Ahmed Maidi

Jean-Pierre Corriou
email: jean-pierre.corriou@univ-lorraine.fr

Distributed Control of Nonlinear Diffusion Systems by Input-Output Linearization

Keywords: distributed parameter system, diffusion system, Cole-Hopf transformation, geometric control, semi-group theory, exponential stability

This paper addresses the distributed control by input-output linearization of a non linear diffusion equation, which describes a particular but important class of distributed parameter systems. Both manipulated and controlled variables are assumed to be distributed in space. The control law is designed using the concept of characteristic index from geometric control by using directly the PDE model without any approximation or reduction. The main idea consists in the control design in assuming an equivalent linear diffusion equation obtained by use of the Cole-Hopf transformation. This framework helps to demonstrate the closed-loop stability using some concepts from the powerful semi-group theory. The performance of the proposed controller is successfully tested, through simulation, by considering a nonlinear heat conduction problem concerning the control of the temperature of a steel plate modeled by a non linear heat equation with Dirichlet boundary conditions.

Introduction

The dynamic behavior of most distributed parameter systems (DPS) is described by partial nonlinear differential equations (PDE). Control of DPS occupies an important place in control theory and constitutes an active research area [START_REF] Christofides | Control of nonlinear distributed process systems: Recent developments and challenges[END_REF][START_REF] Padhi | An account of chronological developments in control of distributed parameter systems[END_REF]. Compared to the control theory for linear DPS, which has been attained a certain level of maturity thanks to semi-group theory [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF][START_REF] Bensoussan | Representation and Control of Infinite Dimensional Systems[END_REF], the control theory of nonlinear DPS remains very challenging and many interesting questions are open concerning control, stabilization and optimization [START_REF] Christofides | Control of nonlinear distributed process systems: Recent developments and challenges[END_REF]. The book by Chen et al. [START_REF]Control of Nonlinear Distributed Parameter Systems[END_REF] gives recent progress in control theory of nonlinear systems and includes many results, applications and literature citations.

Although, in recent years, an extensive effort has been made to investigate the control of nonlinear DPS, however the available results, often developed under some restrictive assumptions, are difficult to generalize and are valid for specified particular classes, which limits their use [START_REF] Christofides | Control of nonlinear distributed process systems: Recent developments and challenges[END_REF][START_REF] Padhi | An account of chronological developments in control of distributed parameter systems[END_REF].

Control design methodologies of a non linear distributed system can be split into two approaches [START_REF] Christofides | Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes[END_REF][START_REF] Ray | Advanced Process Control[END_REF]. The first one called early lumping represents the conventional approach. It consists in performing a reduction of the PDE to derive a set of ordinary differential equations (ODE) that constitute an approximation, or a reduction, of the original PDE model [START_REF] Baker | Finite-dimensional approximation and control of nonlinear parabolic PDE systems[END_REF][START_REF] Christofides | Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds[END_REF], and the controller design is performed in the framework of the classical control theory of lumped parameter systems (LPS) using potentially powerful techniques. It must be noted that through early lumping, the fundamental control theoretical properties (controllability, observability and stability) are lost [START_REF] Christofides | Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes[END_REF][START_REF] Ray | Advanced Process Control[END_REF]. This approach, generally, leads to high dimension controllers which are difficult to implement [START_REF] Christofides | Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes[END_REF]. The second approach, termed as late lumping, uses the PDE model for the controller design without approximation. The approximation is performed only for implementation purposes of the controller. Late lumping allows the control designer to avoid losing the distributed nature of the PDE system and to take full advantage of their natural properties. However, direct handling of PDEs is difficult and the hardest task is related to the proof of the control theoretical properties, which needs sophisticated mathematical tools. In recent years, several control methods that directly take into account the distributed nature of the systems have been developed especially for the linear case [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF][START_REF] Bensoussan | Representation and Control of Infinite Dimensional Systems[END_REF][START_REF] Liu | Elementary feedback stabilization of the linear reactionconvection-diffusion equation and the wave equation[END_REF] and quasi-linear system [START_REF] Christofides | Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes[END_REF].

For nonlinear distributed parameter systems, the examination of the literature dedicated to the control problem of this kind of systems reveals that most contributions come from the early lumping approach and address the controller synthesis problem on the basis of low dimensional ODE approximations of the PDE system (reduced model) using different reduction approaches [START_REF] Li | Modeling of distributed parameter systems for applications-A synthesized review from time-space separation[END_REF] to achieve an accurate reduced model that characterizes the dominant dynamic behavior [START_REF] Christofides | Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds[END_REF][START_REF] Baker | Finite-dimensional approximation and control of nonlinear parabolic PDE systems[END_REF][START_REF] Dubljevic | Distributed nonlinear control of diffusion-reaction processes[END_REF][START_REF] Demetriou | Compensation of spatiotemporally varying disturbances in nonlinear transport processes via actuator scheduling[END_REF][START_REF] Demetriou | A new integrated output feedback controller synthesis and collocated actuator/sensor scheduling framework for distributed parameter processes[END_REF][START_REF] Lou | Feedback control of surface roughness in sputtering processes using the stochastic Kuramoto-Sivashinsky equation[END_REF]. On the other hand, few applications of the late lumping approach have been reported in the literature [START_REF] Godasi | Control of nonlinear distributed parameter processes using symmetry groups and invariance conditions[END_REF] and most of them are developed in the framework of geometric control [START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF][START_REF] Wu | Output regulation of nonisothermal plug-flow reactors with inlet perturbations[END_REF][START_REF] Shang | Feedback control of hyperbolic distributed parameter systems[END_REF][START_REF] Maidi | Boundary geometric control of a countercurrent heat exchanger[END_REF][START_REF] Maidi | Boundary geometric control of a heat equation[END_REF][START_REF] Maidi | Boundary control of a parallel-flow heat exchanger by input-output linearization[END_REF][START_REF] Maidi | Distributed feedback design for systems governed by the wave equation[END_REF], and reveal that geometric control is an interesting and suitable approach for designing controllers for distributed parameter following the late lumping approach. The most important advantages of geometric control is that the control law can be designed directly using the PDE model, which leads to distributed control that increases the performances [START_REF] Christofides | Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes[END_REF]. In addition, the geometric control allows to take advantage of the full potential of an existing control theory for lumped linear systems that offers full powerful controller design techniques. Nevertheless, the difficulty for the geometric control of nonlinear systems is that the closed-loop stability is difficult to prove and needs some sophisticated mathematical tools from functional analysis. Geometric control has been applied with success for linear and quasi-linear systems under some assumptions related to the system characteristics [START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF][START_REF] Wu | Output regulation of nonisothermal plug-flow reactors with inlet perturbations[END_REF][START_REF] Shang | Feedback control of hyperbolic distributed parameter systems[END_REF][START_REF] Maidi | Boundary geometric control of a countercurrent heat exchanger[END_REF][START_REF] Maidi | Boundary geometric control of a heat equation[END_REF][START_REF] Maidi | Boundary control of a parallel-flow heat exchanger by input-output linearization[END_REF][START_REF] Maidi | Distributed feedback design for systems governed by the wave equation[END_REF]. An attempt to extend the geometric approach to nonlinear systems, with boundary actuation and punctual controlled variable, is proposed by Maidi and Corriou [START_REF] Maidi | Boundary control of nonlinear distributed parameter systems by input-output linearization[END_REF]. As the control and observation operators assumed are unbounded, this approach suffers from the lack of efficient tools in proving closed-loop stability.

The investigations of the present work are intended as a contribution to the geometric control of nonlinear diffusive systems, which is one of the most important classes encountered in a wide variety of practical applications [START_REF] Demetriou | A new integrated output feedback controller synthesis and collocated actuator/sensor scheduling framework for distributed parameter processes[END_REF].

Thus, a design approach of the control law that enforces the desired performance and stability is developed. The main idea consists in deriving a linear equivalent model of the nonlinear diffusion system by use of the Cole-Hopf transformation. This linear equivalent model will be used then for control design and in particular for the proof of closed-loop stability using some concepts from semi-group theory. The developed control method is illustrated through a heat conduction problem modeled by a nonlinear diffusion equation.

The present paper is structured as follows. In Section 2, the addressed control problem for a nonlinear diffusion system is presented. In Section 3, the equivalent linear model is derived by means of Cole-Hopf transformation. Section 4 contains the main results associated with the control design approach proposed and the stability analysis based on the semi-group theory. In Section 5, the performance of the designed controller following the proposed approach is evaluated through simulation studies performed in an illustrative example concerning the heating problem of a steel plate with a nonconstant thermal conductivity. Finally, concluding remarks are provided in Section 6.

Control problem formulation

The nonlinear class of DPS, considered here, are described by the onedimensional nonlinear diffusion equation with interior control distributed on the space domain. The corresponding PDE model is

∂x(z, t) ∂t = 1 ρ c p ∂ ∂z k (x(z, t)) ∂x(z, t) ∂z + ϑ(z, t) in Ω×]0, t[ (1)
accompanied by Dirichlet boundary conditions

x(0, t) = x(l, t) = x * in ∂Ω×]0, t[ (2)
and the initial condition

x(z, 0) = φ(z) in Ω (3)
where x(z, t) denotes the state, z ∈ Ω = [0, l] ⊂ ℜ is the spatial domain, ∂Ω are the spatial domain boundaries, t ∈ [0, ∞[ is the time variable, k (x(z, t)) > 0 is the conductivity, and ρ c p the capacity of fluid or solid (in the case of solids, c remplaces c p ). Without loss of generality and to simplify the presentation, the terminology from heat conduction transfer will be used.

Thus, k (x(z, t)) and ρ c p will denote the thermal conductivity and the heat capacity, respectively.

The main space is defined as L 2 (0, l), which is the space of squareintegrable functions on Ω = [0, l], endowed with the usual norm . L 2 (0, l)

and the inner product , L 2 (0, l) defined as

f , g L 2 (0, l) = l 0 f (z) g(z) dz ; ∀f, g ∈ L 2 (0, l) (4)
f L 2 (0, l) = f , f 1 2 L 2 (0, l) (5)
The distributed manipulated variable ϑ(z, t)

∈ L 2 ([0, ∞[ ; L 2 (0, 1)) is
given as follows :

ϑ(z, t) = b(z) u(t) (6)
where b(z) is a known smooth function of z assumed to be square integrable

on the interval [0, l] that is b(z) ∈ L 2 (0, l).
The output variable y(t) to be controlled by manipulating the uniformly

distributed control u(t) ∈ L 2 ([0, ∞[ ; ℜ) is y(t) = Cx(z, t) = l 0 c(z) x(z, t) dz ; C ∈ L L 2 (0, l) ; ℜ (7)
where C is a bounded linear operator, L is the space of all linear bounded operators from L 2 (0, l) into ℜ, and c(z) is a known smooth function of z. In practice, the selection of b(z) and c(z) is typically consistent, therefore the following assumptions will be made

c(z) ∈ H 2 (0, l) (8)
and

c(0) = c(l) = 0 ( 9
)
where H 2 (0, l) denotes a Hilbert space defined as the Sobolev space of order

H 2 (0, l) = c(z) ∈ L 2 (0, l) : d k c(z) dz k ∈ L 2 (0, l) ; k = 1, 2 (10)
The two functions b(z) and c(z) are assumed not orthogonal, thus

b(z) , c(z) L 2 (0, l) = l 0 b(z) c(z) dz = 0 ( 11
)
Remark 1 The control problem is formulated with Dirichlet boundary conditions but the following development still holds for other types of boundary conditions.

Remark 2 In the control problem formulated, a single spatial interval [0, l] is considered but it can be formulated as the problem of controlling the output y(t) at a finite number of spatial intervals as suggested by Christofides [START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF].

For this case, the control law proposed here remains valid.

Cole-Hopf transformation

The Cole-Hopf transformation is widely used for solving nonlinear diffusion equation [START_REF] Vadasz | Analytical solution to nonlinear thermal diffusion: Kirchhoff versus Cole-Hopf transformations[END_REF][START_REF] Taler | Solving Direct and Inverse Heat Conduction Problems[END_REF][START_REF] Arpaci | Conduction Heat Transfer. Mechanics and Thermodynamics Series[END_REF] but for control problems, it has not yet been exploited.

This transformation consists in converting the nonlinear diffusion equation into a linear one if the thermal diffusivity defined by the following ratio

α = k (x(z, t)) ρ c p (12)
is approximately constant. This hypothesis is accepted since in many cases the variation of α with x(z, t) is much less important than that of k (x(z, t)), so that this approximation is reasonable [START_REF] Carslaw | Conduction of Heat in Solids[END_REF], therefore α ≈ constant.

To linearize the nonlinear diffusion equation ( 1) using the Cole-Hopf technique, one seeks a transformation of the form [START_REF] Vadasz | Analytical solution to nonlinear thermal diffusion: Kirchhoff versus Cole-Hopf transformations[END_REF] x(z, t) = h(w(z, t))

where h(x(z, t)) is a continuous bijective function (one-to-one function or mapping).

By using the transform [START_REF] Demetriou | Compensation of spatiotemporally varying disturbances in nonlinear transport processes via actuator scheduling[END_REF], the evaluation of the derivatives of the non-linear diffusion equation [START_REF] Christofides | Control of nonlinear distributed process systems: Recent developments and challenges[END_REF] gives

∂x(z, t) ∂t = ∂h(w(z, t)) ∂t = ∂h(w(z, t)) ∂w(z, t) ∂w(z, t) ∂t (14)
and

∂ ∂z k (x(z, t)) ∂x(z, t) ∂z = ∂ ∂z k (h(w(z, t))) ∂h(w(z, t)) ∂z = k(h(w(z, t))) d 2 h(w(z, t)) dw 2 (z, t) + dk(h(w(z, t))) dh(w(z, t)) dh(w(z, t)) dw(z, t) ∂w(z, t) ∂z 2 + k(h(w(z, t))) dh(w(z, t)) dw(z, t) ∂ 2 w(z, t) ∂z 2 (15)
To make the right hand side of (15) linear, the term between square brackets is set equal to zero,

k(h(w(z, t))) d 2 h(w(z, t)) dw 2 (z, t) + dk(h(w(z, t))) dh(w(z, t)) dh(w(z, t)) dw(z, t) 2 = 0 (16)
this differential equation can be expressed in the following integrable form

d dw(z, t) k(h(w(z, t))) dh(w(z, t)) dw(z, t) = 0 (17) k(h(w(z, t))) dh(w(z, t)) dw(z, t) = c 1 (18)
hence

k(h(w(z, t))) dh(w(z, t)) = c 1 dw(z, t) = c 1 w(z, t) + c 2 (19)
or equivalently

w(z, t) = 1 c 1 k(x(z, t)) dx(z, t) - c 2 c 1 (20)
which can be written under the following form

w(z, t) = h -1 (x(z, t)) = 1 c 1 k(x(z, t)) dx(z, t) - c 2 c 1 (21)
where h -1 ( . ) is the inverse function of h( . ), which is continuous.

In summary, by using the transformation ( 13) with h satisfying ( 16), the nonlinear diffusion equation (1) will be converted to the following linear one

∂h(w(z, t)) ∂w(z, t) ∂w(z, t) ∂t = k(x(z, t)) ρ c p ∂h(w(z, t)) ∂w(z, t) ∂ 2 w(z, t) ∂z 2 + 1 ρ c p ϑ(z, t) ∂w(z, t) ∂t = k(x(z, t)) ρ c p ∂ 2 w(z, t) ∂z 2 + 1 ρ c p ∂h(w(z, t)) ∂w(z, t) ϑ(z, t) (22)
Considering the expressions of diffusivity α and the integration constant c 1 given by ( 12) and ( 18), respectively, equation ( 22) takes the following form

∂w(z, t) ∂t = α ∂ 2 w(z, t) ∂z 2 + α c 1 ϑ(z, t) (23)
and according to (2), the transformation [START_REF] Maidi | Boundary geometric control of a countercurrent heat exchanger[END_REF], with appropriate choice of the integration constants c 1 and c 2 , yields the inhomogeneous boundary conditions w(0, t) = w(l, t) = 0 [START_REF] Maidi | Boundary control of nonlinear distributed parameter systems by input-output linearization[END_REF] with the initial condition w(z, 0) = 0 [START_REF] Renardy | An introduction to partial differential equations[END_REF] and the controlled output is

y(t) = Ch (w(z, t)) (26)
Remark 3 The Kirchhoff transformation is a particular case of the Cole-Hopf transformation [START_REF] Vadasz | Analytical solution to nonlinear thermal diffusion: Kirchhoff versus Cole-Hopf transformations[END_REF]. Compared to the Kirchhoff transformation that uses a definite integral, the Cole-Hopf uses an indefinite integral, which allows the adjustment of the transform by choosing adequate integration constants c 1 and c 2 . Consequently, desirable and simple practical transformation that ensures homogeneous boundary conditions can be obtained using the Cole-Hopf technique. The Kirchhoff transformation is obtained by choosing [START_REF] Vadasz | Analytical solution to nonlinear thermal diffusion: Kirchhoff versus Cole-Hopf transformations[END_REF].

c 1 = k 0 (k 0 = k(x(z, 0)) = k(x 0 )) and c 2 = - 1 k 0 k(x(z, t)) dx(z, t) x(z, t)=x0
Thus,

w(z, t) = 1 k 0 k(x(z, t)) dx(z, t) x(z, t)=x(z, t) - 1 k 0 k(x(z, t)) dx(z, t) x(z, t)=x 0 = 1 k 0 x(z,t) x0 k(x(z, t)) dx(z, t) (27)
Remark 4 The mapping h(x(z, t)) is a continuous bijective function (oneto-one function), thus its inverse function h -1 (x(z, t)) exists and is also con-tinuous.

Remark 5 For control problem design, the determination of transformation h(x(z, t)) is not necessary, but for proving the closed-loop stability, its continuity property will be exploited.

Distributed feedback design

The design of the control law u(t) of the control problem formulated is performed in the framework of geometric control using the concept of characteristic index [START_REF] Christofides | Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes[END_REF][START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF], which is a generalization of the concept of relative degree [START_REF] Isidori | Nonlinear control systems[END_REF][START_REF] Corriou | Process control -Theory and applications[END_REF] used in lumped parameter systems (ODE systems) to PDE systems.

The characteristic index is the smallest order of the time derivative of a given controlled variable which explicitly depends on the manipulated variable.

Considering the distributed control (6), the linear diffusion equation [START_REF] Maidi | Distributed feedback design for systems governed by the wave equation[END_REF] can be written under the following form

∂w(z, t) ∂t = A w(z, t) + B u(t) (28)
where A and B are the following operators

A = α ∂ 2 ∂z 2 ; B = α c 1 b(z) (29)

Control law design

For the synthesis of the geometric control law, the linear diffusion equation ( 28) will be considered. Hence, the first derivative of the controlled output [START_REF] Atkinson | Theoretical numerical Analysis. A functional analysis framework[END_REF] is

dy(t) dt = C ∂h (w(z, t)) ∂t (30) = C dh(w(z, t)) dw(z, t) ∂w(z, t) ∂t = c 1 α ρ c p C ∂w(z, t) ∂t = c 1 α ρ c p C [Aw(z, t) + B u(t)] = c 1 α ρ c p CA w(z, t) + c 1 α ρ c p C B u(t) (31)
The manipulated input u(t) appears linearly in the first time derivative of the output. The development of the second term of the right-hand side of equation ( 31) leads to

c 1 α ρ c p C B u(t) = c 1 α ρ c p l 0 c(z) α c 1 b(z) u(t) dz = 1 ρ c p l 0 c(z) b(z) dz u(t) (32) = 1 ρ c p b(z) , c(z) L 2 (0, l) u(t) (33)
as b(z) and c(z) are not orthogonal, hence

1 ρ c p b(z) , c(z) L 2 (0, l) = 0 (34)
consequently, the characteristic index is σ = 1, which suggests requesting the following input-output response for the closed-loop system (between the controlled output y(t) and the reference input v(t))

τ dy(t) dt + y(t) = v(t) (35)
where τ is the desired time constant of the closed-loop system.

Thus, substituting (31) into equation [START_REF] Kravaris | Geometric methods for nonlinear process control. 2. Controller synthesis[END_REF], the distributed state-feedback control law can be easily deduced as follows

u(t) = ρ c p τ l 0 b(z) c(z) dz v(t) -y(t) - τ c 1 α ρ c p CAw(z, t) (36)
Let us express the control law u(t) according to the state x(z, t). Remember that the expression ( 16) is made equal to zero by the transformation [START_REF] Demetriou | Compensation of spatiotemporally varying disturbances in nonlinear transport processes via actuator scheduling[END_REF], thus considering the relations ( 15) and ( 18), the term Aw(z, t) of the control law (36) can be written as

Aw(z, t) = α ∂ 2 w(z, t) ∂z 2 = α c 1 ∂ ∂z k (x(z, t)) ∂x(z, t) ∂z (37)
and the control law [START_REF] Jai | Automatique des Systèmes Distribués[END_REF] takes the following form

u(t) = ρ c p τ l 0 b(z) c(z) dz v(t) -y(t) - τ ρ c p C ∂ ∂z k (x(z, t)) ∂x(z, t) ∂z = ρ c p τ l 0 b(z) c(z) dz   v(t) -y(t) - τ ρ c p l 0 c(z) ∂ ∂z k (x(z, t)) ∂x(z, t) ∂z dz   (38)
Remark 6 The control law [START_REF] Wouwer | Adaptive method of lines[END_REF] can be directly derived by considering the nonlinear diffusion equation ( 1) in evaluating the first derivative of y(t) given by [START_REF] Ray | Advanced Process Control[END_REF]. In this case, it is difficult to show the closed-loop stability since both system (1) and control law [START_REF] Wouwer | Adaptive method of lines[END_REF] are nonlinear. Thus, the purpose of designing of the control law based on the linear diffusion equation ( 23) is to prove the closed-loop stability using some concepts from semi-group theory as it is shown in the next subsection.

Remark 7

The control law ( 38) is infinite-dimensional, thus for simulation purpose or for on-line implementation, a finite-dimensional approximation of the control law [START_REF] Wouwer | Adaptive method of lines[END_REF] has to be derived using discretization methods, such as finite differences. According to Balas [START_REF] Balas | Finite-dimensional control of distributed parameter systems by Galerkin approximation of infinite dimensional controllers[END_REF], to ensure the convergence of the closed-loop system resulting from the PDE model plus a finite-dimensional approximation of the infinite-dimensional control law to the closed-loop system resulting from the PDE model plus the infinite dimensional control law, the number of discretization points must be increased.

Remark 8

The control law [START_REF] Wouwer | Adaptive method of lines[END_REF] allows the application of the linear control theory to the resulting linear reference input v(t)-controlled output y(t) system (35) using powerful design approaches developed for linear lumped parameter systems. Thus, to handle uncertainty and unmodeled dynamics, the reference input v(t) can be defined by means of a robust controller [START_REF] Isidori | Nonlinear control systems[END_REF][START_REF] Corriou | Process control -Theory and applications[END_REF][START_REF] Kravaris | Geometric methods for nonlinear process control. 1. Background[END_REF][START_REF] Kravaris | Geometric methods for nonlinear process control. 2. Controller synthesis[END_REF], that is

v(t) = t 0 G(t -ξ)[y d (t) -y(t)] dξ (39)
where y d (t) is the desired set point of the controlled output y(t) and the function G( . ), for instance, can be chosen as the inverse of an appropriate transfer function.

This strategy has been applied with success for a counter-current heat exchanger [START_REF] Maidi | Boundary geometric control of a countercurrent heat exchanger[END_REF], a parallel-flow heat exchanger [START_REF] Maidi | Boundary control of a parallel-flow heat exchanger by input-output linearization[END_REF] and for a wave equation [START_REF] Maidi | Distributed feedback design for systems governed by the wave equation[END_REF].

Remark 9

The control law design approach proposed is developed for controlling the output (7) defined as the spatial weighted average. In this case, the control and observation operators B and C, respectively are bounded. The proposed control law is still applicable in the case of a punctual output y p (t)

by adopting the control strategy proposed by [START_REF] Maidi | Boundary control of a parallel-flow heat exchanger by input-output linearization[END_REF][START_REF] Maidi | Boundary control of nonlinear distributed parameter systems by input-output linearization[END_REF] where an external robust controller is introduced to provide the desired reference for the internal controller (see remark 8), that is y d (t), by taking the error between the controlled punctual output y p (t) and its set point y d p (t), i.e.

y d (t) = t 0 G p (t -ξ)[y d p (t) -y p (t)] dξ (40)
For more details about this strategy, the reader is referred to Maidi et al.

[22] and Maidi and Corriou [START_REF] Maidi | Boundary control of nonlinear distributed parameter systems by input-output linearization[END_REF].

Closed-loop stability

The equivalent closed-loop system is given by

∂w(z, t) ∂t = A w(z, t) + B ρ c p τ l 0 b(z) c(z) dz v(t) -y(t) - τ c 1 α ρ c p CA w(z, t) = (A + F ) w(z, t) + ρ c p τ l 0 b(z) c(z) dz B {v(t) -y(t)} (41)
y(t) = Ch(w(z, t)) (42)
where

F = β b(z) CA, β = - l 0 b(z) c(z) dz -1 (43)
By constraining the reference input v(t) and the controlled output y(t) to zero (y(t) = Ch(w(z, t)) = 0, v(t) = 0), the zero dynamics associated with the open-loop system results

∂w(z, t) ∂t

= (A + F )w(z, t) (44)
y(t) ≡ 0 (45)
with boundary conditions [START_REF] Maidi | Boundary control of nonlinear distributed parameter systems by input-output linearization[END_REF].

If the operator A + F generates a stable semi-group, it implies that the zero dynamics is exponentially stable [START_REF] Christofides | Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes[END_REF][START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF]. Thus, according to theorem 1

given in appendix 7.1, the operator A + F generates a semi-group if the operator A is a generator of a semi-group and the operator F is bounded on

L 2 (0, l).
The operator A with the boundary conditions ( 24) and initial condition [START_REF] Renardy | An introduction to partial differential equations[END_REF] generates an exponentially stable semi-group U(t) [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF][START_REF] Jai | Automatique des Systèmes Distribués[END_REF], that is,

U(t) L 2 (0, l) ≤ M e -ω t (46)
with stability constants M = 1 and ω = α π 2 > 0.

Let us, now, demonstrate that the operator F is bounded. According to appendix 7.3, the operator F is bounded if there exists a constant C such that

F w(z, t) L 2 (0, l) ≤ C w(z, t) L 2 (0, l) (47)
F w(z, t) L 2 (0, l) = β b(z) CA w(z, t) L 2 (0, l) = β α b(z) l 0 c(z) ∂ 2 w(z, t) ∂z 2 dz L 2 (0, l) (48)
Now, by integrating by parts and by considering boundary conditions [START_REF] Christofides | Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds[END_REF] and ( 24), the calculus gives

F w(z, t) 2 L 2 (0, l) = β α b(z) l 0 c(z) w(z, t) dz 2 L 2 (0, l) = l 0 β α b(z) l 0 c(z) w(z, t) dz 2 dz = β 2 α 2 l 0 b(z) l 0 c(z) w(z, t) dz 2 dz ( 49
)
and the Cauchy-Schwarz inequality [START_REF] Atkinson | Theoretical numerical Analysis. A functional analysis framework[END_REF] yields

l 0 c(z) w(z, t) dz ≤ c(z) L 2 (0, l) w(z, t) L 2 (0, l) (50)
hence

F w(z, t) 2 L 2 (0, l) ≤ β 2 α 2 l 0 b 2 (z) c(z) L 2 (0, l) w(z, t) L 2 (0, l) 2 dz = β 2 α 2 l 0 b 2 (z) c(z) 2 L 2 (0, l) w(z, t) 2 L 2 (0, l) dz = β 2 α 2 l 0 b 2 (z) l 0 [c(z)] 2 dz l 0 w 2 (z, t) dz dz = β 2 α 2 l 0 [c(z)] 2 dz l 0 w 2 (z, t) dz l 0 b 2 (z) dz F w(z, t) 2 L 2 (0, l) ≤ β 2 α 2 γ l 0 w 2 (z, t) dz ≤ β 2 α 2 γ w(z, t) 2 L 2 (0, l) (51) with γ = l 0 [c(z)] 2 dz l 0 b 2 (z) dz .
From (51), it follows that

F w(z, t) L 2 (0, l) ≤ |β| α γ 1/2 w(z, t) L 2 (0, l) (52)
Recall that b(z) ∈ L 2 (0, l) and c(z) ∈ H 2 (0, l), hence the constant C = |β| α γ 1/2 exists, which means that F is bounded and F L 2 (0, l) = |β| α γ 1/2 . Now, according to the theorem 2 given in appendix 7.2 [37, Theorem 1.1, page 76], the operator A + F generates a semi-group V (t) such that

V (t) L 2 (0, l) ≤ M e (-ω+M F L 2 (0, l) ) t ≤ M e (-ω+M |β| α γ 1/2 ) t ≤ e (-α π 2 +|β| α γ 1/2 ) t ≤ e -(α π 2 -|β| α γ 1/2 ) t (53)
thus the semigroup V (t) is exponentially stable if the following stability constant

ω w = α π 2 -|β| α γ 1/2 (54)
is positive, which yields

|β| γ 1/2 < π 2 (55)
This implies that the exponential stability of V (t) is related to the choice of the functions b(z) and c(z) since both β and γ depend on these functions.

Consequently, in addition to the controllability condition ensured by [START_REF] Li | Modeling of distributed parameter systems for applications-A synthesized review from time-space separation[END_REF], the functions b(z) and c(z) should be chosen so that the stability condition (55) is verified. In this case, the zero dynamics is exponentially stable and the operator A + F generates an exponentially stable semi-group V (t). In the following development, it is assumed that V (t) is stable. Now, the closed-loop system (41-42) can be written in the form of the following interconnected y-subsystem and w-subsystem

ẏ(t) = - 1 τ y(t) + 1 τ v(t) (56) ∂w(z, t) ∂t = (A + F ) w(z, t) + α ρ c p c 1 τ l 0 b(z) c(z) dz b(z) W (t) ( 57
)
where W (t) = v(t) -y(t).

To verify the closed-loop stability, it is equivalent to verify that the cascade interconnection is stable. Thus, it is sufficient to verify the closed-loop stability of each subsystem, which is demonstrated in the following.

According to [START_REF] Kravaris | Geometric methods for nonlinear process control. 2. Controller synthesis[END_REF], as the time constant τ > 0, thus the y-subsystem (56) of the interconnection is exponentially stable, consequently

|W (t)| ≤ K W |W (0)| e -ω W t , K W ≥ 1 and ω W > 0 (58) with W (0) = v(0) -y(0).
Now, as the semi-group V (t), generated by the operator A + F , is exponentially stable, consequently, the state w(z, t) of the closed-loop system (41-42) verifies [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF] w(z, t)

L 2 (0, l) ≤ w(z, 0) L 2 (0, l) e -ωw t + t 0 e -ωw (t-ξ) α ρ c p b(z) L 2 (0, l) c 1 τ l 0 b(z) c(z) dz |W (ξ)| dξ ≤ w(z, 0) L 2 (0, l) e -ωw t + α ρ c p b(z) L 2 (0, l) c 1 τ l 0 b(z) c(z) dz t 0 e -ωw (t-ξ) |W (ξ)| dξ (59)
where

ω w = α π 2 -|β| α γ 1/2 .
Substituting |W (ξ)| by its expression (58) in (59) gives

w(z, t) L 2 (0, l) ≤ w(z, 0) L 2 (0, l) e -ωw t + α ρ c p b(z) L 2 (0, l) c 1 τ l 0 b(z) c(z) dz t 0 e -ωw (t-ξ) K W |W (0)| e -ω W ξ dξ ≤ w(z, 0) L 2 (0, l) e -ωw t + α ρ c p b(z) L 2 (0, l) K W |W (0)| c 1 τ l 0 b(z) c(z) dz e -ωw t t 0 e (ωw-ω W ) ξ dξ (60) Then, if ω w = ω W , w(z, t) L 2 (0, l) ≤ w(z, 0) L 2 (0, l) e -ωw t + α ρ c p b(z) L 2 (0, l) K W |W (0)| c 1 τ l 0 b(z) c(z) dz t e -ωw t ≤ w(z, 0) L 2 (0, l) e -ωw t + α ρ c p b(z) L 2 (0, l) K W |W (0)| c 1 τ l 0 b(z) c(z) dz 1 ω w -ω e -ω t (61)
where 0 < ω < ω w . Thus, the closed-loop system is exponentially stable.

Now, if ω w > ω W , w(z, t) L 2 (0, l) ≤ w(z, 0) L 2 (0, l) e -ωw t + α ρ c p b(z) L 2 (0, l) K W |W (0)| c 1 τ l 0 b(z) c(z) dz e -ωw t e (ωw-ω W ) t -1 ω w -ω W ≤ w(z, 0) L 2 (0, l) e -ωw t + α ρ c p b(z) L 2 (0, l) K W |W (0)| c 1 τ l 0 b(z) c(z) dz e -ω W t -e -ωw t ω w -ω W ≤ w(z, 0) L 2 (0, l) e -ωw t + α ρ c p b(z) L 2 (0, l) K W |W (0)| c 1 τ l 0 b(z) c(z) dz e -ω W t ω w -ω W (62) else if ω w < ω W , w(z, t) L 2 (0, l) ≤ w(z, 0) L 2 (0, l) e -ωw t + α ρ c p b(z) L 2 (0, l) K W |W (0)| c 1 τ l 0 b(z) c(z) dz e -ωw t |ω w -ω W | (63)
in each case, the closed-loop system is exponentially stable. This implies that lim w(z, t) = 0 when t → ∞.

Let us now consider the stability of the state x(z, t) in closed loop, i.e. the nonlinear diffusion equation (1) with the control law [START_REF] Wouwer | Adaptive method of lines[END_REF]. Thus, according to [START_REF] Demetriou | Compensation of spatiotemporally varying disturbances in nonlinear transport processes via actuator scheduling[END_REF], lim x(z, t) when t → ∞ is equal to lim h(w(z, t)) when t → ∞ or equivalently to lim h(w) when w → 0 (w(z, t) is exponentially stable). Since h(w(z, t)) is a continuous bijective function, hence when w → 0, lim h(w) exists that is, according to (13), lim x(z, t) when t → ∞ also exists, consequently the state x(z, t) is bounded which means that x(z, t) is stable and the closed-loop system

∂x(z, t) ∂t = 1 ρ c p ∂ ∂z k (x(z, t)) ∂x(z, t) ∂z + b(z) τ l 0 b(z) c(z) dz   v(t) -y(t) - τ ρ c p l 0 c(z) ∂ ∂z k (x(z, t)) ∂x(z, t) ∂z dz   (64)
is internally stable.

Application example

A steel plate is heated (Fig. 1), before crossing a rolling mill, by thermal radiation applied only at the upper surface of the metallic plate [START_REF] Ray | Advanced Process Control[END_REF] while the lower surface is assumed as adiabatic, i.e. no flux is applied, either for heating or cooling. The vertical sides of the plate are neglected so that finally the heat flux is applied only on the upper part of the perimeter of the cross-section. The heat flux q ′′ (t) is the manipulated variable and it is assumed to be distributed uniformly at the upper surface of the plate. At two positions, considered as z = 0 and z = l, the plate is placed over rolls which are maintained at fixed temperatures and impose these conditions to the temperature of the plate itself. The control of the plate temperature is performed between these two cylinders.

To ensure a rolling of good quality, the heat flux q ′′ (t) is to be controlled in such a way that the temperature distribution, measured by a soft sensor that provides the spatial weighted average temperature T m (t), is kept at a specified temperature T d m (t). The plate has a thickness δ of 10 -2 m. The distance l is taken as l = 1m. The width of the plate is denoted as r.

The thermo-physical properties of the steel [START_REF] Taler | Solving Direct and Inverse Heat Conduction Problems[END_REF] are ρ = 7740 kg . m -3 , c = 470 J . kg -1 . K -1 , taken as constant, and the thermal conductivity k(T ) is a nonlinear function of temperature given as k(T ) = 23.9 + 41 10 -4 T + 10 -7 T 2 (65

)
where T is given in Celsius.

First, to establish the model, consider the general case of a metal plate heated on all faces by a heat flux q ′′ . It is assumed that the heat conduction is observed only along the z axis, i.e. the model has only one dimension, and the temperature is uniform on a cross-section A c = r δ of the plate.

The energy balance formulated for a infinitesimal volume of the plate leads to the following PDE that describes the spatio-temporal evolution of plate temperature subject to the heat flux q ′′ (t)

A c ρ c ∂T (z, t) ∂t = A c ∂ ∂z k (T (z, t)) ∂T (z, t) ∂z + P q ′′ (t) ( 66
)
where P is the heated perimeter corresponding to a cross-section and equal to: P = 2 r + 2 δ.

In the present case, only the upper surface is heated by a radiative flux q ′′ between z = 0 and z = l, the lower surface is adiabatic, the vertical faces are neglected, so that the heated perimeter is reduced to P = r and the energy balance can be written as

∂T (z, t) ∂t = 1 ρ c ∂ ∂z k (T (z, t)) ∂T (z, t) ∂z + q ′′ (t) δ (67)
Both boundary conditions of the plate are assumed identical and of Dirichlet type

T (0, t) = T (l, t) = T r ∀ t ≥ 0 ( 68
)
and the initial temperature distribution of the plate is assumed uniform

T (z, 0) = T r ∀ 0 ≤ z ≤ l (69)
with T r = 800 K.

The temperature distribution of the plate is assessed by the following measurement

T m (t) = l 0 c(z) T (z, t) dz (70)
with c(z) = 0.25 cos(π/2 (1 -z)).

From model (67), it follows that b(z) = δ -1 , thus according to [START_REF] Wouwer | Adaptive method of lines[END_REF], the following control law results

q ′′ (t) = 6 δ ρ c τ T d m (t) -T m (t) - τ ρ c l 0 z (l -z) ∂ ∂z k(T ) ∂T (z, t) ∂z dz (71)
For simulation purpose of the closed-loop system, the method of lines [START_REF] Wouwer | Adaptive method of lines[END_REF] is applied with evaluation of the spatial partial derivatives by means of finite differences based on N = 200 discretization points. The integral terms in the control law are evaluated numerically using the trapezoidal method. The desired constant time τ is taken equal to 300 s. The control is held constant over the sampling period equal to 30 s. To avoid the consequences due to brutal set point steps, the set point T d m (t) has been filtered by a first order filter with a time constant equal to τ f = 600 s. Hence, in the control law (71) instead of set point T d m (t), the filtered set point T d m f (t) defined by

τ f dT d m f (t) dt + T d m f (t) = T d m (t) (72)
is used.

To evaluate the performance of the control strategy, a set point step corresponding to T d m (t) = 160 K of the temperature is specified at t = 500 s.

Fig. 2 shows clearly that the controller behaves adequately and tracks perfectly the desired set point. Also, the control moves of the heat flux q ′′ (t) are smooth and physically acceptable (Fig. 3). An analogous remark could be made about the evolution of the temperature at different positions (Fig. 4).

The two-dimensional and the three-dimensional plots of the temperature profiles given by Figs. 5 and6, respectively, during the tracking of the imposed set point, confirm the good behavior of the controller.

34

In this paper, a design approach of a distributed control, by input-output linearization, for a nonlinear diffusion system is proposed. The design approach is developed based on the Cole-Hopf transformation that converts the original nonlinear problem to a linear one, which allows to take advantage of some concepts from semi-group theory to prove the closed-loop stability.

Thus, the distributed control law is derived using the concept of characteristic index from geometric control and it is shown that under certain assumptions concerning the control problem formulation, a first-order behavior results in closed loop between a desired reference and the controlled output.

The control performances of the proposed design approach are evaluated through numerical simulation by considering the problem of heating a steel plate in view of a rolling process. The obtained simulation results show the effectiveness of the developed control design.

This study demonstrates that the design of the control of PDE system, in the framework of geometric control, is a very successful control approach since it leads to a distributed control law that enhances the control performance by preserving the fundamental control properties, consequently the distributed nature of the PDE system. In addition, the use of the geometric control allows to take full advantage of the advances in control of linear lumped parameter systems to design robust controllers to the obtained linearized model. Nevertheless, the derived control law is infinite-dimensional, which requires to estimate the entire state of the system for on-line implementation.

In practice, it is impossible to have the whole state, so its estimation from available measurements is necessary. The question of state estimation for nonlinear diffusion systems is an interesting challenge presently under the investigation of the authors.

Appendix

Perturbations by bounded linear operators

According to the following theorem [37, Theorem 1.1, page 76], the property of being a generator of a semi-group is preserved by the addition of bounded operators.

Theorem 1 Let X be a Banach space and let A be the infinitesimal generator of a C 0 semi-group U(t) on X, satisfying U(t) X ≤ M e ω t . If B is a bounded linear operator on X, then A + B is the infinitesimal generator of a C 0 semigroup V (t) on X, satisfying V (t) X ≤ M e (ω+M B X ) t .

Perturbation theorem

The following perturbation theorem is a useful criterion to decide whether or not a given family of infinitesimal generators is stable [37, Theorem 1.1, page 76].

Theorem 2 Let A be a stable family of infinitesimal generators with stability constants M and ω. Let B be bounded linear operators on X. If B(t) < K then A + B is a stable family of infinitesimal generators with stability constants M and ω + M K.

Bounded operator

Definition 1 A linear operator A from X to Y is said to be bounded if there exists a constant C such that

A x Y ≤ C x X (73)
If no such C exists, the operator is said to be unbounded.

Figure 1 :

 1 Figure 1: Heating of steel plate (sectional representation).

Figure 2 :

 2 Figure 2: Evolution of the controlled temperature T m (t).

2 ]Figure 3 :

 23 Figure 3: Evolution of the heat flux q ′′ (t).

  01 m z = 0.15 m z = 0.50 m

Figure 4 :

 4 Figure 4: Variation of temperature at different locations along the plate.

t 1 Figure 5 :

 15 Figure 5: Temperature profiles at different time instants.


N Curtis
email: n.curtis@bham.ac.uk

N L Achouri

N I Ashwood

H G Bohlen

W N Catford

N M Clarke

M Freer

P J Haigh

B Laurent

N A Orr

N P Patterson

N Soić

J S Thomas

V Ziman

Breakup reaction study of the Brunnian nucleus 10 C

Keywords: numbers: 23.50.+z, 23.60.+e, 25.60.-t, 27.20.+n

published or not. The documents may come

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The Borromean systems have been the subject of much interest in recent years. These are a class of nuclei that may be described in terms of a three-body structure, with the property that the removal of any one of the three constituents results in an unbound two-body subsystem [1]. Perhaps the best known example is 11 Li, which is believed to possess a dominant 9 Li+n+n structure in its ground state. The two-body subsystems, 10 Li ( 9 Li+n) and the di-neutron (n+n), are unbound by 25 and 66 keV respectively. All three bodies are therefore required to produce the overall bound nuclear system. As such the structure of the Borromean nuclei provides a stringent test of nuclear few body models.

There is only one known example of a four-body nucleus that has Borromean-like properties: 10 C, which may be described as α+α+p+p. This nucleus has the property that the removal of any one of the four constituents results in an unbound three-body system. The removal of a proton from 10 C leads to 9 B, which is unbound to 8 Be+p decay by 185 keV. The removal of an α-particle produces 6 Be. Although the Q-value for 5 Li+p decay is -595 keV, 6 Be is unbound to this channel due to the ∼1.5 MeV width of the 5 Li ground state. In addition, the removal of any two particles from 10 C results in an unbound two-body system. The removal of both protons produces 8 Be (unbound to α+α decay by 92 keV), one α-particle and one proton results in 5 Li (unbound to α+p decay by 1967 keV) and the removal of both αparticles would produce the di-proton (unbound to p+p decay by 93 keV). All four constituents must be present to bind the system, and 10 C may therefore be described as a "Super-Borromean" nucleus. In fact 10 C is the only known nuclear example of a 4 th order Brunnian link [START_REF] Brunn | Über Verkettung[END_REF], a nontrivial link in knot theory that becomes trivial if any component is removed. Three-body Borromean nuclei are examples of 3 rd order Brunnian links.

The aim of this experiment was to study the spectroscopy of 10 C and to determine the relative strengths of the α and proton decay channels: 10 C → 9 B + p, 9 B → 8 Be + p, 8 Be → α + α and 10 C → 6 Be + α, 6 Be → 5 Li + p, 5 Li → α + p. The experiment was performed at the GANIL accelerator facility. A 33.3 MeV/nucleon 10 C beam was produced following the fragmentation of a 60 MeV/nucleon 12 C primary beam in a 3 mm thick Be production target. The secondary beam was purified to 98.6 % 10 C using the LISE3 spectrometer and had an average intensity of 2.5 × 10 4 particles per second. The beam contaminants (observed in the detector telescope) were d (0.2 %), 3 He (0.3 %) and 4 He (0.9 %) at 24.2, 40.4 and 22.3 MeV/nucleon respectively (corresponding to the same magnetic rigidity as the 10 C). The beam was not identified on an event by event basis, but as the contaminants were all light particles they could not give rise to the multiple proton and α-particle channels of interest. The contaminants were only observed in singles events in the detector array, and not in double, triple or quadruple coincidences. Therefore the 4 He contaminant did not contribute to the detection of α-particles from any of the breakup channels considered. The beam was tracked onto the reaction target (20, 45 and 95 mg/cm 2 12 C foils were used) on an event by event basis using a position sensitive drift chamber. A 5 mm thick, 10 mm diameter, Ta beam stop was positioned 106 mm (downstream) from the target. This protected the detector telescope, which was placed at 0 • (on the beam axis), from the direct and small angle scattered 10 C beam. The beam stop was held in place by a 3 mm diameter rod.

The detector telescope consisted of four separate elements, each (50 × 50) mm 2 in active area. The first two elements, at 123 and 130 mm from the target respectively, were silicon resistive strip detectors (RSD). Each RSD was 500 µm thick and segmented into 16 independent 3 mm wide strips. Resistive charge division provided position information along the strip length. The strips on the front detector (RSD1) were vertical and those on the second (RSD2) horizontal. The third element was a 1000 µm thick silicon double sided strip detector (DSSD) placed 148 mm from the target. This was segmented into 32 independent 3 mm wide strips, with 16 horizontal strips on the front face and 16 vertical strips on the back face. The final element, 160 mm from the target, was an array of 64 CsI scintillators, each 20 mm thick and with an area of (6 × 6) mm 2 . The effective solid angle covered by the array, calculated as the difference between the coverage of the CsI detectors and the beam stop and mounting rod, was 85 msr. Used in combination, the four detectors provided ∆E-E particle identification with isotopic separation for H, He and Li ions. The highly segmented nature of the array allowed all four particles from the decay of 10 C (α+α+p+p) to be detected in coincidence. Position resolution for the α-particles was ∼ 1 mm (FWHM) in both X and Y (provided by RSD1 and RSD2). As the proton energy loss in these detectors was minimal the position resolution for these particles was limited to the 3 mm strip pitch of the DSSD.

The telescope was calibrated using an α-particle source and a secondary "cocktail" beam of mixed isotopes at five different magnetic rigidity settings of the LISE3 spectrometer. These covered the energy range of interest. The (FWHM) energy resolution was ∼ 200 keV for the silicon detectors and ∼ 1.5% for the CsI crystals. The average detector counting rates were 3.5 kHz (summed total for the 16 strips of RSD1), 4.3 kHz (summed total for the 16 strips of RSD2), 1.4 kHz (summed total for the 16 front and 16 back strips of the DSSD) and 190 Hz (summed total for the 64 CsI crystals). The higher rates in RSD1 and RSD2 reflect the flux of heavy isotopes (in this case Be and heavier) that were stopped in these detectors and not the DSSD or CsI array. The average data acquisition rate, triggered by events where 3 or more CsI crystals were active, was 140 Hz.

Events were selected for analysis based on the particle multiplicity (M) in the DSSD. Three categories were considered: M = 3 (ααp), M = 3 (αpp) and M = 4 (ααpp). The decay energy (E decay ) for each channel was reconstructed from the kinetic energies and momenta of the decay fragments: P parent = n i=1 P fragmenti (n being the number of decay fragments), E parent = P 2 parent /2m parent (where m parent is the mass of the decaying parent nucleus) and E decay = ( n i=1 E fragmenti )-E parent . The decay energy is related to the excitation energy (E x ) of the decaying nucleus via E x = E decay -Q n , where Q n is the n-body decay Q-value. This approach allows the excitation energy of a decaying nucleus to be determined without considering any intermediate (sequential) decay steps. For example, the excitation energy of 9 B may be obtained directly from the kinetic energies and momenta of the α+α+p decay fragments, without first reconstructing the initial p+ 8 Be decay. In this case m parent is assumed to be 9, and Q 3 , the 3-body decay Q-value, is the Q-value for the α+α+p decay of 9 B (278 keV).

In in the M = 3 (ααp) channel is shown. A strong peak appears at the expected energy for the decay of the 8 Be gs . This energy, Q 2 = 92 keV, is indicated by the arrow. The 8 Be gs events seen in Fig. 1(a) and the proton in the M = 3 (ααp) channel were then used to reconstruct E decay for the 8 Be gs +p decay of 9 B (Fig. 1(b)). Again the arrow indicates the expected ground state position, Q 3 = 278 keV. The cross-sections for the production of 8 Be gs and 9 B gs in the M = 3 (ααp) channel are given in Tab. I. These were calculated by correcting the experimental yields with detection efficiencies obtained from a Monte Carlo simulation of the reaction and detector array. The quoted uncertainties reflect both the statistical errors from the three target data sets and a 20 % uncertainty in the target thicknesses. The absolute crosssections are also expected to be subject to a systematic uncertainty of the order of a factor of two. This arises as the angular distributions for each of the sequential decay processes are unknown. These have therefore been assumed to be isotropic in the centre-of-mass frame of the decays. The relative strength between the two channels is not very sensitive to the angular distributions used in the simulation, however, and is therefore not subject to such a systematic uncertainty. As may be seen in Tab. I, (72 ± 21) % of the 8 Be gs events seen in Fig. 1(a) arise from the decay of the 9 B gs events observed in Fig. 1(b).

In Fig. 1(c) the E decay spectrum for the decay of 5 Li to α+p (Q 2 = 1967 keV) for M = 3 (αpp) events is shown. A clear peak at the 5 Li ground state energy is not seen, indicating little or no yield corresponding to direct α+p removal. However, the Q-value for the decay of 6 Be to

Multiplicity

Decay channel σ (mb) M=3 (ααp) 8 Begs → α + α 42.6 ± 8.8 M=3 (ααp) 9 Bgs → 8 Begs + p 30.5 ± 6.3 M=3 (αpp) ( 5 Li (0-900keV) → α + p) (17.4 ± 3.6) M=3 (αpp) 6 Begs → 5 Li (0-900keV) + p 10.5 ± 2.2 M=4 (ααpp) 10 C → 9 B + p 9.2 ± 1.9 M=4 (ααpp) 10 C → 6 Be + α 3.3 ± 0.7 TABLE I: Cross-sections for the decay channels (see text). 5 Li+p is -595 keV, and hence the available E decay ( 5 Li) corresponding to 5 Li gs decays is 1967 -595 = 1372 keV. This decay proceeds via the tail of the ∼1.5 MeV wide 5 Li gs . A calculation of the effect of the decay phase space and an l=1 centrifugal barrier indicates the yield from the decay is shifted away from 1372 keV towards E decay ( 5 Li) = 0 keV, with 96 % of the 5 Li gs events being in the E decay ( 5 Li) = 0 to 900 keV range. These calculations reproduce the sharp rise in the decay energy spectrum below 500 keV, the predicted decay strength (arbitrarily scaled) being indicated by the dashed line in Fig. 1c.

Although the calculations do describe the data it is recognized that the width of the 5 Li gs is not well known, and there are possible α+p final state interactions. The cross-section for this channel is therefore not unambiguously determined, and hence is only given tentatively in Tab. I. A gate was applied across the 0 to 900 keV range (indicated by the vertical dotted line in Fig. 1(c)) to select the 5 Li gs events resulting from the decay of 6 Be. This gate maximised the 5 Li gs yield and at the same time minimized the background from M = 3 (αpp) events in which the E decay ( 5 Li) was calculated from the α-particle and the incorrect proton. In Fig. 1(d) the E decay spectrum for 6 Be decay, obtained from the M = 3 (αpp) events and gated on the 5 Li gs , is shown. The arrow indicates the expected ground state peak position (Q 3 = 1372 keV). The observed peak indicates the analysis of the intermediate 5 Li decay was successful in selecting the correct events. Without the 5 Li gs gate the 6 Be E decay spectrum contains increased background. The cross-sections for 5 Li gs and 6 Be gs production in the M = 3 (αpp) channel are listed in Tab. I, and are subject to the same factor of two systematic uncertainty as for the 8 Be gs and 9 B gs channels (see above). The cross-sections indicate that (60 ± 18) % of events within the 0 to 900 keV range of the 5 Li E decay arise from 6 Be gs decay.

The 10 C E x spectra, obtained from E x = E decay -Q 4 (Q 4 = -3.727 MeV) for the M = 4 (ααpp) events are shown in Fig. 2, along with the Monte Carlo predicted detection efficiencies (dot-dashed lines). To distinguish the 9 B+p and 6 Be+α decay channels a gate was applied to the 8 Be E decay spectrum reconstructed from the two α-particles. This is not shown, but is very similar in form to that for the M = 3 (ααp) data (Fig. 1 E x spectrum for events involving decay via the 8 Be gs is shown in Fig. 2(a) and corresponds to the 10 C → 9 B + p, 9 B → 8 Be + p, 8 Be → α + α decay channel. Events not decaying via the 8 Be gs are shown in Fig. 2(b). In order to minimize the contribution from decays to the 8 Be excited state at 3.04 MeV, an additional gate, on the 6 Be gs , was applied to this data. The 6 Be E decay spectrum for the M = 4 (ααpp) events is very similar in form to that for the M = 3 (αpp) data shown in Fig. 1(d). The data in Fig. 2(b) therefore correspond to the 10 C → 6 Be + α, 6 Be → 5 Li + p, 5 Li → α + p decay channel. The peaks observed in the spectra have been reproduced using Gaussian line shapes above a smooth (polynomial) background. The overall fits are given by the solid lines in Fig. 2, the background by the dashed lines, and the individual peaks by the dotted lines. Two peaks are seen at energies of (4.20 ± 0.01) and (5.31 ± 0.03) MeV and with widths of (280 ± 10) and (450 ± 90) keV (FWHM) respectively in Fig. 2(a). The peak at 4.20 MeV has not been previously reported and appears to be a new state in 10 C (it does not arise from any threshold effect). The second peak, at 5.31 MeV, corresponds to the known, and in this data, unresolved, states at 5.22 and 5.38 MeV [3]. In Fig. 2(b) one peak is seen at (6.74 ± 0.04) MeV with a width of (570 ± 110) keV. This corresponds to the known state at 6.58 MeV [3]. The observed widths are dominated by experimental resolution, which is proportional to (E x -E threshold ) The structure of the 10 C ground state is, in principle, probed via the α and p-removal processes. Tab. I lists the corresponding cross-sections for the M = 3 and 4 final states. The two multiplicities probe different types of reaction processes. The M = 4 yields reflect "diffractive" breakup, whereas the M = 3 also include absorptive processes, where either the proton or α-particle is "absorbed" by the target. Correspondingly, the M = 3 cross-sections are larger than the M = 4 counterparts. In both the M = 3 and 4 data the 9 B+p cross-section is ∼ 3 times larger than that for the 6 Be+α breakup. It should be noted that the probability of the disintegration of an initial state α-particle is small due to the high breakup threshold of 4 He (19.8 MeV). Thus the favouring of the proton removal correlates with the lower breakup threshold for that channel (Q[ 10 C → 9 B + p] = -4.005 MeV) compared with α-removal (Q[ 10 C → 6 Be + α] = -5.099 MeV). From a clustering perspective, the lower p-decay threshold would permit the 9 B+p configuration to be more strongly developed in the ground state than the 6 Be+α partition.

In Fig. 3 angular distributions are shown for the 4.20 and 5.31 MeV states observed in the 9 B+p decay of 10 C. Here ψ corresponds to the angle between the relative ve-locity vector of the 9 B and p decay fragments and the beam axis. The dotted lines indicate the predicted angular distributions obtained from a Monte Carlo simulation assuming isotropic decay. In Fig. 3(a), the ψ distribution for the 4.20 MeV state, there appears to be little structure, although there is some evidence that the distribution has a minima at ψ = 90 • . This suggests that the data should be described by an odd order Legendre Polynomial. Monte Carlo efficiency corrected Legendre Polynomials of order 1 (dashed line) and 3 (dot-dash line) are shown, but it is clear that neither provides a good description of the experimental yield. The angular distribution for the 5.31 MeV state (Fig. 3(b)) exhibits a much clearer periodic structure, and there is a very good agreement with the Monte Carlo corrected Legendre Polynomial of order 2 (dashed line).

The spectrum of 10 C excited states should closely mirror that of 10 Be (the latest work is summarised in Ref. [4]). In the present region of interest there are four states which are almost degenerate in 10 Be: 5.958 MeV [2 + ], 5.960 MeV [1 -], 6.179 MeV [0 + ] and 6.263 MeV [2 -]. The negative parity states correspond to the excitation of a p 3/2 neutron to the sd-shell, and the 2 + state to the recoupling of the p-shell nucleons [5]. The equivalent states in 10 C should be strongly populated in the 10 B( 3 He,t) 10 C reaction [6]. The 6.179 MeV [0 + ] state in 10 Be corresponds to a (sd) 2 configuration. This state has recently been shown to possess a highly deformed α+2n+α structure [7]. The states observed in the 10 B( 3 He,t) 10 C reaction, at 5.22, 5.38 and 6.58 MeV [6], were also observed in the inelastic scattering studies recently reported in Ref. [8]. It is likely that these correspond to the 1 -, 2 -and 2 + states. The 4.20 MeV state was not observed in Ref. [8], however, although this may be due to a reduced detection efficiency at low E x when compared to the present work. In the current data set the 4.20 MeV state was only seen after "pileup" events in the DSSD were recovered. Such events occur in low E decay breakup, where the small relative laboratory angle between particles often leads to detection of two (or more) particles in the same front DSSD strip but separate back DSSD strips (or vice versa). As the individual particle energies are known from the separate strips from the side of the DSSD where pileup did not occur the event may be reconstructed. Recovery of such events, where the DSSD front and back multiplicities do not both equal M = 4, gave rise to very good detection efficiency at low values of E x . The 4.20 MeV peak was not strongly populated (if at all) in the charge-exchange reaction [6], indicating that it is associated with a more complex configuration, perhaps (sd) 2 . The angular distributions (Fig. 3) for the 4.20 and 5.31 MeV peaks are characteristically different, indicating different spins. The subsequent proton decay of a 1 - (2 -) state would proceed through l-values of 0, 2 (0, 2, 4). A 0 + state would decay via l=1. Interestingly, the data for the 5.31 MeV peak suggests l=2, indicating the peak is associated with the 1 -or 2 -states. The angular distribution for the 4.20 MeV peak, however, indicates odd l, suggesting an association with the 0 + state. The 6.74 MeV state in 10 C could then be the analogue of the 5.958 MeV [2 + ] state in 10 Be.

If the 4.20 MeV state is the analogue of the 6.179 MeV 0 + molecular state in 10 Be [7], it would be expected that the two states should lie at approximately the same energy above the respective cluster decay thresholds. In the case of 10 Be, the 0 + state lies 0.6 MeV below the 9 Be+n decay threshold, whereas the 10 C 4.20 MeV state lies 0.2 MeV above the 9 B+p decay threshold. This relatively small difference could arise from the increased repulsion in the case of 9 B+p. It is possible to calculate the Coulomb energy difference (CED) expected between such molecular states in 10 Be and 10 C. If the present 4.20 MeV state is indeed associated with the 6.17 MeV 10 Be state, then the energy difference is close to 2 MeV. Such a difference is rather large compared to other nuclei but could be enhanced for an extended distribution of the valence particles. The CED of the 10 Be and 10 C ground-states was computed according to the formalism from Ref. [9]

E c = 0.60Z 2 -0.460Z 4/3 -0.15(1 -(-1) Z ) e 2 /R (1)
where E c is the Coulomb energy and R=1.3A 1/3 . The energy difference of the excited states was calculated assuming a 4 He+ 6 He( 6 Be) cluster partition with a separation of 5.9 fm. This separation is that deduced in Ref. [4] from an analysis of the rotational band based on the 0 + 2 state in 10 Be. Such an analysis gives an energy shift of 0.8 MeV (i.e. E x ( 10 C)=5.37 MeV). However, since the two protons are localized at one center, this has a higher energy than two 5 Li clusters with the same separation, with a radius given by R=1.3A 1/3 . In this instance, a reduction of 1.45 MeV in the energy of the 0 + excited state is found, E x ( 10 C)=4.72 MeV. Furthermore, if the valence protons are allowed to occupy extended orbits outside the α-particle cores then the Coulomb energy, and therefore the excitation energy, would be reduced. The Coulomb energy of two concentric spheres of differing radii was calculated in Ref. [START_REF] Wildermuth | A Unified Theory of the Nucleus[END_REF]. With a molecular description of the excited state, the valence particles are exchanged between the α-particle cores. In this instance, the valence protons/neutrons would have a radius approaching the separation of the α-particles. This would induce a Coulomb shift of 1.85 MeV, and place the 0 + state close to 4.3 MeV -consistent with the present measurement. Of course, the present estimates ignore the exchange properties and correlations which may exist in the system and assume spherical distributions. Calcu-lations based or more microscopic approaches would be interesting.

In summary, both the ground state structure and that of the excited states has been explored in the present measurements. The dissociation of the ground state of the Brunnian nucleus 10 C into the constituents 2α + 2p has been measured. The cross-sections indicate that the dominant configuration is 9 B+p, rather than 6 Be+α. In the multiplicity 4 coincidences, the 10 C excitation energy spectra have been reconstructed. A new state was observed at E x = 4.20 MeV, which may be the α+2p+α analogue of the 6.179 MeV [0 + ] α+2n+α molecular state in 10 Be. It should be noted that the ground and excited states should have different structures, based on the understanding which has been developed for 10 Be [4]. The ground state is expected to be more compact. A calculation of the energy difference between the ground and excited states in 10 Be and 10 C shows that, in order to produce a 0 + state at 4.20 MeV, a large degree of clusterization is required and that the protons should occupy delocalised orbits.

The assistance of the GANIL and LPC technical staff is gratefully acknowledged. This work has been supported by the European Community FP6 -Structuring the ERA -Integrated Infrastructure Initiative -contract EURONS n • RII3-CT-2004-506065.

Fig. 1 (FIG. 1 :

 11 FIG.1: Decay energy spectra for (a)8 Be, (b)9 B, (c)5 Li and (d)6 Be. The arrows indicate the known ground state energies. The various lines in (c) are described in the text.

FIG. 2 :

 2 FIG.2: Ex spectra for (a) 9 B + p and (b)6 Be + α decay of 10 C. The various lines are described in the text.

FIG. 3 :

 3 FIG. 3: Angular distribution (ψ) for the (a) 4.20 and (b) 5.31 MeV states in 10 C. The various lines are described in the text.

TABLE II :

 II 1/2 . A resolution of 280 keV at 4.2 MeV would rise to 650 keV at 6.7 MeV following this relationship. The uncertainties quoted in the centroid energies are statistical only. A 200 keV uncertainty across the observed E x range is typical in breakup work of this kind. The cross-sections for the9 B+p and 6 Be+α breakup of 10 C are given in Tab. I. These were obtained by determining the experimental yields over a range of Cross-sections for the 10 C excited states.

	Ex (MeV) Decay channel σ (mb)
	4.20 ± 0.01	9 B + p	2.9 ± 0.6
	5.31 ± 0.03	9 B + p	0.45 ± 0.22
	6.74 ± 0.04	6 Be + α	0.46 ± 0.19