2.3 First order analysis via reduction technique
2.3.7 New necessary conditions
The aim of this paragraph is the derivation of necessary conditions for the outer optimization prob-lem (2.36), which parallels the fourth step of the general recipe ofParagraph 2.3.2.
Theorem 8(Necessary conditions for the outer optimization problemoOP):
Let(A, ¯uI, ¯uA, ¯yI, ¯yA) be the unique solution of the bilevel optimization problem (2.36), (2.37), γ = ∂A and letpmaxmin be given by (2.46).
Then the shape gradient of the reduced objective (2.45a) has a null
∇F(A) =− 1
2λ(pmaxmin|γ−p¯I|γ)2=0.
In particular, there holds
pmaxmin|γ=p¯I|γ onγ. (2.57)
Proof. According toTheorem 6, the bilevel optimization problem (2.36), (2.37) is equivalent to the reduced shape/topology optimization problem (2.45), which can be written compactly as (2.38)
minimize F(B):=J(B;G(B)), subject to
(B ∈ O,
ymin<G3(B)<ymax inJ.
Now omit the strict inequality constraint for a short while and look for necessary conditions of the relaxed optimization problem. In view of the discussion inParagraph 2.2.4, this approach is not unreasonable, since only an inactive constraint is omitted. Again only concentrating on the shape optimization aspect, a necessary condition of the unconstrained, relaxed problem obviously is
0=∇F(A) =− 1
2λ(pmaxmin|γ−p¯I|γ)2.
The definition ofpmaxmin =λ(∆ymaxmin−ymaxmin +ud)(cf. (2.46)) and the optimality conditions ¯uA =−∆ymaxmin+ ymaxmin and−p¯I =λ(u¯I −ud)(cf. step 3 inLemma 7) yield
0=− 1
2λ(pmaxmin|γ−p¯I|γ)2
=− 1
2λ(λ(∆ymaxmin−ymaxmin +ud)|γ+λ(u¯I−ud)|γ)2
=−λ
2((ud−u¯A)|γ+ (u¯I−ud)|γ)2
=−λ
2(u¯I|γ−u¯A|γ)2,
where the last step is due toH1-regularity ofudandLemma 2(mind the twofold meaning of(.)|γ, as trace with respect toI andA, respectively). In other words, the optimal control is weakly continuous across the optimal interface
u¯I|γ=u¯A|γ, onγ.
2.3.7 New necessary conditions 49 A comparison with the necessary conditions ofProposition 3shows, that this condition is a necessary condition for the original state-constrained optimal control problem (2.1): The Dirichlet traces of the adjoint stateptradcoincide on the interface, cf. (2.4g) and (2.4h). With use of the gradient equation (2.2g) this matching is transferred to the optimal control, see (2.5).
The original model problem is equivalent to the considered outer shape optimization problem and hence – following the steps in reversed order – shape gradient equals zero indeed is a necessary optimality condition.
As an easy consequence ofTheorem 8one has
Corollary 1(Local shape derivatives at the optimum):
Let(A, ¯uI, ¯uA, ¯yI, ¯yA)be the unique solution of the bilevel optimization problem (2.36), (2.37), letγ=∂A, letpmaxmin be given by (2.46), and letV∈ V be arbitrarily chosen.
Then the local shape derivativesyI0[V]andpI0[V]defined by (2.48) inLemma 8vanish yI0[V]≡0, pI0[V]≡0.
Proof. According toTheorem 8, there holdspmaxmin|γ = p¯I|γonγ. Hence, the BVP (2.48) is homogeneous and its unique solution isyI0[V] = pI0[V]≡0.
Remarks(on the strict inequality constraint):
SinceyI0[V]≡0 for allV ∈ V, the equationyJ|β =ymaxmin|β holds true forJ “near”I up to first order in perturbationsVofI. Near means here, thatJ =It:=Tt(V)(I)witht>0 sufficiently small, where the transformationTt(V)is defined in the2nditem of the discussion onpage 72.
In view of the reasoning ofParagraph 2.2.4, in particular the first of theRemarkstoLemma 6onpage 28, it turns out to be sufficient to use the constraintyB = ymaxmin onβand not the whole inequality constraint ymin < yJ < ymaxin J in order to derive first order necessary conditions for the set optimal control problem (2.30). That is to say, the interpretation of the interface condition as the “active part” of the inequality constraints, seems to hold true. This result justifies to derive necessary conditions in the given approach while omitting the strict inequality constraint.
In other words, the strict inequality constraint seems to have no impact on the admissible directions of variation. Supposing that this holds true, indeed, directly yields
0=dF(A;V) = (∇F(A),V·nI)L2(J), ∀V∈ V,
⇔ 0=∇F(A).
In particular, the assertion ofTheorem 8would follow without referring to results of [14], cf. Proposi-tion 3, which are not embedded in the approach presented here.
Corollary 2(Higher regularity at the optimum):
Let(A, ¯uI, ¯uA, ¯yI, ¯yA)be the unique solution of the bilevel optimization problem (2.36), (2.37).
Then the adjoint states, the multipliers provided byTheorem 5, and the optimal controls feature higher regularity
¯
uI ∈H2(I), (2.58a) p¯I ∈H2(I), (2.58b)
¯
σI ∈H12(γ), (2.58c)
¯
uA ∈H2(A)˚ , (2.58d) p¯A ∈H2(A)˚ , (2.58e)
¯
qA ∈H2(A)˚ . (2.58f)
Proof. According to the defining equations (2.39a), (2.39b) andTheorem 8the adjoint state ¯pI fulfills
−∆p¯I+p¯I =y¯I −yd a. e. inI,
∂np¯I =0 a. e. onΓ,
¯
pI|γ =pmaxmin|γ a. e. onγ.
This BVP is uniquely solvable inH2(J), since ¯yI−yd∈ H2(J)and sincepmaxmin ∈H2(B˚)(cf. (2.46)), which yieldspmaxmin|β ∈H3/2(γ)(cf.Lemma 1). This is (2.58b). Hence, one obtains (2.58c) via the properties of the Neumann trace operator (cf.Lemma 1) and ¯σI = ∂
I
np¯I ∈ H1/2(γ). Furthermore, the gradient equation (2.39d) yields (2.58a)
¯ uI =−1
λp¯I +ud∈H2(J).
Due to the control law (2.42f) andymaxmin ∈ H4(A)˚ one obtainsH2-regularity of ¯uA, see (2.58d). Moreover, the adjoint state in the active set solves the Dirichlet BVP (2.39e), (2.39f)
−∆p¯A+p¯A =y¯A−yd a. e. in ˚A,
¯
pA|γ =0 a. e. onγ.
Consequently, ¯pA ∈H2(A)˚ , which proves (2.58e). Finally the gradient equation (2.39g) yields (2.58f)
¯
qA =−λ(u¯A−ud)−p¯A ∈H2(B)˚ .
At this point the derivation of the first order necessary conditions of the bilevel optimization problem and its equivalent set optimal control problem is completed. The entire optimality system is repeated for convenience. At this, three different but equivalent formulations, which express that the shape gradient must vanish are given: prescribed inhomogeneous Dirichlet trace of ¯pI onγ, weak continuity across the interface of the optimal control, and prescribed inhomogeneous Dirichlet trace of the multiplier ¯qA onγ.
The first two conditions are known from the proof ofTheorem 8, whereas the last one is a consequence of the first two and the gradient equations (2.59n) and (2.59o).
Corollary 3(Full first order necessary conditions of the set optimal control problem):
LetA ∈ O and(u¯I, ¯uA, ¯yI, ¯yA) ∈ H2(I)×H2(A)˚ ×H2(I)×H2(A)˚ be the unique solution of the set optimal control problem (2.30).
Then there holds
−∆y¯I+y¯I =u¯I inI, (2.59a)
−∆y¯A+y¯A =u¯A in ˚A, (2.59b)
∂ny¯I =0 onΓ, (2.59c)
¯
yI|γ=ymaxmin|γ onγ, (2.59d)
∂
I ny¯I =∂
I
nymaxmin onγ, (2.59e)
−∆ymaxmin+ymaxmin =u¯A in ˚A, (2.59f) ymaxmin|γ=y¯A|γ onγ, (2.59g) ymin<y¯I <ymax inI, (2.59h)
−∆p¯I +p¯I =y¯I−yd inI, (2.59i)
−∆p¯A+p¯A =y¯A−yd in ˚A, (2.59j)
∂np¯I =0 onΓ, (2.59k)
¯
pA|γ=0 onγ, (2.59l)
∂
I
np¯I =σ¯I onγ, (2.59m) λ(u¯I−ud) +p¯I =0 inI, (2.59n) λ(u¯A−ud) +p¯A+q¯A =0 in ˚A, (2.59o)
¯
pI|γ−pmaxmin|γ=0 onγ, (2.59p)
¯
uI|γ−u¯A|γ =0 onγ, (2.59q)
¯
qA|γ−pmaxmin|γ =0 onγ, (2.59r) where the last three equations are equivalent formulations for the condition that the shape gradient must vanish at the optimal configuration.
This optimality system deserves closer attention. Its connection with the necessary condition of [14], cf.Proposition 3, is of major interest.
Proposition 5(Connection between common and new necessary conditions):
LetAbe the (optimal) active set and let ¯pI, ¯pA, ¯qA and ¯σIbe the multipliers of the optimality system (2.59) and letpItrad,pAtrad,µmaxandµγbe given byProposition 3.
51
Then there holds
¯
pI = pItrad inI, (2.60a)
¯
pA+q¯A = pAtrad inA, (2.60b)
−∆q¯A+q¯A =µmax˚
A a. e. in ˚Amax, (2.60c)
−∆q¯A+q¯A =−µmin˚
A a. e. in ˚Amin, (2.60d)
¯ σI+∂
A
n(q¯A+p¯A) =µγ a. e. onγ. (2.60e)
The proof is included in the proof ofCorollary 6.
Remarks(Concluding results):
1. The Bryson-Denham-Dreyfus approach yields an additive decomposition of the adjoint statepAtrad into ¯pA and ¯qA, whereas everything remains unchanged in the inactive set.
2. The relationship between the multipliersµA˚ and ¯qA is directly linked with the BDD reformulation of the state constraint. Instead of the original state constraint, a differentiated counterpart – the control law (2.28) – was used, and, in the end, an analog differential equation holds true for the corresponding multipliers (cf. (2.60c), (2.60d)). In particular, one recognizes that the differential operations made within the BDD ansatz in order to derive the control law (primal regime) finds expression in an inverse manner on the dual regime and consequently yield higher regularity for the multiplier.
3. The new multipliers ¯qA and ¯σI do not feature any sign conditions. On the one hand this fact is not surprising, since they correspond to equality constraints and as such do not have a fixed sign. On the other hand the multipliersµA˚andµγare known to be nonnegative and one might wonder, why this property is not mirrored the new ones, especially if one bares in mind their tight connection shown inProposition 5. This topic is examined inCorollary 6in theAppendix B.