Dejan Brkić
email: dejanrgf@tesla.rcub.bg.ac.rs

Two Efficient Methods for Gas Distributive Network Calculation

Keywords: Calculation methods, Flow rate equation, Hydraulic pipeline systems, Natural gas distribution systems, Pipeline networks

Today, two very efficient methods for calculation of flow distribution per branches of a looped gas pipeline are available. Most common is improved Hardy Cross method, while the second one is so-called unified node-loop method. For a gas pipeline, gas flow rate through a pipe can be determined using Colebrook equation modified by AGA (American Gas Association) for calculation of friction factor accompanied with Darcy-Weisbach equation for pressure drop while second approach is using Renouard equation adopted for gas pipeline calculation. For the development of Renouard equation for gas pipelines some additional thermodynamic properties are involved in comparisons with Colebrook and Darcy-Weisbach model. These differences will be explained. Both equations, the Colebrook's (accompanied with Darcy-Weisbach scheme) and Renouard's will be used for calculation of flow through the pipes of one gas pipeline with eight closed loops which are formed by pipes. Consequently four different cases will be examined because the network is calculated using improved Hardy Cross method and unified node-loop method. Some remarks on optimization in this area of engineering also will be mentioned.

Introduction

A pipeline network is a collection of elements such as pipes, compressors, pumps, valves, regulators, heaters, tanks, and reservoirs interconnected in a specific way. This article is focused on pipes. The behavior of the network is governed by two factors: (i) specific characteristics of the elements and (ii) how elements are connected together. Assumption is that pipes are connected in a smooth way, i.e. so called minor hydraulic loses are neglected. The difficulty to solve the turbulent flow problem in a single pipe lies in the fact that friction factor is a complex function of relative surface roughness and the Reynolds number. Since the value of hydraulic resistance depends on flow rate, problem of flow distribution per pipes in gas distributive looped pipelines have to be solved using some kind of iterative procedures. Similar situation is with electrical resistances when diode is in circuit. With common resistors in electrical circuits where the electrical resistances are not depends on the value of electrical current in a conduit, problem is linear and no iterative procedure has to be used. So problem of flow through single tube is already complex. Despite of it, very efficient procedures are available for solution of flow problem in a complex pipeline such as looped pipeline like natural gas distribution network is. Here has to be noted that in a municipal gas pipeline, natural gas can be treated as incompressible fluid (liquid) i.e. as water. Even under this circumstance, calculation of water pipelines cannot be literary copied and applied for calculation of gas pipelines. Assumption of gas incompressibility means that it is compressed and forced to convey through pipes, but inside the pipeline system pressure drop of already compressed gas is small and hence further changes in gas density can be neglected. This means that gas is compressible fluid in general, but inside a distribution pipeline where the pressure drops can be neglected, natural gas can be treated as incompressible fluid. This is main difference between liquid and incompressible flow. According to this, water flow in pipelines is liquid incompressible flow, while the gas flow is gaseous incompressible flow. Fact is that gas is actually compressed and hence that volume of gas is decreased and then such compressed volume of gas is conveying with constant density through gas distribution pipeline. Hence, mass of gas is constant, but volume is decreased while gas density is according to this, increased.

Operate pressure for distribution gas network is 4·10 5 Pa abs i.e. 3·10 5 Pa gauge and accordingly volume of gas is decreased four times compared to volume of gas at normal conditions. Hence, velocity of gaseous fluids depends on the pressure in pipe since they are compressible (1):

          2 in 2 in st st D Q 4 D p Q p 4 v , (1)

Hydraulics frictions and gas flow rates in pipes

Each pipe is connected to two nodes at its ends. In a pipe network system, pipes are the channels used to convey fluid from one location to another. The physical characteristics of a pipe include the length, inside diameter, roughness coefficient, and minor loss coefficients. The pipe roughness coefficient is associated with the pipe material and age. When fluid is conveyed through the pipe, hydraulic energy is lost due to the friction between the moving fluid and the stationary pipe surface. This friction loss is a major energy loss in pipe flow. Losses of energy or head (pressure) losses depend on the shape, size and roughness of a channel, the velocity density and viscosity of a fluid.

Experiments show that in many cases pressure drop are approximately proportional to the square of the velocity (2):

       2 v D L p p 2 in 2 1 , (2)
Equation ( 2) is called the Darcy-Weisbach equation, named after Henry Darcy, a French engineer of the nineteenth century, and Julius Weisbach, a German mining engineer and the scientist of the same era. In previous equation velocity and gas density must be correlated, since the gas is incompressible fluid, and hence for gas is more suitable equation in next form (3) because Q·ρ=Q st ·ρ st :

st 2 2 st 5 in 2 2 5 in 2 1 Q 8 D L Q 8 D L p p                  , (3)
Density of gas can be noted as (4):

T R z M p      , (4)
Considering that gas density (4) at standard pressure conditions is equal as in average pressure in pipeline, general equation for steady-state flow of gas can be written (5) [START_REF] Coelho | Considerations about equations for steady state flow in natural gas pipelines[END_REF]:

st st st air 2 st 2 5 in 2 st 2 2 2 1 T R z M p D Q L 16 p p C               , (5)
Main parameter related to the hydraulic regime is Darcy's friction factor (λ). Note that the Darcy friction factor is defined in theory as λ=(8·τ)/(ρ·v 2 ). To predict whether flow will be laminar, hydraulically 'smooth', partially turbulent or fully turbulent, it is necessary to explore the characteristics of flow. Hydraulically 'smooth' regime is characteristic of flow through plastic, i.e. polyethylene pipes and it is also sort of turbulent regime. For the steel pipes, partially turbulent regime is most common. In considerations related to the hydraulic frictions has to be very careful because some of the authors use Darcy's friction factor while the others use Fanning's factor. The Darcy's friction coefficient is four times larger than Fanning's while the physical meaning is equal. Graphically, friction factor for known Reynolds number and relative roughness can be determined using well known Moody diagram. The Darcy friction factor and the Moody friction factor are synonyms.

Gas flow through plastic pipes

Inner surface of polyethylene pipes which are almost always used in gas distribution networks are practically smooth and hence flow regime in the typical network is hydraulically 'smooth'. For this regime is suitable Renouard's equation [START_REF] Renouard | Nouvelle méthode pour le calcul des réseaux maillés de conduites de gaz[END_REF] adjusted for natural gas flow (6):

82 . 4 in r 82 . 1 st 2 2 2 1 D L Q 4810 p p C        , (6)
In Renoard's equation adjusted for gas pipelines friction factor is rearranged in the way to be expressed using other flow parameters and also using some thermodynamic properties of natural gas. Using an absolute viscosity of µ=1.0757·10 -5 Pa·s, neglecting the potential energy term and assuming that temperature of natural gas is T avr =T st =288.15 K, pressure is p st =1.01325·10 5 Pa and compressibility factor is Z=1, general steadystate flow equation can be simplified as (8):

82 . 4 in 82 . 0 r 82 . 1 st 2 2 2 1 D L Q 4088 p p C        , (8)
It must be stressed that in the development leading to this more simplified equation ( 8), Renouard used a constant value for city gas kinematic viscosity ν=2.2·10 -5 m 2 /s. Note that kinematic gas viscosity (ν) and dynamic gas viscosity (µ) is connected using gas density (ρ). Relative density of typical natural gas is 0.64. Assumed flow temperature in Renouard's equation is 15 ºC. This means that by fixing the value of gas cinematic viscosity, the density is also kept fixed, which is physically inaccurate when considering compressible gas flows at medium or high pressure, because the cinematic viscosity of gases is highly dependent upon pressure. Every time a gas with a cinematic viscosity different from the city gas is being used, a multiplying correction factor (ν/2.2·10 -5 m 2 /s) 0.18 ·must be applied in [START_REF] Buzzelli | Calculating friction in one step[END_REF]. This product multiplied with gas density is usually called the corrected density. Estimated error in calculation of the pressure differential by using (6) instead of ( 8) is no more than 9%. The use of (6) without the viscosity correction, although quite common in the daily practice, leads to an overestimation on the calculation of the pressure drop, as the quadratic pressure difference of C is about 6% to 9% higher than the value obtained through [START_REF] Epp | Efficient code for steady flows in networks[END_REF]. Regarding to Renouard's formula has to be careful since it does not relate pressure drop but actually difference of the quadratic pressure at the input and the output of pipe. This means that C is not actually pressure drop in spite of the same unit of measurement is as used for the pressure (Pa). But, for gas pipeline calculation, fact that when C →0 that consecutive means that also C → 0 is very useful. Parameter C can be noted as pseudopressure drop.

Gas flow through steel pipes

For commercial steel pipes, Colebrook [START_REF] Colebrook | Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws[END_REF] showed the transition region of turbulence could be described by [START_REF] Brkić | An improvement of Hardy Cross method applied on looped spatial natural gas distribution networks[END_REF]:

                 D 71 . 3 Re 51 . 2 log 2 1 , (9)
This empirical equation is developed using measurements conducted by Colebrook and White [START_REF] Colebrook | Experiments with fluid friction in roughened pipes[END_REF]. Colebrook equation also can be noted as [START_REF] Wood | Hydraulic network analysis using linear theory[END_REF]:

               Re 35 . 9 D log 2 14 . 1 1 , (10)
Colebrook's equation describes a monotonic change in the friction factor from smooth to fully rough (Figure 1).

Fig. 1. Colebrook relation make transitional curve among hydraulically "smooth" regime and turbulent rough regime

It is also the basis for the widely used Moody diagram. Many researchers [START_REF] Coelho | Considerations about equations for steady state flow in natural gas pipelines[END_REF] adopt a modification of the Colebrook equation, using the 2.825 constant instead of 2.51 especially for gas flow calculation (modified Colebrook in Figure 2). Colebrook's equation for determination of hydraulic resistances is implicit in fluid flow friction factor and hence it has to be approximately solved using iterative procedure or using some of the approximate explicit formulas developed by many authors (Table 1). All presented approximations shown in Table 1 are very accurate.

                3 1 6 4 Re 10 D 10 2 1 0.0055 Moody                    Re D 88 D 53 . 0 D 094 . 0 44 . 0 225 . 0   134 . 0 D 62 . 1     Wood         Re / 15 D 715 . 3 / 1 log 2 1        Eck         9 . 0 9 . 0 Re 25 . 21 D log 2 14 . 1 1 Re 74 . 5 D 7 . 3 / 1 log 2 1               Swamee and Jain         9 . 0 Re 7 D 71 . 3 / 1 log 2 1        Churchill         9 . 0 Re 943 . 6 D 715 . 3 / 1 log 2 1        Jain       12 1 5 . 1 2 1 12 Re 8 8                 16 9 . 0 1 D 27 . 0 Re / 7 ln 457 . 2         16 2 Re / 37530   Churchill*                                 8981 . 0 1098 . 1 Re / 8506 . 5 D 8257 . 2 1 log Re 0452 . 5 D 7065 . 3 / 1 log 0 . 2 1 Chen       5 . 6 D Re 135 . 0 Re/ log 8 . 1 1        Round                                   7 . 0 52 . 0 D 29 / Re 1 Re / 7 Re/ log 518 . 4 D 7 . 3 / 1 log 2 1 Barr                                           Re / 13 D 7 . 3 / 1 log Re / 02 . 5 D 7 . 3 / 1 log Re / 02 . 5 D 7 . 3 / 1 log 2 1             Re / 13 D 7 . 3 / 1 log Re / 02 . 5 D 7 . 3 / 1 log 2 1            Zigrang and Sylvester         Re / 9 . 6 D 7 . 3 / 1 log 8 . 1 1 11 . 1        Haaland       2 1 2 3 2 1 2 1 2 /                    2 1 2 2 1 781 . 4 2 / 781 . 4 . 4                 Re / 12 D 7 . 3 / 1 log 2 1                Re / 51 . 2 D 7 . 3 / 1 log 2 1 2                  Re / 51 . 2 D 7 . 3 / 1 log 2 2 3          Serghides       Re / 82 . 96 Re / 95 D 7 . 3 / 1 log 2 1 983 . 0         Manadilli                                                                        
        Re 4587 . 0 ln D Re 124 . 0 S       Sonnad and Goudar                   / 18 . 2 1 / Re / log 2 1       D 32 . 1 1 / 41 . 1 ) Re ln 774 . 0 (                    51 . 2 Re D 7 . 3 / 1 Buzzelli [6]       D 71 . 3 / 1 10 log 2 1 S 4343 . 0                  D 7 . 3 / 1 Re S 18 . 2 log 2 1                 Re 1 . 1 1 ln Re/ 1 . 1 ln 816 . 1 Re/ ln S      Brkić [7]
*also cover laminar regime Percentage error for iterative formulas shown in Table 1 is less than 3% over the entire domain of Reynolds number [START_REF] Gregory | Alternate to standard friction factor equation[END_REF]. Exceptions are Moody, Wood, Eck and Round approximations (Figure 2). These four formulas should not be used.

Fig. 2. Distribution of error for most inaccurate approximations of the Colebrook's equation

While iterative computations are trivial in the context of current computing power, iterative estimation of friction factor can significantly increase the computational burden in complex piping network like here presented where multiple calculations are necessary. So, flow distribution problem in a complex looped gas pipeline has to be solved using an iterative procedure and further to be more complex, when Colebrook equation is used, additional iterative procedure for computing of friction pipe in every pipe has to be performed. Presented approximations (Table 1) are usually used in computer programs to avoid iterative scheme. But some computers codes continue to use Newton-Raphson iteration scheme for solving to the friction factor. For these methods, finding a good starting guess is often difficult. Initial guess and further solution can be very easily done using common software tools like MS Excel 2007. Maximal number of iterations in MS Excel 2007 is 32767. To solve for unknown friction factor λ, one must start by somehow estimating the value of friction factor on the right side of the equation, solve for the new friction factor on the left, enter the new value back on the right side, and continue this process until there is a balance on both sides of the equation within an arbitrary difference. This difference must be small without causing endless computations.

Looped gas pipeline calculations

All methods for looped gas pipeline calculations assume equilibrium among pressure and friction forces in steady and incompressible flow. As a result, they cannot be successfully used in unsteady and compressible flow calculations with large pressure drop where inertia force is important. Minor drop of pressure in the networks for gaseous fluid distribution enables to treat this fluid as incompressible, i.e. as water. Of course, some different approach must exist, but problem is not much different. Since, the resistances in hydraulic networks depend on flow, problem is not linear like in electric circuits, and iterative procedure must be used.

To solve flow distribution problem in the looped pipeline shown in figure 3, maximal consumption for each node including one of more inlet nodes has to be determined. In figure 3 inlet nodes are 1 (through pipe 20) and 5 (through pipe 21) with inlet rates shown also in Figure 3. Four outlet nodes also exist in the example network from Figure 3 and these nodes are 4, 6, 9 and 11. All other nodes are neither inlet nor outlet nodes. First assumed flows are chosen to satisfy first Kirchhoff's law (11). Pipe diameters and node input and output cannot be changed during the iterative procedure. Goal is to find final flow distribution for pipeline system (Figure 3). Second Kirchhoff's law has to be satisfied with demanded accuracy at the end of calculation (12), i.e. L {x} ≈0.

Q Q Q Q 0 Q Q Q Q Q 0 Q Q Q Q 0 Q Q Q Q 0 Q Q Q 0 Q Q Q 0 Q Q Q Q 0 Q Q Q Q 0 Q Q Q Q 0 Q Q Q 0 Q Q Q 0 Q Q Q ) 12 ( ) 11 ( ) 10 ( ) 9 ( ) 8 ( ) 7 ( ) 6 ( ) 5 ( ) 4 ( ) 3 ( ) 2 ( ) 1 ( input
) 1 ( / 9 / / 1 /                                                                           , (11) Loop Loop Loop Loop Loop Loop Loop Loop L C C C L C C C L C C C C L C C C C L C C C C L C C C C L C C C L C C C C                                                , ( 12
)                                                                                                                                                                                                                                                               
) ( ) ( 0 0 0 0 0 ) ( ' 0 ) ( 0 0 0 0 ) ( 0 ' ) ( ) ( 0 0 0 0 ) ( ) ( ' 0 ) ( 0 0 0 0 ) ( 0 ' ) ( ) ( 0 0 0 0 ) ( ) ( ' 0 ) ( 0 0 0 0 ) ( 0 ' ) ( 0 0 0 0 0 ) ( ) ( ' L L L L L L L L Q Q Q Q Q Q Q Q x L Q Q C Q Q C Q Q C L Q Q C Q Q C L Q Q C Q Q C Q Q C Q Q C L Q Q C Q Q C L Q Q C Q Q C Q Q C Q Q C L Q Q C Q Q C L Q Q C Q Q C Q Q C L , (13)

The improved Hardy Cross method

Hardy Cross iterative method with its modification by Epp and Fowler [START_REF] Epp | Efficient code for steady flows in networks[END_REF] today is widely used for calculation of fluid flow through pipes or related pipe diameters in loops-like distribution networks of conduits with known node fluid consumptions. Original Hardy Cross method is a sort of single adjustment methods in which equations are treated one by one, while the improved version treats whole system of equations simultaneously. In both version of the Hardy Cross method, results of calculation per iterations is correction of flow ΔQ rather than flow Q (in optimization problem, results of calculation per iterations is correction of pipe diameter rather than diameter). These corrections (13) should be added to or subtracted from flow (or diameter in inverse problem) calculated in previous iteration using some kind of complex algebraic rules [START_REF] Brkić | An improvement of Hardy Cross method applied on looped spatial natural gas distribution networks[END_REF]. These rules can be implemented in a MS Excel spreadsheet. Lack of space prevents here detail discussion on these rules.

The node-loop method

The node matrix with all nodes included is not linearly independent [START_REF] Wood | Hydraulic network analysis using linear theory[END_REF]. To obtain linear independence any row of the node matrix [N] has to be omitted (15). No information on the topology in that way will be lost. Node 12 will be noted as referential and hence will be virtually omitted from the calculation. For the node-loop method, matrix [V] (14) and the node-loop matrix [NL] are formed to unite both, the node matrix [N] (15) and the loop matrix [L] (16). First eleven rows in [NL] matrix are from the first Kirchhoff's law (matrix [N]), and next eight rows are from the second Kirchhoff's law (matrix [L]) where each term is multiplied with first derivative (for each pipe) of C where Q is as variable.

                                                                                                                                                           output ) 11 ( output ) 9 ( output ) 6 ( input ) 5 ( output ) 4 ( input ) 1 ( Q ' C Q ' C Q ' C L Q ' C Q ' C Q ' C L Q ' C Q ' C Q ' C Q ' C L Q ' C Q ' C Q ' C Q ' C L Q ' C Q ' C Q ' C Q ' C L Q ' C Q ' C Q ' C Q ' C L Q ' C Q ' C Q ' C C Q ' C Q ' C Q ' C Q ' C L Q 0 Q 0 0 Q Q Q 0 0 Q V , (14)                                                          0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 N (15)                                              0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 L , (16)
Now, unknown flows can be calculated directly using (17):

[Q]=inv[NL]x[V], (17)

Optimization problem

In previous text, flow distribution problem for e.g. plastic pipes is solved using (18):

   
L Q 4810 Q Q C Q p p                             , (18)
In the problem of optimization of pipe diameters, flow is not any more treated as variable (19):

   
                             , ( 19
)
Of course, some other adaptations of previously shown methods should be done for optimization problem. As the diameters have to be chosen among a finite set of available nominal values, optimization problem is highly combinatorial.

Conclusions

Compared modified Hardy Cross method and the node loop method, taking as a criterion the number of iteration to achieve final results, both presented methods are equally good. For more complex networks, using the node-loop method, number of required iteration is smaller even compared with the modified Hardy Cross method. But main strength of the node-loop method lays in the fact that it does not require complex numerical scheme for algebraic addition of corrections in each of iterations. In the node-loop method, final results of each of the iterations are flows directly and not correction of flows. Both methods can be used for calculation of gas pipelines made with steel or plastic pipes using the appropriate equation according to discussion in this paper.

Fig. 3 .

 3 Fig. 3. Example of looped gas distributive pipeline network

Nomenclaturep

  pressure, (Pa) L length of pipe, (m) D diameter of pipe, (m) Q flow (m 3 /s) T temperature (K) z gas compressibility factor (-) M relative molecular mass (-) R universal gas constant = 8314.41 J/(kmol•K) Greek symbols ε inside pipe wall roughness (m) λ Darcy (Moody) friction factor or coefficient (-) ρ density (kg/m 3 ) Subscripts and superscripts r relative st standard (T st =288.15 K, p st =101325 Pa) in inner

Table 1 .

 1 Explicit approximations to Colebrook relation.

	Relation

  

				1	/	7065 . 3			D
	1	2	log		0272 . 5	/	Re	log	1	/	567 . 4 827 . 3	/	Re	log D	1	/ 7918 3326 . 5 . 7 /	9924 . 0 Re 815 . 208 D	9345 . 0	Romeo, Royo and Monzón
	1	8686 . 0	  0 . 4587 ln	   1 S / S S / Re 	


Gisèle Mophou
email: gisele.mophou@univ-antilles.frgisele.mophou@aims-cameroon.org

Romario Gildas
email: romario.foko@aims-cameroon.org

Foko Tiomela

Ali Seibou
email: alichaibou10@gmail.com

Optimal control of averaged state of a parabolic equation with missing boundary condition

Keywords: No-regret control, Low-regret control, Average control, Euler-Lagrange first order optimality condition AMS Subject Classification 35Q93, 49J20, 93C05, 93C41

We consider the optimal control of general heat governed by an operator depend on an unknown parameter and with missing boundary condition. Using the notion of no-regret and low-regret control we prove that we can bring the average of the state of our model to a desired state.

Then by means of Euler-Lagrange first order optimality condition, we expressed the optimal control in term of average of an appropriate adjoint state that we characterize by an optimality system. The main tools are the Lebesgue dominated convergence theorem and an appropriate Hilbert space endowed with a norm containing the average of the state.

Introduction

Let N ∈ N * and Ω be a bounded open subset of R N with boundary Γ of class C 2 . Let ω be an open non-empty subset of Ω. For T > 0, we set Q = Ω × (0, T ) , ω T = ω × (0, T ) and Σ = Γ × (0, T ) . We consider the following controlled parabolic problem:

     ∂y ∂t -div(κ(α, x) y) + a 0 y = f + vχ ω in Q, y = g on Σ, y(0) = y 0 in Ω, (1)
where diffusivity κ ∈ C([0, 1] × Ω), the potential a 0 ∈ L ∞ (Q), the function

f ∈ L 2 (Q) is given, v ∈ L 2 (ω T )
is the control and χ ω is the characteristic function of the control set Ω . The initial condition y 0 is known and belongs to L 2 (Ω). The function g belongs to L 2 (Σ).

Under the assumptions on the data, we know that system (1) has a unique solution y(α, v, g) = y(α, x, t; v, g) ∈ L 2 (Q) such that,

- Ω y 0 ϕ(0)dx + Σ κ(α, x)g ∂ϕ ∂ν dσdt + Q y - ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt = Q (f + vχ ω )ϕdxdt
∀ϕ ∈ C ∞ ( Q) such that ϕ| Σ = 0 and ϕ(T ) = 0 in Ω.

System (1) is parameter dependent system with missing boundary condition. We want to act on this system in order to bring the average of the state to a desired state. More precisely, we are interested by the following problem inf

v∈L 2 (ω T ) J(v, g), ∀g ∈ L 2 (Σ), (2)
where the cost function is given by

J(v, g) = 1 0 y(α, v, g)dα -z d 2 L 2 (Q) + N v 2 L 2 (ω T ) , (3)
with z d ∈ L 2 (Q), N > 0 and

1 0 y(α, v, g)dα ∈ L 2 (Q) ∀v ∈ L 2 (ω T ) and ∀ g ∈ L 2 (Σ).
As [START_REF] Lions | Contrôle à moindres regrets des systèmes distribués[END_REF] has no sense, we will study this problem combining the notion of noregret and low-regret control introduce by J.L lion [START_REF] Lions | Contrôle à moindres regrets des systèmes distribués[END_REF] to control problem with incomplete data with the notion of averaged control introduce by E. Zuazua [START_REF] Zuazua | Averaged control[END_REF] to control for parameter dependent system. So we first consider for any γ > 0 the low-regret problem:

inf v∈L 2 (ω T ) sup g∈L 2 (Σ) J(v, g) -J(0, g) -γ||g|| 2 L 2 (Σ)
.

(

) 4
Then we prove that the low-regret control converge to the no-regret control solution of the problem:

inf v∈L 2 (ω T ) sup g∈L 2 (Σ) (J(v, g) -J(0, g)) , (5)
that we characterize by giving the corresponding optimality system. The notion of no-regret and low regret control was introduced by J. L. Lions [START_REF] Lions | Contrôle à moindres regrets des systèmes distribués[END_REF] to control a parabolic equation governed with an operator free of unknown parameter but with unknown initial condition. According to J.L. Lions, by looking for such a control, one looks for the "best possible control" v which do "at least as well" and "not much worse in the worst situation" than doing nothing. This notion was then applied to control some model with incomplete data, including model involving fractional derivative in time. We refer for instance to [START_REF] Nakoulima | Perturbations à moindres regrets dans les systèmes distribués à données manquantes[END_REF][START_REF] Nakoulima | Low-regret control of singular distributed systems: the ill-posed backwards heat problem[END_REF][START_REF] Nakoulima | No-regret control for nonlinear distributed systems with incomplete data[END_REF][START_REF] Nakoulima | Low-regret perturbations in distributed systems with incomplete data[END_REF][START_REF] Jacob | Optimal control for age-structured population dynamics of incomplete data[END_REF][START_REF] Mophou | Optimal control for fractional diffusion equations with incomplete data[END_REF][START_REF] Baleanu | Low-regret control for a fractional wave equation with incomplete data[END_REF]. The averaged control notion was introduced by E. Zuazua [START_REF] Zuazua | Averaged control[END_REF] to analyse the problem of controlling parameter dependent systems.

In this notion, one is interested by the average of the state with respect to the unknown parameter. for more literature on this topic we refer for instance to [START_REF] Lazar | Averaged control and observation of parameterdepending wave equations[END_REF][START_REF] Lü | Averaged controllability for random evolution partial differential equations[END_REF][START_REF] Lohéac | Averaged controllability of parameter dependent conservative semigroups[END_REF] and the reference therein. In this paper we are concerned with the control of a parameter dependent system with missing boundary condition. As far as we know this problem is new. The rest of this paper is stated as follows. In section 2, we reformulate the low-regret and no-regret control problem (4) and [START_REF] Dorville | Low-regret control for singular distributedsystems: the backwards heat ill posed problem[END_REF]. Section 3 is devoted to the proof of the existence of the low-regret control and its characterization. In Section 4, we prove that the low-regret control converges towards the no-regret control that we characterize with an optimality system. Concluding remarks are made in Section 5

Reformulation of the No regret and Low-regret control

In this section we prove using Legendre-Fenchel transform that the inf-sup problem ( 4) is equivalent to a classical optimal control problem.

So, let f ∈ L 2 (Q), v ∈ L 2 (ω T ), g ∈ L 2 (Σ), a 0 ∈ L ∞ (Q), κ ∈ C([0, 1]
× Ω) and y 0 ∈ L 2 (Ω). Let also y(α, v, 0), y(α, 0, g), y(α, 0, 0) be respectively solutions of:

     ∂y(α, v, 0) ∂t -div(κ(α, x) y(α, v, 0) + a 0 y(α, v, 0) = f + vχ ω in Q, y(α, v, 0) = 0 on Σ, y(0; α, v, 0) = y 0 in Ω, (6)      ∂y(α, 0, g) ∂t -div(κ(α, x) y(α, 0, g) + a 0 y(α, 0, g) = f in Q, y(α, 0, g) = g on Σ, y(0; α, 0, g) = y 0 in Ω (7) and      ∂y(α, 0, 0) ∂t -div(κ(α, x) y(α, 0, 0) + a 0 y(α, 0, 0) = f in Q, y(α, 0, 0) = 0 on Σ, y(0; α, 0, 0) = y 0 in Ω. ( 8
)
Then for almost every α ∈ [0, 1], the functions y(α, v, 0) and y(α, 0, 0) belong to L 2 ((0, T );

H 1 0 (Ω)) ∩ C([0, T ]; L 2 (Ω)
) and we can prove by transposition that y(α, 0, g) belongs to L 2 (Q) and is such that,

- Ω y 0 ϕ(0)dx + Σ κ(α, x)g ∂ϕ ∂ν dσdt + Q y - ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt = Q f ϕdxdt, ∀ϕ ∈ C ∞ ( Q) such that ϕ| Σ = 0 and ϕ(T ) = 0 in Ω.
.

Remark 1 Note that 1 0 y(α, v, 0)dα, 1 0
y(α, 0, g)dα and

1 0 y(α, 0, 0)dα be- long to L 2 (Q) because 1 0 y(α, v, g)dα ∈ L 2 (Q) for any v ∈ L 2 (ω T ) and g ∈ L 2 (Σ).
Remark 2 Using Holder inequality, we have that if there exists a constant C independent of α such that y L 2 (Q) ≤ C then 1 0 y(α, ., .)dα

L 2 (Q) ≤ C.
Lemma 2.1 For any v ∈ L 2 (ω T ) and for any g ∈ L 2 (Σ), we have:

J(v, g) -J(0, g) = J(v, 0) -J(0, 0) + 2 Σ 1 0 κ(α, x) ∂ζ(α, v) ∂ν dα gdσdt, ( 9
)
where J is the cost function given by (3) and ζ(α, v)

:= ζ(x, t; α, v) ∈ Z 2,1 (Q) =: L 2 ((0, T ); H 2 (Ω) ∩ H 1 0 (Ω)) ∩ H 1 ((0, T ); L 2 (Ω)) is solution of        - ∂ζ(α, v) ∂t -div(κ(α, x) ζ(α, v) + a 0 ζ(α, v) = - 1 0 η(α, v)dα in Q, ζ(α, v) = 0 on Σ, ζ(T ; α, v) = 0 in Ω, ( 10
) with η(α, v) = y(α, v, 0) -y(α, 0, 0). ( 11
)
Remark 3 Note that in view of Remark 2, we have that

1 0 η(α, v)dα ∈ L 2 (Q).
Proof. Observing that y(α, v, g) = y(α, v, 0) + y(α, 0, g) -y(α, 0, 0), we have

J(v, g) = 1 0 (y(α, v, 0) + y(α, 0, g) -y(α, 0, 0))dα -z d 2 L 2 (Q) + N v 2 L 2 (ω T ) = 1 0 y(α, v, 0)dα -z d 2 L 2 (Q) + N v 2 L 2 (ω T ) + 1 0 y(α, 0, g)dα -z d - 1 0 y(α, 0, 0)dα -z d 2 L 2 (Q) + 2 Q 1 0 y(α, v, 0)dα -z d 1 0
(y(α, 0, g) -y(α, 0, 0))dα dxdt

= J(v, 0) + 1 0 y(α, 0, g)dα -z d 2 L 2 (Q) + 1 0 y(α, 0, 0)dα -z d 2 L 2 (Q) -2 Q 1 0 y(α, 0, g)dα -z d 1 0 y(α, 0, 0)dα -z d dxdt + 2 Q 1 0 (y(α, v, 0) -y(α, 0, 0))dα 1 0 (y(α, 0, g) -y(α, 0, 0))dα dxdt + 2 Q 1 0 (y(α, 0, 0)dα -z d ) 1 0 (y(α, 0, g) -y(α, 0, 0))dα dxdt = J(v, 0) + J(0, g) -J(0, 0) + 2 Q 1 0
(y(α, v, 0) -y(α, 0, 0))dα 1 0 (y(α, 0, g) -y(α, 0, 0))dα dxdt Consequently,

J(v, g) -J(0, g) = J(v, 0) -J(0, 0)+ 2 Q 1 0 (y(α, v, 0) -y(α, 0, 0))dα 1 0
(y(α, 0, g) -y(α, 0, 0))dα dxdt.

(12) In view of Remark 3, system (10) has a unique solution ζ(α, v) ∈ Z 2,1 (Q). Moreover, there exists a constant C > 0 independent of α such that

ζ(α, v) L 2 ((0,T );H 2 (Ω)) ≤ C v L 2 (ω T ) . (13)
Therefore, using the continuity of the trace operator, we have that ∂ζ(α, v) ∂ν

L 2 (Σ) ≤ C v L 2 (ω T ) . (14)
We set π(α, g) = y(α, 0, g) -y(α, 0, 0).

In view of Remark 2, we have that

1 0 π(α, g)dα ∈ L 2 (Q). Observing that π(α, g) is solution of      ∂π ∂t -div(κ(α, x) π) + a 0 π = 0 in Q, π = g on Σ, π(0) = 0 in Ω, ( 15
)
if we multiply the first equation of ( 10) by 1 0 π(α, g)dα and we integrate over Q, we have:

- Q 1 0 (y(α, v, 0) -y(α, 0, 0))dα 1 0 (y(α, 0, g) -y(α, 0, 0))dα dxdt = 1 0 Q - ∂ζ(α, v) ∂t -div(κ(α, x) ζ(α, v) + a 0 ζ(α, v) π(α, g)dxdt dα = - Σ 1 0 κ(α, x) ∂ζ(α, v) ∂ν dα gdσdt Hence, Q 1 0 (y(α, v, 0) -y(α, 0, 0))dα 1 0 (y(α, 0, g) -y(α, 0, 0))dα dxdt = Σ 1 0 κ(α, x) ∂ζ(α, v) ∂ν dα gdσdt. (16)
If we combine ( 16) and ( 12), we obtain that

J(v, g) -J(0, g) = J(v, 0) -J(0, 0) + 2 Σ 1 0 κ(α, x) ∂ζ(α, v) ∂ν dα gdσdt. Remark 4 As κ ∈ C([0, 1]×Ω), using (14), we have that κ ∂ζ(α,v) ∂ν is bounded in L 2 (Σ) independently of α. It then follows from Remark 1 that 1 0 κ(α, x) ∂ζ(α, v) ∂ν dα ∈ L 2 (Σ)
. This means that (9) has a sense.

From now on, we set

∂ ζ(α, v) ∂ν = 1 0 κ(α, x) ∂ζ(α, v) ∂ν dα. (17)
Then according to [START_REF] Lions | Contrôle à moindres regrets des systèmes distribués[END_REF], problem ( 5) is equivalent to:

inf v∈L 2 (ω T ) sup g∈L 2 (Σ) J(v, 0) -J(0, 0) + 2 Σ ∂ ζ(α, v) ∂ν gdσdt . (18)
Consequently, the no-regret control belongs to the set

O = v ∈ L 2 (ω T ) such that ∂ ζ(α, v) ∂ν is orthogonal to L 2 (Σ) . (19)
As such a control is not easy to characterize, we consider the low-regret control problem. Using [START_REF] Lions | Contrôle à moindres regrets des systèmes distribués[END_REF], problem (4) could be rewritten as inf

v∈L 2 (ω T ) (J(v, 0) -J(0, 0) +2 sup g∈L 2 (Σ) Σ ∂ ζ(α, v) ∂ν gdσdt - γ 2 ||g|| 2 L 2 (Σ)
, which by means of Legendre-Fenchel transform is equivalent to the following problem: For any γ > 0, find

u γ ∈ L 2 (Q) such that J γ (u γ ) = inf v∈L 2 (ω T ) J γ (v), (20)
where

J γ (v) = J(v, 0) -J(0, 0) + 1 γ ∂ ζ(α, v) ∂ν 2 L 2 (Σ) . (21)

Existence and characterization of the low-regret control

Our main objective in this section will be to show that the low-regret problem (20) has a unique solution which converges to the no-regret control unique solution of [START_REF] Lazar | Averaged control and observation of parameterdepending wave equations[END_REF]. Furthermore, we will give the equations that characterize the low-regret control.

Remark 5 From now on, we use C(X) to denote a positive constant whose value varies from a line to another but depends on X.

Proposition 3.1 There exists a unique low-regret control u γ ∈ L 2 (ω T ) solution of (20).

Proof. Observing that We have:

-J(0, 0) ≤ J(v, 0) -J(0, 0) + 1 γ 1 0 κ(α, x) ∂ζ(α, v) ∂ν dα 2 L 2 (Σ) ∀v ∈ L 2 (ω T ), we have that the set J γ (v), J γ (v) ≥ -J(0, 0), v ∈ L 2 (ω T ) is not empty. Con- sequently, there exists m γ such that m γ = inf v∈L 2 (ω T ) J γ (v).
Let v n ∈ L 2 (ω T ) be a minimizing sequence such that

J γ (v n ) -→ m γ . (22)
Then, y n := y(x, t; α, v n , 0) satisfies:

     ∂y n ∂t -div(κ(α, x) y n ) + a 0 y n = f + v n χ ω in Q, y n = 0 on Σ, y n (0) = y 0 in Ω. ( 23
) Let us set ζn = ζ(v n ) = 1 0 κ(α, x) ∂ζ(α, v n ) ∂ν dα where ζ n = ζ(α, v n ) is
solution of [START_REF] Nakoulima | Low-regret control of singular distributed systems: the ill-posed backwards heat problem[END_REF]. In view of ( 22),there exists a constant C > 0 independent of n and γ such that

-J(0, 0) ≤ J γ (v n ) ≤ C which implies: 0 ≤ J(v n , 0) + 1 γ ∂ ζn ∂ν 2 L 2 (Σ) ≤ C + J(0, 0).
Hence there exists a non negative constant C independent of n and γ , such that:

0 ≤ J(v n , 0) + 1 γ ∂ ζn ∂ν 2 L 2 (Σ) ≤ C, which implies that J(v n , 0) ≤ C, 1 γ ∂ ζn ∂ν 2 L 2 (Σ) ≤ C.
Hence, using the fact that

J(v n , 0) = 1 0 y(α, v n , 0)dα -z d 2 L 2 (Q) + N v n 2 L 2 (ω T ) ,
we deduce that

||v n || L 2 (ω T ) ≤ C, (24)
∂ ζn ∂ν

L 2 (Σ) ≤ C √ γ, (25) 1 0
y n (α, ., .)dα

L 2 (Q) ≤ C. (26)
In view of ( 23) and (24), there exists C > 0 independent of n and γ such that

||y n || L 2 ((0,T );H 1 0 (Ω)) ≤ C. (27)
Hence, there exist u γ ∈ L 2 (ω T ), y γ ∈ L 2 ((0, T ); H 1 0 (Ω)) and subsequences extracted from (v n ) and (y n ) (still denoted (v n ) and (y n )) such that

v n u γ weakly in L 2 (ω T ), (28)
y n y γ weakly in L 2 ((0, T ); H 1 0 (Ω)). ( 29
)
We set z n (x, t) = 1 0 y n (α, x, t)dα. Then in view of (26) there exist

z γ ∈ L 2 (Q) such that Q z n (x, t)ϕ(x, t)dxdt = 1 0 Q y n (x, t)ϕ(x, t)dxdt → Q z γ (x, t)ϕ(x, t)dxdt ∀ϕ ∈ L 2 (Q).
(30) On the other hand, using the fact that y n is bounded independently of α and the fact that

Q y n (x, t)ϕ(x, t)dxdt → Q y γ (x, t)ϕ(x, t)dxdt ∀ϕ ∈ L 2 (Q), (31)
it follows from the Lebesgue dominated convergence theorem that

1 0 Q y n (α, x, t)ϕ(x, t)dxdt dα = Q 1 0 y n (α, x, t)dα ϕ(x, t)dxdt converges towards 1 0 Q y γ (α, x, t)ϕ(x, t)dxdt dα = Q 1 0 y γ (α, x, t)dα ϕ(x, t)dxdt.
Therefore, from (30) and the uniqueness of the limit, we have that

1 0 y γ (α, x, t)dα = z γ (x, t). (32)
We thus can write

Q 1 0 y(α, v n , 0)ϕdαdxdt → Q 1 0 y(α, u γ , 0)ϕ(x, t)dαdxdt. ( 33
)
The rest of the proof will be divided into steps.

Step 1. We prove that (u γ , y γ ) satisfies [START_REF] Dorville | Contrôle optimal pour les problèmes de contrôlabilité des systèmesdistribués à données manquantes[END_REF]. Let D(Q) being the set of functions of class C ∞ on Q with compact support. Multiplying the first equation in (23) by ϕ ∈ D(Q) and integrating by parts over Q,

Q y n - ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt = Q (f + v n χ ω )ϕdxdt
Passing this latter identity to the limit when n → ∞ while using (28) and (29), we obtain that

Q y γ - ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt = Q (f + u γ χ ω )ϕdxdt
which after integration by parts gives over Q,

Q ∂y γ ∂t -div(κ(α, x) y γ ) + a 0 y γ ϕdxdt = Q (f + u γ χ ω )ϕdxdt.
Hence,we deduce that

∂y γ ∂t -div(κ(α, x) y γ ) + a 0 y γ = f + u γ χ ω in Q. ( 34
)
Now, as, y γ ∈ L 2 ((0, T ), H 1 0 (Ω)), using (34), we have that ∂y γ ∂t ∈ L 2 ((0, T ), H -1 (Ω)). This implies that y γ ∈ C([0, T ], L 2 (Ω)). On the other hand, y γ ∈ L 2 (Q) and div(κ(α, x) y γ ) ∈ H -1 ((0, T ), L 2 (Ω)). This implies that y γ | Σ and ∂y γ ∂ν | Σ exist and belong respectively to H -1 ((0, T ); H -1 2 (Γ)) and H -1 ((0, T ); H -3 2 (Γ)). Now, let ϕ ∈ C ∞ ( Q) with ϕ| Σ = 0 and ϕ(T ) = 0 in Ω. If we multiply the first equation in ( 23) by ϕ and we integrate over Q, we obtain:

T 0 Ω (f + v n χ ω )ϕdxdt = - Ω y 0 ϕ(x, 0)dx + T 0 Ω y n - ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt,
which by passing to the limit when n -→ ∞ while using (28) and (29) gives

T 0 Ω (f + u γ χ ω )ϕdxdt = - Ω y 0 ϕ(x, 0)dx + T 0 Ω y γ - ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt, ∀ϕ ∈ C ∞ ( Q) with ϕ| Σ = 0 and ϕ(T ) = 0 in Ω.
If we integrate the last integral in this latter identity, we obtain that

T 0 Ω (f + u γ χ ω )ϕdxdt = -Ω y 0 ϕ(x, 0)dx + Ω y γ (x, 0)ϕ(x, 0)dx - y γ , κ(α, x) ∂ϕ ∂ν H -3/2 (0,T );H -1 2 (Γ) ,H 3/2 (0,T );H 1 2 (Γ) + T 0 Ω ϕ ∂y γ ∂t -div(κ(α, x) y γ ) + a 0 y γ dxdt, ∀ϕ ∈ C ∞ ( Q) with ϕ| Σ = 0 and ϕ(T ) = 0 in Ω.
which in view of (34) yields 0 = Ω (-y 0 + y γ (0))ϕ(x, 0)dx

- y γ , κ(α, x) ∂ϕ ∂ν H -3/2 (0,T );H -1 2 (Γ) ,H 3/2 (0,T );H 1 2 (Γ) , ∀ϕ ∈ C ∞ ( Q) with ϕ| Σ = 0 and ϕ(T ) = 0 in Ω. ( 35
)
If we take in addition ∂ϕ ∂ν = 0 on Σ, we have Ω (-y 0 + y γ (0))ϕ(x, 0)dx = 0, which implies:

y γ (x, 0) = y 0 (x), x ∈ Ω. ( 36
)
It then follows from (35) that

y γ = 0 on Σ. ( 37
)
Now, combining (34), ( 36) and (37), we find that y γ = y(α, u γ , 0) is solution of ( 6). This means that

     ∂y γ ∂t -div(κ(α, x) y γ ) + a 0 y γ = f + u γ χ ω in Q, y γ = 0 on Σ, y γ (0) = y 0 in Ω . (38)
Step 2. We show that ζ(α, v n ) converges to ζ γ = ζ(α, u γ ) which satisfies [START_REF] Nakoulima | Low-regret control of singular distributed systems: the ill-posed backwards heat problem[END_REF].

We know that

ζ n := ζ(x, t; α, v n ) is solution of        - ∂ζ n ∂t -div(κ(α, x) ζ n + a 0 ζ n = - 1 0 (y n -y(α, 0, 0))dα in Q, ζ n = 0 on Σ, ζ n (T ) = 0
in Ω, (39) Therefore, using (26), we can say that there exists a positive constant C independent of n and γ such that

ζ n L 2 ((0,T );H 2 (Ω)) ≤ C. ( 40
)
And it follows from the continuity of the trace that

∂ζ n ∂ν L 2 (Σ) ≤ C. (41)
Consequently,there exists ζ γ ∈ L 2 ((0, T ); H 2 (Ω)) and a subsequence extracted from

(ζ(α, v n )) (still denoted ζ(α, v n )) such that ζ n ζ γ weakly in L 2 ((0, T ); H 2 (Ω)). ( 42
)
Multiplying the first equation in (39) by ϕ ∈ D(Q) and integrating by parts over Q,

Q ζ n ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt = - Q 1 0 (y n -y(α, 0, 0))dα ϕdxdt
Passing this latter identity to the limit when n → ∞ while using (42), ( 30) and (33), we obtain that

Q ζ γ ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt = - Q 1 0
(y γ -y(α, 0, 0))dα ϕdxdt which after integration by parts gives over Q,

Q - ∂ζ γ ∂t -div(κ(α, x) ζ γ ) + a 0 ζ γ ϕdxdt = - Q 1 0
(y γ -y(α, 0, 0))dα ϕdxdt.

Hence,we deduce that

- ∂ζ γ ∂t -div(κ(α, x) ζ γ ) + a 0 ζ γ = - 1 0 (y γ -y(α, 0, 0))dα in Q. (43)
Now, as ζ γ ∈ L 2 ((0, T ), H 2 (Ω)), using (43), we have that ∂y γ ∂t ∈ L 2 ((0, T ), L 2 (Ω)). This implies that y γ ∈ C([0, T ], L 2 (Ω)). On the other hand, we have that

y γ | Σ ∈ L 2 (Σ).
Now, let ϕ ∈ C ∞ ( Q) with ϕ| Σ = 0 and ϕ(0) = 0 in Ω. If we multiply the first equation in (23) by ϕ and we integrate over Q, we obtain:

- Q 1 0 (y n -y(α, 0, 0))dα ϕdxdt = T 0 Ω
ζ n ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt, which by passing to the limit when n -→ ∞ while using (42), (30) and (33) gives

- Q 1 0 (y γ -y(α, 0, 0))dα ϕdxdt = T 0 Ω ζ γ ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt, ∀ϕ ∈ C ∞ ( Q) with ϕ| Σ = 0 and ϕ(0) = 0 in Ω.
If we integrate the last integral in this latter identity, we obtain that

- Q 1 0 (y γ -y(α, 0, 0))dα ϕdxdt = Ω ζ γ (T )ϕ(x, T )dx - Σ ζ γ κ(α, x) ∂ϕ ∂ν dσdt + T 0 Ω ϕ - ∂ζ γ ∂t -div(κ(α, x) ζ γ ) + a 0 ζ γ dxdt, ∀ϕ ∈ C ∞ ( Q) with ϕ| Σ = 0 and ϕ(0) = 0 in Ω.
which in view of (43) yields

0 = Ω ζ γ (T )ϕ(x, T )dx - Σ ζ γ κ(α, x) ∂ϕ ∂ν dσdt, ∀ϕ ∈ C ∞ ( Q) with ϕ| Σ = 0 and ϕ(0) = 0 in Ω. ( 44
)
If we take in addition ∂ϕ ∂ν = 0 on Σ, we have

Ω ζ γ (T )ϕ(x, T )dx = 0, which implies: ζ γ (x, T ) = 0, x ∈ Ω. ( 45
)
It the follows from (44) that

ζ γ = 0 on Σ. ( 46
)
Now, combining (43), ( 45) and ( 46), we find that

ζ γ = ζ(α, u γ ) is solution of        - ∂ζ γ ∂t -div(κ(α, x) ζ γ ) + a 0 ζ γ = - 1 0 η(α, u γ )dα in Q, ζ γ = 0 on Σ , ζ γ (T ) = 0 in Ω, ( 47
)
where as in [START_REF] Nakoulima | No-regret control for nonlinear distributed systems with incomplete data[END_REF], η(α, u γ ) = y(α, u γ , 0) -y(α, 0, 0).

Moreover, using (41), we prove that

∂ζ n ∂ν ∂ζ γ ∂ν weakly in L 2 (Σ). ( 48
)
In view of (25), there exist β ∈ L 2 (Σ) such that

∂ ζn ∂ν β weakly in L 2 (Σ). ( 49
)
This means that when n → ∞,

Σ 1 0 κ(α, x) ∂ζ n ∂ν dα ϕdσdt → Σ βϕdσdt, ∀ϕ ∈ L 2 (Σ) (50) because ∂ ζn ∂ν = 1 0 κ(α, x) ∂ζ n ∂ν dα.
Using (41) and the fact that κ ∈ C([0, 1] × Ω), we have that

κ(α, x) ∂ζ n ∂ν L 2 (Σ) ≤ C ∂ζ n ∂ν L 2 (Σ) ≤ C. ( 51
)
Hence it follows from (48) that

Σ (κ(α, x) ∂ζ n ∂ν )ϕdσdt → Σ (κ(α, x) ∂ζ γ ∂ν )ϕdσdt, ∀ϕ ∈ L 2 (Σ). ( 52
)
Using the Lebesgue dominated convergence theorem and the uniqueness of the limit, we deduce that

β = ∂ ζγ ∂ν = 1 0 κ(α, x) ∂ζ γ ∂ν dα. ( 53
)
Step 3. We prove that u γ is unique.

Using the weak lower semi-continuity of the function J γ (v), ( 30), ( 32), ( 28), ( 49) and ( 53), we deduce that

J γ (u γ ) ≤ lim inf n→∞ J γ (v n ),
where

J γ (u γ ) = 1 0 y(α, u γ , 0)dα-z d 2 L 2 (Q) -J(0, 0)+ 1 γ ∂ ζγ ∂ν 2 L 2 (Σ) +N u γ 2 L 2 (ω T ) .
As lim inf

n→∞ J γ (v n ) = lim n→∞ J γ (v n ) = inf v∈L 2 (ω T )
J γ (v), we obtain that u γ is solution of (20). The strictly convexity of J γ allows us to conclude that u γ is unique.

We can now characterize the low-regret control u γ .

Proposition 3.2 Let u γ be the solution of (20). Then there exist

p γ ∈ L 2 (Q) and q γ ∈ Z 2,1 (Q) such that {y γ , ζ γ , p γ , q γ } is solution to      ∂y γ ∂t -div(κ(α, x) y γ ) + a 0 y γ = f + u γ χ ω in Q, y γ = 0 on Σ, y γ (0) = y 0 in Ω, (54)
       - ∂ζ γ ∂t -div(κ(α, x) ζ γ ) + a 0 ζ γ = - 1 0 η(α, u γ )dα in Q, ζ γ = 0 on Σ, ζ γ (T ) = 0 in Ω. ( 55
)          ∂p γ ∂t -div(κ(α, x) p γ ) + a 0 p γ = 0 in Q, p γ = 1 √ γ ∂ ζγ ∂ν on Σ, p γ (0) = 0 in Ω, (56)
       - ∂q γ ∂t -div(κ(α, x) q γ ) + a 0 q γ = 1 0 y γ + p γ √ γ dα -z d in Q, q γ = 0 on Σ, q γ (T ) = 0
in Ω (57) and

u γ = - 1 N 1 0 q γ dα in ω T , (58)
where q γ := q γ (α, u γ ) and p γ := p γ (α, u γ ).

Proof. We have already proved in the Step 1 of Proposition 3.1 that the the state y γ = y(α, u γ , 0) associate to optimal control u γ satisfies (54). In Step 2, we obtain that ζ γ = ζ(α, u γ ) satisfies (55). To prove (56)-(58), we write the Euler-Lagrange first order optimality condition which characterizes the optimal control u γ :

lim θ→0 J γ (u γ + θv) -J γ (u γ ) θ = 0 ∀v ∈ L 2 (ω T ). (59)
After some calculations

Q 1 0 y(α, u γ , 0)dα -z d 1 0 ȳ(α, v)dα dxdt + 1 γ Σ 1 0 κ(α, x) ∂ ζ ∂ν (α, v)dα 1 0 κ(α, x) ∂ζ ∂ν (α, u γ )dα dσdt +N ω T u γ v dxdt = 0 ∀ v ∈ L 2 (ω T ), (60)
where ȳ(α, v) is solution to

     ∂ ȳ ∂t -div(κ(α, x) ȳ) + a 0 ȳ = vχ ω in Q, ȳ = 0 on Σ, ȳ(0) = 0 in Ω, (61) and ζ
(α, v) is solution to        - ∂ ζ ∂t -div(κ(α, x) ζ) + a 0 ζ = - 1 0 ȳ(α, v)dα in Q, ζ = 0 on Σ, ζ(T ) = 0 in Ω. (62)
Now, if we multiply the first equation in (61) and in (62) by

1 0 q γ (α, u γ )dα ∈ L 2 (Q) and 1 √ γ 1 0
p γ (α, u γ )dα respectively, then integrate by parts over Q, we obtain

Q 1 0 ȳ(α, v)dα 1 0 y(α, u γ , 0) + 1 √ γ p γ (α, u γ ) dα -z d dxdt = ω T v 1 0 q γ (α, u γ )dα dxdt, (63) and Σ
- 1 γ 1 0 κ(α, x) ∂ ζ ∂ν (α, v) dα 1 0 κ(α, x) ∂ζ ∂ν (α, u γ )dα dσdt = Q 1 0 ȳ(α, v)dα 1 √ γ 1 0 p γ dα dxdt. (64)
Using ( 73), ( 74) and (54), we show as in pages 9-11 that ŷ = y(α, û, 0) satisfies

∂ ŷ ∂t -div(κ(α, x) ŷ) + a 0 ŷ = f + ûχ ω in Q, ŷ = 0 on Σ, ŷ(0) = y 0 in Ω . (75)
Moreover, using (71), (74) and the Lebesgue dominated convergence theorem, we have that

1 0 y(α, u γ , 0)dα 1 0 y(α, û, 0)dα weakly in L 2 (Q). (76)
From ( 71) and (55), we prove that there exists a positive constant

C such that ζ(α, u γ ) L 2 ((0,T );H 2 (Ω)) ≤ C. (77)
Therefore the continuity of the trace allows us to write that

∂ζ γ ∂ν L 2 (Σ) ≤ C. (78)
Consequently, there exists ζ ∈ L 2 ((0, T );

H 2 (Ω)) such that ζ(α, u γ ) ζ weakly in L 2 ((0, T ); H 2 (Ω)). (79)
Therefore proceeding as for ζ n in pages 11-14, we prove that ζ = ζ(α, û) is solution to

       - ∂ ζ ∂t -div(κ(α, x) ζ) + a 0 ζ = - 1 0 (ŷ -y(α, 0, 0))dα in Q, ζ = 0 on Σ , ζ(T ) = 0 in Ω. (80)
Moreover, using (78), we have the following

∂ζ γ ∂ν ∂ ζ ∂ν weakly in L 2 (Σ). (81)
In view of (70),

∂ ζγ ∂ν → 0 strongly in L 2 (Σ). ( 82
) Because ∂ ζγ ∂ν = 1 0 κ(α, x) ∂ζ γ ∂ν dα. Using (78) and the fact that κ ∈ C([0, 1]×Ω),
we have that

κ(α, x) ∂ζ γ ∂ν L 2 (Σ) ≤ C ∂ζ n ∂ν L 2 (Σ) ≤ C. (83)
Hence it follows from (81) that

Σ κ(α, x) ∂ζ γ ∂ν ϕdσdt -→ Σ κ(α, x) ∂ ζ ∂ν ϕdσdt, ∀ϕ ∈ L 2 (Σ). ( 84
)
Using Lebesgue dominated convergence theorem and (82), we deduce that

0 = 1 0 κ(α, x) ∂ ζ ∂ν dα, (85)
which in view of [START_REF] Lü | Averaged controllability for random evolution partial differential equations[END_REF] allows to say that û ∈ O. Therefore, we conclude that the low-regret control u γ converges to the no-regret control û. Proposition 4.2 The no-regret control û, solution of (5) (or equivalently [START_REF] Lazar | Averaged control and observation of parameterdepending wave equations[END_REF]), is characterized by following optimality system:

     ∂ ŷ ∂t -div(κ(α, x) ŷ) + a 0 ŷ = f + ûχ ω in Q, ŷ = 0 on Σ, ŷ(0) = y 0 in Ω, (86)
       - ∂ ζ ∂t -div(κ(α, x) ζ) + a 0 ζ = - 1 0 (ŷ -y(α, 0, 0))dα in Q, ζ = 0 on Σ, ζ(T ) = 0 in Ω, (87)
     ∂ p ∂t -div(κ(α, x) p) + a 0 p = 0 in Q, p = λ 1 on Σ p(0) = 0 in Ω, (88)
       - ∂ q ∂t -div(κ(α, x) q) + a 0 q = 1 0 (ŷ + λ 2 ) dα -z d in Q, q = 0 on Σ, q(T ) = 0
in Ω (89) and

û = - 1 N 1 0 qdα in ω T , (90)
where

λ 1 = lim γ→0 1 √ γ ∂ ζγ ∂ν and λ 2 = lim γ→0 1 √ γ 1 0 p γ (α, u γ )dα.
Proof. we have already relations (86) and (87) ( see ( 75) and ( 80) ). The proof of (88)-( 90) is obtained by passing to the limit in (56) -(58). To this end we will proceed in 3 steps.

Step 1. We show that p satisfies (88).

In view of (70), we have that there exists a positive constant C such that:

1 √ γ ∂ ζγ ∂ν L 2 (Σ) ≤ C, (91)
which in view of (56) allows us to say that

p γ L 2 (Q) ≤ C. (92)
Hence, there exists λ 1 ∈ L 2 (Σ) and p ∈ L 2 (Q) such that:

1 √ γ ∂ ζγ ∂ν λ 1 weakly in L 2 (Σ), ( 93
)
p γ p weakly in L 2 (Q). ( 94
)
If we multiply the first equation in (56) by ϕ ∈ D(Q), we get

0 = Q ( ∂p γ ∂t -div(κ(α, x) p γ ) + a 0 p γ )ϕdxdt = Q p γ - ∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt.
Passing this latter identity to the limit when γ → 0 while using (94), we deduce that 0 = Q p -∂ϕ ∂t -div(κ(α, x) ϕ) + a 0 ϕ dxdt which after integration by parts gives

0 = Q ∂ p ∂t
-div(κ(α, x) p) + a 0 p ϕdxdt, from which we get,

∂ p ∂t -div(κ(α, x) p) + a 0 p = 0 in Q. ( 95
) As p ∈ L 2 (Q) and ∂ p ∂t ∈ L 2 ((0, T ), H -2 (Ω)) we have that p ∈ C([0, T ], H -1 (Ω)).
On the other hand, p ∈ L 2 (Q) and div(κ(α, x) p) ∈ H -1 ((0, T ), L 2 (Ω)). This implies that that p| Σ and ∂ p ∂ν | Σ exist and belong respectively to H -1 ((0, T ); H -1 2 (Γ)) and H -1 ((0, T ); H -3 2 (Γ)). Now, let ϕ ∈ C ∞ ( Q) with ϕ| Σ = 0 and ϕ(T ) = 0 in Ω. If we multiply the first equation in (56) by ϕ and we integrate over Q, we obtain

Σ 1 √ γ ∂ ζγ ∂ν κ(α, x) ∂ϕ ∂ν dσdt + Q p γ - ∂ϕ ∂t -div(κ(α, x) ϕ + a 0 ϕ) dxdt = 0
Passing this latter identity to the limit when γ -→ 0 while using (93) and (94), we obtain

Σ λ 1 κ(α, x) ∂ϕ ∂ν + Q p - ∂ϕ ∂t -div(κ(α, x) ϕ + a 0 ϕ) dxdt = 0, ∀ϕ ∈ C ∞ ( Q) such that ϕ| Σ = 0 and ϕ(T ) = 0,
which after integration by parts yields

Σ κ(α, x)λ 1 ∂ϕ ∂ν + p(0), ϕ(0) H -1 (Ω),H 1 0 (Ω) - p, κ(α, x) ∂ϕ ∂ν H -3/2 ((0,T );H -1 2 (Γ)),H 3/2 ((0,T );H 1 2 (Γ)) + Q ∂ p ∂t -div(κ(α, x) p + a 0 p) ϕdxdt = 0 ∀ϕ ∈ C ∞ ( Q) such that ϕ| Σ = 0 and ϕ(T ) = 0.
In view of (95), we deduce that

p(0), ϕ(0) H -1 (Ω),H 1 0 (Ω) - (λ 1 -p), κ(α, x) ∂ϕ ∂ν H -3/2 ((0,T );H -1 2 (Γ)),H 3/2 ((0,T );H 1 2 (Γ)) = 0, ∀ϕ ∈ C ∞ ( Q) such that ϕ| Σ = 0 and ϕ(T ) = 0. (96)
If we take in addition ∂ϕ ∂ν = 0 on Σ, we have Ω p(0)ϕ(0)dx = 0, which implies that p(0) = 0 in Ω,

and it follows from (96) that

Σ (λ 1 -p)κ(α, x) ∂ϕ ∂ν = 0. This implies p = λ 1 on Σ. (98)
Combining ( 95), ( 97) and (98), we have that p satisfies (88) .

Step 2. We show that q γ converge towards q which satisfies (89). Let q γ 1 and q γ 2 be respectively solution of

       - ∂q γ 1 ∂t -div(κ(α, x) q γ 1 ) + a 0 q γ 1 = 1 0 y γ dα -z d in Q, q γ 1 = 0 on Σ, q γ 1 (T ) = 0 in Ω, (99)
and

       - ∂q γ 2 ∂t -div(κ(α, x) q γ 2 ) + a 0 q γ 2 = 1 0 p γ √ γ dα in Q, q γ 2 = 0 on Σ, q γ 2 (T ) = 0 in Ω. (100)
Then q γ 1 + q γ 2 = q γ , solution of (57). According to (71), there exists a constant C > 0 independent of γ such

q γ 1 L 2 ((0,T );H 2 (Ω)) ≤ C. (101)
Therefore there exists q1 ∈ L 2 ((0, T ); H 2 (Ω)) such that

q γ 1 q1 weakly in L 2 ((0, T ); H 2 (Ω)). ( 102
)
Proceeding as for ζ n in pages 11-14, we prove using ( 102) and ( 76) that q1 is solution to

       - ∂ q1 ∂t -div(κ(α, x) q1 + a 0 q1 = 1 0 ŷdα -z d in Q, q1 = 0 on Σ, q1 (T ) = 0 in Ω. (103)
To complete the proof of the convergence of q γ θ , we need to prove that prove that

1 0 p γ √ γ dα L 2 (Q)
≤ C for some C > 0 independent of γ.

We consider the following set

E = Φ(v) = 1 0 ȳ(α, v, 0)dα| v ∈ L 2 (ω T ) . (104)
In view of Remark 1, E ⊂ L 2 (Q). We define on E × E the inner product:

L 2 (Q) + N 2 u γ 2 L 2 (Q) 1/2 Φ(v) E ≤ C Φ(v) E
because of (69) and (71). We then deduce from (108) that

Q T γ (u γ )Φ(v)dxdt ≤ C Φ(v) E .
This means that

T γ (u γ ) E = 1 √ γ 1 0 p γ dα E ≤ C.
In particular 1 √ γ

1 0 p γ dα L 2 (Q) ≤ C. (109)
This implies that q γ 2 L 2 ((0,T );H 2 (Ω)) ≤ C.

Therefore there exist q2 ∈ L 2 ((0, T ); H 2 (Ω)) λ 2 ∈ L 2 (Q) such that q γ 2 q2 weakly in L 2 ((0, T ); H 2 (Ω)), (111)

1 √ γ 1 0 p γ dα λ 2 weakly in L 2 (Q). ( 112
)
Proceeding as for ζ n in pages 11-14, we prove using (111) and (112) that q2 is solution to

     - ∂ q2 ∂t -div(κ(α, x) q2 ) + a 0 q2 = λ 2 in Q, q2 = 0 on Σ, q2 (T ) = 0 in Ω. (113)
From ( 101) and (110), we have that q γ = q γ 1 + q γ 2 is such that

q γ L 2 (Q) ≤ C (114)
for some C > 0 independent of γ. Hence, q γ q1 + q2 = q weakly in L 2 (Q) (115) and q satisfies (89).

Step 3. Using Remark 2 and (114), we have that

1 0 q γ dα L 2 (Q) ≤ C. ( 116
)
and using Lebesgue dominated convergence theorem we prove that

1 0 q γ dα 1 0 qdα weakly in L 2 (Q). ( 117
)
Finally passing to the limits in (66) while using ( 73) and (117), we deduce (90).

Concluding remarks

In this paper, we proved that averaging the cost function related to our model, the system is still controllable and gives an optimal control which does not depends of the unknown parameter. Using the Lebesgue dominated convergence theorem, we remark also that as γ → 0, the limits of the average state is not far from the average of the limit of the corresponding original state.

Φ 1 0

 1 , w, 0)dα dxdt, ∀Φ(v), φ(w) ∈ E. space. Now if we combine (60) and (64), we obtain that , u γ , 0)dα -z d ȳ(α, v)dα dxdt ∀v ∈ L 2 (ω T ).

p 1 0 1 0 1 0

 111 γ dα. Then in view of (107),Q T γ (u γ )Φ(v)dxdt = N Q u γ v dxdt + Q y(α, u γ , 0)dα -z d ȳ(α, v)dα dxdt ∀ v ∈ L 2 (Q).(108) Observing that there exists a constant C independent of γ such thatQ y(α, u γ , 0)dα -z d 1 0 ȳ(α, v)dα dxdt + N Q u γ v dxdt ≤ 1 0 y(α, u γ , 0)dα -z d 2

Acknowledgment: The first author was supported by the Alexander von Humboldt foundation, under the programme financed by the BMBF entitled "German research Chairs". The second and the third authors are grateful for the facilities provided by the German research Chairs.

Combining (60), ( 63) and (64), we have that

which implies that

4 Existence and characterization of the no-regret control Proposition 4.1 The low-regret control u γ converges in L 2 (Q) to the no-regret control û, solution of (18).

Proof. From (20), we have that

which in view of the expression of J γ given by (21) imples that

Hence, we deduce that

∂ ζγ ∂ν

y γ (α, ., .)dα

In view of ( 69) and (54), there exists C > 0 independent of γ such that

Thus, there exist û ∈ L 2 (ω T ), ŷ ∈ L 2 ((0, T ); H 1 0 (Ω)) such that

y γ ŷ weakly in L 2 ((0, T ); H 1 0 (Ω)).

(74)