Rajesh Koduri

Sorin Olaru

Pedro Rodriguez-Ayerbe

Sensitivity of Piecewise control laws with respect to perturbation of the state-space partition

The control design techniques for linear or hybrid systems (e.g. anti-wind up, MPC) lead often to off-line state-space partitions with non-overlapping convex polyhedral regions. This corresponds to a piecewise state feedback control laws associated to polyhedral partition of the state-space. In this work, we consider the perturbation in the representation of the vertices of the polyhedral regions. For such control laws, this problem is of particular interest with respect to the reduced precision representation of the state-space partition. The perturbed state-space partitions might lose one of the important property of the explicit controllers: the "non-overlapping" characterization. We derive a set called vertex sensitivity region to determine the admissible perturbation independently for each vertex of the polyhedral partition. A perturbation is deemed admissible if the non-overlapping property of the polyhedral regions is preserved. In the present work, the analysis of the sensitivity of each vertex is done under the assumption that the rest of the partition remains on the nominal configuration.

I. INTRODUCTION

Model Predictive Control (MPC) is one example of control design methodologies which has the ability to handle multivariate systems in the presence of constraints on inputs, states and outputs. MPC algorithms have been widely used to control plants or processes related to chemical, electrical and mechanical applications [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF]. However, implementing an MPC algorithm to obtain the optimal control input on-line by solving a Quadratic Programming (QP) problem for every time step using the available measurement is time consuming, particularly when it comes to system with fast dynamics. This on-line computational complexity can be overcome by transforming the QP problem to multi-parametric Quadratic Programming (mp-QP) problem and solving it off-line at the control design stage. This approach of solving the mp-QP problem off-line is also called Explicit MPC (EMPC), where the computation of the optimal control input is reduced to a simple evaluation of algebraic functions stored in a look-up table [START_REF] Alessio | A survey on explicit model predictive control[END_REF], [START_REF] Bemporard | The explicit linear quadratic regulator for constrained systems[END_REF]. EMPC control laws can be easily evaluated and implemented on-line for systems with extremely fast dynamics and as long as the state-space models are of small dimensions.

Such control laws can be effectively implemented on micro-controller circuits or on Field-Programmable Gate *This work is supported by TEMPO Marie-Curie European Project rajesh.koduri@centralesupelec.fr, sorin.olaru@centralesupelec.fr, pedro.rodriguez@centralesupelec.fr Array (FPGA) for a wide-range of control applications [START_REF] Johansen | Hardware Synthesis of Explicit Model Predictive Controllers[END_REF], [START_REF] Ingole | FPGA Implementation of Explicit Model Predictive Control for Closed Loop Control of Depth of Anesthesia[END_REF]. The cost of the micro-controller circuits or FPGA depend on the size of the memory unit or Arithmetic and Logic Unit (ALU) associated with it. For storing a relatively small state-space partitions and the PWA control laws associated with each partition, the cost of the microcontroller required might be reasonable. But for storing considerably large state-space partitions and PWA functions, a truncation (interpreted also as a quantization) operation should be performed on the representation of the state-space regions and the associated PWA control laws in order to reduce the hardware costs [START_REF] Knyazev | Explicit model predictive control accuracy analysis[END_REF]. The quantized state-space regions and quantized affine control laws associated with the regions adversely impact the accuracy of the control laws. Moreover, the modified regions might lose the nonoverlapping property of the state-space partition. Such a phenomenon will lead to non-uniqueness of the feedback control, discontinuities and the associated loss in stability, performance and other real-time certifications.

The authors of [START_REF] Nguyen | Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law[END_REF] proposed a geometrical approach to adopt the robustness margin for PWA control law with respect to the positive invariance characteristics of a nominal discrete-time linear system. This work has been extended in [START_REF] Koduri | Robustness margin for piecewise affine explicit control law[END_REF] to handle the robustness of the contractivity and not only the invariance. It is worth to be mentioned that these studies handle the robustness/fragility with respect to the model parameters and the control law gains but does not extend the analysis to the quantized explicit control law or quantized state-space regions. A recent study on the accuracy of explicit model predictive control law for quantized state-space partitions and quantized explicit control law is proposed in [START_REF] Knyazev | Explicit model predictive control accuracy analysis[END_REF] analyzing in general the amount of bits required in order to achieve a relative degree of control input accuracy. However, the paper constructs the analysis based on the assumption that the quantized statespace regions are non-overlapping and thus do not answer to the essential question of the admissible representation errors in the state space domain. The present work deals with this essential property of non-overapping regions in the state space partitioning, thus completing all these previous studies on the rigorous theoretical ground.

In order to analyze the impact on the closed-loop, the present work starts from the existence of a discrete-time linear system controlled by an explicit piecewise affine control law. We show how the polyhedral partitions change when perturbations on the vertex representation of the poly-hedral partitions or regions take place. This perturbed vertex transforms the original state-space regions into new regions. The goal is to analyze to what extent the non-overlapping property of the polyhedral partition can be preserved in event of an perturbation in the representation of the vertices of the regions. We characterize the vertex perturbation by determining a set called vertex sensitivity region which preserves the non-overlapping property of the polyhedral partition.

The paper is organized as follows, After the definition of the basic notations, the background of the explicit MPC is presented and the main problem is stated in a mathematical form. In section III, the main results concerning with the formulation of overlapping in the change of partition is shown and the vertex sensitivity margin is provided with related theorem, proof and implementation algorithm. In section IV, the numerical example for vertex sensitivity is provided.

BASIC NOTATIONS AND PRELIMINARIES

This section addresses some basic notations and definitions. The sets R, R + , Z, N and N + denote set of real numbers, set of non-negative real numbers, set of integers, set of non-negative integers, set of positive integers, respectively. We denote R n a Euclidean space and x ∈ R n a vector with n elements.

A set S ⊂ R n is a proper C-set if it is convex, closed, compact and contains the origin in its interior. A polyhedron is the (convex) intersection of a finite number of open or closed half-spaces and a polytope is a bounded and closed polyhedron. The set of vertices of a polytope M ⊂ R n is denoted V(M ). For a N ∈ N + , I N denotes the set of integers, I N := { i ∈ N + | i ≤ N }. For a given set P, int(P) denotes the interior of P and Conv denotes the convex hull.

II. BACKGROUND

A. System Description

Let us consider for the statement of the problem a discretetime linear system given by,

x(k + 1) = Ax(k) + Bu(k), (1)
here, x(k) ∈ R n is the state vector at time k and u(k) ∈ R m is the control input vector. The system states and inputs variables are subject to constraints, with state constraints given by

X = {x : H x x ≤ h x , H x ∈ R p×n , h x ∈ R p }, (2)
And input constraints by,

U = {u : H u u ≤ h u , H u ∈ R pu×m , h u ∈ R pu }. ( 3
)
where the matrices H x , H u and the vectors h x , h u are assumed to be constant, and X ⊂ R n and U ⊂ R m . The state and input constraints sets X and U are proper C-sets.

Definition 1: A closed and bounded set P ⊂ X is called controlled positively invariant with respect to (1) if there exists a control law u * (x(k)), such that ∀ x(0) ∈ P, then x(k) ∈ P, ∀ k ∈ Z.

Consider the discrete-time linear prediction model given by (1) subject to the state and input constraints (2), (3). The constrained finite-time optimal control problem for the standard MPC is given by,

J(x(k), U ) = min u∈U x(k + N p ) T P x(k + N p ) + Np-1 j=0 (x(k + j) T Qx(k + j) + u(k + j) T Ru(k + j)), (4a) s.t. x(k + j + 1) = Ax(k + j) + Bu(k + j), j = 0, • • • , N p -1, (4b) x(0) = x(k), x(N p ) ∈ X f , (4c) u(k + j) ∈ U, x(k + j) ∈ X , j = 0, • • • , N p -1. (4d)
Here, N p is the prediction horizon and the optimal control sequence obtained from the open-loop finite-time optimal control problem can be written as:

U * = [u(0) * T • • • u(N p -1) * T ] T ∈ R m . (5)
The weighting matrices Q = Q T ≥ 0 and P =P T ≥ 0 are positive semi-definite matrices, and R = R T > 0 is a positive definite matrix, all of them contributing to the definition of the performance index of the optimization problem. X f is a controlled invariant set admissible with respect to the input and state constraints [START_REF] Olaru | Avoiding constraints redundancy in predictive control optimization routines[END_REF].

The MPC problem (4) can be equivalently written in terms of a multi-parametric Quadratic Programming (mp-QP) problem [START_REF] Bemporard | The explicit linear quadratic regulator for constrained systems[END_REF].

J(x, U * ) = min U * 1 2 U * T HU * + x T F T U * + 1 2 x T Ȳ x, (6a) s.t ḠU * ≤ W + Ēx. (6b)
In ( 6), x is the current state supposed to be measured. The matrices H, F , Ȳ , Ḡ, W , Ē are obtained after the transformation (see [START_REF] Olaru | Avoiding constraints redundancy in predictive control optimization routines[END_REF] for further details). Definition 2: Consider a set of sets P N (R). This will define a polyhedral partition of the C-set R ⊂ R n , with

P N (R) = {R 1 , R 2 • • • R N }, N ∈ N + and R i ⊂ R n if 1) R = ∪ i∈I N R i , 2) int(R i ) = ∅, ∀i ∈ I N , 3) int(R i ) ∩ int(R j ) = ∅, ∀ (i, j) ∈ I 2
N , and i = j. Definition 3: The solution to the EMPC problem is a piecewise affine function f pwa defined over a polyhedral partition P N (R):

f pwa (x) = F i x k + g i , if x k ∈ R i . (7)
with i ∈ I N , F i ∈ R m×n and g i ∈ R m , and the regions R i are polytopic and bounded. The feedback law ( 7) is continuous and by substituting in the equation [START_REF] Mayne | Constrained model predictive control: Stability and optimality[END_REF], it leads to the PWA closed-loop dynamics:

x(k +1) = (A+BF i )x(k)+Bg i ∀x(k) ∈ R i , i ∈ I N . (8)

III. MAIN RESULTS

The polyhedral partition P N (R) considered next is the one resulting from the solution of the EMPC problem [START_REF] Nguyen | Explicit robustness and fragility margins for linear discrete systems with piecewise affine control law[END_REF]. In order to develop the sensitivity analysis, let us consider a polytopic region R i ⊂ R ∈ R n within the polyhedral partition given by the half space representation,

R i = {x |H i x ≤ k i }, i ∈ I N . ( 9
)
For each such region, an equivalent vertex representation exists in the virtue of Motzkin duality [START_REF] Motzkin | The double description method[END_REF]:

R i = Conv{v i,1 , • • • , v i,ri }, i ∈ I N , (10)
where r i is the number of vertices of R i .

In order to analyze the sensitivity of this set representation and its implication on the PWA control, a perturbation in the representation of the vertex v i,j , j ∈ I ri of the region Ri :

vi,j = v i,j + ∆v i,j (11)
will be considered leading to a new polyhedral set:

Ri = Conv{v i,1 , v i,2 , . . . , v i,j + ∆v i,j , • • • , v i,ri }. ( 12
)
The comparative analysis of the set ( 9) and ( 10) with respect to (12) will represent the main object of study.

Remark : The analysis will privilege the vertex representation [START_REF] Motzkin | The double description method[END_REF] with respect to the half-space representation (9) for two reasons. The first is related to the fact that the perturbation is concentrated on a single element of the set of generators with obvious computation advantages. A second argument is related to the non-emptiness of the set Ri in [START_REF] Herceg | Multi-Parametric Toolbox 3.0[END_REF] independent on the magnitude of the perturbation ∆v i,j . As a term of comparison, the equivalent half-space representation of Ri is

Ri = {x | Ĥi x ≤ k}, i ∈ I N , (13)
where several entries of the inequalities are affected and thus the analysis of the perturbation will have to deal with structural problems with respect to the Definition 2 of the polyhedral partitions. Indeed considering a perturbation directly with respect to the half-space representation (9) can lead to an empty set Ri posing well-possessedness issues in the characterization of the partition.

With respect to the perturbed sets in [START_REF] Herceg | Multi-Parametric Toolbox 3.0[END_REF], the main issue is the overlapping of the regions occurred due to representation errors in the original vertices of state-space partitions.

In order to provide a graphical interpretation of the problem under study let us consider the example of a simple continuous PWA function f pwa (x) defined over the interval In Fig. 1, the regions representation before and after the perturbation on the vertices is presented. In Fig. 1 (a), the nominal three regions obtained from EMPC denoted by

[-µ, µ] = [v 2,1 , v 3,2 ] ⊂ R. f pwa (x) =    αx for x ∈ R 1 = Conv{v 1,1 , v 1,2 }, β for x ∈ R 2 = Conv{v 2,1 , v 2,2 }, γ for x ∈ R 3 = Conv{v 3,1 , v 3,2 }.    (
R 1 , R 2 and R 3 , R = R 1 ∪ R 2 ∪ R 3 satisfy by definition the property int(R i ) ∩ int(R j ) = ∅, ∀i, j ∈ I 3 , i = j.
The vertices of the regions are denoted by v i,ri , here i and r i denote the index of regions and the index of vertices for each region respectively. After introducing a perturbation ∆v on the vertex v 1,2 and v 3,1 (recall that v 1,2 = v 3,1 ) which falls in the regions R 1 and R 3 , the perturbed vertex is denoted by v1,2 and v3,1 (v 1,2 = v 1,2 + ∆v 1,2 , v3,1 = v 3,1 + ∆v 3,1 , v1,2 = v3,1 ) as depicted in Fig. 1 (b).

Consequently, one vertex displacement will influence the topology of the regions R 1 and R 3 and leading to R1 and R3 , while region R 2 = R2 remains unchanged. It can be noticed from Fig. 1 (b) that even after the change in the regions, R = R1 ∪ R2 ∪ R3 and particularly the property int( Ri ) ∩ int( Rj ) = ∅, ∀i, j ∈ I 3 , i = j is preserved. Thus the nonoverlapping property of the polyhedral partition holds.

A very important consequence of the perturbation is the lost of the continuity for the PWA function f pwa (x) defined over R = R1 ∪ R2 ∪ R3 . Moreover, the control input depends on which region the state vector x falls into and in particular with the regions R1 and R3 , the control action being multivalued at the frontier. It is obvious that the control input modifications will lead to a loss of control continuity and consequently of the performances. This phenomenon can be seen however as a singularity and is out of the scope of the present work.

In the same framework, in Fig. 1 (c), the vertex v 1,2 and v 3,1 is perturbed to v1,2 and v3,1 and jumped into the region R 2 adversely affecting all the three regions. The new regions denoted by R1 , R2 and R3 are overlapping, i.e., int(

R1 ) ∩ int( R2 ) = ∅, int( R2 ) ∩ int( R3 ) = ∅ and int( R1 ) ∩ int( R3 ) = ∅.
This possible overlapping due to changes in the vertices of the polyhedral partition represents a critical structural change because the unicity of the control law is lost on a compact full-dimensional region of the state space. The nonuniqueness of the control action leads to behaviors which are difficult to characterized in terms of determinedness and lose of performance and thus should be avoided in the first place. This issue forms the basis for investigation in the present work and can be resumed by the need to characterize the limits of the perturbation which preserve the "non-overlapping" property of the polyhedral partition.

In the following, a formal definition of the vertex sensitivity is provided in the perspective of non-overlapping property of the polyhedral regions.

Definition 4: Consider the polyhedral partition P

N (R) ∈ R n with R i = Conv{v i,1 , • • • , v i,ri }, i ∈ I N . Let v ∈ R n
be a vertex within the polyhedral partition P N (R) and denote J v as the subset of indexes of regions having v as a vertex:

J v = {j ∈ I N | v ∈ V(R j )}. ( 15
)
The set V ⊂ R n is describing the vertex sensitivity region for v if ∀v ∈ V the collection of sets

Rj = Conv{V(R j ) \ {v}, v}, ∀j ∈ J v , Rj = R j , ∀j ∈ I N \ J v (16)
represents a polyhedral partition:

PN (R) = { R1 , • • • , RN }. Theorem 1: Consider the subset of regions R j , j ∈ J v of P N (R) such that v ∈ V(R j ), ∀j ∈ J v , then the perturbed vertex sensitivity is represented by a polyhedral set V ⊂ R n .
Proof: The proof will be constructive based on the enumeration of the possible overlapping scenarios and superposing the linear constraints imposed to avoid such configurations. The indexes within J v identify the regions containing v as a vertex. For each j ∈ J v , the half-space representation can be exploited:

R j = {x | H j x ≤ k j } or explicitly, (17a) R j = {x | h j,ri x ≤ k j,ri , r i = 1, • • • , r j } (17b)
here r j is the total number of closed half-spaces of R j . It is straight forward to identify the subset of those half-spaces passing through the vertex v.

Consider n v to be the total number of hyperplanes h j,ri x = k j,ri , j ∈ J v , r i = {1, • • • , r j } from the closed half-spaces that construct the regions R j , ∀j ∈ J v based on the definition (17b). Let us denote m v with m v < n v the number of hyperplanes that pass through the vertex v. Let us reorder the complete set of half-spaces of R j , ∀j ∈ J v such that the first m v pass by v. We have then

h i v = k i , ∀i ∈ I mv .
Scenario I: Let us construct a set denoted as R which includes a first set of constraints imposed on the vertex,

R = {x ∈ R n |h i x ≤ k i , ∀i ∈ I mv \ I nv }. (18)
The set R ⊂

j∈J v
R j is a closed convex subset of the union of the regions R j . Supposing that v / ∈ R, then the perturbed vertex characterize an overlapping phenomena. Such excessive perturbation leads to v / ∈

j∈J v R j and v ∈ j∈I N \J v
R j and this overlapping violation makes some closed half-spaces of the regions

R j redundant. Considering a case v / ∈ j∈J v R j and v / ∈ j∈I N \J v
R j , the perturbed vertex implodes the feasible set R and such violations will lead to the invalidation of the polyhedral partition P N (R).

There after, in (18) some constraints are placed to make the set R closed and bounded and positioning the perturbed vertex such that v ∈ R will overcome such possibilities of overlapping phenomena.

Scenario II: Let us introduce the set Φ and initialize it Φ ← R n . Consider a subset of the regions R j and denote it by Rj , ∀j ∈ J v ,

Rj = Conv{{v j,1 , • • • , v j,rj } \ v}. (19)
By duality, the set Rj is equivalently given by the half-space representation:

Rj = {x | hj,r b x ≤ kj,r b , r b = 1, • • • , rj } (20)
here, rj denotes the total number of closed half-space inequalities of the region Rj . In the following, we aim to impose a set of linear constraints on v to avoid the overlapping violations caused by the modifications on the m v number of hyperplanes that pass through the vertex v from (18). Consider nv be the total number of closed halfspaces from the regions Rj , ∀j ∈ J v as defined in (20). Now analyzing each of the nv closed half-spaces with respect to the closed and bounded set R one can construct the set:

Φ = { R ∩ {-hi x ≤ -ki }, ∀i ∈ I nv R ∩ { hi x ≤ ki } = R}. (21)
Whenever a perturbation takes place such that v ∈ int( Rj ) and v ∈ R, for some j ∈ J v , it leads to an overlapping scenario. Indeed the regions Rj = Rj for all the indexes j such that v ∈ int( Rj ). All the remaining regions with indexes j such that v / ∈ int( Rj ) are containing the point v as a vertex and the overlapping is proved. To avoid this phenomenon, the restriction to the region Φ described in (21) needs to be imposed in order to avoid v ∈ int( Rj ).

( Ri) ∩ int( Rj) = ∅, ∀i, j ∈ J v , i = j. (b) After perturbation v / ∈ V int( Ri) ∩ int( Rj) = ∅, ∀i, j ∈ J v , i = j.
Finally, the set that describes for all possible admissible perturbation for a given vertex v preserving the overlapping property of the polyhedral partition can be given as:

V ← Φ. (22)
Fig. 2 (a) shows a polyhedral partition with four regions

R i , i = 1, • • • , 4 and the vertex of interest v = [1 -1] T
denoted by a black dot. The vertex v is one of the generators of three among the four regions of the partition. In Fig. 2 (b), the sensitivity of the vertex v is represented by V with a blue polytope. By consequence the vertex denoted by v must be settled to any of the points in the polytope V in the event of loss of precision in the representation.

The vertex sensitivity region V is constrained by four closed half-space inequalities. For any point outside the blue polytope, the non-overlapping property of the EMPC is lost. The new regions formed with the displaced vertex v V guarantees the "non-overlapping" property of the polyhedral partition. This observation is better validated with the help of the Fig. 3 (a) and Fig. 3 (b), where the polyhedral regions are recreated by the displacement of vertex v. In Fig. 3 (a) the vertex v = [1 -1] T is displaced to vertex v = [0 0] T ∈ V which alters all the four regions and still preserves the overlapping property i.e., int( Ri ) ∩ int( Rj ) = ∅, ∀i, j ∈ I 4 , i = j. In Fig. 3 (b), it is clearly visible that the overlapping of the regions takes place since

v = [0 -2] T / ∈ V .
The procedure for computing the vertex sensitivity region V is resumed in Algorithm 1.

Algorithm 1 Algorithm for computing the vertex sensitivity set V The prediction horizon chosen is 2, the input constraint -5 ≤ u(k) ≤ 5 and the state constraint -5 ≤ [1 0] x(k) ≤ 5 are considered in the receding horizon optimization. Solving the EMPC problem yields 13 affine controllers and their validity regions obtained using MPT Toolbox 3.0 to compute the polyhedral regions and their associated PWA control laws [START_REF] Herceg | Multi-Parametric Toolbox 3.0[END_REF]. Fig. 4 (a), presents the state-space partition P N (R) with 13 regions denoted as R = ∪ 13 i=1 R i each of it associated with a gain matrix F i and an offset g i . In Fig. 4 (b), the vertex of interest v is denoted by a black dot and the regions R j , j ∈ J v containing the point v are highlighted in terms of a union of polyhedra depicted in white color. By applying the main result of the present paper and the Algorithm 1, the vertex sensitivity region is obtained as presented in Fig. 5 in gray color. In Fig. 6 (a), the vertex v is displaced to v within the sensitivity region V affecting the shapes of four regions and the perturbed collection of regions Rj , j = 1, • • • , 13 does not violate the non-overlapping property. The plot is recreated with four changed regions whereas the unaffected regions kept unchanged.

Input: R = ∪ N i=1 R i , i ∈ I N , v ∈ R. Output: V Initialization : M = [ ], W = [ ]. 1: Find the regions R j that contain v, such that ∀j ∈ J v . 2: % Compute the half space representation of R j . R j = {x | h j,ri x ≤ k j,ri , r i = 1, • • • , r j } LOOP Process 3: for j = 1 to J v do 4: V = [ ] 5: for r i = 1 to r j do 6: if h j,ri × v < k j,ri then 7: M = [M ; h j,ri ]; W = [W ; k j,ri ] 8: end if 9: if v j,ri = v then 10: V = [V ; v j,ri
In Fig. 6 (b), we can see the overlapping takes places among the affected regions caused by a perturbation in the vertex which exceeded the admissible sensitivity margin for the respective vertex v / ∈ V .

V. CONCLUSIONS

In the present work, we derived a vertex sensitivity set characterizing the admissible disturbance with respect to a given vertex all by preserving the non-overlapping property of the polyhedral regions. This element is fundamental in the characterization of the sensitivity of the representation of a polyhedral partition. The next step towards this goal is to consider simultaneous disturbance in all the vertices of the polyhedral partition. Further work is needed to analyze the condition such that the trajectories converge to the origin despite the perturbation on the vertex of the polyhedral regions.

  14)which can be simplified by the continuity between the neighboring regions: v 1,2 = v 3,1 and v 2,2 = v 1,1 .

(a) 1 -

 1 D Polyhedral with three regions R1, R2 and R3 and the corresponding PWA functions before perturbation. (b) 1-D Polyhedral with three regions R1, R2 and R3 before perturbation of the vertex representation. Illustration of regions R1, R2 and R3 after perturbation preserving the property, int( Ri)∩int( Rj) = ∅, ∀i, j ∈ I3, i = j. (c) 1-D Polyhedral with three regions R1, R2 and R3 before perturbation. Illustration of the regions R1, R2 and R3 after perturbation with the particular configuration leading to: int( Ri)∩int( Rj) = ∅, ∀i, j ∈ I3, i = j.

Fig. 1 :

 1 Fig. 1: 1-D Polyhedral partition with three regions before and after perturbation and the corresponding PWA function.

  (a) Polyhedral with four regions R1, R2, R3, R4 and the black dot represents v. (b) The vertex of interest v and the vertex sensitivity region V is shown.

Fig. 2 :

 2 Fig. 2: Polyhedral partition with four regions, the vertex of interest v and the vertex sensitivity region V are shown.

  (a) After perturbation v ∈ V the regions are changed and int

Fig. 3 :

 3 Fig. 3: The vertex v denoted by a black dot in Fig. 2 (a) is perturbed to v changing the regions R 1 , R 2 , R 3 , R 4 to R1 , R2 , R3 , R4 .

  ]

%

  Compute Rj by vertex representation 14: Rj = P olyhedron(V ) 15: end for 16: % Compute R by half-space representation 17: R = P olyhedron(M, W ) 18: % t j is the number of closed half-spaces of Rj 19: for j = 1 to J v do 20: for b i = 1 to t j do 21: if ({ hj,bi , kj,bi } ∩ R)! = R then 22: M = [M ; -hj,bi ] 23: W = [W ; -kj,bi ] for 27: % Compute V by half-space representation 28: V = P olyhedron(M, W ) 29: return V IV. NUMERICAL EXAMPLES Consider a discrete-time linear system, x(k + 1) = 1.4 0 0.8 -1.1 x(k) + 0.5 0.7 u(k).Consider the finite time optimal control based on quadratic cost with weighting matrices Q = 1 0 0 1 and R = 1.

  The vertex of interest v shown with a black dot and the polyhedrals containing v is colored with white.

Fig. 4 :Fig. 5 :

 45 Fig. 4: State-space partition with 13 regions

  (a) The vertex v ∈ V preserves the non-overlapping property of the regions. (b) The vertex v / ∈ V violates the non-overlapping property of the regions, the zoomed part confirms the violation does happen.

Fig. 6 :

 6 Fig. 6: The black dot in both figures show the perturbed vertex v.


Salim Yasmineh
email: sayasmineh@gmail.com

Simple interpretation of the double-slit experiment

Keywords: wave-particle duality, time, measurement, interference

This paper aims to understand how sequential, separate and independent impacts of particles on the screen of a double-slit experiment produce an interference pattern when both slits are open and how no such pattern is produced when detectors are placed at the slits. These strange phenomena can be explained by conjecturing the existence of a certain "thickness" of time. This conjecture is falsifiable, fits the observed results and maximizes simplicity for explaining other quantum phenomena without contradicting the established principles and laws of quantum mechanics.

Introduction

The double-slit experiment illustrates well some of the startling features of quantum mechanics. When particles such as electrons are sent one at a time through a double-slit plate (hereafter called slit A and slit B), single random impacts are observed on a screen behind the plate as expected out of individual particles. However, when the electrons are allowed to build up one by one, the cumulative effect of a great number of impacts on the screen reveals an interference pattern of light and dark bands characteristic of waves arriving at the screen from the two slits. Meanwhile, the interference pattern is made up of individual and sequential impacts and although these sequential impacts are separate and independent, yet it seems as if the electrons work together to produce the interference pattern on the screen. This phenomenon seems to entail that the electrons embody a wave-like feature in addition to their particle nature hence illustrating a particle-wave duality structure.

When the electrons are made to build up one by one while detectors DA and DB are placed at slits A and B respectively to find out through which slit each electron went, the interference pattern disappears and the electrons behave solely as particles. It seems thus impossible to observe interference and to simultaneously know through which slit the particle has passed. The best explanation that can be made out of these strange features is that the same electron seems to pass simultaneously through both slits when no detectors are present and through only one slit when detectors are present [START_REF] Cohen-Tannoudji | Laloë Mécanique quantique I[END_REF][START_REF] Rickles | The Philosophy of Physics[END_REF]. This seemingly paradoxical statement is in conformity with the experimental data.

The state vector of an electron passing through slit A may be denoted |𝐴⟩, similarly, the state vector of an electron passing through slit B may be denoted |𝐵⟩. An electron passing through both slits A and B at the same time is said to be in a superposition state and its state-vector is denoted |𝜓⟩ = 𝑎|𝐴⟩ + 𝑏|𝐵⟩, where "a" and "b" are called the probability amplitudes. The modsquare of "𝑎" represents the probability of the particle to be measured by the DA detector at the slit A and likewise the mod-square of "𝑏" represents the probability of the particle to be measured by the DB detector at the slit B.

Conventionally, when no detectors are present, the state-vector |𝜓⟩ = 𝑎|𝐴⟩ + 𝑏|𝐵⟩ of the electron is said to evolve per a deterministic continuous unitary evolution U whereas, when detectors DA and DB measure from which slit the electron passes, the deterministic evolution of the state-vector |𝜓⟩ is transformed into a probabilistic discontinuous and non-linear state reduction R as explained by Penrose [START_REF] Rickles | The Philosophy of Physics[END_REF]. The two processes U and R create a conflict in the formalism of quantum mechanics. Different ontologies have been proposed to interpret the strange combination of the deterministic continuous U process with the probabilistic discontinuous R process.

According to the Copenhagen interpretation [START_REF]Penrose The Road to Reality[END_REF][START_REF] Wimmel | Quantum physics & observed reality[END_REF], the state-vector |𝜓⟩ and the U and R processes should be regarded as a description of the experimenter's knowledge. There exist several other interpretations amongst which the Everett interpretation or what is more commonly known as the many-world interpretation [START_REF] Everett | The Theory of the Universal Wave function[END_REF][START_REF] Everett | [END_REF], according to which there is no wave function collapse and all measurement results exist but in different worlds. In line with this interpretation, it is claimed [START_REF] Wallace | Emergent Multiverse[END_REF] that when a measurement is conducted on an electron in the superposition state 𝑎|𝐴⟩ + 𝑏|𝐵⟩, a deterministic branching takes place where on one branch detector A detects the electron while detector B doesn't and at the same time but on the other branch (i.e. another world), detector A doesn't detect the electron while detector B does detect it. However, this interpretation pauses some probabilistic as well as ontological problems. In particular, the axioms of quantum mechanics say nothing about the existence of multiple physical worlds [START_REF] Kent | [END_REF].

Another interpretation is the De Broglie-Bohm deterministic theory according to which particles interact via a quantum potential and are assumed to have existing trajectories at all times. This model seems to make more sense of quantum mechanics than the other interpretations as discussed in detail by Jean Bricmont in his book "Making Sense of Quantum Mechanics [START_REF] Bricmont | Making Sense of Quantum Mechanics[END_REF].

In this paper, it is intended to introduce an alternative explanatory hypothesis that makes sense of the double-slit experiment and in which the process of measurement does not come in conflict with the deterministic evolution of the states of a particle.

It is learned from the above double-slit experiment that when no measurement is conducted, the state-vector is a bloc of two states |𝐴⟩ and |𝐵⟩ whereas, an act of measurement reveals only one of these two states. In this article both states |𝐴⟩ and |𝐵⟩ are considered as separate and equally real events. Indeed, it is conjectured the existence of a three-dimensional-time composed of the usual one-dimensional-time augmented with an unnoticeably small two-dimensional cross section. According to this model, time can be viewed as a non-geometrical line presenting a thread-like-form having a certain tiny "thickness" such that the two states |𝐴⟩ and |𝐵⟩ form separate events with respect to the cross-section (or thickness) of this three-dimensional-thread of time. Meanwhile, in the conventional picture of time, the cross-section of the time-thread is considered as a geometrical point and thus, the two states |𝐴⟩ and |𝐵⟩ are astonishingly considered to form a single event.

Formalism

Time-threads

Time is considered to have a three-dimensional thread-like-form, hereinafter referred to as "elementary-time-thread". The "longitudinal direction" of the elementary-time-thread corresponds to the usual physical-time-axis where each point is specified by a physical-timeindex. However, the "cross-section" of the elementary-time-thread is referred to as a "statetime-plane" where each point is specified by a "state-time-index" defined by a couple of statetime-coordinates. Finally, each point -referred to as "elementary-time-instant" -of the elementary-time-thread is specified by a triplet of time-coordinates (one physical-timecoordinate and two state-time-coordinates).

Each elementary-time-thread can be defined in a reference frame consisting of a threedimensional coordinate system in R 3 (or in R-C) composed of the ordinary "physical-time-axis" (𝑡 -𝑎𝑥𝑖𝑠) along a real coordinate axis R, a "first-state-time-axis" (𝑢 -𝑎𝑥𝑖𝑠) and a "secondstate-time-axis" (𝑣 -𝑎𝑥𝑖𝑠). The elementary-time-thread has thus its cross-section comprised in a "state-time-plane" (𝑢, 𝑣) defined by the 𝑢 -𝑎𝑥𝑖𝑠 and 𝑣 -𝑎𝑥𝑖𝑠 and its longitudinal orientation defined along the ordinary physical 𝑡 -𝑎𝑥𝑖𝑠. Each "elementary-time-instant" of the elementary-time-thread is specified by a point 𝑤 = (𝑡, 𝑢, 𝑣) where 𝑡, 𝑢 and 𝑣 are real numbers. For simplicity, the coordinates in the state-time-plane are defined by a single symbol s = (𝑢, 𝑣) called a "state-time-index" which can also be defined by a complex number of the form:

s = 𝑢 + 𝑖𝑣 = |s|𝑒 𝑖𝜑 (1)
where |s| and 𝜑 are the magnitude and argument of the state-time-index s. Thus, each "elementary-time-instant" 𝑤 of the elementary-time-thread is specified by the point 𝑤 = (𝑡, s ).

Quantum features with respect to dynamical time-threads

Conventionally, a quantum system (e.g. spin of a particle) can be defined by a state-vector in an orthonormal eigenvector basis. For any observable Q, the state-vector |𝜓⟩ is defined by a superposition of vector projections in an eigenbasis {|𝜓 𝑖 ⟩}. In other words, the state-vector |𝜓⟩ is defined as a linear combination of the different possible states. The normalized conventional state-vector of the quantum system is expressed as follows:

|𝜓⟩ = ∑ 𝑐 𝑖 𝑖 |𝜓 𝑖 ⟩ (2)
where |𝜓 𝑖 ⟩ are orthonormal states of the quantum system verifying ⟨𝜓 𝑖 |𝜓 𝑗 ⟩ = 𝛿 𝑖𝑗 (Kronecker delta) and the coefficients 𝑐 𝑖 of the state-vector |𝜓⟩ define the "probability amplitudes" in the specific orthonormal eigenvector basis {|𝜓 𝑖 ⟩}.

In contrast, according to the present three-dimensional-time-model, the quantum system can be represented by a fundamental-state-vector -noted hereafter |𝜓⟩ 𝐹 . The different points (i.e. the different state-time-indices s(𝑢, 𝑣) belonging to the state-time-plane) visited by the different states of a given fundamental-state-vector |𝜓⟩ 𝐹 at a given physical-time-index 𝑡 form a "statetime-domain" denoted "𝐷s" belonging to the "state-time-plane" (𝑢, 𝑣). The "state-time-domain 𝐷s" is an open subset of the "state-time-plane" (𝑢, 𝑣). The area (denoted "𝐿s") of the state-time-domain 𝐷s is a measure 𝜇 of 𝐷s (i.e. 𝐿s = 𝜇(𝐷s(t))) which represents the "state-timeduration" (hereafter called the "state-time-term") of the fundamental-state-vector |𝜓⟩ 𝐹 at a given physical-time-index 𝑡.

The fundamental-state-vector |𝜓⟩ 𝐹 may thus be viewed as evolving with respect to a threedimensional-time-manifold (i.e. a "time-volume" referred hereafter "3d-time-manifold") embedded in the elementary-time-thread and made up of elementary-time-instants 𝑤 = (𝑡, 𝑢, 𝑣) = (t, s) wherein, each physical-time-index 𝑡 is mapped to a corresponding state-timedomain 𝐷s(t) whose measure 𝐿s(t) represents the "state-time-term" of the fundamental-statevector |𝜓⟩ 𝐹 at that physical-time-index 𝑡. Schematically, a state-time-domain 𝐷s(t) belonging to a 3d-time-manifold is formed by the intersection between the 3d-time-manifold and the cross-section (i.e. the state-time-plane (𝑢, 𝑣)) of the elementary-time-thread at the physicaltime-index 𝑡. In the following text, the suffix "𝑡" is sometimes dropped as it is implicitly clear that any state-time-domain 𝐷s is defined with respect to a corresponding physical-time-index 𝑡.

The fundamental-state-vector |𝜓⟩ 𝐹 associated to a given observable and evolving with respect to its corresponding 3d-time-manifold (t, s) can thus be expressed by a complex valued statevector |𝜓(𝑥, 𝑡, s)⟩ as follows:

|𝜓⟩ 𝐹 = |𝜓(𝑥, 𝑡, s)⟩ 𝑤ℎ𝑒𝑟𝑒 𝑥 ∈ 𝑹 𝟑 , 𝑠 𝜖𝐷𝑠 𝑎𝑛𝑑 𝑡 ∈ 𝑹 (3)
The above expression (3) indicates that at any current physical-time-index 𝑡, the different states |𝜓(𝑥, 𝑡, s)⟩ with respect to s can be viewed as forming a "state-time-block" wherein, all potential state-outcomes coexist but do not occur at once, or "simultaneously", with respect to the corresponding state-time-domain 𝐷𝑠. In the "state-time-block" all states are labelled by different "state-dates" and thus, at any given time-index 𝑡, the different states do not occur simultaneously with respect to the state-time-domain 𝐷𝑠, but may be considered to occur "at once" only with respect to the physical-time 𝑡. In this case these states may be called "physicaltime-simultaneous" or simply "partially simultaneous" implicitly with respect to only the physical-time 𝑡.

Therefore, a given fundamental-state-vector |𝜓⟩ 𝐹 evolving with respect to its state-time-domain 𝐷s at a given physical-time-index 𝑡, can be expressed by its different states |𝜓 𝑘 (𝑥)⟩, as follows:

|𝜓⟩ 𝐹 = |𝜓(𝑥, 𝑡, s)⟩ = { |𝜓 𝑘 (𝑥)⟩ 𝑖𝑓 s 𝜖𝐷s 𝑘 (t) 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (5)
The above equation is defined at any given physical-time-index 𝑡 and may also be expressed as follows:

|𝜓⟩ 𝐹 = |𝜓(𝑥, 𝑡, s)⟩ = ∑ 𝛿 s (𝐷s 𝑘 (t))|𝜓 𝑘 (𝑥)⟩ 𝑘 (6)
where 𝛿 s (𝐷s 𝑘 (t)) is the Dirac measure (or indicator function) defined by:

𝛿 s (𝐷s 𝑘 ) = { 1 𝑖𝑓 s 𝜖𝐷s 𝑘 0 𝑖𝑓 𝑛𝑜𝑡 (7)
The dynamics of the fundamental-state-vector |𝜓⟩ 𝐹 of the present model obeys a deterministic process with respect to the 3d-time-manifold and can be decomposed into a set of two equations comprising the above equation ( 6) defined with respect to the state-time as well as Schrodinger equation defined with respect to the physical time:

𝑖ℏ 𝜕 𝜕t |𝜓(𝑥, 𝑡)⟩ = 𝐻|𝜓(𝑥, 𝑡)⟩ (8)
Equation ( 6) expresses the form of the fundamental-state-vector |𝜓(𝑥, 𝑡, s)⟩ with respect to the "state-time-plane" (𝑢, 𝑣) and in particular, with respect to the state-time-domain 𝐷s at each physical-time-index 𝑡. Thus, at each 𝑡, the state-vector |𝜓(𝑥, 𝑡, s)⟩ can be viewed as an "intrinsic signal" in function of s bounded by the state-time-domain 𝐷s. In contrast, equation ( 8) is the conventional Schrodinger equation that governs the evolution of the intrinsic signal as a bloc with respect to the physical-time-axis (𝑡 -𝑎𝑥𝑖𝑠).

Equation ( 6) can also be expressed in a continuous spectrum as follows:

|𝜓(𝑥, 𝑡, 𝑠)⟩ = ∫ 𝑑𝑥𝛿 𝑠 (𝐷𝑠(𝑥, 𝑡)) |𝜓(𝑥)⟩ (9)
Where for a given 𝑡, 𝐷𝑠(𝑥, 𝑡) represents the set of all the state-time-indices 𝑠 during which the quantum system is in the state |𝜓(𝑥)⟩ and where:

𝛿 𝑠 (𝐷𝑠(𝑥, 𝑡)) = { 1 𝑖𝑓 s 𝜖𝐷𝑠(𝑥, 𝑡) 0 𝑖𝑓 𝑛𝑜𝑡 (10)
Conventionally, the solution of the Schrodinger equation ( 8) has the following form:

|𝜓(𝑥, 𝑡)⟩ = ∑ 𝑐 𝑗 (𝑥, 𝑡)|𝜓 𝑗 (𝑥)⟩ 𝑗 (11)
where the mod-square of 𝑐 𝑗 (i.e. ‖𝑐 𝑗 ‖ 2 ) represents the probability associated to the state |𝜓 𝑗 (𝑥)⟩ and where:

∑ ‖𝑐 𝑗 ‖ 2 𝑗 = 1 (12)
All the states {|𝜓 𝑗 (𝑥)⟩ } are visited by the state-time-indices s while spanning 𝐷s and thus, each state |𝜓 𝑗 (𝑥)⟩ should be the image of a corresponding state-time-sub-domain 𝐷𝑠 𝑘 ⊂ 𝐷s. In other terms, for each 𝑡, there exists a bijective correspondence between the set of state-time-subdomains {𝐷𝑠 𝑘 ; ⋃ 𝐷𝑠 𝑘 𝑘 = 𝐷s} and the set of states {|𝜓 𝑘 (𝑥)⟩ }. This bijective correspondence is indeed expressed by equation [START_REF] Everett | [END_REF].

On the other hand, when the state-time-index s is made to span the state-time-domain 𝐷s, then at each physical-time-index 𝑡, the state-vector |𝜓(𝑥, 𝑡, s)⟩ should satisfy the normalisation property:

‖|𝜓(𝑥, 𝑡, s)⟩‖ 2 = ⟨𝜓(𝑥, 𝑡, s)|𝜓(𝑥, 𝑡, s)⟩ = ∑ (𝛿 s (𝐷s 𝑘 )) 2 = 𝑘 1 (13)
The last equality in equation ( 13) is justified because when the whole state-time-domain 𝐷s is spanned by the state-time-index s, then necessarily 𝛿 s (𝐷s 𝑘 ) = 1 for each 𝑘 and thus, equation ( 13) is consistent with equation (12).

It is to be noted that when the state-time-index s spans the state-time-domain 𝐷s, the state of the system at a given 𝑡 is defined by a set of states:

(𝜓(𝑥, 𝑡, s)⟩) s∈𝐷s = {|𝜓 𝑘 (𝑥)⟩ ; 𝑘 = 1, 2, … . . } (14)
The average value of the above set of states can be expressed as follows:

∑ 𝑃 𝑘 |𝜓 𝑘 (𝑥)⟩ 𝑘 ( 15
)
where 𝑃 𝑘 is the probability associated to the state |𝜓 𝑘 (𝑥)⟩ which is also equal to ‖𝑐 𝑘 ‖ 2 as expressed in equation ( 12).

On the other hand, the average value of the above set of states (𝜓(𝑥, 𝑡, s)⟩) s∈𝐷s of equation ( 13) can also be expressed by normalising the sum of the fundamental-state-vector |𝜓⟩ 𝐹 over all the state-time-domain 𝐷s as follows:

∫|𝜓⟩ 𝐹 𝑑s ∫ 𝑑s = ∫|𝜓(𝑥,𝑡,s)⟩ 𝑑s ∫ 𝑑s = ∑ 𝑃 𝑘 |𝜓 𝑘 (𝑥)⟩ 𝑘 = ∑ ‖𝑐 𝑘 ‖ 2 |𝜓 𝑘 (𝑥)⟩ 𝑘 (16)
However, the left-hand side of equation ( 16) can be expressed as follows:

Comparing equation (18) to equation ( 16), it is deduced that the probability 𝑃 𝑘 is:

𝑃 𝑘 = ‖𝑐 𝑘 ‖ 2 = 𝑙 sk 𝐿 s (19)
However, ∑ 𝑙 sk 𝑘 = 𝐿 s and thus ∑ 𝑙 sk 𝐿 s 𝑘 = 1 which is also consistent with equation (12).

Therefore, the probability amplitude can be expressed as follows:

𝑐 𝑘 = √𝑙 sk 𝐿 s ⁄ 𝑒 𝑖𝜑 𝑘 (𝑡,s) (20)
where 𝜑 𝑘 (𝑡, s) is a phase function that may depend on 𝑡 and s:

Thus, equation (11) -solution of the Schrodinger equation -can be expressed as follows:

|𝜓(𝑥, 𝑡)⟩ = ∑ 𝑐 𝑗 (𝑥, 𝑡)|𝜓 𝑗 (𝑥)⟩ 𝑗 = ∑ √𝑙 sk 𝐿 s ⁄ 𝑒 𝑖𝜑 𝑘 (𝑡,s) |𝜓 𝑘 (𝑥)⟩ 𝑘 (21)
The state of a quantum system can be therefore specified by the set of equations ( 6) and ( 21 𝑙 sk during which the state has been visited. Indeed, when the structure and details in the statetime-domain 𝐷s are not taken into consideration, it is evident that all the states |𝜓 𝑘 (𝑥)⟩ would be seen as superposed and would seem to occur at once when in fact it is not the case. However, the dependence with respect to the state-time-domain 𝐷s remains intrinsically in-built in the probability amplitudes which depend on 𝑙 sk and 𝐿 s . Nevertheless, the intrinsic dynamics is occulted by equation ( 21) leading to the seemingly mysterious measurement problem.

Indeed, if we take conventional equation (11) where the state-time-indices are completely ignored and their existence is not even suspected, the different states |𝜓 𝑘 (𝑥)⟩ occur at once and thus, a measurement at any physical-time instant 𝑡, would seem to impose a collapse of the state-vector |𝜓(𝑥, 𝑡)⟩ into an arbitrary outcome.

However, even if we don't have any knowledge about the state-time-indices and if a measurement is to take place at any physical-time-instant 𝑡, then equation (21) shows that the outcome could be any state |𝜓 𝑘 (𝑥)⟩ with a corresponding probability 𝑙 sk 𝐿 s ⁄ without any collapse whatsoever because the different states |𝜓 𝑘 (𝑥)⟩ as shown by equation [START_REF] Everett | [END_REF] do not occur at once with respect to the state-time-indices. It should be noted, that the outcome can be deterministically predicted if in addition to the physical-time-index, the state-time-index is also known. Nevertheless, once the act of measurement is carried out, suppose at the elementary time-instant 𝑤 𝑀 = (𝑡 𝑀 , s 𝑀 ) then, the state |𝜓 𝑀 (𝑥, 𝑡 𝑀 , s 𝑀 )⟩ that happened to exist at that elementary instant is selected and this outcome is "trapped", "frozen" or "captured" by the interaction between the system and the measuring device. Thus, just after the measurement, the state-vector |𝜓(𝑥, 𝑡, s)⟩ becomes stationary with respect to the state-time-indices s (i.e. 𝛿 s (𝐷s 𝑀 (t)) = 1, ∀ s 𝜖𝐷s ) and can be expressed as follows:

|𝜓(𝑥, 𝑡, s)⟩ = ∑ 𝛿 s (𝐷s 𝑘 (t)) 𝑘 |𝜓 𝑘 (𝑥)⟩ = |𝜓 𝑀 (𝑥)⟩ (25)
On the other hand, immediately after the measurement the state-time-term 𝑙 sM of the subdomain 𝐷𝑠 𝑀 corresponding to the measured state |𝜓 𝑀 ⟩ becomes equal to the state-time-term 𝐿 s of the whole state-time-domain 𝐷s. In other terms the state-vector |𝜓(𝑥, 𝑡)⟩ also becomes stationary (i.e. 𝑙 sM = 𝐿 s and 𝑙 sk = 0, 𝑘 ≠ 𝑀) and can be expressed as follows:

|𝜓(𝑥, 𝑡)⟩ = ∑ √𝑙 sk 𝐿 s ⁄ 𝑒 𝑖𝜑 𝑘 (𝑡,s) |𝜓 𝑘 (𝑥)⟩ 𝑘 = 𝑒 𝑖𝜑 𝑀 (𝑡) |𝜓 𝑀 (𝑥)⟩ (26)
It should be noted that the phase term in equation ( 26) expresses the dependence on the physical-time-indices 𝑡.

Free particle

The state of a free particle can be specified by the Schrodinger equation ( 8) in combination to anyone of equations ( 9) and (24) which can be expressed in a one-dimensional position representation along an x-axis as follows:

|𝑥 where the kets |𝑥⟩ are the states in which the particle is definitely at the position 𝑥 and where 𝜓(𝑥, 𝑡, s) and 𝜓(𝑥, 𝑡) are the probability amplitudes. In particular, the probability amplitude 𝜓(𝑥, 𝑡, s) = 𝛿 𝑠 (𝐷𝑠(𝑥)) is a binary function which can only be equal to either 1 or 0 and hence, specifies whether the quantum system is in the position 𝑥 (i.e. state |𝑥⟩) or not at a given 𝑠 and a given 𝑡.

In contrast, the probability amplitude 𝜓(𝑥, 𝑡) = √𝑙 𝑠 (𝑥, 𝑡) 𝐿 𝑠 ⁄ 𝑒 𝑖𝜑(𝑡,s) indicates the "rate of presence" of a given state |𝑥⟩ at a specific physical-time-index 𝑡. In particular, ‖𝜓(𝑥, 𝑡)‖ 2 = 𝑙 𝑠 (𝑥, 𝑡) 𝐿 𝑠 ⁄ represents for a given physical-time-index 𝑡, the total rate of the state-time presence of the particle at the position 𝑥, which is also equal to the probability of finding the particle at that position. Indeed, it is straightforwardly evident that the probability of detecting the particle at the position 𝑥 is proportional to the state-time spent by the particle at that position.

The free particle can be viewed at a given physical instant 𝑡, as a spatial cloud formed by the different positions visited by the particle at the different state-time-indices of its state-timedomain. Indeed, at each 𝑡, the particle may be considered as having an oscillatory movement in space with respect to the state-time and thus, at each 𝑡, the particle has multiple positions as if there is not only one particle but plenty of particles that can be represented by a wavelet with respect to 𝑥. The oscillatory-like movement of the particle with respect to the state-time-indices 𝑠 in combination with its proper dynamics with respect to the physical-time-indices 𝑡 create the wavelike movement of the particle.

For simplicity, it can be assumed as described in Binney et al [START_REF] Binney | The Physics of Quantum Mechanics[END_REF], that the free particle has a well-defined energy 𝐸. Thus, the phase term 𝑒 𝑖𝜑(𝑡,s) in equation ( 28) can be expressed by 𝑒 -𝑖𝐸𝑡 ℏ ⁄ which can be approximated by a state of well-defined momentum |𝑝⟩ and can be described by a plane wave of wavelength 𝜆 = ℎ 𝑝 ⁄ as follows:

𝑒 𝑖𝜑(𝑡,s) = 𝑒 𝑖𝑝𝑑 ℏ ⁄ ( 29
)
where 𝑑 is the distance covered by the particle with respect to the physical-time 𝑡.

On the other hand, it is reasonable to suppose that the spatial points of the cloud furthest away from its centre are visited less "frequently" by the particle than those nearest to the centre. Here, the term "frequently" is to be understood with respect to the state-time and not with respect to the physical-time. Thus, it is sensible to suppose that at a given physical-time 𝑡, the probability amplitude of the particle's positions 𝑥 is the square-root of a Gaussian:

𝜓(𝑥, 𝑡) = √𝑙 𝑠 (𝑥, 𝑡) 𝐿 𝑠 ⁄ = 1 (2𝜋𝜎 2 ) 1 4 ⁄ 𝑒 -x 2 4𝜎 2 ⁄ (30)
where 𝜎 2 is the variance of the distribution.

Inserting expressions (29) and (30) into equation ( 28), one gets:

|𝑥(𝑡)⟩ = ∫ 𝑑𝑥 𝜓(𝑥, 𝑡) |𝑥⟩ = 1 (2𝜋𝜎 2 ) 1 4 ⁄ ∫ 𝑑𝑥𝑒 -x 2 4𝜎 2 ⁄ 𝑒 𝑖𝑝𝑑 ℏ ⁄ |𝑥⟩ (31)
Equation ( 31) expresses the state of the particle at a physical-time-index 𝑡.

However, concerning measurement, equation ( 27) shows that at a given physical-time-instant 𝑡, all the different positions of the particle do not occur at once with respect to the state-timeindices and thus, if a measurement is to take place, the outcome could be any position 𝑥 with a corresponding probability 𝑙 𝑠 (𝑥, 𝑡) 𝐿 𝑠 ⁄ as expressed in equation ( 28) without any collapse whatsoever. Now, suppose that at the elementary time-instant 𝑤 𝑀 = (𝑡 𝑀 , s 𝑀 ), the particle has been detected to be at the position 𝑥 𝑀 . Then, just after the measurement, the position becomes stationary with respect to the state-time-indices s (i.e. 𝛿 s (𝐷𝑠(𝑥, t)) = 1, ∀ s 𝜖𝐷s ) but not necessarily with respect to the physical-time 𝑡. In other words, just after the measurement the position of the particle may evolve only with respect to 𝑡 in the same way as a classical evolution of a point particle. At each physical-time 𝑡 immediately after the measurement, the particle would have only one definite position knowing that the particle's position remains stationary with respect to the state-time-indices s (i.e. the position of the particle with respect to the state-time becomes a rectangular function 𝑥(𝑠, 𝑡) = 𝑥(𝑡), ∀ 𝑠 𝜖𝐷𝑠 ). Thus, immediately after the measurement, equation ( 27 On the other hand, the state-vector |𝑥(𝑡)⟩ of equation (31) also becomes stationary (i.e. 𝑙 𝑠 (𝑥, 𝑡) = 𝐿 s ) and thus immediately after the measurement, the outcome probability of finding the particle at the position 𝑥(𝑡) is:

𝑃(𝑥) = ‖𝜓(𝑥, 𝑡)‖ 2 = ‖√𝑙 𝑠 (𝑥, 𝑡) 𝐿 𝑠 ⁄ 𝑒 𝑖𝜑(𝑡,s) ‖ 2 = ‖√𝐿 𝑠 𝐿 𝑠 ⁄ 𝑒 𝑖𝜑(𝑡,s) ‖ 2 = 1 (33)

Revisiting the double-slit experiment

In this section, we consider a double-slit apparatus comprising a source S of particles, a doubleslit-plate and an impact-screen according to the representation described for example in Binney et al. [START_REF] Binney | The Physics of Quantum Mechanics[END_REF]. Let the double-slit plate and the impact screen be disposed on parallel (x,y) planes with respect to a cartesian coordinate system (x,y,z) having its origin at the source S. Let 2𝛼 be the distance between the slits along the x-axis and let the source S be equidistant from the slits A and B. Let the x-coordinates of slits A and B be 𝑥 𝐴 and 𝑥 𝐵 respectively. Finally, let 𝐿 be the distance along the z-axis between the impact screen and the double-slit plate. We are essentially interested in the particle's impact on the screen along the x-axis and thus, it is sufficient, without much loss of generality, to define the state-vector of a particle only with respect to the xcoordinates.

As described in section 2.3, a free particle emanating out of the source S can be viewed at a given physical-time-instant 𝑡, as a wavelet formed by the different positions visited by the particle at the different state-time-indices 𝑠 of its state-time-domain 𝐷𝑠 whose dynamics is defined by equations ( 27) and (31).

According to equation (31), the probability amplitude 𝜓(𝑥, 𝑡) of the particle can be expressed as follows:

𝜓(𝑥, 𝑡) = 1 (2𝜋𝜎 2 ) 1 4 ⁄ 𝑒 -x 2 2𝜎 2 ⁄ 𝑒 𝑖𝑝𝑑 ℏ ⁄ (34)
When the particle (represented at each physical-time-instant 𝑡, by a wavelet of different positions) strikes the double-slit-plate, two outgoing wavelets emerge out of the slits. In other words, the primary wavelet coming from the source is separated into two secondary wavelets spanned by the same particle at each given physical-time-instant 𝑡.

The state-vectors |𝑥 𝐴 (𝑡)⟩ and |𝑥 𝐵 (𝑡)⟩ represent the different positions of the same particle around slits A and B respectively, at different state-time-indices 𝑠 but at the same physicaltime-index 𝑡.

Similarly, in accordance to equation (28), the main wavelet represented by the state-vector |𝑥(𝑡)⟩ can be expressed as follows: where 𝑙 sA and 𝑙 sB represent the state-time-terms during which the particle is in the state of being around slits A and B respectively. In other terms, at a given physical-time-instant 𝑡, the particle spends the fraction 𝑙 sA of its state-time-term 𝐿 s around slit A and the fraction 𝑙 sB of its state-time-term 𝐿 s around slit B.

Meanwhile, 𝐿 s = 𝑙 sA + 𝑙 sB and it is realistic to suppose that both states have equal state-timeterms (i.e. 𝑙 sA = 𝑙 sB = 𝐿 s 2 ⁄ ). Moreover, it is reasonably supposed that the probability distribution of the particle around each slit with respect to 𝑥 is a Gaussian. In addition, it is assumed that the particle coming out of the slits retains its former wavelength and thus, the particle around each slit can be described (as in section 2.3) by a plane wave of wavelength 𝜆 = ℎ 𝑝 ⁄ such that equation (37) can be expressed as follows:

|𝑥(𝑡)⟩ = 1 (2𝜋𝜎 2 ) 1 4 ⁄ √2 2 [∫ 𝑑𝑥 𝑒 -(𝑥-𝑥 𝐴 ) 2 4𝜎 2 ⁄ 𝑒 𝑖𝑝𝑑 𝐴 ℏ ⁄ |𝑥 𝐴 ⟩ + ∫ 𝑑𝑥 𝑒 -(𝑥-𝑥 𝐵 ) 2 4𝜎 2 ⁄ 𝑒 𝑖𝑝𝑑 𝐵 ℏ ⁄ |𝑥 𝐵 ⟩] (38)
According to equation (38), the probability amplitudes 𝜓 𝐴 (𝑥, 𝑡) and 𝜓 𝐵 (𝑥, 𝑡) arriving at the point 𝑥 on the screen after covering distances 𝑑 𝐴 and 𝑑 𝐵 from slits A and B respectively are given by:

𝜓 𝐴 (𝑥, 𝑡) = 1 (2𝜋𝜎 2 ) 1 4 ⁄ √2 2 𝑒 -(𝑥-𝑥 𝐴 ) 2 4𝜎 2 ⁄ 𝑒 𝑖𝑝𝑑 𝐴 ℏ ⁄ (39) 𝜓 𝐵 (𝑥, 𝑡) = 1 (2𝜋𝜎 2 ) 1 4 ⁄ √2 2 𝑒 -(𝑥-𝑥 𝐵 ) 2 4𝜎 2 ⁄ 𝑒 𝑖𝑝𝑑 𝐵 ℏ ⁄ (40)
Thus, the same particle has a movement at each physical-time-instant 𝑡 that can be assimilated to two waves and thus, the particle interferes with itself or in other words, viewed from the perspective of the physical-time, the particle seems to be formed of an infinite number of pointparticles moving in a wave-like structure. Each point-particle represents the particle at a specific position at a corresponding state-time-index.

The probability of the particle to be at the position 𝑥 on the screen is thus given by:

𝑃(𝑥) = |𝜓 𝐴 (𝑥, 𝑡) + 𝜓 𝐵 (𝑥, 𝑡)| 2 = |𝜓 𝐴 | 2 + |𝜓 𝐵 | 2 + 2𝑅𝑒(𝜓 𝐴 * 𝜓 𝐵 ) (41)
The last term refers to the interference 𝐼(𝑥) between the two probability amplitudes:

𝐼(𝑥) = 2𝑅𝑒(𝜓 𝐴 * 𝜓 𝐵 ) = 1 (2𝜋𝜎 2 ) 1 2 ⁄ 𝑒 -[(𝑥-𝑥 𝐴 ) 2 +(𝑥-𝑥 𝐵 ) 2 ] 2𝜎 2 ⁄ 𝑒 𝑖𝑝(𝑑 𝐵 -𝑑 𝐴 ) ℏ ⁄ (42)
However, by using Pythagoras' theorem and by assuming that both 𝑥 and 𝛼 are much smaller than 𝐿, it is deduced:

𝑑 𝐴 = √𝐿 2 + (𝑥 -𝛼) 2 ≈ 𝐿 2 ⁄ + (𝑥 -𝛼) 2 2𝐿 ⁄ (43)
𝑑 𝐵 = √𝐿 2 + (𝑥 + 𝛼) 2 ≈ 𝐿 2 ⁄ + (𝑥 + 𝛼) 2 2𝐿 ⁄ (44)
Therefore:

𝑑 𝐵 -𝑑 𝐴 ≈ 2𝑥𝛼 𝐿 ⁄ (45)
Equation (42) can thus be expressed as follows:

𝐼(𝑥) = 1 (2𝜋𝜎 2 ) 1 2 ⁄ 𝑒 -[(𝑥-𝑥 𝐴 ) 2 +(𝑥-𝑥 𝐵 ) 2 ] 2𝜎 2 ⁄ cos (2𝑝𝛼𝑥 ℏ𝐿 ⁄ ) (46)
The above equation (39) illustrates a Gaussian distribution of the interference phenomenon produced on the screen as confirmed by experiment.

On the other hand, once detectors are placed at the slits, the particle is detected at either slit A or slit B. Indeed, suppose that the particle at the elementary instant 𝑤 𝑀 = (𝑡 𝑀 , s 𝑀 ) happened to be at the position |𝑥 𝐴 ⟩, then this position is trapped by the interaction between the particle and the detector at that elementary instant 𝑤 𝑀 and thus, the particle is observed at slit A.

Immediately after the detection, the position at which the particle was trapped becomes stationary with respect to the state-time-indices s. That is, 𝛿 s (𝐷𝑠(𝑥, t)) = 1, ∀ s 𝜖𝐷s or equivalently the state-time-term 𝑙 𝑠 (𝑥, 𝑡) during which the particle is at the position 𝑥(𝑡) is equal to the total state-time-term 𝐿 s . Therefore, the particle may evolve only with respect to 𝑡 having only one definite position at each 𝑡 and thus, no interference can take place. After detection the electron simply behaves as a simple particle coming from slit A and impacts the screen at a specific point as confirmed by experiment.

Uncertainty

The uncertainty principle comes from a statistical distribution of the different states of a particle with respect to the state-time-indices 𝑠 at any given physical-time-index 𝑡.

Indeed, consider a particle whose position |𝑥⟩ is defined by a one-dimensional representation along the x-axis according to equations ( 27) and (28). Equation ( 27) implies that when the statetime-index s spans the state-time-domain 𝐷s, the position of the particle at a given 𝑡 can be considered as a continuous random variable that takes its values in R. Equation (28) implies that the probability distribution density of the position is given by:

𝑃(𝑥) = ‖𝜓(𝑥)‖ 2 = 𝑙 𝑠 (𝑥, 𝑡) 𝐿 𝑠 ⁄ (47)
where 𝜓(𝑥) is the wave function defining the position of the particle. It should be noted that the probability distribution 𝑃(𝑥) reflects the statistical distribution of the different positions of the particle in the state-time-domain 𝐷s at a given physical-time-index 𝑡.

Fourier analysis shows that the probability distribution of the momentum |𝑝⟩ of the particle is given by ‖𝜓 ̂(𝑝)‖ 2 where 𝜓 ̂ is the Fourier transform of 𝜓. A variance 𝑉𝑎𝑟(𝑥) for the distribution of position 𝑥 and a variance 𝑉𝑎𝑟(𝑝) for the distribution of momentum 𝑝 satisfy the following well-known [START_REF] Cohen-Tannoudji | Laloë Mécanique quantique I[END_REF][START_REF] Bricmont | Making Sense of Quantum Mechanics[END_REF] classical mathematical relation:

𝑉𝑎𝑟(𝑥)𝑉𝑎𝑟(𝑝) ≥ ℎ 4 ⁄ (48)
It should be emphasised that the above uncertainty relation (48) comes from the statistical distribution of the different positions and momenta of a single particle at a given 𝑡 taken by the different state-time-indices belonging to the time thickness. It does not come from a statistical distribution of the different positions and momenta of different particles.

The down bound of the uncertainty relation (48) is attained when the probability distribution is Gaussian. Indeed, let the probability distribution 𝑃(𝑥) of the different positions of a particle at a given 𝑡 taken with respect to different state-time-indices 𝑠 be defined by equation (30). Thus, by making abstraction of the phase term, the probability amplitude 𝜓(𝑥, 𝑡) of the particle at a given 𝑡 can be expressed as follows:

𝜓(𝑥, 𝑡) = 1 (2𝜋𝜎 𝑥 2 ) 1 4 ⁄ 𝑒 -x 2 4𝜎 𝑥 2 ⁄ ( 49
)
where 𝜎 𝑥 is the dispersion with respect to the position 𝑥 of the particle around an origin.

On the other hand, it is known [START_REF] Binney | The Physics of Quantum Mechanics[END_REF] that the normalised wave-function of a particle of momentum 𝑝 in the one-dimensional space representation is given by: ⟨𝑥|𝑝⟩ = where 𝜎 𝑝 = ℏ 2𝜎 𝑥 ⁄ and thus:

𝜎 𝑥 𝜎 𝑝 = ℏ 2 ⁄ (53)
Equation ( 53) indicates that at a given 𝑡, a single particle has a spectrum of different positions scattered with a dispersion 𝜎 𝑥 and a spectrum of different momenta scattered with a dispersion 𝜎 𝑝 such that their product is a constant equal to ℏ 2 ⁄ .

The particle has a definite position and momentum at each elementary instant 𝑤 = (𝑡, 𝑠) but of course, for each physical time-index 𝑡, there exists a plurality of state-time-indices 𝑠 at which the particle has its position and momentum scattered according to the dispersions 𝜎 𝑥 and 𝜎 𝑝 which are related in accordance to equation (53). If the position of the particle is measured, its position is confined to a small delta of space with respect to the state-time-indices whereas, the dispersion of its momentum is enlarged.

Indeed, the thickness of time provides a particle with a wave-like nature that creates interference as already discussed in section 3. Thus, a particle that has a wide range of momenta taken by the different state-time-indices 𝑠 at a given 𝑡, creates a high interference pattern leading the particle to be confined into a smaller space. Indeed, for a given 𝑡, a particle that keeps changing its momentum with respect to the state-time-indices 𝑠 stagnates around its position. In contrast, for a given 𝑡, a particle that has a definite momentum creates less interference and thus spreads out in space.

Conclusion

By assuming the existence of a time-thickness, the superposition principle and measurement become logically simple and the inconsistencies of quantum mechanics seem to vanish. Indeed, from the viewpoint of the physical-time, the particle has a plurality of equally real positions at each physical-time-index and thus seems to behave as a plurality of different particles having a wave-like movement. However, when a measurement is conducted, the particle is "captured" at one specific position with respect to the state-time-indices and thus after the measurement the position of the particle may vary only with respect to the physical-time and not with respect to the state-time-indices.

The concept of the above interpretation may be summarised by readapting the above Wheeler's citation by saying that thickness of time is particle's way to be all over "at once".

  (𝑡, s)⟩ = ∫ 𝑑𝑥 𝜓(𝑥, 𝑡, s) |𝑥⟩ = ∫ 𝑑𝑥 𝛿 𝑠 (𝐷𝑠(𝑥, 𝑡))|𝑥⟩ (27) |𝑥(𝑡)⟩ = ∫ 𝑑𝑥 𝜓(𝑥, 𝑡) |𝑥⟩ = ∫ 𝑑𝑥√𝑙 𝑠 (𝑥, 𝑡) 𝐿 𝑠 ⁄ 𝑒 𝑖𝜑(𝑡,s) |𝑥⟩ (28)

  ) becomes: |𝑥(𝑡, s)⟩ = ∫ 𝑑𝑥 𝜓(𝑥, 𝑡, s) |𝑥⟩ = ∫ 𝑑𝑥 𝛿 𝑠 (𝐷𝑠(𝑥, 𝑡))|𝑥⟩ = |𝑥⟩ (32)

  |𝑥(𝑡)⟩ = ∫ 𝑑𝑥 𝜓(𝑥, 𝑡) |𝑥⟩ = ∫ 𝑑𝑥 𝜓 𝐴 (𝑥, 𝑡) |𝑥 𝐴 ⟩ + ∫ 𝑑𝑥 𝜓 𝐵 (𝑥, 𝑡) |𝑥 𝐵 ⟩ (36) |𝑥(𝑡)⟩ = ∫ 𝑑𝑥√𝑙 𝑠𝐴 (𝑥, 𝑡) 𝐿 𝑠 ⁄ 𝑒 𝑖𝜑 𝐴 (𝑡,s) |𝑥 𝐴 ⟩ + ∫ 𝑑𝑥√𝑙 𝑠𝐵 (𝑥, 𝑡) 𝐿 𝑠 ⁄ 𝑒 𝑖𝜑 𝐵 (𝑡,s) |𝑥 𝐵 ⟩ (37)

  of momentum 𝜓(𝑝, 𝑡) of the particle at a given 𝑡 can thus be expressed as follows: 𝜓(𝑝, 𝑡) = ∫ 𝑑𝑥 above Gauss integral, the above equation (51) becomes:

  At each physical-time-index 𝑡, equation (6) (i.e. first equation of system (22)) expresses the state-vector |𝜓(𝑥, 𝑡, s)⟩ with respect to all eigenvectors |𝜓 𝑘 (𝑥)⟩ while having a single outcome at each state-time-index s. However, equation (21) (i.e. second equation of system (22)) describes the state-vector |𝜓(𝑥, 𝑡)⟩ as a superposition of eigenvectors |𝜓 𝑘 (𝑥)⟩ in a conventional manner but where the probability of a given state |𝜓 𝑘 (𝑥)⟩ is proportional to the state-time-term

				) as
	follows:
	{	|𝜓(𝑥, 𝑡, s)⟩ = ∑ 𝛿 s (𝐷s 𝑘 (t))|𝜓 𝑘 (𝑥)⟩ 𝑘 |𝜓(𝑥, 𝑡)⟩ = ∑ 𝑐 𝑗 (𝑡)|𝜓 𝑗 (𝑥)⟩ 𝑗 = ∑ √𝑙 sk 𝐿 s 𝑘 ⁄ 𝑒 𝑖𝜑 𝑘 (𝑡,s) |𝜓 𝑘 (𝑥)⟩	(22)
	In a continuous spectrum, the above set of equations becomes:
	|𝜓(𝑥, 𝑡, s)⟩ = ∫ 𝑑𝑥 𝛿 𝑠 (𝐷𝑠(𝑥, 𝑡))|𝜓(𝑥)⟩	(23)
	|𝜓(𝑥, 𝑡)⟩ = ∫ 𝑑𝑥 𝜓(𝑥, 𝑡) |𝜓(𝑥)⟩ = ∫ 𝑑𝑥√𝑙 𝑠 (𝑥, 𝑡) 𝐿 𝑠 ⁄ 𝑒 𝑖𝜑(𝑡,s) |𝜓(𝑥)⟩	(24)

  The primary wavelet represented by the state-vector |𝑥(𝑡, s)⟩ can be thus expressed as the sum of the secondary wavelets represented by the state-vectors |𝑥 𝐴 (𝑡, 𝑠)⟩ and |𝑥 𝐵 (𝑡, 𝑠)⟩ in accordance with equation (27) as follows: |𝑥(𝑡, s)⟩ = 𝛿 s (𝐷s 𝐴 (t))|𝑥 𝐴 (𝑡)⟩ + 𝛿 s (𝐷s 𝐵 (t))|𝑥 𝐵 (𝑡)⟩ = { |𝑥 𝐴 (𝑡)⟩ 𝑖𝑓 s𝜖𝐷s 𝐴 |𝑥 𝐵 (𝑡)⟩ 𝑖𝑓 s𝜖𝐷s 𝐵


Vladimir Parezanović

Olivier Cadot

published or not. The documents may come

The impact of a local perturbation on global properties of a turbulent wake

Stationary perturbation techniques are used to investigate the sensitivity of the global properties of the wake behind a bluff body at moderate Re. The entire recirculation bubble is found to be a sensitive region for the global frequency selection and the quality of the synchronization. A striking position is found at its center where synchronization is destroyed and the frequency is increased. High speed particle image velocimetry sheds some light into the underlying mechanisms, which are interpreted in terms of vorticity interaction, flow reattachment, and flow deviation. Strong mean flow modifications are observed to correspond to drastic changes in drag and its fluctuations.

The effects of a local perturbation on the global mode of two-dimensional ͑2D͒ wakes [START_REF] Strykowski | On the formation and suppression of vortex shedding at low Reynolds number[END_REF][START_REF] Hill | A theoretical approach for analyzing the restabilization of wakes[END_REF][START_REF] Chomaz | Global instabilities in spatially developing flows: Nonnormality and nonlinearity[END_REF] has received much attention since the theoretical work of Giannetti and Luchini. [START_REF] Giannetti | Structural sensitivity to the first instability of the cylinder wake[END_REF] For Reynolds number close to the first instability, the computation of the direct and adjoint global modes of the basic flow is able to retrieve the spatial structure of the sensitivity of the Bénard-von Kármán instability, obtained experimentally by Strykowski and Sreenivasan. [START_REF] Strykowski | On the formation and suppression of vortex shedding at low Reynolds number[END_REF] In the experiment, the sensitivity is deduced from the response of the global mode around the critical Reynolds number to a perturbation created by a small control cylinder. Recent theoretical developments [START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF][START_REF] Luchini | Structural sensitivity of the finiteamplitude vortex shedding behind a circular cylinder[END_REF] have shown that the basic flow modification introduced by the local perturbation represents the dominant contribution to the global mode sensitivity. The steady disturbance technique is also very successful at large Reynolds numbers to change the wake frequency and reduce the aerodynamics forces. [START_REF] Sakamoto | An optimum suppression of fluid forces by controlling a shear layer separated from a square prism[END_REF][START_REF] Thiria | Passive drag control of a blunt trailing edged cylinder[END_REF] Although the underlying mechanisms for these effects still remain unclear, the possibilities for practical applications are numerous. Using the same technique of stationary perturbations, we investigate the nature of mechanisms responsible for the modification of global properties of a large Reynolds number wake. The area explored with the control cylinder entirely covers the recirculation bubble with high spatial resolution. This enables us to discern regions of the wake, where different mechanisms assume the dominant role. We use high speed particle image velocimetry ͑PIV͒ measurements to identify these processes through vorticity dynamics of the wake. The present results are compared to the previous sensitivity analysis [START_REF] Strykowski | On the formation and suppression of vortex shedding at low Reynolds number[END_REF][START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF][START_REF] Luchini | Structural sensitivity of the finiteamplitude vortex shedding behind a circular cylinder[END_REF] performed around the critical Reynolds number.

The wake is produced by a symmetric cylinder ͑that will be referred to as the main cylinder͒ with a leading edge profiled as a semicircle with a diameter of D = 25 mm and a flat trailing edge at a right angle to the flow ͑Fig. 1͒. The shape has been chosen to avoid the displacement of the separation points due to the presence of the control cylinder ͑see Ref. 8 for a detailed description of this flow͒. The primary cylinder is placed in an Eiffel-type wind tunnel and blown over a spanwise length of 400 mm, while its total span is 600 mm. On both ends of the primary cylinder, the main velocity decreases to zero through a 1.5 cm thick boundary layer. The main velocity is 7.5 m/s and the Reynolds number of the flow is Re= UD / ϳ 12 500. The secondary ͑control͒ cylinder has a circular cross section with diameter of d =3 mm and is mounted on a 2D microcontrol displacement console. Its total span is 800 mm. Displacement is performed in 110 steps of 0.5 mm in the y direction and 45 steps of 1 mm in the x direction and covers an area located at the rear of the main body ͑Fig. 1͒. For each of the 110ϫ 45 steps, we perform simultaneous, 30 s, measurements of local pressure around the main cylinder and local velocity in the wake. A complete experiment takes about 2 days. For pressure measurements, the main cylinder is equipped with 32 pressure taps at its central section connected to two Scanivalve DSA 3217/16px devices. The form drag C D is then classically estimated by integrating the projected pressure stress over the cylinder perimeter. A hot wire anemometer, sensitive to the modulus of the velocity in the ͑x , y͒ plane, is placed at the spanwise center, 6D = 150 mm downstream of the trailing edge of the main cylinder and level with its upper edge. Autopower spectra of the velocity time series are averaged over windows of 4 s and have a resolution of 0.25 Hz. The spectra exhibit a strong peak at the vortex shedding frequency from which the mean frequency f ͑referred to as the global mode frequency͒ and the width ⌬f of the peak are extracted. The measured quantities are the global mode frequency f, the quality factor of the peak Q = f / ⌬f, the mean form drag C D , and its rate of fluctuations ␦ D = C D rms / C D . The natural wake without the control cylinder is taken as the reference case for which all corresponding quantities are indexed with 0: f 0 = 74.1 Hz, Q 0 = 35.7, C D0 = 0.76, and ␦ D0 = 0.028. The measured quantities m are nondimensionalized by taking the difference relative to the reference case ͓m = ͑mm 0 ͒ / m 0 ͔ and are mapped as a function of the control cylinder's position in Fig. 2. For all of these maps, red corresponds to an increase ͑m ˜Ͼ 0͒ and blue to a decrease ͑m Ͻ 0͒ in values. For five locations of the control cylinder, labeled as configurations 1-5 in Fig. 2͑a͒ and indicated in a͒ Electronic mail: vladimir.parezanovic@ensta.fr.

1

The impact of a local perturbation on global properties of a turbulent wake other maps, high speed PIV acquisitions of 500 fields/s are performed. In order to emphasize the coherent structure dynamics, participating in the global mode, velocity fields are phase averaged over 10 cycles. The PIV results are presented in Fig. 3 as color maps of the phase-averaged vorticity superimposed on a contour plot of the mean vorticity and the zero isoline of the mean longitudinal velocity u ¯= 0 to localize the backflow region.

The f ˜-map is shown in Fig. 2͑a͒. We can see global frequency changes in several distinct regions. Although the probe is not placed on the axis of symmetry Ox of the bluff body, the resulting map is symmetric. This shows that the frequency of the wake is space independent and is still gov-erned by a global mode. The main causes of the frequency changes are revealed by the PIV measurements and essentially consist of three mechanisms: vorticity interaction, flow deviation, and boundary layer reattachment. The vorticity interaction mechanism is clearly observable for locations 1 ͓Fig. 3͑b͔͒ and 3 ͓Fig. 3͑f͔͒ of the control cylinder. In configuration 1, the positive vorticity created by the primary cylinder is canceled by the negative one created by the control cylinder. This can be seen in the mean vorticity contours in Fig. 3͑b͒. The positive vorticity in the lower sheet is then reduced while the upper sheet remains unaffected. The coupling between them both, through the Biot and Savart law responsible for vortex shedding, [START_REF] Gerrard | The mechanics of the formation region of vortices behind bluff bodies[END_REF] is then weakened and the global frequency is reduced ͑f ˜Ͻ 0͒ ͓see location 1 in Fig. 2͑a͔͒. In configuration 3 ͓Fig. 3͑f͔͒, the positive vorticity created by the control cylinder feeds the positive vorticity sheet created by the primary cylinder. Similarly as before, this constructive interaction leads to an increase in the global frequency ͑f ˜Ͼ 0͒ ͓see location 3 in Fig. 2͑a͔͒. The fact that regions with large positive f ˜extend up to 1D downstream from the body ͑the bubble length with no control cylinder is 0.84D͒ is a consequence of the reattachment of the primary vorticity sheet on the control cylinder. It is due to the entrainment velocity from the low velocity region inside the bubble FIG. 2. ͑Color͒ Sensitivity of the global properties relative to the natural case ͑see text͒. Blue and red are, respectively, negative and positive values, white is zero. ͑a͒ Global frequency sensitivity displayed as f ˜-map; the interval between the isolines is 3% with values ranging from Ϫ10% to +20%. ͑b͒ Quality factor sensitivity Q ˜-map; values are ranging from Ϫ60% to +74%, the two displayed isolines are at Ϫ20% and +20%. ͑c͒ Mean drag change displayed as C ˜D-map; the interval between the isolines is 5% with values ranging from Ϫ45% to +8%. ͑d͒ The rate of fluctuation of drag ␦ ˜D-map; values are ranging from Ϫ90% to +350%, the four displayed isolines are at Ϫ50%, Ϫ20%, +30%, and +150%. Numbered positions correspond to the flow configurations, as shown in Fig. toward the large velocity in the shear layers. [START_REF] Gerrard | The mechanics of the formation region of vortices behind bluff bodies[END_REF] When the control cylinder is at the inner edge of the shear, as in Fig. 3͑f͒, the entrained velocity forces the shear to stick to the control cylinder. If so, the global frequency can only increase due to constructive vorticity interaction. The sheet reattachment on the control cylinder allows the separated region to be extended downstream ͓note the increase in the backflow region length in Fig. 3͑f͔͒. The frequency increase in the region around location 2 may be interpreted by flow deviation. In this case, the deviation of the flow by the control cylinder pushes the primary vortex sheet ͓Fig. 3͑d͔͒ without direct vorticity interaction. The blockage due to the presence of the control cylinder accelerates the flow and changes the vorticity distribution in the shear. For location 4, where the frequency is increased, the PIV measurements ͓Fig. 3͑c͔͒ show that the control cylinder is in the middle of the backflow region. Here the mechanism of frequency increase involves a periodic feedback effect. The vorticity created by the control cylinder is convected upstream and periodically affects the primary vortex sheet before the rolling-up process. When the distance x C is increased downstream, the frequency reduces gradually to its natural value; however, major effects are still observable on the mean flow, as, for instance, in configuration 5, for which f ˜Ϸ 0, the bubble is stretched due to a flow reattachment on the control cylinder ͓Fig. 3͑e͔͒. For positions of the control cylinder further downstream, the mean bubble shrinks gradually to its natural size.

As for sensitivity analysis obtained around the critical Reynolds number, [START_REF] Strykowski | On the formation and suppression of vortex shedding at low Reynolds number[END_REF][START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF][START_REF] Luchini | Structural sensitivity of the finiteamplitude vortex shedding behind a circular cylinder[END_REF] the global mode frequency selection is found to be sensitive to perturbations inside the bubble ͑lo-cation 4͒, in and around the shears ͑location 1 and 3͒, and outside the bubble in location 2. On the other hand, only reductions in frequency are found in Refs. 1, 5, and 6, while both increases and decreases are found in the present experiment. Near the threshold of the instability, the role of the perturbation, whatever its position is, is to stabilize the instability [START_REF] Marquet | Sensitivity analysis and passive control of cylinder flow[END_REF] and the main resulting effect reduces the frequency. For larger Reynolds numbers, one may wonder what a theoretical sensitivity analysis would predict if performed on the mean flow as done in Refs. 10 and 11 for the natural wake.

We now turn to the sharpness of the flow synchronization measured by the quality factor. It is shown by the Q ˜-map in Fig. 2͑b͒. The quality factor is 60% reduced in the blue regions and 74% increased in the red ones. The reattached flow configurations 3 and 5 have a small quality factor certainly because the reattachment to a fixed point inhibits oscillations of the velocity field. The quality is also reduced when the control cylinder is right in the middle of the bubble ͑configuration 4͒, where the vorticity feedback effect occurs and also, but less pronounced, in the two symmetric regions located at y C = Ϯ 0.33D in the vicinity of the rear of the main body. Except for these three positions, Q ˜is increased when the control cylinder is in the recirculation bubble.

Consequences on the form drag are drastic as we can notice in the C ˜D-map ͓Fig. 2͑c͔͒. The larger drag reduction ͑up to 45%͒ is associated with the interaction of the control cylinder with the primary shear layers ͑two symmetric streaks around location 3͒. For these regions, the roll-up pro- cess takes place far downstream causing an increase in the recirculation bubble length ͓Fig. 3͑f͔͒. A similar effect is observable around the position y C =0,x C = 1.26D ͓see configuration 5 in Fig. 3͑e͔͒, where the drag is also reduced. The downstream displacement of the roll-up region together with the increase in the bubble length increases the base pressure which reduces the drag. [START_REF] Roshko | Perspectives on bluff body aerodynamics[END_REF] We can notice two symmetric regions that correspond to local drag reduction maxima around location 2. As said before, the acceleration associated with the flow deviation might also increase the bubble length. Finally, we can observe two regions of small, but nevertheless significant drag increase of 8% when the control cylinder is right in the middle of one of the two eddies constituting the recirculation bubble.

The drag fluctuation rate shown in the ␦ ˜D-map in Fig. 2͑d͒ is mostly reduced except in much localized regions. For those regions ͑around location 5 and beside location 3͒, the high speed PIV shows that the recirculation bubble alternates between two states; it can either be attached to the control cylinder, having a large size with low drag, or detached with its much smaller natural size and high drag. This alternative creates a strong crisis on the drag fluctuations. For the other blue regions, the reduction in the fluctuation rate is strongly correlated with the reduction in the mean drag. The lower the drag, the larger is the bubble size, causing the unsteady roll-up process to take place further downstream. For large bubble lengths, the near wake of the main body simply becomes less unsteady.

In conclusion, the global properties of a wake at large Reynolds number are sensitive to a perturbation located in and around the shear layers or inside the recirculation bubble. This is in qualitative agreement with both previous theoretical and experimental sensitivity analyses at low Reynolds number. However, there is a disagreement with the frequency selection. A theoretical approach based on the mean flow stability analysis might be more relevant for applications at large Reynolds number. From our PIV measurements, the global mode frequency selection is attributed to vorticity interaction between the control cylinder and the pri-mary shear layers. The flow reattachment on the control cylinder can drastically increase the recirculation bubble size, which could explain the origin of mean and fluctuating drag modifications. Our experimental work creates avenues to other studies, especially about the selection of the quality factor of the shedding mode displayed in Fig. 2͑b͒. The quality factor is intimately related to the three-dimensional properties of the wake 13 that we will further investigate. This work was supported by the ANR Project No. 06-BLAN-0363-01 "HighSpeedPIV."

FIG. 1 .

 1 FIG. 1. Sketch of the experimental layout at the spanwise center section ͑D = 25 mm, d = 3 mm͒. The dashed rectangle represents the area explored by the control cylinder. The hot wire probe is depicted in the right hand side of the image.

5 FIG. 3 .

 53 FIG.3. ͑Color͒ PIV results showing the instantaneous vorticity, the mean vorticity, and the backflow region for configurations 1-5 referenced in Fig.2. Color maps display phase-averaged vorticity, with red for positive and blue for negative vorticity. The continuous lines are isolines of the mean vorticity at Ϯ2000 and Ϯ3000 s -1 . The dashed line is the zero isoline of the mean longitudinal velocity. The region inside the dashed line contains the backflow and its total length corresponds to the recirculation bubble length. ͑a͒ shows the flow without the control cylinder; the dimensionless coordinates ͑x C / D , y C / D͒ of various configurations, when the control cylinder is present, are ͑b͒ ͑0.12,Ϫ0.76͒, ͑c͒ ͑0.44,0͒, ͑d͒ ͑0.78,0.68͒, ͑e͒ ͑1.26,0͒, and ͑f͒ ͑0.78,0.4͒.