In this paper, the distributed control for non-cooperative elliptic systems under conjugation conditions is established. First, the existence and uniqueness of the state for these systems with Dirichlet and conjugation conditions is proved, then the set of equations and inequalities that characterizes the distributed control of these systems is found. The non-cooperative Neumann systems with conjugation conditions is also discussed

Abdelrhman, Lamiaa Mohamed

Alsaban, A. A.

Serag, H. M.

English

Scitech Research Journals

Journal of Progressive Research in MathematicsVol 18,1 - 2021 — E-ISSN: 2395-0218http://www.scitecresearch.com/journals/index.php/jprmDistributed Control for Non-Cooperative Sys-tems Under Conjugation ConditionsA. A. Alsaban 1, H. M. Serag 2, L. M. Abd-Elrhman 31Department of Mathematics, Faculty of Science, Ibb University, Ibb, Yemen.Email: ahmedalsaban@yahoo.com2Department of Mathematics, Faculty of Science, Al-Azhar University,Nasr City (11884), Cairo, Egypt.Email: serraghm@yahoo.com3Department of Mathematics, Faculty of Science, Al-Azhar University [for girls],Nasr City, Cairo, Egypt.Email: lamia.abdelrhman3461@yahoo.comReceived: January 23, 2021; Accepted: February 7, 2021; Published: March 3, 2021Copyright © 2021 by author(s) and Scitech Research Organisation(SRO).This work is licensed under the Creative Commons Attribution International License (CC BY).http://creativecommons.org/licenses/by/4.0/AbstractIn this paper, the distributed control for non-cooperative elliptic systems under conjugationconditions is established. First, the existence and uniqueness of the state for these systemswith Dirichlet and conjugation conditions is proved, then the set of equations and inequali-ties that characterizes the distributed control of these systems is found. The non-cooperativeNeumann systems with conjugation conditions is also discussed.KeywordsNon cooperative elliptic systems - Conjugation conditions - Dirichlet and Neumann conditions- Existence and uniqueness of solutions - Distributed control1. IntroductionThe necessary and sufficient conditions of optimality for systems governed by partial differential equa-tions have been studied by Lions [11]. The control problems described by either infinite order operators oroperators with an infinite number of variables were established by Gali et. al. [3, 4, 5]. These results havebeen extended to cooperative systems [1, 2, 6, 13, 16] or non-cooperative systems [10, 17].Serag et. al. discussed the optimal control for systems involving schrodinger operators [12, 14]. Theexistence results have been proved for some non linear systems in [8, 9, 14, 15]. Some applications for controlproblems have been introduced for example in [7, 10].Cite this paper: A. A. Alsaban, H. M. Serag and L. M. Abd-Elrhman (2021). Distributed Control forNon-Cooperative Systems Under Conjugation Conditions. Journal of Progressive Research in Mathematics,18(1), 55-63. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/20221A. A. Alsaban, H. M. Serag, L. M. Abd-ElrhmanNew optimal control problems of distributed systems described by an elliptic, parabolic and hyperbolicoperators with conjugation conditions and by a quadratic cost functional have been studied by Sergienkoand Deineka [18-20].In the present work, using the theory of Lions [11], Sergienko and Deineka [18-20], the distributed controlfor n× n non cooperative Dirichlet elliptic systems is discussed . First the existence and uniqueness of thestate for these systems is proved, then the set of equations and inequalities that characterizes the distributedcontrol of these systems is found. The optimal control of distributed type for non-cooperative Neumannproblems under conjugation conditions is also studied.2 Distributed control for non-cooperative Dirichlet elliptic sys-tems under conjugation conditionsIn this section, we study the distributed control for the following n × n non cooperative Dirichlet ellipticsystems: −∆hi +∑nj=1 aijhj = fi in Ω,hi = 0 on Γ, i = 1, 2, ..., n,(1)under conjugation conditions:R1{∂hi∂vA}−+R2{∂hi∂vA}+= [hi] + δ on γ,[∂hi∂vA]=[n∑i,j=1∂hi∂xjcos(v, xi)]= wi on γ, i = 1, 2, ...n,(2)where Ω1 and Ω2, with boundary ∂Ω1 and ∂Ω2 respectively, are bounded, continuous and strictly Lipchitzdomains from Rn such that :Ω = Ω1 ∪ Ω2, Ω1 ∩ Ω2 = φ,Γ = (∂Ω1 ∪ ∂Ω2)/γ, is boundary of Ω, γ = ∂Ω1 ∩ ∂Ω2 6= φ, γ = γ+ ∪ γ−∂Ω1 ∩ γ = γ+, ∂Ω2 ∩ γ = γ−, fi ∈ L2(Ω), (i = 1, 2, ..., n),R1, R2, w, δ ∈ C (γ) , R1, R2 ≥ 0, R1 +R2 ≥ R0 > 0, R0 = constant, (3)−→n is an ort of an outer normal toγ, [ϕ] = ϕ+ − ϕ−,ϕ+ = {ϕ}+ = ϕ(x) for x ∈ γ+ϕ− = {ϕ}− = ϕ(x) for x ∈ γ−.System (1) is called cooperative system if aij > 0 ∀ i 6= j , otherwise is called non-cooperative system.In our work, we assumeaij = 1 if i ≥ j,−1 if i < j, i, j = 1, 2, ..., n.(i.e. non-cooperative systems).56A. A. Alsaban, H. M. Serag, L. M. Abd-ElrhmanWe first prove the existence of the state for system (1) under conjugation conditions (2). Then, we discusthe existence of distributed control for this system; and we find the set of equations and inequalities thatcharacterizes this distributed control.Existence and uniqueness of the stateSinceH10 (Ω) ⊆ L2(Ω) ⊆ H−1(Ω),then by Cartesian product, we have chain of the form(H10 (Ω))n ⊆ (L2(Ω))n ⊆ (H−1(Ω))n.On (H10 (Ω))n, we introduce the bilinear form:a(h, ψ) =n∑i=1∫Ω∇hi∇ψidx+n∑i,j=1∫Ωaijhiψidx+n∑i=1∫γ[hi][ψi]R1 +R2dγ.(4)It is easy to check that|a(h, ψ)| ≤ k1‖h‖‖ψ‖. (5)The bilinear form (4) is coercive on (H10 (Ω))n; that is, there exists a positive constant C such thata(h, h) ≥ C‖h‖2(H10 (Ω))n ∀h = {h1, h2, ..., hn} ∈ (H10 (Ω))n (6)Proof.a(h, h) =n∑i=1∫Ω|∇hi|2 dx+n∑i=1∫Ω|hi|2 dx+n∑i=1∫γ[hi]2R1 +R2dγ=n∑i=1∫Ω(|∇hi|2 + |hi|2)dx+n∑i=1∫γ[hi]2R1 +R2dγ(3), impliesa(h, h) ≥Cn∑i=1∫Ω(|∇hi|2 + |hi|2)dx,thereforea(h, h) ≥Cn∑i=1‖hi‖2H10 (Ω)=C‖h‖2(H10 (Ω))n ,57A. A. Alsaban, H. M. Serag, L. M. Abd-Elrhmanwhich proves the coerciveness condition of the bilinear form (4) on (H10 (Ω))n .Now, letL(ψ) =n∑i=1∫Ωfi(x)ψi(x)dx+n∑i=1∫γ(R2w − δ)ψiR1 +R2dγ−n∑i=1∫γwψ+i dγ(7)be a linear form on (H10 (Ω))n, this linear form is continuous, since :|L(ψ)| ≤ K‖ψ‖(H10 (Ω))n ∀ψ ∈ (H10 (Ω))n,K is a constant. (8)Then using Lax Milgram lemma, there exists a unique solution h ∈ (H10 (Ω))n such that:a(h, ψ) = L(ψ) ∀ψ = (ψi)ni=1 ∈ (H10 (Ω))n. (9)Then, we have proved the following theoremFor a given f= {fi}ni=1 ∈ (L2(Ω))n there exists a unique solution h = {hi}ni=1 ∈ (H10 (Ω))n for non-cooperative Dirichlet system (1) with conjugation conditions (2)Formulation of the control problemThe space U=(L2(Ω))n is the space of controls. For a control u= {u1, u2, ..., un} ∈ (L2(Ω))n, the stateh(u)= {h1(u), h2(u), ..., hn(u)} of the system is given by the solution of−∆hi(u) +∑nj=1 aijhj(u) = fi(u) + ui in Ω,hi(u) = 0 on Γ, i = 1, 2, ..., n,(10)under conjugation conditions:R1{∂hi(u)∂vA}−+R2{∂hi(u)∂vA}+= [hi(u)] + δ on γ,[∂hi(u)∂vA]=[n∑i,j=1∂hi(u)∂xjcos(v, xi)]= wi on γ, i = 1, 2, ...n.(11)Specify the observation equation byz(u) = {z1(u), z2(u), ..., zn(u)} =Ch(u) = C{h1(u), h2(u), ..., hn(u)}={h1(u), h2(u), ..., hn(u)}.For a given zd = {z1d, z2d, ..., znd} ∈ (L2(Ω))n,the cost functional is given byJ(v) =n∑i=1‖hi(v)− zid‖2L2(Ω) + (Nv, v)(L2(Ω))n , (12)where N is a hermitian positive definite operator such that :(Nv, v)(L2(Ω))n ≥M‖v‖2(L2(Ω))n , M > 0, ∀ v ∈ U. (13)The control problem then is to find :{u = {u1, u2, ..., un} ∈ Uad such that:J(u) = inf J(v) ∀v ∈ Uad,(14)58A. A. Alsaban, H. M. Serag, L. M. Abd-Elrhmanwhere Uad is a closed convex subset of (L2(Ω))n.The cost functional (12)can be written asJ(v) = π(v, v)− 2H(v) +n∑i=1‖zid − hi(0))‖2L2(Ω),whereπ(u, v) =n∑i=1(hi(u)− hi(0), hi(v)− hi(0))L2(Ω) + (Nv, v)(L2(Ω))n , (15)is a continuous bilinear form and from (13), it is coercive, that is:π(v, v) ≥ N‖v‖2(L2(Ω))n andH(v) =n∑i=1(zid − hi(0), hi(v)− hi(0))L2(Ω), (16)is a continuous linear form on (L2(Ω))n. Then, using the theory of Lions [11], there exists a unique optimalcontrol of problem (14); Moreover it is characterized by Let us suppose that (6) holds and the cost functionalis given by (12), then the distributed control u is characterized by−∆pi(u) +∑nj=1 aijpi(u) = hi(u)− zid in Ω,pi(u) = 0 on Γ,[∂pi(u)∂vA∗]= 0 on γ,{∂pi(u)∂vA∗}±= 1R1+R2 [pi(u)] on γ,∑ni=1(pi(u), vi − ui) + (Nu, v − u)(L2(Ω))n ≥ 0, i = 1, 2, ..., n,(17)together with(10),where p(u) = {p1(u), p2(u), ..., pn(u)} is the adjoint state.Proof. The optimal control u = {ui}ni=1 ∈ (L2(Ω))n is characterized by [11] :π(u, v − u) ≥ H(v − u) ∀ v = {v1, v2, ..., vn} ∈ Uad.From (15), and (16):π(u, v − u)−H(v − u) =n∑i=1(hi(u)− zid, hi(v)− hi(u))L2(Ω)+n∑i=1(Nui, vi − ui)L2(Ω) ≥ 0.(18)Since the model A of the system is given byAh(u) = A(h1(u), h2(u), ..., hn(u)) =n∑i=1(−∆hi(u) +n∑j=1aijhj(u)),and since(p(u), Ah(u)) = (A∗p(u), h(u)),59A. A. Alsaban, H. M. Serag, L. M. Abd-Elrhmanthen(p(u), Ah(u))(L2(Ω))n =n∑i=1(pi(u),−∆hi(u) +n∑j=1aijhj(u))=n∑i=1(−∆pi(u) +n∑j=1ajipj(u), hi(u))L2(Ω),hence (18) is equivalent ton∑i=1(−∆pi(u)−n∑j=1ajipj(u), hi(v − u)− hi(0))L2(Ω)+n∑i=1(Nui, vi − ui)L2(Ω) ≥ 0.Thereforen∑i=1(A∗pi(u), hi(v − u)− hi(0))L2(Ω)+n∑i=1(Nui, vi − ui)L2(Ω) ≥ 0.Applying Green’s formula, we obtainn∑i=1(pi(u), Ahi(v − u)−Ahi(0))L2(Ω)+n∑i=1(Nui, vi − ui)L2(Ω) ≥ 0.Using equation (10), we getn∑i=1(pi(u), vi − ui) +n∑i=1(Nui, vi − ui)L2(Ω) ≥ 0,i.en∑i=1∫Ω(pi(u) +Nui)(vi − ui)dx ≥ 0.3 Distributed control for non-cooperative Neumann elliptic sys-tems with conjugation conditionsIn this section, we consider the following non-cooperative Neumann elliptic system−∆hi +∑nj=1 aijhj = fi in Ω,∂hi∂νA= gi on Γ,(19)60A. A. Alsaban, H. M. Serag, L. M. Abd-Elrhmanwith conjugation conditions (2), where g = {g1, g2, ..., gn} ∈ (L2(Γ))n is given function. We introduce againthe bilinear form (4) which is coercive on (H1(Ω))n, since((H10 (Ω))n) ⊆ ((H1(Ω))n).Then based on (6), (8) and Lax-Milgram lemma, there exists a unique solution h for system (19) such that :a(h, ψ) = LN (ψ), ∀ψ ∈ (H1(Ω))n,whereLN (ψ) =n∑i=1∫Ωfi(x)ψi(x)dx+n∑i=1∫Γgi(x)ψi(x)dΓ+n∑i=1∫γ(R2w − δ)ψiR1 +R2dγ −n∑i=1∫γwψ+i dγ,is a continuous linear form defined on (H1(Ω))n.Let us multiply both sides of first equation of (19) by ψ ∈ (H1(Ω))n and integrate over Ω , we obtainn∑i=1∫Ω(−∆hi +n∑j=1aijhj)ψi(x)dx =n∑i=1∫Ωfiψidx.Applying Green’s formula,n∑i=1∫Ω(−∆hi +n∑j=1aijhj)ψi(x)dx+n∑i=1∫Γ(∂hi∂νA)ψi(x)dΓ+n∑i=1∫γ(∂hi∂νA)ψi(x)dγ + a(h, ψ) =n∑i=1∫Ωfiψidx.Then, froma(h, ψ) = LN (ψ) ,we deduce the Neumann conditions∂hi∂νA= gi, on Γ.So we can formulate the corresponding control problem:The space U=(L2(Ω))n is the space of controls. For a control u= {u1, u2, ..., un} ∈ (L2(Ω))n, the stateh(u)= {h1(u), h2(u), ..., hn(u)} of the system is given by the solution of−∆hi(u) +∑nj=1 aijhj(u) = fi(u) + ui in Ω,∂hi(u)∂νA= gi on Γ,(20)under conjugation conditions (11). For a given zd = {z1d, z2d, ..., znd} ∈ (L2(Ω))n, the cost functional isagain given by (12), then there exists a unique optimal control u ∈ Uad such that:{u = {u1, u2, ..., un} ∈ Uad such that:J(u) = inf J(v) ∀v ∈ Uad.Moreover it is characterized by the following equations and inequalities61A. A. Alsaban, H. M. Serag, L. M. Abd-Elrhman−∆pi(u) +∑nj=1 aijpi(u) = hi(u)− zid in Ω,∂pi(u)∂vA∗ = 0 on Γ,[∂pi(u)∂vA∗]= 0 on γ,{∂pi(u)∂vA∗}±= 1R1+R2 [pi(u)] on γ∑ni=1(pi(u), vi − ui) + (Nu, v − u)(L2(Ω))n ≥ 0, i = 1, 2, ..., n,together with(20), where p(u) = {p1(u), p2(u), ..., pn(u)} is the adjoint state.AcknowledgementThe authors would like to express their gratitude to Professor Dr. I. M. Gali, Mathematics Department,Faculty of Science, Al-Azhar University, for suggesting the problem and critically reading the manuscript.References[1] El-Saify , H. A. , Serag. , H. M. , and Shehata, M. A. , Time optimal control problem for cooperativehyperbolic systems involving the Laplace operator, J. of Dynamics and Control Systems, Vol. 15, No.3, (2009), pp. 405-423.[2] Fleckinger J., and Serag, H. M., Semilinear cooperative elliptic systems on Rn , Rend. Mat. Appl., Vol.15 , No.1, (1995), pp. 89-108.[3] Gali, I.M., Optimal control of system governed by elliptic operators of infinite order, Ordinary andPartial Diff. Eqns., Proc. Dundee Scotland ,Springer-Verlag Ser. Lecture Notes in Maths., Vol. 964,(1982), pp. 263-272.[4] Gali, I. M. and El-Saify, H. A. , Optimal control of a system governed by hyperbolic operator with aninfinite number of variables, J. of Mathematical Analysis and Applications, Vol. 85, No. 1, (1982), pp.24-30.[5] Gali, I. M. and EL-Saify H. 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Distributed Control for Non-Cooperative Systems Under Conjugation Conditions

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Distributed Control for Non-Cooperative Systems Under Conjugation Conditions

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