Final remarks

One can consider other optimal control problems than (P), with a different objective function, for example, a problem of tracking type:

𝒥˜ 𝑦, 𝑢

=𝒥 𝑦, 𝑢₊

𝑁

Õ

𝛼=1

𝑏_𝛼 2

^{𝑓𝛼−𝑓}_𝑑^𝛼

2

𝐿2([0,𝑇]×Ω×R³⁾

+𝑏_𝐸

2 k𝐸−𝐸_𝑑k²_𝐿2

([0,𝑇]×R³;R³⁾

+𝑏_𝐻

2 k𝐻−𝐻_𝑑k²_𝐿2

([0,𝑇]×R³;R³⁾

, where 𝑏_𝛼, 𝑏_𝐸, 𝑏_𝐻 > 0 are parameters and 𝑓^𝛼

𝑑 , 𝐸_𝑑, 𝐻_𝑑 are desired states. Since this new objective function already grants coercivity in 𝑓^𝛼,𝐸, and𝐻with respect to the 𝐿²-norm, at first sight it seems that the artificial constraint (2.1.2) can be abolished.

However, without this constraint, we cannot pass to the limit in the term of (1.1.3a) with𝑗^int during an analog proof of Theorem 2.2.1 since for this an𝐿¹

𝛼kin-estimate on 𝑓^𝛼 is necessary; cf. Lemma 2.1.3. Thus, imposing (2.1.1) and (2.1.2) is still necessary.

Analogues of Theorems 2.2.1, 2.4.3, 2.4.11, and 2.4.13 can be proved, and in Theo-rem 2.4.13 the convergences of 𝑓_𝑠^𝛼

𝑘,𝐸_𝑠𝑘, and𝐻_𝑠𝑘 are also strong in𝐿² because of the tracking terms in the new objective function.

We could also consider the case that we additionally try to control the system by inserting particles from outside, that is, considering nonvanishing 𝑔^𝛼 in the right-hand side of (VM.2) and treating them as controls as well. Then we add some norm of the𝑔^𝛼to the objective function as a penalization term. There occur two problems:

Firstly, since (2.1.1) is still necessary and since we have to include 𝐿^∞-norms of the 𝑔^𝛼 there on the right-hand side, the set of functions satisfying this new constraint is no longer convex. We can bypass this problem by imposing 𝐿^∞-bounds on the 𝑔^𝛼 a priori, for example, by imposing box constraints. Secondly, we have to add the 𝐿¹

𝛼kin-norms of the 𝑔^𝛼 to the right-hand side of (2.1.2). To be then able to pass to the limit in (2.1.2), we need that the space the 𝑔^𝛼 lie in is compactly embedded in 𝐿¹

𝛼kin

𝛾𝑇⁻_•, 𝑑𝛾_𝛼

—this is analogous to the consideration of𝒰 as the control space instead of simply 𝐿². That compact embedding is, for example, guaranteed by the restriction𝑔^𝛼 ∈ 𝐻¹

𝛾𝑇⁻•

∩ {|𝑣|<𝑅}

and𝑔^𝛼 =0 for|𝑣| >𝑅with𝑅>0 fixed. Another possibility is to impose an a priori bound on the𝐿¹

𝛼kin-norms of the𝑔^𝛼, for example, by imposing box constraints as above and a bound on the support of the𝑔^𝛼with respect to𝑣, and then adding this a priori bound to the right-hand side of (2.1.2) instead of the𝐿¹

𝛼kin-norms of the𝑔^𝛼.

2.5 Final remarks 107 In Theorem 2.4.13, a suitable sequence of optimal points of (Ps) converges to an optimal point of (P), at least weakly, some components even strongly. However, we do not know ifallminimizers of (P) can be “obtained” in this way. In [Lio85], usually an approximate problem with an adaptive objective function is considered, in order to derive first order optimality conditions foranygiven, fixed minimizer of (P). Here, this means adding norms of 𝑓𝛼−𝑓𝛼

∗ , 𝑓₊^𝛼− 𝑓_∗,+^𝛼 ,𝐸−𝐸_∗,𝐻−𝐻_∗, and𝑢−𝑢_∗to𝒥. With an analogue of Theorem 2.4.13, one can then show that 𝑦_𝑠, 𝑢_𝑠

converges strongly to 𝑦_∗, 𝑢_∗

in a suitable norm, and this holds for thefulllimit𝑠 → ∞. However, this method is not constructive since one has to know 𝑦_∗, 𝑢_∗

a priori to consider the approximate problem, and thus in our case not reasonable; in general it is reasonable if one can pass to the limit in the first order optimality conditions.

CHAPTER 3

Confined steady states in an infinitely long cylinder

3.1 The set-up

The previous chapter was devoted to the question how to adjust the currents (and thus the external electromagnetic fields) in some external electric coils to confine the plasma as best as possible. With “good confinement” we meant that the amount of the plasma particles hitting the boundary ofΩ are to be kept as small as possible, while the control costs should be not too exhaustive. However, one might ask two questions: Firstly, as they were given and fixed through these considerations, what is a reasonable choice of the initial data for the particle densities and the electromagnetic fields? Secondly, is there really a choice of initial data and external currents such that the plasma is really confined during the whole time interval[0, 𝑇_•], i.e., such that there are no hits on the boundary? This leads to another question, which we will deal with in this chapter: Is there a configuration, that is independent of time and where the plasma particles are away from the boundary of the fusion reactor?

Before we analyze this problem about the existence of such a configuration, which we henceforth call a “confined steady state”, we first discuss the basic ideas for plasma confinement—more information on fusion plasma physics can be found in the classical book of Stacey [Sta12]. The physical basis for confinement is the fact that charged particles spiral about magnetic field lines. The so-called gyroradius, that is, the radius of such a spiral, is inversely proportional to the strength of the magnetic field. This gives rise to the idea of linear confinement devices: The fusion reactor is a long cylinder and the external magnetic field points in the direction of the symmetry axis of this cylinder. If this external magnetic field is sufficiently strong, the gyroradii of the plasma particles will be smaller than the radius of the cylinder, whence the plasma is confined in the fusion device. However, this setting cannot prevent the plasma current from having a nonvanishing component in the direction of the symmetry axis.

Thus, there will be losses at the ends of the long cylinder. In practice, one can try to overcome this problem by one of the two following modifications: Firstly, so-called

109

magnetic mirrors are added at these ends. Secondly, the long cylinder is bent into a torus. This second idea is pursued typically in modern research. Toroidal geometry has the advantage of avoiding such losses but has the disadvantage that it gives rise to drifts of the plasma particles, which finally cause the particles to move radially outwards and thus make confinement impossible. Therefore, the external magnetic field needs to have a poloidal component additional to its toroidal one. This approach then leads to Tokamak devices.

However, analyzing the problem of existence of confined steady states from a mathe-matics point of view in toroidal geometry seems quite hard. We discuss the difficulties in Section 3.6. As a first step towards this, we consider the set-up of a linear confine-ment device instead. For mathematical reasons, it will be convenient to assume that the cylinder is infinitely long (which is of course not conceivable from a practical point of view). Thus, we fix𝑅₀ >0 and let

ΩB _{𝑥 ∈}

R^{3 | 𝑥2}₁^+𝑥₂²<𝑅²

0 .

In contrast to the previous chapters,Ωis no longer bounded since it extends infinitely in the 𝑥₃-direction. Because of the axial symmetry of the set-up, it is natural to work with cylindrical coordinates 𝑟,𝜑, 𝑥₃

. In these coordinates, we simply have Ω = _𝑥∈

R^{3 |𝑟}<𝑅₀ . Furthermore, we now consider purely reflecting boundary conditions for the particles and perfect conductor boundary conditions for the fields on𝜕Ω. Due to perfect conductor boundary conditions, Maxwell’s equations are only imposed onΩ, where𝜀=𝜇=Id by assumption. Hence, we no longer distinguish the 𝐸- and𝐷-, and the𝐻- and𝐵-field, respectively, and use𝐸and𝐵for denotation of the electromagnetic fields. Moreover, we consider an external magnetic field𝐵^ext, which is supposed to be divergence free, as given and thus no longer consider an external current density 𝑢(whence we neglect an external electric field). Therefore, the only charge and current densities are the internal ones, i.e.,

𝜌=𝜌^int=

𝑁

Õ

𝛼=1

𝑞_𝛼

∫

R³

𝑓^𝛼𝑑𝑣, 𝑗=𝑗^int=

𝑁

Õ

𝛼=1

𝑞_𝛼

∫

R³

b^𝑣𝛼𝑓^𝛼𝑑𝑣.

In the following, there often occur cylindrical coordinates and the corresponding local, orthonormal coordinate basis 𝑒_𝑟, 𝑒_𝜑, 𝑒₃

, where 𝑒_𝑟 = cos𝜑,sin𝜑,0_{, 𝑒}

𝜑= −sin𝜑,cos𝜑,0_{, 𝑒}

3=(0,0,1).

For a vector𝑤∈R³, we denote with𝑤_𝑟,𝑤_𝜑, and𝑤₃the coordinates of𝑤in this local coordinate system, i.e.,

𝑤_𝑟 =𝑤·𝑒_𝑟, 𝑤_𝜑=𝑤·𝑒_𝜑, 𝑤₃ =𝑤·𝑒₃. Altogether, the whole Vlasov–Maxwell system in this set-up reads

𝜕^𝑡𝑓^𝛼+

b^𝑣𝛼·𝜕^𝑥𝑓^𝛼+𝑞_𝛼 𝐸+

b^𝑣𝛼× 𝐵+𝐵^ext ·𝜕^𝑣𝑓^𝛼 =0 on𝐼_𝑇

•×Ω×R^3, 𝑓₋^𝛼 =𝐾 𝑓₊^𝛼 on𝛾𝑇⁻_•,

3.1 The set-up 111 𝑓^𝛼(0)= 𝑓˚𝛼 onΩ×R^3,

𝜕^𝑡𝐸−curl^𝑥𝐵=−4𝜋𝑗 on𝐼_𝑇

•×Ω,

𝜕^𝑡𝐵+curl^𝑥𝐸=0 on𝐼_𝑇

•×Ω, div^𝑥𝐸=4𝜋𝜌 on𝐼_𝑇

•×Ω,

div^𝑥𝐵=0 on𝐼_𝑇

•×Ω, 𝐸_𝜑 =𝐸₃=𝐵_𝑟+𝐵^ext_𝑟 =0 on𝐼_𝑇•×𝜕Ω,

(𝐸, 𝐵)(0)= 𝐸,˚ 𝐵˚

onΩ. Note that in the following the divergence part of Maxwell’s equations will be important and not neglected anymore. Furthermore, the perfect conductor boundary condition reads𝐸×𝑛 = 0 = 𝐵^tot·𝑛in the general case (where 𝐵^tot is the total magnetic field;

here, 𝐵^tot = 𝐵+𝐵^ext) and reduces to𝐸_𝜑 = 𝐸₃ = 𝐵^tot_𝑟 = 0 in the case ofΩbeing an infinitely long cylinder since here𝑛=𝑒_𝑟.

It is convenient to introduce electromagnetic potentials, which will be the functions we work with mostly, namely, the electric scalar potential𝜙and the magnetic vector potential𝐴^tot=𝐴+𝐴^ext, which splits into the internal and external potentials𝐴and 𝐴^ext. The electromagnetic fields and potentials are related via

𝐸=−𝜕^𝑥𝜙−𝜕^𝑡𝐴, 𝐵=curl^𝑥𝐴, 𝐵^ext =curl^𝑥𝐴^ext. (3.1.1) Then, Gauss’s law for magnetism (div^𝑥𝐵=0) and Faraday’s law (𝜕^𝑡𝐵+curl^𝑥𝐸=0) are automatically satisfied. There is some freedom to demand a certain gauge condition on the potentials. We will consider Lorenz gauge for the internal potentials

𝜕^𝑡𝜙+div^𝑥𝐴=0, (3.1.2)

which of course is the same as Coulomb gauge div^𝑥𝐴=0

if the potentials are independent of time, and similarly div^𝑥𝐴^ext =0 for the external potential. Using the gauge (3.1.2), the remaining Maxwell’s equations, i.e., 𝜕^𝑡𝐸− curl^𝑥𝐵=−4𝜋𝑗and div^𝑥𝐸=4𝜋𝜌, become

𝜕²𝑡𝜙−Δ𝑥𝜙=4𝜋𝜌, 𝜕𝑡²𝐴−Δ𝑥𝐴=4𝜋𝑗, (3.1.3) where the latter equation is to be understood componentwise (in Cartesian coordi-nates).

Similar set-ups have already been studied earlier, for example, in [Pou92; Rei92]. The basic strategy to obtain steady states was first mentioned in [Deg90]. Closely related to our considerations is [BF93], where (among other set-ups) existence of steady states in an infinitely long cylinder without external magnetic field was proved. However, an important condition there is that there is only one particle species and thus only a fixed sign of particle charges appears. Therefore, 𝜌 has a fixed sign and 𝜙 is

monotone, which is crucial for the considerations in [BF93]. As opposed to this, we allow positively and negatively charged particles.

The question about existence of confined steady states for a Vlasov–Poisson plasma (that is, 𝐵 = 0) was considered in [Sku14] and [Kno19]. The approach of the latter work is similar to ours but needs some smallness assumption on the ansatz functions, which we can avoid, and is restricted to homogeneous external magnetic fields parallel to the symmetry axis.

Im Dokument The Relativistic Vlasov-Maxwell System with External Electromagnetic Fields (Seite 116-122)