Optimal control of the time-periodic MHD equations

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Nonlinear Analysis 63 (25) e1687 e1699 www.elsevier.

Library, Tallahassee, FL 3236-412, USA Abstract We consider the mathematical formulation and analysis of an optimal control problem associated with the tracking of the velocity and the magnetic field

Keywords: Magnetohydrodynamic equations; Optimal control 1.

1 Nonlinear Analysis 63 (25) e1687 e Optimal control of the time-periodic MHD equations Max Gunzburger, Catalin Trenchea School of Computational Science and Information Technology, Florida State University, 4 Dirac Science Library, Tallahassee, FL , USA Abstract We consider the mathematical formulation and analysis of an optimal control problem associated with the tracking of the velocity and the magnetic field of a viscous, incompressible, electrically conducting fluid in a bounded two-dimensional domain through the adjustment of distributed controls. 25 Elsevier Ltd. All rights reserved. Keywords: Magnetohydrodynamic equations; Optimal control 1. Introduction The paper studies the optimal control problem: minimize 1 ( (u(x, t) u (x, t)) 2 + curl(b(x, t) B (x, t)) 2 + l( ψ 2 1 (x, t) 2 Q + ψ 2 (x, t) 2 )) dt dx (1.1) over ψ 1, ψ 2,u,B (L 2 (Q)) 2 subject to the nondimensional magnetohydrodynamic equations (MHD equations) for a viscous incompressible resistive fluid (see [5,6]) u t + (u )u 1 Re Δu + p + S ( 1 2 B2 ) S(B )B = f + ψ 1 in Ω R, Corresponding author. address: trenchea@csit.fsu.edu (C. Trenchea) X/$ - see front matter 25 Elsevier Ltd. All rights reserved. doi:1.116/j.na

2 e1688 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 B + (u )B (B )u + 1 t Rm cūrl(curl B) = ψ 2 in Ω R, (1.2) div u =, div B = inω R, u = on Ω R, B n = and curl B = on Ω R, u(x, ) = u(x, T ), B(x, ) = B(x,T ) (x, t) Ω R. Here Q = Ω (,T), Ω is an open bounded simply connected subset of R 2 with smooth boundary Ω, f is a T-periodic (nondimensional) volume density force, u = (u 1 (x, t), u 2 (x, t)) is the velocity of the particle of fluid which is at point x at time t, B = (B 1 (x, t), B 2 (x, t)) is the magnetic field at point x at time t, p = p(x, t) stands for the pressure of the fluid while ψ 1, ψ 2 L 2 loc (R; L2 (Ω)) are divergence free T-periodic inputs, and u,b L 2 loc (R; H 1 (Ω)) are the T-periodic reference velocity and magnetic field, respectively. The nondimensional quantities p, u, B correspond to the normalization by reference units denoted by L,T,U = L /T,B, for lengths, times, velocities, and magnetic fields. There are three nondimensional numbers in the equation which represent the Reynolds number Re = L u /ν (where ν is the kinematic viscosity), the magnetic Reynolds number Rm = L u σμ (where μ is the magnetic permeability and σ the conductivity of the fluid, assumed to be constant), S = M 2 /ReRm = B 2/μρ u 2 (where M is the Hartman number) and l>. We recall the definitions of the curl and cūrl operators in 2-dimensions curl u = u 2 u 1 for every vector u = (u 1,u 2 ), x 1 x 2 ( cūrl =, ) for every scalar function x 2 x 1 and the following formula: cūrl curl u = grad div u Δu. (1.3) 2. Weak formulation and existence Let us briefly recall the way we can represent the MHD equations (1.2) as an infinitedimensional equation (see [6,8,2]). The spaces used are a combination of spaces for the Navier Stokes equations (denoted with subscript 1) and spaces used in the theory of Maxwell equations (denoted with subscript 2). They are V 1 ={v (C (Ω))2, div v = }, V 1 ={v H 1 (Ω), div v = } (the closure of V 1 in H 1 (Ω) = (H 1 (Ω))2 ), H 1 ={v L 2 (Ω), div v = and v n Ω = } (the closure of V 1 in L 2 (Ω) = (L 2 (Ω)) 2 ), V 2 ={C (C (Ω)) 2, div C = and C n Ω = },

3 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 e1689 V 2 ={v H 1 (Ω), div C = and C n Ω = } (the closure of V 2 in H 1 (Ω) = (H 1 (Ω)) 2 ), H 2 = (the closure of V 2 in L 2 (Ω)) = H 1. The space V 1 is endowed with the scalar product ((u, v)) 1 = ( u, v ) = x i x i Ω 1 i 2 1 i 2 u x i v x i dx, which is the scalar product on H 1 (Ω). The dual space of V 1 is characterized by (see [8]) V 1 ={v H 1 (Ω), div v = }. The space V 2 is endowed with the scalar product ((u, v)) 2 = (curl u, curl v) which is equivalent to the usual scalar product induced by H 1 (Ω) on V 2. We set (see [8]) V = V 1 V 2, H = H 1 H 2,V the dual space of V, and by identifying H with its own dual we have V H V. The space H will be endowed with the following scalar products: (Φ, Ψ) = (u, v) + (B, C) for all Φ = (u, B), Ψ = (v, C) H, [Φ, Ψ]=(u, v) + S(B,C) and the induced (equivalent) norms Φ =(Φ, Φ) 1/2, [Φ]=[Φ, Φ] 1/2. The space V will be endowed with three scalar products ((Φ, Ψ)) = 1 Re ((u, v)) Rm ((B, C)) 2, Φ, Ψ = 1 Re ((u, v)) 1 + S Rm ((B, C)) 2, ((Φ, Φ)) J = ((u, v)) 1 + ((B, C)) 2 and the equivalent norms Φ =((Φ, Φ)) 1/2, Φ = Φ, Φ 1/2, Φ J = ((Φ, Φ)) 1/2 J. Let A 1 L(V 1,V 1 ), A 2 L(V 2,V 2 ), A L(V, V ), A J L(V, V ) be defined by A 1 u, v =((u, v)) 1 for all u, v V 1,

4 e169 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 A 2 B,C =((B, C)) 2 for all B,C V 2, AΦ, Ψ =((Φ, Ψ)), A J Φ, Ψ =((Φ, Ψ)) J for all Φ, Ψ V. As in [8] we consider A 1 L(V 1,V 1 ), A 2 L(V 2,V 2 ), A L(V, V ) as unbounded operators on H 1,H 2,H, for which the domains are D(A 1 ) ={u V 1, A 1 u H 1 }=H 2 (Ω) V 1, D(A 2 ) ={B V 2, A 2 B H 2 }=H 2 (Ω) V 2, D(A) = D(A 2 ) D(A 2 ) = (H 2 (Ω)) 2 V. Let b : L 1 (Ω) W 1,1 (Ω) L 1 (Ω) R be defined by b(u, v, w) = u i D i v j w j Ω 1 i,j 2 whenever the integrals make sense. We recall that, for m i satisfying m 1 + m 2 + m 3 > 1 or m 1 + m 2 + m 3 = 1 where at least two m i are nonzero, we have b(u, v, w) c 1 u H m 1 v H m 2 +1 w H m 3, (u, v, w) H m 1 (Ω) H m2+1 (Ω) H m 3 (Ω). (2.1) For m 1 = m 3 = 1,m 2 = we find that the trilinear form b is continuous on (H 1 (Ω)) 3 and satisfies b(u, v, v) =, u V α (α = 1, 2), v H 1 (Ω), b(u, v, w) = b(u, w, v), u V α, v, w H 1 (Ω). (2.2) We also define the trilinear form B : V V V R by setting B (Φ 1, Φ 2, Φ 3 ) = b(u 1,u 2,u 3 ) Sb(B 1,B 2,u 3 ) + b(u 1,B 2,B 3 ) b(b 1,u 2,B 3 ) for all Φ i = (u i,b i ) V, and the bilinear continuous operator B : V V V B(Φ 1, Φ 2 ), Φ 3 =B (Φ 1, Φ 2, Φ 3 ) Φ i V. From (2.1) we get B (Φ 1, Φ 2, Φ 3 ) c 2 max(1,s) Φ 1 H m 1 Φ 2 H m 2 +1 Φ 3 H m 3, (Φ 1, Φ 2, Φ 3 ) H m 1 (Ω) H m2+1 (Ω) H m 3 (Ω). (2.3) This yields for m 1 = m 2 = 1/2,m 3 = that B (Φ 1, Φ 2, Φ 3 ) c 3 ( Φ 1 Φ 1 Φ 2 AΦ 2 ) 1/2 Φ 3, (Φ 1, Φ 2, Φ 3 ) V D(A) H, (2.4) where c 3 = c 3 (Ω,S,Re,Rm). Let M M 4 (R) denote the diagonal matrix m ii = 1 for 1 i 2, m ii = S for 3 i 4. (2.5)

5 From (2.1) and the identity M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 e1691 B (Φ 1, Φ 2,MΦ 2 ) = b(u 1,u 2,u 2 ) + Sb(u 1,B 2,B 2 ) S(b(B 1,B 2,u 2 ) + b(b 1,u 2,B 2 )) we finally get B (Φ 1, Φ 2,MΦ 2 ) = Φ 1, Φ 2 V, B (Φ 1, Φ 2,MΦ 3 ) = B (Φ 1, Φ 3,MΦ 2 ) Φ i V. (2.6) Let f(t)=p(f (t), ), Ψ(t)=P(ψ 1 (t), ψ 2 (t)) where P : (L 2 (Ω)) 2 H is the projection on H. Then we rewrite the state equation (1.2) as dφ (t) + AΦ(t) + B(Φ(t), Φ(t)) = f(t)+ Ψ(t), dt t Φ() = Φ(T ) (,T), (2.7) and confine to the strong solutions Φ L 2 (,T; D(A)) W 1,2 ([,T]; H). Assume that Φ = (u,b ) L 2 (,T; V). Then we may reformulate problem (1.1) as ( 1 Minimize J(Φ, Ψ) = 2 Φ(t) Φ (t) 2 J + l ) 2 Ψ(t) 2 dt(p) over (Φ, Ψ) (L 2 (,T; D(A)) W 1,2 ([,T]; H)) L 2 (,T; H)subject to (2.7). Theorem 2.1. There is at least one solution (Φ, Ψ ) to problem (P ). Proof. Let {Φ n, Ψ n } be a minimizing sequence in problem (P ), i.e., inf (P ) J(Φ n, Ψ n ) inf (P ) + 1 n, (2.8) Φ n + AΦ n + B(Φ n, Φ n ) = f + Ψ n, a.e. t (,T); Φ n () = Φ n (T ). (2.9) By (2.8) it follows that {Φ n } is bounded in L 2 (,T; V), {Ψ n } is bounded in L 2 (,T; H) and therefore on a subsequence, again denoted by n, wehave Ψ n Ψ weakly in L 2 (,T; H). If we multiply (2.9) by tmφ n, integrate on Ω we get by (2.6) that This yields 1 d 2 dt (t[φ n(t)] 2 ) 1 2 [Φ n(t)] 2 + t Φ n (t) 2 = t[f(t)+ Ψ n (t), Φ n (t)], a.e. t (,T). (2.1) t[φ n (t)] 2 + τ Φ n (s) 2 ds C, t [,T]

6 e1692 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 and therefore [Φ n ()]=[Φ n (T )] C, n N. Here C denotes several positive constants independent of Φ and n. Next we multiply (2.9) by ta n and obtain after some calculus involving Young s inequality and (2.4) that 1 d ( t Φ n (t) 2) 1 2 dt 2 Φ n(t) 2 + t AΦ n (t) 2 = (f (t) + Φ n (t), taφ n ) B (Φ n (t), Φ n (t), taφ n (t)) t 4 AΦ n(t) 2 + t f(t)+ Φ n (t) 2 + tc 3 Φ n (t) 1/2 Φ n (t) AΦ n 3/2 t 2 AΦ n(t) 2 + tc Φ n (t) 2 Φ n (t) 4 + t f(t)+ Φ n (t) 2. Now integrating on (,t)and using the above estimates we get t Φ n (t) 2 + s AΦ n (s) 2 ds C which by Grönwall s lemma gives t Φ n (t) 2 C, t (,T]. (1 + s Φ n (t) 4 ) ds Since Φ n () = Φ n (T ) we infer that Φ n () C. Finally, multiplying (2.9) by AΦ n and integrating on Ω (,t)we obtain as above ( ) Φ n (t) 2 + AΦ n (s) 2 ds C Φ n () 2 + Φ n (s) 4 ds and therefore This yields Φ n (t) 2 + AΦ n (s) 2 ds C, t [,T]. Φ n L 2 (,T ;H) + B(Φ n, Φ n ) L 2 (,T ;H) C. Since V H we infer that {Φ n } is compact in C([,T]; H) L 2 (,T; V) and on subsequences we have Φ n Φ strongly in L 2 (,T; V) C([,T]; H), AΦ n AΦ weakly in L 2 (,T; H), Φ n (Φ ) weakly in L 2 (,T; H). By (2.4) we have (B(Φ n, Φ n ) B(Φ, Φ ), Φ) B (Φ n Φ, Φ n, Φ) +B (Φ, Φ n Φ, Φ) C Φ n Φ 1/2 ( Φ n Φ 1/2 Φ n 1/2 AΦ n 1/2 + Φ 1/2 Φ 1/2 A(Φ n Φ ) 1/2 ) Φ (2.11)

7 for all Φ H, and therefore M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 e1693 B(Φ n, Φ n ) B(Φ, Φ ) strongly in L 2 (,T; H). Letting n go to in (2.8), (2.9) we see that (Φ, Ψ ) satisfies system (2.7) and J(Φ, Ψ )= inf(p ). 3. Optimality conditions Let (Φ, Ψ ) be an optimal pair in problem (P ). For each ε > consider the approximating problem: minimize ( 1 2 Φ Φ 2 J + l 2 Ψ ) 2ε ξ 2 dt(p ε ) over Φ L 2 (,T; D(A)) W 1,2 ([,T]; H), Ψ, ξ L 2 (,T; H)subject to Φ (t) + AΦ(t) + B(Φ(t), Φ(t)) = f(t)+ Ψ(t) + ξ(t), t (,T); Φ() = Φ(T ). (3.1) By Theorem 2.1 for each ε > problem (P ε ) has at least one solution (Φ ε, Ψ ε, ξ ε ). Lemma 3.1. For ε we have Φ ε Φ strongly in L 2 (,T; V) C([,T]; H), Φ ε (Φ ), AΦ ε AΦ weakly in L 2 (,T; H), Ψ ε Ψ, ε 1/2 ξ ε weakly in L 2 (,T; H), { } lim ε inf Φ ε,ψ ε,ξ ε (P ε ) = inf (P ). (3.2) Φ,Ψ Proof. By taking (Φ, Ψ, ξ) = (Φ, Ψ, ) in (P ε ) we get ( 1 inf (P ε) Φ,Ψ,ξ 2 Φ Φ 2 J + l ) 2 Ψ 2 dt inf (P ). Φ,Ψ If multiply (3.1) with MΦ ε,tmφ ε and integrate on (, T ), (,t) respectively, we get by (2.4), (2.6) that t[φ ε (t)] 2 + Φ ε (t) 2 dt C. Now if we multiply (3.1) by taφ ε, integrate on (,t), we see as above that t Φ ε (t) 2 C, and therefore (,T] Φ ε () = Φ ε (T ) C, ε >.

8 e1694 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 When we multiply (3.1) by AΦ ε we obtain Φ ε (t) 2 + AΦ ε (s) 2 ds C, t [,T] and from (3.1) we have that Φ ε L 2 (,T ;H) + B(Φ ε, Φ ε ) L 2 (,T ;H) C, ε >. Hence on a subsequence we have Φ ε Φ strongly in C([,T]; H) L 2 (,T; V) Φ ε Φ, AΦ ε A Φ weakly in L 2 (,T; H) Ψ ε Ψ, ξ ε weakly in L 2 (,T; H). On the other hand, by (2.11) we see that B(Φ ε, Φ ε ) B( Φ, Φ) strongly in L 2 (,T; H) and therefore ( Φ, Ψ) is a solution to the state system (2.7). Finally, taking the limit in (P ε ), by the weak lower semicontinuity of the H-norm we obtain that inf (P ) Φ,Ψ ( 1 2 Φ Φ 2 J + l ) { } 2 Ψ 2 dt lim inf (P ε) ε Φ,Ψ,ξ hence Φ = Φ, Ψ = Ψ and the conclusions of Lemma 3.1 follow. In the space L 2 (,T; H)we define the operators (see [1]) A ε = + A + B(Φ ε, ) + B(, Φ ε ), D(A ε ) = X, A ε = + A + B (Φ ε,, ) + B (, Φ ε, ), X, (3.3) where X ={ W 1,2 ([,T]; H) L 2 (,T; D(A)), () = (T )}. It is easily seen that (A ε Υ, ) dt = (A ε,υ)dt,,υ D(A ε ) = D(A ε ) = X. The operators A and A are defined by the same formulae (3.3) where Φ ε = Φ.

9 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 e1695 Lemma 3.2. The operators A ε, A ε, A, A are closed, densely defined and have closed ranges in L 2 (,T; H). Moreover, dim N(A ε ), dim N(A ε ) n, independent of ε, A ε is the adjoint of A ε and the following estimates hold: A 1 ε g L 2 (,T ;D(A)) W 1,2 ([,T ];H) C g L 2 (,T ;H), g R(A ε ), (A ε ) 1 g L 2 (,T ;D(A)) W 1,2 ([,T ];H) C g L 2 (,T ;H), g R(A ε ). (3.4) Similarly, the operators A, A are mutually adjoint and estimates (3.4) remain true for A, A. We have used the symbols N and R to denote the null space and the range of the corresponding operators. (For hints on the proof see the case of Navier Stokes equations in [1]). For λ R,Φ X, Ψ L 2 (,T; H)we set ξ λ = (Φ ε + λφ) + A(Φ ε + λφ) + B(Φ ε + λφ, Φ ε + λφ) (f + Ψ ε + λψ). We may write ξ λ as ξ λ = ξ ε + λ(φ + AΦ + B(Φ ε, Φ) + B(Φ, Φ ε ) + λb(φ, Φ) Ψ) and so by the optimality of (Φ ε, Ψ ε, ξ ε ) in (P ε ) we have ( ((Φ ε Φ, Φ)) J + l(ψ ε, Ψ) + 1 ε (ξ ε, Φ + AΦ + B(Φ ε, Φ) ) +B(Φ, Φ ε ) Ψ) dt (3.5) for all X, Ψ L 2 (,T; H). We set q ε = 1 ε ξ ε, and for Ψ = we get from above ( ((Φε Φ, Φ)) J + (q ε, A ε Φ) ) dt =. Hence q ε D(A ε ) and A ε q ε = A J (Φ ε Φ ). (3.6) Therefore by (3.5) we obtain Ψ ε = 1 l q ε a.e. in (,T). (3.7) Then by Lemma 3.1 it follows that q ε L 2 (,T ;H) C, ε >. Now we may write q ε as qε 1 + q2 ε where q1 ε R(A ε), qε 2 know that N(A ε ). By Lemma 3.2 we q 1 ε L 2 (,T ;D(A)) W 1,2 ([,T ];H) C, ε >

10 e1696 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 hence on a subsequence, again denoted {ε},wehave q 1 ε q1 weakly in L 2 (,T; D(A)) W 1,2 ([,T]; H), q 2 ε q2 strongly in L 2 (,T; D(A)) W 1,2 ([,T]; H) because {q 2 ε } N(A ε ) and dim N(A ε ) n. Now letting ε tend to into (3.6), (3.7) it follows by Lemma 3.1 that A (q 1 + q 2 ) = A J (Φ Φ ); Ψ = 1 l (q1 + q 2 ) a.e. t (,T). Let denote q = q 1 + q 2. We have established the following maximum principle result for problem (P ): Theorem 3.1. If the pair (Φ, Ψ ) is optimal in problem (P ) then there is q L 2 (,T; D(A)) W 1,2 ([,T]; H)such that q (t) Aq B (Φ,,q) B (, Φ,q)= A J (Φ Φ ), a.e. t (,T), q() = q(t), (3.8) Ψ (t) = 1 q(t), a.e. t (,T). (3.9) l If q = (q u,q B ) and Ψ = (ψ 1, ψ 2 ) the adjoint system can be written as q u + 1 t Re Δq u + u q u B q B + q u u + q B B + p = Δ(u u ) in Q, q B 1 t Rm cūrl(curl q B) SB q u + u q B Sq u B q B u = cūrl(curl(b B )), div q u =, div q B = in Q, q u = onσ, q B n = and curl q B = on Σ, q u (x, ) = q u (x, T ), q B (x, ) = q B (x, T ) in Q and the optimality condition ψ 1 = 1 l q u, ψ 2 = 1 l q B in Q. 4. Semidiscrete-in-time approximations Let σ N ={t n } N n= be a partition of [,T] into equal intervals of duration Δt = T/N with t = and t N = T. We will denote by v the vector (v (1),v (2),...,v (N) ) of functions

11 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 e1697 belonging to a space Y = Y N. We associate the following approximate function: v N (t, x) = v (n) (x), t [t n 1,t n ],n= 1, 2,...,N, where v () = v (N), and a continuous, piecewise (in time t) linear function vpl N = vn pl (t, x) defined by the interpolating conditions v N pl (t n,x)= v (n) (x), n = 1, 2,...,N. On this partition we define the discrete target Φ (n) (x) = Φ (t n,x)for n =, 1,...,N. The state variables Φ (n) D(A) are constrained to satisfy the semidiscrete MHD equation 1 Δt (Φ(n) Φ (n 1) ) + AΦ (n) + B(Φ (n), Φ (n) ) = f (n) + Ψ (n) (4.1) obtained from (2.7) by a backward Euler discretization in time and the periodic condition Φ () = Φ (N). (4.2) Optimization is achieved by means of the minimization of the discretized-in-time functional J N (Φ, Ψ) = 1 N 2 Δt Φ (n) Φ (n) 2 J + l N 2 Δt Ψ (n) 2. (4.3) This functional results from applying the right-point discretization rule in time to the continuous functional J. The discrete-in-time approximate optimal control problem is then given by given Δt = T/N, L 2 (,T; V) find (Φ, Ψ) in D(A) H such that (P N ) (Φ, Ψ) is the solution of (4.1) and the functional (4.3) is minimized. As in the continuous case we have the following result on the existence of a optimal pair: Theorem 4.1. Given T>, Δt =T/Nthere exists at least one optimal solution (Φ, Ψ ) D(A) H of the semidiscrete optimal control problem. Now we can prove the convergence of the semidiscrete optimal control problem. Theorem 4.2. For Δt the solution {(Φ (n), Ψ (n) )} N of the semidiscrete-in-time optimal control problem tends to the solution (Φ, Ψ ) of the corresponding continuous optimal control problem. Proof. Using the similar computations as in the previous section (see [3,4,8]) we obtain easily that {(Φ N, Ψ N )} N=1 is uniformly bounded in L2 (,T; D(A)) L (,T; V) L 2 (,T; H)and { d dt Φ N pl } is uniformly bounded in L2 (,T; V). Moreover, we have that Φ N Φ N pl 2 L 2 (,T ;V) = Δt 3 N Φ (n) Φ (n 1) 2 when Δt.

12 e1698 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 Hence on subsequences we have that Ψ N Ψ weakly in L 2 (,T; H), Φ N Φ strongly in L 2 (,T; V), weak-* in L 2 (,T; D(A)). (4.4) Eq. (4.1) can be interpreted as dφ N pl dt + AΦ N + B(Φ N, Φ N ) = f N + Ψ N and as we pass N we find that the solution of the semidiscrete problem (P N ) converges to the corresponding solution of the continuous optimal control problem (P ). Due to the lack of differentiability in the application Ψ Φ(Ψ) (see e.g. [7]) we will replace problem (P N ) by a sequence of approximating problems (Pε N ), for which we can compute necessary conditions of optimality. For each ε consider the following optimization problem: minimize J N ε (Φ, Ψ, ξ) = Δt 2 N Φ (n) Φ (n) 2 J + l N 2 Δt Ψ (n) 2 + Δt 2ε N ξ (n) 2 (Pε N ) over (Φ, Ψ, ξ) D(A) H H satisfying 1 Δt (Φ(n) Φ (n 1) ) + AΦ (n) + B(Φ (n), Φ (n) ) = f (n) + Ψ (n) + ξ (n), Φ () = Φ (N). (4.5) By Theorem 4.1, for each ε > problem (P N ε ) has at least one solution (Φ ε, Ψ ε, ξ ε ). Lemma 4.1. For ε we have Φ (n) ε Φ (n) weakly in D(A), strongly in V, Ψ (n) ε Ψ (n) weakly in V, strongly in H, ε 1/2 ξ (n) ε weakly in H (4.6) for all n = 1,...,N and { } lim ε inf (Φ,Ψ,ξ) (P N ε ) = inf (Φ,Ψ) (P N ). For λ R, Φ (n) D(A), n =,...,N with Φ () = Φ (N) we denote ( Φ (n) Φ (n 1) ζ (n) λ = ζ (n) ε + λ Δt + λb(φ (n), Φ (n) ) Ψ (n) ) + AΦ (n) + B(Φ (n) ε, Φ (n) ) + B(Φ (n), Φ (n) ε )

13 M. Gunzburger, C. Trenchea / Nonlinear Analysis 63 (25) e1687 e1699 e1699 and by the optimality of (Φ (n) ε, Ψ (n) ε, ζ (n) ε ) we get Δt ((Φ (n) ε Φ (n), Φ (n) )) J + lδt (Ψ (n) ε, Ψ (n) ) + Δt ε ( N ζ (n) ε, Φ(n) Φ (n 1) Δt + AΦ (n) + B(Φ (n) ε, Φ (n) ) + B(Φ (n), Φ (n) ε ) + λb(φ (n), Φ (n) ) Ψ (n) ) (4.7) for all {Φ (n) } N n= D(A)N+1 with Φ () =Φ (N), and for all Ψ (n) H,,...,N. We set q ε (n) = 1 ε ζ(n) ε. Using an argument similar to the continuous case, it can be easily proved the following maximum principle for the semidiscrete-in-time optimal control problem (P N ). Theorem 4.3. If the pair (Φ, Ψ ) is optimal in problem (P N ) then there is q D(A) such that 1 Δt (q(n) q (n 1) ) Aq (n) B (Φ (n) ε,q (n), Ψ (n) ) B (Ψ (n),q (n), Φ (n) ε ) = A J (Φ (n) Φ (n) ), Ψ (n) = 1 l q(n) for all n = 1,...,N, with q () = q (N). References [1] V. Barbu, Optimal control of Navier Stokes equations with periodic inputs, Nonlinear Appl. Theory, Methods Appl. 31 (1998) [2] M. Gunzburger, A. Meir, J. Peterson, On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics, Math. Comput. 56 (194) (1991) [3] M. Gunzburger, The velocity tracking problem for Navier Stokes flows with bounded distributed controls, SIAM J. Control. Optim. 37 (6) (1999) [4] M. Gunzburger, Analysis and approximation of the velocity tracking problem for Navier Stokes flows with distributed control, SIAM J. Numer. Anal. 37 (5) (2) [5] L. Landau, E. Lifschitz, Electrodynamique des milieux continus, Physique théorique, tomeviii, MIR, Moscow [6] M. Sermange, R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math. 6 (1983) [7] R. Temam, Une proprieté générique des solutions stationnaries ou périodiques des équations de Navier Stokes, in: H. Fujita (ed.), Symposium Franco-Japonais, Septembre 1976 FunctionalAnalysis and NumericalAnalysis, Japan Society for the Promotion of Science, [8] R. Temam, Navier Stokes Equations, North-Holland, Amsterdam, 1979.

The Navier-Stokes Equations with Time Delay. Werner Varnhorn. Faculty of Mathematics University of Kassel, Germany

The Navier-Stokes Equations with Time Delay Werner Varnhorn Faculty of Mathematics University of Kassel, Germany AMS: 35 (A 35, D 5, K 55, Q 1), 65 M 1, 76 D 5 Abstract In the present paper we use a time