An approximate optimization problem

Following the outlined strategy, we introduce a penalization parameter𝑠>0 (which will be driven to∞later) and consider the approximate problem

𝑦∈𝒴,𝑢∈𝒰min 𝒥_𝑠 𝑦, 𝑢_,

s.t. (2.1.1), (2.1.2), and (2.4.1) hold )

(Ps) where the objective function is

𝒥_𝑠 𝑦, 𝑢

and the additional constraint is

𝒩_𝛼 𝑓^𝛼_{≤ ℒ}

being some minimizer of (P), and𝐶_Γis the (optimal) constant corresponding to the continuous embedding 𝒰 ⊂ 𝐿² [0, 𝑇_•] ×Γ;R³

. On the one hand, (2.4.1) is automatically satisfied for any minimizer 𝑦_∗, 𝑢_∗

of (P)—in particular, there are feasible points for (Ps)—which can be verified as follows: Due to (2.1.2) it holds that

k(𝐸_∗, 𝐻_∗)k_𝐿2([

which yields (2.4.1) in view of (2.1.1) and (2.1.6).

On the other hand, (2.4.1) ensures a certain weak lower semicontinuity ofk 𝒢 k_Λ∗by the following lemma—and this is conversely the very reason why we impose (2.4.1).

2.4 First order optimality conditions 93 Lemma 2.4.1. Let 𝑦_𝑘, 𝑢_𝑘 ⊂ 𝒴 × 𝒰 with 𝑓_𝑘^𝛼 ≥ 0and limit functions 𝑢 ∈ 𝒰, 𝑓^𝛼 ∈ 𝐿^∞ [0, 𝑇_•] ×Ω×R³

, 𝑓𝛼 + ∈ 𝐿^𝑞

𝛾𝑇⁺_•, 𝑑𝛾𝛼

,(𝐸, 𝐻) ∈ 𝐿² [0, 𝑇_•] ×Ω;R⁶

such that for 𝑘→

∞it holds that𝑢_𝑘⇀𝑢in𝒰,𝑓^𝛼

𝑘

⇀∗ 𝑓𝛼in𝐿^∞ [0, 𝑇_•] ×Ω×R³ ,𝑓^𝛼

𝑘,+⇀ 𝑓₊^𝛼in𝐿^𝑞 𝛾𝑇⁺_•, 𝑑𝛾𝛼

, (𝐸_𝑘, 𝐻_𝑘)⇀(𝐸, 𝐻)in𝐿² [0, 𝑇_•] ×Ω;R⁶

. Furthermore, assume that(2.1.2)and(2.4.1)are satisfied along the sequence. Then, 𝑦, 𝑢 _{∈ 𝒴 × 𝒰}

,(2.1.2)and(2.4.1)are preserved in the limit, and

^{𝒢 𝑦, 𝑢} _Λ∗ ≤lim inf

𝑘→∞

^{𝒢 𝑦}𝑘, 𝑢_{𝑘 Λ}∗. (2.4.2) Proof. Note that(𝑢_𝑘)converges to𝑢strongly in𝐿² [0, 𝑇_•] ×Γ;R³

. Step 1. 𝑓^𝛼 ∈ 𝒴^𝛼

pd and (2.1.2) and (2.4.1) are preserved in the limit: Take 𝜂 ∈ 𝐶_𝑐^∞ ]0, 𝑇_•[ ×Ω×R³

and consider

𝑔_𝑘 B𝜕^𝑡 𝜂𝑓_𝑘^𝛼₊

b^𝑣𝛼·𝜕^𝑥 𝜂𝑓_𝑘^𝛼_. In light of (2.4.1), the sequence 𝑔_𝑘

is bounded in𝐿² [0, 𝑇_•] ×Ω;𝐻⁻¹ R³ . Therefore, 𝑔_𝑘

converges, after possibly extracting a suitable subsequence, to some 𝑔weak-* in 𝐿² [0, 𝑇_•] ×Ω;𝐻⁻¹ R³ . Since for all𝜑∈𝐶^∞_𝑐 ]0, 𝑇_•[ ×Ω×R³

𝑔 𝜑

= lim

𝑘→∞ 𝜕^𝑡 𝜂𝑓_𝑘^𝛼₊

b^𝑣𝛼·𝜕^𝑥 𝜂𝑓_𝑘^𝛼 𝜑

= lim

𝑘→∞

−

∫ ^𝑇• 0

∫

Ω

∫

R³

𝜂𝑓_𝑘^𝛼𝜕^𝑡𝜑+

b^𝑣𝛼𝜂𝑓_𝑘^𝛼·𝜕^𝑥𝜑_{𝑑𝑣𝑑𝑥𝑑𝑡}

=−

∫ ^𝑇• 0

∫

Ω

∫

R³

𝜂𝑓^𝛼𝜕^𝑡𝜑+

b^𝑣𝛼𝜂𝑓^𝛼·𝜕^𝑥𝜑_{𝑑𝑣𝑑𝑥𝑑𝑡}

= 𝜕^𝑡 𝜂𝑓^𝛼₊

b^𝑣𝛼·𝜕^𝑥 𝜂𝑓^𝛼 𝜑 and since𝐶_𝑐^∞ ]0, 𝑇_•[ ×Ω×R³

is dense in𝐿² [0, 𝑇_•] ×Ω;𝐻¹ R³ , we have

𝜕^𝑡 𝜂𝑓^𝛼₊

b^𝑣𝛼·𝜕^𝑥 𝜂𝑓^𝛼

=𝑔 ∈𝐿² [0, 𝑇_•] ×Ω;𝐻⁻¹ R^{3 .} Furthermore, by weak-*-convergence it holds that

_𝜕𝑡 𝜂𝑓^𝛼₊

b^𝑣𝛼·𝜕^𝑥 𝜂𝑓^𝛼 _𝐿2([

0,𝑇_•]×Ω;𝐻⁻1(R³⁾⁾

≤lim inf

𝑘→∞

_𝜕𝑡 𝜂𝑓_𝑘^𝛼₊

b^𝑣𝛼·𝜕^𝑥 𝜂𝑓_𝑘^𝛼 _𝐿2([

0,𝑇_•]×Ω;𝐻⁻1(R³⁾⁾

≤ ℒ_𝛼

if𝜂satisfies (2.1.4). Thus, (2.4.1) is preserved in the limit. Moreover, as in the proof of Theorem 2.2.1, we also see that 𝑓^𝛼 ∈

𝐿¹

𝛼kin∩𝐿^∞

[0, 𝑇_•] ×Ω×R³

and that (2.1.2) is preserved in the limit. Altogether, 𝑦, 𝑢_{∈ 𝒴 × 𝒰}

.

Step 2. Proof of (2.4.2): To this end, we have to pass to the limit in the right-hand sides of (1.1.2) and (1.1.3); this procedure has already been carried out a few times in similar, yet not identical situations. As a consequence of Lemma 2.1.3, we may assume that

𝑗^int

𝑘

converges weakly to𝑗^intin𝐿⁴3 [0, 𝑇_•] ×Ω;R³

; in order to verify that

this weak limit indeed is𝑗^int, we recall that an energy estimate like (2.1.2) is sufficient.

Hence, we can easily pass to the limit in all terms but the nonlinear one, first for 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ_∈ Ψ^𝑁_𝑇

•

×Θ²_𝑇

•and then for arbitrary 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ _∈

Λwith the help of Section 2.3.1. Regarding the nonlinear term, we first consider𝜓^𝛼 ∈Ψ𝑇_•that factorizes, as in Section 1.4. For some𝑙 ∈Nand𝜁∈𝐶^∞_𝑐 R³

with supp𝜁⊂𝐵_𝑅(for some𝑅>0), we find an𝜂^𝑙 ∈𝐶^∞_𝑐 (]0, 𝑇_•[ ×Ω×𝐵_𝑅), similarly to (1.4.17), such that

∫

R³

𝜁(𝑣) 1−𝜂^𝑙 _𝑓𝛼

𝑘 −𝑓^𝛼 (·,·, 𝑣)𝑑𝑣 _𝐿2([

0,𝑇_•]×Ω)

< 1

𝑙; (2.4.3)

note that the𝐿²-norms of the 𝑓𝛼

𝑘 are uniformly bounded. For this fixed𝜂^𝑙it holds that

_𝜕𝑡 𝜂^𝑙𝑓_𝑘^𝛼₊

b^𝑣𝛼·𝜕^𝑥 𝜂^𝑙𝑓_𝑘^𝛼 _𝐿2(

R^×R³;𝐻⁻1(R³⁾⁾=

_𝜕𝑡 𝜂^𝑙𝑓_𝑘^𝛼₊

b^𝑣𝛼·𝜕^𝑥 𝜂^𝑙𝑓_𝑘^𝛼 _𝐿2([

0,𝑇_•]×Ω;𝐻⁻1(R³⁾⁾

≤ 𝒩_𝛼 𝑓^𝛼

𝑘

_𝜂𝑙

_𝐻1(]

0,𝑇_•[×Ω×R³⁾

+ _𝜂𝑙

_𝐿∞([0,𝑇_•]×Ω;𝐻1(R³⁾⁾

.

By virtue of (2.4.1), the right-hand side is uniformly bounded in𝑘, whence we have for a subsequence possibly depending on𝑙,

∫

R³

𝜁(𝑣) 𝜂^𝑙𝑓_𝑘^𝛼

𝑗

(·,·, 𝑣)𝑑𝑣^𝑗→∞−→

∫

R³

𝜁(𝑣) 𝜂^𝑙𝑓^𝛼_{(·,·, 𝑣)𝑑𝑣}

in 𝐿²([0, 𝑇_•] ×Ω) due to Lemma 1.4.2. Assuming that all 𝜓^𝛼 ∈ Ψ𝑇• factorize, i.e., 𝜓^𝛼(𝑡, 𝑥, 𝑣) = 𝜓₁^𝛼(𝑡, 𝑥)𝜓^𝛼₂(𝑣), and using (2.4.3), we may now pass to the limit in all terms along a common subsequence, that is,

𝒢 𝑦, 𝑢 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ

= lim

𝑗→∞𝒢

𝑦_𝑘𝑗, 𝑢_𝑘𝑗 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ ,

via the same diagonal sequence argument as in Section 1.4.2 or the proof of Theo-rem 2.2.1. Since the limit on the left-hand side does not depend on the extraction of this subsequence, we conclude that the equality above even holds for the full limit 𝑘→ ∞by using the standard subsubsequence argument. Thus,

^{𝒢 𝑦, 𝑢}

𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ

^{≤lim inf}_𝑘→∞

^{𝒢 𝑦}𝑘, 𝑢_𝑘

Λ^∗

𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ Λ

.

This inequality then also holds for general 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ_∈

Λby a density argument;

see Section 1.4 and the definition ofΛ. Altogether, (2.4.2) is proved.

Remark 2.4.2. It is important to understand the necessity of (2.4.1) for Lemma 2.4.1 and for later treating (P_s): In the proof of Theorem 2.2.1, we applied the momentum averaging lemma 1.4.2 to a sequence where any 𝑓^𝛼

𝑘 already solvesa Vlasov equation in the sense of distributions, that is,

𝜕^𝑡𝑓_𝑘^𝛼+

b^𝑣𝛼 ·𝜕^𝑥𝑓_𝑘^𝛼 =−div^𝑣 𝐹_𝑘𝑓_𝑘^𝛼_,

2.4 First order optimality conditions 95 which gave us a direct estimate on the𝐿² [0, 𝑇_•] ×Ω;𝐻⁻¹ R³ -norm of some𝜂𝑓_𝑘^𝛼by the corresponding a priori𝐿^𝑝-bounds on𝐹_𝑘 and 𝑓^𝛼

𝑘. However, the 𝑓^𝛼 of some 𝑦, 𝑢 that is feasible for (Ps) do not necessarily solve a Vlasov equation as above. Thus, suitable estimates on the𝐿² [0, 𝑇_•] ×Ω;𝐻⁻¹ R³ -norm along some sequence cannot be obtained without imposing them a priori, that is, imposing (2.4.1). Without this, we would not be able to pass to the limit as in the proof above, and the important weak lower semicontinuity ofk 𝒢 k_Λ∗ could not be proved.

Now we are able to prove existence of minimizers of (Ps).

Theorem 2.4.3. There is a (not necessarily unique) minimizer of (Ps).

Proof. This is proved in much the same way as Theorem 2.2.1 was proved. We no longer have to show that (VM) has to be preserved in the limit. Instead, we apply Lemma 2.4.1: The assumptions there are satisfied for a minimizing sequence (after extracting a suitable subsequence) and the respective weak limits. Thus, the new constraint (2.4.1) is also preserved in the limit, and the new objective function 𝒥_𝑠 indeed attains its minimum at the limit tuple 𝑦, 𝑢

.

Later, we will need that𝒴 × 𝒰is complete; this is proved in the following lemma.

Lemma 2.4.4. 𝒴 × 𝒰is a Banach space.

Proof. We only have to show completeness of𝒴^𝛼

pd: Let 𝑓_𝑘

be a Cauchy sequence in 𝒴^𝛼

pd. Clearly, this sequence converges to some 𝑓 with respect to the 𝐿¹

𝛼kin- and 𝐿^∞-norm. For some𝜂 ∈ 𝐶^∞_𝑐 ]0, 𝑇_•[ ×Ω×R³

, the sequence 𝜕^𝑡 𝜂𝑓_𝑘+

b^𝑣𝛼·𝜕^𝑥 𝜂𝑓_𝑘 converges to some 𝑔 in 𝐿² [0, 𝑇_•] ×Ω;𝐻⁻¹ R³ since this space is complete. As in Step 1 of the proof of Lemma 2.4.1, we see that𝑔 =𝜕^𝑡 𝜂𝑓₊

b^𝑣𝛼·𝜕^𝑥 𝜂𝑓

. If𝜂satisfies (2.1.4), then

_𝜕𝑡 𝜂 𝑓 − 𝑓_𝑘 +

b^𝑣𝛼·𝜕^𝑥 𝜂 𝑓 − 𝑓_{𝑘 𝐿2([}

0,𝑇•]×Ω;𝐻⁻1(R³⁾⁾

≤

_𝜕𝑡 𝜂 𝑓 − 𝑓_𝑚 +

b^𝑣𝛼·𝜕^𝑥 𝜂 𝑓 −𝑓_{𝑚 𝐿2([}

0,𝑇_•]×Ω;𝐻⁻1(R³⁾⁾

+

_𝜕𝑡 𝜂 𝑓_𝑚− 𝑓_𝑘 +

b^𝑣𝛼·𝜕^𝑥 𝜂 𝑓_𝑚 −𝑓_{𝑘 𝐿2([0,𝑇}

•]×Ω;𝐻⁻1(R³⁾⁾

≤

_𝜕𝑡 𝜂 𝑓 − 𝑓_𝑚 +

b^𝑣𝛼·𝜕^𝑥 𝜂 𝑓 −𝑓_{𝑚 𝐿2([}

0,𝑇_•]×Ω;𝐻⁻1(R³⁾⁾

+ 𝒩_𝛼 𝑓_𝑚− 𝑓_𝑘

for any 𝑘, 𝑚 ∈ N. Here, the second summand of the right-hand side can be made arbitrarily small (uniformly in𝜂) for large𝑘and 𝑚because of the Cauchy property, and the first summand is arbitrarily small if 𝑚 = 𝑚 𝜂

is large enough. Thus, 𝑓_𝑘 converges to 𝑓 in the whole𝒴^𝛼

pd-norm altogether.

Next, we want to derive first order optimality conditions for a minimizer of (Ps).

To this end, we consider the differentiability of the objective function𝒥_𝑠. Clearly, the only difficult term is

^{𝒢 𝑦, 𝑢}

2

Λ^∗. To tackle this one, we state a duality result, which links differentiability of a norm to uniform convexity of the dual space.

Proposition 2.4.5. A Banach space 𝑋 is uniformly smooth if and only if𝑋^∗ is uniformly convex. In this case, for each unit vector𝑥 ∈ 𝑋there is exactly one𝑥^∗∈ 𝑋^∗withk𝑥^∗k_𝑋∗ =1 satisfying𝑥^∗𝑥 =1. Furthermore, this𝑥^∗is the derivative of the norm at𝑥.

Here, “uniformly smooth” means that lim𝑡→0

^{𝑥+𝑡 𝑦}

_𝑋^{− k𝑥k}𝑋

𝑡

exists and is uniform in𝑥, 𝑦 ∈ {𝑧∈𝑋 | k𝑧k_𝑋 =1}. The original work in this subject was done by Day [Day44]; see also [Lin04, Chapter 2] and [Bre11, Section 3.7, Problem 13] for an overview of different concepts of and relations between convexity and smoothness of normed spaces.

From Proposition 2.4.5 we easily get the following corollary, which we will need in the following.

Corollary 2.4.6. Let𝑋be a Banach space such that𝑋^∗is uniformly convex. Then the map 𝑧:𝑋 → R,𝑧(𝑥)= ¹₂k𝑥k²_𝑋 is differentiable on𝑋with derivative𝑧⁰(𝑥) =𝑥^∗where𝑥^∗is the unique element of𝑋^∗satisfyingk𝑥^∗k_𝑋∗ =k𝑥k_𝑋and𝑥^∗𝑥= k𝑥k²_𝑋. (The map𝑧⁰:𝑋 →𝑋^∗is often referred to as the duality map.)

Proof. By Proposition 2.4.5, the norm is differentiable on the unit sphere of𝑋. Since the norm is positive homogeneous, this holds true on𝑋except in𝑥=0, and the derivative is𝑥^∗such thatk𝑥^∗k_𝑋∗ =1 and𝑥^∗𝑥=k𝑥k_𝑋(still this𝑥^∗is uniquely determined by these two properties). Applying the chain rule we see that𝑧is differentiable on𝑋\ {0}and has the asserted derivative.

That𝑧is differentiable in𝑥 =0 and𝑧⁰(0)=0 is clear.

With this corollary we see that the objective function𝒥_𝑠is differentiable.

Lemma 2.4.7. The objective function𝒥_𝑠is differentiable, and its derivative is given by 𝒥_𝑠⁰ 𝑦, 𝑢

𝛿𝑦,𝛿𝑢

=

𝑁

Õ

𝛼=1

𝑤_𝛼

∫

𝛾𝑇•⁺

sign 𝑓₊^{𝛼 𝑓}+^𝛼

𝑞−1

𝛿𝑓₊^𝛼𝑑𝛾_𝛼

+ Õ3

𝑗=1

∫ ^𝑇• 0

∫

Γ

sign 𝑢_𝑗 ^𝑢𝑗

𝑟−1

𝛿𝑢_𝑗+𝜅1sign 𝜕^𝑡𝑢_{𝑗 𝜕}𝑡𝑢_𝑗

𝑟−1

𝜕^𝑡𝛿𝑢_𝑗

+𝜅2

Õ3

𝑖=1

sign 𝜕^𝑥𝑖𝑢_{𝑗 𝜕}𝑥_𝑖𝑢_𝑗

𝑟−1

𝜕^𝑥𝑖𝛿𝑢_𝑗

! 𝑑𝑥𝑑𝑡

+

𝑁

Õ

𝛼=1

−

∫ ^𝑇• 0

∫

Ω

∫

R³

𝜕^𝑡𝜓^𝛼+

b^𝑣𝛼·𝜕^𝑥𝜓^𝛼+𝑞_𝛼(𝐸+

b^𝑣𝛼×𝐻) ·𝜕^𝑣𝜓^𝛼 𝛿𝑓^𝛼 +𝑞_𝛼(𝛿𝐸+

b^𝑣𝛼×𝛿𝐻)𝑓^𝛼·𝜕^𝑣𝜓^𝛼_{𝑑𝑣𝑑𝑥𝑑𝑡}

2.4 First order optimality conditions 97

+

∫

𝛾⁺𝑇•

𝛿𝑓₊^𝛼𝜓^𝛼𝑑𝛾𝛼−

∫

𝛾𝑇•⁻

𝒦_𝛼𝛿𝑓₊^𝛼_{𝜓𝛼𝑑𝛾}

𝛼

!

+

∫ ^𝑇• 0

∫

R³

𝜀𝛿𝐸·𝜕^𝑡𝜗^𝑒−𝛿𝐻·curl^𝑥𝜗^𝑒−4𝜋 𝛿𝑗^int+𝛿𝑢_·

𝜗^𝑒_{𝑑𝑥𝑑𝑡}

+

∫ ^𝑇• 0

∫

R³

𝜇𝛿𝐻·𝜕^𝑡𝜗^ℎ+𝛿𝐸·curl^𝑥𝜗^ℎ

𝑑𝑥𝑑𝑡, (2.4.4)

where 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ_∈

Λis the unique element inΛsatisfying

𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ Λ

=𝑠

^{𝒢 𝑦, 𝑢} _Λ∗, 𝒢 𝑦, 𝑢 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ

=𝑠

^{𝒢 𝑦, 𝑢}

2

Λ^∗. (2.4.5) Proof. The only difficult term is₂^𝑠

^{𝒢 𝑦, 𝑢}

2

Λ^∗. The other terms are easy to handle in a standard way.

Denoting𝑍 𝑦, 𝑢

= ^𝑠₂

^{𝒢 𝑦, 𝑢}

2

Λ^∗ we apply Lemma 2.3.4 and Corollary 2.4.6. The latter is applicable since the dual ofΛ^∗, that is,Λ^{∗∗ Λ}, is uniformly convex due to Lemma 2.3.3. At this point we should mention that this step is exactly the reason why we work with a uniformly convex, reflexive test function space. Hence, additionally using the chain rule, we see that𝑍is differentiable with

𝑍⁰ 𝑦, 𝑢

𝛿𝑦,𝛿𝑢

=𝑠𝜆^∗∗𝒢⁰ 𝑦, 𝑢

𝛿𝑦,𝛿𝑢

(2.4.6) where𝜆^∗∗∈Λ^∗∗uniquely satisfies

k𝜆^∗∗k_Λ∗∗ =

^{𝒢 𝑦, 𝑢} _Λ∗, 𝜆^∗∗𝒢 𝑦, 𝑢

=

^{𝒢 𝑦, 𝑢}

2

Λ^∗. (2.4.7) SinceΛis reflexive, we can regard𝜆^∗∗ as a𝜆∈Λvia the canonical isomorphism. We define 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ

by multiplying this𝜆with the positive number𝑠. On the one hand, from (2.4.6) we get the remaining part of (2.4.4), that is,

𝒢⁰ 𝑦, 𝑢

𝛿𝑦,𝛿𝑢 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ ,

which is given by (2.3.4). On the other hand, (2.4.7) instantly yields (2.4.5).

Remark 2.4.8. Such a 𝜓^𝛼

𝛼,𝜗^𝑒,𝜗^ℎ

will later act as a Lagrangian multiplier with respect to the Vlasov–Maxwell system, that is, a solution of the adjoint system, if the point 𝑦, 𝑢

is a minimizer of (Ps) or, later, of (P). In general, when one has a differentiable control-to-state operator 𝑢 ↦→ 𝑦(𝑢)at hand (which we do not have in our case), computing the adjoint state as the solution of the adjoint system, which is a part of the first order optimality conditions, is an efficient way to compute the total derivative 𝑑𝑢^𝑑 𝒥 𝑦(𝑢), 𝑢

when trying to find a minimizer numerically; see [Hin+09, Section 1.6.2], for example.

Next, we derive necessary first order optimality conditions for (Ps). To tackle an optimization problem with certain constraints and to prove existence of Lagrangian

multipliers with respect to them, one has to verify some constraint qualification. To this end, we state a famous result of Zowe and Kurcyusz [ZK79], which is based on a fundamental work of Robinson [Rob76].

Proposition 2.4.9. Let𝑋,𝑌be Banach spaces,𝑆⊂𝑋nonempty, closed, and convex,𝑄 ⊂𝑌 a closed convex cone (𝑄is a “cone” means0∈𝑄,𝑥 ∈𝑄⇒ ∀𝜆 >0 :𝜆𝑥 ∈𝑄),𝜙:𝑋→R differentiable, and 𝑔:𝑋 →𝑌continuously differentiable. Denote for𝐴 ⊂𝑋 (and similarly for𝐴⊂𝑌)

𝐴⁺={𝑥^∗∈𝑋^∗ | ∀𝑎∈𝐴:𝑥^∗𝑎≥0} and denote for𝑥 ∈𝑋and𝑦∈𝑌

𝑆_𝑥 ={𝜆(𝑐−𝑥) |𝑐∈𝑆,𝜆≥0}, 𝑄_𝑦 =_𝑘−

𝜆𝑦 | 𝑘∈𝑄,𝜆≥0 .

Let𝑥_∗ ∈ 𝑋be a local minimizer (i.e., a local minimizer of the objective function restricted to all feasible points) of the problem

min𝑥∈𝑋 𝜙(𝑥)

s.t. 𝑥 ∈𝑆, 𝑔(𝑥) ∈𝑄, and let the constraint qualification

𝑔⁰(𝑥_∗)𝑆_𝑥

∗−𝑄_𝑔(𝑥

∗)=𝑌 (CQ)

hold.

Then there is a Lagrange multiplier𝑦^∗∈𝑌^∗at𝑥_∗for the problem above, i.e., (i) 𝑦^∗∈𝑄⁺,

(ii) 𝑦^∗𝑔(𝑥_∗)=0,

(iii) 𝜙⁰(𝑥_∗) −𝑦^∗◦𝑔⁰(𝑥_∗) ∈𝑆⁺_𝑥

∗.

We apply this result to our problem (Ps). As we have shown in Lemma 2.4.7, the objective function is differentiable. In the following, let

𝑆 B

𝑦, 𝑢 _{∈ 𝒴 × 𝒰 |}

0≤ 𝑓^𝛼 ≤

𝑓˚𝛼

_𝐿∞(Ω×R³⁾

a.e.,𝒩_𝛼 𝑓^𝛼 _{≤ ℒ}

𝛼,𝛼=1, . . . , 𝑁

⊂ 𝒴 × 𝒰 C^{𝑋 ,} 𝑄 BR^≥0 ⊂RC^𝑌.

Clearly, 𝑆is nonempty, closed, and convex, and𝑄is a closed convex cone. Further-more, the constraints (2.1.1), (2.1.2), and (2.4.1) are equivalent to

𝑦, 𝑢_{∈𝑆, 𝑔 𝑦, 𝑢∈ 𝑄,}

2.4 First order optimality conditions 99

It is easy to see that𝑔is continuously differentiable with 𝑔⁰ 𝑦, 𝑢 _{𝛿𝑦,𝛿𝑢} We verify the constraint qualification (CQ).

Lemma 2.4.10. Let 𝑦_𝑠, 𝑢_𝑠

be a (global) minimizer of (Ps). Then,(CQ)is satisfied if𝑠 is sufficiently large.

Proof. First, we exclude the possibility that some𝑓𝑠𝛼is identically zero for𝑠sufficiently large (since then the term ₂^𝑠

^{𝒢 𝑦}𝑠, 𝑢_𝑠

2

Λ^∗is too large for 𝑦_𝑠, 𝑢_𝑠

to be a minimizer of (P_s)): For each 𝛼, let𝜓∗^𝛼:[0, 𝑇_•] ×Ω×R^{3 →} R, 𝜓^𝛼∗(𝑡, 𝑥, 𝑣) = 𝜂(𝑡)𝜑^𝛼(𝑥, 𝑣), where

where 𝑦_∗, 𝑢_∗

is a minimizer of (P) and where the strict inequality holds for𝑠 suffi-ciently large, i.e.,

𝑠> max

𝛼=1,...,𝑁

8 _𝜓^𝛼_∗

2

𝑊1,𝑝,𝑞˜𝒥 𝑦_∗, 𝑢_∗

𝑓˚𝛼

4 𝐿2(Ω×R³⁾

;

note that the right-hand side does not depend on𝑠and𝛼0and that no 𝑓˚𝛼is identically zero. Since 𝑦_∗, 𝑢_∗

is feasible for (Ps), (2.4.8) is a contradiction to 𝑦_𝑠, 𝑢_𝑠

being a minimizer of (Ps).

To prove the lemma, we have to show that for each𝑑 ∈Rthere are𝜆1,𝜆2 ≥0,𝑘≥0, and 𝛿𝑦,𝛿𝑢_∈𝑆

satisfying

𝜆1𝑔⁰ 𝑦_𝑠, 𝑢_{𝑠 𝛿𝑦−𝑦}

𝑠,𝛿𝑢−𝑢_{𝑠−𝑘+𝜆}

2𝑔 𝑦_𝑠, 𝑢_𝑠

=𝑑. (2.4.9)

We choose𝛿𝑓𝛼

+ = 𝑓𝑠,+𝛼 for all𝛼,𝛿𝐸 =𝐸_𝑠,𝛿𝐻 =𝐻_𝑠,𝛿𝑢 =𝑢_𝑠, and consider two cases;

note that in the following it always holds that𝜆1,𝜆2 ≥0,𝑘≥0, and 𝛿𝑦,𝛿𝑢_{∈ 𝑆} : Case 1. 𝑑 ≤0: Choose𝜆1 =𝜆2=0,𝛿𝑓𝛼 = 𝑓𝑠𝛼for all𝛼,𝑘=−𝑑.

Case 2. 𝑑 >0: Choose𝜆2=0,𝛿𝑓¹=0,𝛿𝑓^𝛼 = 𝑓_𝑠^𝛼for𝛼≥2,𝑘=0. Since 𝑔⁰ 𝑦_𝑠, 𝑢_𝑠

𝛿𝑦−𝑦_𝑠,𝛿𝑢−𝑢_𝑠

=

∫ ^𝑇• 0

∫

Ω

∫

R³

𝑣⁰

1𝑓_𝑠¹𝑑𝑣𝑑𝑥𝑑𝑡 >0, we can choose𝜆1 >0 such that (2.4.9) is satisfied.

In all cases (2.4.9) holds; the proof is complete.

Now, Proposition 2.4.9 gives us the following theorem.

Theorem 2.4.11. Let𝑠be sufficiently large and 𝑦_𝑠, 𝑢_𝑠

a minimizer of(Ps). Then there exist 𝜈^𝑠 ≥0and𝜏^𝛼𝑠 ∈

𝒴𝛼 pd

^∗

,𝛼=1, . . . , 𝑁, such that:

(i) 𝜈^𝑠 =0or𝑔 𝑦_𝑠, 𝑢_𝑠

=0.

(ii)

𝑁

Õ

𝛼=1

𝜏𝑠^𝛼𝑓_𝑠^𝛼 ≤

𝑁

Õ

𝛼=1

𝜏^𝛼𝑠𝛿𝑓^𝛼

for all𝛿𝑓^𝛼 ∈ 𝒴^𝛼

pdsatisfying0≤𝛿𝑓^𝛼 ≤

𝑓˚𝛼

_𝐿∞(Ω×R³⁾

a.e. and𝒩_𝛼 𝛿𝑓^𝛼 _{≤ ℒ}

𝛼. (iii) For all 𝛿𝑦,𝛿𝑢_{∈ 𝒴 × 𝒰}

it holds that

0=

𝑁

Õ

𝛼=1

𝑤_𝛼

∫

𝛾𝑇•⁺

sign 𝑓_𝑠,+^{𝛼 𝑓}_𝑠,+^𝛼

𝑞−1

𝛿𝑓₊^𝛼𝑑𝛾𝛼

+

3

Õ

𝑗=1

∫ ^𝑇• 0

∫

Γ

sign 𝑢_𝑠,𝑗 ^𝑢𝑠,𝑗

𝑟−1

𝛿𝑢_𝑗+𝜅1sign 𝜕^𝑡𝑢_{𝑠,𝑗 𝜕}𝑡𝑢_𝑠,𝑗

𝑟−1

𝜕^𝑡𝛿𝑢_𝑗

2.4 First order optimality conditions 101

Λis, in accordance with(2.4.5), given by

being a solution of the adjoint system

and the stationarity condition

0= Õ3

𝑗=1

∫ ^𝑇• 0

∫

Γ

sign 𝑢_𝑠,𝑗 ^𝑢𝑠,𝑗

𝑟−1

𝛿𝑢_𝑗+𝜅1sign 𝜕^𝑡𝑢_{𝑠,𝑗 𝜕}𝑡𝑢_𝑠,𝑗

𝑟−1

𝜕^𝑡𝛿𝑢_𝑗

+𝜅2

Õ3

𝑖=1

sign 𝜕^𝑥𝑖𝑢_{𝑠,𝑗 𝜕}𝑥_𝑖𝑢_𝑠,𝑗

𝑟−1

𝜕^𝑥𝑖𝛿𝑢_𝑗

! 𝑑𝑥𝑑𝑡

−

∫ ^𝑇• 0

∫

Γ

4𝜋𝜗^𝑒𝑠−𝜈^𝑠𝛽𝑢_𝑠·

𝛿𝑢 𝑑𝑥𝑑𝑡 for all𝛿𝑢∈ 𝒰 (SCs) being satisfied.

Proof. Since (CQ) holds due to Lemma 2.4.10 and𝒴 × 𝒰 is a Banach space due to Lemma 2.4.4, by Proposition 2.4.9 there is𝜈^𝑠 ∈ Racting as a Lagrangian multiplier with respect to (2.1.2). Proposition 2.4.9.(i) implies 𝜈^𝑠 ≥ 0, and Proposition 2.4.9.(ii) yields part 2.4.11.(i).

With Proposition 2.4.9.(iii) and the notation used there we see that 𝜏^𝑠 B^{𝒥𝑠0 𝑦𝑠, 𝑢𝑠}₋

𝜈^𝑠·𝑔⁰ 𝑦_𝑠, 𝑢_{𝑠 ∈𝑆}+

(^𝑦𝑠,𝑢_𝑠) ⊂ (𝒴 × 𝒰 )^∗. (2.4.12) Consequently,𝜏^𝑠can be decomposed into

𝜏^𝑠≡ (𝜏^𝛼𝑠)

𝛼, 𝜏^𝛼𝑠,+

𝛼,𝜏𝑠^𝑒,𝜏𝑠^ℎ,𝜏^𝑢𝑠

∈

?𝑁 𝛼=1

𝒴^𝛼

pd

^∗

×𝐿^𝑞 𝛾𝑇⁺_•, 𝑑𝛾𝛼

^∗

!

×

𝐿² [0, 𝑇_•] ×R³;R^3∗2

× 𝒰^∗.

Since the set𝑆(^𝑦𝑠,𝑢_𝑠)only limits the directions𝛿𝑓^𝛼and not the directions𝛿𝑓₊^𝛼,𝛿𝐸,𝛿𝐻, and𝛿𝑢, the property𝜏^𝑠 ∈𝑆⁺

(𝑦_𝑠,𝑢_𝑠)yields that all𝜏^𝑠,+𝛼 and moreover𝜏^𝑠𝑒,𝜏^𝑠ℎ, and𝜏^𝑢𝑠 have to vanish. Thus,𝜏^𝑠 ≡ (𝜏^𝑠𝛼)

𝛼via

𝜏^𝑠 𝛿𝑦,𝛿𝑢

=

𝑁

Õ

𝛼=1

𝜏^𝑠𝛼𝛿𝑓^𝛼. (2.4.13)

On the one hand, by𝜏^𝑠 ∈𝑆⁺

(^𝑦𝑠,𝑢_𝑠)and the identification (2.4.13) we have for all𝛿𝑓𝛼 ∈ 𝒴𝛼

pdsatisfying 0≤𝛿𝑓𝛼 ≤

𝑓˚𝛼

_𝐿∞(Ω×R³⁾

a.e. and𝒩_𝛼 𝛿𝑓𝛼 _{≤ ℒ}

𝛼,

𝑁

Õ

𝛼=1

𝜏^𝛼𝑠 𝛿𝑓^𝛼−𝑓_𝑠^𝛼 _≥ 0,

which is part 2.4.11.(ii). On the other hand, (2.4.12) and (2.4.13) instantly yields (2.4.10) recalling the formula for𝒥_𝑠⁰from Lemma 2.4.7.

2.4 First order optimality conditions 103 Setting𝛿𝑢and all but one of the directions𝛿𝑓^𝛼,𝛿𝑓₊^𝛼,𝛿𝐸, and𝛿𝐻to zero and the one remaining arbitrary, we conclude that the adjoint system (Ads) holds. Note that a priori the𝜓^𝛼𝑠,𝜗^𝑒𝑠, and𝜗^ℎ𝑠 vanish for𝑡=𝑇_•by definition of the test function spaceΛ. Finally, setting all directions but𝛿𝑢to zero yields (SCs). Thus, also the proof of part 2.4.11.(iii) is complete.

Remark 2.4.12. If, for example,𝑟=2 and the boundary ofΓis smooth, (SCs) can easily be interpreted as the weak form of the second order PDE

𝜅1𝜕²𝑡𝑢_𝑠+𝜅2Δ𝑥𝑢_𝑠 =−4𝜋𝜗^𝑒𝑠 + 𝜈^𝑠𝛽+1_𝑢

𝑠 on[0, 𝑇_•] ×Γ,

𝜕^𝑡𝑢_𝑠(0)=𝜕^𝑡𝑢_𝑠(𝑇_•)=0 onΓ,

𝜕^𝑛Γ𝑢_𝑠 =0 on[0, 𝑇_•] ×𝜕Γ.

Here,𝜕^𝑛Γdenotes the directional derivative in the direction of the outer unit normal 𝑛_Γof𝜕Γ.

Im Dokument The Relativistic Vlasov-Maxwell System with External Electromagnetic Fields (Seite 102-113)