Dan Tiba
email: dan.tiba@imar.ro

Finite element discretization in shape optimization problems for the stationary Navier-Stokes equation

Keywords: shape optimization, full discretization, finite elements, convergence

For shape optimization problems associated to stationary Navier-Stokes equations, we introduce the corresponding finite element approximation and we prove convergence results.

Introduction

Optimal design and optimal control problems for partial differential equations are extensively studied in the recent mathematical literature. In the case of stationary Navier-Stokes equations, we quote the works Casas, Mateos and Raymond [2007], [START_REF] Rösch | Optimal control of the Stokes equations : a priori error analyses for finite element discretization with postprocessing[END_REF], Los [START_REF] Reyes | Optimal control of the stationary Navier-Stokes equations with mixed control-state constraints[END_REF] devoted to optimal control problems or to approximation procedures. Shape optimization problems related to fluid mechanics have been discussed in Borrvall and Petersson [2003], Mohammadi and Pironneau [2001], [START_REF] Posta | Optimal control of Navier-Stokes equations by Oseen aproximations[END_REF], [START_REF] Roubicek | Lipschitz stability of optimal controls for steady-state Navier-Stokes equation[END_REF], [START_REF] Halanay | Shape optimization for stationary Navier-Stokes equations[END_REF]. See as well [START_REF] Liu | On the structural optimization problems[END_REF], [START_REF] Liu | Error estimates in the approximation of optimization problems governed by nonlinear operators[END_REF] for related problems and arguments. This work is concerned with the discretization and the associated convergence analysis, in the spirit of general shape optimization problems for linear elliptic systems, as discussed in [START_REF] Chenais | Finite-element approximation of 2 D elliptic optimal design[END_REF] and in [START_REF] Tiba | Finite element approximation for shape optimization problems with Neumann and mixed boundary conditions[END_REF]. Another approximation procedure for such problems is due to Neittaamäki, Pennanen and Tiba [2009].

In the next section we formulate the problem and review briefly some preliminaries, necessary in the subsequent parts. Section 3 investigates some approximation properties of the stationary Navier-Stokes equation under our discretization approach. The last section introduces the fully discretized optimization problem and studies its convergence.

Problem formulation and preliminaries

Let Ω ⊂ R d be an (unknown) lipschitzian domain, such that E ⊂ Ω ⊂ D ⊂ R d with E ⊂ D some given bounded domains and d an arbitrary natural number. We recall from [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF] the definition of the following spaces :

V(Ω) = {y ∈ D(Ω) d ; div y = 0}, (1)
V (Ω) = closure of V(Ω) in H 1 0 (Ω) d . (2)
Then, it is known that V (Ω) = {y ∈ H 1 0 (Ω) d , div y = 0}, as Ω is assumed lipschitzian. For any y ∈ V (Ω), if y is its extension by 0 to D, then y ∈ V (D) and conversely, if z ∈ V (D) and z = 0 a.e. in D\Ω ; then z = z| Ω ∈ V (Ω). Such properties may be partially extended to domains with the segment property, Wang and Yang [2008].

The weak formulation of the stationary Navier-Stokes equation with Dirichlet (no-slip) boundary conditions is

Ω (ν d i,j=1 ∂y j ∂x i ∂v j ∂x i + d i,j=1 y i ∂y j ∂x i v j )dx = Ω d j=1 f j v j dx, ∀v ∈ V (Ω) (3)
where We associate to (3) an integral cost functional of the form

f = (f 1 , • • • , f d ) ∈ H -1 (D)
Λ j(x, y(x))dx ( 4
)
where Λ is either E ⊂ Ω or Ω and y is one of the weak solutions of (3). The integrand j : D × R d → R satisfies measurability and continuity properties to be precised later.

The shape optimization problem considered in this paper consists in the minimization of the performance index (4) subject to the state system (3) and to the constraints

E ⊂ Ω ⊂ D, (5)
for any Ω ∈ O, where O is a prescribed family of domains. If the Lipschitz assumption is valid for any Ω ∈ O with a uniform constant, then O is compact with respect to the Hausdorff-Pompeiu complementary metric. A similar compactness result holds for domains with the uniform segment property according to Theorem A3.9, Neittaanmäki, Sprekels and Tiba [2006]. The following existence result is a simplified version of Theorem 1 in Halanay and Tiba [2009].

Theorem 1. Assume that j(x, y n (x)) → j(x, y(x)) weakly in L 2 (E) if y n → y strongly in L 2 (E) and O is compact. Then, the shape optimization problem (3)-( 5), with Λ = E has at least one optimal pair [Ω * , y * ] ∈ O × V (Ω * ) if it has an admissible pair.

Remark This theorem should be understood in the sense of singular control problems [START_REF] Lions | Contrôle des systèmes distribuiés singuliers[END_REF], Neittaanmäki, Sprekels and Tiba [2006, 3.1.3.1]. The state system is ill-posed (nonuniqueness), but the optimization problem (3)-( 5) is well defined as minimization over admissible pairs [Ω, y], Ω ∈ O satisfying (5) and y ∈ V (Ω) being one of the weak solutions of (3).

Discretization of the state equation

We assume now that D is a smooth bounded subdomain of R 2 and we consider a family of uniformly regular finite element meshes

{T h } h>0 in D with h = max T h ∈T h diam(T h ).
For any admissible Ω ∈ O, we define its discrete approximation as follows [START_REF] Chenais | Finite-element approximation of 2 D elliptic optimal design[END_REF] or [START_REF] Tiba | Finite element approximation for shape optimization problems with Neumann and mixed boundary conditions[END_REF] where other variants are also discussed) :

Ω h = int ∪ {T h ; T h ∈ T h , T h ⊂ Ω} (6)
According, for instance, to [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF], there are many possibilities to introduce a finite element space V h in Ω h approximating [START_REF] Casas | Error estimates for the numerical approximation of a distributed control problem for the steady-state Navier-Stokes equations[END_REF], that is approximating H 1 0 (Ω) and the divergence free condition. In particular, the piecewise linear finite elements are not possible to be used in this setting. One also has to impose null values on ∂Ω h in order to take account the Dirichlet boundary condition and any y h ∈ V h may be extended by 0 to Ω, respectively to D. We shall also write V h (Ω) or V h (D) in order to avoid possible confusions.

One example of space V h (in dimension 2 as assumed here) is the space of continuous functions, vanishing outside Ω h , that are polynomials of degree less or equal two on any simplex T ∈ T h and satisfy :

T divy h dx = 0, ∀T ∈ T h , ∀y h ∈ V h (7)
On V h we take the scalar product (•, •) h induced by H 1 0 (Ω). Note that V h is an external approximation of V due to [START_REF] Liu | On the structural optimization problems[END_REF]. The discrete approximation of (3) is

ν(y h , v h ) h + b h (y h , y h , v h ) = Ω f • v h dx, ∀v h ∈ V h (8)
Notice that the last integral in ( 8) is over Ω h in fact, as v h vanishes outside Ω h . We have denoted by "•" the scalar product in R 2 and b

h (•, •, •) is the trilinear form approximating b(y, v, w) = 2 i,j=1 Ω y i D i v j w j dx, ∀y, v, w ∈ H 1 0 (Ω). A detailed construction of b h (•, •, •) and the proof of b h (u h , u h , r h v) → b(u, u, v), ∀v ∈ V(Ω) (9)
if u h → u weakly in H 1 0 (Ω) can be found in Teman [1979], Ch. II.3. Here r h v ∈ V h is given by a term that takes the same values as v ∈ V(Ω) in the interior nodes and edge midpoints of Ω h plus a correction term defined in Temam [1979, p.81]. On ∂Ω h , r h v should be zero.

Then, the following convergence property is also valid. Proposition 3.1 Under the above conditions, there exists at least one u h ∈ V h , solution of (8) , for each h > 0.

The family {u h } in H 1 0 (Ω) has strong accumulation points, denoted ū, which are solutions of (3)

Remark. If the uniqueness property is valid for (3), the convergence is valid without taking subsequences. In Casas, Mateos and Raymond [2007] and in Girault and Raviart [1989] Ch. II 4, finite element approximations with uniform convergence properties are indicated, including error estimates.

Approximation of the shape optimization problem

We also discretize the cost functional (4) and the constraint (5) :

J h (y h ) = E h j(x, y h (x))dx ( 10
)
where y h is any of the solutions of (8), associated to Ω h and E h is obtained as in (6), starting from E ;

E h ⊂ Ω h ⊂ D. (11)
Notice that for any admissible Ω ∈ O, restriction ( 11) is automatically fulfilled by our discretization construction. The collection of all admissible discretized open sets is denoted by O h . The discrete shape optimization problem is defined by ( 8), [START_REF] Mohhamadi | Applied shape optimization for fluids[END_REF], [START_REF] Neittaanmäki | Optimization of elliptic systems. Theory and applications[END_REF]. By [START_REF] Lions | Contrôle des systèmes distribuiés singuliers[END_REF], the family O h is always finite, for any given h > 0. Then, the discrete minimization problem has at least one discrete optimal solution denoted by Ω * h ∈ O h . Since (8) may have, in principle, an infinity of solutions y n h , we remark that in each T ∈ T h , T ⊂ Ω h , the corresponding coefficients of y n h are bounded, by the construction of the finite elements. This is a consequence of |y n h | V h bounded and it is enough to pass to the limit in ( 8), [START_REF] Mohhamadi | Applied shape optimization for fluids[END_REF] on a minimizing sequence (with respect to n) of admissible states (h and Ω h are fixed here). The minimization in [START_REF] Mohhamadi | Applied shape optimization for fluids[END_REF] should be understood as minimization over pairs [Ω h , y h ] ∈ O h × V h (Ω h ), similar to the situation in Theorem 1.

We recall first some convergence properties of the admissible pairs [Ω h , y

h ] ∈ O h × V h (Ω h ), when h → 0. Proposition 4.1 i) If Ω ∈ O, then Ω h ∈ O h and Ω h → Ω in the Hausdorff- Pompeiu complementary topology. ii) If Ω h ∈ O h and Ω h → Ω in the Hausdorff-Pompeiu complementary topology, then Ω ∈ O.
Remark At point ii), the discrete sets Ω h are not necessarily constructed via (6) starting from Ω. Point i) also applies to the discretization of E and E h → E in the Hausdorff-Pompeiu complementary topology. The proof of this proposition and other related properties may be found in [START_REF] Chenais | Finite-element approximation of 2 D elliptic optimal design[END_REF] and in [START_REF] Tiba | Finite element approximation for shape optimization problems with Neumann and mixed boundary conditions[END_REF].

In the sequel, a crucial role is played by the following result which is an extension of Proposition 3.1.

Theorem 2.

If Ω h ∈ O h and y h ∈ V h is any solution of ( 8) and if Ω h → Ω in the Hausdorff-Pompeiu complementary topology, then for any subdomain K, compactly included in Ω there is h 0 > 0 such that K ⊂ Ω h , h < h 0 and

y h | K → ŷ| K (12)
weakly in H 1 (K), on a subsequence, where

ŷ ∈ V ( Ω) is a solution of (3) in Ω ∈ O.

Proof

The fact that Ω ∈ O is a consequence of P4.1. The inclusion K ⊂ Ω h for h < h 0 is known as the Γ -property of the Hausdorff-Pompeiu complementary convergence, Neittaanmäki, Sprekels and [START_REF] Neittaanmäki | Optimization of elliptic systems. Theory and applications[END_REF], p. 63.

Extend y h by 0 to D and denote it by ỹh ∈ H 1 0 (D). By [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF], p. 209, we have

b h (u h , v h , v h ) = 0, |b h (u h , v h , w h ) ≤ c|u h | V h |v h | V h |w h | V h (13)
for any u h , v h , w h in V h , where c > 0 is an absolute constant.

Fixing

v h = y h ∈ V h in (8) we get that {|y h | V h } is bounded, due to (13)
, and {ỹ h } is bounded in H 1 0 (D). On a subsequence, we have ỹh → ỹ ∈ H 1 0 (D). A simple distributions argument gives that ỹ| D\ Ω = 0 almost everywhere. Then ỹ| Ω ∈ H 1 0 ( Ω) as we have assumed that any admissible domain Ω ∈ O is lipschitzian and the trace theorem may be applied.We also get y ∈ V ( Ω) by an adaptation of Proposition 4.3, Temam[1979], p.83. In particular y h | K → ỹ| K weakly in H 1 (K), on a subsequence.

We have to show that ỹ| Ω is a solution of (3). We fix in (8) v h = r h v for any v ∈ V( Ω). In particular supp v ⊂ Ω is a compact subset and the Γ -property gives that supp v ⊂ Ω h for h < h 0 . Consequently r h v ∈ V h for h < h 0 and may be used in [START_REF] Liu | Error estimates in the approximation of optimization problems governed by nonlinear operators[END_REF]. Moreover, by [START_REF] Reyes | Optimal control of the stationary Navier-Stokes equations with mixed control-state constraints[END_REF] we have b h (y h , y h , r h v) → b(ỹ, ỹ, v), ∀v ∈ V( Ω). ( 14)

Relation ( 14) is obtained by applying [START_REF] Reyes | Optimal control of the stationary Navier-Stokes equations with mixed control-state constraints[END_REF] in D as ỹ ∈ H 1 0 (D), v ∈ V(D) by extending it with 0 outside Ω and since ỹh → ỹ weakly in H 1 0 (D). The formulas for b(•, •, •) and b h (•, •, •) are not affected by these extensions. One can pass to the limit in ( 8) by ( 14) and the strong convergence r h v → ṽ in H 1 0 (D) due to the regularity of v ∈ V( Ω). This ends the proof since V( Ω) is dense in V ( Ω) and ( 3) may be obtained.

Remark In fact, we have shown that the extensions ỹh → ỹ weakly in H 1 0 (D), on a subsequence. If the solution of ( 3) is unique, the convergence is valid on the whole sequence. Theorem 3. i) Any accumulation point of any sequence {Ω * h } h→0 of discrete minimizers of ( 10) is a continuous minimizer Ω * of (4).

ii) J h (Ω * h ) → J(Ω * ) for h → 0, on the initial sequence.

Proof i) Clearly {Ω * h }, h > 0 is relatively compact in the Hausdorff-Pompeiu complementary metric and we may assume that Ω * h → Ω on a subsequence; where Ω ∈ O by Proposition 4.1.

By Theorem 2, we get ỹh | E → ŷ| E strongly in L 2 (E), where ỹh is the extension by 0 of y h and ŷ is a solution of (3) in Ω. The convergence is valid on a subsequence.

We have J h (Ω * h ) → J( Ω). This is a consequence of j(x, ỹh ) → j(x, ŷ) weakly in L 2 (E) (see the assumption on j(•, •) in Theorem 1 ) and of

J h (Ω * h ) = E h j(x, y h )dx = E j(x, ỹh )dx - E\E h j(x, ỹh )dx (15)
The last integral in [START_REF] Rösch | Optimal control of the Stokes equations : a priori error analyses for finite element discretization with postprocessing[END_REF] converges to 0 as meas(E\E h ) → 0, Tiba [2011], and j(x, ỹh ) is bounded in L 2 (E), which is argued above.

For any Ω ∈ O, we can construct Ω h as in [START_REF] Lions | Contrôle des systèmes distribuiés singuliers[END_REF] and again by Theorem 2 and Proposition 4.1 we obtain that J h (Ω h ) → J(Ω). Taking into account that J h (Ω * h ) ≤ J h (Ω h ) we infer that J( Ω) ≤ J(Ω) for any Ω ∈ O, i.e. Ω is optimal for the problem (3)-( 5) and we redenote it by Ω * .

ii) This is a consequence of i) as the minimal value J(Ω * ) is uniquely associated to O.

Remark The results of this section may be extended to the cost functional corresponding to the choice Λ = Ω by using supplementary arguments as in Neittaanmäki, Sprekels and [START_REF] Neittaanmäki | Optimization of elliptic systems. Theory and applications[END_REF], p. 472.

Remark The approach of this paper is based on a fixed grid given in the whole domain D, i.e. it is a fixed domain method. It should be noticed that the finite dimensional optimization problem is nonconvex and it is not easy to find a global minimum Ω * h , h > 0. Starting with some initial guess Ω ∈ O, one can define Ωh ∈ O h by ( 6) and use it as initial iteration in some descent algorithm for the finite dimensional problem. Denote by Ωh the obtained finite dimensional "solution" (which is not necessarily a global minimum of J h ). Then, reading [START_REF] Lions | Contrôle des systèmes distribuiés singuliers[END_REF] in the converse sense, we get at least one Ω ∈ O, corresponding to Ωh . If the descent property for J h "dominates" the approximation error between (3) and ( 8), then J( Ω) < J( Ω), i.e. the method may find a better admissible domain from the point of view of the cost J.

  d and ν > 0 is the viscosity. By Theorem 1.2 from Temam [1979], the equation (3) has at least one solution y ∈ V (Ω). If d > 4, the supplementary condition y ∈ [L d (Ω)] d should be included in the definition (1), (2) of V (Ω).

Acknowledgement This work was supported by Grant 145/2011 of CNCS, Romania.


Mykhailo Potomkin
email: mika_potemkin@mail.ru

B Verkin

Asymptotic Behavior of Nonlinear Transmission Plate Problem

Keywords: Transmission problems, long-time behavior, asymptotic smoothness, attractor

We study a nonlinear transmission problem for a plate which consists of thermoelastic and isothermal parts. The problem generates a dynamical system in a suitable Hilbert space. Main result is the proof of asymptotic smoothness of this dynamical system and existence of a compact global attractor in special cases.

Introduction

In this work we deal with a partially thermoelastic plate: one part is of isothermal material, the second one is of material whose structure does not neglect the thermal dissipation. Due to the thermal dissipation, purely thermoelastic plate is exponentially stable in linear case (see, e.g., survey in [START_REF] Lasiecka | Control Theory for PDEs[END_REF]Chapter 3A]) or possesses a compact global attractor in cases of different kind of nonlinearities (see, e.g., [START_REF] Chueshov | Long-time behavior of second order evolution equations with nonlinear damping[END_REF]). On the other hand, in the case of purely isothermal plate the energy is constant, thus there could not be any decay to zero point in the linear model and global attractor in the nonlinear model. Here we investigate whether the thermal dissipation on a part of the plate is enough to have any stabilization. Exponential stability of the linear problem of this type was established in [START_REF] Rivera | A transmission problem for thermoelastic plates[END_REF].

Let Ω 1 , Ω 2 and Ω be bounded open sets in R 2 , Γ 0 = Ω 1 ∩ Ω 2 , Γ 1 = ∂Ω 1 /Γ 0 and Γ 2 = ∂Ω 2 /Γ 0 be smooth surfaces. We also assume that Ω = Ω 1 ∪ Ω 2 ∪ Γ 0 and Γ 1 ∩Γ 2 = ∅. In the model under consideration the plate (its middle surface), in equilibrium, occupies the domain Ω which consists of two parts Ω 1 and Ω 2 with common boundary Γ 0 . In what follows below ν denotes the outward normal vector on Γ 1 and Γ 2 , in cases of common boundary Γ 0 the vector ν is outward to Ω 2 . We consider the following system of equations:

ρ 1 u tt + β 1 ∆ 2 u + µ∆θ + F 1 (u, v) = 0 in Ω 1 × R + , (1)
ρ 0 θ t -β 0 ∆θ -µ∆u t = 0 in Ω 1 × R + , (2)
ρ 2 v tt + β 2 ∆ 2 v + F 2 (u, v) = 0 in Ω 2 × R + .
(3)

Fig. 1.

We impose the following boundary conditions:

u = ∂u ∂ν = 0 on Γ 1 , v = ∂v ∂ν = 0 on Γ 2 , (4)
u = v, ∂u ∂ν = ∂v ∂ν , β 1 ∆u = β 2 ∆v, β 1 ∂∆u ∂ν + µ ∂θ ∂ν = β 2 ∂∆v ∂ν on Γ 0 , (5)
θ = 0 on Γ 0 , ∂θ ∂ν + λθ = 0 on Γ 1 . (6)
The equations above are equipped with the following initial data:

u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), θ(x, 0) = θ 0 (x) in Ω 1 , v(x, 0) = v 0 (x), v t (x, 0) = v 1 (x) in Ω 2 . (7)
Coefficients ρ i , β i and µ are strictly positive, the functions

F i : H 2 (Ω 1 ) × H 2 (Ω 2 ) -→ L 2 (Ω i ), i = 1, 2
are nonlinear. Functions u(x, t) and v(x, t) describe the vertical displacement of the plate in Ω 1 and Ω 2 , respectively, and function θ(x, t) describes the temperature regime. Equations ( 1) and (3) are plate equations, equation ( 1) is a heat equation. Equalities in (4) mean that the plate is clamped along Γ 1 and Γ 2 . Boundary conditions (5) are transmission boundary conditions. Temperature function θ satisfies the Newton law of cooling on Γ 1 with some positive coefficient λ and vanishes on Γ 0 (see equalities [START_REF] Raugel | Global attractors in partial differential equations[END_REF]).

Let us introduce four problems which are concrete examples of abstract problem (1)- [START_REF] Khanmamedov | Global attractors for von Karman equations with nonlinear dissipation[END_REF].

Problem A corresponds to oscillations of a plate in the Berger approach. In this case

F 1 (u, v) = -M (u, v)∆u, F 2 (u, v) = -M (u, v)∆v, where M (u, v) = Γ + γ   Ω1 |∇u| 2 dx + Ω2 |∇v| 2 dx   . ( 8
)
Here Γ is a real number, γ is strictly positive.

In problem B we consider scalar nonlinearities, namely,

F 1 (u, v) = f 1 (u), F 2 (u, v) = f 2 (v),
where the scalar functions f i ∈ C 2 satisfy the following conditions: there exist such p > 1 and C > 0 that

|f ′ i (s)| ≤ C(1 + |s| p ), lim inf |s|→∞ f i (s) s > 0.
In problem C we deal with the von Karman nonlinearity. Here we set

Γ 2 = ∅ and F 1 (u, v) = -[u, F 1 ], F 2 (u, v) = -[v, F 2 ],
where [ψ, ϕ] = ψ xx ϕ yy + ψ yy ϕ xx -2ψ xy ϕ xy is the von Karman brackets; the Airy stress functions F 1 and F 2 solve the following equations (parameters γ i are strictly positive):

γ 1 ∆ 2 F 1 + [u, u] = 0 in Ω 1 × R + and γ 2 ∆ 2 F 2 + [v, v] = 0 in Ω 2 × R + (9)
with the boundary conditions:

on Γ 1 : F 1 = ∂ ∂ν F 1 = 0, on Γ 0 : F 1 = F 2 , ∂ ∂ν F 1 = ∂ ∂ν F 2 , γ 1 ∆F 1 = γ 2 ∆F 2 , γ 1 ∂ ∂ν ∆F 1 = γ 2 ∂ ∂ν ∆F 2 .
Problem D corresponds to the problem of oscillations of the Berger plate on an elastic base. Mathematically, this problem is a a generalization of problems A and B. Nonlinearities F i are given by the following equalities:

F 1 (u, v) = -M (||∇u|| 2 Ω1 + ||∇v|| 2 Ω2 )∆u + a 1 (x)u|u| p-1 + g 1 (x, u), F 2 (u, v) = -M (||∇u|| 2 Ω1 + ||∇v|| 2 Ω2 )∆v + a 2 (x)v|v| p-1 + g 2 (x, v),
where M (s) = s 1+α with α > 0, a 1 (x) ∈ L ∞ (Ω 1 ) and a 2 (x) ∈ L ∞ (Ω 2 ). We assume that the following condition holds:

either a(x) ≥ c 0 ∀x ∈ Ω or 2(α + 2) > p + 1, p ≥ 1. ( 10
)
Here a = {a 1 , a 2 } and c 0 > 0 is a small number. The functions g 1 (x, u) and g 2 (x, v) are scalar and satisfy the growth condition for some ε 0 > 0 and any

x i ∈ Ω i : ∂ ∂u g 1 (x 1 , u) + ∂ ∂v g 2 (x 2 , v) ≤ C(1+|u| max{0,p-1-ε0} +|v| max{0,p-1-ε0} ), ( 11
)
Our main result is the property of asymptotic smoothness of the dynamical system generated by weak solutions of problem (1)- [START_REF] Khanmamedov | Global attractors for von Karman equations with nonlinear dissipation[END_REF]. To achieve it we use method of so-called compensated compactness function first introduced in [START_REF] Khanmamedov | Global attractors for von Karman equations with nonlinear dissipation[END_REF] (see also [START_REF] Chueshov | Long-time behavior of second order evolution equations with nonlinear damping[END_REF]Proposition 2.10]). The method for various types of nonlinearities was developed in [START_REF] Chueshov | Long-time behavior of second order evolution equations with nonlinear damping[END_REF]. We also need to impose the following conditions on parameters:

ρ 1 ≥ ρ 2 and β 1 ≤ β 2 (12)
and on geometric structure of Ω i :

(x -x 0 ) • ν(x) ≥ δ 0 on Γ 0 , (13)
(x -x 0 ) • ν(x) ≤ 0 on Γ 2 (14)
for some x 0 ∈ R 2 and δ 0 > 0. Imposing these conditions authors of work [START_REF] Rivera | A transmission problem for thermoelastic plates[END_REF] proved the exponential stability of linear problem (F 1 = F 2 = 0). Asymptotic smoothness is important property of a dynamical system if one wants to prove the existence of a compact global attractor. In particular, asymptotically smooth dynamical system possesses a compact global attractor, if it possesses an appropriate Lyapunov function (for details we refer to [2, Chapter 2.4]).

Our result on asymptotic smoothness is applicable for each of the concrete problem listed above. For problems A, B and D we proved that the corresponding dynamical system possesses an appropriate Lyapunov function and, thus, there exists a compact global attractor.

Up to our best knowledge asymptotic behavior in transmission problem for a plate of types A, B, C and D was not considered before.

One can find formulations of our results in the next section. For proofs and other details we refer to our works [START_REF] Potomkin | A nonlinear transmission problem for a compound plate with thermoelastic part[END_REF] and [START_REF] Potomkin | On Transmission Problem for Berger Plates on an Elastic Base[END_REF].

Formulation of Main Result

Dynamical System

Below notation ψ = {ψ 1 , ψ 2 } means that ψ(x) defined for x ∈ Ω is equal to

ψ i (x), if x ∈ Ω i , for i = 1, 2.
To formulate a well-posedness result we need to impose the following conditions:

Ω1 |F 1 (w 1 1 , w 1 2 ) -F 1 (w 2 1 , w 2 2 )| 2 dx + Ω2 |F 2 (w 1 1 , w 1 2 ) -F 2 (w 2 1 , w 2 2 )| 2 dx (15) ≤ C(r)|| w 1 1 -w 2 1 , w 1 2 -w 2 2 || 2 H 2 T for all || w i 1 , w i 2 || H 2 0 (Ω) ≤ r, i = 1, 2.
We also assume that there exists such continuous functional Π :

H 2 0 (Ω) → R that d dt Π(w 1 , w 2 ) = Ω1 F 1 (w 1 , w 2 )w 1,t dx + Ω2 F 2 (w 1 , w 2 )w 2,t dx, ( 16
) Π(w 1 , w 2 ) ≥ -C, ∃C > 0, (17)
Π(w 1 , w 2 ) ≤ G || {w 1 , w 2 } || H 2 0 (Ω) . (18)
Condition ( 16) holds for

{w 1 , w 2 } ∈ L 2 (0, T ; H 2 0 (Ω)) and {w 1,t , w 2,t } ∈ L 2 ((0, T ) × Ω).
Conditions ( 17) and ( 18) hold for all {w 1 , w 2 } ∈ H 2 0 (Ω). The scalar function G : R + -→ R + is supposed to be bounded on bounded intervals. The condition ( 16) also means that feedback forces

{F 1 (u, v), F 2 (u, v)} are potential, i.e., {F 1 (u, v), F 2 (u, v)} is a Freshet derivative of Π(u, v).
For problem A Π(w 1 , w 2 ) = 1 4 M 2 (w 1 , w 2 ), for problem B:

Π(w 1 , w 2 ) = Ω1 w1(x) 0 f 1 (s)dsdx + Ω1 w2(x) 0 f 2 (s)dsdx.
For problem C, if calculate F i according to (9):

Π(w 1 , w 2 ) = γ 1 2 Ω1 |∆F 1 | 2 dx + γ 2 2 Ω2 |∆F 2 | 2 dx. ( 19
)
For problem D:

Π(w) = 1 2(α + 2) ||∇w|| 2(α+2) L 2 (Ω) + 1 p + 1 Ω a(x)|w(x)| p+1 dx+ Ω w (x) 0 g(x, s)dsdx,
where a = {a 1 , a 2 } and g = {g 1 , g 2 }.

We introduce phase space H = H 2 0 (Ω)×L 2 (Ω)×L 2 (Ω 2 ) and energy function E : H -→ R which we define for an argument w = (w 1 , w 2 , w 3 , w 4 , w 5 ) (here {w 1 , w 2 } ∈ H 2 0 (Ω), {w 3 , w 4 } ∈ L 2 (Ω) and w 5 ∈ L 2 (Ω)) as follows

E(w) = 1 2 Ω1 β 1 |∆w 1 | 2 + ρ 1 |w 3 | 2 + ρ 0 |w 5 | 2 dx + Ω2 β 2 |∆w 2 | 2 + ρ 2 |w 4 | 2 dx + 2Π(w 1 , w 2 ) . (20)
Now we are in position to formulate the theorem on well-posedness of problem (1)- [START_REF] Khanmamedov | Global attractors for von Karman equations with nonlinear dissipation[END_REF].

Theorem 1. Let (15),( 16), ( 17) and (18) hold. Then for any initial w 0 ∈ H and T > 0 there exists a unique weak solution w(t) ∈ C([0, T ]; H). Moreover, it satisfies the energy equality:

E(w(T )) -E(w(t)) = - T t Ω1 β 0 |∇w 5 | 2 dxdτ - T t Γ1 β 0 λ|w 5 | 2 dΓ dτ (21)
for all 0 ≤ t ≤ T . If one sets S t w 0 = w(t), then (H, S t ) is a continuous dynamical system.

Asymptotic Smoothness

In this subsection we define the notion of asymptotic smoothness of a dynamical system, impose additional conditions on nonlinear functions F i and formulate our result on asymptotic smoothness of the dynamical system (H, S t ).

Definition 1. Let (X, S t ) be a dynamical system. Assume that X is a complete metric space and S t is a semigroup of operators on X.The dynamical system (X, S t ) is said to be asymptotically smooth if for any positively invariant bounded set D ⊂ X there exists a compact K in the closure D of D such that

lim t→+∞ sup x∈D dist X (S t x, K) = 0.
For the detailed discussion and applications of asymptotic smoothness we refer to, e.g., [START_REF] Raugel | Global attractors in partial differential equations[END_REF] and [START_REF] Chueshov | Long-time behavior of second order evolution equations with nonlinear damping[END_REF].

To obtain asymptotic smoothness we need to impose additional conditions on nonlinear functions F i .

Assume that there exist such δ, σ > 0 that

w -→ Π(w) : H 2-δ -→ R is a continuous mapping, (22)
w -→ Π ′ Φ (w) : H 2-δ -→ H -σ is a continuous mapping. (23)
Our main result is the following theorem.

Theorem 2. Let ( 12),( 13), ( 14), ( 15), ( 17), ( 18), ( 22) and ( 23) hold. Then the dynamical system (H, S t ) is asymptotically smooth.

The method of the proof is based on idea of compensated compactness function (see [START_REF] Khanmamedov | Global attractors for von Karman equations with nonlinear dissipation[END_REF] and [START_REF] Chueshov | Long-time behavior of second order evolution equations with nonlinear damping[END_REF]). This result is applicable for all concrete problems, A,B,C and D, listed in introduction.

Compact Global Attractor and Its Properties

The direct application of [2, Corollary 2.29] gives the following theorem. Theorem 3. Let conditions of theorem 2 hold. Assume also that for all solution w(t) to problem ( 1)-( 7) the following statement holds: if E(w(t)) does not depend on t, then w(t) does not depend on t.

(

) 24
In other words, energy E is constant only on stationary trajectories. Then the dynamical system (H, S t ) possesses a compact global attractor.

Verification of condition (24) could be reduced to the problem which is relative to unique continuation problems. Let w(t) = (u(t), v(t), u t (t), v t (t), θ(t)) be a solution of problem (1)- [START_REF] Khanmamedov | Global attractors for von Karman equations with nonlinear dissipation[END_REF] such that E(w(t)) is constant. Let us denote v h (t) := v(t + h)v(t), B(t)v h := F 2 (u, v(t + h)) -F 2 (u, v(t)) for some h > 0. If E(w(t)) does not depend on t, then v h solves the following problem:

ρ 2 v h tt + β 2 ∆ 2 v h = B(t)v h , v h | Γ0 = ∂v h ∂ν | Γ0 = ∆v h | Γ0 = ∂∆v h ∂ν | Γ0 = 0.
Therefore, if we prove that v h (t) ≡ 0, then we will have that w(t) ≡ w 0 . Using Pochozhaev multiplicator (xx 0 ) • ∇v h , where vector x 0 is the same as in conditions ( 13) and ( 14), for concrete problems A and B with f 2 ≡ 0 and Carleman estimates obtained in [START_REF] Albano | Carleman estimates for the Euler-Bernoulli plate operator[END_REF] for concrete problems B and D, we manage to verify (24). Thus, we obtain the following corollary.

Corollary 1. Let ( 12),( 13), ( 14) hold. The dynamical system (H, S t ) corresponding to one of the concrete problems A, B or D possesses a compact global attractor.


Thomas Leduc
email: thomas.leduc@lip6.fr

One-dimensional discrete computer model of the subduction erosion phenomenon (plate tectonics process) Multiconference CESA'98

In this proceeding, we present a discrete computer model of a tectonic process : the subduction erosion (without accretion) phenomenon. Up to now, models that have been studied were too inflexible. Here, we describe an original and simple approach, that has not been explored yet, based on overlapping cellular automata. Problems concerning implementation are exposed, as well as some results. At least we present a parallel code implementing the model.

Introduction

This work is the result of a cooperation with the Laboratoire de Géodynamique Tectonique et Environnement (Paris). It refers to the modeling of a geotectonics phenomenon : that of the subduction erosion on the contact between two convergent margins (or tectonic plates). We start initially by presenting the context geotectonics, then we approach our discrete model and finally we expose the results of our simulation. We show, in particular, that we have to develop a software adapted to our own simulation.

Subduction of oceanic crust beneath continent

In the theory of plate tectonics, the lithosphere is divided into a network of blocks (plates), floating upon viscous asthenosphere.

Plates are mechanically decoupled from each other, although plate margins are in intimate contact. There are relative plates motions of approximately 10 cm a year due to the convection process in the mantle. The boundaries of plates can take three forms : mid-oceanic ridge (where plates are diverging ; accretive or constructive plate margins), trenches (where plates are converging ; destructive plate margins) and transform faults (relative motions of adjacent plates are tangential ; conservative plate margins). Plates are considered to be internally rigid, that is to say : when a stress is applied to a margin of a plate, it is transmitted to its opposite margin with no deformation of the plate interior [START_REF] Kearey | Global Tectonics[END_REF]. The deformation, then, only takes place at plate margins.

At convergent margins, the oceanic plate (the lower plate) is subducted beneath a continental crust (the upper plate). The subduction zone is characterized by an oceanic trench, an overpressure of fluids along the subduction interface, a landward dipping zone of earthquakes (Figure 1). . . In this kind of kinematics, two types of major processes can occur [START_REF] Jacques | La fosse d'Amérique Centrale : convergence, accrétion, érosion tectonique[END_REF] : a compression (type 1 margins, accreting process) and an extension (type 2 margins, subduction erosion process). At type 1 margins, accreting ones, sedimentary material that has been laid down the outer slope, is progressively accumulated to the front of the inner slope. At type 2 margins (21,000 km in global length, all over the world), all incoming sediment is subducted beneath the upper plate and sunk to the viscous asthenosphere.

Up to now, some experimental Sandbox modeling and some global numerical models (Finite Elements methods) have already been made [START_REF] Lallemand | Coulomb theory applied to accretionary and nonaccretionary wedges : possible causes for tectonic erosion and/or frontal accretion[END_REF]. The goal of such studies is to estimate the amount of solid mass of sediment subducted at convergent ocean margins, that is not frontally accreted.

Cellular Automata as an alternative to differential equations in modeling physics

Introduction

In the last two decades, a wide variety of cellular automata applications has been proposed, including modeling of reaction-diffusion systems, hydrodynamics, biological modeling. . . Sandbox modeling approaches of this geophysical phenomenon and simplicity, efficiency, but also easiness to implement, of models of physics of the granular state [START_REF] Heinrich | Physics of the granular state[END_REF], have finally convinced us to develop an original model based on fully discrete dynamical systems. Space, time and even dynamical variables are defined to be discrete. Moreover, the simplicity of the computational model means that it is very easy to incorporate different physical effects. We really think that "discrete dynamical systems, in this view, attempt not to simulate specific physical phenomena but rather embody general physical ideas" [START_REF] Gérard | Simulating physics with cellular automata[END_REF]. As it is simpler to simulate avalanches and self-organized criticality in a sandpile with a one-dimensional cellular automaton rather than with a two-dimensional one, we have first decided to study a one-dimensional model of the subduction-erosion phenomenon.

Modeling

Topology : in our model, "space" is a discrete universe, a regular array of sites (cells), as shown in figure 2. Each cell is a vertical portion of "space" the value of which belongs to a finite set. Therefore, we consider a finite set, in one-dimensional space, of finite automata. The state of a cell is determined by a set of seven cross-section thicknesses. Each of those thicknesses is an integer less than the global height. Dynamics of the discrete system : starting from an initial configuration, the set of cells evolves at each discrete time step according to a local transition (updating) rule. The sampling of the motion is periodic and the update is local, parallel and synchronous. The rule of transition can be seen as a function whose arguments are the states of the cell itself and its neighboring cells, at the previous time step. The evolution of a cell depends only on a local neighborhood of three cells (running cell included). About our automaton : The subducted lithosphere is mechanically smooth and passes from an almost horizontal level in front of the pit to a constant dip under the volcanic arc (beyond 50-100 km). It is generally estimated that the subduction speed is a constant. With regard to the boundaries problem, it is necessary to compute separately the cells located at the two limits of the unidi-mensional automaton (the first introduces material into the system, whereas the last makes possible to eliminate it). The engine of "our" basal erosion is not the subduction of asperities (such as the underwater mounts, mid-oceanic ridge, grabens. . . ) on the oceanic plate. It is rather necessary to take into account overpressures of fluid at the top of the subduction interface, which can induce a hydrofracturation.

Specificity of our global transition function :

Our cellular automaton is, in fact, an overlap of three global functions of transition. These three functions of transition can be regarded as generalizations of the one-dimensional Sand Pile Model. They represent three different physical phenomena, themselves, on three quite distinct scales of time. In the circumstances : a dive of the oceanic plate (oblique downward translation under the constraint of the weighty effort exerted by the upper plate), a subcrustal erosion (by hydro-fracturing and overpressures on the outline level of subduction) as well as the subsidences which result from it on all the height from the overlapping plate, and, finally, surface avalanches at the top of the higher plate.

These last phenomena, true rockslide on the surface of the overlapping margin, occur when the local dip of this one is too important. It is important to distinguish three different scales of time. That of the "instantaneous" phenomena (relative to the other phenomena at least) initially, which occur on a "fast" scale of time (it is the case of the surface avalanches). That of the phenomena "with great width", then, which occur on a "slow" time scale (it is the case of the subduction even of the oceanic margin with the "generation of steps" which is correlated to it, but it is also the case of "ageing" and the subcrustal erosion of the overlapping continental margin). Lastly, we introduce an "average" scale of time to represent physical phenomena intercalating themselves between the two classes of above mentioned phenomena (it is the case of the general translation of the face of the oceanic margin, of the levelling of the oceanic trench and the subsidences within the overlapping margin).

Possibilities and limits of the model

Our local transition function [START_REF] Thomas | Modélisation par un système dynamique discret du processus de subduction-érosion en tectonique des plaques : première approche unidimensionnelle[END_REF], is able to represent phenomena on a wide time scale such as : a dip of the lower subducted plate (80 km / 1 million years), a landward migration of the front of the upper plate and the trench axis (20 km / 1 million years), a subcrustally erosion of the continental margin basement, an underthrust of sediment material and avalanches at the top of the overriding plate (almost instantaneous phenomena compared with the dip or the landward migration). On the other hand, in a cellular automaton, physical reality is only approached by simplistic rules of evolution that have to be modified according to the experiment. The physical rules are replaced by rules of evolution. The interpretation of the results can thus be as well as possible only qualitative.

Simulation

Development of a new dedicated software

There already exists, in the public domain, software tools making possible to visualize [START_REF] Michael | Xtoys : cellular automata on Xwindows[END_REF] or handle cellular automata. However, it seems us necessary to develop our own platform. Indeed, although very effective and very useful for a two-dimensional simulation, The Cellular Automata Simulation System [START_REF] Dana | The Cellular Automata Simulation System : Language Reference Manual[END_REF], developed around the Cellang language, cannot absolutely be appropriate to us here, since it does not allow us a two-dimensional visualization in the case of a uni-dimensional automaton. We chose the tandem C and Parallel Virtual Machine [START_REF] Al | PVM: Parallel Virtual Machine (A Users' Guide and Tutorial for Networked Parallel Computing[END_REF] library for questions of portability. Thus our application functions as well, except for the performances, on a network of heterogeneous workstations as on a Massively Parallel Processors computer like the CRAY T3E (concerning the processing and the first graphic postprocessing at least).

A parallel implementation on the CRAY T3E computer

The parallel strategy we use is the following one : work and data are distributed among the processor elements. We use a simple and natural one-dimensional domain decomposition : the array of cells has been divided into a number of parts (subdomains of contiguous cells) equal to the number of PEs. After each iteration, solution values on the boundaries of a subdomain need to be exchanged with the adjacent subdomain. We can see that each processor will exchange messages with its left and right neighbors except the first and last ones which have respectively no predecessor and no successor. That's why there is an increase of communication time as the number of processors increases (figure 4). There is a unified framework for parallel and sequential simulation (speedup measures are consequently more fair).

Parallel algorithm

We call "supervisor" the first of the processes to have been included in the group named "workers". As we may notice in the program below, there is transmission of an aggregate variable called C oef fT r between the "slow" global transition function and the "average" global transition function from the dynamic system. This aggregate variable is recovered at output of the "slow" global transition function. It is then passed in parameter of the "average" global transition function. It makes possible the whole of the cells to take note, immediately and in a synchronous way, of a phenomenon of translation of the oceanic margin.

procedure Fglobal(c,cc,

Initialize the local array c do

Fglobal(c,cc,fslow ,Coef fT r) until "fast condition" is achieved until "average condition" is achieved If "saving condition" is achieved then

The supervisor gather all the locals arrays distributed on the workers If the current process is the supervisor then save the current image until "slow condition" is achieved exit PVM

Results, visualization and future improvements

A two-dimensional motion picture is available at http://quartz.dgs.jussieu.fr:8080/ANIM/ using a MPEG-player software. It has been made using the convert tool of the Im-ageMagick software. These images were generated, for the processing, on a CRAY T3D with 256 processors DEC ALPHA EV5, and, for postprocessing, on a four-processors Silicon Graphics Power Challenge XL.

As shown in the figure 3, the dynamical behavior of our model is proper. We are expecting from a further improvement of the two dimensional cellular automaton we are developing, that it will also represent a frontal accretion, a subcrustally underplating and an internal fluid overpressure causing hydrofracturing effects in the upper plate.

Figure 1 :Figure 2 :

 12 Figure 1: Subduction of oceanic crust beneath continent (8 cm/year, 8 0 dip).

Figure 3 :Figure 4 :

 34 Figure 3: 1100 th time step of the motion picture (MPEG format).

  flocal ,coeff)

	cc	c
	Send the values of c1 and clength to the
	left and right neighboring processors
	Receive the updated values of c0 and
	clength+1 (the "ghost cells") from the
	corresponding processors
	Compute the global boundaries cells (on
	the two boundaries PEs)
	For i	1 to length, do
	ci = flocale(cci 1; cci; cci+1; coef f )
	end
	Notes :
	in order to optimize the performance of the previous pro-
	cedure, we have overlapped time of communications by
	time of computations (non-blocking sending of bound-
	aries values, computation of local data, receiving of up-
	dated values from the neighboring PEs, last computations
	of boundaries cells),
	the "saving condition" of the following program depends
	on three parameter : whole number of images, period be-
	tween two images, number of the first image,
	the number of cells must be a multiple of the number of
	processor elements.
	program
	PVM initialization
	Join the group "workers"
	If the current process is the supervisor
	then
	Read the files of parameters
	Broadcast the values of parameters to
	all the tasks of the group "workers"
	else
	Receive the values of parameters sent by
	the supervisor
	Let c and cc be two arrays of length + 2 cells, where length = NBC nbproc (N BC 2 IN)

Acknowledgement :

This work has been partially supported by the Institut du Développement et des Ressources en Informatique Scientifique (IDRIS, France).


GENETIC MODIFICATION OF THE HUMAN GERM LINE: THE REASONS WHY THIS PROJECT HAS FADED CHANGER LES GENES? UN PROJET QUI N'EST PLUS PRIORITAIRE

Keywords: Amélioration de l'être humain, biologie moléculaire, édition du génome, eugénisme, recombinaison homologue Eugenics, genome editing: homologous recombination, human enhancement, molecular biology

La modification ciblée de la lignée germinale (et donc de l'espèce humaine) est restée un objectif distant mais raisonnable, depuis l'émergence de la génétique (et même avant) jusqu'à ces dernières années. J'ai choisi trois temps pour étudier les évolutions historiques de ce projetdans les années 1930, au sommet du mouvement eugéniste, autour de 1974 quand la biologie moléculaire triomphait, et aujourd'hui -et j'ai sélectionné trois critères pour estimer la faisabilité d'un tel projet -l'état des connaissances scientifiques, l'existence de techniques adaptées, et les demandes de la société. Bien que les techniques longtemps espérées pour modifier la lignée germinale soient aujourd'hui disponibles, je montrerai que la plupart des attentes qui soutenaient ce projet ont disparu, ou sont considérées comme pouvant être atteintes par des stratégies totalement différentes.

The Colloquium -Biologie et devenir de l'homme‖ was organized in Paris in 1974, at a time when the new discipline of molecular biology had a very high profile. In the preceding years, the chemical nature and structure of the genes had been unveiled, the origin of mutations understood, and the precise relation between genes and proteins (the genetic code) discovered. Molecular biology had rapidly acquired a dominant position within scientific institutions: the recent appointment of Jacques Monod to Director of the Pasteur Institute was a sign of this newly acquired power. In addition, molecular biology was on the eve of a new revolution-the rise of genetic engineering. The projects were already there, and the first steps had been accomplished in US laboratories. However, these early achievements had been acknowledged by a very small number of French biologists.

The objective of the 1974 Colloquium was to discuss the new powers of biology, and the new duties of biologists. Within this framework, I have decided to examine how the project to modify the human germ line genetically was reconsidered after the rise of molecular biology, and what it has become forty years later, with the huge amount of biological information acquired since the beginnings of molecular biology. This project has a very long historical background, even if the name given to it changed with the state of knowledge, and the tools at its disposal. What would in the past have been called -transformation of the human species‖ is now considered as -genetic enhancement‖, or more neutrally as -genome editing‖. Despite these changes in vocabulary, the objective has remained similar, with its two projects-the correction of genetic defects and the enhancement of human genetic abilities. I will compare these two projects and the contrasting attitudes towards them, in the 1970s and today. I needed a point of reference, which I have chosen as the 1930s, at the pinnacle of eugenics. In the first part, I will present the criteria that I have selected to estimate the feasibility of these projects at a given time. Quite surprisingly, I will provide evidence for an inverse relation between the extent of knowledge and the availability of techniques permitting the modification of the genome, and the priority accorded to these projects. Today, the technologies are there, but the motivation has disappeared! CRITERIA TO ESTIMATE THE FEASIBILITY OF THESE PROJECTS Three criteria must be fulfilled for such projects to be developed. The first is a sufficient state of knowledge. The second is the availability of tools permitting their realization. And the third is that such projects have to be considered as valuable, a priority not only for specialists, but for a large fraction of society. These criteria are obviously of relative value. Scientific knowledge can be considered sufficient at a given time, and only later shown to have been insufficient to support the projects that were proposed. The social consensus is never perfect and is particularly difficult to gauge in authoritarian societies. The notion of -scientific knowledge‖ is not as simple as might be thought at first glance: to appreciate the consequences of a genetic modification of the germ line, the skills of molecular biologists are not sufficient: population geneticists and evolutionary biologists are needed to estimate the long-term consequences of these modifications.

PROJECTS IN THE 1930S

The idea that it was necessary to control (and to improve) human reproduction is not new. Plato, and Cabanis at the beginning of the 19 th century, were advocates. After the acceptance of Darwinian evolutionary theory, this ambition dramatically evolved into the idea of replacing the action of natural selection, which had disappeared in human societies because of the development of social and medical care, by artificial selection. It was deemed necessary both to improve the reproduction of the best and to prevent the reproduction of individuals likely to transmit their physical and mental deficiencies to their progeny.

Eugenic methods of forced sterilization were not unanimously accepted in the first decades of the 20 th century, but there was a wide consensus on the necessity and possibility to improve the human species. The talk given by the physical chemist Jean Perrin at the inauguration of the newly constructed Institute of Physical-Chemical Biology (IBPC) in Paris in 1927 bears witness to these expectations: -The issue is to modify, maybe to a prodigious degree, the type of equilibrium, the organs, the hereditary basis of organisms. This search for an experimental transformation of species will play for the biologist a role analogous to that played for the chemist for centuries by the transmutation of elements... This research may lead us, must lead us, to transform current human beings, unchanged for millennia, into higher and higher beings, richer in sensations, feelings, and thoughts, and more generally richer in what will correspond for consciousness to a wider and more complex development of the brain‖ (1).

The stimulating role of physics in future developments in biology is obvious in this quotation: the transformation of elements has become feasible for the chemist, as the transformation of species will be for the biologist in the near future. Experimental transformism refers to the neo-Lamarckian tradition dominant among French biologists (2), according to which organisms can be directly modified through changes in the environment.

In the following years, under the impetus given by population geneticists, the Modern Synthesis between genetics and Darwinism was elaborated by the evolutionary biologists Julian Huxley, Ernst Mayr, Theodosius Dobzhansky, George Simpson and others. Most of the founders of the Modern Synthesis accepted the idea that human beings were at the top of evolution, the first to have had access to its rules. For this reason, they were now in charge of evolution, of the future transformations of organisms and human beings (3). Even George Simpson, the most committed of evolutionary biologists in the fight against finalism, nevertheless admitted that -the fact that man knows that he evolves entails the possibility that he can do something to influence his own biological destiny‖ [START_REF] Simpson | The world into which Darwin led us[END_REF].

By using our criterion of feasibility, it is obvious that these projects were beyond reach. The experimental transformism never worked, i.e. changes in the environment never directly produced stable modifications of the progeny. The models used by eugenicists to develop their projects were rapidly shown to be not only simplistic, but also scientifically incorrect. Feeble-mindedness, one of the major incentives for forced sterilization, was not due to one unique recessive mutation as initially proposed by H. Goddard [START_REF] Barker | The biology of stupidity: genetics, eugenics and mental deficiency in the interwar years[END_REF]. And if most of the defects result from recessive mutations, forced sterilization will have a limited effect since it does not prevent the transmission of -bad‖ copies of the genes through generations.

The consensus in favour of a genetic modification was initially strong, but it progressively faded because of the way the eugenic measures had been applied in the US, and later in Germany. As was argued by Thomas Morgan as early as 1934 in his Nobel lecture, there are other more human ways to address these problems through medicine [START_REF] Morgan | Nobel lecture[END_REF].

PROJECTS AT THE TIME OF THE PARIS CONFERENCE

Molecular biologists had contrasting attitudes towards the projects of gene modification burgeoning in the 1960s and 1970s. Some were enthusiastic. Such was the case of Rollin Hotchkiss, a specialist in bacterial transformation: -The wealthy and other royal families as always can even hope to purchase special advantage, such as determinants of musical ability, linkage groups providing skill in political oratoryor will they prefer skill in such gentlemanly pursuits as polo, or (somewhat less expensive) single factors enabling one to ride graceful and sure on an appropriately well-bred horse?‖ (7, p.199) Edward Tatum, the discoverer with George Beadle of the one gene-one enzyme relation, with Joshua Lederberg of sexuality in bacteria, was also convinced that with progress in our understanding of functioning and regulation of gene activity, it will be possible -to exclude structural or metabolic errors in the developing organism but also to produce better organisms‖ [START_REF] Tatum | A case history in biological research[END_REF]. Bernard Davis was much more cautious in his -prospects for genetic intervention in man‖, pointing to the difficulties stemming from the polygenic control of most human traits [START_REF] Davis | Prospects for genetic intervention in man[END_REF]. Jacques Monod was even more pessimistic, considering that the complexity of the genomes of higher organisms prevented their modification, maybe forever [START_REF] Monod | Le hasard et la nécessité[END_REF].

This absence of consensus among scientists reflected the particular situation that prevailed at the end of the 1960s, the contrast between the rapid discovery of the major principles guiding gene action in the preceding period and the complete absence of suitable tools for the isolation and characterization of the genes present in complex organisms. These tools were developed in the 1970s and by the 1980s were widely used in labs.

But there was also no social consensus on the need for such projects. Eugenics has now become a frightening word. Rollin Hotchkiss felt constrained to give credit to the altruism of the exponents of eugenics since the time of Galtonsomething that has been far from obvious (7, p.197). In the 1960s, in relation to the war in Vietnam, there was a generalized lack of confidence in science and technology.

FROM THE 1970S TO TODAY In the last forty years, knowledge of genes and of the way they contribute to the construction of organisms and the realization of their functions has progressed dramatically. It has become possible, at least in the most favourable cases, to describe fully the causal chain that relates the product of a gene to the complex functions in the elaboration of which it participates; and to explain, in the case of mutations, the functional modification from the alteration of the gene.

In parallel, Bernard Davis's anticipations have been fully confirmed: in most cases, there is no simple relation between a gene product and a function of the organism. Gene products cooperate in the formation of complex systems. In addition, the actions of genes cannot be understood without a precise description of the hierarchy of the structural levels forming an organism.

What is most remarkable is the technological progress leading to the replacement at will of a gene in a genome by a different, modified version of the gene. As I have mentioned, the first step was the development of genetic engineering in the 1970s. This opened the door to animal and plant transgenesis, and to the first attempts at gene therapy. The main problem was that the position of gene insertion was uncontrolled. It remained impossible to replace a non-functional copy of a gene by a functional one; what was possible was simply to add to the genome a functional copy of the non-functional gene. The insertion might occur close to an oncogene or in a tumour-suppressor gene, facilitating the emergence of tumour cells. This occurred in the first attempts at gene therapy in France at the end of the 1990s in immune deficient babies (the so-called -bubble babies‖ forced to live in a bubble protecting them from infection) [START_REF] Cavazzana-Calvo | Gene therapy of human severe combined immunodeficiency (SCID)-X1 disease[END_REF].

The only way to target a precise site in the genome was to insert the gene by homologous recombination. This strategy led at the end of the 1980s to the development of the knockout technology permitting the specific inactivation of a gene. The result was achieved not by an increase in the level of homologous recombination, but by the selection of the rare embryonic stem cells in which homologous recombination had occurred [START_REF] Thomas | Site-directed mutagenesis by gene targeting in mouse embryo-derived stem cells[END_REF]. These cells were injected into a blastocyst and transgenic animals were obtained at the next generation. This strategy could obviously not be used for modification of the human germ line. Different methods to increase the proportion of homologous recombination were tried, with results insufficient to permit any application to humans. The breakthrough occurred a few years ago with the adaptation to animals of a system used by bacteria to protect themselves against bacteriophages and foreign plasmidsthe CRISPR-Cas system [START_REF] Horvath | CRISPR/Cas, the immune system of bacteria and archaea[END_REF]. In this system, the action of a nuclease is targeted towards a specific position in the genome by a guiding RNA, in presence of a template for homologous recombination.

The efficiency is now sufficient to permit the correction of a genetic disease such as muscular dystrophy in mice by injection of the three components at the one-cell stage, after fertilization. The percentage of homologous recombination is not yet 100%, but already sufficient to correct the disease [START_REF] Long | Prevention of muscular dystrophy in mice by CRISPR/Cas9-mediated editing of germline DNA[END_REF]. It is a completely new result that opens the door to precise editing of the genome.

A second line of research reached the point in its development where a modification in the human germ line seemed not only desirable, but also achievable. The first steps were done by biologists looking for a way to palliate alterations of the cytoplasm in the egg [START_REF] Callaway | The power of three[END_REF]. They showed that the injection of cytoplasmic extracts from normal individuals could correct the cytoplasmic abnormalities. It was hypothesized that these abnormalities probably originated in a dysfunction of mitochondria, and that it was the mitochondria present in the cytoplasmic extracts that corrected the deficiency.

This approach was resumed in the United Kingdom to the point that a therapeutic project was submitted to the Human Fertilisation and Embryology Authority, debated, and approved. Opponents underlined that, using the protocol, this would be the first time a genetic change had been deliberately made to the human germ line [START_REF] Darnovsky | A slippery slope to human germline modification[END_REF].

The prospect of deriving sperm cells and eggs from stem cells was also considered to be achievable in the near future (17). This opened the door to the application of the strategy for knocking out genes in animals to genetic modification of the human germ line.

So, in a few years parallel and independent research lines converged, demonstrating the possibility of genetically modifying the human germ line.

Independently, various transgenesis experiments on animals showed that some of these modifications could lead to performance enhancement [START_REF] Manabe | Facilitation of longterm potentiation and memory in mice lacking nociceptin receptors[END_REF]. One of the most emblematic examples was the enhancement of learning and memory obtained by a genetic modification of one receptor of neurotransmitters, the glutamate receptor [START_REF] Tang | Genetic enhancement of learning and memory in mice[END_REF].

In fact, since the 1970s, the hopes expressed by Rollin Hotchkiss and Edward Tatum have never been completely put aside. In 1998, a meeting was organized at the University of California Los Angeles that recommended brushing aside legal obstacles preventing the modification of the human germ line [START_REF] Wadman | Germline gene therapy ‗must be spared excessive regulation[END_REF]. In 2001, Jonathan King described this issue, and the apparently unanimous opposition to these experiments, as -biology's last taboo‖ [START_REF] Knight | Biology's last taboo[END_REF].

What might appear as an irreversible move towards modification of the human genome has over recent years had to contend with a growing number of arguments against going in this direction, or suggesting radically different directions to address the same issues.

Some of these arguments are not new. As already argued by Thomas Morgan, some genetic disorders can be addressed using drugs. More recently, Arnold Munnich has strongly argued that it is far too restrictive to limit the fight against genetic diseases to gene therapy. Like many diseases, they could be controlled, or even cured, by well-chosen drugs, and he has afforded examples confirming the efficiency of these indirect strategies (22).

The importance given to the study of epigenetic modifications suggests also that another road-modification of the environment-might be followed to attenuate the effects of gene dysfunction.

Similarly, although it was supported by a very different spirit, the myth of the convergence of nanotechnology, biotechnology, information technology and cognitive science (NBIC) led to radically different ways of enhancing humans, by fitting them with electronic devices to replace the functions of deficient organs. The creation of artificial retinas is progressing rapidly and in a few years will offer new ways to counter the consequences of some genetic defects.

Of the two objectives always sought by supporters of intervention in human reproduction elimination of defects and enhancement of human abilitiesthe first may be reached, more simply and efficiently, by early diagnosis (before implantation or prenatally) and elimination of the affected embryos.

For a policy of germline modification to be efficient, it would have to be applied to individuals in whom only one copy of the gene is mutated (heterozygotes). They do not suffer from disease in recessive genetic disorders, which are the most frequent. Therefore, gene editing would create a risk for these individuals, without any direct benefit for them. In addition, a significant proportion of genetic defects are not transmitted by the parents, but arise de novo at each generation. The hope to eradicate forever genetic defects in humans is an illusion. Whatever the strategy used, the efforts will have to be permanently reinitiated.

Probably the most significant blow to the vision of a new world in which genome editing would have a major place came from the transformations of evolutionary biology. The idea that humans are at the top of evolution and have the responsibility to prolong its action on themselves and on other organisms has totally disappeared from the writings of evolutionary biologists, and probably also, at least partially, from their thoughts. The idea that mutations are good or bad is simplistic: the effect of a mutation depends upon the environment. A mutation with a deleterious effect can afford a benefit in particular conditions: such is the case for the mutation responsible for sickle cell anaemia which prevents the development of the agent of malaria. It is difficult to predict the short-term effects of a genetic modificationit is often impossible to anticipate its side effectsbut it is even more difficult to foresee the long-term effects in future environmental conditions that are unknown.

CONCLUSIONS

There has been a progressive separation between the therapeutic projects of genome editing and the more ambitious projects of genetic enhancement. Some of the former, limited in their objectives, will probably be developed, such as the replacement of mitochondria. Many other issues will find another solution, consisting for instance in earlier and more efficient diagnosis of affected embryos and their elimination.

The place of genetic modification in human enhancement will be limited. Genetic modifications to obtain -superhuman athletes‖ are still discussed, at least as a possibility that some will try to exploit (24), but it is obvious that this will be a modification of somatic cells in individuals, not of the germ cells. There is no longer any ambition to produce a -race of athletes‖.

Genetic modification of the germ line would require a consensus on the biological future of mankind that does not exist! A bigger brain is an objective that is no longer considered valuable. And what was sought through the creation of a bigger brain could be achieved by increased interconnection between the human body and electronic devices.

Priorities have changed dramatically since the time when human beings were seen as the masters of evolution. Today, our aim is far less ambitious: to ensure the survival of humans and other species endangered by the uncontrolled human actions of previous centuries!

ACKNOWLEDGEMENTS

I am indebted to David Marsh for critical reading of the manuscript, and to Jean-Noël Missa and Laurence Perbal for the wonderful organization of the Brussels meeting.

Michel Morange

Centre Cavaillès, République des savoirs: Lettres, sciences, philosophie, USR3608, Ecole normale supérieure, 29 rue d'Ulm, 75230 Paris Cedex 05, France Tel: 0144322961 Fax: 0144323941 email: michel.morange@ens.fr

The author declares that he has no conflict of interest

Bibliography

(1) Perrin J., "L'Institut Edmond de Rothschild", La Revue de France, 1927, p.3-12


Matthias Gerdts
email: matthias.gerdts@unibw.de

Ilaria Xausa
email: ilaria.xausa@unibw.de

Avoidance Trajectories Using Reachable Sets and Parametric Sensitivity Analysis

Keywords: driver assistance, collision avoidance, optimal control, reachable sets, parametric sensitivity analysis

The article suggests a conceptual model-based simulation method with the aim to detect collision of cars in all-day road traffic. The benefit of the method within a driver assistance system would be twofold. Firstly, unavoidable accidents could be detected and appropriate actions like full braking maneuvers could be initiated in due course. Secondly, in case of an avoidable accident the algorithm is able to suggest an evasion trajectory that could be tracked by a future active steering driver assistance system. The algorithm exploits numerical optimal control techniques and reachable set analysis. A parametric sensitivity analysis is employed to investigate the influence of inaccurate sensor measurements.

Introduction

Over the years many passive and active safety systems have been developed for modern passenger cars with the aim to reduce the number of casualties in traffic accidents. Passive safety systems contain amongst others improvements of the chassis, airbags, seat belts, and seat belt tighteners. These safety systems help to reduce the severeness of accidents once an accident has occured. In contrast, semi-active safety systems and driver assistance systems, for instance anti-blocking system, braking assistant, anti-slip regulation, electronic stability control, adaptive cruise control, lane departure warning, or blind spot intervention, become active in critical situations before an accident occurs and intend to prevent accidents. In future, active driver assistance systems that actively initiate braking maneuvers or even active steering maneuvers to avoid obstacles will become relevant in order to detect potential collisions and reduce severeness of collisions. The availability of high performance sensors will play a central role in future collision avoidance systems. But next to the required technical devices, intelligent software systems and algorithms will play a crucial role as well. The main tasks in collision avoidance are to reliably indicate future collisions andif possible -to provide escape trajectories if such exist. This paper suggests an optimal control based method that has the potential of fulfilling these two tasks.

Model Scenarios

We investigate two model scenarios that are likely to occur in all-day traffic. According to [9] wrong velocity, short distance, and overtaking maneuvers are responsible for approximately 29.7 % of accidents with injuries to persons. The following model scenarios address these situations, see Figure 1. The typical time to collision ranges from 0.5 to 3 seconds. For simplicity we assume a straight road throughout. A reference coordinate system is used with the x-axis pointing into the longitudinal direction of the road and the y-axis pointing in the crossdirection of the road.

Scenario 1: A stationary obstacle at a given distance to an approaching car, which drives at a prescribed speed, has to be avoided.

Scenario 2: An overtaking maneuver on a highway is considered. One car (car A) has initiated an overtaking maneuver to overtake car B while another car (car C) is approaching at a prescribed speed. For these two model scenarios we aim at answering the following questions:

-Can a collision be avoided? -If a collision can be avoided, how can it be avoided?

Model of the Car

In this article the single-track car model is used. It is a simplified car model, which is commonly used in the automobile industry for basic investigations of the dynamical behavior of cars. It is based on the simplifying assumptions that rolling and pitching behavior of the car body can be neglected. The car model includes two control variables for the driver: the steering angle velocity |w δ | ≤ 0.5 [rad/s] and a function

F B with values in [F Bmin , F Bmax ], F Bmin = -5000 [N ],
F Bmax = 15000 [N ], which models a combined brake (if F B > 0) and acceleration (if F B < 0) assembly. Details on the model can be found in [START_REF] Gerdts | Solving Mixed-Integer Optimal Control Problems by Branch&Bound: A Case Study from Automobile Test-Driving with Gear Shift[END_REF][START_REF] Gerdts | A variable time transformation method for mixed-integer optimal control problems[END_REF][START_REF] Gerdts | Generating Optimal Trajectories for an Automatically Driven Car[END_REF]. The dynamics are given by the following system of differential equations for the car's center of gravity (x, y) in the plane, the yaw angle ψ, the velocities v x and v y in x-and y-direction, respectively, the yaw angle rate w ψ , and the steering angle δ:

x ′ = v x , y ′ = v y , ψ ′ = w ψ , δ ′ = w δ , (1)
v ′ x = 1 m [F x cos ψ -F y sin ψ] , (2)
v ′ y = 1 m [F x sin ψ + F y cos ψ] , (3)
w ′ ψ = 1 I zz [F sf • l v • cos δ -F sr • l h + F lf • l v • sin δ] , (4)
The functions F x , F y , F sf , F sr , F lf denote forces (in x-, y-direction, as well as lateral and longitudinal tyre forces at front and rear wheels) and are smooth nonlinear functions of the state (x, y, ψ, v x , v y , w ψ , δ) and m, I zz , l v , l h are constants. For further details please refer to [START_REF] Gerdts | Solving Mixed-Integer Optimal Control Problems by Branch&Bound: A Case Study from Automobile Test-Driving with Gear Shift[END_REF][START_REF] Gerdts | A variable time transformation method for mixed-integer optimal control problems[END_REF][START_REF] Gerdts | Generating Optimal Trajectories for an Automatically Driven Car[END_REF]. For the following numerical computations we used realistic data for the various parameters involved in this model. Unfortunately, these parameter values are proprietary and may not be published. For a different parameter set which is quite realistic please refer to [START_REF] Gerdts | Solving Mixed-Integer Optimal Control Problems by Branch&Bound: A Case Study from Automobile Test-Driving with Gear Shift[END_REF].

Collision Detection and Collision Avoidance

Once an obstacle has been detected by suitable sensors, e.g. radar or lidar, we use the following approaches to decide whether a collision is going to happen or not. As we intend to use optimal control to model the scenarios, as a by-product we obtain evasion trajectories if such exist at all. We investigate three different approaches. Herein, it is assumed for simplicity that the obstacle in scenario 1 is fixed close to the right boundary of a straight road as in the left picture in Figure 1. Moreover, the following approaches assume that the constellation of car and obstacle is such that a collision cannot be avoided by just applying a full braking maneuver.

Approach 1: Reaching a Safe Target Position for Scenario 1

The first approach aims at reaching a safe target state, which should be defined such that the evading car is able to avoid the obstacle and moreover is able to continue its drive after the obstacle has been passed. This approach is modeled by the following optimal control problem OCP(y 0 , v x,0 ):

Minimize c 1 t f + c 2 d + c 3 t f 0 w δ (t) 2 dt
subject to the equations of motion ( 1)-( 4) with initial condition

(x(0), y(0), ψ(0), v x (0), v y (0), w ψ (0), δ(0)) = (0, y 0 , 0, v x,0 , 0, 0, 0), the control constraints |w δ | ≤ w δ,max , F B ∈ [F B,min , F B,max ], the pure state constraint y min ≤ y(t) ≤ y max ,
and boundary conditions

x(t f ) = d, v y (t f ) = 0, y(t f ) ≥ y target .
Herein, the final time t f is supposed to be free and c 1 , c 2 , c 3 ≥ 0 are suitable constants. y 0 is the initial y-position of the evading car on the road, v x,0 is the initial velocity in x-direction of the evading car. y min and y max define the boundaries of the road. The terminal constraint v y (t f ) = 0 shall ensure that the evading car can continue its drive beyond t f without leaving the road immediately. y target defines a y-position sufficiently far away from the obstacle's y-position (safe target position).

d is the initial distance of the evading car to the obstacle. If c 2 > 0, then d is assumed to be an additional optimization parameter. The resulting optimal control problem then aims at finding the minimal distance to the obstacle that still allows to avoid a collision. If c 2 = 0, then d is supposed to be a fixed distance. In this case, it might happen that the problem becomes infeasible owing to the constraint y(t f ) ≥ y target . A remedy in this case is approach 2 below. Approach 1 yields one single optimal trajectory provided an avoidance trajectory exists. This avoidance trajectory could be tracked by a real car.

However, the solution depends on the definition of the safe target position y target and it does not exist if a collision is unavoidable. Therefore it would be nicer to have full information about what points on the road can actually be reached by the evading car. This leads to the following reachable set approach.

Approach 2: Computing the Projected Reachable Set

The second approach aims at providing all points on the road that can be reached by the evading car in finite time t f from a given initial state with boundary condition v y (t f ) = 0. More precisely, we aim at computing the projected reachable set

PR := d∈[dmin,dmax] y∈PR(d) {(d, y)},
where

PR(d) := ŷ ∈ R | ∃ final time t f > 0, controls w δ , F B ,
and states x, y, ψ, v x , v y , w δ , δ such that dynamics and constraints are satisfied

and ŷ = y(t f ), x(t f ) = d, v y (t f ) = 0
denotes the projected reachable set at initial distance d. Note that we are not interested in the reachable set at t f for the full state vector but only for the components x and y. In order to approximate the projected reachable set we employ the optimal control technique in [START_REF] Baier | A computational method for non-convex reachable sets using optimal control[END_REF][START_REF] Baier | Approximation of Reachable Sets using Optimal Control Algorithms[END_REF], which for a simplified setting allows a first order approximation. The set PR is approximated as follows. For N, M ∈ N and step-sizes h = (d max -d min )/N and k = (y max -y min )/M let

G h,k = {(d i , y j ) ∈ R 2 | d i = d min
+ih, y j = y min +jk, i = 0, . . . , N, j = 0, . . . , M } denote a grid covering the road region of interest. Then for each grid point (d i , y j ) ∈ G h,k the following optimal control problem is solved:

Minimize 1 2 (y(t f )) -y j ) 2
subject to the equations of motion ( 1)-( 4) with initial condition (x(0), y(0), ψ(0), v x (0), v y (0), w ψ (0), δ(0)) = (0, y 0 , 0, v x,0 , 0, 0, 0),

the control constraints |w δ | ≤ w δ,max , F B ∈ [F B,
f ) = d i , v y (t f ) = 0.
Let x * i,j (•) and y * i,j (•) denote the optimal solution components of the state vector. The projected reachable set is approximated by collecting all grid points in G h,k with distance of order O(h + k) to end points of trajectories:

PR h,k := (d i ,y j )∈G h,k : (x * i,j (t f ),y * i,j (t f ))-(d i ,y j ) ≤C(h+k) {(d i , y j )}.
Herein, C > 0 is a constant. In [START_REF] Baier | Approximation of Reachable Sets using Optimal Control Algorithms[END_REF] it is shown that the approximation PR h,k convergences in the Hausdorff distance to PR of order O(h + k) as h and k approach zero, if PR is closed and non-empty. Direct discretization techniques for the numerical solution of the optimal control problem introduce a further approximation to PR h,k whose convergence properties for a special setting are analyzed in [START_REF] Baier | Approximation of Reachable Sets using Optimal Control Algorithms[END_REF] as well.

The projected reachable set approximations PR h,k are depicted in Figure 3 for different initial velocities and the data y min = 1.3, y max = 5.7, d min = 10, d max = 200, y 0 = 1.75, F B,min = -5000, F B,max = 15000. The optimal control problems have been solved by the software OCPID-DAE1 [START_REF] Gerdts | OCPID-DAE1 -Optimal Control and Parameter Identification with Differential-Algebraic Equations of Index 1: User's Guide[END_REF].

Note that the obstacle car is not taken into account in the optimal control problems. But once the projected reachable set is known, it can be decided for a given obstacle position whether a collision can be avoided or not by investigating the remaining space in the projected reachable set outside the obstacle at the x-position of the obstacle.

Approach 3: Feasibility Problem for Scenario 2

For a fixed obstacle it is comparatively simple to define a safe target position or to approximate the projected reachable set, but for moving objects as in the overtaking maneuver in Figure 1 it is not, since a collision with all other moving cars has to be avoided at all times. In the overtaking scenario in Figure 1 let car A denote the car that overtakes a car called car B and car C is the car approaching car A in opposite direction. Cars B and C are supposed to drive at constant velocity in a straight line. Anti-collision constraints lead to the following pure state constraints, where W denotes the maximum width of the cars (for simplicity we use balls to model the anti-collision constraints):

(x A (t) -x B (t)) 2 + (y A (t) -y B (t)) 2 ≥ W 2 , (don't hit car B) (x A (t) -x C (t)) 2 + (y A (t) -y C (t)) 2 ≥ W 2 , (don't hit car C)
Unfortunately, these constraints will be infeasible if there is no way to avoid a collision. Of course, in this case the resulting optimal control problems do not have a solution and numerical methods will fail. In order to circumvent this problem the relaxed constraints are considered, where α denotes the maximal constraint violation. Now, an optimal control problem with the aim to minimize the constraint violation α is solved subject to the above constraints. A collision detection algorithm is then given by considering the minimal constraint violation α * . If α * > 0, then a collision cannot be avoided (the anti-collision constraints cannot be satisfied). If α * ≤ 0, then a collision can be avoided with a trajectory that is produced by the optimal control problem. We illustrate the outcome for the following data: Table 1 summarizes the results for different initial distances of cars A and C obtained with OCPID-DAE1 [START_REF] Gerdts | OCPID-DAE1 -Optimal Control and Parameter Identification with Differential-Algebraic Equations of Index 1: User's Guide[END_REF]. A movie that visualizes the overtaking maneuver with initial distance of 60 m can be downloaded on the homepage of the first author.

(x A (t) -x B (t)) 2 + (y A (t) -y B (t)) 2 + α ≥ W 2 , (x A (t) -x C (t)) 2 + (y A (t) -y C (t)) 2 + α ≥ W 2

Sensor Influence

In the collision avoidance scenarios, the initial position, i.e the constellation of evading car and obstacles, is determined by sensor measurements. These sensor measurements are subject to measurement errors and hence it is important to investigate how the optimal solution depends on these sensor measurement errors. We outline this for approach 1 and consider the initial values y 0 and v x,0 to be parameters in the optimal control problem OCP(y 0 , v x,0 ). We apply the sensitivity analysis in [START_REF] Büskens | Sensitivity Analysis and Real-Time Control of Parametric Optimal Control Problems Using Nonlinear Programming Methods[END_REF], which exploits the sensitivity results in [START_REF] Fiacco | Introduction to Sensitivity and Stability Analysis in Nonlinear Programming[END_REF] for finite dimensional optimization problems. To this end, OCP(y 0 , v x,0 ) is discretized using piecewise constant control approximations w δ,j ≈ w δ (t j ) and F B,j ≈ F B (t j ), j = 0, . . . , K, and Runge-Kutta approximations for the state on a grid with grid points t j , j = 0, . . . , K. The discretized problem is a finite dimensional nonlinear optimization problem, which is solved for nominal parameters ŷ0 and vx,0 .

If the nominal optimal solution satisfies the assumptions of the sensitivity theorem in [START_REF] Fiacco | Introduction to Sensitivity and Stability Analysis in Nonlinear Programming[END_REF], i.e. second order sufficient conditions, linear independence constraint qualification and strict complementarity, then it was shown that the solution locally depends continuously differentiable on the parameters y 0 and v x,0 and the sensitivities of the optimal control discretization with respect to the initial values y 0 and v x,0 can be computed, that is we obtain the sensitivities dw δ,j (y 0 , v x,0 ) (ŷ 0 , vx,0 ), dF B,j (y 0 , v x,0 ) (ŷ 0 , vx,0 ), j = 0, . . . , K.

The sensitivities indicate how sensitive the solution depends on perturbations in the initial values and hence can help to specify tolerances for sensors. We omit the details here since this sensitivity approach became quite standard in the meanwhile. Details can be found, e.g., in [START_REF] Büskens | Sensitivity Analysis and Real-Time Control of Parametric Optimal Control Problems Using Nonlinear Programming Methods[END_REF].

Figure 4 shows the sensitivity differentials for the controls w δ and F B with respect to y 0 and v x,0 at the nominal parameters ŷ0 = 1.75 and vx,0 = 35. From the pictures it can be concluded that a perturbation in the initial y-position has the highest influence on the controls w δ and F B . A perturbation of the initial velocity v x,0 has less influence. Hence, the sensor measurement of the y-position (respectively, the offset to the obstacle) should be more accurate than the sensor measurement of the velocity. More elaborate investigations regarding the definition of sensor tolerances that are necessary to achieve a certain performance are currently under investigation. Please note that the above sensitivity approach does not work for the optimal control problems in approach 2 as those do not satisfy the assumptions of the sensitivity theorem in [START_REF] Fiacco | Introduction to Sensitivity and Stability Analysis in Nonlinear Programming[END_REF] whenever a grid point is in the projected reachable set. Adding a regularization term in the objective function might help to overcome this difficulty and would allow to investigate the dependence of the projected reachable set on sensor measurements.

Outlook

The paper suggests different approaches to an avoidance trajectory system based on optimal control techniques, reachable set computations, and sensitivity analysis. Many extensions are possible, e.g. computation of driver-friendly trajectories for active steering driver assistance systems, more complicated road geometries, real-time approximations, investigation of worst-case scenarios or cooperative control in the presence of many moving objects, and the investigation of parameter dependence of the projected reachable set. These issues are currently under investigation.

Fig. 1 .

 1 Fig. 1. Collision avoidance model scenarios: stationary obstacle (left) and overtaking maneuver (right)

Fig. 2 .

 2 Fig. 2. Avoidance trajectory for vx,0 = 35, y0 = 1.75, ytarget = 3.85: center of gravity and velocity in x-direction (top), steering angle velocity and braking/acceleration force (bottom). The distance d computes to 35.6798 and the final time t f to 1.1399.

Figure 2

 2 Figure2shows the result for y min = 0.9, y max = 6.1, y 0 = 1.75, v x,0 = 35, c 1 = c 2 = 0.1, c 2 = 0.2, w δ,max = 0.5, F B,min = -5000, F B,max = 15000, y target = 3.85. The minimal distance computes to d = 35.6798 [m] and the final time (time to collision) is t f = 1.1399 [s].Approach 1 yields one single optimal trajectory provided an avoidance trajectory exists. This avoidance trajectory could be tracked by a real car.However, the solution depends on the definition of the safe target position y target and it does not exist if a collision is unavoidable. Therefore it would be nicer to have full information about what points on the road can actually be reached by the evading car. This leads to the following reachable set approach.

vx, 0 Fig. 3 .

 03 Fig. 3. Projected reachable sets for initial velocities vx,0 = 75, 100, 150, 250 [km/h]. The approaching car can avoid a collision with the obstacle in the first three settings, but not in the final setting, if the measurements of the obstacle and the approaching car are taken into account.

-

  car A: 100 [km/h], car B: 75 [km/h], car C: 100 [km/h] -car width 2.6 [m], road width 7 [m] -initial y-position of car A : 5.25 [m] initial y-position of car B : 1.75 [m] initial y-position of car C : 5.25 [m]

  (scaled by 10^-4) t (normalized time) sensitivity dFB/dvx0 vs time

Fig. 4 .

 4 Fig. 4. Sensitivity differentials for the optimal controls of OCP(ŷ0, vx,0) at ŷ0 = 1.75 and vx,0 = 35: Control and sensitivities of w δ w.r.t. y0 and vx,0 (left) and of FB w.r.t. y0 and vx,0 (right).

Table 1 .

 1 Results for the constraint minimization problem for the overtaking maneuver.

	initial distance constraint violation α * collision
	[m]	[m]
	10	0.24780E+01	yes
	20	0.22789E+01	yes
	30	0.21355E+01	yes
	40	0.19351E+01	yes
	50	0.94517E-01	yes
	60	0.74140E-08	no
	. . .	. . .	. . .
	190	0.74593E-08	no
	200	0.74760E-08	no

⋆ The work is supported by the Initial Training Network Sensitivity Analysis for Deterministic Controller Design (ITN SADCO), which is funded by 7th Framework Programme of the EU.