Sophie Jan
email: sophie.jan@math.univ-toulouse.fr

MINIMIZING THE FUEL CONSUMPTION OF A VEHICLE FROM THE SHELL ECO-MARATHON: A NUMERICAL STUDY *

Keywords: Mathematics Subject Classification. 49J15, 90C90, 49N90 Optimal control, singular arcs, nonlinear programming, continuation method, indirect multiple shooting

We apply four different methods to study an intrinsically bang-bang optimal control problem. We study first a relaxed problem that we solve with a naive nonlinear programming approach. Since these preliminary results reveal singular arcs, we then use Pontryagin's Minimum Principle and apply multiple indirect shooting methods combined with homotopy approach to obtain an accurate solution of the relaxed problem. Finally, in order to recover a purely bang-bang solution for the original problem, we use once again a nonlinear programming approach.

Introduction

In the context of environment preservation, Shell company organizes each year a competition named Shell Eco-marathon. The aim is to design and build a vehicle, and to optimize the driving strategy in order to use as little fuel as possible to travel a given distance L in a time less than a given duration T .

For example, for a competition on the race track displayed in Figure 1, L ≈ 25.5 km (7 laps) and T ≈ 51 min. This quantity T corresponds to the time needed to travel the distance L with a mean speed of 30 km/h. This paper is organized in four sections. In Section 1, we describe the problem and formulate it as an intrinsically bang-bang optimal control problem. Then, we analyze, in Section 2, a relaxed version of this optimal control problem using Pontryagin's Minimum Principle. In the third section, we report numerical experiments on both the relaxed and the original optimal control problems. Finally, we conclude in Section 4.

Position of the problem

In this section, we first describe the vehicle and the race track. Then we make explicit the dynamics of the vehicle, and we introduce some scaling quantities that are numerically interesting. Finally, we model the problem of minimizing the fuel consumption of the vehicle as an optimal control problem.

The vehicle

The vehicle is represented in Figure 1. It has three identical wheels, two at the front and one at the rear, whose radius and moment of inertia are respectively denoted by R and I r . We assume that the engine is well-balanced and Γ denotes its torque. We denote by C x and S the drag coefficient and frontal surface of the vehicle. The quantity M depicts the total mass of the vehicle and its pilot.

In the sequel, we use d(t) and v(t) to denote respectively the distance covered by the vehicle from the starting point and its speed at time t. We use k to represent the friction coefficient of the wheels on the road. It seems physically quite reasonable to consider k as a function of the speed of the vehicle v. However, in this study, we shall assume that the friction coefficient k(v) is essentially constant, equal to a certain k 0 . Since k(v) is physically zero when the speed is zero, the function k(v) is assumed to be a function of class C 2 that satisfies the following constraints:

k(0) = 0, k(v) = k 0 for all v ≥ ε, k (ε) = 0, k (ε) = 0
for some given ε. The choice of ε is discussed in Section 3.2. The representative curve of k is displayed in Figure 2.

In the eco-marathon competition, the allowed driving strategy consists in alternating periods where the engine is "on" with periods where it is off. We use the variable u(t) to describe this strategy: u(t) is equal to one when the engine is "on", and it is zero otherwise.

The race track

The profile of the race track is only known through altitudes measured at some points. With these tabulated data, we build a function α of class C 2 using piecewise-cubic spline interpolation. The value α(d) represents the angle between the horizontal axis and the slope of the road when the vehicle has covered a distance d on the race track.

Dynamics of the vehicle

We describe now the dynamics of the vehicle using the second Newton's law in the longitudinal direction. This gives, for all t > 0,

3I r R 2 + M v(t) = -M g sin(α(d(t))) (gravitation) - 1 2 ρC x Sv(t) 2 (aerodynamics) -M gk(v(t)) cos(α(d(t))) (rolling friction) +u(t) Γ R (engine),
where g is the gravity constant and ρ is the air density. Then, denoting

A = 3I r R 2 + M > 0, B = - ρC x S 2A < 0, C = - M g A < 0, D = Γ RA > 0,
and

C(d, v) = C sin α(d) + k(v) cos α(d) ,
we may rewrite the preceding equation under the form of the following system of differential equations of order one:

ḋ(t) v(t) = v(t) Bv(t) 2 + C(d(t), v(t)) + Du(t) .

Rescaling

For numerical purposes, we perform the following rescaling:

t = t c τ, d(t) = x c δ(τ ), v(t) = v c ν(τ ), u(t) = U (τ ),
where τ ∈ [0, 1] and where t c := T , x c and v c are characteristic quantities of time, length and speed (respectively).

With these new variables, the equations for the dynamics become

δ(τ ) ν(τ ) = Ãν(τ ) Bν(τ ) 2 + C(δ(τ ), ν(τ )) + DU (τ )
,

where à := t c v c x c , B := Bv c t c , C(δ, ν) := t c v c C(x c δ, v c ν), D := D t c v c •
Let us also introduce the following notations:

t f = t c τ f , T = t c Υ, L = x c Δ.

The optimal control problem

The eco-marathon challenge can be modeled under the form of the following optimal control problem (Lagrange form):

Minimize fuel consumption := τ f 0 U (τ )ν(τ )dτ subject to 0 ≤ τ f ≤ Υ, (final time), U (τ ) ∈ {0, 1} ∀τ ∈ [0, τ f ],
(engine on or off),

δ(0) = 0, ν(0) = 0, (initial conditions), ∀τ ∈ [0, τ f ], δ(τ ) = Ãν(τ ), ν(τ ) = Bν(τ ) 2 + C(δ(τ ), ν(τ )) + DU (τ ), (dynamics)
,

δ(τ f ) = Δ, (final condition),
where the unknowns are the function U and the final time τ f . This problem may be reformulated into the following Mayer form:

Minimize fuel consumption := X 3 (τ f ) subject to 0 ≤ τ f ≤ Υ, U (τ ) ∈ {0, 1} ∀τ ∈ [0, τ f ], X(0) = (0, 0, 0), ∀τ ∈]0, τ f ], Ẋ(τ ) = f (X(τ ), U(τ )), X 1 (τ f ) = Δ, (2.1)
where

X : [0, 1] → R 3 τ → δ(τ ), ν(τ ), e(τ ) ,
and

f (X, U ) = ⎛ ⎝ Ãν, Bν 2 + C(δ, ν) + DU U ν ⎞ ⎠ .
In our application context, for τ ∈ [0, τ f ] ⊂ [0, 1], it is obvious that both the distance δ covered by the vehicle and its speed ν are bounded. Then, it is immediate to prove, using the Cauchy-Lipschitz theorem [START_REF] Stoer | Introduction to numerical analysis[END_REF], that there exists a unique solution for the problem:

X(0) = (0, 0, 0), ∀τ ∈]0, τ f ], Ẋ(τ ) = f (X(τ ), U(τ )).
(2.2)

We study the following relaxed optimal control problem:

Minimize fuel consumption := X 3 (τ f ) subject to 0 ≤ τ f ≤ Υ, U (τ ) ∈ [0, 1] ∀τ ∈ [0, τ f ], X(0) = (0, 0, 0), ∀τ ∈]0, τ f ], Ẋ(τ ) = f (X(τ ), U(τ )), X 1 (τ f ) = Δ, (2.3)
where the control function U now takes values in the interval [0, 1]. This "continuous relaxation" means that we are not restricted to on/off engine switches anymore: we are also allowed to accelerate more or less.

It seems natural that the final time t f (respectively τ f ) be equal to the maximal authorized time T (respectively Υ ). This will be numerically demonstrated in Section 4.1.

As a consequence, in the next section, we shall analyse the relaxed optimal control problem (2.3) assuming that the final time τ f is fixed and equal to Υ .

Analysis of the optimal control problem

In this section, we analyze the following problem using Pontryagin's Minimum Principle:

Minimize the consumption := X 3 (Υ ) subject to U (τ ) ∈ [0, 1] ∀τ ∈ [0, Υ ] , X(0) = (0, 0, 0), ∀τ ∈ ]0, Υ ] , Ẋ(τ ) = f (X(τ ), U(τ )), X 1 (Υ ) = Δ. ( 3.1)
Assuming that we are not in a pathological case, the Hamiltonian associated with Problem (3.1) is defined by

H(X, U, Λ) := p Ãν + q Bν 2 + C(δ, ν) + DU + rU ν,
where Λ = (p, q, r) represents the vector of adjoint variables or costates.

Consequently, the adjoint equations are

Λ(τ ) = g(X(τ ), U(τ ), Λ(τ )), (3.2)
where

g(X, U, Λ) := - ⎛ ⎜ ⎜ ⎜ ⎝ q ∂ C ∂δ (δ, ν) p à + q 2 Bν + ∂ C ∂ν (δ, ν) + rU 0 ⎞ ⎟ ⎟ ⎟ ⎠ .

Application of Pontryagin's minimum principle

Applying Pontryagin's Minimum Principle [START_REF] Bonnans | Singular arcs in the generalized Goddard's problem[END_REF][START_REF] Hiriart-Urruty | Les mathématiques du mieux faire, Ellipses[END_REF][START_REF] Hocking | Optimal control, An introduction to the theory with applications[END_REF][START_REF] Locatelli | Optimal control, An introduction[END_REF][START_REF] Trélat | Contrôle optimal, Mathématiques Concrètes. Vuibert[END_REF], we have that a necessary condition for (X(.), U(.)) being a solution of (3.1) is the existence of a piecewise smooth vector Λ(.) := (p(.), q(.), r(.)) such that:

1. (X(.), Λ(.)) solves (2.2) and (3.2); 2. ∀U ∈ [0, 1], H(X(τ ), U(τ ), Λ(τ )) ≤ H(X(τ ), U, Λ(τ )), for almost every τ ∈ [0, Υ ]; 3. Λ(.) = 0; 4. q(Υ ) = 0, and r(Υ ) = 1.
Using the so-called transversality condition r(Υ ) = 1 and the third equation of (3.2), we have that

r(τ ) = 1, ∀τ ∈ [0, Υ ].
Since the Hamiltonian is linear with respect to the control, we have

H(X, U, Λ) := β(X, Λ)U + γ(X, Λ),
with β(X, Λ) := ν + Dq and γ(X, Λ) := Ãpν + q( Bν 2 + C(δ, ν)). We obtain easily that an optimal control for (3.1) is given by

U (τ ) = 0 if β(X(τ ), Λ(τ )) > 0, 1 if β(X(τ ), Λ(τ )) < 0.
The function Ψ (τ ) := β(X(τ ), Λ(τ )) is called the switching function.

Singular arcs

We analyze now the case where the switching function vanishes on a closed interval [τ 0 , τ 1 ] ⊂ [0, Υ ]:

Ψ (τ ) := ν(τ ) + Dq(τ ) = 0, ∀τ ∈ [τ 0 , τ 1 ]. (3.3)
In order to compute the singular control, we use a classical technique, recalled in [START_REF] Bonnans | Singular arcs in the generalized Goddard's problem[END_REF] for example: we differentiate relation (3.3) with respect to time until U appears explicitly. Remark 3.1. For the sake of notational simplicity, we do not write explicitly the dependence on τ in the sequel of this section.

= Ψ = ν + D q = Bν 2 + C(δ, ν) + DU -D U + p à + q 2 Bν + ∂ C ∂ν (δ, ν) = Bν 2 + C(δ, ν) -D Ãp + ν 2 Bν + ∂ C ∂ν (δ, ν) = 3 Bν 2 + C(δ, ν) -D Ãp + ν ∂ C ∂ν (δ, ν). ( 3.4)
This relation does not depend on the control U . We differentiate once more to obtain, almost everywhere on [τ 0 , τ 1 ]:

0 = Ψ = 6 Bν ν + ∂ C ∂δ (δ, ν) δ + ∂ C ∂ν (δ, ν) ν -Ã D ṗ + ν ∂ C ∂ν (δ, ν) + ν ∂ 2 C ∂ν∂δ (δ, ν) δ + ∂ 2 C ∂ν 2 (δ, ν) ν = ∂ C ∂δ (δ, ν) + ν ∂ 2 C ∂ν∂δ (δ, ν) δ -Ã D ṗ + 6 Bν + 2 ∂ C ∂ν (δ, ν) + ν ∂ 2 C ∂ν 2 (δ, ν) ν.
Using now the Hamiltonian equations and relation (3.3), we have

0 = Ψ = ∂ C ∂δ (δ, ν) + ν ∂ 2 C ∂ν∂δ (δ, ν) Ãν + Ã Dq ∂ C ∂δ (δ, ν) + 6 Bν + 2 ∂ C ∂ν (δ, ν) + ν ∂ 2 C ∂ν 2 (δ, ν) ( Bν 2 + C(δ, ν) + DU ) = 6 B + ∂ 2 C ∂ν 2 (δ, ν) ν + 2 ∂ C ∂ν (δ, ν) ( Bν 2 + C(δ, ν) + DU ) + Ãν 2 ∂ 2 C ∂ν∂δ (δ, ν). (3.5)
The way the parameter ε has been chosen in the definition of the function k renders negative the first factor in (3.5) for all δ ∈ [0, Δ], for all ν ∈ 0, 3v m v c (v m is the mean speed of 30 km/h). Thus, the Legendre-Clebsch necessary optimality condition for the existence of a singular arc is satisfied [START_REF] Robbins | A generalized Legendre-Clebsch condition for the singular cases of optimal control[END_REF]. More precisely:

- ∂ ∂U d 2 dt 2 ∂H ∂U ≥ 0.
Using (3.4) and (3.3), we first obtain, almost everywhere on [τ 0 , τ 1 ]:

p(τ ) = 3 Bν(τ ) 2 + C(δ, ν) + ν ∂ C ∂ν (δ, ν) Ã D and q(τ ) = - ν(τ ) D •
We then derive U by using (3.5):

U (τ ) = - 1 D ⎛ ⎜ ⎜ ⎝ Ãν 2 ∂ 2 C ∂ν∂δ (δ, ν) 6 B + ∂ 2 C ∂ν 2 (δ, ν) ν + 2 ∂ C ∂ν (δ, ν) + Bν(τ ) 2 + C(δ, ν) ⎞ ⎟ ⎟ ⎠ . (3.6)

Summary

A necessary condition for (X(.), U(.)) to be a solution of (3.1) is the existence of a nontrivial vector Λ(.) := (p(.), q(.)) such that Y (.) := (X(.), Λ(.)) is solution of

Ẏ (τ ) = h(Y (τ ), U(τ )) (3.7) with h(Y, U ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Ãν, Bν 2 + C(δ, ν) + DU U ν -q ∂ C ∂δ (δ, ν) -Ãp(τ ) + q(τ ) 2 Bν + ∂ C ∂ν (δ, ν) + U ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, with the two-point boundary conditions:

Y 1 (0) = 0, Y 2 (0) = 0, Y 3 (0) = 0, Y 1 (Υ ) = Δ, Y 5 (Υ ) = 0, and
U (τ ) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 i f Ψ (τ ) > 0, - 1 D ⎛ ⎜ ⎜ ⎝ Ãν 2 ∂ 2 C ∂ν∂δ (δ, ν) 6 B + ∂ 2 C ∂ν 2 (δ, ν) ν + 2 ∂ C ∂ν (δ, ν) + Bν(τ ) 2 + C(δ, ν) ⎞ ⎟ ⎟ ⎠ if Ψ (τ ) = 0 for τ ∈ [τ 0 , τ 1 ] ⊂ [0, Υ ], 1 i f Ψ (τ ) < 0,
with the switching function Ψ (τ ) := ν(τ ) + Dq(τ ).

Numerical experiments

In this section, we apply the same techniques as in [START_REF] Bonnans | Singular arcs in the generalized Goddard's problem[END_REF]. In a first step, in order to have an idea of the difficulty of the problem, we use a naive and simple nonlinear-programming approach. Then, since it is well known that indirect shooting methods are more accurate, we use such techniques to obtain more precise results. However, the major drawback of indirect methods lies in its initialization. That is why we combine indirect methods with a continuation process. Finally, we compare the preceding results with a purely bang-bang solution that we obtain via another nonlinear-programming approach.

To compute the results depicted in this section, we have considered the race track from Figure 1 and we have chosen the following characteristic quantities:

t c := T, x c := L and v c := v m ,
where v m is the mean speed of 30 km/h. With this choice, we have Υ = 1, and Δ = 1.

The other parameters defining our test problem here are summarized in Table 1.

In the sequel, we report results for two cases: the general one and the simplified one. In the latter, we assume that the race track is flat. This means that the function α that represents the angle between the horizontal plane and the slope of the road is identically zero. Consequently, C only depends on ν:

C(δ, ν) := t c v c Ck(v c ν).

A naive approach to solve the relaxed optimal control problem with τ f free in [0, Υ ]

In this section, we report results for the problem (2.3). First, we consider that the final time τ f is free on the interval [0, Υ ]. Then, we discretize the time interval [0, τ f ] into N subintervals of equal size Δτ := τ f /N , and we consider that the control remains constant on each of these subintervals. Let us set the following notation: • U i is the value of the control on the subinterval I i , for i = 1, . . . , N; • X(τ, 1) is the solution of (2.2) on I 1 with U (τ ) ≡ U 1 and X(τ 0 , 1) = (0, 0, 0);

• τ i := iΔτ , for i = 0, . . . , N; • I i := [τ i-1 , τ i ] , for i = 1, . . . , N;
• X(τ, i) is the solution of (2.2) on I i with U (τ ) ≡ U i and X(τ i-1 , i) = X(τ i-1 , i -1) for i = 2, . . . , N.
Then, we consider the following problem:

Minimize X 3 (τ N , N) subject to 0 ≤ U i ≤ 1, i = 1, . . . , N, 0 ≤ τ f ≤ Υ, X 1 (τ N , N) = Δ, (4.1)
where the unknowns are the N values of U i , and the value of the final time τ f . To solve problem (4.1), we use the SQP fmincon (with the integrator ode113) routine of Matlab for various values of N , and various random initial guesses of the unknowns. Our results are displayed in Figure 3.

A first observation is that these results confirm numerically the fact that τ f = Υ . Consequently, from now on, we assume that the final time τ f is equal to Υ and we consider problem (3.1).

These results also reveal the existence of singular arc(s). For the simplified case, we observe one singular arc. In the general case, it is more difficult to determine both the number of singular arcs, and their junction points with regular arcs.

Structure of the control for the relaxed optimal control problem with τ f = Υ

Since we have numerically established in the preceding section that the final time is equal to the maximum authorized time, we concentrate now on problem (3.1). Our aim is to identify accurately the structure of the control solution for this problem. For this purpose, we use multiple indirect shooting methods. In order to avoid the well-known difficulties to initialize such indirect methods, we combine them with a continuation technique.

Quadratic penalty

Let us add to problem (3.1) a term quadratic in U [START_REF] Bonnans | Singular arcs in the generalized Goddard's problem[END_REF]. We obtain, for λ ∈ [0, 1]:

minimize the consumption := X 3 (Υ ) subject to U (τ ) ∈ [0, 1] ∀τ ∈ [0, Υ ], X(0) = (0, 0, 0), ∀τ ∈]0, Υ ], Ẋ(τ ) = f λ (X(τ ), U(τ )), X 1 (Υ ) = Δ, ( 4.2)
where

f λ (X, U ) = f (X, U ) + ⎛ ⎝ 0 0 (1 -λ)U 2 ⎞ ⎠ .
The Hamiltonian associated with problem (4.2) is

H λ (X, U, Λ) := H(X, U, Λ) + (1 -λ)U 2
and consequently the Hamiltonian equations are the same as for (3.1), i.e. (3.7).

For λ = 1, (4.2) reduces to the problem (3.1) that we want to solve. The interesting point is that for λ ∈ [0, 1[, problem (4.2) is smooth, U → H λ (X, U, Λ) is convex, and the control U is continuous.

Indirect shooting method to solve the penalized problem

We consider here that λ ∈ [0, 1[ is fixed. Using the analysis performed in Section 3.1, we have that a necessary condition for (X(.), U(.)) to be a solution of (4.2) is the existence of a non-zero vector Λ(.) := (p(.), q(.))

such that Y := (X, Λ) is solution, for τ ∈]0, Υ [, of Ẏ (τ ) = h λ (Y (τ ), U(τ )), (4.3)
where

h λ (Y, U ) = h(Y, U ) + ⎛ ⎜ ⎜ ⎜ ⎝ 0 0 (1 -λ)U 2 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ ,
with the two-point boundary values

X(0) = (0, 0, 0), X 1 (Υ ) = Δ, Λ 2 (Υ ) = 0,
and

U := ⎧ ⎨ ⎩ 0 if U * < 0, 1 if U * > 1, U * otherwise, (4.4)
where

U * = - ν + Dq 2(1 -λ) solution of ∂H λ ∂U (U * ) = 0 .
Indirect shooting methods consist in searching a zero of the following shooting function

Z λ : R 2 → R 2 (p 0 , q 0 ) → (Y 1 (Υ ) -Δ, Y 5 (Υ )),
where Y solves (4.3) with Y (0) = (0, 0, 0, p 0 , q 0 ) as initial conditions, and where U is defined in (4.4).

Multiple indirect shooting method to solve the penalized problem

For computational efficiency and robustness, we use a multiple shooting method to solve problem (4.2). Let us introduce the following notation:

• M is the number of subintervals of [0, Υ ];

• Δτ := Υ/M is the length of each of these subintervals;

• τ i = iΔτ , for i = 0, 1, . . . , M; • z 0 := (0, 0, 0, p 0 , q 0 ) and z i := (δ i , ν i , e i , p i , q i ), for i = 1, . . . , M -1;

• for i = 0, . . . , M -1: z λ (τ, τ i , z i ) is the solution of (4.3) for τ ∈]τ i , τ i+1 ], with Y (τ i ) = z i and U defined in (4.4).
Thus, here the multiple indirect shooting method consists in finding a zero of the function Z λ,M defined from R 2+5(M-1) to R 5(M-1)+2 as follows:

Z λ,M (p 0 , q 0 , δ 1 , ν 1 , e 1 , p 1 , q 1 , . . . , δ M-1 , ν M-1 , e M-1 , p M-1 , q M-1 ) := (z λ (τ 1 , τ 0 , z 0 ) -z 1 , . . . , z λ (τ M-1 , τ M-2 , z M-2 ) -z M-1 , δ(τ M , τ M-1 , z M-1 ) -Δ, q(τ M , τ M-1 , z M-1 )).
The first 5(M -1) relations impose the continuity of the states and co-states on the whole trajectory, and the last two relations are the final conditions given by Pontryagin's Minimum Principle.

Continuation

The idea of the continuation method is to go smoothly from the solution of (4.2) with λ = 0, which is easy to compute, to the solution of (4.2) with λ ≈ 1.

More precisely, for a fixed number of subintervals M , we solve a chain of sub-problems Z λ k ,M (ζ) = 0 for an increasing sequence (λ k ) k∈N . This sequence is chosen such that 1. λ 0 = 0 and it is easy to find a solution ζ 0 of Z 0,M (ζ) = 0; 2. the solution ζ k of Z λ k ,M (ζ) = 0 is used as a starting point for computing ζ k+1 ; 3. after P steps, ζ P should be a good starting point for computing a solution of the relaxed optimal control problem (3.1) (with τ f = Υ ).

Remark 4.1. In the general case, the above first step (with λ 0 = 0) is not straightforward. For this reason, we propose performing a first continuation on the length of the race track in order to solve initial problem (with

λ 0 = 0) Z 0,M (ζ) = 0.
We implement the above way of doing in Matlab, using

• the ode113 routine to solve the ordinary differential equations;

• the fsolve routine, with the ('Jacobian','on') option, to find a solution to Z λ k ,M (ζ) = 0. We compute the exact Jacobian of the shooting function using the variational equations.

Figure 4 displays the results. They confirm the existence of singular arc(s). Indeed, in the case of the flat race track, there is one singular arc for τ between around 0.15 and 0.65. In the general case, it remains difficult to evaluate the number of singular arcs.

Accurate solution of the relaxed optimal control problem with τ f = Υ in the case of a flat race track

Thanks to the preceding numerical experiments, the structure of the optimal control for problem (3.1) on a flat race track seems to be of type "on-singular-off". Let us now determine precisely the switching times. For this purpose, we use once more a multiple indirect shooting method.

Let us introduce the following notation:

• z on (τ ) denotes the solution of (3.7), for τ ∈]0, τ 1 ] with Y (0) = z 0 and U = 1;

• z sing (τ ) is the solution of (3.7) for τ ∈]τ 1 , τ 2 ] with Y (τ 1 ) = z 1 and U is as defined in (3.6);

• z off (τ ) is the solution of (3.7) for τ ∈]τ 2 , Υ ] with Y (τ 2 ) = z 2 and U = 0.

The multiple indirect shooting method therefore consists in finding a zero of the function Z defined from R 2+1+5+1+5 to R 5+5+2+2 as follows: The first 10 relations impose the continuity of the states and co-states on the whole trajectory. The last two relations are the final conditions given by Pontryagin's Minimum Principle. The other two relations correspond to the zero values at the first switching time of the switching function and its first derivative. As for starting points, we have used approximate values obtained at the end of the continuation process.

Z(p 0 , q 0 , τ 1 , δ 1 , ν 1 , e 1 , p 1 , q 1 , τ 2 , δ 2 , ν 2 , e 2 , p 2 , q 2 ) := (z on (τ 1 ) -z 1 , z sing (τ 2 ) -z 2 , ν 1 + Dq 1 , 3 Bν 2 1 + Ck 0 -Ã Dp 1 , δ off (Υ ) -1, q off (Υ )).
Using Matlab and the same routines as before (but with the ('Jacobian','off') option), we obtain the results displayed in Figure 5. These give us precise values for the switching times and optimal consumption: τ 1 = 0.069, τ 2 = 0.6368 and e off (Υ ) = 0.10165. Remark 4.2. In the simplified case of a flat race track, since the function C only depends on ν, the singular control is given by (3.6):

U (τ ) = - 1 D Bν(τ ) 2 + C(δ, ν) .
This implies that the second component of the function h used in (3.7) is zero. Consequently, the speed ν and the control are constant between τ 1 and τ 2 . This can be seen in Figure 5.

A first heuristics to solve the original intrinsically bang-bang optimal control problem

In the preceding sections, we have studied a relaxed optimal control problem in which the control is allowed to take any value between 0 and 1. However, our original problem is concerned with purely 0-1 solutions. In this section, we describe a first heuristics that allows us to find purely bang-bang solutions to the original problem (2.1).

First, we assume that there is a given number N of switching times. Let us introduce the following notation:

• 0 = τ 0 ≤ τ 1 ≤ τ 2 ≤ . . . ≤ τ N ≤ τ N +1 = τ f ≤ Υ ; • I i := [τ i-1 , τ i ] , for i = 1, . . . , N + 1; • for i = 1, . . . , N + 1, U i = 1
for odd i, 0 for even i; • X(τ, 1) is the solution of (2.2) on I 1 with U (τ ) ≡ U 1 and X(τ 0 , 1) = (0, 0, 0);

• X(τ, i) is the solution of (2.2) on I i with U (τ ) ≡ U i and X(τ i-1 , i) = X(τ i-1 , i -1) for i = 2, . . . , N + 1.
We consider the following problem:

Minimize X 3 (τ N +1 , N + 1) subject to 0 = τ 0 ≤ τ 1 ≤ τ 2 ≤ . . . ≤ τ N ≤ τ N +1 = τ f ≤ Υ, X 1 (τ N +1 , N + 1) = Δ, ( 4.5)
where the unknowns are the N + 1 values of τ i . We tackle problem (4.5) with the interior-point fmincon (with the integrator ode113) routine of Matlab for N = 1, 2, . . . , 19. We perform fmincon for different random initial guesses of the unknowns, and this for each of considered values of N . An example of results we can obtain with this method is displayed in Figure 6.

In both cases, we observe in Figure 7 that the cost of the solution of (4.5) decreases when the number of switching times N increases. When the race track is flat, this cost seems to converge to the optimal cost of the relaxed problem. In the general case, the existence of local minima may explain the small fluctuations we can observe on the cost.

A comparison is made in Figure 8 between the results obtained with this heuristics for N = 20 and the singular arc obtained for the relaxed problem when the race track is flat. We can observe that the consumption and the speed obtained with these two different methods are very similar. This is obtained thanks to an accumulation of switching times during the singular arc.

Finally, we show in Figure 9 the impact of the final time on the criterion. As expected, the cost of the solution of (4.5) increases when the final time decreases.

With this methodology, we confirm once more the intuitive fact that τ f = Υ . However, these results are not completely satisfactory because we have no guarantee of optimality. It would be interesting to study whether the guaranteed global optimization techniques used in [START_REF] Merakeb | On minimizing the energy consumption of an electrical vehicle[END_REF] are applicable here.

Conclusions and perspectives

We have studied the behaviour of an intrinsically bang-bang optimal control problem by considering its continuous relaxation. Using different methodologies (nonlinear programming and indirect shooting methods), we have numerically established that the relaxed problem involves singular arcs. Finally, we could recover purely bang-bang solutions of the original problem of minimizing the fuel consumption of a vehicle from the Shell Eco-marathon using a nonlinear programming approach.

It would be interesting to extend our study in several ways. First, it could be interesting to study other techniques such as branch-and-bound based global optimization methods using interval analysis [START_REF] Merakeb | On minimizing the energy consumption of an electrical vehicle[END_REF]. Then, in this paper, the race track is considered as a straight line. We should study the influence of the curves of the race track on the numerical results. Another issue involves considering an engine torque that depends on the vehicle speed v.

Figure 1 .

 1 Figure 1. Left: Nogaro's race circuit (South-West of France). Right: the vehicle of the Toulouse Ingénierie Mécanique team (Université Paul Sabatier and Institut National des Sciences Appliquées de Toulouse).

Figure 2 .

 2 Figure 2. Representative curve of the function k(v).

First, we

  have Ψ (τ ) = 0 for almost every τ ∈ [τ 0 , τ 1 ]. This gives, using the Hamiltonian equations (2.2)-(3.2) and relation (3.3), 0

Figure 3 .

 3 Figure 3. Results obtained by the naive nonlinear-programming approach (4.1) with N := 40 subintervals of [0, τ f ]. Top: flat race track. Bottom: general case.

Figure 4 .

 4 Figure 4. Results obtained by the quadratic penalty continuation method with M := 20 subintervals of [0, Υ ]. Top: flat race track. Bottom: general case.

Figure 5 .

 5 Figure 5. Results obtained by a multiple indirect shooting method on problem (3.1) in the simplified case.

Figure 6 .

 6 Figure 6. Results obtained by the nonlinear programming approach (4.5) with N = 11 switching times into [0, τ f ]. Top: flat race track. Bottom: general case.

Figure 7 .Figure 8 .

 78 Figure 7. Decreasing of the consumption (plain) with the number of switching times and comparison with the optimal consumption of the relaxed problem (dashed). Top: flat race track. Bottom: general case.

Figure 9 .

 9 Figure 9. Impact of the final time on the consumption, in the case of the flat race track.

Table 1 .

 1 Parameters of the numerical tests.

	Vehicle
	Vehicle drag coefficient		Cx = 0.1
	Frontal vehicle surface	m 2	S = 0.35
	Wheel inertia moment	kg m 2	Ir = 0.041
	Wheel radius	m	R = 0.2375
	Vehicle (including pilot) mass kg	M = 69
	Engine torque	kg m 2 s -2 Γ = 5.5
	Race track
	Mean speed	m s -1	vm = 30/3.6 (30 km h -1 )
	Distance to be covered	m	L = 3636 (one lap)
	Maximum time allowed	s	T = L/vm
	Universal constants
	Air density	kg m -3 ρ = 1.23
	Gravity constant	m s -2	g = 9.81

Acknowledgements. This study was prompted by Yves Mégret, the leader of the Toulouse Ingénierie Mécanique team. The author is also thankful to Bonnans [2] and Epenoy [1] for interesting discussions about logarithmic continuation, Jean-Baptiste Hiriart-Urruty for his encouragement, Marcel Mongeau and Aude Rondepierre for their help, Sager [9] for a test of his code, and Emmanuel Trélat for more discussions about the continuation approach (quadratic versus logarithmic).

* This work has been partially supported by French National Research Agency (ANR) through COSINUS program (project ID4CS No. ANR-09-COSI-005).


Laurent Véron
email: veronl@univ-tours.fr

Nonlinear boundary value problems relative to the one dimensional heat equation

Keywords: Nonlinear heat flux, Singularities, Radon measures, Marcinkiewicz spaces. MS Classification 2010: 35J65, 35L71

We consider the problem of existence of a solution u to ∂tu-∂xxu = 0 in (0, T )×R+ subject to the boundary condition -ux(t, 0)+g(u(t, 0)) = µ on (0, T ) where µ is a measure on (0, T ) and g a continuous nondecreasing function. When p > 1 we study the set of self-similar solutions of ∂tu -∂xxu = 0 in R+ × R+ such that -ux(t, 0) + u p = 0 on (0, ∞). At end, we present various extensions to a higher dimensional framework.

Introduction

Let g : R → R be a continuous nondecreasing function. Set Q T R + = (0, T )×R+ for 0 < T ≤ ∞ and ∂ ℓ Q T R + = R+ ×{0}. The aim of this article is to study the following 1-dimensional heat equation with a nonlinear flux on the parabolic boundary

ut -uxx = 0 in Q T R + -ux(., 0) + g(u(., 0)) = µ in [0, T ) u(0, .) = ν in R+, (1)
where ν, µ are Radon measures in R+ and [0, T ) respectively. A related problem in Q ∞ R + for which there exist explicit solutions is the following,

ut -uxx = 0 in Q ∞ R +
ux(t, 0) + |u| p-1 u(t, 0) = 0 for all t > 0 lim t→0 u(t, x) = 0 for all x > 0,

where p > 1. Problem ( 2) is invariant under the transformation T k defined for all k > 0 by

T k [u](t, x) = k 1 p-1 u(k 2 t, kx). (3)
This leads naturaly to look for existence of self-similar solutions under the form us(t, x) = t

- 1 2(p-1) ω x √ t . (4)
Putting η = x √ t , ω satisfies

-ω ′′ - 1 2 ηω ′ - 1 2(p -1) ω = 0 in R+ -ω ′ (0) + |ω| p-1 ω(0) = 0 lim η→∞ η 1 p-1 ω(η) = 0.
(5)

Self-similar solutions of non-linear diffusion equations such as porousmedia or fast-diffusion equation were discovered long time ago by Kompaneets and Zeldovich and a thourougful study was made by Barenblatt, reducing the study to the one of integrable ordinary differential equations with explicit solutions. Concerning semilinear heat equation Brezis, Terman and Peletier opened the study of self-similar solutions of semilinear heat equations in proving in [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF] the existence of a positive strongly singular function satisfying

ut -∆u + |u| p-1 u = 0 in R+ × R n , (6)
and vanishing at t = 0 on R n \ {0}. They called it the very singular solution. Their method of construction is based upon the study of an ordinary differential equation with a phase space analysis. A new and more flexible method based upon variational analysis has been provided by [START_REF] Escobedo | Variational problems related to the self-similar solutions of the heat equations[END_REF]. Other singular solutions of [START_REF] Chen | Fractional heat equations with subcritical absorption having a measure as initial data[END_REF] in different configurations such as boundary singularities have been studied in [START_REF] Marcus | Semilinear parabolic equations with measure boundary data and isolated singularities[END_REF]. We set K(η) = e η 2 /4 and

L 2 K (R+) = φ ∈ L 1 loc (R+) : R + φ 2 Kdx := φ 2 L 2 K < ∞ , (7)
and, for k ≥ 1,

H k K (R+) = φ ∈ L 2 K (R+) : k α=0 φ (α) 2 L 2 K := φ 2 H k K < ∞ . (8)
Let us denote by E the subset of H 1 K (R+) of weak solutions of (5) that is the set of functions satisfying

∞ 0 ω ′ ζ ′ - 1 2(p -1) ωζ K(η)dη + |ω| p-1 ωζ (0) = 0, (9)
and by E+ the subset of nonnegative solutions. The next result gives the structure of E .

Theorem 1.1. 1-If p ≥ 2, then E = {0}. 2-If 1 < p ≤ 3 2 , then E+ = {0} 3 -If 3 2 < p < 2 then E = {ωs, -ωs, 0}
where ωs is the unique positive solution of [START_REF] Brezis | A very singular solution of the heat equation with absorption[END_REF]. Furthermore there exists c > 1 such that

c -1 η 1 p-1 -1 ≤ e η 2 4 ωs(η) ≤ cη 1 p-1 -1 for all η > 0. ( 10
)
Whenever it exists the function us defined in ( 4) is the limit, when ℓ → ∞ of the positive solutions u ℓδ 0 of

ut -uxx = 0 in Q ∞ R + -ux(t, .) + |u| p-1 u(t, .) = ℓδ0 in [0, T ) lim t→0 u(t, x) = 0 for all x ∈ R+. (11)
When such a function us does not exits the sequence {u ℓδ 0 } tends to infinity. This is a charateristic phenomenon of an underlying fractional diffusion associated to the linear equation

ut -uxx = 0 in Q ∞ R + -ux(., 0) = µ in [0, ∞) u(0, .) = 0 in R+. (12)
More generaly we consider problem (1). We define the set X(Q T R + ) of test functions by

X(Q T R + ) = ζ ∈ C 1,2 c ([0, T ) × [0, ∞)) : ζx(t, 0) = 0 for t ∈ [0, T ] . ( 13
)
Definition 1.2. Let ν, µ be Radon measures in R+ and [0, T ) respectively. A function u defined in Q T R + and belonging to

L 1 loc (Q T R + ) ∩ L 1 (∂ ℓ Q T R + ; dt) such that g(u) ∈ L 1 (∂ ℓ Q T R + ; dt) is a weak solution of (1) if for every ζ ∈ X(Q T R + ) there holds - T 0 ∞ 0 (ζt + ζxx)udxdt + T 0 (g(u)ζ) (t, 0)dt = ∞ 0 ζdν(x) + T 0 ζ(t, 0)dµ(t). (14)
We denote by E(t, x) the Gaussian kernel in R+ × R. The solution of

vt -vxx = 0 in Q ∞ R + -vx = δ0 in R+ v(0, .) = 0 in R+, (15)
has explicit expression

v(t, x) = 2E(t, x) = 1 √ πt e -x 2 4t . (16)
If x, y > 0 and s < t we set Ẽ(ts, x, y) = E(ts, xy) + E(ts, x + y).

When ν ∈ M b (R+) and µ ∈ M b (R+) the solution of

vt -vxx = 0 in Q ∞ R + -vx(., 0) = µ in R+ u(0, .) = ν in R+, (17)
is given by

vν,µ(t, x) = ∞ 0 Ẽ(t, x, y)dν(y) + 2 t 0 E(t -s, x)dµ(s) = E R + [ν](t, x) + E R + ×{0} [µ](t, x) = E Q ∞ R + [(ν, µ)](t, x). (18)
We prove the following existence and uniqueness result.

Theorem 1.3. Let g : R → R be a continuous nondecreasing function such that g(0

) = 0. If g satisfies ∞ 1 (g(s) -g(-s))s -3 ds < ∞, (19)
then for any bounded Borel measures ν in R+ and µ in [0, T ), there exists a unique weak solution u := uν,µ ∈ L 1 (Q T R + ) of (1). Furthermore the mapping (ν, µ) → uν,µ is nondecreasing.

When g(s) = |s| p-1 s, condition (19) is satisfied if 0 < p < 2. ( 20
)
The above result is still valid under minor modifications if R+ is replaced by a bounded interval I := (a, b), and problem (1) by

ut -uxx = 0 in Q T I ux(., b) + g(u(., b)) = µ1 in [0, T ) -ux(., a) + g(u(., a)) = µ2 in [0, T ) u(0, .) = ν in (a, b), (21)
where ν, µj (j = 1, 2) are Radon measures in I and (0, T ) respectively.

In the last section we present the scheme of the natural extensions of this problem to a multidimensional framework

ut -∆u = 0 in Q T R n + -ux n + g(u) = µ in ∂ ℓ Q T R n + u(0, .) = ν in R n + , (22)

BOUNDARY VALUE PROBLEMS RELATIVE TO HEAT EQUATION 5

The construction of solutions with measure data can be generalized but there are some difficulties in the obtention of self-similar solutions. The equation with a source flux

ut -∆u = 0 in Q T R n + ux n + g(u) = 0 in ∂ ℓ Q T R n + u(0, .) = ν in R n + , (23)
has been studied by several authors, in particular Fila, Ishige, Kawakami and Sato [START_REF] Fila | Existence of positive solutions of a semilinear elliptic equation with a dynamical boundary condition[END_REF], [START_REF] Ishige | Global solutions of the heat equation with a nonlinear boundary condition[END_REF], [START_REF] Ishige | Heat equation with a nonlinear boundary condition and uniformly local L r spaces[END_REF]. Their main concern deals with global existence of solutions.

Aknowledgements. The author is grateful to the reviewer for mentioning reference [START_REF] Fila | The blow-up rate for the heat equation with a non-linear boundary condition[END_REF] which pointed out the role of Whittaker's equation which was used for analyzing the blow-up of positive solutions of (23) when g(u) = u p when n = 1.

Self-similar solutions

The symmetrization

We define the operator LK in C 2 0 (R) by

LK(φ) = -K -1 (Kφ ′ ) ′ .
The operator LK has been thouroughly studied in [START_REF] Escobedo | Variational problems related to the self-similar solutions of the heat equations[END_REF]. In particular inf

∞ -∞ φ ′2 K(η)η : ∞ -∞ φ 2 K(η)dη = 1 = 1 2 . ( 24
)
The above infimum is achieved by φ1 = (4π) -1 2 K -1 and LK is an isomorphism from

H 1 K (R) onto its dual (H 1 K (R)) ′ ∼ H -1 K (R). Finally L -1 K is compact from L 2 K (R) into H 1 K (R), which implies that LK is a Fredholm self-adjoint operator with σ(LK ) = λj = 1+j-1 2 : j = 1, 2, ... , and
ker (LK -λjI d ) = span φ (j) 1 . If φ is defined in R+, φ(x) = φ(-x)
is the symmetric with respect to 0 while φ * (x) = -φ(-x) is the antisymmetric with respect to 0. The operator LK restricted to R+ is denoted by L + K . The operator L +,N K with Neumann condition at x = 0 is again a Fredholm operator. This is also valid for the operator

L +,D K with Dirichlet condition at x = 0. Hence, if φ is an eigenfunction of L +,N K , then φ is an eigenfunction of LK in L 2 K (R). Similarly, if φ is an eigenfunction of L +,D K , then φ * is an eigenfunction of LK in L 2 K (R). Conversely, any even (resp. odd) eigenfunction of LK in L 2 K (R) satisfies Neumann (resp. Dirichlet) boundary condition at x = 0. Hence its restiction to L 2 K (R+) is an eigenfunction of L +,N K (resp. L +,D K ). Since φ (j) 1
is even (resp. odd) if and only if j is even (resp. odd), we derive

H 1,0 K (R+) = ∞ ℓ=1 span φ (2ℓ+1) 1 , (25)
and

H 1 K (R+) = ∞ ℓ=0 span φ (2ℓ) 1 . ( 26
) Note that φ ∈ H 1 K (R+) such that φx(0) = 0 (resp. φ(0) = 0) implies φ ∈ H 1 K (R) (resp. φ * ∈ H 1 K (R)). Furthermore, φ1 is an eigenfunction of L + K in H 1 K (R n + ) with Neumann boundary condition on ∂R n + while ∂x n φ1 is an eigenfunction of L + K in H 1 K (R n + )

with Dirichlet boundary condition on ∂R n

+ . We list below two important properties of H 1 K (R+) valid for any β > 0. Actually they are proved in [START_REF] Escobedo | Variational problems related to the self-similar solutions of the heat equations[END_REF]Prop. 1.12] with

H 1 K β (R) but the proof is valid with H 1 K β (R+). (i) φ ∈ H 1 K β (R+) =⇒ K β 2 φ ∈ C 0, 1 2 (R+) (ii) H 1 K β (R+) ֒→ L 2 K β (R+) is compact for all n ≥ 1. (27) 2.2. Proof of Theorem 1.1-(i)-(ii) Assume p ≥ 2, then 1 2(p-1) ≤ 1 2 . If ω is a weak solution, then ∞ 0 ω ′2 - 1 2(p -1) ω 2 Kdη + |ω| p+1 (0) = 0. If 1 2 > 1 2(p-1)
we deduce that ω = 0. Furthermore, when 1 2 = 1 2(p-1) then

|ω| p+1 (0) = 0.
If ω is nonzero, it is an eigenfunction of L +,D K . Since the first eigenvalue is 1 it would imply 1 = 1 2(p-1) ≤ 1 2 , contradiction. Assume 1 < p ≤ 3 2 and ω is a nonnegative weak solution. We take

ζ(η) = ηe -η 2 4 = -2φ ′ 1 (η), then ∞ 0 -ζ ′′ - 1 2(p -1) ζ ωK(η)dη + ζ ′ (0)ω p (0) = 0. Since -ζ ′′ = ζ⌊ R + > 0 and ζ ′ (0) = φ1(0) = 1, we derive ωζ = 0 if 1 > 1 2(p-1) and ω(0) = 0 if 1 = 1 2(p-1)
. Hence ω ′ (0) = 0 by the equation and ω ≡ 0 by the Cauchy-Lipschitz theorem.

BOUNDARY VALUE PROBLEMS RELATIVE TO HEAT EQUATION 7

Proof of Theorem 1.1-(iii)

We define the following functional on

H 1 K (R n + ) J(φ) = 1 2 ∞ 0 φ ′2 - 1 2(p -1) φ 2 Kdη + 1 p + 1 |φ(0)| p+1 . ( 28
)
Lemma 2.1. The functional J is lower semicontinuous in H 1 K (R+). It tends to infinity at infinity and achieves negative values.

Proof. We write

J(ψ) = J1(ψ) -J2(ψ) = J1(ψ) - 1 2(p -1) ψ 2 L 2 K .
Clearly J1 is convex and J2 is continuous in the weak topology of

H 1 K (R+) since the imbedding of H 1 K (R+) into L 2 K (R+) is compact. Hence J is weakly semicontinuous in H 1 K (R+). Let ǫ > 0, then J(ǫφ1) = 1 4 - 1 4(p -1)
ǫ 2 √ π 2 + ǫ p+1 p + 1 . Since 1 < p < 2, 1 4 -1 4(p-1) < 0. Hence J(ǫφ1) < 0 for ǫ small enough, thus J achieves negative values on H 1 K (R+). If ψ ∈ H 1 K (R+)
it can be written in a unique way under the form ψ = aφ1 + ψ1 where a = 2 √ πψ(0) and ψ1 ∈ H 1,0 K (R+). Hence, for any ǫ > 0,

J(ψ) = 1 2 ∞ 0 ψ ′2 1 - 1 2(p -1) ψ 2 1 Kdη + a 2 2 ∞ 0 φ ′2 1 - 1 2(p -1) φ 2 1 Kdη + a ∞ 0 ψ ′ 1 φ ′ 1 - 1 2(p -1) ψ1φ1 Kdη + 1 p + 1 |a| p+1 ≥ 2p -3 4(p -1) ∞ 0 ψ ′2 1 Kdη - aǫ 2 ∞ 0 ψ ′2 1 + 1 2(p -1) ψ 2 1 Kdη + a 2 (p -2) √ π 4(p -1) - ap √ π 4(p -1)ǫ + 1 p + 1 |a| p+1 .
Note that ψ 2

H 1 K ≤ 4 ψ ′ 1 2 L 2 K + a 2 .
Since 2p -3 > 0, we can take ǫ > 0 small enough in order that lim

ψ H 1 K →∞ J(ψ) = ∞. ( 29
)
End of the proof of Theorem 1.1-(iii). By Lemma 2.1 the functional J achieves its minimum in H 1 K (R+) at some ωs = 0, and ωs can be assumed to be nonnegative since J is even. By the strong maximum principle ωs > 0, and by the method used in the proof of [START_REF] Moutoussamy | Isolated singularities and asymptotic behaviour of the solutions of a semilinear heat equation[END_REF]Proposition 1] is is easy to prove that positive solutions belong to H 2 K (R+). Assume that ω is another positive solution, then

∞ 0 (Kω ′ s ) ′ ωs - (K ω′ s ) ′ ωs (ω 2 s -ω2 s )dη = 0.
Integration by parts, easily justified by regularity, yields

∞ 0 (Kω ′ s ) ′ ωs - (K ω′ s ) ′ ωs (ω 2 s -ω2 s )dη = Kω ′ s ωs - ω2 s ωs -K ω′ s ω 2 s ωs -ωs ∞ 0 - ∞ 0 ωs - ω2 s ωs ′ Kω ′ s dη + ∞ 0 ω 2 s ωs -ωs ′ Kω ′ s dη = -ω p-1 s -ωp-1 s ω 2 s -ω2 s (0) - ∞ 0 ω ′ s ωs -ωs ω′ s ωs 2 + ωs ω′ s -ωsω ′ s ωs 2 dη.
This implies that ωs = ωs. The proof of ( 10) is similar as the proof of estimate (2.5) in [13, Theorem 4.1].

The explicit approach

This part is an adaptation to our problem of what has been done in [START_REF] Fila | The blow-up rate for the heat equation with a non-linear boundary condition[END_REF] concerning the blow-up problem in equation ( 23). Let ω be a solution of

ω ′′ + 1 2 ηω ′ + 1 2(p -1) ω = 0 in R+. ( 30
)
We set r = η 2 4 and ω(η) = r -1 4 e -r 2 Z(r).

Then Z satisfies the Whittaker equation (with the standard notations)

Zrr + - 1 4 + k r + 1 -4µ 2 4r 2 Z = 0 (31)
where k = 1 2(p-1) -1 4 and µ = 1 4 . Notice that the only difference with the expression in [9, Lemma 3.1] is the value of the coefficient k. This equation admits two linearly independent solutions

Z1(r) = e -r 2 r 1 2 +µ U 1 2 + µ -k, 1 + 2µ, r , and
Z2(r) = e -r 2 r 1 2 +µ M 1 2 + µ -k, 1 + 2µ, r .
The functions U and M are the Whittaker functions which play an important role not only in analysis but also in group theory. The have the following asymptotic expansion as r → ∞ (see e.g. [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]),

U 1 2 + µ -k, 1 + 2µ, r = r k-µ-1 2 1 + O(r -1 = r 1 2(p-1) -1 1 + O(r -1 ,
and

M 1 2 + µ -k, 1 + 2µ, r = Γ(1 + 2µ) Γ( 1 2 + µ -k) e r r -(µ+ 1 2 +k) 1 + O(r -1 = Γ( 3 2 ) Γ(1 -1 2(p-1) ) e r r - p 2(p-1) 1 + O(r -1 . Then Z1(r) = r 1 2(p-1) -1 4 e -r 2 1 + O(r -1 , and
Z2(r) = Γ( 3 2 ) Γ(1 -1 2(p-1) ) r 1 4 - 1 2(p-1) -e r 2 1 + O(r -1 .
To this corresponds the two linearly independent solutions ω1 and ω2 of (30) with the following behaviour as η → ∞,

(i) ω1(η) = c1η 1 p-1 -1 e -η 2 4 1 + O(η -2 , (ii) ω2(η) = c2η -1 p-1 1 + O(η -2 . (32)
Clearly only ω1 satisfies the decay estimate ω(η) = o(η -1 p-1 ) as η → ∞. Hence the solution ω is a multiple of ω1 and the multiplicative constant c is adjusted in order to fit the condition ω ′ (0) = ω p (0).

Problem with measure data

The regular problem

Set G(r) = r 0 g(s)ds. We consider the functional J in L 2 (R+) with domain D(J) = H 1 (R+) defined by

J(u) = 1 2 ∞ 0 u 2 x dx + G(v(0)).
It is convex and lower semicontinuous in L 2 (R+) and its subdifferential ∂J sastisfies

∞ 0 ∂J(u)ζdx = ∞ 0 uxζxdx + g(u(0))ζ(0), for all ζ ∈ H 1 (R+). Therefore ∞ 0 ∂J(u)ζdx = - ∞ 0 uxxζdx + (g(u(0)) -ux(0))ζ(0). Hence ∂J(u) = -uxx for all u ∈ D(∂J) = {v ∈ H 1 (R+) : vx(0) = g(v(0))}.
(33) The operator ∂J is maximal monotone, hence it generates a semi-group of contractions. Furthermore, for any u0 ∈ L 2 (R+) and F ∈ L 2 (0, T ; L 2 (L 2 (R+)) there exists a unique strong solution to Ut + ∂J(U ) = F a.e. on (0, T )

U (0) = u0. ( 34
)
Proposition 3.1. Let µ ∈ H 1 (0, T ) and ν ∈ L 2 (R+). Then there exists a unique function u ∈ C([0, T ]; L 2 (R+) such that √ tuxx ∈ L 2 ((0, T ) × R+) which satisfies (35). The mapping (µ, ν) → u := uµ,ν is non-decreasing and u is a weak solution in the sense that it satisfies [START_REF] Marcus | Isolated boundary singularities of signed solutions of some nonlinear parabolic equations[END_REF].

Proof. Let η ∈ C 2 0 ([0, ∞)) such that η(0) = 0, η ′ (0) = 1. If f ∈ H 1 (0, T ), ν ∈ L 2 (R+), and u is a solution of ut -uxx = 0 in Q T R + -ux(., 0) + g(u(., 0)) = µ(t) in [0, T ) u(0, .) = ν in R+, (35)
where ν ∈ L 2 (R+), then the function v(t, x) = u(t, x)µ(t)η(x) satisfies

vt -vxx = F in Q T R + -vx(., 0) + g(v(., 0)) = 0 in [0, T ) v(0, .) = ν -µ(0)η in R+, (36)
with F (t, x) = -(µ ′ (t)η(x) + µ(t)η ′′ (x)). The proof of the existence follows by using [3, Theorem 3.6].

Next, let (μ, ν) ∈ H 1 (0, T ) × L 2 (R+) such that μ ≤ µ and ν ≤ ν and let ũ = uμ,ν , then 1 2

d dt ∞ 0 (ũ -u) 2 + dx + ∞ 0 (∂x(ũ -u)+) 2 dx -(μ(t) -µ(t)) (ũ(t, 0) -u(t, 0))+ + (g(ũ(t, 0)) -g(u(t, 0))))(ũ(t, 0) -u(t, 0)) = 0. Then ∞ 0 (ũ -u) 2 + dx⌊t=0 =⇒ ∞ 0 (ũ -u) 2 + dx = 0 on [0, T ].
We can also use (18) to express the solution of (35):

u(t, x) = ∞ 0 Ẽ(t, x, y)ν(y)dy + 2 t 0 E(t -s, x)(µ(s) -g(u(s, 0)))ds.

BOUNDARY VALUE PROBLEMS RELATIVE TO HEAT EQUATION 11

In particular, if g(0) = 0, then

|u(t, x)| ≤ ∞ 0 Ẽ(t, x, y)|ν(y)|dy + 2 t 0 E(t -s, x)|µ(s)|ds.
The proof of ( 14) follows since u is a strong solution.

Next, we prove that the problem is well-posed if µ ∈ L 1 (0, T ).

Proposition 3.2. Assume {νn} ⊂ Cc(R+) and {µn} ⊂ C 1 ([0, T ]) are Cauchy sequences in L 1 (R+) and L 1 (0, T ) respectively. Then the sequence {un} of solutions of

un t -un xx = 0 in Q T R + -un x(., 0) + g(un(., 0)) = µn(t) in [0, T ) un(0, .) = νn in R+, (37)
converges in C([0, T ]; L 1 (R+) to a function u which satisfies [START_REF] Marcus | Isolated boundary singularities of signed solutions of some nonlinear parabolic equations[END_REF].

Proof. For ǫ > 0 let pǫ be an odd C 1 function defined on R such that p ′ ǫ ≥ 0 and pǫ(r) = 1 on [ǫ, ∞), and put jǫ(r

) = r 0 pǫ(s)ds. Then d dt ∞ 0 jǫ(un -um)dx + ∞ 0 (un x -um x) 2 p ′ ǫ (un -um)dx + (g(un(t, 0)) -g(um(t, 0))) pǫ(un(t, 0) -um(t, 0)) = (µn(t) -µm(t)) pǫ(un(t, 0) -um(t, 0)). Hence ∞ 0 jǫ(un -um)(t, x)dx + (g(un(t, 0)) -g(um(t, 0))) pǫ(un(t, 0) -um(t, 0)) ≤ ∞ 0
jǫ(νnνm)dx + (µn(t)µm(t)) pǫ(un(t, 0)um(t, 0)).

Letting ǫ → 0 implies pǫ → sgn0, hence for any t ∈ [0, T ],

∞ 0 |un -um|(t, x)dx + |g(un(t, 0)) -g(um(t, 0)| ≤ ∞ 0 |νn -νm|dx + |µn(t) -µm(t)|. (38)
Therefore {un} and {g(un(., 0)} are Cauchy sequences in C([0, T ]; L 1 (R+)) and C([0, T ]) respectively with limit u and g(u) and u = uν,µ satisfies [START_REF] Marcus | Isolated boundary singularities of signed solutions of some nonlinear parabolic equations[END_REF]. If we assume that (ν, ν) and (µ, μ) are couples of elements of L 1 (R+) and L 1 (0, T ) respectively and if u = uν,µ and ũ = uν,μ, there holds by the above technique,

∞ 0 |u -ũ|(t, x)dx + |g(u(t, 0)) -g(ũ(t, 0)| ≤ ∞ 0 |ν -ν|dx + |μ(t) -μ(t)| for all t ∈ [0, T ]. ( 39
)
The following lemma is a parabolic version of an inequality due to Brezis.

Lemma 3.3. Let ν ∈ L 1 (R+) and µ ∈ L 1 (0, T ) and v be a function defined in [0, T ) × R+, belonging to L 1 (Q T R + ) ∩ L 1 (∂ ℓ Q T R + ) and satisfying - T 0 ∞ 0 (ζt + ζxx)vdxdt = T 0 ζ(., 0)µdt + ∞ 0 νζdx. (40)
Then for any

ζ ∈ X(Q T R + ), ζ ≥ 0, there holds - T 0 ∞ 0 (ζt + ζxx)|v|dxdt ≤ ∞ 0 ζ(., 0)sign(v)µdt + ∞ 0 |ν|ζdx. (41) Similarly - T 0 ∞ 0 (ζt + ζxx)v+dxdt ≤ ∞ 0 ζ(., 0)sign+(v)µdt + ∞ 0 ν+ζdx. (42)
Proof. Let pǫ be the approximation of sign0 used in Proposition 3.2 and ηǫ be the solution of

-ηǫ t -ηǫ xx = pǫ(v) in Q T R + ηǫ x(., 0) = 0 in [0, T ] ηǫ(0, .) = 0 in R+.
Then |ηǫ| ≤ η * where η * satisfies

-η * t -η * xx = 1 in Q T R + η * x (., 0) = 0 in [0, T ] η * (0, .) = 0 in R+.
Although ηǫ does not belong to

X(Q T R + ) (it is not in C 1,2 ([0, T ) × R+)
, it is an admissible test function and we deduce that there exists a unique solution to (40). Thus v is given by expression (18).

In order to prove (41), we can assume that µ and ν are smooth,

ζ ∈ X(Q T R + ), ζ ≥ 0 and set hǫ = pǫ(v)ζ and wǫ = vpǫ(v), then ∞ 0 hǫ xxvdx = ∞ 0 (2p ′ ǫ (v)vxζx + pǫ(v)ζxx + ζ(pǫ(v))xx) vdx = ∞ 0 (2vp ′ ǫ (v)vxζx -wǫ xζx -(vζ)x(pǫ(v))x) dx -ζ(t, 0)v(t, 0)p ′ ǫ (v(t, 0))vx(t, 0) = - ∞ 0 ζx(jǫ(v))x + ζp ′ (v)ǫv 2 x dx -ζ(t, 0)v(t, 0)p ′ ǫ (v(t, 0))vx(t, 0) = - ∞ 0 ζp ′ (v)ǫv 2 x -jǫ(v)ζxx dx -ζ(t, 0)v(t, 0)p ′ ǫ (v(t, 0))vx(t, 0), (43)

BOUNDARY VALUE PROBLEMS RELATIVE TO HEAT EQUATION 13

and

T 0 hǫ tvdt = T 0 (pǫ(v)ζt + p ′ ǫ (v)ζvt)vdt. ( 44
) Since v is smooth 0 = T 0 ∞ 0 (vt -vxx)hǫdxdt = - T 0 ∞ 0 (hǫ t + hǫ xx)vdxdt - ∞ 0 hǫ(0, x)ν(x)dx - T 0 [pǫ(v(t, 0)) -v(t, 0)p ′ ǫ (v(t, 0))] ζ(t, 0)µ(t)dt.
Therefore, using (41) and ( 42),

- T 0 ∞ 0 (jǫv)ζxx + vpǫ(v)ζt) dxdt + T 0 ∞ 0 ζp ′ ǫ (v)v 2 x -vp ′ ǫ (v)vtζ dxdt = ∞ 0 hǫ(0, x)ν(x)dx + T 0 hǫ(t, 0)µ(t)dt. ( 45
) Put ℓǫ(s) = s 0 rp ′ ǫ (r)dr, then |ℓǫ(s) ≤ cǫ -1 s 2 χ [-ǫ,ǫ] (s)|. Since T 0 ∞ 0 ζvp ′ ǫ (v)vtdxdt = - ∞ 0 ℓǫ(v(0, x))ζ(x)dx - T 0 ∞ 0 ζtℓǫ(v)dxdt,
and ζ has compact support, it follows that

lim ǫ→0 T 0 ∞ 0 ζvp ′ ǫ (v)vtdxdt = 0.
Letting ǫ → 0 in (45), we derive (41) for smooth v. Using Proposition 3.2 completes the proof of (41). The proof of (42) is similar.

Remark

∈ C ∞ c (R+) with 0 ≤ ηn ≤ 1, ηn(x) = 1 on [0, n], ηn(x) = 0 on [n + 1, ∞), |η ′ n | ≤ 2, |η ′′ n | ≤ 4. Then ηnζ ∈ X(Q T R +
) by letting n → ∞ and the proof follows by letting n → ∞.

Proof of Theorem 1.3

We give first some heat-ball estimates relative to our problem. For r > 0, x ∈ R+ and t ∈ R we set e(t, x; r) = (s, y) ∈ (0, T ) × R+ : s ≤ t, Ẽ(ts, x, y) ≥ r .

(47) Since e(t, x; r)

⊂ [t -1 4πer 2 , t] × [x -1 r √ πe , x + 1 r √ πe ], there holds |e(t, x; r)| ≤ 1 2r 3 (πe) 3 2 , ( 48
)
and if e * (t; r) = {s ∈ (0, T ) :

s ≤ t, E(t -s, 0, 0) ≥ r} , (49)
then we have

e * (t; r) ⊂ [t -1 4πer 2 , t] =⇒ |e * (t; r)| ≤ 1 4r 2 πe . ( 50
)
If G is a measured space, λ a positive measure on G and q > 1, M q (G, λ) is the Marcinkiewicz space of measurable functions f : G → R satisfying for some constant c > 0 and all measurable set E ⊂ G,

E |f |dλ ≤ c (λ(E)) 1 p ′ , (51)
and f M q (G,λ) = inf{c > 0 s.t. (50) holds}.

Lemma 3.4. Assume µ,ν are bounded measure in R+ and R+ respectively and u is the solution of (17) given by (18) and vν,µ is the solution of (17). Then

vν,µ M 3 (Q T R + ) + vν,µ⌊ ∂Q T R + M 2 (∂Q T R + ) ≤ c µ M(∂Q T R + ) + ν M(Q T R +
) .

(52)

Proof. First we consider v0,µ v0,µ(t, x) = 2 t 0 E(ts, x)dµ(s).

BOUNDARY VALUE PROBLEMS RELATIVE TO HEAT EQUATION 15

If F ⊂ [0, T ] is a Borel set, than for any τ > 0

F E(t -s, 0)ds = F ∩{E≤τ } E(t -s, 0)ds + F ∩{E>τ } E(t -s, 0)ds ≤ τ |F | + {E>τ } E(t -s, 0)ds ≤ τ |F | - ∞ τ λd|e * (t, λ)| ≤ τ |F | + ∞ τ λd|e * (t, λ)| ≤ τ |F | + 1 4πeτ
.

If we choose τ 2 = 1 4πe|F | , we derive F E(t -s, 0)ds ≤ |F | 1 2 √ πe . ( 53
) If F ⊂ (0, T ) is a Borel set then F v0,µ(t, 0)dt = 2 t 0 F E(t -s, 0)dtdµ(s) ≤ 2|F | 1 2 √ πe µ M(∂Q T R +
) .

This proves that

v0,µ⌊ ∂Q T R + M 2 (∂Q T R + ) ≤ c µ M(∂Q T R + ) . (54)
Similarly

, if G ⊂ [0, T ] × [0, ∞) is a Borel set, then G Ẽ(t -s, x, 0)ds ≤ 2|G| 1 3 √ πe , (55)
and v0,µ

M 3 (Q T R + ) ≤ c µ M(∂Q T R + ) . (56)
In the same way we prove that

vν,0 M 3 (Q T R + ) + vν,0⌊ ∂Q T R + M 2 (∂Q T R + ) ≤ c ν M(Q T R + ) . (57)
This ends the proof.

Proof of Theorem 1.3 Uniqueness. Assume u and ũ are solutions of (1), then w = u-ũ satisfies

wt -wxx = 0 in Q T R + -wx(., 0) + g(u(., 0)) -g(ũ(., 0)) = 0 in [0, T ) w(0, .) = 0 in R+. (58)
Applying (41), we obtain

- T 0 ∞ 0 (ζt+ζxx)|w|dxdt+ ∞ 0 (g(u(., 0))-g(ũ(., 0)))sign(w)ζ(t, 0)dt ≤ 0, for any ζ ∈ X T R + with ζ ≥ 0. Let θ ∈ C 1 c (Q T R + ), η ≥ 0, we take ζ to be the solution of -ζt -ζxx = θ in (0, T ) × R+ ζx(t, 0) = 0 in (0, T ) ζ(T, x) = 0 in (0, ∞).
Then ζ satisfies (46), hence

T 0 ∞ 0 θ|w|dxdt + ∞ 0 (g(u(., 0)) -g(ũ(., 0)))sign(w)ζ(t, 0)dt ≤ 0.
This implies w = 0.

Existence. Without loss of generality we can assume that µ and ν are nonnegative. Let {νn} ⊂ Cc(R+) and {µn} ⊂ Cc([R+]0, T )) converging to ν and µ in the sense of measures and let un be the solution of (37).

Then from (39),

T 0 ∞ 0 |un|dxdt + T 0 |g(un(t, 0))|dt ≤ T ∞ 0 |νn|dx + T 0 |µn|dt.
(59) Therefore un and g(un(., 0)) remain bounded respectively in L 1 (Q T R + ) and in L 1 (0, T ). Furthermore, by Lemma 3.4, un remains bounded in M 3 (Q T R + ) and in M 2 (∂Q T R + ). We can also write un under the form un(t, x) = ∞ 0 Ẽ(t, x, y)µn(y)dy + 2 t 0 E(ts, x)(νn(t)g(un(t, 0)))ds = An(t, x) + Bn(t, x).

(60) Since we can perform the even reflexion through y = 0, the mapping (t, x) → An(t, x) := ∞ 0 Ẽ(t, x, y)µn(y)dy, is relatively compact in C m loc (Q T R + ) for any m ∈ N * . Hence we can extract a subsequence {un k } which converges uniformly on every compact subset of (0, T ] × [0, ∞), hence a.e. on (0, T ] for the 1-dimensional Lebesque measure. Concerning the boundary term (t, x) → Bn(t, x) := t 0 E(ts, x)(νn(t)g(un(t, 0)))ds, it is relatively compact on every compact subset of [0, T ] × (0, ∞). If x = 0, then Bn(t, 0) = t 0 (νn(t)g(un(t, 0))) ds π(ts) .

Since νn(.)g(un(., 0)) L 1 (0,T ) , t → Bn(t, 0) is uniformly integrable on (0, T ), hence relatively compact by the Frechet-Kolmogorov Theorem. Therefore there exists a subsequence, still denoted by {n k } such that Bn k (t, 0) converges for almost all t ∈ (0, T ). This implies that the sequence of function {un k } defined by (60) converges in Q T R + up to a set Θ∪Λ where Θ ⊂ Q T R + is neglectable for the 2-dimensional Lebesgue measure and Λ ⊂ ∂ ℓ Q T R + neglectable for the 1-dimensional Lebesgue measure. From Lemma 3.4, (u n,k ) . For ǫ > 0 given, we chose λ large enough so that the integral term above is smaller than ǫ and then |F | such that g(λ)|F |+ ≤ ǫ. Hence {g(u n,k (., 0))} is uniformly integrable. Therefore up to a subsequence, it converges to g(u(., 0)) in L 1 (0, T ). Clearly u satisfies Monotonicity. If ν ≥ ν and µ ≥ μ; we can choose the approximations such that νn ≥ νn and µn ≥ μn. It follows from (42) that uν n ,µn ≥

⌊ Q T R + , u⌊ ∂ ℓ Q T R + ) converges in L 1 loc (Q T R + )×L 1 (∂ ℓ Q T R + )

  and the convergence of each of the components holds also almost everywhere (up to a subsequence). Since u n,k is a weak solution, it satisfies for anyζ ∈ X(Q T R + ) ζxx)u n,k dxdt + T 0 (g(u n,k )ζ) (t, 0)dt = ∞ 0 ζν n,k (x)dx + T 0 ζ(t, 0)µ n,k (t)dt.(61)In order to prove the convergence of g(u n,k (t, 0)), we use Vitali's convergence theorem and the assumption (19). Let F ⊂ [0, T ] be a Borel set.Using the fact that 0 ≤ u n,k ≤ vν n,k ,µ n,k and the estimate of Lemma 3.4, we have for any λ > 0,F |g(u n,k (t, 0))|dt ≤ F ∩{u n,k(t,0)≤λ} |g(u n,k (t, 0))|dt +{u n,k (t,0)>λ} |g(u n,k (t, 0))|dt ≤ g(λ)|F | -∞ λ σd|{t : |g(u n,k (t, 0))| > σ}| ≤ g(λ)|F | + c ∞ λ |g(σ)|σ -3 ds, where c depends of µ M(∂Q T R + ) + ν M(Q T R +

LAURENT V ÉRON

uν n , μn . Choosing the same subsequence {n k }, the limits u, ũ are in the same order. The conclusion follows by uniqueness.

3.3. The case g(u) = |u| p-1 u Condition (19) is satisfied if p < 2. If this condition holds there exists a solution u ℓδ 0 = u 0,ℓδ 0 and the mapping ℓ → u ℓδ 0 is increasing. Theorem 3.5. (i) If 1 < p ≤ 3 2 , u ℓδ 0 tends to ∞ when k → ∞. (ii) If 3 2 < p < 2, u ℓδ 0 converges to Uω s defined by

when k → ∞.

Proof. By uniqueness and using (3), there holds

for any k, ℓ > 0. Since ℓ → u ℓδ 0 is increasing, its limit u∞, when ℓ → ∞, satisfies

Hence u∞ is a positive self-similar solution of (2), provided it exists.

, u kδ 0 admits no finite limit when k → ∞ which ends the proof.

Remark. As a consequence of this result, no a priori estimate of Brezis-Friedman type (parabolic Keller-Osserman) exists for a nonnegative func-

ux(., 0) + |u| p-1 u(., 0) = 0 for all t > 0 u(0, x) = 0 for all x > 0.

(65)

.

(66)

The type of phenomenon (i) in Theorem 3.5 is characteristic of fractional diffusion. It has already been observed in [6, Theorem 1.3] with equations

when 0 < α < 1 is small and p > 1 is close to 1.

BOUNDARY VALUE PROBLEMS RELATIVE TO HEAT EQUATION 19

Extension and open problems

The natural extension is to replace a one dimensional domain by a mutidimenional one. The main open problem is the question of a priori estimate as stated in the last remark above.

Self-similar solutions

Let η = (η1, ..., ηn) be the coordinates in R n and denote R n

and

. Similarly to Section 2 we define LK in C 2 0 (R n ) by

If α = (α1, ..., αn) ∈ N n , we set |α| = α1 + α2 + ... + αn. We denote by φ1 the function K -1 . Then the set of eigenvalues of LK is the set of numbers λ k = n+k 2 : k ∈ N with corresponding set of eigenspaces

The operators L +,N K and L +,D K are defined acoordingly in

We define the following functional on

If ω is a solution of (73), the function

Here we have set R n + = {x = (x1, ..., xn) = (x ′ , xn) : xn > 0}. We denote by E the subset H 1 K (R n + )∩L p (∂R n + ; dη ′ ) of solutions of (73) and by E+ the subset of positive solutions. As for the case n = 1 we have the following non-existence result

The proof is similar to the one of Theorem 1.1. Hence the existence is to be found in the range 1

tends to infinity.

Problem with measure data

The method for proving Theorem 1.3 can be adapted to prove the following n-dimensional result Theorem 4.2. Let g : R → R be a nondecreasing continuous function such that g(0) = 0 and

then for any bounded Radon measures ν in R n + and µ in (0, T ) × ∂R n + , there exists a unique Borel function u := uν,µ defined in Q

in the sense that

) such that ζx n = 0 on (0, T ) × ∂R n + and ζ(T, .) = 0. Furthermore (ν, µ) → uν,µ) is nondecreasing.