(1) -û(1) L 1 ≤ u(T τ ) -û(T τ ) L 1 ≤ q 1 u 0 -û(0) L 1 , (2.16)
where

q 1 = 1 + C 1 ε -M -1 θδ ϕ L 1 . Hence, if ε = (2C 1 M ) -1 θδ ϕ L 1 , then q 1 = 1 -1 2 M -1 θδ ϕ L 1 < 1.
Iteration of this procedure will give the required exponential decay of the L 1 -norm of the difference u(t) -û(t).

Proof of Proposition 2.3

The proof builds on the argument used in [Shi17, Section 3.2] for the 1D Burgers equation. Since the result we establish here is stronger, we provide all the details.

Step 1. We first show that for any q ∈ (0, 1) there is ε = ε(Π, T, q, ρ) > 0 such that if w ∈ L ∞ (Q T ) is a non-negative solution of problem (2.6), then either (2.9) holds, or inf

x∈Π w(T, x) ≥ ε w(0) L 1 .
(2.17) Indeed, there is no loss of generality in assuming that w(0) L 1 = 1. Denoting by G(t, x, y) Green's function for (2.6) and writing w in the form

w(t, x) = D G(t, x, y)w(0, y) dy, (2.18)
we note that, in view of inequality (3.2) with j = |α| = 0, sup

(t,x)∈[ T 2 ,T ]×D w(t, x) ≤ M, (2.19)
where we set (2.20)

M = sup t∈[ T 2 ,T ] sup x,y∈D G(t, x, y). Let K ⊂ D be a connected compact set such that Π ⊂ K and L (D \ K) ≤ q 2M ,
We now set ε = q(2CL (K))

w(T ) L 1 ≤ w(T /2) L 1 ≤ D\K w(T /2) dx + K w(T /2) dx ≤ M L (D \ K) + CεL (K) ≤ q.
This coincides with (2.9) in the case under study.

Step 2 . We now consider the case of an arbitrary solution, assuming again that w(0) L 1 = 1. Let us denote by w + 0 and w - 0 the positive and negative parts of w(0) and write w ± for the solution of (2.6) issued from w ± 0 . We set r = w + 0 L 1 and assume without loss of generality that r ≥ 1 2 w 0 L 1 = 1 2 . Suppose first that w + (T ) L 1 ≤ r/2. Then, in view of the contraction of the L 1 -norm of solutions for (2.6), we have

w(T ) L 1 ≤ w + (T ) L 1 + w -(T ) L 1 ≤ r 2 + w -(0) L 1 = 1 - r 2 ≤ 3 4 .
This coincides with the first inequality in (2.9) with q = 3 4 and w(0) L 1 = 1. We now assume that w + (T ) L 1 > r 2 = 1 2 w + 0 L 1 , so that inequality (2.9) with q = 1 2 does not hold for the positive solution w + . Then we can find δ > 0 such that

inf x∈Π w + (T, x) ≥ δ w + 0 L 1 ≥ δ 2 . (2.22) If, in addition, sup x∈Π w -(T, x) ≤ δ 4 ,
then inequality (2.10) holds with δ = δ 4 . Thus, we can assume that

w -(T, x 0 ) ≥ δ 4 (2.23)
for some point x 0 ∈ Π. Using representation (2.18) with w = w -and inequality (3.2) with j = 0 and |α| = 1, we derive

sup x∈D |∇w -(T, x)| ≤ M 1
where M 1 does not depend on w. It follows that

w -(T, x) = w -(T, x 0 ) + w -(T, x) -w -(T, x 0 ) ≥ δ 4 -M 1 |x -x 0 | ≥ δ 8 , provided that |x -x 0 | ≤ r 0 := δ (8M 1 ) -1 .
Denoting by B the ball of radius r 0 centred at x 0 and assuming without loss of generality 1 that B ⊂ Π, we derive

inf x∈B w -(T, x) ≥ κ := δ 8 . (2.24)
In this case, denoting by 1 B the indicator function of B and using inequalities (2.22) and (2.24), we can write

w(T ) L 1 = D |w + (T, x) -w -(T, x)| dx = D w + (T, x) -κ1 B (x) -w -(T, x) -κ1 B (x) dx ≤ D w + (T, x) -κ1 B (x) dx + D w + (T, x) -κ1 B (x) dx ≤ w + (T ) L 1 + w -(T ) L 1 -2κ L (B) ≤ 1 -2κ L (B) =: q,
1 We can repeat the above argument for a small subset of Π and find a point x 0 that satisfies (2.23) and lies at some positive distance from the boundary of Π.

where the last inequality follows from the contraction of the L 1 -norm of solutions for (2.6). This completes the proof of the proposition.

Proof of Theorem 2.1

Step 1: Reduction. Choosing η ≡ 0 on the time interval [0, 1), we see that, in view of Corollary 1.5 and Proposition 3.2,

u(1) C 2+γ + û(1) C 2+γ ≤ C 1 , u(1) -û(1) L 1 ≤ u 0 -û(0) L 1 ,
where C 1 ≥ 1 is a number not depending on u 0 and û. Therefore, we can assume from the very beginning that the C 2+γ norms of u 0 and û(0) are bounded by some universal constant and prove inequality (2.2) for t ≥ 0.

Suppose we found a control η(t, x) = ξ(t)ϕ(x), with a piece-wise constant function ξ satisfying (2.3) and (2.4), such that, for any t ≥ 0,

u(t) -û(t) C 2+γ ≤ C 2 ,
(2.25)

u(t) -û(t) L 1 ≤ C 2 e -at u 0 -û(0) L 1 , (2.26)
where C 2 and a are positive numbers not depending on u 0 and û. In this case, by the interpolation inequality (2.5), for any σ ∈ (0, γ) we can write

u(t) -û(t) C 2+σ ≤ C C 2 e -at u 0 -û(0) L 1 θ ,
where θ = γ-σ d+γ+2 . We thus obtain inequality (2.2) with α = θa and β = θ.

Step 2: Uniform bound of the C 2+γ -norm. We claim that if ξ has the form described in the theorem, then inequality (2.25) holds with a sufficiently large constant C 2 depending only on |||h||| γ . Indeed, a straightforward consequence of Corollary 1.5 is that û(t) C 2+γ ≤ C 3 for all t ≥ 0 and

u(k) C 2+γ ≤ C 3 (2.27)
for all integers k ≥ 0, where we write C i for unessential positive numbers that may depend only on |||h||| γ . Thus, the required bound (2.25) will be established if we prove that, for any k ≥ 0, inequality (2.27) implies that

u(t) C 2+γ ≤ C 4 for t ∈ [k, k + 1].
(2.28)

In view of translation invariance, there is no loss of generality in assuming that k = 0. We thus consider the solution u(t, x) of problem (2.1), (1.2), (1.3) on the time interval J = [0, 1], with an initial condition u 0 ∈ C 2+γ (D) and a control function of the form

η(t, x) = ξ(t)ϕ(x), ξ(t) = 0 for t ∈ 0, 1 2 ∪ [T τ , 1], ξ for t ∈ 1 2 , T τ , (2.29)
where T τ = 1 2 + τ , τ ∈ (0, 1 4 ), and τ | ξ| ≤ C 5 . The function u is constructed by applying Remark 1.2 in the domain Q = J × D. This results in a continuous curve u : J → C 2+γ (D) whose restrictions to the cylinders

Ω 1 := (0, 1 2 ) × D, Ω 2 := ( 1 2 , T τ ) × D, Ω 3 = (T τ , 1) × D belong to the spaces X γ (Ω i ), i = 1,
∂ t g -ν∆g + d j=1 ∂ j A j (g + ζ(t, x)) = h + ν∆ζ.
Rewriting this as the inhomogeneous heat equation

∂ t g -ν∆g = f (t, x) := h + ν∆ζ - d j=1 ∂ j A j (g + ζ(t, x))
and applying Theorem 10.1 in [LSU68, Chapter III], we conclude that, for some γ ∈ (0, γ] the norm |||g||| γ is bounded by a universal constant. A simple calculation shows that if τ 1-γ | ξ| ≤ 1, (2.30) then for any multi-index α ∈ Z d + there is a universal constant C α such that

|||∂ α x ζ||| γ ≤ C α .
(2.31)

Using now Theorem 5.2 in [LSU68, Chapter IV] with l = γ , we obtain a universal bound for g X γ (Q) . Applying again the same result with l = γ, we derive a universal bound for g X γ (Q) . The required bound (2.28) follows from the representation u = g + ζ and inequality (2.31).

Step 3: Decay of the L 1 -norm. We now prove (2.26). To this end, suppose we have established the following estimate for all integers k ≥ 1:

u(k) -û(k) L 1 ≤ q 1 u(k -1) -û(k -1) L 1 .
(2.32)

In this case, inequality (2.26) will follow immediately if we prove that

u(t) -û(t) L 1 ≤ C u(k) -û(k) L 1 for t ∈ [k, k + 1]. (2.33)
By homogeneity, it suffices to consider the case k = 0. The difference w = u -û is the solution of the problem ∂ t w -ν∆w + div x (a(t, x)w) = η(t, x), w ∂D = 0, w(0) = w 0 ,

(2.34)

where w 0 = u(0) -û(0), η is defined by (2.29), and the vector function a is given by (2.7). Let ψ : R → R be a smooth non-decreasing function vanishing at zero and such that ψ(s) = ±1 for ±s ≥ 1, and let ψ ε (s) = ψ(s/ε). We multiply the first relation in (2.34) by ψ ε (w) and integrate over D. Denoting by Ψ ε the primitive of ψ ε vanishing at zero, after some simple transformations, we derive

∂ t D Ψ ε (w)dx + ν ε D |∇w| 2 ψ ( w ε )dx = 1 ε D w a, ∇w ψ ( w ε )dx + D ηψ( w ε )dx.
Using the boundedness of ψ and the Cauchy-Schwarz inequality, we obtain

∂ t D Ψ ε (w)dx ≤ C ν D |w| 2 ε ψ ( w ε ) dx + ξ(t) ϕ L 1 . (2.35)
Since the integral on the right-hand side is bounded by w L 1 , integrating (2.35) in time and passing to the limit as ε → 0 + , we obtain

w(t) L 1 ≤ C ν t 0 w(s) L 1 ds + d + τ | ξ| ϕ L 1 , where d = u(0) -û(0) L 1 . Assuming that τ | ξ| ≤ K 1 d, (2.36)
where K 1 > 0 is fixed, and applying Gronwall's inequality, we arrive at (2.33) with k = 0. It remains to prove (2.32). There is no loss of generality in assuming that k = 0, so that we prove (2.16). We shall need the following auxiliary result.

Lemma 2.4. Let τ, R > 0 and ξ ∈ R be some numbers such that τ | ξ| ≤ 1, let η(t, x) = ξϕ(x) for 0 ≤ t ≤ τ , and let u, û ∈ C 1,2 (Q τ ) be solutions of Eqs. (2.1) and (1.1) respectively such that u C 1,2 + û C 1,2 ≤ R. Then there is C > 0 depending only on R such that the difference z = u -û satisfies the inequality

z(t) -(z(0) + t ξϕ) L ∞ (D) ≤ C t for 0 ≤ t ≤ τ .
(2.37)

Taking this lemma for granted, let us complete the proof of (2.32). As was established at the end of Section 2.1, the required inequality (2.16) will be proved if we show that (2.14) holds, and the restrictions imposed on τ and ξ are compatible. The validity of (2.14) follows from inequality (2.37) rewritten on the interval [ 1 2 , 1 2 + τ ]. To see the compatibility of restrictions, let us recall that we seek θ > 0 such that the numbers

τ = (2C 1 M ) -1 θδd ϕ L 1 , | ξ| = (τ M ) -1 θδd = 2C 1 ϕ -1 L 1
satisfy inequalities (2.30) and (2.36). Since d is bounded above by a universal number and γ ∈ (0, 1), it is easy to see that both (2.30) and (2.36) hold for θ 1, and relations (2.3) are valid for K = 2C 1 ϕ -1 L 1 and κ = (2C 1 M ) -1 θδ ϕ L 1 . We have thus completed the proof of Theorem 2.1.

Proof of Lemma 2.4. The difference z = u -û belongs to the space C 1,2 (Q τ ) and satisfies Eqs. (2.11), in which the initial time t = 1 2 is replaced by

t = 0. Setting f (t, x) = ν∆z -div A(u) -A(û) ,
we see that the function z 1 (t) = z(t) -z(0) -t ξϕ is a solution of the problem

∂ t z 1 = f (t, x), z 1 (0) = 0. (2.38)
The hypotheses imposed on u and û imply that the L ∞ norm of f is bounded by a constant depending only on R. Integrating (2.38) in time, we arrive at the required estimate (2.37).

3 Appendix

Green's function for linear parabolic equations

Let D ⊂ R d be a bounded domain with C 3 -smooth boundary, let J T = [0, T ] be an interval, and let G(t, x, y) be the Green function for the second-order parabolic PDE

∂ t w -ν∆ x w + b(t, x), ∇ w + b 0 (t, x)w = 0, x ∈ D, (3.1)
where the functions b = (b 1 , . . . , b d ) and b 0 belong to the Hölder class C γ 2 ,γ (Q T ) for some γ ∈ (0, 1). In other words, G is the solution of (3.1) satisfying the initial and boundary conditons

G t=0 = δ y , G x∈∂D = 0,
where δ y stands for the Dirac mass at y. A proof of the existence of G, as well as the following inequalities, can be found in [START_REF] Ladyženskaja | Linear and Quasilinear Equations of Parabolic Type[END_REF]

|∂ j t ∂ α x G(t, x 1 , y) -∂ j t ∂ α x G(t, x 2 , y)| ≤ C |x 1 -x 2 | γ t -d+2+|α|

Harnack's inequality and L 1 -contraction

We first recall a fundamental estimate for positive solutions of (3.1). The following proposition is a simple particular case of more general results established in [START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF][START_REF] Landis | Second Order Equations of Elliptic and Parabolic Type[END_REF][START_REF] Krylov | A property of the solutions of parabolic equations with measurable coefficients[END_REF]. (3.4)

Let us note that [START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF][START_REF] Landis | Second Order Equations of Elliptic and Parabolic Type[END_REF] deal with parabolic equations with no lowerorder terms, while the paper [START_REF] Krylov | A property of the solutions of parabolic equations with measurable coefficients[END_REF] (see Theorem 1.1) considers the general case, assuming that the coefficient in front of the zeroth-order term satisfies a sign condition. That condition can easily be removed by the change of unknown function u = e λt v, with a suitable choice of λ ∈ R. Moreover, it follows from (3.2) that any solution w ∈ L ∞ loc (R + × D) for Eq. (3.1) with the Dirichlet boundary condition is a continuous function of its arguments for t > 0, so that the leftand right-hand sides of (3.4) are well defined.

We now turn to the particular case of (3.1). Namely, consider the linear problem (2.6), in which a ∈ C(Q T ) is a given function. The following result is a simple consequence of the maximum principle and a duality argument, and its proof can be found in [Hör97, Lemma 3.2.2]. Proposition 3.2. Let w ∈ C 1,2 (Q T ) be a solution of problem (2.6) in which a ∈ C(Q T ). Then w(T ) L 1 ≤ w(0) L 1 . (3.5)

Maximum principle for the nonlinear problem

Proof. Since the function -u satisfies an equation of the same form, it suffices to prove that u(t, x) does not exceed the right-hand side of (3.5) for (t, x) ∈ Q T .

To this end, we introduce the functions

H(t) = t 0 h(s) L ∞ (D)
ds, a j (t, x) = A j (u(t, x))

and note that v := u -H ∈ C 1,2 (Q T ) satisfies the differential inequality

∂ t v + d j=1 a j (t, x)∂ j v -ν∆v ≤ 0.
By the maximum principle, we have v(t, x) ≤ max(u(0), 0) ≤ u(0) L ∞ (D) , (t, x) ∈ Q T .

Combining this with the definition of v, we obtain the required inequality.

  where L stands for the Lebesgue measure on R d . By Harnack's inequality (see Proposition 3.1), we can find C = C(K, T ) > 0 such that sup x∈K w T 2 , x ≤ C inf x∈K w(T, x).

2e

  -c d(x 1 ,x 2 ,y) 2 t , (3.3) where 0 < t ≤ T , x, x 1 , x 2 , y ∈ D, d(x 1 , x 2 , y) = |x 1 -y| ∧ |x 2 -y|, j ≥ 0 and α ∈ Z d + are such that 2j + |α| ≤ 2,and the constants C and c on the right-hand side depend only ν, T , D, γ, and the norms of the functions b and b 0 in the space C γ 2 ,γ (Q T ).

Proposition 3. 1 .

 1 Let T > 0 and let b 0 , b ∈ L ∞ (Q T ). Then, for any compact set K ⊂ D and any numbers 0 < s < t ≤ T , there isC > 0 depending on b 0 L ∞ (Qt) + b L ∞ (Qt) such that, if w ∈ L ∞ loc (R + × D)is a non-negative solution of (3.1) satisfying the Dirichlet boundary condition, then sup x∈K w(s, x) ≤ C inf x∈K w(t, x).

Proposition 3. 3 .

 3 Let u ∈ C 1,2 (Q T ) be a solution of (1.1) with A ∈ C 1 (R, R d ) and h ∈ C(Q T ). Then u(t) L ∞ (Qt) ≤ u(0) L ∞ (D) + t 0 h(s) L ∞ (D) ds, 0 ≤ t ≤ T.

  2, 3. By the maximum principle applied consecutively to u| Ωi (see Proposition 3.3), the L ∞ norm of u is bounded by a universal constant (i.e., a number not depending on τ and ξ). Setting g = u -ζ with ζ(t, x) =

	t 0 η(s, x) ds, we see that g(t) is a continuous C 2+γ (D)-valued
	function satisfying the equation

  (see Theorems 16.2 and 16.3 in Chapter IV):

	|∂ j t ∂ α x G(t, x, y)| ≤ C t -d+2j+|α| 2	e -c |x-y| 2 t	,	(3.2)


D Dauvergne
email: d.dauvergne@ipnl.in2p3.fr

M Battaglia

G Montarou

E Testa

New methods of real-time control imaging for ion therapy

We present the ongoing studies aiming at providing a real-time control of the dose distribution during ion therapy. These studies are undertaken in the frame of the National and Rhône-Alpes Regional Research Programs for Hadontherapy. We aim at implementing combined modalities for real-time quality control of the deposited dose for future therapy centers. Several modalities are under development within this research program. Improvements on Positron Emission Tomography (PET) can be obtained by means of Time of Flight, using fast scintillators or resistive plate chambers, with dedicated readout and electronics. The in-bream prompt gamma imaging looks very promising in view of recent preliminary results, provided Time of Flight is used to discriminate gamma from neutrons and scattered particles. A collimated detection setup or a Compton camera is envisaged. Prompt emission of light charged particles like protons is also discussed.

Introduction

The main distinctive characteristic of ion-therapy, as compared to conventional radiotherapy, is the highly non-uniform profile of dose deposition inside the irradiated body. The dose increases gradually along the ion path to reach a steep maximum at the Bragg peak. Beyond this point, the longitudinal dose drops down quite abruptly, since only a small flux of secondary particles contribute to it. The lateral dose distribution is also quite narrow. Indeed, the transverse dispersion comes mostly from multiple scattering of the charged projectiles before stopping.

Therefore, in vivo 3D-imaging of the dose is of utmost importance for quality control during patient treatment, and drives a diversified R&D program which includes a variety of techniques. The uncertainty on the location of the Bragg peak in the patient may be large, as much as 1 cm in some cases, due to the stoichiometric calibration of the CT images used in treatment planning, as well as to possible morphologic changes occuring between CT and treatment and during the treatment itself, such as patient mispositioning tumour shrinkage, weight change and organ motion. Since no primary radiation emerges from the patient, in vivo imaging must be performed exploiting secondary radiations. To be useful, such radiation must emerge from the patient, i.e. to have a weak interaction probability, and be correlated to the primary beam dose distribution. This information on the electromagnetic energy deposition (the LET) can be extracted from hadronic interactions [START_REF] Schardt | Nuclear fragmentation of high-energy heavy-ion beams in water[END_REF][START_REF] Inaniwa | Experimental determination of particle range and dose distribution in thick targets through fragmentation reactions of stable heavy ions[END_REF][START_REF] Matsufuji | Influence of fragment reaction of relativistic heavy charged particles on heavy-ion radiotherapy[END_REF].

A significant effort has been already deployed to address these issues and several techniques have been proposed and tested. The present paper reviews the various developments underway on the on-line beam dose deposition imaging that are performed within the Rhône-Alpes Regional Program for Research in Hadrontherapy, with the motivation of providing new methods for quality control in future therapy centers like ETOILE in Lyon. These developments include improvement of in-beam Positron Emission Tomography (IBPET) by means of time of flight, in-beam imaging by prompt gamma-rays and secondary charged particles.

資料3

2

Positron Emission Tomography

So far the only operating control system available during ion therapy is PET, which is based on the observation of the radiation from radioisotopes formed in the nuclei fragmentation process, namely the annihilation of β + particles into two photons [START_REF] Matsufuji | Influence of fragment reaction of relativistic heavy charged particles on heavy-ion radiotherapy[END_REF][START_REF] Enghardt | Charged hadron tumour therapy monitoring by means of PET[END_REF][START_REF] Parodi | PET imaging for treatment verification of ion therapy: Implementation and experience at GSI Darmstadt and MGH Boston[END_REF][START_REF] Solevi | Study of an in-beam PET system for CNAO, the National Centre for Oncological Hadrontherapy[END_REF][START_REF] Nishio | Experimental verification of proton beam monitoring in a human body by use of activity image of positron-emitting nuclei generated by nuclear fragmentation reaction[END_REF]. The main β + radioactive isotopes formed during proton or carbon irradiation are 11 C, 15 O and 10 C, with respective half-lives of 20 min, 2 min and 20 s. The suitability of various light ion beams for PET imaging has been studied at GSI and HIMAC [START_REF] Iseki | Positron camera for range verification of heavy-ion radiotherapy[END_REF][START_REF] Pawelke | In-beam PET imaging for the control of heavy-ion tumour therapy[END_REF][START_REF] Inaniwa | Quantitative comparison of suitability of various beams for range monitoring with induced β+ activity in hadron therapy[END_REF]. Although only the target nuclei undergo fragmentation during proton therapy, the absolute rate of β + emission induced during a treatment is higher by a factor of 3-5 for proton than for carbon irradiation, where also projectile fragmentation occurs, due to the larger fluence of protons required to provide the same effective dose.

The relatively short radioisotope lifetimes cited above make IBPET very attractive. This is the solution that was promoted by Enghardt et al. [START_REF] Enghardt | The application of PET to quality assurance of heavy-ion tumor therapy[END_REF] with the dual-head PET BASTEI (Beta Activity Measurements at the Therapy with Energetic Ions) that has been installed at the GSI pilot project and has demonstrated its ability to provide a control of neck and head irradiations with carbon ions. The IBPET was used at GSI until 2008, whereas most of PET devices used in ion-therapy centers are offline, which necessitates the transport of the patient immediately after the treatment. However, the relatively low induced activities [START_REF] Enghardt | Dose Quantification from In-Beam Positron Emission Tomography[END_REF] require the acquisition of PET data for about 10 minutes or more, which limits considerably the patient flow in routine clinical environment in the case of IBPET. In addition, the integration of an IBPET device into a treatment site has to be arranged as a limited angle scanner, in order to avoid interference with the patient positioning. This causes artefacts in the reconstructed β + activity distributions which partly destroy the relationship between dose and activity. Moreover, over such a period of acquisition time, the metabolism causes a wash-out of the radioactive nuclei, which blurs the image of the implanted radioactive isotopes [13]. The low statistics problem could be circumvented by using radioactive 11 C ion beams instead of stable ions which, in addition, provide additional therapeutic dose after implantation due to the projectile radioactivity [START_REF] Kitagawa | Medical application of radioactive nuclear beams at HIMAC[END_REF]. Tomitani et al have studied the biologic lifetime, i.e. the metabolic wash-out, of the implanted radio-elements in clinical conditions with rabbits. They reported that more than 50% of the β+ activity is lost within 4 minutes or less, depending on the tissue irradiated [START_REF] Tomitani | Washout studies of 11C in rabbit thigh muscle implanted by secondary beams of HIMAC[END_REF].

Improvements of IBPET involve mainly the use of TOF between the two detected gammas. Indeed, during heavy ion treatments, a considerable amount of activity is coming from outside the field of view of the IBPET camera. An accuracy below 200 ps FWHM would reduce considerably the region of interest to a size of the same order as the tumor volume, and therefore decrease the number of background events registered. An important issue of PET imaging is the computer procedure needed to reconstruct an image and extract information about the deposited dose [START_REF] Inaniwa | Maximum likelihood estimation of proton irradiated field and deposited dose distribution[END_REF][START_REF] Shakirin | A Method for System Matrix Construction and Processing for Reconstruction of In-Beam PET Data[END_REF]. A significant gain on the reconstruction time could be obtained by increasing the coincidence accuracy by means of TOF between the two detected photons. This increase in image processing speed arises from the possibility to calculate at once the point of positron annihilation as soon as the coordinates of the hit detectors and the TOF information are known. Such an algorithm needs to process one single iteration through the collected data, unlike standard iterative reconstruction algorithms.

Thus this would also reduce limited solid angle artifacts and open the way to quantitative imaging. Lanthanum halide (LaBr 3 ) scintillating crystals can in principle reach such 10 -11 s time resolution. For the readout of the scintillating material, two alternatives could be explored: fast, position-sensitive photomultipliers having sub-ns rise times, or the recently developed solid state Silicon PhotoMultiplier (SiPM).

The SiPMs operate in the Geiger mode due to the avalanche nature of their signal, and they have a fast response time with nominal resolution of 50 ps or better. Unlike photomultipliers, SiPMs are not sensitive to magnetic fields, which makes them candidates for use in a multimodal TOF-PET -MRI. But they have not been used in large systems to date.

資料3

3

Prompt gamma imaging

Since the probability for prompt gamma emission is expected to be of the same order as that for fragmentation, gamma imaging may be regarded as a competitive technique to provide real-time information about the local dose [START_REF] Kim | Prompt gamma measurements for locating the dose falloff region in the proton therapy[END_REF][START_REF] Testa | Monitoring the Bragg peak location of 73 MeV/u carbon ions by means of prompt gamma-ray measurements[END_REF][START_REF] Polf | Prompt gamma-ray emission from biological tissues during proton irradiation: a preliminary study[END_REF] both for proton and carbon ion therapy.

Prompt gamma-rays are emitted by excited fragments with sufficiently small decay times so that the largest part of the photons are emitted in flight, keeping somewhat the information on the nuclear fragmentation location.

However, measuring prompt gamma during a therapy treatment is not straightforward, since a major source of background arises from the emission of other secondary species like neutrons and light fragments, and also from Compton scattering of photons emitted in all directions. In particular, neutrons have a high multiplicity and high scattering probability in surrounding materials, and they may not be correlated to the ion path.

Therefore, various alternatives are offered to provide selective information about the prompt gamma emission profile.

Collimated gamma-camera

The first one consists in a selective collimation and shielding. This method was followed by Min et al [START_REF] Kim | Prompt gamma measurements for locating the dose falloff region in the proton therapy[END_REF] during proton irradiation of a water phantom target. Using a scanning system based on a thick lead and paraffin collimator, the authors could demonstrate that the gamma emission profile keeps correlated with the proton range in water up to projectile incident energy of 200 MeV. However, the limited solid angle imposed by the heavy and oversized collimation setup may prevent this method from being operated in a clinical environment.

Time Of Flight-gamma camera with beam tagging

In order to improve the detection efficiency and the background rejection, our group proposed a prompt gamma detection method [START_REF] Testa | Monitoring the Bragg peak location of 73 MeV/u carbon ions by means of prompt gamma-ray measurements[END_REF][START_REF] Testa | Dose profile monitoring with carbon ions by means of prompt-gamma measurements[END_REF] which uses time of flight information to discriminate between photons and massive particles. Indeed, massive particles and scattered radiations in the surrounding materials are delayed before reaching the photon detector, as compared to direct photons. The concept was tested on a carbon ion beam at GANIL, taking advantage of the pulsed structure of the incident beam, 1 ns pulse each 80 ns. Figure 1 shows a 2D-spectrum of the energy detected by a collimated scintillator as a function of TOF for a 13 C ion 4 beam energy of 74 MeV/u impinging a thick PMMA target. The detector was located at 90° from the beam axis, at a distance of 60 cm from the target, with a narrow collimation slit (2 mm). The detected prompt gamma photons appear as a vertical band at short TOF values (the small drift with energy is due to electronics artifacts). However they account for only a minor fraction of the detected particles, which are dominated by neutrons.

The feasibility of the method is shown in Figure 2 which gives the longitudinal profile of the measured rates corresponding to the selections of prompt gammas only and of neutrons, above a given energy threshold, corresponding to the selection cuts highlighted in Figure 1. A photography of the irradiated sample shows clearly the beam range by means of the visible damage in the target. The Bragg peak corresponds to the sharp defect accumulation at the end of the range. The photon profile is well correlated to the ion range, whereas that for neutrons steadily increases with depth. Further studies performed at higher beam energies at GSI confirm that the correlation between prompt gamma yields and dose profile is preserved for carbon ions with longer ranges (in this case, the TOF was triggered by a thin scintillator intercepting the low intensity beam). This makes the proposed method feasible for the typical energies of carbon ion therapy.

The absolute photon yields measured are encouraging in view of designing a real-time control device during therapy. Although the solid angle of the collimated detector was small in these experiments, one can extrapolate the observed yields in view of a more efficient detection device. By increasing the solid angle by two orders of magnitude or more -which is made possible by minimizing the passive shielding against neutrons -enough contrast between the irradiated and non-irradiated zones should be observed within a few

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5 seconds exposure, thus allowing a real-time control during patient treatments. These feasibility studies are combined with GEANT 4 simulations which, in a first step, provide direct comparison with theoretical models, and, in a second step, will help for designing an optimized detection setup.

A 2D-transverse position information on the incoming ions is required to complete the longitudinal profile.

The incident beam tagging, in both time and transverse position, requires the development of a fast position-sensitive detector, with a count rate capability of 10 8 ions/s and 1 ns time resolution. A prototype, consisting in a scintillating fibers hodoscope read out by flat-panel PMs, is under development at IPN Lyon.

Since scintillating fibers are affected by radiation damage on the intense primary beam, an alternative technology under consideration consists in polycrystalline diamond detectors, produced by Chemical Vapor Deposition (CVD) [START_REF] Descamps | Clinical studies of optimised single crystal and polycrystalline diamonds for radiotherapy dosimetry[END_REF][START_REF] Bergonzo | Diamond as a tool for synchrotron radiation monitoring: beam position, profile, and temporal distribution[END_REF][START_REF] Rebisz | Alternative methods for heavy-ion therapy dosimetry[END_REF]. Such detectors exhibit quite good radiation tolerance. Diamond detectors can be obtained with large area, 1 ns or less timing resolution, and with x and y segmentation in order to cope with the high count rate and provide position information in an event by event mode.

Although 3D-data are recorded by this method, the limited statistics imposed by the collimation may restrict the quantitative imaging to 1D or 2D projections. Nevertheless, this represents a considerable breakthrough in the quality control for ion-therapy, since this information is not yet available.

Compton camera

A considerable improvement in the detection of prompt photons can be achieved with a Compton Camera.

The system can be considered to function like a standard γ-camera where the performance-limiting absorbing collimator is replaced by an electronically-operating collimator. A gamma ray emitted undergoes a Compton scattering in a first (scatter) detector, and the scattered photon which interacts in a secondary (absorption) detector conveys information on the direction of the incoming gamma ray. Information on energy deposition and interaction position in these detectors confines the possible incoming path of the initial gamma ray to lie on a cone, whose apex is the interaction position in the scatter detector. Compton cameras are currently under development worldwide, for application in gamma astronomy, medical imaging or homeland security [START_REF] Tanimori | Advanced Compton Camera with the ability in electron tracking based on Micro Pixel Gas Detector for Medical Imaging Nuclear Science Symposium Conference Record[END_REF][START_REF] Protic | Large volume Si(Li) orthogonal strip detectors for Compton effect -based instruments[END_REF].

For the case of prompt gamma imaging, where the source is extended and the energy spectrum is broad, the position and energy resolutions of the scatter detector are the key issues. A stack of Si(Li) detectors is envisaged [START_REF] Walenta | Vertex determination in a stack of Si-drift Detectors for High Resolution Gamma-ray Imaging[END_REF]. In parallel, a specific reconstruction strategy, based on a precise iterative and fast analytic reconstruction algorithm, is under development.

The Compton camera solution, with an efficiency increase of nearly two orders of magnitude compared to a collimated gamma-camera, is a much more ambitious and promising technology allowing online real 3D-imaging of the dose. Beyond the R&D challenges, a successful development of such a device would represent a major breakthrough in ion therapy imaging [START_REF] Feng | A Design of Compton Cameras for Imaging Gamma Emission in Proton Therapy[END_REF].

Charged Particle Imaging

The analysis of charged particles, created in the ion interaction and fragmentation processes, which emerge from the patient offer a potential complementary opportunity to determine the points of interaction of the primary beam. The principle, which resembles that of vertex identification in fixed target particle physics experiments, is to reconstruct the trajectories of the particles emerging form the interactions by a precise charged particle hodoscope and extrapolate them back to their production point. The intercept of the trajectory of individual particle with the line of flight of the incoming ion, has been proposed in the context of the CNAO-sponsored AQUA quality assurance program and should provide a determination of the point of interaction with a resolution of a few mm, limited by multiple scattering and the effect of secondary interactions. Further, the possibility to use multiple particle tracks produced at the same interaction points would open the way to a real vertex reconstruction, intrinsically able to give a three-dimensional measurement 資料3 6 of the point of interaction with accuracies of the order of 1 mm. A program of simulation validation using data for the secondary proton yield and energy spectra and of secondary proton tracking and vertexing is currently being carried out jointly by IPN Lyon and TERA. In a second phase, a demonstration of the 3D reconstruction capabilities of these methods on a carbon ion beam using a small scale charged particle tracker and calorimeter is envisaged.

Conclusion

A diversified research program on online dose monitoring during ion therapy is currently ongoing, supported by the Rhône Alpes Regional Program for Research in Hadrontherapy. The program is carried out in the perspective of providing advanced and combined solutions for the treatment quality insurance to the ETOILE ion therapy, which is expected to treat the first patients by late 2012. These studies are also supported by CNRS-GDR MI2B. They are part of the European project ENVISION proposed by the ENLIGHT++ research network in the frame of the FP7.

Figure 1 .

 1 Figure 1. Two-dimensional spectrum of the γ-equivalent energy deposited in the NaI(Tl) detector as a function of the time of flight. The collimated scintillator aims at a location at the middle of the ion path (detection angle θ=90°) for a 74 MeV/u carbon ion beam inside a PMMA target. The left rectangle corresponds to a selection of prompt γ-rays, and the right one to a neutron selection [21].

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Figure 2 :

 2 Figure 2: Detection rates as a function of the longitudinal position z obtained for the two different selections on the time of flight TOF and the energy deposited E in the detector indicated in Figure 1: photon selection (squares) and neutron selection (circles) [19]. Bottom: picture of the irradiated PMMA target.