Pierre Lissy
email: lissy@ceremade.jussieu.fr

The cost of the control in the case of a minimal time of control: the example of the one-dimensional heat equation

Keywords: null controllability, parabolic equations, minimal time, controllability cost, non-harmonic Fourier series, moment method AMS Classification: 35K05, 93B05, 42A70

In this article, we consider the controllability of the one-dimensional heat equation with an internal control depending only on the time variable and an imposed profile depending on the space variable. It is well-known that in this context, there might exist a minimal time of null-controllability T0, depending on the behavior of the Fourier coefficients of the profile. We prove two different results. The first one, which is surprising, is that the cost of the controllability in time T > T0 close to T0 may explode in an arbitrary way. On the other hand, we prove as a second result that for a large class of profiles, the cost of controllability at time T > T0 is bounded from above by exp(C(T0)/(T -T0)) for some constant C(T0) > 0 depending on T0. The main method used here is the moment method and some tools coming from complex analysis.

Introduction

Let T > 0. In what follows, we will consider the following controlled heat equation on (0, T ) × (0, π): y t -y xx = f (x)u(t) in (0, T ) × (0, π), y(0, •) = y 0 in (0, π),

where y 0 ∈ L 2 (0, π), u ∈ L 2 (0, T ) is the control and f ∈ H -1 (0, π) is an imposed profile for this control.

It is well-known that equation ( 1) is well-posed in the sense that there exists a unique solution y ∈ C 0 ([0, T ], L 2 (0, π))∩L 2 ([(0, T ), H 0 1 (0, π)) verifying moreover that there exists a constant C > 0 such that for every y 0 ∈ L 2 (0, π), every f ∈ H -1 (0, π) and every v ∈ L 2 (0, T ), we have ||y|| C 0 ([0,T ],L 2 (0,π)) + ||y|| L 2 ((0,T ),H 0 1 (0,π)) C(||y 0 || L 2 (0,π) + ||f || H -1 (0,π) ||v|| L 2 (0,T ) ), which implies notably that the control operator u ∈ R → f (•)u is admissible for the semigroup e t∆ with domain D(∆). The controllability properties of this equation has been widely studied (see notably [START_REF] Fattorini | Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations[END_REF], [START_REF] Dolecki | Observability for the one-dimensional heat equation[END_REF] and [START_REF] Ammar-Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF]). The approximate controllability can be easily characterized by the condition

f k = 0, ∀k ∈ N * , (2)
where f k := f, e k H -1 (0,π),H 1 0 (0,π) .

Assume from now on and until the end of the article that the condition (2) is satisfied. Concerning the study of the null-controllability of (1), one very efficient tool in the one-dimensional case is the celebrated moment method introduced in [START_REF] Fattorini | Exact controllability theorems for linear parabolic equations in one space dimension[END_REF]. Let us consider the 1-D Laplace operator ∂ xx with domain D(∆) := H 2 (0, π)∩H 1 0 (0, π) and state space H := L 2 (0, π). It is well-known that -∆ : D(∆) → L 2 (0, π) is a positive definite operator with compact resolvent, the k-th eigenvalue is λ k = k 2 , an associated normalized eigenvector is

e k (x) := √ 2 √ π sin (kπx) .
Let us decompose the initial condition y 0 on the Hilbert basis e k :

y 0 (x) = ∞ k=1 a k e k (x), (4)
where

(a k ) k∈N * ∈ l 2 (N * ). Then, it is classical that imposing y(T, •) = 0 is equivalent to saying that for every k ∈ N * we have T 0 e λ k t u(t)dt = - a k f k . (5)
u is then solution of a moment problem which can be solved by finding a bi-orthogonal family to the family of exponentials {exp(λ k t)} on [0, T ]. Let us introduce the following quantities:

I k (f ) := - log(|f k |) k 2 (6)
and

T 0 := lim sup k→∞ I k (f ) ∈ [0, ∞]. (7)
It is then proved in [START_REF] Ammar-Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF] that:

1. System (1) is null-controllable at any time T > T 0 .

2. System (1) is not null-controllable at any time T < T 0 .

Hence, there might exist a minimal time of controllability, depending on the action of the control through the profile f . Let us mention that in [START_REF] Dolecki | Observability for the one-dimensional heat equation[END_REF], a comprehensive study of ( 1) is performed in the particular case where f (x) := δ x0 ∈ H -1 (0, π), where x 0 ∈ (0, π). In this particular case, one readily obtain that the minimal time of controllability is given by

T 0 (x 0 ) := lim sup k→∞ - log(| sin(kx 0 )|) k 2 .
The dependance with respect to x 0 of T 0 is then carefully studied and it can notably be proved that:

1. For almost all x 0 ∈ (0, π), T 0 (x 0 ) = 0,

2. For every τ ∈ [0, ∞], {x 0 ∈ (0, π)|T 0 (x 0 ) = τ } is dense in (0, π).
This notably means that the minimal time of controllability can take any value between 0 and ∞.

Let us mention that the existence of a minimal time of control for parabolic equations or systems may occur in many other situations, for other examples see notably [START_REF] Ammar-Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF] and [START_REF] Ammar-Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF].

In the case where T > T 0 , one can easily prove (see for example [7, Chapter 2, Section 2.3]) that for every y 0 ∈ L 2 (0, π), there exists a unique optimal (for the L 2 (0, T )-norm) control u opt ∈ L 2 (0, T ) bringing y 0 to 0, the map y 0 → u opt is then linear continuous. The norm of this operator is called the optimal null control cost at time T (or in a more concise form the cost of the control), denoted by C H (T ). By definition, C H (T ) is the smallest constant amongst the constants C > 0 such that for every y 0 ∈ L 2 (0, π), there exists some control u driving y 0 to 0 at time T with ||u|| L 2 (0,T ) C||y 0 || L 2 (0,π) .

In the case where there is no minimal time of control and under appropriate conditions on the f k that should not decrease too fast (what we will call "the usual case" from now on), the cost of fast controls for linear parabolic equations or systems with distributed (or equivalently in this case boundary) control has been widely studied (see notably [START_REF] Güichal | A lower bound of the norm of the control operator for the heat equation[END_REF], [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF], [START_REF] Ervedoza | Sharp observability estimates for the heat equations[END_REF], [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], [START_REF] Benabdallah | Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the N-dimensional boundary null controllability in cylindrical domains[END_REF] and [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF]) and is now quite well-understood in the one-dimensional case, even if there are still some open problems. We mention that understanding the behavior of the cost of controllability is crucial because it can be used to deduce some results in higher dimension (see [START_REF] Benabdallah | Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the N-dimensional boundary null controllability in cylindrical domains[END_REF]) and to obtain nonlinear results (see notably [START_REF] Fernández-Cara | Null and approximate controllability for weakly blowing up semilinear heat equations[END_REF]). It is shown that in the context of the usual case, the cost of fast controls is roughly of the form exp(C/T ) as T → 0, were C is some appropriate constant depending on the geometry. Let us mention that for all the cases studied up to now, the cost of fast controls was a purely high-frequency phenomena, depending only on the asymptotic behavior of the eigenvalues at infinity (for a precise study, see notably [START_REF] Lissy | On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension[END_REF], [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] and [START_REF] Lissy | Construction of Gevrey functions with compact support using the Bray-Mandelbrojt iterative process and applications to the moment method in control theory[END_REF]).

Let us also mention that the multi-dimensional case seems to be out of reach for the moment: there are only few results of existence of minimal time, always in particular geometries (see notably for the case of hypoelliptic diffusions [START_REF] Beauchard | Degenerate parabolic operators of Kolmogorov type with a geometric control condition[END_REF] or [START_REF] Beauchard | 2D Grushin-type equations: minimal time and null controllable data[END_REF]) and sometimes this minimal time is not even known precisely. Concerning the cost of the control, even in the usual case of the heat equation with boundary or distributed control, there are only some partial results coming from [START_REF] Miller | Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time[END_REF], the upper bound being obtained under very strong geometric restrictions. Hence, taking into account the lack of comprehension of the multi-dimensional case, the author thinks that it is very reasonable and interesting to have a look first at the one-dimensional case, which is maybe simpler from the multi-dimensional case but cannot be considered as trivial though.

In this context, a natural question arising is the following: for equation [START_REF] Ammar-Khodja | The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials[END_REF], what is the behavior of C H (T ) when T → T + 0 ? Up to our knowledge, this question has never been investigated in the context of the existence of a minimal time on control in the parabolic case. One could expect that the cost is of the form exp(C(T 0 )/(T -T 0 )) as T → T + 0 , by analogy with the usual case. However, it is not always the case, as highlighted by the following Theorem: Theorem 1.1 For every function g : R + * → R + * supposed to be increasing and verifying g(x) → +∞ for x → +∞, for every T 0 ∈ [0, ∞), there exists f ∈ H -1 (0, π) such that for any T close enough to T 0 , one has

C H (T ) 1 √ T g 1 T -T 0 .
This theorem means that the cost of the control can increase arbritrarily fast as T goes to the minimal time T 0 , which is very surprising. This can be explained by the fact that contrary to the usual case, the cost of the control depends not only on the behavior of I k (f ) (defined in [START_REF] Coron | Singular optimal control: A linear 1-D parabolic-hyperbolic example[END_REF]) at infinity but also on how it differs from its limit superior T 0 . The proof relies on tools coming from complex analysis in the spirit of [START_REF] Coron | Singular optimal control: A linear 1-D parabolic-hyperbolic example[END_REF] (see also [START_REF] Lissy | Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation[END_REF] and [START_REF] Gueye | Singular Optimal Control of a 1-D Parabolic-Hyperbolic Degenerate Equation[END_REF]). The main idea (that differs from what was done in the previous references) is to consider the optimal control associated to an initial condition with one pure eigenmode which is not necessarily the first one and that is adapted to the time of control T . f is then chosen is such a way that the sequence f k decreases "very slowly" to the minimal time T 0 , the rate of convergence depending on the choice of the function g. One serious consequence of this result is that it may be hard to obtain sharp results in the semi-linear case in this context.

Our second theorem provides an upper bound on C H (T ) under some adequate assumption on the sequence (f k ) k∈N .

Theorem 1.2 Assume that f is chosen such that

I k (f ) T 0 , ∀k ∈ N * . ( 8
)
Then there exists a constant C(T 0 ) > 0, depending only on T 0 , such that

C H (T ) exp C(T 0 ) T -T 0 .
Here we use the moment method in the spirit of what was done in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF]. The idea is to use the Paley-Wiener strategy, the main difficulty is to "catch" precisely the minimal time of control, which requires a careful study and a new estimate on the multiplier of [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF]. Let us mention that in [START_REF] Ammar-Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF], the strategy used by the authors was slightly different since it was based on an idea coming from [START_REF] Schwartz | Étude des Sommes d'Exponentielles, 2ième éd[END_REF], the main drawback of the argument being that we cannot estimate precisely the constants appearing and hence Schwartz's strategy is useless in order to estimate precisely the cost of the control. Let us mention that condition ( 8) is likely far from being sharp to obtain the conclusion of Theorem 1.2 since for T 0 = 0, we obtain a different (and in general more restrictive) condition than in the usual case, where the cost of the control is always of the form exp(C/T ).

Concerning some extensions and open problems arising after this study, we can mention the following:

-For a given profile f , can we obtain precise estimates on C H (T ) for T close to T 0 ? Notably, is it possible to give a lower bound that is true for every profile f ? If this lower bound exists, is it of the form exp(C/(T -T 0 ))? -We chose in this paper to study a very particular, which is of interest because of the unexpected behavior it highlights. An interesting question would be: can we generalize the study to other cases where there exists a minimal time of control, notably in the case where the minimal time of control occurs because of the condensation of the eigenvalues as in the system presented in [2, Section 6.2]? More generally, can we extend the study in the abstract case given in [START_REF] Ammar-Khodja | Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences[END_REF]? (this is more challenging since even in the case where there is no minimal time of control, it seems to the author that there is no general results concerning the cost of fast controls)

2 Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1

In all what follows, C will always be a numerical constant independent of the parameter T . Let us fix some T > T 0 , where T 0 ∈ [0, ∞) is given by [START_REF] Coron | Control and nonlinearity[END_REF].

Let n ∈ N * to be chosen later (depending on T ). We define y 0 ∈ L 2 (0, π) as follows:

y 0 (x) := sin (nx) . (9)
One readily verifies that there exists some numerical constant C (independent on n) such that

||y 0 || L 2 (0,π) C. ( 10
)
We consider u the optimal control associated to this initial condition, which verifies by definition and thanks to estimate (10)

||u|| L 2 (0,π) C H (T )||y 0 || L 2 (0,π) CC H (T ). (11)
Thanks to the moment problem verified by any control, we obtain that for any k ∈ N we have

f k T 0 u(t) exp k 2 t dt = - π 0 sin (nx) sin (kx) dx. ( 12
)
Let us define the complex function v by

v(z) := T /2 -T /2 u t + T 2 exp(-izt)dt. (13)
Using ( 12) and ( 13), we deduce that

v(in 2 ) = - π 2f n exp - n 2 T 2 , (14)
and for every k ∈ N with k = n we have

v(ik 2 ) = 0. (15)
We deduce, using ( 13), ( 11) and the Cauchy-Schwarz inequality, that

|v(z)| exp T |Im(z)| 2 T 0 |u(t)|dt C H (T ) √ T exp T |Im(z)| 2 ||y 0 || L 2 (0,π) CC H (T ) √ T exp T |Im(z)| 2 . (16)
In what follows, we call

b k := ik 2 . ( 17
)
Using the usual representation of the functions of exponential type given for example in [15, Theorem p.56], we have, for every z such that Im(z) > 0,

ln(|v(z)|) = ∞ 1 ln |z -a l | |z -a l | + σx 2 + x 2 π R ln(|v(τ )|) |τ -z| 2 dτ,
where the a k are all the roots of v of positive imaginary part and σ is the type of v, which verifies thanks to (16) that

σ T 2 . ( 18
)
We apply this equality at point b n = in 2 , then we use (17) (remark that b n is a pure imaginary number) and ( 18) to obtain

ln(|v(b n )|) ∞ l=1 ln |b n -a l | |b n -a l | + T n 2 2 + |b n | π R ln(|f (τ )|) τ 2 + |b n | 2 dτ. ( 19
)
Let us study the right-hand side of this equality.

1. First term of the right-hand side: We study

∞ l=1 ln |b n -a l | |b n -a l | .
It is easy to prove that, if (z 1 , z 2 ) ∈ C 2 , then

|z 1 -z 2 | |z 1 -z 2 | 1 if and only if Im(z 1 )Im(z 2 ) 0.
Hence, we readily infer that

∞ l=1 ln |b n -a l | |b n -a l | 0. ( 20
)
2. Concerning the last term of the right-hand-side, an easy change of variables gives

|b n | R dτ τ 2 + |b n | 2 = π.
Hence, using the fact that τ is real and ( 16), we deduce that

|b n | π R ln |v(τ )| τ 2 + b 2 n dτ ln(CC H (T ) √ T ). (21)
Using ( 14), ( 19), ( 20) and ( 21), we deduce that

ln π 2|f n | - n 2 T 2 n 2 T 2 + ln(CC H (T ) √ T ), (22)
hence there exists a numerical constant C > 0 such that

C H (T ) C |f n | √ T exp -n 2 T . ( 23
)
Now, let us consider any positive and increasing function h : R + * → R + * such that h(x) → ∞ as x → ∞. Such a function is necessarily bijective and we call h -1 its reciprocal function. Let us consider (f n ) n∈N * ∈ l 2 (N * ) defined in such a way that

I n (f ) = T 0 + 1 h -1 (n 2 ) .
One can for example consider

f n := exp -n 2 T 0 + 1 h -1 (n 2 ) ∈ l 2 (N * ).
In this case it is clear that lim sup n→∞

I n (f ) = lim n→∞ log( 1 |fn| ) n 2
= T 0 . Then we have thanks to (23)

C H (T ) C √ T exp n 2 T 0 -T + 1 h -1 (n 2 ) . ( 24
)
Let us now explain how to choose n. We assume that T is close enough to T 0 . Now, we choose n such a way that (for example)

1 2(T -T 0 ) h -1 (n 2 ) 1 4(T -T 0 ) , (25)
which is always possible (at least for T close enough to T 0 ) since h -1 is increasing and goes to ∞ at ∞. Hence, we deduce using ( 24) and ( 25) that

C H (T ) C √ T exp (T -T 0 )h 1 4(T -T 0 )
.

One then easily obtain the desired result by choosing h in such a way that

g(x) = C exp 1 x h x 4 , i.e. h(x) = 4x log g(4x) C ,
because it is clear that if g is positive, increasing and goes to ∞ at ∞ then h is well-defined at least for large enough x (which is sufficient for our purpose), is increasing and goes to ∞ at ∞.

Proof of Theorem 1.2

In all what follows, C will always be a numerical constant independent of all parameters. We consider some time T > T 0 . We will construct our bi-orthogonal family by using the celebrated Paley-Wiener Theorem. Let us recall that T 0 is given by [START_REF] Coron | Control and nonlinearity[END_REF].

First of all, we define

F (z) := ∞ k=1 (1 + iz k 2 ) = sin(π √ -iz) π √ -iz . (26)
F will be used in what follows for the construction of the biorthogonal family to (e λ k t ) k∈N * . Now, we introduce the multiplier, which is very similar to the one studied in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF]. Let ν > 0 and δ ∈ (0, 1) some parameters. From now on we call

β := T (1 -δ) 2 . ( 27
)
We introduce

σ ν (t) := exp - ν 1 -t 2
prolonged by 0 outside (-1; 1). We call

H β (z) := C ν 1 -1 σ ν (t)e -itβz dt, (28)
where

C ν := 1/||σ ν || 1 .
Looking carefully at the proof of [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF]Lemma 4.3], we can easily deduce

1 2 e ν C ν 3 2 √ ν + 1e ν , (29)
|H β (z)| e T 2 |Im(z)| , (30)
H β (x) C νβ|x|| √ ν + 1e 3ν/4- √ νβ|x| . (31)
The main new estimate that will interest us (and that differs from what is done in [START_REF] Tenenbaum | New blow-up rates of fast controls for the Schrödinger and heat equations[END_REF]) is the following: Lemma 2.1 For any x ∈ R + and any r ∈ (1/2, 1), we have

H β (ix) C(1 - √ r)e -ν 1-r +βrx . (32)
Proof of Lemma 2.1 Let r ∈ (1/2, 1) be some parameter destined to tend to 1, let η ∈ (0, 1) and µ ∈ (0, 1) some other parameters that will be linked to r afterwards. Then, using the expression of H β given in (28), we obtain by restricting the integral over ((1 -µ)η, η) that

|H β (ix)| µηC ν e -ν 1-η 2 +βxη(1-µ) . (33)
Now, we choose η = √ r ∈ (0, 1) and µ = 1 -√ r ∈ (0, 1), so that η(1 -µ) = r. We obtain thanks to (33) and ( 29) that

H β (ix) C √ r(1 - √ r)e -ν 1-r +βrx C(1 - √ r)e -ν 1-r +βrx .
Let us know define what will be the Fourier transform of our bi-orthogonal family. We set

Φ k (z) := F (z) (z + ik 2 )F (ik 2 ) H β (z) H β (ik 2 ) . ( 34
)
Using the definition of F given in (26), it is clear that

Φ k (iλ n ) = δ k,n . (35)
Let us prove that Φ k is of exponential type T /2. This is the purpose of the next lemma.

Lemma 2.2 There exists some constant C k > 0 such that for every z ∈ C, one has

F (z) (z + ik 2 )F (ik 2 ) C k e π √ |z| . (36)
Consequently, Φ k is of exponential type T /2.

Proof of Lemma 2.2 Since

z → F (z) (z + ik 2 )F (ik 2 ) is continuous on C,
it is enough to prove inequality (36) for |z| large enough. For instance, for |z| 2k 2 , we have

|F (z)| |z + ik 2 ||F (ik 2 )| C|F (z)| Ce π √ |z| .
Using (36) together with (30) and the definition of β given in (27), we deduce that Φ k is of exponential type T /2, and the proof is complete.

Let us know give a precise estimate of z → F (z) (z+ik 2 )F (ik 2 ) on the real axis.

Lemma 2.3 For x ∈ R, one has |F (x)| |(z + ik 2 )F (ik 2 )| Ce π |x| 2 . ( 37
)
Proof of Lemma 2.3 Since x ∈ R, we have Let us know give the final estimate of our multiplier.

|F (x)| |(x + ik 2 )F (iλ k )| |F (x)| k 2 |F (ik 2 )| . ( 38
Proposition 2.1 For some well-chosen ν (depending on δ and T ),

Φ k ∈ L 1 (R) ∩ L 2 (R) and ||Φ k || L 1 (R) C √ ν + 1 1 - √ r e ( 3 4 + 1 1-r )ν-rβλ k . ( 39
)
Proof of Proposition 2.1 Putting together (37), ( 31) and (32), we obtain that for x ∈ R, we have

|Φ k (x)| C √ ν + 1 νβ|x|| 1 - √ r e π |x| 2 +( 3 4 + 1 1-r )ν- √ νβ|x|-rβλ k .
Let us now choose ν. We choose ν in such a way that

βν = 1 √ 2 + 1, i.e. ν := (1 + √ 2) 2 (1 -δ)T . (40)
We deduce that

|Φ k (x)| C √ ν + 1 |x|| 1 - √ r e ( 3 √ |x|-rβλ k .
Hence, we have that

Φ k ∈ L 1 (R) ∩ L 2 (R) and ||Φ k || L 1 (R) C √ ν + 1 1 - √ r e ( 3 4 + 1 1-r )ν-rβλ k .
Proof of Theorem 1.2. We are now able to construct the control. Using the version of the Paley-Wiener Theorem given in [START_REF] Rudin | Real and complex analysis[END_REF]Th. 19.3,p. 370], we can state that for every k ∈ N * , Φ k is the Fourier transform of a function w k ∈ L 2 (R) with compact support [-T /2, T /2]. Moreover, by construction {w k } is biorthogonal to the family {e -k 2 t } on [-T /2, T /2]. Then, one can create the control thanks to the family {h k }. Going back to expression (5), we consider a control u defined by

u(t) := - ∞ k=1 a k f k exp - T k 2 2 w k t - T 2 . ( 41
)
Let us remark that going back to the expression of T 0 given in [START_REF] Coron | Control and nonlinearity[END_REF], the expression of u is meaningful as soon as δ is small enough and r is close enough to 1(depending on T -T 0 ), thanks to ( 27) and (39), which will be assumed from now on. By construction, the corresponding solution y of (1) verifies y(T, •) ≡ 0. Moreover, one easily verifies that u ∈ C 0 ([0, T ], R). Using ( 41 Hence, using condition (8), we deduce

||u(t)|| L ∞ (0,T ) C √ ν + 1 1 - √ r e ( 3 4 + 1 1-r )ν k |a k |e k 2 (T0-rT (1-δ) 2 -T 2 ) . ( 42
)
The equation (in the variable r)

T 0 - rT (1 -δ) 2 - T 2 = - T -T 0 2
has a unique solution r 0 ∈ (0, 1) given by

r 0 := T 0 T (1 -δ) (43)
as soon as

0 < δ < T -T 0 T . ( 44
)
Going back to (42) and using the particular value of r given in (43) together with the Cauchy-Schwarz inequality we obtain that

||u(t)|| L ∞ (0,T ) C √ T -T 0 √ ν + 1 1 - √ r 0 e ( 3 4 + 1 1-r 0 )ν ( k |a k | 2 ) 1/2 . ( 45
)
We now choose δ = T -T0 2T , which verifies condition (44), so that by (43) we obtain

r 0 = 2T 0 T + T 0 . (46)
Hence, for T close enough to T 0 , taking into account (45), the definition of ν given in (40), (46), and the fact that all the terms appearing in the right-hand side of (45) in front of the exponential are at most powers of T -T 0 , we obtain

Funding

The author is partially supported by the project IFSMACS funded by the french Agence Nationale de la Recherche, 2015-2019 (Reference: ANR-15-CE40-0010).

|a k |e k 2 (

 2 ), (27) and inequality (39), we obtain||u(t)|| L ∞ (0,T ) Ik(f)-rT (1

  ||u(t)|| L ∞ (0,T ) e C(T 0 ) T -T 0 ||y 0 || H .

)

  Let us estimate |F (x)|. One more time we use that we know explicitly the form of F and the

	fact that Re	√	i = 1/	√	2 to deduce that
					|F (x)| e π |x| 2 .
	It remains us to estimate |F (iλ k )|.
	From (26), one easily infer that
					|F (ik 2 )| =	1 2k 2
	which enables us to conclude as wished thanks to (38).

+ 1 1-r )ν-


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Sedimentary Patterns of Moving Sand Dunes in Orinoca District, Bolivia

Keywords: Dune motion, Sedimentary patterns, Satellite images, Google Earth

The satellite images of Google Earth show an interesting feature of some dune fields of the region near Orinoca Canton, close to the Lake Poopo, Bolivia. They are leaving, during their motion, some sedimentary patterns, such as the coastal dunes of Brazil.

In two previous papers, we have discussed that some sand dunes are existing, which have a peculiarity with respect to the sand dunes of the deserts [START_REF] Sparavigna | Patterns of Moving Brazilian Coastal Dunes[END_REF][START_REF] Sparavigna | Analysis of the Motion of Some Brazilian Coastal Dunes[END_REF]. The peculiarity is the following: during their motion, these dunes are leaving "footprints" behind, footprints which are the sedimentary patterns due to the seasonal processes occurring to them. Remarkable examples of sedimentary patterns are those of the coastal dunes of Brazil, created under an environment rich of humidity and with heavy rainfall [START_REF] Tsoar | The effect of climate change on the mobility and stability of coastal sand dunes in Ceará State (NE Brazil)[END_REF][START_REF] Mckee | Deformational structures in Brazilian coastal dunes[END_REF][START_REF] Hilbert | Morphologic and sedimentologic patterns of active aeolian dune-fields on the east coast of Maranhão, northeast Brazil[END_REF]. As discussed in [START_REF] Hilbert | Morphologic and sedimentologic patterns of active aeolian dune-fields on the east coast of Maranhão, northeast Brazil[END_REF], the observed patterns on the soil are formed due to seasonal phenomena. During the wet season, interdune plains are flooded and the soil is reworked by intermittent drainages; during the dry season, the deposits formed by the drainages and interdune lakes become temporary internal sedimentary sources for the system [START_REF] Hilbert | Morphologic and sedimentologic patterns of active aeolian dune-fields on the east coast of Maranhão, northeast Brazil[END_REF]. Like the dunes of the desert, the coastal dunes can move, as we can easily see from the time series of satellite images. We can use Google Earth for instance, as discussed in [START_REF] Sparavigna | A study of moving sand dunes by means of satellite images[END_REF][START_REF] Sparavigna | Moving dunes on the Google Earth[END_REF][START_REF] Sparavigna | The GNU Image Manipulation Program applied to study the sand dunes[END_REF][START_REF] Sparavigna | Peruvian transverse dunes in the Google Earth images[END_REF], to observe the motion of the Brazilian coastal dunes [START_REF] Sparavigna | Patterns of Moving Brazilian Coastal Dunes[END_REF][START_REF] Sparavigna | Analysis of the Motion of Some Brazilian Coastal Dunes[END_REF]. In the given references, we have shown and compared the "footprints" these dunes are producing. Here, we will give some examples of sedimentary patterns -observed by means of the Google Earth images -produced by sand dunes that are not coastal dunes such as those we can find near the ocean, but are dunes in a quite different environment. These dunes are in the Orinoca district (cantón) (see Figure 1), in the Andamarca Municipality in the Oruro Department, a mining region in southwest Bolivia. The Orinoca District is located at 3,800 m above sea-level, on the western shore of Lake Poopo. This lake was a large saline lake located in a shallow depression in the Altiplano of Bolivia. Let us note that in [START_REF] Vv | Wikipedia[END_REF], it is told that Lake Poopo "was" a lake. "In December 2015, the former lake had completely dried up, leaving only a few marshy areas. ... Suggested causes of the current drying include drought due to climate change, as well as diversion of water for mining and agriculture". In the Figures 2 and3 we can see details of two of the dune fields in the map of Figure 1.

We can easily observe the sedimentary patterns the dunes are leaving because of seasonal effects. Supposing that each year, the dune is producing a footprint, we could deduce its motion, as in the Figure 4, using the scale in the image. For instance, the dune in the Figure 4 moved of about 14 meters per year.

For the dunes of Orinoca district, Google Earth is giving only images recorded in 2012. Therefore, we have not information on the recent behaviour of the dunes, on the effect of drought on them and on a consequent variation of their migration rates.

Let us conclude stressing the fact that not only the coastal dunes near the oceans can have sedimentary patterns. As we have shown in this paper, also the sand dunes of a quite different environment can have them. The dunes here shown are on the Altiplano, that is the Andean Plateau or Bolivian Plateau. There the climate is cool with a diurnal cycle of temperature which is very wide. However, as told in [START_REF] Vv | Wikipedia[END_REF], the Altiplano has a marked seasonal cycle of rainfall, with the rainy season concentrated between December and March. The rest of the year tends to be very dry, cool, windy and sunny. Then, it happens that during the wet season, the interdune plains can be flooded. During the dry season, these interdune spaces become temporary areas for sedimentary deposits. In this manner, with a mechanism similar to that of the coastal dunes of Brazil, footprints are formed by the dunes on the Bolivian Altiplano.

Figure 4: Using the patterns we can estimate the motion of the dune. In this case, it was of about 14 meters per year.

Figure 1 :

 1 Figure 1: Dune fields in Orinoca Canton, Bolivia.

Figure 2 .

 2 Figure 2. One of the dunes fields of Figure 1. Note the "footprints" of the dunes. The image was recorded in 2012.

Figure 3 .

 3 Figure 3. A detail of another dune fields of the Figure 2. The image was recorded in 2012.


H H Hesselink
email: henk.hesselink@nlr.nl

J M Nibourg
email: joyce.nibourg@nlr.nl

L D'estampes
email: estampes@recherche.enac.fr

P Lezaud
email: lezaud@recherche.enac.fr

Airport Capacity Forecast Short-term Forecasting of Runway Capacity

Keywords: component, airport, capacity, foreasting, runway management I

Airports expect major benefits from increasing predictability of the operation. This paper will investigate the use of forecast information to determine forecast airport capacity, which will allow airport stakeholders to optimize the use of their resources. The focus will be on forecasting runway capacity, at most airports the major factor for the overall airport capacity. The possibility to model forecast runway capacity, based on probabilistic inputs, will be investigated with a lead time up to two days. Inputs from meteorological services (wind speed, wind direction and visibility) are the major factors for determining which runways to use. Other inputs are runway availability, operational procedures and demand. The runway configuration of Amsterdam Airport Schiphol, with a complex layout of six runways, has been chosen for the evaluation. A large dataset, covering runway use of 2012 and half 2013, was used to evaluate the model, which forecasts runway configurations first and then calculates capacity forecasts. The paper will show that a high quality runway capacity forecast is feasible.

INTRODUCTION

For preparing operations at airports, plans are made by each stakeholder to ensure that operations are running smoothly. These plans concern the allocation of personnel and the use of infrastructure and other resources to cater for expected demand. In case of disrupted operations, caused by e.g. bad weather or strikes, it is important to have precise information on the available capacity so that actions can be taken in advance to ensure that aircraft and passengers experience as little as possible from the disruption and that the infrastructure and resources that are still available, will be used in an optimum way. This paper will focus on forecasting runway availability, hence forecasting runway capacity, through an automated method, based on the weather forecast and other relevant parameters. The paper will discuss the runway and capacity forecasting method and present results from an implementation of the method.

The capacity of the runway configuration depends on the lay out and visibility conditions as separation increases with lower visibility. The first step in determining runway capacity is the determination of the runway configuration that will be used. Tailwind and crosswind constraints limit the use of runways, while the configuration of parallel, dependent and crossing runways also has significant influence on capacity of the complete system. Furthermore, the expected demand of arrival and departure traffic is a factor in deciding to use a specific configuration of runways. The second step is the runway capacity forecasting, as presented in this paper, which has been set up and evaluated taking Amsterdam Airport Schiphol as example. The work presented in this paper builds on earlier research toward runway configuration forecast, as described in [START_REF] Hesselink | Probabilistic 2-Day Forecast of Runway Use[END_REF] and [START_REF] Hesselink | Environment-Aware Runway Allocation Advise System[END_REF].

To evaluate the accuracy of the models, we have used the MAPE (Mean Absolute Percent Error) criterion and examined the distribution of the errors through the use of histograms. The forecasted runway configuration and capacity has been compared with the actual runway use (configuration and capacity) at Schiphol over the period from January 2012 to June 2013.

The research question we are addressing is to find out whether it is possible to use the probabilistic weather forecast to provide a probabilistic forecast of runway capacity, with a certainty that will allow stakeholders to make well informed decisions.

II. RUNWAY CONFIGURATION CHANGE MANAGEMENT

For efficient operations at airports, it is necessary that the runway configuration does not change too often. Runway changes are costly operations; moreover ATC developments in Continuous Descent Operations (CDO), Collaborative Pre-Departure Sequencing (CPDS) and Airport Collaborative Decision Making (A-CDM) require an efficient traffic flow and predictable runway allocation. An aircraft on route to an airport will be able to better plan its landing time from early knowledge on the runway in use at the time of arrival. Depending on the runway that will be used, the aircraft may take up to ten minutes more; an unacceptable uncertainty in terms of planning of CDOs. Just as well, taxi times differ significantly from one runway to another, which has a serious impact on the planning systems in A-CDM.

The work in this paper is based on work carried out in the project Airport Capacity Forecast (ACF), carried out by NLR, ENAC and GESAC. Air traffic control bases the decision for using runways on traffic demand (one or more runways necessary), meteorological conditions (wind direction, wind speed, gust, and visibility), and the availability of runways and ILS systems. An evenly important factor is the agreement with local communities on noise limits. Several airports operate a preferential runway system: when more than one runway combination satisfies all weather criteria, the one that is most preferred with respect to noise load management will be used.

From these factors, meteorology is the most uncertain parameter as it changes continuously over time and large changes may occur in the weather in brief time periods. Air traffic control will therefore not only look at current weather conditions, but will also take into account the weather forecast for the next hours in their decision on which runways to use.

A. Runway configuration forecast

To forecast the runway capacity, it is necessary to first make a forecast of the runway configuration that will be used.

In [START_REF] Hesselink | Probabilistic 2-Day Forecast of Runway Use[END_REF] and [START_REF] Hesselink | Environment-Aware Runway Allocation Advise System[END_REF], a method for determining an optimum runway configuration has been described. Based on the direction of available runways and the wind direction, all available runways, where cross-and tailwind are not above limits, can be determined, see Figure 2. Next, an overlay with the preferential runway system and information on runway availability determines which runway configuration will be best to use. ATC will eventually take a decision based on this but also considers factors like changing wind over time and whether the calculated runway configuration is well within limits for crossand tailwind or just at the boundaries of safety.

For forecasting runway configurations, another factor plays a role: uncertainty. The most uncertain factor is the weather. We can use the weather forecast to make a forecast of the runway configuration, just as described above, but the uncertainty in the weather will have its reflection in the configuration forecast. Other uncertain factors include the start time of maintenance (depending on e.g. weather) and information on demand (leading to expected time of runway configuration changes).

For the study presented here, we will use the example of the runway system and the mode of operation of Amsterdam Airport Schiphol. The airport operates at most times several runways simultaneously. The airport's lay out is presented in Figure 1. Schiphol's preference is to use runways in segregated mode as the use of separate runways does not require special measures for separating traffic at the runways. For indicating runway combinations, in the examples used below, we will use standard runway numbers, where first arrival runway numbers will be given and then departure runway numbers. Arrivals and departures will be delimited by a slash. For example 06 36R/36L means that three runways are indicated: runways 06 and 36R are for arrivals and runway 36L for departures. When two departure runways are in use, this will be indicated as for example 06/36L 36C.

Based on traffic demand, five different peak periods can be distinguished: the arrival peak period with two arrival and one departure runways (2+1), the departure period with one arrival and two departure runways (1+2), the off-peak period (1+1), the night (1+1) and the inter-peak period (2+2). The latter may only be used in limited periods of the day to cater for handling extra demand during the switch from one peak to another and cannot be used for planning purposes. The opposite occurs as well: if during an arrival or departure peak, demand is not too high, ATC may decide to temporarily close one of the arrival or departure runways.

We have developed a model to determine a probabilistic forecast of the runway configuration that will be used. The model uses forecasted wind direction and wind speed with a given variance and forecasted visibility conditions with a given uncertainty in percentages. TABLE I 13:00 100 0 0 0 0

1 In order to make a forecast of the runway configuration that will most probably be used, a method has been designed, based on the scores in the table, together with the knowledge that some configurations are preferred over others because of noise considerations. The method incorporates non-availability of runways because of planned maintenance and availability of equipment, such as ILS. The method for determining the runway configuration has been described in [START_REF] Hesselink | Probabilistic 2-Day Forecast of Runway Use[END_REF]. In the example in TABLE I, at 15:00, the highest score is 77, and we can predict that runway configuration 18R/18L 18C will be used; while at 21:00, the airport will have changed to 18R/24 18L (a score of 94 is sufficiently high to select this combination, even though a higher score exists). Somewhere in between 15:00 and 21:00, the airport will change configuration.

B. Runway capacity forecast

TABLE II provides the capacity of runway configurations for different peak periods (inbound peak, outbound peak, off peak and night) and for different visibility conditions (Good Visibility, Marginal Visibility, Low Visibility Procedures phases A, B, C/D, and (N)UDP for (Non) Uniform Daylight Period) [START_REF]Improved Low visibility and Ceiling Forecasts at Schiphol Airport[END_REF]. The visibility conditions as used by Schiphol are given in TABLE IV. 1 Because of commercial sensitivity of these figures, the ones presented in this table are not the real values, however the order of magnitude is comparable to the actual ones. The actual table has been used in the study.

The method described below applies for all peak periods. We selected January, 2012 to June, 2013 as evaluation period. The weather forecast gives us the percentage for each visibility category (good, marginal, A, B, C/D). TABLE III shows this information for the considered period. As can be seen, the total of the columns of visibility is equal to 100 %.

All figures indicate the declared capacity, which is the capacity per hour used to specify the number of slots available for schedule coordination purposes, taking into account airport infrastructure, typical operating conditions, accepted delay and political issues [START_REF] Iata | Airport Development Reference Manual[END_REF] [START_REF] Iata | Airport Development Reference Manual[END_REF]. Declared capacity can vary throughout the day accounting for inbound or outbound peak periods, offpeak periods or night time. Actual runway performance will slightly differ from this value, as the traffic realisation depends on the traffic mix: different aircraft types on one runway will require larger inter-aircraft separation, leading to lower performance. Besides, demand can be below capacity, leading to actual performance lower than capacity. Using the following equation, we are able to compute the arrival and departure capacity figures for the considered period. A forecast per hour will be used here.

( , ) = P(G) × G ( , ) + P(M) × M ( , ) + P(A) × A ( , ) + P(B) × B ( , ) + P(C/D) × C/D ( , )
where is the estimated configuration, is the estimated peak period and for each couple of ( , ), G is the capacity of the runway configuration with good visibility, M is the capacity of the runway configuration with marginal visibility, A is the capacity of the runway configuration with low visibility procedures, phase A, B is the capacity of the runway configuration with low visibility procedures, phase B and C/D is the capacity of the runway configuration with low visibility procedures, phase C/D. TABLE V presents this computation. For each column of visibility (good, marginal,…), we compute a capacity, which is the product of the percentage of visibility (from TABLE III) and the capacity of the runway in this condition of visibility (from TABLE II). The column, named "Total", is the sum of these columns and represents the weighted forecasted capacity for arrivals and for departures. Because of the weighted values (percentage = numbers between 0 and 1), the weighted forecasted capacity is not necessary an integer value. The value the will eventually be presented on the display and used by the decision maker will need further study.

III. EVALUATION

In this section, accuracy of the forecast of the runway configuration and accuracy of the forecast of runway capacity are evaluated using statistical methods.

A. Evaluation of runway configuration forecast

In [START_REF] Belgocontrol | [END_REF], the algorithm to forecast the configuration (column "Forecasted configuration" in TABLE VI) is described. The actual configuration is the configuration chosen by the controller (column "Actual configuration" in TABLE VI).

In TABLE VII, the number 1 for "Actual configuration" (or for "Forecasted configuration") represents the most preferred configuration for each peak period. For example, "Configuration 1" is 06 36R/36L for the departure peak, 06/36L 36C for the arrival peak, 06/36L for the off-peak and 06/36L for the night period. The table therefore contains the union of these four possible runway configurations.

TABLE VII shows the differences between the forecasted and the actual configuration in a contingency table, which it displays the frequency distribution of two qualitative variables. In the table, the non-diagonal numbers (every number except the grey-marked numbers) are non-matching forecasts and the "Actual configuration" is the one that is observed in the operation from January 2012 to June 2013. For example, the number 2741 is the number of good forecasts for the first configuration (actual configuration = forecasted configuration = 1); the number 134 indicates the number where the first configuration was forecasted, while the second configuration was actually used (forecasted configuration=1; actual configuration=2).

We can see in the TABLE VII that for the forecasted configuration or for the actual configuration, the first two preference configurations (1 and 2) represent a large majority (7914 for the forecasted and 6941 for the actual). ATC is able to use the first and second preference from the preferential system almost 70% of the time. We can compute for qualitative variables an equivalent to the Mean Square Error (MSE) to monitor the accuracy. For example, we can compute the score = number of good forecasts divide by total number of observations (the number of good forecasts is equal to the sum of figures in the diagonal of the Table VII, the grey cells). Score = 6909/9120 = 0.758.

(1)

This score seems acceptable, since it means that three times out of four, the configuration is forecasted correctly.

We can compute the different errors in the forecasts, as illustrated in TABLE VIII and we can observe that the higher preferences have smaller errors. The reason for the smaller error in the 7 th runway configuration is an operational one. In the 7 th configuration, in some of the periods, a western configuration is specified, which can be forecasted relatively easily with western wind. Higher other configurations are north or south.

B. Forecasted capacity vs actual capacity

To evaluate the capacity forecast, we need the actual capacity, which is computed from the observed configuration and the observed visibility condition (good, marginal…). This information is available, as we are monitoring the situation over 2012 and the first half of 2013. From the weather observation and the information from the airport on the actual runway configurations over this period, we can use TABLE IX as an indication of the realised capacity over the one and a half year.

With the values of TABLE IX, we can draw for arrival and departure the histograms of the errors: 'actual capacity' -'forecasted capacity' (for each hour and each date), see Figure 3. We can make some remarks:

• The peak in the histogram at value 0 indicates that the forecast can be made with high quality.

• The cumulative distribution function does not appear to be a Gaussian distribution, because of the second, smaller peak at around minus 30. This bimodality, where the forecasted capacity is larger than the actual capacity, could be explained by the choice of the controller to use two runways instead of three, during an arrival or departure peak period.

To examine the accuracy of the forecast, different criteria exist. The Mean Square Error is the most commonly used but it is not well-adapted to our study. In fact, a runway capacity with an error of 5 aircraft compared to the realised capacity of 10 aircraft per hour has more impact than an error of 5 aircraft with a capacity of 70 aircraft per hour. That is why we decide to divide the error by the actual capacity to give more emphasis to the error of 5 out of 10. For this reason, we will compute the Mean Absolute Percent Error (MAPE), which is the mean of these values (|actual-forecasted| / actual).

We have the following values: MAPE (arrival) = 12.88 % MAPE (departure) = 15.32 %

C. Discussion

This section will discuss the results of the runway configuration forecast and the capacity forecast. Score of the forecast for runway configuration is 0.758, while the MAPE for the forecast for runway capacity is 12.88% for the arrival and 15.32% for the departure capacity. These figures are considered sufficiently accurate to use in operation. Surprisingly, capacity scores are higher than those for the runway configuration, while the capacity forecast is based on the configuration figures. This seems contradictory. The reason is that some different runway configurations provide (almost) the same capacity, so that a small error in the determination of the runway configuration may have no effect on the capacity figure.

Just as well, we can note that an erroneous forecast of the runway configuration does not mean that all runways in the configuration have been forecasted wrongly. Some runway configurations have overlapping runways, e.g. the inbound configuration 18R 18C / 18L uses two landing runways: 18R and 18C and one runway for take-off: 18L. A similar configuration is 18R 18C / 24, with the same arrival configuration but another departure runway; in the situation where one of the configurations has been forecasted, but the other was actually used, we have considered the runway configuration to be forecasted wrongly, even though two of the three runways have been forecasted correctly and all inbound runways have been forecasted correctly. The resulting capacity for both runway configurations shows indeed the same inbound capacity (68 movements per hour) and almost similar outbound capacity (40 for the three parallel runways and 35 when using runway 24, because of the slight dependency between the runways). This motivates why the runway configuration forecast can be off, while capacity forecast can be correct, after all.

Several reasons for erroneous runway configuration forecasts have been examined:

• Weather forecast accuracy.

• Methodological issue.

• ATC makes another decision.

Weather forecast: From [START_REF]Improved Low visibility and Ceiling Forecasts at Schiphol Airport[END_REF], it can be seen that specifically visibility is difficult to forecast and about 10% of visibility forecast does not classify the forecast in the correct category.

Through use of our probabilistic approach, the margin in error reduces but still exists. Analysis has shown that the impact of weather forecast is in the magnitude of 5%.

Methodology:

The method used can be improved and the analysis given above will help to further understand where changes to the method can be made. It must be noted again here that in many cases the arrival configuration has been forecasted correctly, while the departure is not, in which case the runway configuration has been considered to be forecasted incorrectly. Analysis has shown that some wind directions show better results, specifically a northern or southern wind will demonstrate a highly accurate runway configuration forecast, while a north-eastern or south-western wind performs significantly worse. The night period at Schiphol is more restrictive, leading to very accurate forecasts.

ATC:

The air traffic control supervisor eventually will decide on what runway configuration to use. He will consider more aspects than is possible in our forecast, like local phenomena in the weather (rain or thunderstorms) and incidents on runways or taxiways. Just as well, when reconfiguring the airport from northern to southern runway use, sometimes some intermediate configurations are used for a brief period of time. These cannot be forecasted.

Further work towards runway capacity forecast will focus on the elements described above to improve quality of the forecasts.

One element already mentioned to be further studied, is the presentation of the information. Operators will either have to get acquainted to the probabilistic presentation of forecast data or will have to be presented with figures they can apply directly. In the latter case, the figures will have to be rounded off to the nearest integer value that represents one possible runway configuration with a corresponding capacity figure. Obviously, information will get lost but this will better fit to the operation.

The presentation will largely depend on the environment in which the forecasting will be integrated. In a larger operating environment, such as Total Airport Management (TAM), the forecasting element will become part of the cooperation and planning systems. Methods and presentation will have to connect to that of TAM.

In a larger context, the runway capacity forecast may become part of a larger airport capacity forecast system. Although at most airports, the runway system mostly represents the capacity of the airport as a whole as well, other elements may become bottlenecks in specific situations. For example, at some airports, the taxiway system becomes the capacity bottleneck in low visibility operations. Other major factors in airport capacity are de-icing operations and snow removal. At landside, the passenger's security passage is a capacity issue that will benefit from good forecast information. In fact, de-icing and security passage are already under investigation by the consortium.

IV. RELATED WORK

Most work on runway capacity is concerned with more efficient use of existing infrastructure, while enhancing capacity. Examples of studies concern those for Arrival Management, Departure Management, and A-CDM. When forecasted runway capacity is studied, this is mostly for airport expansions or for periods in which maintenance is scheduled. Examples of studies concern those with fast time simulation studies. As the scope of these studies is different from the capacity forecast that we present in this project, their relevance to the work described is considered limited.

The "other side" of capacity forecasting is considered by us to be demand forecasting. This type of forecasting takes the current situation as basis and will extend the situation through prediction or simulation towards a moment in the future, either long term (months or years) or short term (minutes to hours). Contrary to capacity forecasting, demand forecasting does not consider resource availability to be the basis. When both are put together, Airport Capacity and Demand Balancing, is considered a powerful mechanism, where future capacity is matched with future demand in an optimum manner. Cooperation on planning systems is one of the topics in Total Airport Management and preparation of planning, e.g. through capacity forecast will be one element in the complete chain of processes [START_REF] Günter | Total Airport Management -Operational Concept And Logical Architecture[END_REF].

Runway configuration advice is given to controllers amongst others at Amsterdam Airport Schiphol, Basle Euro Airport and Brussels Zaventem airport [START_REF] Hesselink | Environment-Aware Runway Allocation Advise System[END_REF][3], through systems that present the air traffic control supervisor the most appropriate runway configuration to use or show what runways cannot be used, because of exceeding wind limits. Runway configuration forecasting is performed at Frankfurt airport, where the environment is informed on expected eastern or western runway use for the following days [8].

V. CONCLUSION

We have presented a runway configuration forecasting model, based on meteorological information, and evaluated the model with recorded data over 2012 and 2013. Based on the forecasted runway configuration, a capacity forecast has been made and evaluated as well. Runway capacity is at most airport the main factor in airport capacity.

The runway configuration forecast has been proven to be possible with high accuracy, whereas the capacity forecast can be modelled with even higher accuracy. Although capacity is based on the runway configuration, the fact that different configurations allow the same capacity causes this to have higher accuracy. Just as well, some runway configurations consist of partly the same runways and where the configurations could be forecasted wrongly, the capacity of the runways will yield a reasonably correct figure.

The work described here will be beneficial to airports and airspace users, who will be able to better plan their resources. It will bring automation of airport operations to a higher level and enable further automated planning of the operations, where stakeholders will get a role to oversee the operations at a higher level than is currently the case. A higher predictability of the operations can be expected as the adherence levels to target times will lead to increased predictability for runway planning functions in AMAN, CPDS, A-DCB and A-CDM.

Further work is ongoing to improve quality of the runway configuration forecasting and to apply the methodology of capacity forecasting to other airport planning processes, such as planning for de-icing and in-terminal planning of passenger processes. The human machine interface will need to be designed and integrated in a larger set of related functions such as for A-DCB and A-CDM.

Figure 2 .Figure 1 .

 21 Figure 2. Runway crosswind and tailwind

  shows the different possible runway configurations in the rows, together with their forecasted scores of use in the columns. The columns indicate the lead time of the forecasts. The scores in the cells indicate per runway configuration the possibility of being able to use within the given variance of the wind and visibility limits. Colours are a value indication: from yellow to indicate a high score to red to indicate a low score. For example, the score of the runway configuration 18R/18L 18C is equal to 77 %, which indicates there is a 77 % possibility to use this runway without having to change runway configuration in the next hour.

Figure 3 .

 3 Figure 3. Histograms of errors for arrival capacity (top) and for departure capacity (bottom)

TABLE I .

 I RUNWAY CONFIGURATION SCORES (FOR DEPARTURE PEAK, ONLY)

TABLE II .

 II PART OF CAPACITY OF CONFIGURATIONS AVAILABLE AT AMSTERDAM AIRPORT SCHIPHOL 1 . AN EMPTY CELL MEANS THAT THE CONFIGURATION IS NOT AVAILABLE UNDER THE SPECIFIED CONDITIONS.

	Runway

configuration Arrival Departure Good Visibility Marginal Visibility Low Visibility Procedures Good Visibility Marginal Visibility Low Visibility Procedures Peak period Arrival Departure UDP NUPD Phase A Phase B Phase C/D UDP NUDP Phase A Phase B Phase C/D

		06+36R	36L	68	65					40	40
		18R+18C 24	68	65	65	55	45	35	37	30	30	25	25	20
	Arrival peak	36R+36C 36L	65	60	65	55	45		40	40	35	25	25
		18R+18C 18L	68	68	68	55			40	40	35	25
		...
		06	36L+36C 38	38					70	70
		18R	24+18L	38	38	38	25			74	74	70	50
	Departure peak	36R	36L+36C 38	38					70	70
		18R	18L+18C 38	38					70	70
		...
										TABLE III. PERCENTAGES OF VISIBILITY FORECASTS
	TABLE IV. VISIBILITY CONDITIONS USED BY SCHIPHOL
	Classification Visibility (VIS/RVR)	Ceiling
	Good		>5 km VIS		and > 1000 ft
	Marginal	1.5 -5 km VIS	or	300 -1000 ft
	LVP phase A	550 -1500 m RVR	or	200 -300 ft
	LVP phase B	350 -550 RVR	or	< 200 ft
	LVP phase C	200 -350 RVR
	LVP phase D	< 200 RVR

Date Hour Visibility Good Marginal LVP A LVP B LVP C/D

	10/01/2012	5:00 70	15	15	0	0
	10/01/2012	6:00 65	15	15	5	0
	10/01/2012	7:00 65	30	5	0	0
	…	…	…	…	…	…	…
	18/07/2012	22:00 70	30	0	0	0
	18/07/2012	23:00 70	30	0	0	0
	19/07/2012	0:00 60	40	0	0	0
	…		…	…	…	…	…
	30/06/2013	11:00 100	0	0	0	0
	30/06/2013	12:00 100	0	0	0	0
	30/06/2013

TABLE V .

 V ARRIVAL AND DEPARTURE CAPACITY

			Arrival							Departure
	Date	Hour	Good Visibility Marginal Visibility UDP NUDP	Low Visibility Phase A Phase B Phase C	Total	Good Visibility Marginal Visibility UDP NUDP	Low Visibility Phase A Phase B Phase C	Total
	10/01/2012 5:00 0	16,8	3,6		3,6	0	0	24	0	17,5	3,8	3	0	0	24,3
	10/01/2012 6:00 0	43,6	10,1		8,4	2,2	0	64,2 0	20,8	4,8	3,6	1,3	0	30,5
	10/01/2012 7:00 0	43,6	20,1		2,8	0	0	66,5 0	20,8	9,6	1,2	0	0	31,6
	…	…	…	…	…		…	…	…	…	…	…	…	…	…	…	…
	18/07/2012 22:00 16,8 0	7,2		0	0	0	24	17,5 0	7,5	0	0	0	25
	18/07/2012 23:00 0	16,8	7,2		0	0	0	24	0	17,5	7,5	0	0	0	25
	19/07/2012 0:00 0	14,4	9,6		0	0	0	24	0	15	10	0	0	0	25
	…	…	…	…	…		…	…	…	…	…	…	…	…	…	…	…
	30/06/2013 11:00 68	0	0		0	0	0	68	37	0	0	0	0	0	37
	30/06/2013 12:00 38	0	0		0	0	0	38	74	0	0	0	0	0	74
	30/06/2013 13:00 68	0	0		0	0	0	68	37	0	0	0	0	0	37
	TABLE VI. FORECASTED AND ACTUAL CONFIGURATION
	Date		Hour	Forecasted configuration	Actual configuration
	10/01/2012 5:00	18R/24		18R/24
	10/01/2012 6:00	18R 18C/24	18R/24
	10/01/2012 7:00	18R 18C/24	18R 18C/24
	…		…
	18/07/2012 18:00 27/24 18L	18R/24
	18/07/2012 19:00 18R/24 18L	18R/24 18L
	18/07/2012 20:00 18R/18L		18R/24
	…		…
	30/06/2013 11:00 18R 18C/24	18R 18C/24
	30/06/2013 12:00 18R/24 18L	18R/24
	30/06/2013 13:00 18R 18C/24	18R 18C/24

TABLE VII .

 VII CONTINGENCY TABLE OF FORECASTED/ACTUAL CONFIGURATION

			Actual configuration
			1	2	3	4	5	6	7	8	...
		1	2741	134 149	59 135 64	10 27
		2	110 3328	34 276 165 43 101	9
		3	107	6 183	2	10 81	10 25
	Forecasted configuration	4 5 6	10 4 18	223 36 3	31 293 0 0 1 4	13 98 4 59 4 2	19 5 0	1 0 0
		7	1	189	8	1	0	1 205	9
		8	1	30	2	0	0 25	9	2

...

TABLE VIII .

 VIII ERRORS OF POOR FORECASTS

				TABLE IX. PREDICTED AND ACTUAL CAPACITY
	Actual configuration	Error	Date	Hour	Arrival Forecasted capacity	Actual capacity	Departure Forecasted capacity	Actual capacity
	1	1 -2741 / 2992 = 0.08	10/01/2012 5:00	24	24	24,3	25
	2	1 -3328 / 3949 = 0.16	10/01/2012 6:00 10/01/2012 7:00	64,2 66,5	38 68	30,5 31,6	40 37
	3	1 -183 / 408 = 0.55	… 18/07/2012 18:00 32 …	38	71,8	40
	4	1 -293 / 635 = 0.54	18/07/2012 19:00 38	38	73,8	74
	5	1 -98 / 425 = 0.77	18/07/2012 20:00 38 … …	38	38,5	40
	6	1 -59 / 279 = 0.79	30/06/2013 11:00 68	68	37	37
			30/06/2013 12:00 38	38	74	40
	7	1 -205 / 359 = 0.43	30/06/2013 13:00 68	68	37	37
	8	1 -2 / 73 = 0.97
	...

ACKNOWLEDGMENT

This work is co-financed by EUROCONTROL acting on behalf of the SESAR Joint Undertaking (the SJU) and the EUROPEAN UNION as part of Work Package E in the SESAR Programme. Opinions expressed in this work reflect the authors' views only and EUROCONTROL and/or the SJU shall not be considered liable for them or for any use that may be made of the information contained herein.

We would like to thank the people at GESAC for their support to the modelling for Airport Capacity Forecast. We also thank Stéphane Catteau for his help and advice.


Laurence Cholvy
email: laurence.cholvy@onera.fr

Influence-Based Opinion Diffusion (Extended Abstract)

Keywords:

INTRODUCTION

Understanding the dynamics of opinion among agents is an important question which has recently received large attention in the community of autonomous agents and multiagent systems [START_REF] Pryymak | Efficient Opinion Sharing in Large Decentralised Teams[END_REF][START_REF] Crawford | Opposites Repel: The Effect of Incorporating Repulsion on Opinion Dynamics in the Bounded Confidence Model[END_REF][START_REF] Chatterjee | Predicting Migration and Opinion Adoption Patterns in Agent Communities[END_REF][START_REF] Chau | Social Judgment theory based model on opinion formation, polarization and evolution[END_REF][START_REF] Tsang | Opinion dynamics of skeptical agents[END_REF][START_REF] Christoff | A logic for diffusion in social networks[END_REF][START_REF] Grandi | Propositional Opinion Diffusion[END_REF][START_REF] Schwind | Belief Revision Games[END_REF]. It requires to model several parameters such as the different relations which exist between agents, the opinions agents have and the way they make them evolve. This present work inspires from [START_REF] Schwind | Belief Revision Games[END_REF] and propose a formal model of opinion diffusion in a population of agents by assuming that: (i) opinions are propositional formulas or, equivalently, sets of propositional interpretations; (ii) the population of agents is structured according to a binary relation of influence which relates two agents when one influences the other; (iii) any agent orders its influencers (i.e. agents which influence it) according to the strength of the influence relation; (iv) any agent changes its opinion by merging the opinions of its influencers, from the most influential one to the least; (v) there is a special formula, called integrity constraint which expresses something true in the world and which has to be taken into account by the merging operator.

INFLUENCE-BASED BELIEF REVISION GAMES

Consider a finite propositional language L. If ϕ is a formula of L, M od(ϕ) denotes the set of models of ϕ i.e., the set of interpretations in which ϕ is true. A multi-set of formulas {ϕ1, ..., ϕn} equipped with a total order ≺ s.t. ϕi ≺ ϕi+1 (i = 1...n -1) is called an ordered multi-set of formulas and denoted ϕ1 ≺ ... ≺ ϕn. definition 1 (Importance-Based Merging Operator). An Importance-Based Merging Operator, is a function ∆, which associates a formula µ and a non empty ordered multiset of consistent formulas ϕ1 ≺ ... ≺ ϕn with a formula denoted ∆µ(ϕ1 ≺ ... ≺ ϕn) so that: with: • [D(w, ϕ1), ..., D(w, ϕn)] is a vector whose k th element is D(w, ϕ k )

• w ≤ d,
• D(w, ϕ) = min w ∈M od(ϕ) d(w, w ), w and w being two interpretations and d is a pseudo-distance between interpretations.

• ≤ lex is a lexicographic comparison of vectors of reals defined by: [v1, ..

.vn] ≤ lex [v 1 , ...v n ] iff (i) ∀k v k = v k or (ii) ∃k v k < v k and ∀j < k vj = v j
By definition, Importance-Based Merging Operators are Distance-Based Merging Operators [START_REF] Konieczny | Marquis: DA2 merging operators[END_REF].

In the following, we introduce Influence-Based Belief Revision Games and Influence-Based Belief Sequences. These definitions are adapted from [START_REF] Schwind | Belief Revision Games[END_REF]. definition 2 (Influence-Based Belief Revision Game). An Influence-Based Belief Revision Game (IBRG) is a quadruplet G = (A, µ, B, Inf ) where: A = {1, ..., n} is a finite set of agents; µ is a consistent formula of L; B is a function which associates any agent i of A with a consistent formula of L denoted for short Bi such that Bi |= µ; Inf is a function which associates any agent i of A with a nonempty set of agents {i1, ...in i } equipped with a total order ≺i s.t. i k ≺i i k+1 for i = 1...(ni -1). For short, we denote

Inf (i) = {i1 ≺i ... ≺i in i }.
A is the finite set of agents. The formula µ represents the information which is true in the world. It is called integrity constraint. For any agent i, the formula Bi represents its initial beliefs. It is called the initial belief state of i. We assume that agents are rational and thus that Bi is consistent and satisfies the integrity constraint µ. For any agent i, Inf (i) is the non-empty set of agents i considers as influent i.e., i considers that its own opinion is influenced by the opinions of agents in Inf (i). With the total order ≺i, i ranks its influencers according to their degree of influence: for any agents j and k in Inf (i), j ≺i k means that, according to i, its own opinion is more influenced by j's opinion than by k's opinion. Notice that i may or may not belong to Inf (i). definition 3 (Influence-Based Belief sequence). Consider an (IBRG) G = (A, µ, B, Inf ) and an agent i ∈ A with Inf (i) = {i1 ≺i ... ≺i in i }. The Influence-Based Belief Sequence of i, denoted (B s i ) s∈N , is defined by:

B 0 i = Bi and ∀s ∈ N, B s+1 i = ∆µ(B s i 1 ≺ ... ≺ B s in i ).
The Influence-Based Belief Sequence (or Belief Sequence for short) of agent i, (B s i ) s∈N , represents i's belief state all along the game: B 0 i is the initial belief state of i; B s i is the belief state of i after s moves i.e., the opinion of i after s steps. The evolution of i's opinion is done according to the importance-based merging operator ∆µ: i's opinion at step s is the result of ∆µ applied to the ordered multi-set of opinions:

B s i 1 ≺ ... ≺ B s in i .
proposition 1. In an (IBRG), the belief sequence of any agent is cyclic i.e., the belief sequence of any agent i is characterized by an initial segment B 0 i ...B b-1 i and a belief cycle, Cyc(Bi) = B b i ...B e i which will be repeated up as infinitum.

definition 4. An (IBRG) G = (A, µ, B, Inf ) converges iff ∀i ∈ A | Cyc(Bi) |= 1.
definition 5. Let G = (A, µ, B, Inf ) be an (IBRG) and ϕ a formula of L. ϕ is accepted by agent i of A iff for all B s i ∈ Cyc(Bi), we have B s i |= ϕ. ϕ is unanimously accepted in G iff ϕ is accepted by all i in A. ϕ is majoritary accepted if the number of agents who accept it is strictly greater than the number of agents who do not.

Let us now introduce three more definitions related to (IBRG). The first one is adapted from [START_REF] Grandi | Propositional Opinion Diffusion[END_REF], the two others are original. definition 6 (DAG with loops). From any (IBRG) G = (A, µ, B, Inf ), we can build a graph whose nodes are agents of A and edges are i → j iff i ∈ Inf (j). We say that G is a DAG with loops if this graph is a directed graph where the only permitted cycles are of type i → i. definition 7 (Dogmatic agent). Consider the (IBRG) G = (A, µ, B, Inf ) and i ∈ A. i is a dogmatic agent iff

Inf (i) = {i}.
An agent is dogmatic when it is not influenced by other agents. As a consequence, a dogmatic agent i never changes its opinion i.e., ∀s ≥ 0

B s i = B 0 i .
definition 8 (Sphere of Influence of an agent).

Let G = (A, µ, B, Inf ) an IBRG. Let i ∈ A. Sphere(i) = {j : Inf (j) = {i ≺ ...}} ∪ {j k : ∃j0...j k-1 ∀m = 1...(k - 1) Inf (jm) = {jm-1 ≺ ..

.} and j0 = i}

The following proposition identifies a case where IBRG converge.

proposition 2. DAG with loops converges.

The following proposition identifies some sufficient conditions for accepting a given formula. proposition 3. Let G be an IBRG and i a dogmatic agent. If Sphere(i) = A then Bi is unanimously accepted. If | Sphere(i) |> |A| 2 then Bi is majoritary accepted. Obviously, if an agent is the most infuential one for any agent in the population, then its initial opinion is unanimously accepted. If an agent is the most infuential one for more than a half population, then its initial opinion is majoritary accepted.

CONCLUDING REMARKS

This work suggested to model influence-based diffusion of opinion with Influenced-Based Belief Revision Games. It also presented some conditions on the Influence-Based Revision Game which guarantee the convergence of the diffusion process or the emergence of a majoritary or unanimous opinion. This work can be extended in several directions.

First, it could be interesting to change our assumption about the way agents orders their influencers. In this present work, we have assumed that agents order their influencers according to a strict order. Thus, two agents cannot be considered as equally influential for a given agent. Changing this assumption will lead us to consider preorders instead of orders. Defining a merging operator which takes into account preorders is an open question.

Secondly, an interesting extension is to consider that there is no global integrity constraint but only some local ones, shared by the agents who, in some way, belong to a commn community. Studying the acceptability in such a context is challenging.

Finally, to make this work more realistic, we have to consider the fact that an agent may influence another one not for any type of information but for only some kind of information, related to a given topic. Extending this work to topic-dependent influence is the next step of our research.


Vladimir Zhukov

Timkov S V Zhukov

The Office of

Keywords: fractal, structure, Universe, expansion, rotation, space, dark energy, dark matter, the Hubble constant, Planck's constant, universal proportions of Planck . В.Ф. Тимков, Аппарат Совета национальной безопасности и обороны Украины С.В. Тимков, В.А. Жуков, структура, Вселенная, расширение, вращение, пространство

published or not. The documents may come

Introduction

Consider some modern ideas about the structure of the Universe. Professor Martin Rees (Institute of Astronomy, University of Cambridge) argues that the Universe has a cellular structure , each such cell in the trillions of times smaller than an atom [1,2,3]. Professor Michael Longo (University of Michigan in Ann Arbor), having studied [4,5] in the framework of Sloan Digital Sky Survey (SDSS) database project SDSS DR5, which contains about 40,000 galaxies ( of them -more than 15,000 spiral galaxies ) , for where the value of redshift z <0,04, came to the conclusion that left twisted spiral galaxies is much larger than spiral galaxies swirling right. Conclusions of professor Michael Longo confirmed a group of scientists led by Professor Lior Shamir (Lawrence Technological University) [6] . Was investigated about 250 thousand spiral galaxies , for which the value of redshift z <0,3. Professor Lior Shamir also found that galaxies left more than right. Symmetry breaking between the right-and left -twisted spiral galaxies is about seven percent, but the probability that is a cosmic accident is very low -claims Professor Michael Longo. The results of research professors Michael Longo and Lior Shamir contradict the notion that the Universe is homogeneous and symmetric. Scientists believe that the asymmetry of the Universe emerged in the Big Bang at the expense of the initial rotation of the Universe space counterclockwise. This means, firstly , that the space of the Universe could have existed before the Big Bang , and secondly , as claims Professor Michael Longo universe revolves now. The rotation of the space of the universe is also supported by research Professors S.-C. Su and M.-C. Chu (Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China) [7]. In 2006, Professor Laura Mersini-Houghton (University of North Carolina at Chapel Hill), Richard Holman (Carnegie Mellon University) and Tomo Takahashi (Saga University), in joint papers [8,9] , predicted the existence of anomalies relic microwave background of the Universe and suggested that the cause of these anomalies may be an impact on observable of the Universe other of the Universes, which are located close by. On the possible impact on observable of the Universe other of the Universes expressed [10] Dr. Stephen M. Feeney (University College London), Dr. Matthew C. Johnson (Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada), Dr. Daniel J. Mortlock (California Institute of Technology, Pasadena), Dr. Hiranya V. (Astrophysics Group, Imperial College London) and Dr. Peiris (Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge). Examining data on the microwave background of the Universe , which are obtained based on the results of seven years of observations WMAP, the researchers found four spots with abnormal values CMB , which scientists believe are the result of collisions ( close approach ) of the observable Universe and other Universes . Professor Douglas Scott (University of British Columbia), investigating anomalies CMB [11,12] concluded that the observed Universe has a spherical shape. Professor Roger Penrose (Oxford University) together with Professor Vahe Gurzadyan (Yerevan Physics Institute in Armenia) conducted a study of the cosmic background radiation anomalies [13]. Professors claim to have found evidence of the existence of the Universe before the Big Bang, Proposes to consolidate all of the above findings of theoretical and experimental studies in one hypothesis -a fractal structure of the Universe. The Universe is an infinite aggregate of spatial hierarchic fractals of a spherical shape. Spatial hierarchy of fractals means that the smaller by size (and by hierarchy) fractals lie within the greater fractals, i.e., fractals are nested within each other. A certain linear dimension of the space length makes the similarity basis of the Universe fractals. This dimension tends to zero, however our knowledge about Nature does not still allow us of making analysis with the use of the linear dimensions which are smaller than the Planck dimension l p [18]:

-35 1.616199 10 p lm  . Let us consider that the Planck dimension is the universal constant for the all level fractals of the Universe. Therefore, the basis of the spatial fractal structure of the Universe is selected as follows:

-for vacuum -the Planck cube. Its volume equals: Let us consider the most accessible for observation electromagnetic fractal of the Universeit is the observable Universe . We assume that it is shaped as a limited-size sphere. The sphere rotates in a single plane, as a minimum. Electromagnetic fractal of the Universe is located within the gravitational fractal of the Universe. In its turn electromagnetic fractal is senior in relation to the atomic and nuclear fractals.

The assumption that the sphere of the electromagnetic fractal of the Universe rotates makes useless the hypotheses about the expanding and accelerating Universe and dark energy. Really. Any theory that is not confirmed by experience, or on the basis of which it is possible to make ambiguous judgments and conclusions, is a hypothesis. In terms of conclusions about the character and nature of the red shift from of distant objects of observable Universe, and depending on their speed from the Hubble constant, both hypotheses: expanding with an acceleration of the Universe and the rotating Universe [14] is equal. Their fundamental difference -in the choice of the reference system and the conditions of observation. The hypothesis of the existence of dark energy is incorrect due to incorrectly chosen frame of reference, and not properly defined the conditions of observation. Apologists for dark energy to repeat the mistakes that once made Archimedes and Ptolemy, and who chose the reference system with the Earth at the center of the Universe, and then Copernicus, who replaced in the reference frame the Earth to the Sun. Really. Solution of Einstein's equations of general relativity is carried out in the reference system , which is linked to the observable Universe, and which is declared as a global and absolute. In accordance with the choice of the reference system [14] of the observable Universe becomes the center of the whole Universe -Mega Universe. Further, by solving the equations of general relativity reference system actually breaks down into two parts: one of them -relatively immobile, and the second is removed rapidly from the first (if the chosen of reference system is global -the question is -where is removed?). The source of this accelerated movement of the second reference system is declared negative vacuum pressure (lambda constant in Einstein's equations of general relativity): -PΔV, where (-P) -is a negative vacuum pressure, ΔV -this volume of space, increased under negative vacuum pressure. However, this formula is valid only for closed systems. That is, systems in which the vacuum pressure inside and outside are different, in other words in the formula -PΔV instinctively understood that the vacuum pressure in the observable Universe, and the vacuum pressure in the rest of the Universe, vary. This violates the requirement that the lambda -it is constant for the entire Universe. The observable Universe is not closed, it is limited. For a limited system to change the position of their boundaries (eg, extension), you must have a vacuum pressure differential ΔP, that is, must: -ΔPΔV. The differential vacuum pressure at the limited observable Universe means that the lambda -it is not a constant.

Conclusion: the negative vacuum pressure can not be the source of dark energy.

The hypothesis [14] of a rotating space of the observable Universe, i.e. her of electromagnetic of fractal of the Universe, is considered in a global of reference system, which is associated with Mega Universei.e. with her of gravitational of fractal of the Universe, and that includes the observable Universe as a part of it. Currently, the expansion of the observable Universe since the Big Bang is finished. The observable Universe is limited. Expansion of the observable Universe ended up at a time when its gravity factor (vacuum pressure) was equal gravity factor (vacuum pressure) the part of the Mega Universe in which is located the observable Universe. Equal gravity factors derived from the opposite hypothesis of a rotating Universe, which admitted the existence of dark energy force and the lack of rotation of the observable Universe space, and where it was found that the gravity factor observable Universe has become equal to the acceleration of dark energy force, and actually gravity factor that part Mega Universe in which is located the observable Universe. Conclusion: the assumption of the rotation of the observable Universe of space allows us to solve the question of the sources as dark energy and dark matter.

Further evidence of the lack of dark energy is the new law of physics [15]: "Universal Plank Proportions" (UPP), which establishes significant, sustained and repeated relations and connections between the three self-sufficient characteristics of any an object of the observable Universe (including itself), which is endowed with the mass, namely, the actual mass m, radius of curvature space S, a time delay t dm signal light.

According to this law, in the observable Universe anybody having mass , m creates a gravitational field that curves the surrounding space with a radius of curvature S (actually S -is the length of a gravitational wave) and introducing into t at a distance R from each other, the law of gravity is given by: .

2 2 1 2 2 2 1 2 2 1 R t t c F R S S F R m m G F dm dm p p   
Where G -is the gravitational constant, c -is the speed of light in vacuum.

Before this law, we know only one characteristic of the object -its mass. On the basis of the law UPP openly and confirmed by experiments a new property of matterits gravity-electromagnetic resonance (GER) [15,16]. It is known that today in astrophysics is not possible to accurately determine the mass of distant objects observable Universe, and using the law UPP and the GER we can accurately calculate their mass. It is sufficient to measure the frequency of the envelope of the radiation spectrum [16]. On the basis of the UPP and GER, we can create new sources of energy. The versatility, efficiency, stability and repeatability of Planck of proportions also lies in the fact that for the gravitational constant G, of constants of Planck: l p , m p , t p , of the relevant characteristics of the observable Universe: R Ue , M Ue , T Ue , as well as of the relevant characteristics of any her body: S, m, t dm is true [17]:

3 33 11 3 1 2 2 2 2
6.67408 10 .

p Ue

p p Ue Ue dm l RS G m kg s m t M T mt        
On the basis of the Planck of proportions, we can also go to the geometrical system of units in physics, that additionally underscores their versatility.

Most dynamic physical processes have two basic states -steady state and transient state. In the steady state (when there is no movement of the masses with an acceleration when there is no merging of massive bodies) the gravitational field of the body, which has a mass, bends space around itself. This forms a kind of standing gravitational waves that are can be found only indirect methods, for example, based on gravitational-electromagnetic resonance. The LIGO experiment observed the process of transition, which occurred as a result of deformation of surrounding curved space formed to move in space gravitational wave. The parameters of curved space in steady state (standing gravitational waves) and the parameters of gravitational waves in the transition process to the same bodies are different.

As the fractal hierarchy grows, the Universe does expand but that expansion is fractal and stationary.

Evaluation of main spatial and energy parameters of the electromagnetic fractal of the Universe

The observable Universe is electromagnetic of fractal of the Universe.

         (1)
According to Planck's mission [19] Hubble constant H is equal to:

Mpc s km H / ) ( 77 . 0 80 . 67 1 1    or . x10 197248345 . 2 1 -18   s H (2)
In [14,15,16,17] it is shown that the Gravity factor

According to[20] of the International Astronomical Union, a light year is equal to m 580800 9460730472 .Then Ue R radius of the observable Universe is14.42 billion light-years, which correlates with the experimental data [19].

From (4) it follows that the value of the inverse of the Hubble constant -this is a temporary Ue T delay [15]

Obviously,

Ue

T -is the period of rotation of the sphere electromagnetic of the fractal of the Universe.

Evaluation of the volume

Ue Q of the sphere of the electromagnetic fractal of the Universe equals: 1.727 10 .

Ue

Ue Ue M kg m Q     (11)
The Hubble constant value for 2010 [21] which was obtained on the basis of seven years measurements made by NASA (Wilkinson Microwave Anisotropy Probe -WMAP) in combination with other methods of measurement is equal to: (km/s)/mpc . As WMAP reported, the Hubble constant value for 2012 equals: 69.32 ± 0.80 (km/s)/mpc [22,23]. According to the observation data published by the Planck Mission of the European Space Agency in 2014, the Hubble constant value for 4 years (from 2009 to 2013) comprise: 67.80 ± 0.77 (km/s)/mpc [19]. Based on this data it can be supposed that the Hubble constant scattering for various methods of measurement and by years can also be caused by precession of the rotating sphere of the electromagnetic fractal of the Universe in the plane being at any a small angle to the major rotation axis. The main kinds of matter motion within the electromagnetic fractal of the Universe are: 1. Motion of matter along the rotation axis of the electromagnetic fractal of the Universe caused by the Big Bang. This motion is similar to the existing vision about motion of the matter after the Big Bang, however without space expansion. 2. Rotational motion of matter around one of the axis of the sphere of the electromagnetic fractal of the Universe. 3. Motion of matter caused by precession of the sphere of the electromagnetic fractal of the Universe. 4. Motion of matter caused by centrifugal forces in the rotating and precessing sphere of the electromagnetic fractal of the Universe (effect of a centrifuge with precession). 5. Motion of matter caused by the own gravitational interaction forces. 6. Motion of matter caused by dark matter. All above motions are independent. After superposition of all kinds of motion the resultant vectors of total velocity of matter in each point of space in the sphere of the electromagnetic fractal of the Universe within the near field of view zone will have a chaotic distribution of directions and meanings of the absolute values. As the distance from the observation point (Earth) increases, the component of the total velocity vector conditioned by the rotating motion of the sphere of the electromagnetic fractal of the Universe will grow proportionate to the Hubble constant and the distance. From the distance of 5 billion light-years and more the contribution of this component will become prevailing, which is confirmed by the experimental Hubble law. The effect of the imaginary presence of dark energy is most pronounced at a distance of 5 billion light-years too.

Evaluation of main spatial and energy parameters of the nuclear and atomic fractals of the Universe

As hydrogen comprises more than 75 percent of the baryon mass of the electromagnetic fractal of the Universe [24,25], the hydrogen nucleus and atom were selected as the model. Correspondingly, the proton mass and radius are [18]: The mass and radius of the nuclear fractal of the Universe equals:

-27 16

1.672621777 10 kg; R 8.768 10 .

Un pr

Un pr M m r m        (12)
Then the volume of the sphere of the nuclear fractal of the Universe equals:

-45 3 2,823516 10 m .

Un Q  (13)
The volumetric density of the nuclear fractal of the Universe equals:

The mass of the atomic fractal can be determined at a first approximation as the sum of masses of the proton

The radius of the sphere of the atomic fractal of the Universe Ua R is the Bohr radius, [1] i.e.:

-11

5, 2917721092 10 m

Ua b Rr    . ( 16
)
The volume of the sphere of the atomic fractal of the Universe equals:

-31 3

6.2071466797 10 m

Ua Q  . ( 17
)
The volumetric density of the atomic fractal of the Universe equals:

Let us construct a graph (log-log plot) which reflects dependence between the volumetric (spatial) density of the mass of nuclear, atomic and electromagnetic fractals of the Universe and their volumes (Fig. 1). The analysis of the graph and numeric characteristics of the atomic fractal of the Universe leads to a conclusion that this fractal is transitional, boundary between two basic fractals of the Universe -electromagnetic and nuclear. It can also be assumed that there exists a transitional fractal between the electromagnetic and gravitational fractals of the Universe. Based on the analysis of the atomic fractal characteristics it can be assumed that the fractal structure of the Universe can not have well-defined transitions between the main fractals. That is, the transitions between the main fractals Universe is not clear, fuzzy and "smoky" (sfumato), and the spatial structure of the Universe is actually sfumato-fractal. It is obvious that evaluation of the radius (and volume) of the electromagnetic of fractal of the universe -it is their bottom verge.

Evaluation of main spatial and energy parameters of the gravitational fractal of the Universe

As fractals are self-similar events, processes and spatial forms, then the geometrical dimensions of the gravitational fractal of the Universe can be evaluated after we find the spatial analogue coefficient which correlates the electromagnetic and gravitational fractals of the Universe. The spatial analogue coefficient linking the gravitational fractal of the Universe and its electromagnetic fractal is determined similar to Coulomb laws and the universal law of gravitation:

12 12 1 2 1 2 22

;

,

eg kq q Gm m FF rr  ( 19
)
where k is the electrical constant,

-
R R k    . ( 23
)
The volume of the gravitational fractal of the Universe Q equals:

            (
To evaluate the gravity factor (for upper border evaluation of mass Ug M  ) of the gravitational fractal of the Universe g g , we apply its simplified model: let all mass of the fractal lies in the centre of its sphere and let the gravitational constant value is the same as for the electromagnetic fractal of the Universe, then (with formula 26):

2 Ug Ug g GM g R   , - 11 1 2
1.0446 10

g g m s   . ( 27
)
The magnitudes of the gravity factor and volumetric density of the mass of gravitational of fractal of the Universe excludes the possibility for him to become a source of dark energy in the electromagnetic of fractal of the Universe. Should the gravitational fractal of the Universe rotate, its period g T can be determined by the formula of mathematical pendulum, then with formula (27): (

) 28
There are fractals that are older than of the gravitational. Their scale factor is so big that we can not detect their presence by modern devices. If you assign serial numbers fractals (excluding transition fractals): nuclear -1, electromagnetic -2, gravitational -3, ..., and take into account that the ratio of logarithms of the masses and the logarithms of the volume of fractals, and too the ratio of logarithms of masses and logarithms volumes junior and senior fractals is range of about 0.618 -the basis of the golden fractal, it can be assumed that the spatial and energy structure of the Universe is an algebraic multiplicative gold fractal, i.e. logarithms of masses and logarithms of volumes of fractals of the Universe are sequences of Fibonacci numbers . Then the fractal with the serial number 4, and older than the gravitational fractal, taking into account the upper estimate of its mass, is true:

U Ue Ug Lg Q Lg Q Lg Q      , 4 ( ) ( ) (
) 53.26 89.9 143.16

U Ue Ug Lg M Lg M Lg M       .
Proof of the fractal structure of the universe is also the fact that the fundamental constants are also algebraic structure the gold fractal [17].

Conclusion

In each fractal of the Universe there is a dominant kind of energy (force interaction) while the "traces" of any and all kinds of energy of other fractals are also present. The energy value of these "traces" decreases with the increase of the difference of fractal numbers and is inversely proportionate to the product of similarity of fractal coefficients by energy between the intermediate fractals.

According to the adopted hypothesis the Universe has a sfumato-fractal structure with spatial and energy characteristics which is based on multiplicative golden fractal. Inflation of space and of energy of the Universe is a stationary sfumato-fractal process that does not require an assumption concerning expansion of the Universe due to dark energy. There exists no dark energy. The effect interpreted as a presence of dark energy is created by a rotary motion of the electromagnetic fractal spherical space.

  space where a mass is -the Planck sphere. Its volume equals:



  of the mass of the electromagnetic fractal of the Universe equals:

Fig. 1 .

 1 Fig. 1. Volumetric density mass of fractals of the Universe versus its volume for the nuclear, atomic and electromagnetic fractals of the Universe.

  and bottom border evaluation of the mass and evaluation of volumetric density mass of the gravitational of fractal of the Universe. The bottom border evaluation of the mass Ug M  and evaluation of volumetric density mass Ug   of the gravitational of fractal of the Universe determined on the basis of the law [

  25) Define the upper border evaluation of the mass Ug M  and evaluation of volumetric density mass Ug   of gravitational of fractal of the Universe on the basis of fractal similarity logarithms volumetric density of the mass of nuclear, electromagnetic and gravitational fractals of the Universe. Let us construct a graph (log-log plot), which reflects dependence between the volumetric (spatial) density of the mass of nuclear and electromagnetic fractals of the Universe and their volumes (Fig.2), and extrapolate this graph to the common logarithm value of the volume of the gravitational fractal of the Universe.

Fig. 2 .

 2 Fig. 2. Volumetric density mass of fractals of the Universe versus its volume for the nuclear, electromagnetic and gravitational fractals of the Universe.

  this space time delay dm t in the dissemination of signal. Body characteristics

																																			S m , ,	t	dm	interconnected universal
	proportions Planck:
										m		p m l p	S	;	m		t m p p	t	dm	;	S		t l	p p	t	dm	;	S		m l p p	; m	t	dm		l t	p p	S	;	t	dm		m t p p	, m
	where	l	p	,	m	p	,	t	p	-					is			the		Planck		constant,	respectively	-	length,	mass	and	time.
		Each body characteristics:	S m , ,	t	dm
																									E	mc  2		p F	S		e h	t	dm	,
		where:	e h 	p p E t	-is the quantum of Planck energy, where p E -Planck energy:	E	p 	m	p	c	2	; p F -is the
	Planck power:	p F 	m	p	a	p	, where p a -is Planck accelerating:	p a 	t l	2 p p	, and for two bodies with weight 1 m and
	2 m , length of a gravitational wave 1 S and 2 S , the time delay 1 dm t and 2 dm

separately from other uniquely determines him the energy parameters:

  According to [18] National Institute of Standards and Technology (NIST) for 2015 constants

																								G	,	c	,	l	p	,	m	p	,	t	p	have the following meanings:
	G		67408 . 6	(	) 31	x10	-11	m	3	kg	1	s	2	;	c	299792458	ms	1	;	l	p	616229 . 1	(	38	)	x10	-35	; m
	m	p	176470 . 2	(	) 51	x10	-	8	; kg	t	p			39116 . 5	13 (	)	x10	-	44	. s

The authors express gratitude to Professor Krivosheyev O.V.for critical comments and help in the work on this article.