Uniqueness and stability of solutions of boundary control problems

Boundary Control Problems for Oberbeck–Boussinesq Model of

4. Uniqueness and stability of solutions of boundary control problems

In this section we firstly consider the problem (21) in the case whereI=I1andu=g∈K1, i.e.

we consider one-parameter boundary control problem J(x,g)≡μ0

2 v−vd2Q+μ1

2g21/2,Γ→inf,F(x,g) =0,x= (v,q,S)∈X,g∈K1. (49) Let(x1,u1)≡(u1,p1,C1,g1) be a solution to problem (49) which corresponds to a function vd≡u(1)d ∈L2(Q),(x2,u2)≡(u2,p2,C2,g2)be a solution to problem (49) which corresponds to another function ˜vd≡u(2)d ∈L2(Q). Settingg=g1−g2,ud=u(1)d −u(2)d in addition to (34), we note that

(I1)u(xi),w=2(ui−ud,w)Q,(I1)u(x1)−(I1)u(x2),u=2(u−ud,u)Q=

2(u2Q−(u,ud)Q), (I1)C=0. (50)

The relations (35), (37) for problem (49) do not change while identities (36), (32), (33) and the main inequality (43) take by (50) and remark 1 a form

λa1(C,S) + (kC,S) +c1(u,C1,S) +c1(u2,C,S) =0 ∀S∈ T, (51)

νa0(w,ξi) +c(ui,w,ξi) +c(w,ui,ξi) +κc1(w,Ci,θi) +b(w,σi) +ζi,wΓ=

−μ0(ui−u(i)d ,w)Q∀w∈H˜1(Ω), ξi∈V, i=1, 2, (52) κ[λa˜(τ,θi) + (kτ,θi) +c1(ui,τ,θi) +ζi,τΓD]−b1(τ,ξi) =0∀τ∈H1(Ω), (53) c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2) +μ0(u2Q−(u,ud)Q)≤ −μ1g21/2,Γ. (54) It follows from (37) thatC∈ T. SetS=Cin (51). Using (9) we obtain that

λa1(C,C) + (kC,C) =−c1(u,C1,C). (55) It follows from (13), (14), (29) that

|c1(u,C1,C)| ≤γ1MC0u1C1, |b1(C,u)| ≤β1u1C1. (56) Taking into account (11), (56), we obtain from (55) thatδ1λC2≤γ1M0Cu1C1. From this inequality we deduce the following estimate forC1:

C1≤γ1δM0C

1λ u1. (57)

Using (46), (57) and (30) we have u1≤c0(2R+1)g1/2,Γ+2βδ 1

0ν γ1MC0

δ1λ u1=c0(2R+1)g1/2,Γ+2Rau1. (58) It follows from (58) and (30) that(1−2Ra)u1≤c0(2R+1)g1/2,Γ. Taking into account that 2Ra<1 by (31) we obtain from this estimate, (57) and (48) that

u1≤c0(2R+1)

1−2Ra g1/2,Γ, (59)

C1≤c0(2R+1) (1−2Ra)

γ1M0C

δ1λ g1/2,Γ,p ≤ δ0νc0(2R+1)(R+Ra)

β0(1−2Ra) g1/2,Γ. (60) Setw=ξi,τ=θiin (52), (53). Using (8), (9) and conditionsξi∈V,θi∈ T we deduce that

νa0(ξi,ξi) =−c(ξi,ui,ξi)−κc1(ξi,Ci,θi)−μ0(ui−u(i)d ,ξi)Q, (61) κ[λa1(θi,θi) + (kθi,θi)] =b1(θi,ξi),i=1, 2. (62) It follows from (10)–(14), (7), (29) that

a0(ξi,ξi)≥δ0ξi21,|c(ξi,ui,ξi)| ≤γ0ui1ξi21≤γ0M0uξi21, (63)

496 Advanced Topics in Mass Transfer

a1(θi,θi)≥δ1θi21, |b1(θi,ξi)| ≤β1θi1ξi1,|c1(ξi,Ci,θi)| ≤γ1M0Cξi1θi1, (64)

|(ui−u(i)d ,ξi)Q| ≤ ui−u(i)d QξiQ≤cQ(cQM0u+u(i)d Q)ξi1. (65) Taking into account (63)–(65) we deduce from (61) and (62) that

θi1≤δβ1

1λκξi1,

δ0ν−γ0Mu0−βδ1γ1

1λ M0C

ξi21≤μ0cQ(cQM0u+u(i)d Q)ξi1. Combining these inequalities with (45) and (30) gives

ξi1≤2μγ0γ3

0 (Re+Re0), θi1≤δβ1

1λκ 2μ0γ3

γ0 (Re+Re0), (66) where

γ3=c2Q, Re0=δγ0

0νcQmax(u(1)d Q,u(2)d Q). (67) Taking into account (12), (13), (57), (66) and (30) we have

|c(u,u,ξ1+ξ2)| ≤γ0u21(ξ11+ξ21)≤4μ0γ3(Re+Re0)u21,

κ|c1(u,C,θ1+θ2)| ≤κγ1u1C1(θ11+θ21)≤ γ1M0C

δ1λ γ1β1

δ1λ

4μ0γ3(Re+Re0)

γ0 u21=4μ0γ3(Re+Re0)γγ1

0PRau21 which yields

|c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2)| ≤4μ0γ3(Re+Re0) [1+ (γ1/γ0)PRa]u21. (68) Using (59) we deduce from (68) that

|c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2)| ≤

≤4μ0γ3c20(2R+1)2(Re+Re0)[1+ (γ1/γ0)PRa]

(1−2Ra)2 g21/2,Γ. (69)

Let input data for problem (49) and parametersμ0,μ1be such that (1−ε)μ1≥4μ0γ3c20(2R+1)2(Re+Re0)[1+ (γ1/γ0)PRa]

(1−2Ra)2 ,ε=const>0. (70) Here and furtherε>0 is a (small) constant. In view of (70) we find from (69) that

|c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2)| ≤(1−ε)μ1g21/2,Γ. (71) Taking into account (71) we come from (54) to the inequality

μ0(u2Q−(u,ud)Q)≤ −c(u,u,ξ1+ξ2)−κc1(u,C,θ1+θ2)−μ1g21/2,Γ≤ −εμ1g21/2,Γ. (72)

It follows from (72) thatu2Q≤(u,ud)Q≤ uQudQ, which yieldsuQ≤ udQ. As u=u1−u2,ud=u(1)d −u(2)d we deduce the following estimate:

u1−u2Q≤ u(1)d −u(2)d Q. (73) The estimate (73) in the case where Q=Ω has the sense of the stability estimate of the component ˆuof the solution(u, ˆˆ p, ˆC, ˆg)to problem (49) with respect to small disturbances in theL2(Ω)-norm of the functionvd∈L2(Ω)which enters into the expression for the functional I1in (22). In the case whereu(1)d =u(2)d it follows from (73) thatu1=u2. This yields together with (57), (47) and conditionu|Γ=g≡g1−g2in (35) thatC1=C2,p1=p2andg1=g2. The latter means the uniqueness of the solution to problem (49) whenQ=Ωand (70) holds.

We note that the uniqueness and stability of the solution to problem (49) under condition (70) take place and in the case whereQ⊂Ω, i.e.Qis a part ofΩ. In order to prove this fact let us consider the inequality (72). Using (73) rewrite it in the form

εμ1g21/2,Γ≤c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2) +μ1g21/2,Γ≤

≤ −μ0u2Q+μ0uQudQ≤μ0ud2Q.

From this relation, (59) and (60) we deduce the following stability estimates:

g1−g21/2,Γ≤ μ0

εμ1u(1)d −u(2)d Hs(Q),u1−u21≤c0(2R+1) 1−2Ra

μ0

εμ1u(1)d −u(2)d Hs(Q), C1−C21≤γ1M0Cc0(2R+1)

δ1λ(1−2Ra) μ0

εμ1u(1)d −u(2)d Hs(Q), p1−p2 ≤δ0νc0(2R+1)(R+Ra)

β0(1−2Ra)

μ0

εμ1u(1)d −u(2)d Hs(Q)(R ≡δ−10 +2Re) (74) wheres=0. Thus we have proved the following theorem.

Theorem 7. Let under conditions (i), (ii), (j) and (31) the quadruple(ui,pi,Ci,gi)be the solution to problem (49) corresponding to a given functionu(i)d ∈L2(Q), i=1, 2, and the condition (70) holds whereγ3andRe0are defined in (67). Then stability estimates (73) and (74) under s=0hold true.

We emphasize that the uniqueness and stability of the solution to problem (49) both under Q=Ω, and underQ⊂Ωis proved only if parameterμ1in (49) is positive and satisfies (70).

This means that term(μ1/2)g21/2,Γin the expression for the minimized functionalJin (49) has a regularizing effect on control problem (49).

In the same manner one can study uniqueness and stability of solutions to boundary control problems for another cost functionals depending on the velocity u. Let us consider for example the control problem

J(x,g)≡μ0

2v−vd21,Q+μ1

2g21/2,Γ→inf,F(x,g) =0,x= (v,q,S)∈X,g∈K1, (75) which corresponds to the cost functional I2(v) =v−vd21,Q. Denoting by (xi,ui) ≡ (ui,pi,Ci,gi) the solution to problem (75) corresponding to a function vd ≡u(i)d ∈H1(Q), i=1, 2, and settingg=g1−g2,ud=u(1)d −u(2)d in addition to (34) we note that

(I2)u(xi),w=2(ui−u(i)d ,w)1,Q, (I2)u(x1)−(I2)u(x2),u=2(u−ud,u)1,Q

498

=2(u21,Q−(u,ud)1,Q), (I2)C=0. (76) In view of (76) relations (35), (37), (51), (53), (62) and estimates (57), (59), (60) do not change while (43) and (32) underw=ξitake instead of (54), (61) the form

c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2) +μ0(u1,Q−(u,ud)1,Q)≤ −μ1g21/2,Γ, (77) νa(ξi,ξi) =−c(ξi,ui,ξi)−κc1(ξi,Ci,θi)−μ0(ui−u(i)d ,ξi)1,Q, ξi∈V. (78) Using the estimates (29) we deduce (instead of (65)) that

|(ui−u(i)d ,ξi)1,Q| ≤ ui−u(i)d 1,Qξi1,Q≤(Mu0+u(i)d 1,Q)ξi1. Proceeding as above we obtain estimates (66) forξi1,θi1and inequality (69) where

γ3=1, Re0= (γ0/δ0ν)max(u(1)d 1,Q,u(2)d 1,Q). (79) Let us assume that condition (70) takes place whereγ3andRe0are defined in (79). Using (70) we deduce (71). Taking into account (71) we obtain from (77) that

μ0(u21,Q−(u,ud)1,Q)≤ −c(u,u,ξ1+ξ2)−κc1(u,C,θ1+θ2)−μ1g21/2,Γ≤ −εμ1g21/2,Γ. (80) It follows from (80) thatu21,Q≤(u,ud)1,Qwhich yieldsu1,Q≤ ud1,Qor

u1−u21,Q≤ u(1)d −u(2)d 1,Q. (81) In the case whereQ=Ωwe deduce from (81), relationu|Γ=g=g1−g2, (7), (57) and (47) the following estimates:

u1−u21≤ u(1)d −u(2)d 1,g1−g21/2,Γ≤cΓu(1)d −u(2)d 1,

C1−C21≤γ1δM0C

1λ u(1)d −u(2)d 1,p1−p2 ≤ δ0ν(R+Ra)

β0 u(1)d −u(2)d 1. (82) The estimates (82) have the sense of stability estimates for the solution(u, ˆˆ p, ˆC, ˆg)to problem (75) underQ=Ωwith respect to small disturbances inH1(Ω)-norm of the functionvdwhich enters into the expression for the functionalI2. In the case whereu(1)d =u(2)d we deduce from (82) thatu1=u2,g1=g2,C1=C2,p1=p2which means the uniqueness of solution to control problem (75) underQ=Ω. IfQ⊂Ωthe estimates (82) do not hold true but using (80) one can obtain more rough estimates of kind (74) instead of them. In fact rewriting (80) in view of (81) in the form

εμ1g21/2,Γ≤ −μ0u21,Q+μ0u1,Qud1,Q≤μ0ud21,Q

and using (59), (60) we come to to the estimates (74) unders=1. Thus we have proved the following result.

Theorem 8. Let under conditions (i), (ii), (j) and (31) the quadruple(ui,pi,Ci,gi)be a solution to problem (75) corresponding to a given functionu(i)d ∈H1(Q), i=1, 2, and the condition (70) holds

whereγ3 andRe0are defined in (79). Then stability estimates (81) and (74) under s=1hold true.

Furthermore estimates (82) hold if Q=Ω.

We again note that the uniqueness and stability of the solution to problem (75) both under Q=Ωand underQ⊂Ωis proved above under condition that the parameterμ1in (75) satisfies (70). We can not prove the stability of the solution to problem (75) as well as to problem (49) in the case whereμ1=0. But we can establish the local uniqueness of the solution to problem (75) underμ1=0 in the case whereQ=Ω. In fact settingμ1=0,Q=Ω,u(1)d =u(2)d in (77) we obtain the inequality

c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2)≤ −μ0u21. (83) Let input data for problem (75) be such that

4(Re+Re0)[1+ (γ1/γ0)PRa]<1. (84) It follows from (68) underγ3=1 and (83) thatu=0, and from (57), (47) and relationu|Γ=g we deduce thatC1=C2,p1=p2,g1=g2. So the next theorem holds.

Theorem 9.Let under conditions (i), (ii), (j) and (31)vd∈H1(Ω)be a given function,μ0>0,μ1≥0 and the condition (84) takes place whereRe0= (γ0/δ0ν)vd1. Then the solution(u, ˆˆ p, ˆC, ˆg)to problem (75) under Q=Ωis unique.

Let us consider the one-parameter control problem J(x,g)≡μ0

2 rotv−ηd2Q+μ1

2g21/2,Γ→inf,F(x,g) =0,x= (v,g,S)∈X,g∈K1, (85) corresponding to the cost functional I3(v) = rotv −ηd2Q. Denoting by (xi,ui) = (ui,pi,Ci,gi),i=1, 2, the solution to problem (85) corresponding to the functionηd=ζ(i)d ∈ L2(Q),i=1, 2, and settingg=g1−g2,ζd=ζ(1)d −ζ(2)d in addition to (34) we note that

(I3)u(xi),w=2(rotui−ζ(i)d , rotw)Q,(I3)u(x1)−(I3)u(x2),u=

=2(rotu−ζd, rotu)Q, (I3)C=0. (86) In view of (86) relations (35), (37), (51), (53), (62) and estimates (57), (59), (60) do not change while (43) and (32) underw=ξitransform to

c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2) +μ0(rotu2Q−(ζd, rotu)Q)≤ −μ1g21/2,Γ, (87) νa0(ξi,ξi) =−c(ξi,ui,ξi)−κc1(ξi,Ci,θi)−μ0(rotui−ζ(i)d , rotξi)Q, ξi∈V. (88) Using (7) and (29) we have

|(rotui−ζ(i)d , rotξi)Q| ≤(rotuiQ+ζ(i)d Q)rotξiQ≤cr(crM0u+ζ(i)d Q)ξi1. (89) Taking into account (63), (64), (89) we deduce from (88) and (62) that

θi1≤β1ξi1

δ1λκ ,(δ0ν−γ0M0u−βδ1γ1

1λ MC0)ξi21≤μ0cr(crM0u+ζ(i)d Q)ξi1.

In view of (45) we obtain from this inequality thatξi1 ≤(2μ0/δ0ν)c2r(M0u+c−1r ζd(i)Q) which yields (66), (68) and (69) where

γ3=c2r, Re0=δγ0

0νcrmax(ζ(1)d Q,ζ(2)d Q). (90) Let us assume that the condition (70) takes place whereγ3andRe0are defined in (90). Using (70) we deduce (71). Taking into account (71) we obtain from (87) that

μ0(rotu2Q−(rotu,ζd)Q)≤ −εμ1g21/2,Γ. (91) It follows from (91) thatrotu2Q≤(rotu,ζd)Qwhich yieldsrotuQ≤ ζdQor

rotu1−rotu2Q≤ ζd(1)−ζ(2)d Q. (92) Rewriting (91) by (92) in the formεμ1g21/2,Γ≤μ0ζd2Qand using (59), (60) we obtain the following stability estimates:

g1−g21/2,Γ≤ μ0

εμ1ζ(1)d −ζ(2)d Q,u1−u21≤c0(2R+1) 1−2Ra

μ0

εμ1ζ(1)d −ζ(2)d Q, C1−C21≤γ1δM0C

1λ

c0(2R+1) 1−2Ra

μ0

εμ1ζ(1)d −ζ(2)d Q, p1−p2 ≤ δ0νc0(2R+1)(R+Ra)

β0(1−2Ra)

μ0

εμ1ζ(1)d −ζ(2)d Q. (93) Thus we have proved the following theorem.

Theorem 10. Let under conditions (i), (ii), (j) and (31) the quadruple(ui,pi,Ci,gi)be a solution to problem (85) corresponding to a given functionζ(i)d ∈L2(Q), i=1, 2, and condition (70) holds where γ3andRe0are defined in (90). Then stability estimates (92) and (93) hold true.

We can not prove the stability of the solution to problem (85) in the case whereμ1=0. But we can establish the local uniqueness of the solution to problem (85) under more strict conditions onΩand boundary vectorgif we replace condition (j) by the next condition:

(j)Ωis a simply connected domain with the boundaryΓ∈C1,1;K1⊂H˜1/2(Γ)is a convex closed set consisting of functionsgwhich satisfy the conditiongãn|Γ=qwhereq∈H1/2(Γ) is a given function.

Indeed, let us note that (87) takes underμ1=0,ζ(1)d =ζ(2)d ,Q=Ωa form

c(u,u,ξ1+ξ2) +κc2(u,C,θ1+θ2)≤ −μ0rotu2. (94) Under the first condition in (j) the differenceg=g1−g2has the zero normal component onΓ.

Therefore taking into account the simple connectedness of the domainΩwe have the estimate u1≤c3rotuwith the constantc3depending onΩ(Girault & Raviart, 1986). Using this estimate we deduce from (68) whereγ3andRe0are given in (90) that

|c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2)| ≤4μ0c2rc23(Re+Re0)[1+ (γ1/γ0)PRa]rotu2. (95) Let input data for problem (85) be such that

4c2rc23(Re+Re0)[1+ (γ1/γ0)PRa]<1. (96)

It follows from (94) and (95) that rotu=0which yieldsu=0oru1=u2. From (57), (47) and the conditionu|Γ=gwe deduce thatC1=C2, p1=p2,g1=g2. Thus we have proved the following theorem.

Theorem 11. Let under conditions (i), (ii), (j) and (31)ηd∈L2(Ω)be a given function,μ0>0, μ1≥0and the condition (96) takes place whereRe0= (γ0/δ0νcr)ηd. Then the solution(u, ˆˆ p, ˆC, ˆg) to problem (85) under Q=Ωis unique.

In conclusion we consider two-parameter control problem corresponding to the cost functionalI1, i.e. we consider the control problem

J(x,u)≡μ0

2v−vd2Q+μ1

2 g21/2,Γ+μ2

2 χ2ΓN→inf, F(x,u) =0,

x= (v,q,S)∈X, u= (g,χ)∈K1ìK2. (97) Let (x1,u1) ≡(u1,p1,C1,g1,χ1) be a solution to problem (97) which corresponds to a functionvd≡u(1)d ∈L2(Q),(x2,u2)≡(u2,p2,C2,g2,χ2)be a solution to problem (97) which corresponds to another function ˜vd≡u(2)d ∈L2(Q). Settingg=g1−g2,ud=u(1)d −u(2)d in addition to (34), we note that the relations (50) hold true for problem (97). In view of (50) relations (52), (53), (61), (62) and estimates (66), (67) do not change while (43) takes the form

c(u,u,ξ1+ξ2) +κc1(u,C,η1+η2) +μ0(u2Q−(u,ud)Q)≤ −μ1g21/2,Γ−μ2χ2ΓN. (98) Moreover instead of (51) we have to use the original identity (36). It follows from (37) that C∈ T. SettingS=Cin (36) and using (9) we obtain that

λa1(C,C) + (kC,C) =−c1(u,C1,C) + (χ,C)ΓN. (99) Using (11), (15) and the first estimate in (56) we deduce from (99) that δ1λC2 ≤ γ1M0Cu1C1+γ2χΓNC1. This yields the following estimate forC1:

C1≤γ1δM0C

1λ u1+δγ2

1λχΓN. (100)

Taking into account (46) we obtain from (100) that C1≤2βδ 1

0ν γ1MC

δ1λ C1+c0(2R+1)γ1δM0C

1λ g1/2,Γ+δγ2

1λχΓN. (101) Using (30) and (46), (47) we deduce the following estimates forC,uandp:

C1≤c0(2R+1) (1−2Ra)

γ1M0Cg1/2,Γ

δ1λ + γ2χΓN δ1λ(1−2Ra), u1≤c0(2R+1)g1/2,Γ

1−2Ra + 2β1γ2χΓN δ0νδ1λ(1−2Ra), p ≤ δ0νc0(2R+1)(R+Ra)g1/2,Γ

β0(1−2Ra) +β1(2R+1)γ2χΓN

β0δ1λ(1−2Ra) . (102)

It follows from (12), (13), (66) and (102) that

|c(u,u,ξ1+ξ2)| ≤γ0u21(ξ11+ξ21)≤4μ0γ3(Re+Re0) (1−2Ra)2

c0(2R+1)g1/2,Γ+

2β1 δ0ν

γ2

δ1λχΓN

, κ|c1(u,C,θ1+θ2)| ≤κγ1u1C1(θ11+θ21)≤4μ0γ3γ1 γ0

β1 δ1λì (Re+Re0)

γ0(1−2Ra)2

c0(2R+1)g1/2,Γ+2βδ 1

0ν

γ2χΓN δ1λ

c0(2R+1)γ1M0Cg1/2,Γ

δ1λ +γ2χΓN δ1λ

. Hereγ3andRe0are defined in (67). From these inequalities and (30) we deduce that

|c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2)| ≤μ0(a1g21/2,Γ+a2χ2ΓN). (103) Here constantsa1anda2are given by

a1=2μ0γ0γ(Re+Re0)

(1−2Ra)2 C02(2R+1)2

6+γγ1

0(5Ra+1)P2

,

a2=4μ0γ0γ(Re+Re0) (δ0ν)2(1−2Ra)2

2β1rγ2 δ1λ

2 3+γγ1

0(Ra+2)P2

. (104)

Let input data for problem (97) and parametersμ0,μ1,μ2be such that

(1−ε)μ1≥μ0a1, (1−ε)μ2≥μ0a2, ε=const>0. (105) In view of (105) we deduce from (103) that

|c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2)| ≤(1−ε)μ1g21/2,Γ+ (1−ε)μ2χ2ΓN. (106) Combining (98) and (106) we obtain the inequality

μ0(u2Q−(u,ud)Q)≤ −c(u,u,ξ1+ξ2)−κc1(u,C,θ1+θ2)−μ1g21/2,Γ−μ2χΓN≤

−εμ1g21/2,Γ−εμ2χΓN. (107)

Using (107) we deduce (73) which takes place under conditions (105). The uniqueness of the solution to problem (97) follows from this estimate and (107), (102) whenQ=Ω.

Rewriting (107) in view of (73) in the form

εμ1g21/2,Γ+εμ2χ2ΓN≤c(u,u,ξ1+ξ2) +κc1(u,C,θ1+θ2) +μ1g21/2,Γ+μ2χ2ΓN≤

≤ −μ0u2Q+μ0uQudQ≤μ0ud2Q, and using (102) we obtain the following stability estimates:

g1−g2ΓN≤εμμ0

1u(1)d −u(2)d Q, χ1−χ2ΓN≤εμμ0

2u(1)d −u(2)d Q,

u1−u21≤

c0(2R+1) (1−2Ra)√μ

1 +δ 2β1γ2

0νδ1λ(1−2Ra)√μ

2

μ0

ε u(1)d −u(2)d Q, C1−C21≤ c0(2R+1)

(1−2Ra) γ1M0C δ1λ√μ

1 +δ γ2

1λ(1−2Ra)√μ

2

μ0

ε u(1)d −u(2)d Q, p1−p2 ≤ (2R+1)

β0(1−2Ra)

c0δ0ν(R+Ra)

√μ

1 +δβ1γ2

1λ√μ

2

μ0

ε u(1)d −u(2)d Q. (108) So the next theorem holds.

Theorem 12.Let under conditions (i), (ii), (j) and (31) the quintuple(ui,pi,Ci,gi,χi)be a solution to problem (97) corresponding to a given functionu(i)d ∈L2(Q), i=1, 2, and conditions (105) hold where parameters a1and a3are given by relations (104) in whichγ3andRe0are defined in (67). Then stability estimates (73) and (108) hold true.

In the same manner one can study two-parameter control problems for another cost functionals entering into (22). Consider for example the following control problem:

J(x,g,χ)≡μ0

2p−pd2Q+μ1

2g21/2,Γ+μ2

2 χ2ΓN→inf,

F(x,u) =0, x= (v,q,S)∈X, u= (g,χ)∈K1ìK2, (109) corresponding to the functionalI4. The following theorem holds.

Theorem 13.Let under conditions (i), (ii), (j) and (31) the quintuple(ui,pi,Ci,gi,χi)be a solution to problem (109) corresponding to a given function p(i)d ∈L2(Q), i=1, 2, and the conditions (105) hold where parameters a1and a3are given by relations

a1=2μ0γ0(R+Ra)M˜p

(1−2Ra)2 C20(2R+1)2

6+γγ1

0(5Ra+1)P2

,

a2=4μ0γ0(R+Ra)M˜p (δ0ν)2(1−2Ra)2

2β1rγ2

δ1λ 2

3+γγ1

0(Ra+2)P2

in which

M˜p=β−10

M0p+max(p(1)d Q,p(2)d Q). Then the estimate

p1−p2Q≤ p(1)d −p(2)d Q and stability estimates

g1−g21/2,Γ≤ μ0

εμ1p(1)d −p(2)d Q, χ1−χ2ΓN≤ μ0

εμ2p(1)d −p(2)d Q, u1−u21≤

c0(2R+1) (1−2Ra)√μ

1 +δ 2β1γ2

0νδ1λ(1−2Ra)√μ

2

μ0

ε p(1)d −p(2)d Q, C1−C21≤ c0(2R+1)

(1−2Ra) γ1M0T δ1λ√μ

1+δ γ2

1λ(1−2Ra)√μ

2

μ0

ε p(1)d −p(2)d Q,

p1−p2 ≤ (2R+1) β0(1−2Ra)

c0δ0ν(R+Ra)

√μ

1 +δβ1γ2

1λ√μ

2

μ0

ε p(1)d −p(2)d Q

hold true.

Similar theorems can be formulated and proved for Models 2 and 3. Details can be found in (Alekseev, 2006; 2007a; Alekseev & Soboleva, 2009; Alekseev & Tereshko, 2010a; Alekseev &

Khludnev, 2010).