OPTIMAL CONTROL OF A PARTIALLY KNOWN COUPLED SYSTEM OF BOD AND DO

The work presented in this paper is concerned with the organic pollution problem and water quality valuation. Biochemical oxygen demand has been used to evaluate the quality of water. If organic matter is present the dissolved oxygen is consumed. This article considers an optimal control problem of coupled system with missing initial conditions, which presents the relation between the biochemical oxygen demand and the dissolved oxygen. The main objective is to control the concentration of dissolved oxygen using the information given in the biochemical oxygen demand equation. The main tool used to characterize the optimal control of the investigate system under the Pareto control formulation.


Introduction
The environmental pollution problem is one most serious problems faced by the world. It is always linked to some terrible problems, which are unable to find a solution and causing irreparable nature damage. The presence of a sufficient concentration of dissolved oxygen (DO) is all-important and necessary to preserve water life. If more oxygen is consumed than is produced, dissolved oxygen levels decline and some sensitive animals may move away, weaken, or die. Oxygen is gained from the atmosphere and plants as a result of photosynthesis. Running water, because of its churning, dissolves more oxygen than still water. Respiration by aquatic animals, decomposition, and various chemical reactions consume oxygen. The required quantity of dissolved oxygen by aerobic biological organisms which is used for decomposing organic material under aerobic conditions at a specified temperature is called Biochemical oxygen demand (BOD). The decrease of BOD is the way for judging the effectiveness of water purification [1][2][3]. Many studies have been published in the context of improving the quality of the method and procedure can use to reduce the BOD level, we refer to works by D.M. Reynolds, S.R. Ahmad in [12], Salguero, Jazmin and Valverde, Jhonny in [13], Magdalena Zajda and Urszula Aleksander-Kwaterczak in [14], etc.
This article provides the main insights into the debate on optimal control choice of an evolution coupled system that presents the relation between biochemical oxygen demand and dissolved oxygen. Because the concentration of dissolved oxygen is of prime importance in considering the quality of water, we try to control its level by giving an assessment of the biochemical oxygen demand and for studying too its physicochemical characteristics. Elsewhere, the posed coupled systems are given with unknown initial conditions that present some barriers. The main aim of our work is to characterize the optimal control. For finding the characterization of this optimal control, we dispose of the incomplete data by introducing the concepts of no-regret control and the sequence of least regret controls. The optimality coupled systems of the no regret control are formed by passing to the limit.

Setting the problem
In this section, we present a mathematical model that is used for studying the pollution problem. This considered model is not standard because it contains some missing initial conditions. We consider a fixed final time T > 0, and Ω a bounded open subset of R N , N ∈ 1, 2, 3 of smooth boundary Γ.
We denote by Q = Ω×]0, T [ the space-time cylinder and by Σ = Γ×]0, T [ her boundary. We are interested in an evolutionary organic pollution problem in surface waters for example lakes or estuaries which is reduced to this reaction-dispersion/diffusion problem with uncertainly where y, z are BOD and DO in a given water sample at a certain temperature over a specific time period.
The control function ω presents the sources of dissolved oxygen from the atmosphere and photosynthesis of plants on the control region O, and χ O is the characteristic function of O. We suppose that for all ω ∈ U ad , we have were ω min and ω max present the minimum and the maximum concentrations of dissolved oxygen that would be present in water at a specific temperature, in the absence of other factors. The initial conditions (g 1 , g 2 ) ∈ G ⊂ H − 1 2 (Ω) × H 1 0 (Ω) assumed to be unknown. The boundary conditions (z, The functions r,r and d are reaction coefficients. The coupled systems (2.1) has a unique pair solution (y, g) = (y(ω, g), z(ω, g)) where There are many factors that can be reduced the level of dissolved oxygen like the respiration of the plant life and the animal life, decomposition of organic matter, reduction due to other gases, temperature increase, and others. For these reasons, the main goal of this work is to control the evolutionary organic pollution problem. Exactly, we control the level of dissolved oxygen, where we point out here that we did not insert any control function in the biochemical oxygen demand equations.
For fixed pair (y d , z d ) ∈ (L 2 (Ω)) 2 and for N > 0, g = (g 1 , g 2 ) ∈ G, we define the quadratic cost function associated to (2.1) Let us minimize the following optimal control with incomplete data: The problem (2.4) has no sense in the case of the space G is infinite. Then, if G is finite we try to solve the inf -sup problem However, in this situation, it is very difficult to ensure that sup g∈G J(ω, g) is bounded.
In (1992), J.L.Lions has done a good idea by adding an additional concept which is called "no regret control. The concept of no-regret control (or, equivalently, Pareto control ) of distributed systems with missing data is used by J.L. Lions in [4, Pareto control of distributed systems, page 90]. In [4][5][6][7], J.L. Lions applied the Pareto control and he associated it with a sequence of low-regret controls defined by a quadratic perturbation for deterministic distributed systems with incomplete data. In [10], O. Nakolima, R. Dorville, and A. Omrane studied how the no regret control can be extended to the hyperbolic case. They also generalized these concepts in the case of ill-posed deterministic problems, without assuming Slater's condition [8,9]. In [11], Hafdallah A, and Ayadi A applied no regret and low regret concepts to control a thermoelastic body with missing initial conditions. A. Hafdallah, A. Ayadi, and C. Laouar applied the no-regret control notion to control an ill-posed wave equation, see [15].
The principle of this idea is based on looking for controls such that In that case, we solve the following problem In the following, we are defining the no regret control for the partially known problem (2.1).

Defining the no-regret control
We say thatω ∈ U ad defines a no-regret control for (2.1) if it is the optimal solution of (2.7).
Lemma 3.1. For every ω ∈ U ad the problem (2.7) is equivalent to ) satisfies the following backward coupled equations Proof. By linearity, we can write the solution to (2.7) in the form y(ω, g) = y(ω, 0) + y(0, g), z(ω, g) = z(ω, 0) + z(0, g). Then, the functional J(ω, g) can be written We introduce (ζ(ω), ξ(ω)) the solution of (3.2). Then, we use integration by parts, we obtain and Remark 3.1. The no regret control exist only if g 1 and ζ(ω)(x, 0) (respectively g 2 and ζ(ω)(x, 0)) are perpendicular to each other in H − 1 2 (Ω) (respectively in H 1 0 (Ω)). For this reason, we consider the following set of admissible controlŝ In [4][5][6][7], J.L. Lions applied the no control and he associated it with a sequence of low-regret controls defined by a quadratic perturbation for deterministic distributed systems with incomplete data. The sequence of low-regret controls is expected to converge to the no regret control.

Defining the sequence of low-regret controls (Least regret controls)
For every γ > 0, we relax the problem (3.1) by introducing a quadratic perturbation such that We say thatω γ ∈ U ad is the sequence of low-regret controls for (2.1) ifω γ is the solution to Lemma 4.1. Problem (4.1) can be written as Proof. From (3.1) and (3.2), the problem (4.1) is written as , sup Lemma 4.2. The problem (4.2)-(4.3) has a unique solutionω γ , which called sequence of least regret controls.
Proof. Since the set of admissible controls U ad is non-empty closed and bounded, we have Then there exists Let (ω γ n ) ∈ U ad be a minimizing sequence such that d γ = lim n→∞ J γ (ω γ n ) = J γ (ω γ ). Then, we get So, we deduce the bounds where C γ is a positive constant and (y γ n , z γ n ) = (y(ω γ n , 0), z(ω γ n , 0)) solves the coupled systems Multiplying the first equality of (4.5) by y γ n and the second equality by z γ n . We integrate over Ω, we find By integrating over [0, T ] and by applying the Gronwall lemma we obtain where C γ is a positive constant. From (4.4), we deduce ∂z γ n ∂t − div(d(x) z γ n ) +r(x)z γ n + r(x)y γ n L 2 (0,T ;O) ≤ C γ .
At last, we have The functional J γ is quadratic coercive, thusω γ is unique.
The characterization of the sequence of least regret controls is given in the following Proposition 4.1.
We need some a priori estimations, which we make in the following Lemma.
Lemma 4.3. There exist some positive constants C independent of γ satisfy the following estimations: and, Proof. Sinceω γ is the sequence of least regret controls, we have In particular case when ω = 0, we get Thus, we have So (4.18) holds. Multiplying the first equality of (4.8) byŷ γ and the second equality byẑ γ and we integrate over Ω, we find Ωŷ By integrating over [0, T ] and by applying the Gronwall lemma we obtain where C γ is a positive constant. From the last estimations we get (4.19). We follow a similar method to demonstrate (4.19) for finding (4.20).
When we pass to the limit when γ → 0, the sequence of least regret controls convergesω γ to the no regret controlω.
In close, the inequality (4.28) can be deduced using the weak convergence ofp γ ,q γ andω γ .

Conclusion
This work examined an evolution coupled system, with missing initial conditions that presented the relation between biochemical oxygen demand (BOD) and dissolved oxygen (DO). Since the decrease of BOD is a good way for judging the effectiveness of water purification, our main objective was to control the level of dissolved oxygen, for giving more information about it. We gave the characterization of the optimal control through applied the idea of no regret, where we solved an optimal control problem with uncertainly. The obtained method optimization problem is to be transformed into classical optimal control via the notion of low regret control. Finally, the coupled optimality system for the least regret control converges weakly to the coupled optimality systems for no regret control or the optimal control.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.