International Journal of Analysis and Applications Volume 19, Number 6 (2021), 984-996 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-984 OPTIMAL CONTROL OF A PARTIALLY KNOWN COUPLED SYSTEM OF BOD AND DO LAOUAR CHAFIA1,2,∗, AYADI ABDELHAMID1 AND HAFDALLAH ABDELHAK3 1Laboratoire des Systèmes Dynamiques et Contrôle, Larbi Ben M’hidi University, Oum El Bouaghi, Algeria 2LAbbes Laghrour University, BP 1252 Route of Batna Khenchela, 40004 khenchela, Algeria 3Laboratory of Mathematics, Informatics and Systems (LAMIS), Larbi Tebessi University, Tébessa, Algeria ∗Corresponding author: laouar chafia@univ-khenchela.dz Abstract. 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. 1. 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 Received August 23rd, 2021; accepted October 7th, 2021; published November 25th, 2021. 2010 Mathematics Subject Classification. 49J21, 93C20, 93C41. Key words and phrases. optimal control; problem with missing data; pareto control; no regret control; least regret controls; optimality system. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 984 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-984 Int. J. Anal. Appl. 19 (6) (2021) 985 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 [1–3]. 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 character- ization 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. 2. 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 RN, 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 (2.1)   ∂y ∂t − div(d(x)Oy) + r(x)y = 0 ∂z ∂t − div(d(x)Oz) + r̃(x)z + r(x)y = ωχO in Q, y(x, 0) = g1, z(x, 0) = g2 in Ω, z = 0, ∂z ∂ν = 0 on Σ, 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 χO is the characteristic function of O. We suppose that for all ω ∈Uad, we have (2.2) Uad = {ω ∈ L2(Q) : ωmin ≤ ω ≤ ωmax} is non-empty closed, convex, Int. J. Anal. Appl. 19 (6) (2021) 986 were ωmin and ωmax 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 (g1,g2) ∈ G ⊂ H− 1 2 (Ω) × H10 (Ω) assumed to be unknown. The boundary conditions (z, ∂z ∂ν ) ∈ H− 1 2 (Σ) × H− 1 2 (Σ). The functions r, r̃ and d are reaction coefficients. The coupled systems (2.1) has a unique pair solution (y,g) = (y(ω,g),z(ω,g)) where (y,z) ∈ L2(Q) ∩C∞(0,T; H− 1 2 (Ω)) ×L2(0,T; H10 (Ω) ∩H 2(Ω)) ∩C∞(0,T; H10 (Ω)). 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 (yd,zd) ∈ (L2(Ω))2 and for N > 0,g = (g1,g2) ∈ G, we define the quadratic cost function associated to (2.1) (2.3) J(ω,g) = ‖y(ω,g) −yd‖2L2(Q) + ‖z(ω,g) −zd‖ 2 L2(Q) + ∫ T 0 ∫ O Nω2dtdx. Let us minimize the following optimal control with incomplete data: (2.4) inf ω∈Uad J(ω,g) ∀g ∈G. 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 (2.5) inf ω∈Uad sup g∈G J(ω,g). However, in this situation, it is very difficult to ensure that supg∈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 [4–7], 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]. Int. J. Anal. Appl. 19 (6) (2021) 987 The principle of this idea is based on looking for controls such that (2.6) J(ω,g) ≤ J(0,g) ∀g ∈G. Condition (2.6) implies sup g∈G [ J(ω,g) −J(0,g) ] is bounded. In that case, we solve the following problem (2.7) inf ω∈Uad sup g∈G [ J(ω,g) −J(0,g) ] . In the following, we are defining the no regret control for the partially known problem (2.1). 3. Defining the no-regret control We say that ω̂ ∈Uad defines a no-regret control for (2.1) if it is the optimal solution of (2.7). Lemma 3.1. For every ω ∈Uad the problem (2.7) is equivalent to (3.1) inf ω∈Uad ( J(ω, 0) −J(0, 0) + 2 sup g∈G ∫ Ω [ g1.ζ(ω)(x, 0) + g2.ξ(ω)(x, 0) ] dx, g = (g1,g2) ∈G, where (ζ,ξ) = (ζ(ω, 0)(x,t),ξ(ω, 0)(x,t)) satisfies the following backward coupled equations (3.2)   − ∂ζ ∂t − div(d(x)Oζ) + r(x)ζ + r(x)ξ = y(ω, 0) − ∂ξ ∂t − div(d(x)Oξ) + r̃(x)ξ = z(ω, 0) in Q, ζ(x,T) = 0, ξ(x,T) = 0 in Ω, ζ = 0, ∂ζ ∂ν = 0 on Σ. 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 J(ω,g) = J(ω, 0) −J(0, 0) + 2 ∫∫ Q [ y(ω, 0)y(0,g) + z(ω, 0)z(0,g) ] dtdx. We introduce (ζ(ω),ξ(ω)) the solution of (3.2). Then, we use integration by parts, we obtain (3.3) ∫∫ Q −( ∂ζ ∂t + div(d(x)Oζ) + r(x)ζ + r(x)ξ)y(0,g)dtdx = ∫ Ω g1ζ(ω)(x, 0)dx + ∫∫ Q ξr(x)y(0,g)dtdx, and (3.4) ∫∫ Q (− ∂ξ ∂t − div(d(x)Oξ) + r̃(x)ξ)z(0,g)dtdx = ∫ Ω g2.ξ(ω)(x, 0)dx + ∫∫ Q ξ( ∂z ∂t (0,g) − div(d(x)Oz(0,g)) + r̃(x)z(0,g))dtdx. Int. J. Anal. Appl. 19 (6) (2021) 988 Adding (3.3) to (3.4), we get∫∫ Q [ y(ω, 0)y(0,g) + z(ω, 0)z(0,g) ] dtdx = ∫ Ω [ g1.ζ(ω)(x, 0) + g2.ξ(ω)(x, 0) ] dx. � Remark 3.1. The no regret control exist only if g1 and ζ(ω)(x, 0) (respectively g2 and ζ(ω)(x, 0)) are perpendicular to each other in H− 1 2 (Ω) (respectively in H10 (Ω)). For this reason, we consider the following set of admissible controls Ûad = {ω ∈Uad : 〈g1,ζ(ω)(x, 0)〉 H −1 2 (Ω) = 0, 〈g2,ξ(ω)(x, 0)〉H10 (Ω) = 0}. In [4–7], 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. 4. 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 J(ω,g) −J(0,g) ≤ γ‖g‖2G, ∀g ∈G. We say that ω̂γ ∈Uad is the sequence of low-regret controls for (2.1) if ω̂γ is the solution to (4.1) inf ω∈Uad sup g∈G [ J(ω,g) −J(0,g) −γ ( ‖g1‖2 H −1 2 (Ω) + ‖g2‖2H10 (Ω) )] . Lemma 4.1. Problem (4.1) can be written as (4.2) inf ω∈Uad Jγ(ω), where (4.3) Jγ(ω) = J(ω, 0) −J(0, 0) + 1 γ ‖ζ(ω)(x, 0)‖2 H −1 2 (Ω) + 1 γ ‖ξ(ω)(x, 0)‖2H10 (Ω). Proof. From (3.1) and (3.2), the problem (4.1) is written as inf ω∈Uad ( J(ω, 0) − J(0, 0) + sup g∈G ∫ Ω [ (2g1ζ(ω)(x, 0) − γg21 ) + (2g2ξ(ω)(x, 0) − γg 2 2 ) ] dx ) . The functions f : g1 7→ (2g1ζ(ω)(x, 0) − γg21 ) and f̃ : g2 7→ (2g2ξ(ω)(x, 0) − γg22 ) are concave. Then, it’s absolutely clear that sup g1∈H −1 2 (Ω) f(g1) = 1 γ ‖ζ(ω)(x, 0)‖2 H −1 2 (Ω) , sup g2∈H10 (Ω) f̃(g2) = 1 γ ‖ξ(ω)(x, 0)‖2H10 (Ω). � Lemma 4.2. The problem (4.2)-(4.3) has a unique solution ω̂γ, which called sequence of least regret controls. Furthermore, when γ → 0, the control ω̂γ converges weakly to the unique no regret control ω̂. Int. J. Anal. Appl. 19 (6) (2021) 989 Proof. Since the set of admissible controls Uad is non-empty closed and bounded, we have Jγ(ω) ≥−J(0, 0) = −‖yd‖2L2(Ω) −‖zd‖ 2 L2(Ω), ∀ω ∈Uad. Then there exists dγ := inf ω∈Uad Jγ(ω) ≥ 0. Let (ωγn) ∈Uad be a minimizing sequence such that dγ = limn→∞Jγ(ωγn) = Jγ(ωγ). Then, we get dγ ≤Jγ(ωγn) < d γ + 1 n < dγ + 1. So, we deduce the bounds (4.4) ‖ωγn‖L2(0,T;O) ≤ C γ, ‖y(ωγn, 0)‖L2(Q) ≤ C γ, ‖z(ωγn, 0)‖L2(Q) ≤ C γ, 1 √ γ ‖ζ(ωγn)(x, 0)‖H−12 (Ω) ≤ C γ, 1 √ γ ‖ξ(ωγn)(x, 0)‖H10 (Ω) ≤ C γ. where Cγ is a positive constant and (yγn,z γ n) = (y(ω γ n, 0),z(ω γ n, 0)) solves the coupled systems (4.5)   ∂yγn ∂t − div(d(x)Oyγn) + r(x)yγn = 0 ∂zγn ∂t − div(d(x)Ozγn) + r̃(x)zγn + r(x)yγn = ωγnχO in Q, yγn(x, 0) = 0, z γ n(x, 0) = 0, in Ω, zγn = 0, ∂zγn ∂ν = 0 on Σ. Multiplying the first equality of (4.5) by yγn and the second equality by z γ n. We integrate over Ω, we find 1 2 d dt ∫ Ω |yγn(t)| 2dx + ∫ Ω r(x)|yγn(t)| 2dx− ∫ Ω div (d(x)yγn(t))y γ n(t)dx = 0, and 1 2 d dt ∫ Ω |zγn(t)| 2dx + ∫ Ω r̃(x)|zγn(t)| 2 − div( d(x).zγn(t))z γ n(t) + r(x)y γ n(t)z γ n(t)dx = ∫ O ωγn(t)z γ n(t)dx. By integrating over [0,T] and by applying the Gronwall lemma we obtain ‖yγn‖L∞(0,T;Y) ≤ C γ, ‖zγn‖L∞(0,T;H10 (Ω)) ≤ C γ, where Cγ is a positive constant. From (4.4), we deduce ‖ ∂zγn ∂t − div(d(x)Ozγn) + r̃(x)z γ n + r(x)y γ n‖L2(0,T;O) ≤ C γ. Int. J. Anal. Appl. 19 (6) (2021) 990 Then, there exists a subsequence of (ωγn), that we denote with the same indices such that, when n goes to +∞, ωγn ⇀ ω̂ γ weakly in L2(0,T;O), yγn ⇀ ŷ γ weakly in L∞(0,T;Y), zγn ⇀ ẑ γ weakly in L∞(0,T; H10 (Ω)), ∂zγn ∂t − div(d(x)Ozγn) + r̃(x)z γ n + r(x)y γ n ⇀ f weakly in L 2(0,T;O). The space L∞(Q) (respectively L∞(0,T; H10 (Ω))) is continuously embedded in L 2(Q) (respectively L2(0,T; H10 (Ω))). Clearly, we have (4.6) yγn ⇀ ŷ γ weakly in L2(Q), zγn ⇀ ẑ γ weakly in L2(0,T; H10 (Ω)). Multiplying two equalities in (4.5) by two test functions ϕ,ψ ∈ D(Q), we obtain 〈yγn,− ∂ϕ ∂t − div(d(x)Oϕ) + r(x)ϕ〉L2(Q) = 0, 〈zγn,− ∂ψ ∂t − div(d(x)Oψ) + r̃(x)ψ〉L2(Q) + 〈yγn,r(x)ψ〉L2(Q) = 〈ω γ n,ψ〉L2(Q). By adding the last two equalities and passing to the limit, we get (4.7)   ∂ŷγ ∂t − div(d(x)Oŷγ) + r(x)ŷγ = 0 ∂ẑγ ∂t − div(d(x)Oẑγ) + r̃(x)ẑγ + r(x)ŷγ = ω̂γχO in L2(0,T,O). From (4.6) and (4.7), we get ŷγ(x, 0) = 0, ẑγ(x, 0) = 0. Now, we have to prove that (ζγn,ξ γ n) converges to (ζ̂ γ, ξ̂γ). Let ζγn = ζ(ω γ n) and ξ γ n = ξ(ω γ n). Reverse time variable by taking ζ̃γn(x,t) = ζ γ n(x,T − t), ξ̃γn(x,t) = ξγn(x,T − t), ỹγn(x,t) = yγn(x,T − t) and z̃γn(x,t) = zγn(x,T − t). Then, we have  − ∂ζ̃γn ∂t − div(d(x)Oζ̃γn) + r(x)ζ̃γn + r(x)ξ̃γn = ỹγn − ∂ξ̃γn ∂t − div(d(x)Oξ̃γn) + r̃(x)ξ̃γn = z̃γn in Q, ζ̃γn(x, 0) = 0, ξ̃ γ n(x, 0) = 0 in Ω, ζ̃γn = 0, ∂ζ̃γn ∂ν = 0 on Σ. Then, we deduce that ζ̃γn ⇀ ζ̂ γ weakly in L2(Q), ξ̃γn ⇀ ξ̂ γ weakly in L2(0,T; H10 (Ω)). Hence, ζ̃γn(x, 0) ⇀ ζ̂ γ(x, 0) weakly in H− 1 2 (Ω), ξ̃γn(x, 0) ⇀ ξ̂ γ(x, 0) weakly in H10 (Ω). Int. J. Anal. Appl. 19 (6) (2021) 991 At last, we have lim n→∞ Jγ(ωγn) = J γ(ωγ) = inf ω∈Uad Jγ(ω). The functional Jγ is quadratic coercive, thus ω̂γ is unique. � The characterization of the sequence of least regret controls is given in the following Proposition 4.1. Proposition 4.1. The unique sequence of least regret controls ω̂γ is characterized by the following coupled system (4.8)   ∂ŷγ ∂t − div(d(x)Oŷγ) + r(x)ŷγ = 0 ∂ẑγ ∂t − div(d(x)Oẑγ) + r̃(x)ẑγ + r(x)ŷγ = ω̂γχO in Q, ŷγ(x, 0) = 0, ẑγ(x, 0) = 0, in Ω, ẑγ = 0, ∂ẑγ ∂ν = 0 on Σ, (4.9)   − ∂ζ̂γ ∂t − div(d(x)Oζ̂γ) + r(x)ζ̂γ + r(x)ξ̂γ = y(ω − ω̂γ) − ∂ξ̂γ ∂t − div(d(x)Oξ̂γ) + r̃(x)ξ̂γ = z(ω − ω̂γ) in Q, ζ̂γ(x,T) = 0, ξ̂γ(x,T) = 0 in Ω, ζ̂γ = 0, ∂ζ̂γ ∂ν = 0 on Σ, (4.10)   ∂ρ̂γ ∂t − div(d(x)Oρ̂γ) + r(x)ρ̂γ = 0 ∂σ̂γ ∂t − div(d(x)Oσ̂γ) + r̃(x)σ̂γ + r(x)ρ̂γ = 0 in Q, ρ̂γ(x, 0) = − 1 γ ζ(ω − ω̂γ)(x, 0), σ̂γ(x, 0) = − 1 γ ξ(ω − ω̂γ)(x, 0) in Ω, ẑγ = 0, ∂ẑγ ∂ν = 0 on Σ, and (4.11)   − ∂p̂γ ∂t − div(d(x)Op̂γ) + r(x)p̂γ + r(x)q̂γ = ŷγ −yd + ρ̂γ − ∂q̂γ ∂t − div(d(x)Oq̂γ) + r̃(x)q̂γ = ẑγ −zd + σ̂γ in Q, p̂γ(x,T) = 0, q̂γ(x,T) = 0 in Ω, p̂γ = 0, ∂q̂γ ∂ν = 0 on Σ. Furthermore, for all ω ∈Uad, we have (4.12) ∫ T 0 ∫ O (q̂γ + Nω̂γ)(ω − ω̂γ)dxdt ≥ 0. Proof. The functional Jγ is quadratic coercive, thus it possesses a unique minimum ω̂γ. This minimum is a solution to the Euler equation, thus for all ω ∈Uad, we have lim h→0 Jγ ( ω̂γ + h(ω − ω̂γ) ) −Jγ(ω̂γ) h ≥ 0. Int. J. Anal. Appl. 19 (6) (2021) 992 So, we have (4.13) ∫∫ Q [ y(ω − ω̂γ)(ŷγ −yd) + z(ω − ω̂γ)(ẑγ −zd) ] dtdx + ∫ T 0 ∫ O Nω̂γ(ω − ω̂γ)dtdx + 1 γ ∫ Ω [ ζ̂γ(x, 0)ζ(ω − ω̂γ)(x, 0) + ξ̂γ(x, 0)ξ(ω − ω̂γ)(x, 0) ] dx ≥ 0. We introduce (ρ̂γ, σ̂γ) = (ρ(ω̂γ, 0)(x,t),σ(ω̂γ, 0)(x,t)) solution to (4.10). By integration by parts, we get (4.14) ∫∫ Q ζ̂γ( ∂ρ̂γ ∂t − div(d(x)Oρ̂γ) + r(x)ρ̂γ)dtdx = ∫∫ Q ρ̂γ(− ∂ζ̂γ ∂t − div(d(x)Oζ̂γ) + r(x)ζ̂γ)dtdx + 1 γ ∫ Ω ζ̂γ(x, 0)ζ(ω − ω̂γ)(x, 0)dx = 0, and (4.15) ∫∫ Q ξ̂γ( ∂σ̂γ ∂t − div(d(x)Oσ̂γ) + r̃(x)σ̂γ + r(x)ρ̂γ)dtdx = ∫∫ Q σ̂γz(ω − ω̂γ)dtdx + ∫∫ Q ρ̂γr̃(x)ξ̂γdtdx + 1 γ ∫ Ω ξ̂γ(x, 0)ξ(ω − ω̂γ)(x, 0)dx = 0. Adding (4.14) to (4.15) amounts to (4.16) 1 γ ∫ Ω [ ζ̂γ(x, 0)ζ(ω − ω̂γ)(x, 0) + ξ̂γ(x, 0)ξ(ω − ω̂γ)(x, 0) ] dx = ∫∫ Q [ρ̂γy(ω − ω̂γ) + σ̂γz(ω − ω̂γ)]dtdx. Replacing (4.16) in (4.13), we find (4.17) ∫∫ Q [ y(ω − ω̂γ)(ŷγ −yd + ρ̂γ) + z(ω − ω̂γ)(ẑγ −zd + σ̂γ) ] dtdx + ∫ T 0 ∫ O Nω̂γ(ω − ω̂γ)dtdx ≥ 0. We introduce now the coupled adjoint state (p̂γ, q̂γ) = (p(ω̂γ, 0)(x,t),q(ω̂γ, 0)(x,t)) solution to (4.11). Fi- nally, we obtain (4.12) by replacing (4.11) in (4.17) and integration by parts. � 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: (4.18) ‖ω̂γ‖L2(0,T,O) ≤C, ‖ŷγ‖L2(Q) ≤C, ‖ẑγ‖L2(Q) ≤C, 1 √ γ ‖ζ̂γ(x, 0)‖ H −1 2 (Ω) ≤C, 1 √ γ ‖ξ̂γ(x, 0)‖H10 (Ω) ≤C, Int. J. Anal. Appl. 19 (6) (2021) 993 and, ‖ŷγ‖L2(Q) ≤C, ‖ ∂ŷγ ∂t ‖L2(Q) ≤C, ‖ẑγ‖L2(0,T;H10 (Ω)) ≤C, ‖ ∂ẑγ ∂t ‖L2(Q) ≤C,(4.19) ‖ζ̂γ‖L2(Q) ≤C, ‖ ∂ζ̂γ ∂t ‖L2(Q) ≤C, ‖ξ̂γ‖L2(0,T;H10 (Ω)) ≤C, ‖ ∂ξ̂γ ∂t ‖L2(Q) ≤C,(4.20) ‖ρ̂γ‖L2(Q) ≤C, ‖ ∂ρ̂γ ∂t ‖L2(Q) ≤C, ‖σ̂γ‖L2(0,T;H10 (Ω)) ≤C, ‖ ∂σ̂γ ∂t ‖L2(Q) ≤C,(4.21) ‖p̂γ‖L2(Q) ≤C, ‖ ∂p̂γ ∂t ‖L2(Q) ≤C, ‖q̂γ‖L2(0,T;H10 (Ω)) ≤C, ‖ ∂q̂γ ∂t ‖L2(Q) ≤C.(4.22) Proof. Since ω̂γ is the sequence of least regret controls, we have Jγ(ω̂γ) ≤Jγ(ω) ∀ω ∈Uad. In particular case when ω = 0, we get J(ω̂γ, 0) −J(0, 0) + 1 γ ‖ζ̂γ(x, 0)‖2 H −1 2 (Ω) + 1 γ ‖ξ̂γ(x, 0)‖2H10 (Ω) ≤ 0. Thus, we have (4.23) ‖ŷγ −yd‖2L2(Q) + ‖ẑ γ −zd‖2L2(Q) + N‖|ω̂ γ‖2L2(Q) + 1 γ ‖ζ̂γ(x, 0)‖2 H −1 2 (Ω) + 1 γ ‖ξ̂γ(x, 0)‖2H10 (Ω) ≤‖yd‖2L2(Ω) + ‖zd‖ 2 L2(Ω) = Constant. So (4.18) holds. Multiplying the first equality of (4.8) by ŷγ and the second equality by ẑγ and we integrate over Ω, we find ∫ Ω ŷγ(t) (∂ŷγ(t) ∂t − div(d(x)Oŷγ(t)) + r(x)ŷγ(t) ) dx = 1 2 d dt ∫ Ω |ŷγ(t)|2dx + ∫ Ω r(x)|ŷγ(t)|2dx− ∫ Ω div (d(x)ŷγ(t))ŷγ(t)dx = 0, and ∫ Ω ẑγ(t) (∂ẑγ(t) ∂t − div(d(x)Oẑγ(t)) + r̃(x)ẑγ(t) + r(x)ŷγ(t) ) dx = 1 2 d dt ∫ Ω |ẑγ(t)|2dx + ∫ Ω r̃(x)|ẑγ(t)|2dx− ∫ Ω div( d(x).ẑγ(t))ẑγ(t) + ∫ Ω r(x)ŷγ(t)ẑγ(t)dx = ∫ O ω̂γ(t)ẑγ(t)dx. By integrating over [0,T] and by applying the Gronwall lemma we obtain ‖ŷγ‖L2(Q) ≤Cγ, ‖ ∂ŷγ ∂t ‖L2(Q) ≤Cγ, ‖ẑγ‖L2(0,T;H10 (Ω)) ≤Cγ, ‖ ∂ẑγ ∂t ‖L2(Q) ≤Cγ, where Cγ 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). � Int. J. Anal. Appl. 19 (6) (2021) 994 When we pass to the limit when γ → 0, the sequence of least regret controls converges ω̂γ to the no regret control ω̂. Theorem 4.1. The no regret control ω̂ = limγ→0 ω̂ γ is characterized by the unique set {(ŷ, ẑ), (ζ̂, ξ̂), (ρ̂, σ̂), (p̂, q̂)} solution to the following coupled optimality system (SO) (4.24)   ∂ŷ ∂t − div(d(x)Oŷ) + r(x)ŷ = 0 ∂ẑ ∂t − div(d(x)Oẑ) + r̃(x)ẑ + r(x)ŷ = ω̂χO in Q, ŷ(x, 0) = 0, ẑ(x, 0) = 0, in Ω, ẑ = 0, ∂ẑ ∂ν = 0 on Σ, (4.25)   − ∂ζ̂ ∂t − div(d(x)Oζ̂) + r(x)ζ̂ + r(x)ξ̂ = y(ω − ω̂) − ∂ξ̂ ∂t − div(d(x)Oξ̂) + r̃(x)ξ̂ = z(ω − ω̂) in Q, ζ̂(x,T) = 0, ξ̂(x,T) = 0 in Ω, ζ̂ = 0, ∂ζ̂ ∂ν = 0 on Σ, (4.26)   ∂ρ̂ ∂t − div(d(x)Oρ̂) + r(x)ρ̂ = 0 ∂σ̂ ∂t − div(d(x)Oσ̂) + r̃(x)σ̂ + r(x)ρ̂ = 0 in Q, ρ̂(x, 0) = ρ0, σ̂(x, 0) = σ0 in Ω, σ̂ = 0, ∂σ̂ ∂ν = 0 on Σ, and (4.27)   − ∂p̂ ∂t − div(d(x)Op̂) + r(x)p̂ + r(x)q̂ = ŷ −yd + ρ̂ − ∂q̂ ∂t − div(d(x)Oq̂) + r̃(x)q̂ = ẑ −zd + σ̂ in Q, p̂(x,T) = 0, q̂(x,T) = 0 in Ω, p̂ = 0, ∂p̂ ∂ν = 0 on Σ, with the variational inequality (4.28) ∫ T 0 ∫ O (q̂ + Nω̂)(ω − ω̂)dxdt ≥ 0, with the following limits: ρ0 = − lim γ→0 1 γ ζ(ω − ω̂γ)(x, 0), σ0 = − lim γ→0 1 γ ξ(ω − ω̂γ)(x, 0). Proof. From inequality (4.23), we may extract some some subsequence of (ω̂γ, ŷγ, ẑγ)γ that we denote with the same indices such that, when γ goes to 0, we have ω̂γ ⇀ ω̂ weakly in L2(0,T;O), (ŷγ, ẑγ) ⇀ (ŷ, ẑ) weakly in L2(Q) × L2(0,T; H10 (Ω)). Int. J. Anal. Appl. 19 (6) (2021) 995 On the other hand, from the optimality coupled systems in Proposition 4.1, the sequences ( ∂ŷγ ∂t −div(d(x)Oŷγ)+ r(x)ŷγ)γ and ( ∂ẑγ ∂t − div(d(x)Oẑγ) + r̃(x)ẑγ + r(x)ŷγ)γ are bounded in L2(Q). So, we have ∂ŷγ ∂t − div(d(x)Oŷγ) + r(x)ŷγ ⇀ ∂ŷ ∂t − div(d(x)Oŷ) + r(x)ŷ weakly in L2(Q), ∂ẑγ ∂t − div(d(x)Oẑγ) + r̃(x)ẑγ + r(x)ŷγ ⇀ ∂ẑ ∂t − div(d(x)Oẑ) + r̃(x)ẑ + r(x)ŷ in L2(Q). Taking to the limit γ → 0, we get (4.24). Also, from to a priori estimates of Lemma 4.3, and by using the same method we found (4.25)-(4.27). From (4.18), we have − 1 γ ( ζ(ω − ω̂γ)(x, 0),ξ(ω − ω̂γ)(x, 0) ) ⇀ (ρ0,σ0) weakly in H −1 2 (Ω) ×H10 (Ω). In close, the inequality (4.28) can be deduced using the weak convergence of p̂γ, q̂γ and ω̂γ. � 5. 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. 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