Necessary Optimality Conditions

Since the reduced functional j is convex due to the (affine) linear-quadratic structure of the considered control problem, the necessary and sufficient optimality condition for the optimal solution ¯q ∈Q_ad of (3.29) reads as

j⁰(¯q)(δq−q)¯ ≥0 ∀δq∈Q_ad. (3.32) Based on this, we derive in the sequel an optimality system separately for the configurations C1 and C2. Thereby, we make use of the self-adjoint formulation (3.20). Due to this symmetry,

3. Optimal Control of Linear Fluid-Structure Interaction

the derivation of an optimality system for configuration C1 is straightforward. For configu-ration C2 however, this is not directly possible since for q ∈ Q = L²(I)^N, the right-hand sides B_fq and Bsq do not fulfill the prerequisites of Theorem 3.3. Therefore, an additional approximation step will be necessary.

Control Distributed in Space (Configuration C1)

Here, the control q ∈ Q = (L²(Ω)^d)^N acts as volume force through the linear operators B_f and B_s, as described in configuration C1. Since in this case it holds B_fq ∈ H¹(I;H_f) and B_sq ∈ H¹(I;H_s), the weak formulation (3.20) is applicable for f_f = B_fq and f_s = B_sq by Theorem 3.3. For the derivative of the reduced functional given by (3.28), we directly obtain the following representation:

Lemma 3.8. Let the initial datau₀,u₁, andv₀ satisfy Assumption 1 and let the desired states fulfill v_d ∈ H¹(I;V_f^∗)∩L²(I;H_f) and u_d ∈L²(I;H_s). Let for given q ∈Q the triple (v, u, p) be the solution of (3.20) with f_f = B_fq, fs = Bsq, and g = 0 guaranteed by Theorem 3.3.

Furthermore, let(z^v, z^u, z^p)be the solution of the adjoint equation (3.31)guaranteed by Theo-rem 3.7. Then, the directional derivative of the reduced cost functional atq in directionδq∈Q is given by

j⁰(q)(δq) =

N

X

i=1

((gⁱ_fδqⁱ, z^v))_f+ ((gⁱ_sδqⁱ, z^v))s+α(qⁱ, δqⁱ) .

Proof. By Theorem 3.3, the control to state mapGcan be understood as mapping fromQto L²(I;V)×L²(I;Vs)×L²(I;L_f). Similar to the proof of Lemma 3.5, let (ˆv,u,ˆ p) be the solutionˆ of (3.20) for f_f = f_s =g = 0. Furthermore, we denote byG₀:Q →L²(I;H_f)×L²(I;H_s)× L²(I;L_f) the linear partGgiven by (3.20) for zero initial data andf_f =B_fq,f_s=B_sq,g= 0.

Then,Gcan be written for q∈Qas

(v(q), u(q), p(q)) =Gq= (ˆv,u,ˆ p) +ˆ G0q.

Hence, we get directly

j⁰(q)(δq) =γ_f((v−v_d, δv))_f+γ_s((u−u_d, δu))_s+α

N

X

i=1

(qⁱ, δqⁱ) (3.33) for all δq∈Q, where (δv, δu, δp) =G₀δq.

By construction, (δv, δu, δp) solves for the right-hand sides f_f = B_fδq, fs = Bsδq, g = 0, and for zero initial data the equation (3.20). Thus, we obtain by testing this equation with (ϕ, ψ, ξ) = (z^v, z^u, z^p)∈L²(I;V)×L²(I;V_s)×L²(I;L_f) the identity

ρ_f((δv_t, z^v))_f−((δp,divz^v))_f +ν_f((∇δv,∇z^v))_f +ρ_s((δv_t, z^v))_s+µ_s((∇δu,∇z^v))_s =

N

X

i=1

((gⁱ_fδqⁱ, z^v))_f + ((gⁱ_sδqⁱ, z^v))_s , µs((∇δv,∇z^u))s−µs((∇δut,∇z^u))s = 0,

((z^p,divδv))_f = 0.

26

3.3. Optimal Control Problem zero initial conditions att= 0, the boundary terms vanish when using integration by parts in time. If we insert the equations into each other, then we obtain for any δq∈Q

γ_f((v−v_d, δv))_f +γs((u−u_d, δu))s=

Together with (3.33), this implies the assertion.

Combining the condition (3.32) and Lemma 3.8 implies the following representation and reg-ularity for the optimal control ¯q in terms of the pointwise projection P_Qad on the admissible set Q_ad, given by

P_Qad:L²(Ω)^d→L²(Ω)^d, P_Qad(r)(x) := max(q_a,min(q_b, r(x)))

for almost allx∈Ω, where the projection has to be applied componentwise forr ∈L²(Ω)^d. Lemma 3.9. Let the assumptions of Lemma 3.8 be fulfilled. Then, for configuration C1 the optimal solution q¯ ∈ Q_ad of the considered optimal control problem (3.29) fulfills for i= 1,2, . . . , N: Thus, for the optimal control holds q¯

Ωf ∈(H¹(Ω_f)^d)^N and q¯

Ωs ∈(H¹(Ω_s)^d)^N. Proof. The necessary optimality condition (3.32) can be written as

Z

3. Optimal Control of Linear Fluid-Structure Interaction

The optimal solution ¯q ∈ Q_ad solves the optimality system presented in the following theo-rem:

Theorem 3.10. Let the initial data u₀, u₁, and v₀ satisfy Assumption 1. Furthermore, let v_d∈H¹(I;V_f^∗)∩L²(I;H_f) andu_d∈L²(I;Hs). Then, the optimal solutionq¯∈Qad of optimal control problem (3.29)for configuration C1 fulfillsq¯

Ωf ∈(H¹(Ω_f)^d)^N and q¯

Ωs ∈(H¹(Ω_s)^d)^N as well as the following necessary optimality condition:

1. The optimal state (¯v,u,¯ p) = (v(¯¯ q), u(¯q), p(¯q))solves

In the following, the control q ∈ Q = (L²(I))^N is controlling the volume force through the linear operatorsB_f andB_s described in configuration C2. As Theorem 3.3 does not guarantee existence of a unique solution of (3.20) for a right-hand side f_f = B_fq ∈ L²(I;H_f) and f_s =B_sq ∈L²(I;H_s), we cannot directly proceed as for Configuration 1. Therefore, we will make use of a smooth sequence inQ∩(H¹(I))^N converging against the optimal solution. For smooth controls, the symmetric formulation (3.20) can be utilized, and a priori estimates for the adjoint then lead to higher regularity also for the limit. Then, we are able to derive the optimality system similar to the configuration C1.

Lemma 3.11. Let the initial datau₀,u₁, and v₀ satisfy Assumption 1. Furthermore, letv_d ∈ H¹(I;V_f^∗)∩L²(I;H_f) andu_d ∈L²(I;Hs). Additionally, for a given control q∈Q∩(H¹(I))^N, let the triple(v, u, p) be the solution of (3.20) with f_f =B_fq, f_s=B_sq, and g= 0 guaranteed by Theorem 3.3. Furthermore, let (z^v, z^u, z^p) be the solution of the adjoint equation (3.31)

28

3.3. Optimal Control Problem

guaranteed by Theorem 3.7. Then, the directional derivative of the reduced cost functional j at q in directionδq∈Q is given by (L²(I))^N topology, we obtain the assertion.

In the next lemma, we prove that the optimal control ¯q lies in Q∩(H¹(I))^N such that the representation derived in Lemma 3.11 is also valid for ¯q. Therefore, we will introduce also for configuration C2 the pointwise projection P_Qad on the admissible setQ_ad given here by

P_Qad:L²(I)→L²(I), P_Qad(r)(t) := max(q_a,min(q_b, r(t))) for almost allt∈I.

Lemma 3.12. Let the assumptions of Lemma 3.11 be fulfilled. Then, the optimal solution

¯

q ∈Q_adof the considered optimal control problem (3.29)for configurationC2lies in(H¹(I))^N.

Proof. Let ¯q ∈ Q_ad be the optimal solution. We consider a smooth sequence (q_n) with qn∈Q∩(H¹(I))^N andqn→q¯inQ. As in the proof of Lemma 3.8, according to Theorem 3.3, we have that (v_n, u_n, p_n) =Gq_nsolves (3.20) with right-hand sides f_f =B_fq_n,f_s =B_sq_n, and g = 0. The velocity and displacement have at least the regularities v_n ∈ H¹(I;H_f) and un∈L²(I;Vs). Therefore, Theorem 3.7 guarantees the existence of a unique adjoint solution (z_n^v, z_n^u, zn^p) of (3.31) with v_n−v_d and u_n−u_d in the right-hand side. By Lemma 3.11, we

for allδq ∈Q. Due to estimate a) in Theorem 3.3, the linearity of (3.20), and the boundedness of hⁱ inV_div, we get form, n∈N the estimate

3. Optimal Control of Linear Fluid-Structure Interaction

Furthermore, due to estimate a) in Theorem 3.7, the adjoint variables fulfill the estimate kz_n^v−z_m^vk²_L^∞_(I;L²_(Ω))+kz^v_n−z^v_mk²_L²_(I;H¹_(Ωf))+kz^u_n−z_m^uk²_L²_(I;H¹_(Ωs))

If we consider in (3.31) test functions ϕ ∈ L²(I;V_div) that are divergence-free in the fluid domain Ω_f, we get the estimate

By combining the above estimates, we derive for the adjointz_n^v−z_m^v the bound k∂_tz_n^v−∂_tz_m^vk²_L²_(I;V^∗ In addition, the directional derivative of the reduced cost functional j⁰(·)(δq) is continuous as the control to state mappingG:Q→L²(I;H_f)×L²(I;H_s) is affine-linear and continuous accordingly to Lemma 3.5. Therefore, the convergence q_n → q¯ in Q implies in addition j⁰(q_n)(δq)→j⁰(¯q)(δq) and we obtain the identity

As the optimal solution ¯q fulfills the necessary optimality condition (3.32), we get the opti-mality condition Using the projection P_Qad on the admissible set Q_ad, this can be expressed as

¯

3.3. Optimal Control Problem

The time regularity of the limit ˜z^v ∈H¹(I;V_div^∗ ) and the assumed regularity ofh∈V_div imply

∂trⁱ(t) =−1

αhhⁱ, ∂tz˜^v(t,·)iVdiv×V_div^∗ , i= 1,2, . . . , N

for almost all t∈I, and consequently thatrⁱ ∈H¹(I). Using the stability of the projection kP_Qad(r)kH¹(I)≤ krkH¹(I),

we obtain the asserted regularity ¯q∈(H¹(I))^N.

Then, the optimal solution ¯q∈Q_ad of the considered optimal control problem in configuration C2 fulfills the following theorem:

Theorem 3.13. Let the initial data u₀, u₁, and v₀ satisfy Assumption 1. Furthermore, let the desired states fulfill v_d ∈ H¹(I;V_f^∗)∩L²(I;H_f) and u_d ∈ L²(I;H_s). Then, the optimal solution q¯∈Q_ad of the considered optimal control problem (3.29) for configuration C2 fulfills

¯

q ∈(H¹(I))^N and the following necessary optimality condition:

1. The optimal state (¯v,u,¯ p) = (v(¯¯ q), u(¯q), p(¯q)) solves ρ_f((∂_t¯v, ϕ))_f −((¯p,divϕ))_f+ν_f((∇¯v,∇ϕ))_f

+ρs((∂tv, ϕ))¯ s+µs((∇u,¯ ∇ϕ))_f = ((B_fq, ϕ))¯ _f+ ((Bsq, ϕ))¯ s ∀ϕ∈L²(I;V), µ_s((∇v,¯ ∇ψ))_s−µ_s((∇∂_tu,¯ ∇ψ))_s= 0 ∀ψ∈L²(I;V_s), ((ξ,div ¯v))_f = 0 ∀ξ ∈L²(I;L_f).

2. The optimal adjoint (¯z^v,z¯^u,z¯^p) = (z^v(¯q), z^u(¯q), z^p(¯q)) solves

−ρ_f((ϕ, ∂_tz¯^v))_f +ν_f((∇ϕ,∇z¯^v))_f+ ((¯z^p,divϕ))_f

−ρ_s((ϕ, ∂_tz¯^v))_s+µ_s((∇ϕ,∇z¯^u))_s=γ_f((¯v−v_d, ϕ))_f ∀ϕ∈L²(I;V), µs((∇ψ,∇z¯^v))s+µs((∇ψ,∇∂tz¯^u))s=γs((¯u−ud, ψ))s ∀ψ∈L²(I;Vs),

−((ξ,div ¯z^v))_f = 0 ∀ξ ∈L²(I;L_f).

3. It holds for i= 1,2, . . . , N that

¯

qⁱ= P_Qad

−1 α

Z

Ω

hⁱ(x)¯z^v(·, x) dx

.

Proof. As ¯q ∈Q∩(H¹(I))^N, one can choose in the proof of Lemma 3.12 the sequenceqn= ¯q.

This immediately implies that ˜z^v = ¯z^v.

Remark 3.7. Thereby, the optimal state variable (¯v,u,¯ p) = (v(¯¯ q), u(¯q), p(¯q)) and adjoint variable (¯z^v,z¯^u,z¯^p) = (z^v(¯q), z^u(¯q), z^p(¯q)) fulfill the a priori estimates in Theorem 3.3 and Theorem 3.7.

4. Optimal Control of Nonlinear

Im Dokument Optimal Control of Time-Dependent Nonlinear Fluid-Structure Interaction  (Seite 31-39)