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We now turn to the analysis of a velocity boundary control problem for the Navier-Stokes equations, which is of particular interest in view of an application to optimal design of bypass grafts (see Sect. 5.2.4). We consider the Lagrangian multiplier method [16] for the treatment of the Dirichlet boundary conditions, which includes the control function itself. Let us consider a state system similar to (1.17), given by:

                 −ν∆v + (v · ∇)v + ∇p = 0 in Ω ∇ · v = 0 in Ω v = uc on Γc v = vin on Γin v = 0 on Γw −pn + ν∂v ∂n = 0 on Γout, (4.3)

where uc ∈ Uc := (H1/2(Γc))2 is the boundary control function and uin ∈ Uin(H1/2(Γin))2 a

second inflow condition. The domain Ω ⊂Rd for d = 2, 3 is assumed to be piecewise C2 with convex corners, while the Dirichlet portion of ∂Ω is ΓD= Γc∪ Γin∪ Γw. Moreover, let us define

uD∈ (H1/2(ΓD))d : uD=    uc on Γc vin on Γin 0 on Γw.

Let us consider a cost functional ˜J : V × Uc× Uin→R+ which represents the objective of the

optimization. It depends both on velocity and inflow conditions, and will be chosen according to the specific applications (see Sect. 5.2.4). Let us denote by Y = (v, p, ξ) ∈ X = V × Q × G the state variable, given by the velocity, the pressure and the Lagrange multiplier accounting for the Dirichlet conditions, with V = (H1_(Ω))d_{, Q = L}2_{(Ω) and G = (H}−1/2_(Γ

D))d, respectively. If the

function uin∈ Uin is known, the first problem of interest – denoted as deterministic design (OC)

problem – can be formulated as follows: given vin, find the boundary control function ˆuc solving

ˆ uc= arg min uc∈Uc, ad J (uc) s.t. Y = (v, p, ξ) ∈ X : ˜ A(Y, W ) = F (W ; uD), ∀ W = (z, q, λ) ∈ X (DD-OC)

where Uc, ad ⊆ Uc, J (uc) = ˜J (v, uc, vin) and the operators appearing in the state problem is

˜

A(Y, W ) = a(v, z) + b(p, z) + b(q, y) + c(v, v, z) + gD(ξ, z) + gD(v, λ),

F (w; uD) = −gD(uD, λ), (4.4)

where (using the same notation of Sect. 3.1)

a(v, z) := ν Z Ω ∇u : ∇z dΩ, b(p, z) := Z Ω p∇ · z dΩ, c(v, w, z) := Z Ω (v · ∇)w · z dΩ,

and the Dirichlet conditions on ΓD with boundary data uD are enforced through the bilinear

form

gD(v, w) :=

Z

ΓD

We recall that the state problem (4.4) is well-posed under the conditions stated in Theorem 3.4. Moreover, the velocity field satisfies the following stability estimate:

kvkV ≤

2

νkF (·; uD)k(H−1(Ω))d,

where kF (·; uD)k(H−1_(Ω))d is defined like in (3.39) and depends on the Dirichlet boundary condi-

tions uD. In particular, the solution v is unique and depends continuously on the data uD.

Let us consider ˜J : V × Uc× Uin→R+ as a functional of the velocity field v ∈ V , and denote

J (v) = J (v, uc, vin). A general existence result for the first optimality problem (DD-OC) can

be found in [130] (see Lemma 2.1 and the related proof):

Theorem 4.4. Let us assume that the cost functional J (v):

1. is bounded, i.e. there exists C0> 0 s.t. J (v) ≤ C0kvk2V;

2. is convex, i.e. for any u1, u2 ∈ V and γ ∈ [0, 1] it holds that (1 − γ)J (u) + γJ (u) ≥

J ((1 − γ)u + γu);

3. satisfies for some constants C1, C2, C3> 0 the weak coercivity inequality

J (v) ≥ C1kvk2V − C2kvkV − C3 for all v ∈ V. (4.5)

Let the admissible set Uc, ad for the control function be a closed and convex subset of Uc =

(H1/2(Γc))d. Then the problem (DD-OC) admits at least one solution.

The proof of Theorem 4.4 is a direct application of the abstract result stated in Theorem 4.1. A second optimal flow control problem of remarkable interest for the sake of applications is a worst-case OC problem, where we are interested to seek for the optimal control ˆuc that minimizes

the worst-case value of J (uc, vin) = ˜J (v, uc, vin) over all admissible values of the function vin.

Hence, we are interested in studying the so-called robust design problem: find the boundary control function ˆuc solving the worst-case optimization problem

ˆ uc= arg min uc∈Uc, ad max vin∈Uin, ad J (uc, vin) s.t. Y = (v, p, ξ) ∈ X : ˜ A(Y, W ) = F (W ; uc, vin), ∀ W = (z, q, λ) ∈ X (RD-OC)

The robust design problem (RD-OC) can be understood as a one-shot game, where the optimizer plays first and chooses the control function uc to minimize the cost functional J . The second

player then follows by choosing the function vinto maximize the cost function J . The payoff for

the designer is −J and for the second player J . Thus the optimal strategy for the optimizer is given as the solution of a min-max type of strategy obtained by solving (RD-OC), while the second player will choose his response by solving another problem. We call this the complementary

uncertainty problem, and it is defined as: given a known boundary control function uc, find the

function ˆvin solving ˆ vin= arg max vin∈Uin, ad J (uc, vin) s.t. Y = (v, p, ξ) ∈ X : ˜ A(Y, W ) = F (W ; vin), ∀ W = (z, q, λ) ∈ X (CU)

Theorem 4.5. Let us assume that Γinis an open and connected, non-empty subset of ∂Ω, and

that the cost functional J (v):

1. is bounded (see (i), Theorem. 4.4); 2. is upper semicontinuous, i.e. lim sup

v→v∗ J (v) ≤ J (v

∗_{) for all v}∗_{∈ V .}

Let the admissible set Uin, ad⊆ UC4 be a closed subset of

UC4:=u ∈ (H

2_(Γ

in))d : kuk(H2_(Γ

in))d≤ C4 , (4.6)

for some C4> 0 small enough such that (3.37) is satisfied, and furthermore that the viscosity is

large enough to satisfy (3.39). Then the problem (CU) admits at least one optimal solution.

Proof. Since Γin is bounded, the embedding H2(Γin) ,→ H1(Γin) is compact by Rellich’s theorem

and Uin is compact in (H1(Γin))d. According to Theorem 3.4, the solution map vin7→ v(vin)

is continuous in the H1_{-topology under our assumptions. Thus the image of U}

C4 through the

resolvent operator (4.4), restricted to velocity field, is a compact set in V . By Theorem 4.1, a bounded upper semicontinuous functional attains its maximum in a compact set.

Remark 4.6. For coercive cost functionals satisfying (4.5) the maximizer of (CU) will be found

on the boundary of the set of admissible functions Uin, ad. Thus we expect to find maximizers that

become increasingly singular as we increase C4 in (4.6).

Existence of solutions for the worst-case problem (RD-OC) in the infinite-dimensional case has not been extensively studied. In a recent paper [140], the concept of weak lower semi-continuity for set-valued functions is used to prove existence results for OC problems governed by PDEs for functionals of the min-max type. In the case that the admissible set Uin, ad does not depend on

the control variable uc, and therefore a sufficient condition for the weak lower semi-continuity of

b

J (uc) := sup

vin∈Uin, ad

˜

J (v, uc, vin)

is that ˜J ( · , · , vin) is weakly lower semi-continuous for all admissible vin ∈ Uin, ad (Theorem

2.5 of [140]). This assumption of independence does not strictly hold for problems discussed in chapter 6 (see eq. (5.3)), so further study on well-posedness of the min-max formulation would be needed.

4.2

Shape optimization problems

We discuss in this section the main results assuring the well-posedness of the shape optimization (SO) problems relevant to the applications presented in Part III. These problems are more difficult than the optimal control case, because of the shape dependence and the need of a suitable definition of shape variations. After treating the general case of an abstract SO problem, we characterize the case of problems governed by Navier-Stokes equations.

4.2.1

Abstract formulation

Recalling Sect. 1.3.2, a shape optimization problem can be expressed as a constrained optimization problem in the general form (1.11), here rewritten for the reader’s convenience:

ˆ

Ωo= arg min

Ωo∈Oad

y(Ωo) ∈ X(Ωo) is the state variable, X(Ωo) a reflexive Banach space defined on Ωo, Oad⊆ O

the set of admissible shapes, where O is a generic class of shapes (to be specified later). Here the state equation representing the constraint is defined over the domain Ωo – which for the

sake of convenience is identified with the original domain of the parametrized framework – whereas J : Oad → R is the cost functional, depending on the state variable as well, i.e.

J (Ωo) = ˜J (y(Ωo), Ωo), where ˜J : X × Oad → R. Let us recall that we denote by Γw the

free-boundary which can be displaced in order to optimize the objective.

Provided that the state problem admits a unique solution in any domain Ω, we can introduce a mapping y that with any Ω ∈ O associates the state solution y(Ω) ∈ X(Ω), i.e. y : Ω 7→ y(Ω) ∈

X(Ω). Moreover, let {Ωn}∞n=1 ⊂ O be a sequence of shapes and {yn}∞n=1 a sequence of state

solutions, being yn≡ y(Ωn) ∈ X(Ωn). Let us denote Ωn τ

→ ˆΩ and yn y two suitable notions of

convergence1_{, of the sequence of shapes and of state solutions, respectively; remark that in the}

latter case convergence involves different functional spaces, defined on the sequence {Ωn}∞n=1.

Then, the following abstract existence result holds (see for example [137], Theorem 2.10):

Theorem 4.7. Let E = {(Ω, y(Ω)), ∀ Ω ∈ Oad} be the graph of the mapping y(·) restricted to

Oad. Assume that

i) E is compact, i.e. for any sequence {(Ωn, y(Ωn)) ∈ E }∞n=1, there exists a subsequence

{(Ωnk, y(Ωnk)) ∈ E }

∞

k=1and ( ˆΩ, y( ˆΩ)) ∈ E , such that Ωnk τ

→ ˆΩ, y(Ωnk) y( ˆΩ) for k → ∞;

ii) the cost functional J (Ω) is lower semicontinuous, i.e. if Ωn τ

→ ˆΩ and yn ˆy, then

lim infn→∞ J (y˜ n, Ωn) ≥ ˜J (ˆy, ˆΩ).

Then, if ˜J is bounded from below, problem (4.7) has at least one solution.

Remark 4.8. The usual way to verity the first hypothesis stands on the two following assumptions:

1. compactness of Oad in O: for any sequence {Ωn}n≥1 ⊂ Oad there exists a subsequence

{Ωnk}f ≥1⊂ Oad such that Ωnk τ

→ Ω when k → ∞;

2. continuity of y(Ω) with respect to the domain: for any sequence {Ωn}n≥1⊂ Oad, Ω ⊂ Oad,

the following implication holds: Ωn τ

→ Ω ⇒ y(Ωn) y(Ω).

With respect to OC problems, verifying the well-posedness of SO problems involves additional assumptions of regularity on admissible shapes and continuity of state solution with respect to shape deformations. Thanks to Remark 4.8, we can decouple the dependence of the shape on the state solution: the first assumption is in fact just related to the set of admissible shapes. Usually, further regularity assumptions on the set of admissible shapes or admissible deformations ensure the compactness of Oad. For instance, Lipschitz domains (or, equivalently, domains satisfying the

so-called uniform cone condition [139]) yield to compact sets of admissible shapes; additional constraints, for instance on the volume of the admissible domains might also be imposed. Another class of domains yielding this property are the ones obtained through a perturbation of the identity map; this concept is useful not only to frame the analysis of problem (4.7), but also to express optimality conditions as well.

Let us denote

O ≡ O(Ω) := {Ωo= T (Ω) for some T ∈ T }, (4.8)

the space of shapes obtained by deforming Ω through the mapping T , where T = {T :Rd _→

Rd_{, (T − I) ∈ W}1,∞_(Rd_;Rd_{), (T}−1_{− I) ∈ W}1,∞_(Rd_;Rd_{)} is a space of diffeomorphisms}2 _inRd_;

in other words, T ∈ T is a differentiable mapping whose inverse is also differentiable. It can be shown that such a map is also continuous, so that the topology of the shapes obtained by deformation through T ∈ T is the same as the one of Ω.

As a set Oadof admissible shapes for the SO problem (4.7), we can thus define

Oad= {Ωo∈ O(Ω) : ∂Ωo\ Γwis fixed, |Ωo| ≤ V }.

Nevertheless, to ensure a compactness property, we need to enforce a uniform regularity condition – exploited e.g. in the forthcoming analysis of the FFD mappings – by introducing the following

(pseudo) distance over the set of shapes O:

d(Ω1, Ω2) = inf

T ∈T : T (Ω1_)=Ω2 kT − IkW1,∞(R

d_;Rd₎+ kT−1− Ik_W1,∞_(Rd_;Rd₎
. (4.9)

Thus, it can be shown (see e.g. [221]) that the set of admissible shapes defined by Oad= {Ωo∈ O(Ω) : d(Ω, Ωo) ≤ R : ∂Ωo\ Γw is fixed, |Ωo| ≤ V }

verifies a suitable compactness3_{property. This ensures the existence of (at least) one optimal}

shape, provided the other conditions required by Theorem 4.7 are verified.

Thanks to previous results, we thus can focus on perturbation of the identity maps, given by

T (·; θ) = I + θ, with θ ∈ W1,∞(Rd;Rd);

θ :Rd_→Rd _{can be regarded as a vectorial field which deforms the reference domain Ω to obtain}

the original domain Ωo. In particular, (i) T (Ω; θ) ∈ O(Ω) if the displacement field θ is sufficiently

small and (ii) the set of shapes obtained through T (Ω; θ) is compact, as stated by the following Lemma (see e.g. [7], Lemma 6.13 and [221], Theorem 2.4):

Lemma 4.9. Let θ ∈ W1,∞_(Rd_;Rd_{) be a vectorial field such that kθk}

W1,∞_(Rd_;Rd₎ ≤ 1. Then

T (·; θ) = I + θ ∈ T , i.e. it is a perturbation of the identity. Moreover:

1. if kθkW1,∞_(Rd_;Rd₎≤ 1 − α for some α ∈ (0, 1), then kT−1(·; θ) − Ik_W1,∞_(Rd_;Rd₎≤ 1/α;

2. the following family of shapes is compact (with respect to the pseudo-distance (4.9)):

OT(Ω) := {Ωo= T (Ω; θ) : kθkW1,∞≤ 1 − α, |Ωo| ≤ V }. (4.10)

Not only, we can also define the notion of shape derivative, unavoidable to formulate the optimality conditions, by using the derivative operation with respect to θ. We just point out that, as in the OC case, also for a shape optimization problem the optimality conditions can be expressed by means of the Lagrangian functional framework (see e.g. [7], Chapter 6). Following the same notation of Sect. 4.1.1, let us define the Lagrangian functional associated to (4.7) as

L(y, z, Ω) := ˜J (y, Ω) + F (z; Ω) − ˜A(y, z; Ω),

2_{Recall that W}1,∞_(Rd_{; R}d_{) is the space of Lipschitz functions ϕ : R}d_{→ R}d_{such that ϕ and ∇ϕ are bounded} in Rd_{, which can be equipped with the norm kϕk}

W1,∞_(Rd_;Rd₎= sup_x∈Rd(kϕk2+ |||∇ϕ|||2) where k · k2 (resp.

||| · |||2) is the Euclidean norm over R2 (resp. the induced Euclidean matrix norm over Rd×d).

3_{This is another important topology which can be used in conjunction with mapping techniques. Whenever}

domain perturbations are described by a family of bijective mappings having some regularities (for example, T ∈ T ), convergence of domains can be defined using the pseudo-distance (4.9) among the mappings T s.t. T (Ω) ∈ T .

where z ∈ X(Ω) is the Lagrangian multiplier. The optimum ( ˆΩ, ˆy( ˆΩ), ˆz( ˆΩ)) ∈ Oad× X( ˆΩ) × X( ˆΩ)

is a stationary point of the Lagrangian functional and fulfills the following optimality conditions4_:

∂pL(Ω, y(Ω), z(Ω))[φ] = 0 ∀ φ ∈ V state equation,

∂yL(Ω, y(Ω), z(Ω))[ψ] = 0 ∀ ψ ∈ V adjoint equation,

∂ΩL(Ω, y(Ω), z(Ω))[θ] ≥ 0 ∀ θ ∈ W1,∞(Rd;Rd), optimality condition.

(4.11)

As in the OC case, by differentiating L(y, z, Ω) with respect to z we find the state equation, while by differentiating L(y, z, Ω) with respect to y we find the adjoint equation:

find p(Ω) ∈ V (Ω) : a0(y(Ω))(z(Ω), ψ) = ∂yJ (Ω, y(Ω))(ψ)˜ ∀ψ ∈ V (Ω). (4.12)

Finally, by differentiating L(y, z, Ω) with respect to the shape we obtain

∂ΩL(Ω, y(Ω), z(Ω))[θ] = dJ (Ω; θ), (4.13)

where dJ (Ω; θ) = limt→0(J (Ωto) − J (Ω))/t is the derivative of J in Ω and direction θ, and

Ωt

o= T (Ω; tθ); in the same way, we can define the derivative of the Lagrangian. Last equation of

system (4.11) thus provides the expression of the gradient of the cost functional, needed e.g. for gradient-based numerical optimization procedures.

In document JULIO 2016 ESTUDIO DE SEGURIDAD Y SALUD (página 77-80)