Optimal control of a Vlasov-Poisson plasma by an external magnetic field

The aim of various technical applications (for example fusion research) is to control a plasma by magnetic fields in a desired fashion. In our model the plasma is described by the Vlasov-Poisson system that is equipped with an external magnetic field. We will prove that this model satisfies some basic properties that are necessary for calculus of variations. After that, we will analyze an optimal control problem with a tracking type cost functional with respect to the following topics: Necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality, uniqueness of the optimal control under certain conditions.


Introduction
The three dimensional Vlasov-Poisson system in the plasma physical case is given by the following system of partial differential equations: (1) Here f = f (t, x, v) ≥ 0 denotes the distribution function of the particle ensemble that is a scalar function representing the density in phase space. Its time evolution is described by the first line of (1) which is a first order partial differential equation that is referred to as the Vlasov equation. For any measurable set M ⊂ R 6 , M f (t, x, v) d(x, v) yields the charge of the particles that have space coordinates x ∈ R 3 and velocity coordinates v ∈ R 3 with (x, v) ∈ M at time t ≥ 0. The function ψ is the electrostatic potential that is induced by the 1 arXiv:1808.00547v2 [math.AP] 28 Sep 2018 charge of the particles. It is given by Poisson's equation −∆ψ = 4πρ with an homogeneous boundary condition where ρ denotes the volume charge density. The self-consistent electric field is then given by −∂ x ψ. Note that both ψ and −∂ x ψ depend linearly on f . Hence the Vlasov-Poisson system is nonlinear due to the bilinear term −∂ x ψ · ∂ v f in the Vlasov equation. Assuming f to be sufficiently regular (e.g., f (t) := f (t, ·, ·) ∈ C 1 c (R 6 ) for all t ≥ 0), we can solve Poisson's equation explicitly and obtain Considering f → ψ f as a linear operator we can formally rewrite the Vlasov-Poisson system as Combined with the condition f | t=0 =f (4) for some functionf ∈ C 1 c (R 6 ) we obtain an initial value problem. A first local existence and uniqueness result to this initial value problem was proved by R. Kurth [5]. Later J. Batt [1] established a continuation criterion which claims that a local solution can be extended as long as its velocity support is under control. Finally, two different proofs for global existence of classical solutions were established independently and almost simultaneously, one by K. Pfaffelmoser [10] and one by P.-L. Lions and B. Perthame [8]. Later, a greatly simplified version of Pfaffelmoser's proof was published by J. Schaeffer [12]. This means that the follwing result is established: Any nonnegative initial datum f ∈ C 1 c (R 6 ) launches a global classical solution f ∈ C 1 ([0, ∞[×R 6 ) of the Vlasov-Poisson system (1) satisfying the initial condition (4). Moreover, for every time t ∈ [0, ∞[, f (t) = f (t, ·, ·) is compactly supported in R 6 . Hence equation (2) and the reformulation of the Vlasov-Poisson system (3) are well-defined iff ∈ C 1 c (R 6 ). For more information we recommend to consider the article [11] by G. Rein that gives an overview on the most important results.
To control the distribution function f we will add an external magnetic field B to the Vlasov equation: The cross product v × B occurs since, unlike the electric field, the magnetic field interacts with the particles via Lorentz force. If we want to discuss an optimal control problem where the PDE-constraint is given by (5) we must firstly establish the basics for variational calculus: • We will introduce a set B K such that any field B ∈ B K induces a unique and sufficiently regular strong solution f = f B of the initial value problem (5) that exists on any given time interval [0, T ]. The set B K will be referred to as the set of admissible fields.
• We will show that the solution f B depends Lipschitz-continuously on the field B while the partial derivatives ∂ t f B , ∂ x f B and ∂ v f B depend Hölder-continuously on B.
• We will prove that the operator B → f B is compact/weakly compact and Fréchet differentiable in some suitable sense.
With this foundations we can discuss a model problem of optimal control: Letf ∈ C 2 c (R 6 ) be any given initial datum and let T > 0 be any given final time. The aim is to control the distribution function f B in such a way that its value at time T matches a desired distribution f d as closely as possible. We will consider the following optimal control problem Minimize J(B) : It will be analyzed with respect to the following topics: • Existence of a globally optimal solution, • necessary conditions of first order for locally optimal solutions, • derivation of an optimality system, • sufficient conditions of second order for locally optimal solutions, • uniqueness of the optimal control for small values of T λ .

Notation and preliminaries
Our notation is mostly standard or self-explaining. However, to avoid misunderstandings, we fix some of it here. We will also present some basic results that are necessary for the later approach.
Let d ∈ N, U ⊂ R d be any open subset, k ∈ N and 1 ≤ p ≤ ∞ be arbitrary. C k (U ) denotes the space of k times continuously differentiable functions on U , C c (U ) denotes the space of C k (U )-functions having compact support in U and C k b (U ) denotes the space of C k (U )-functions that are bounded with respect to the norm For any γ ∈]0, 1], C k,γ (U ) denotes the space of Hölder-continuous C k (U )-functions, i.e., where for any u ∈ C k (U ), Note that C k b (U ), · C k b (U ) and C k,γ (U ), · C k,γ (U ) are Banach spaces. L p (U ) denotes the standard L p -space on U and W k,p (U ) denotes the standard Sobolov space on U as, for instance, defined by E. Lieb and M. Loss in [7, s. 2.1,6.7]. If U = R d we will sometimes omit the argument "(R d )". For example, we will just write L p , W k,p or C k b instead of L p (R d ), W k,p (R d ) or C k b (R d ). If U = R d we will not use this abbreviation. We will also use Banach space-valued Sobolev spaces as defined by L. C. Evans [3, p. 301-305]. For any Banach space X, L p (0, T ; X) denotes the space of Banach space-valued L p -functions [0, T ] t → u(t) ∈ X. Analogously, W k,p (0, T ; X) denotes the Banach space-valued Sobolev space. The following properties are essential: Lemma 1. Let T > 0, d, m ∈ N, U ⊂ R d be any open subset and 1 ≤ p, q < ∞ be arbitrary. Then the following holds: (a) For any function u ∈ L p (0, T ; W k,q (R d )) there exists some sequence (u j ) ⊂ C ∞ (]0, T [×R d ) such that ∀j ∈ N ∃r j > 0 ∀t ∈ [0, T ] : supp u j (t) ⊂ B r j (0) and u j → u in L p (0, T ; W k,q (R d )).
(b) For any function u ∈ W k,p (0, T ; L q (R d )) there exists some sequence and u j → u in W k,p (0, T ; L q (R d )).  As these results are very common, we will not give a proof of this lemma in this paper. For more information on this topic we recommend to consider [14], [9] and [6].
In order to write down the three dimensional Vlasov-Poisson system concisely we will first define some operators and notations: For d ∈ N, 1 ≤ p ≤ ∞ and r > 0 let L p r (R d ) denote the set of functions ϕ ∈ L p (R d ) having compact support supp ϕ ⊂ B r (0) ⊂ R d . Then the operator is linear and bounded. It also holds that ρ ϕ ∈ L 2 r (R 3 ) for any ϕ ∈ L 2 r (R 6 ). Let now R > 0 be any arbitrary radius. From the Calderon-Zygmund inequality [4, p. 230] we can conclude that is a linear and bounded operator. According to E. Lieb and M. Loss [7, s. 6.21], the gradient of ψ ϕ is given by and then, because of (7), the operator is also linear and bounded. Some more properties of the potential ψ ϕ and its field ∂ x ψ ϕ are given by the following lemma.
(b) Let 1 < p, q < ∞ be any real numbers and suppose that additionally ϕ ∈ L p (R 6 ). Then ρ ϕ ∈ L p (R 3 ) and there exists some constant C > 0 depending only on p and r such that (c) Suppose that additionally ϕ ∈ L ∞ (R 6 ). Then ρ ϕ ∈ L ∞ (R 3 ) and there exists some constant C > 0 depending only on r such that Moreover ∂ x ψ ϕ ∈ C 0,γ (R 3 ; R 3 ) for any γ ∈]0, 1[ and there exists some constant C > 0 depending only on r such that Proof Item (a): The Calderon-Zygmund lemma ([4, p. 230]) states that ψ ϕ is in H 2 loc (R 3 ) and satisfies −∆ψ ϕ = 4πρ ϕ almost everywhere on R 3 . By Sobolev's inequality, ψ ϕ has a continuous representative and one can easily show that ψ ϕ (x) → 0 if |x| → ∞, i.e., ψ ϕ satisfies the boundary value problem. Any other solution is then given by ψ ϕ + h where ∆h = 0 almost everywhere and h satisfies the boundary condition. Then, by Weyl's lemma, h is a harmonic function and thus h = 0 which means uniqueness. This proves (a).
Item (b): ρ ϕ ∈ L p (R 6 ) with ρ ϕ L p ≤ C ϕ L p is a direct consequence of Jensen's inequality. The first two inequalities are already established by E. Stein [13, p. 119]. Note that the Calderon-Zygmund inequality also yields D 2 ψ ϕ L p (B 2r (0)) ≤ C ϕ L p . Moreover, if |x| ≥ 2r, Thus D 2 ψ ϕ L p (R 3 \B 2r (0)) ≤ C ϕ L p which completes the proof of (b). We will also use the notation In this case we will write for any t and x. As already mentioned in the introduction we consider the following initial value problem: In the following let T > 0 andf ∈ C 2 c (R 6 ; R + 0 ) be arbitrary but fixed. Let B = B(t, x) be a given external magnetic field and let f = f (t, x, v) denote the distribution function that is supposed to be controlled. Its electric field ∂ x ψ f = ∂ x ψ f (t, x) is formally defined as described above. In the following we will show that the solution f satisfies the required condition "f (t) ∈ L 2 r (R 6 )" that ensures that ρ f , ψ f and ∂ x ψ f are well-defined. Of course this is possible only if the magnetic field B is regular enough. The regularity of those fields will be specified in the following section.
3 Admissible fields and the field-state operator

The set of admissible fields
We will now introduce the set our magnetic fields will belong to: The set of admissible fields. For T > 0 and β > 3 let W = W(β) denote the reflexive Banach space L 2 0, T ; W 2,β (R 3 ; R 3 ) , let H denote the Hilbert space L 2 0, T ; H 1 (R 3 ; R 3 ) and let · W and · H denote their standard norms. Then V := W ∩ H with · V := · W + · H is also a Banach space. Definition 3. Let K > 0 and 3 < β < ∞ be arbitrary fixed constants. Then is called the set of admissible fields.

Remark 4.
In the approach of Section 3 and 4 it would be sufficient to consider fields B ∈ W with B W ≤ K. However, in the model that is discussed in Sections 5-7 the regularity B ∈ V will be necessary, especially for Fréchet differentiability of the cost functional. Therefore we will use this condition right from the beginning.
The most important properties of the set of admissible fields are listed in the following lemma.
Lemma 5. The set of admissible fields B K has the following properties: (a) B K is a bounded, convex and closed subset of V.
Thus there exist constants k 0 , k 1 > 0 depending only on β such that for all B ∈ B K , for almost all t ∈ [0, T ]. Moreover for any r > 0 there exist constants k 2 , k 3 > 0 depending only on β and r such that for all B ∈ B K , (c) The space W is continuously embedded in L 2 (0, T ; C 1,γ ). Thus for all B ∈ B K it holds that B ∈ L 2 (0, T ; C 1,γ ) with B L 2 (0,T ;C 1,γ ) ≤ k 1 K .
Then for any B ∈ B K , there exists a sequence (B k ) k∈N ⊂ M such that (e) B K ⊂ V is weakly compact, i.e., any sequence in B K contains a subsequence converging weakly in V to some limit in B K .
Proof (a) is obvious and (b) is a direct consequence of Sobolev's embedding theorem and the fact that for all B ∈ B K , B(t) ∈ W 2,β (R 3 ; R 3 ) for almost all t ∈ [0, T ]. Then the k 1inequality of (b) immediately implies (c). (d) follows instantly from Lemma 1(a). Without loss of generality, we can assume that B W ≤ 2K. As B K is a bounded subset of W the Banach-Alaoglu theorem implies that any sequence (B k ) ⊂ B K contains a subsequence (B * k ) converging weakly to some limit B ∈ W. Now, (B * k ) is a bounded sequence in H and thus it has a subsequence (B * * k ) that converges weakly to some limit in H. Because of uniqueness, this limit must be B. Hence B * * k B in V and since the norm · V is weakly lower semicountinuous it follows that B V ≤ K. That is (e). 7

The characteristic flow of the Vlasov equation
Since the Vlasov equation is a first-order partial differential equation, it suggests itself to consider the characteristic system. On that point, we will consider a general version of the Vlasov equation, with given fields F = F (t, x) and G = G(t, x). Then the following holds: Lemma 6. Let I ⊂ R be an interval and let F, G ∈ C(I × R 3 ; R 3 ) be continuously differentiable with respect to x and bounded on J × R 3 for every compact subinterval J ⊂ I. Then for every t ∈ I and z = (x, v) ∈ R 6 there exists a unique solution I s → (X, V )(s, t, x, v) of the characteristic systemẋ The characteristic flow Z := (X, V ) has the following properties: (a) Z : I × I × R 6 → R 6 is continuously differentiable.
The relation between the characteristic flow and the solution f of the Vlasov equation (10) is described by the following lemma. (a) A function f ∈ C 1 (I × R 6 ) satifies the Vlasov equation (10) iff it is constant along every solution of the characteristic system (6).
is the unique solution of (10) in the space C 1 (I ×R 6 ) with f (0) =f . Iff is nonnegative then so is f. For all t ∈ I and 1 ≤ p ≤ ∞, For G = 0 the proofs of both lemmata are presented by G. Rein [11, p. 394]. The proofs for a general field G ∈ C(I × R 3 ; R 3 ) proceed analogously. 8

Classical solutions for smooth external fields
As already mentioned in the introduction, the standard initial value problem ((3), (4)) posesses a unique global classical solution. This result holds true if the Vlasov equation is equipped with an external magnetic field B in C [0, T ]; C 1 b (R 3 ; R 3 ) which will be established in the next theorem. Unfortunately the proof does not work if the field is merely an element of B K . Since such fields are only L 2 in time, the same holds for the right-hand side of the characteristic system. This makes it impossible to determine a solution in the classical sense of ordinary differential equations. However, we can approximate any field B ∈ B K by a sequence Lemma 5. This allows us to construct a certain kind of strong solution to the field B as a limit of the classical solutions induced by the fields B k .
is compactly supported in R 6 in such a way that there exists some constant R > 0 depending only on T ,f , K and β such that for all t ∈ [0, T ], Proof Step 1 -Local existence and uniqueness: For the standard Vlasov-Poisson system ((3), (4)) the existence and uniqueness of a local classical solution was firstly established by R. Kurth [5]. As the field B ∈ C [0, T ]; C 1 b (R 3 ; R 3 ) is regular enough the existence and uniqueness of a local classical solution to our problem can be proved analogously. In this paper we will only sketch the most important steps of that proof. The idea is to define a recursive sequence by f 0 (t, z) :=f (z) and f k+1 (t, z) :=f Z k (0, t, z) , k ∈ N 0 for any t ≥ 0 and z = (x, v) ∈ R 6 where Z k denotes the solution oḟ By induction we obtain that for any k ∈ N 0 , Z k is continuously differentiable with respect to all its variables and f k ∈ C 1 ([0, T ] × R 6 ). Moreover, according to Lemma 7, it holds that is the unique solution of the initial value problem We intend to prove that the sequence (f k ) converges to the solution of the initial value problem (9) if k tends to infinity. Analogously to Kurth's proof we can show that there exists δ > 0 and functions Z, f with Z ∈ C([0, δ 0 ] 2 × R 6 ), f ∈ C([0, δ 0 ] × R 6 ) for any δ 0 < δ such that Z(s, t, z) = lim k→∞ Z k (s, t, z) and f (t, z) =f Z(0, t, z) = lim k→∞ f k (t, z) uniformely in s, t and z. For any arbitrary δ 0 < δ it turns out that (∂ x ψ f k ) and (D 2 . As δ 0 was arbitrary this yields f ∈ C 1 ([0, δ[×R 6 ). Thus f is a local solution of the initial value problem (9) on the time interval [0, δ[ according to Lemma 7 as it is constant along any characteristic curve.
Step 2 -Higher regularity: If B ∈ C([0, T ]; C 2 b ) it additionally follows by induction that for all k ∈ N 0 and s, t ∈ [0, T ], for any δ 0 < δ. Thus we can conclude that Step 3 -Continuation onto the interval [0, T ]: Obviously Batt's continuation criterion (cf. J. Batt [1]) also holds true in our case. This means that we can show that the solution exists on [0, T ] by the following argumentation: We assume that [0, T * [ with T * ≤ T is the right maximal time interval of the local solution and we show that is bounded on [0, T * [. But then, according to Batt, the solution f can be extended beyond T * which is a contradiction as T * was chosen to be maximal. Hence we can conclude that the solution exists on the whole time interval [0, T ].
For the standard Vlasov-Poisson system (without an external field) such a bound on P (t) is established in the Pfaffelmoser-Schaeffer proof [10,12]. We will proceed analogously and single out one particle in our distribution. Mathematically, this means to fix a characteristic (X, V )(s) = (X, V )(s, 0, x, v) with (X, V )(0) = (x, v) ∈ suppf . Now suppose that 0 ≤ δ ≤ t < T * . In the following, constants denoted by C may depend only onf , T , K and β. The aim in the Pfaffelmoser-Schaeffer proof is to bound the difference |V (t) − V (t − δ)| from above by an expression in the shape of CδP (t) α where α < 1 is essential. In our case an analogous approach would merely yield some bound that is ideally in the fashion of CδP (t) α +C √ δP (t) because of the additional field term in thev equation of the characteristic system. However, we can use the fact that an external magnetic field changes only the direction of a particle's velocity vector but not its modulus. This is reflected in the following computation: For The quadratic version of Gronwall's lemma (cf. Dragomir [2, p. 4]) and the definition of ∂ x ψ f then impliy that Using this inequality, the rest of the proof proceeds very similarly to the Pfaffelmoser-Schaeffer proof.
Temporarily we will write f B to denote the classical solution that is induced by the field B. In order to prove that any field B ∈ B K still induces a strong solution of the initial value problem the following two lemmata are essential. For fields B ∈ M Lemma 10 asserts that f B depends Lipschitz continuously on B while its derivatives ∂ z f B and ∂ t f B are Hölder continuous with respect to B. In the course of the construction of a strong solution to some field B ∈ B K we will approximate B by a sequence (B k ) ⊂ M and then Lemma 10 will ensure are Cauchy sequences in some sense. To prove Lemma 10 we will need some uniform bounds that are established in Lemma 9.
Z B and f B are twice continuously differentiable with respect to z and there exist constants c 5 , c 6 > 0 depending only onf , T , K, and β such that for all s ∈ [0, T ], and z ∈ B R (0) be arbitrary (without loss of generality s ≤ t) and let i, j ∈ 1, ..., 6 be arbitrary indices. Let B ∈ M be an arbitrary field and let Z B : [0, T ] × [0, T ] × R 6 → R 6 denote the induced solution of the characteristic system satisfying the initial condition Z B (t, t, z) = z. For brevity, we will use the notation Z B (s) = Z B (s, t, z). The letter C will denote a positive generic constant depending only onf , K, T and β. Using the fundamental theorem of calculus we obtain that Hence applying first the standard version and then the quadratic version of Gronwall's lemma provides that The partial derivatives ∂ z i Z B , i = 1, ..., 6 can be bounded by and consequently, by Gronwall's lemma, According to G. Rein in [11, p. 389], and now, by (15) and Gronwall's lemma, and we can finally conclude that ∂ t f B L 2 (0,T ;C b ) ≤ C =: c 4 by expressing ∂ t f by the Vlasov equation. In Step 2 of the proof of Theorem 8 we have already showed that for all s, t ∈ [0, T ], .., 6} be arbitrary. Recall that, according to Lemma 2, Now, for all s, t ∈ [0, T ] (without loss of generality s ≤ t), Note that for all s, t ∈ [0, T ] and z ∈ B R (0), Now, using Hölder's inequality with exponents p = β−1 β and q = β, estimate (17) and a nonlinear generalization of Gronwall's lemma (cf. [2, p. 11]) with exponent β−1 β ∈]0, 1[, we obtain that ∂ z i ∂ z j Z k (s, t, ·) L β ≤ C. This finally implies the c 5 -estimate and the c 6 -estimate easily follows. Lemma 10. Let B, H ∈ M and let f B , f H be the induced classical solutions. Moreover, let Z B denote the solution of the characteristic system to the field B satisfying Z B (t, t, z) = z and let Z H be defined analogously. Then, there exist constants 1 , 2 , L 1 , L 2 , L 3 > 0 depending only onf , T , K, β such that where γ = γ(β) is the Hölder exponent from Lemma 5.
Proof Let B, H ∈ M, s, t ∈ [0, T ] and z ∈ B R (0) be arbitrary. Without loss of generality s ≤ t. Moreover, let C > 0 denote a generic constant depending only onf , T and K and let Z B and Z H denote the solutions of the characteristic system to the fields B and H satisfying Z B (t, t, z) = z and Z H (t, t, z) = z. Using the fundamental theorem of calculus, Lemma 9 and Lemma 5 we obtain by a straightforward computation. Thus by Lemma 2 (c) and Lemma 9, and then by chain rule, for every τ ∈ [0, T ] and every i, j ∈ {1, ..., 6}. This estimate can be used to prove (19) and (21) similarly to the above procedure. Finally (22) follows directly from (20) and (21) by representing ∂ t f B and ∂ t f H by their corresponding Vlasov equation. 13

Strong solutions for admissible external fields
Now we will show that any field B ∈ B K still induces a unique strong solution which can be constructed as the limit of solutions Such a strong solution is defined as follows: Definition 11. Let B ∈ B K be any admissible field. We call f a strong solution of the initial value problem (9) to the field B, iff the following holds: for some constant C > 0 depending only onf , T , K and β.
(ii) f satisfies the Vlasov equation (iii) f satisfies the initial condition f t=0 =f almost everywhere on R 6 , First of all one can easily establish that such a strong solution is unique.
Proposition 12. Let B ∈ B K be any field and suppose that there exists a strong solution f of the initial value problem (9) to the field B. Then this solution is unique.
Proof Suppose that there exists another strong solution g to the field B. Then the difference h := f − g satisfies almost everywhere on [0, T ] × R 6 . Thus by integration by parts, almost everywhere on R 6 which means uniqueness. Now we will show that any admissible field B ∈ B K actually induces a unique strong solution. Note that this solution is even more regular than it was demanded in the definition. However the weaker requirements of the definition will be essential in the later approach (see Proposition 16) and it will also be important that uniqueness was established under those weaker conditions. Theorem 13. Let B ∈ B K . Then there exists a unique strong solution f of the initial value problem (9) to the field B. Moreover this solution satisfies the following properties which are even stronger than the conditions that are demanded in Definition 11: for some constant C > 0 depending only onf , T , K and β.
(b) f satisfies the initial condition f t=0 =f everywhere on R 6 , Proof Let B ∈ B K arbitrary. According to Lemma 5, we can choose some sequence Now Lemma 10 and Lemma 9 provide that for all t ∈ [0, T ] and j, k ∈ N, Due to completeness there exists a unique function is also bounded in L ∞ (0, T ; W 2,β ) by some constant depending only onf , T , K and β, the Banach-Alaoglu theorem states that there exists some functionf ∈ L ∞ (0, T ; W 2,β ) such that f B k * f up to a subsequence. This means that for any α ≤ 2, the sequence (D α z f B k ) converges to D α zf with respect to the weak-*-topology on Because of uniqueness of the limit it holds that D α z f = D α zf and thus To show that f is a strong solution to the field B, we have to verify the conditions from Definition 11. The strong convergence of ( Thus f (0) =f everywhere on R 6 that is (b) which directly implies (iii). Due to uniform convergence and continuity of f , it is evident that Moreover by the weak-* lower semicontinuity of the norm, This proves (a) which includes condition (i). Finally, uniqueness follows directly from Proposition 12.
Now that we have showed that any magnetic field B ∈ B K induces a unique strong solution of the initial value problem (9), we can define an operator mapping every admissible field onto its induced state.
is called the field-state operator. At this point f B denotes the unique strong solution of (9) that is induced by the field B ∈ B K . From now on the notation f B is to be understood as the value of the field-state operator at point B ∈ B K .

Continuity and compactness of the field-state operator
Obviously the Lipschitz estimates of Lemma 10 hold true for the strong solutions by approximation.
The proof of this Corollary is obvious. I states that the field-state operator is globally Lipschitz-continuous with respect to the norm on C [0, T ]; C b and globally Höldercontinuous with exponent γ = γ(β) with respect to the norm on W 1,2 0, T ; C b and the norm on C [0, T ]; C 1 b . The following proposition provides (weak) compactness of the field-state operator that will be very useful in terms of variational calculus.
if j tends to infinity.
We will now show that f is a strong solution to the field B by verifying the conditions from Definition 11.
Condition (iv): Let ε > 0 be arbitrary. We will now assume that there exists some The inequality f W 1,2 (0,T ;L p (R 6 )) + f L 2 (0,T ;W 1,p ) ≤ C where C > 0 depends only onf , T , K and β follows directly from the weak convergence and the weak lower semicontinuity of the norm. Since C does not depend on p this inequality holds true for p = ∞.
Condition (iii): It holds that f k f in W 1,2 (0, T ; L 2 ) with f k (0) =f almost everywhere on R 6 for all k ∈ N. By Mazur's lemma we can construct some sequence (f * k ) k∈N such that f * k → f in W 1,2 (0, T ; L 2 ) where for any k ∈ N, f * k is a convex combination of f 1 , ..., f k . Then of course f * k (0) =f almost everywhere on R 6 as well and hence Thus f (0) =f almost everywhere on R 6 .
) due to Lemma 1. Then, because of the compact support, the Rellich-Kondrachov theorem implies that f k → f in L 2 ([0, T ] × R 6 ), up to a subsequence. From Lemma 2 (b) we can conclude that for any t ∈ [0, T ], For brevity, we will now use the notation may depend on ϕ). It also holds that By integration by parts, and the right-hand side converges to zero as k → ∞. As ϕ was arbitrary this implies that Consequently f is a strong solution to the field B and thus f = f B because of uniqueness. Furthermore we have showed that there exists a subsequence (B k j ) of (B k ) such that (f B k j ) is converging in the demanded fashion.

A general inhomogenous linear Vlasov equation
Since a Fréchet derivative is a linear approximation, we will find out later that the derivative of the field-state operator is determined by an inhomogenous linear Vlasov equation. In this section we will analyze those linear Vlasov equations in general, i.e., we will establish some existence and uniqueness results. The type and the regularity of the solution will depend on the regularity of the coefficients.
Let r 0 ≥ 0 and r 2 > r 1 ≥ 0 be arbitrary. We consider the following inhomogenous linear version of the Vlasov equation: The coefficients are supposed to have the following regularity Moreover Φ a,f is given by for all (t, x) ∈ [0, T ] × R 3 . We will also use the notation As a ∈ C [0, T ]; C 1 b (R 6 ) with compact support supp a(t) ⊂ B r 0 (0) for all t ∈ [0, T ], Lemma 2 provides the following inequalities: For any r > 0 there exists some constant c > 0 that may depend only on r and r 0 such that for almost all t ∈ [0, T ], If Because of density this result holds true if . If merely f ∈ L 2 (0, T ; H 1 ) the result holds true in the weak sense.
For any t ∈ [0, T ] and z ∈ R 6 the characteristic systemẋ Moreover, there exists some constant C(r) > 0 depending only on A L 2 (0,T ; and r such that for all s, t ∈ [0, T ], The proof is simple and very similar to the proof of Lemma 9. Therefore it will not be presented. Now we can establish an existence and uniqueness result for classical solutions of the system (24) if the regularity conditions (25)-(31) hold. Unfortunately the coefficients of the systems that will occur in this paper do not satisfy those strong conditions. However, we will still be able to prove an existence and uniqueness result for strong solutions of (24) if the regularity conditions are slightly weaker.
(a) If we use a final value condition f t=T =f instead of the initial value condition f t=0 = f the problem can be treated completely analogously. The results of Proposition 18 and Corollary 21 hold true in this case. Only the implicit depiction of a classical solution must be replaced by (b) Suppose that C = 0 and recall that Φ a,f depends only on f Br 0 (0) . Hence, if we choose r 1 = ζ(r 0 ) then for all t ∈ [0, T ] and z ∈ B r 0 (0), because in this case χ Z(s, t, z) = 1 as Z(s, t, B r 0 (0)) ⊂ B r 1 (0). This means that the values of f Br 0 (0) do not depend on the choice of χ as long as (31) and (33) hold.
Proof of Proposition 18 Let c > 0 denote a generic constant depending only on r 0 , r 2 , T and the norms of the coefficients. For t ∈ [0, T ] and z ∈ R 6 let Z = (X, V )(s, t, z) denote the solution of the characteristic system with Z(t, t, z) = z. Moreover, for t ∈ [0, T ] and z ∈ R 6 , we define a recursive sequence by f 0 (t, z) :=f (z) and f n+1 (t, z) :=f (Z(0, t, z)) + t 0 ∂ x ψ fn · C + χΦ a,fn + b s, Z(s, t, z) ds.
By induction we can conclude that all f n are continuous. Then for any fixed τ ∈ [0, T ] and n ∈ N the functionsf , ∂ x ψ fn · C (τ ), χΦ a,fn (τ ) and b(τ ) are continuous and compactly supported in B r (0) with r = max{r 0 , r 2 }. This directly implies that f 0 (t) is compactly supported with supp f 0 (t) ⊂ B r (0) for all t ∈ [0, T ]. Moreover, for any τ ∈ [0, T ], Lemma 17 implies that suppf (Z(s, t, ·)) = Z(t, s, suppf ) supp ∂ x ψ f · C](τ, Z(s, t, ·)) = Z t, s, supp ∂ x ψ fn · C(τ ) supp χΦ a,fn (τ, Z(s, t, ·)) = Z t, s, supp χΦ a,fn (τ ) If we choose τ = s we can inductively deduce that supp f n (t) ⊂ B ζ(r) (0) for all t ∈ [0, T ] and all n ∈ N. Finally, by another induction, f n ∈ C 1 (]0, T [×R 6 ) as the partial derivatives can be described recursively. Using Lemma 17, (38), (39) and Lemma 2, we con conclude by a straightforward computation that there exists some constant c * > 0 such that for all t ∈ [0, T ], for m, n ∈ N 0 . Thus by induction, and hence for m, n ∈ N with n < m, One can easily show that f is a classical solution of (24) by differentiating both sides of (43) with respect to t. We will finally prove uniqueness by assuming that there exists another solutionf of the initial value problem and define d := f −f . Then for any t ∈ [0, T ], and hence d(t) L 2 = 0 for all t ∈ [0, T ] by Gronwall's lemma. This directly implies that f =f which means uniqueness.
Definition 20. We call f a strong solution of the initial value problem (24) iff the following holds: (ii) f satisfies (iii) f satisfies the initial condition f t=0 =f almost everywhere on R 6 .
Corollary 21. We define r := max{r 0 , r 2 } and let C > 0 denote some constant depending only on r 0 , r 2 and the norms of the coefficients.

Proof To prove (a) we can choose
and for all t ∈ [0, T ], supp b k (t), suppf k , and supp C(t) ⊂ B r 0 +1 (0). Then, due to Proposition 18, for every k ∈ N there exists a unique classical solution f k of (24) to the coefficients a, b k ,f k , A, B k , C k and χ. Moreover for all t ∈ [0, T ], supp f k (t) ⊂ B (0) with := ζ(2 + max{r 0 , r 2 }) = ζ(2 + r). Now let Z k denote the solution of the characteristic system to A and B k satisfying Z k (t, t, z) = z and let c > 0 denote some generic constant depending only on T , r 0 , r 2 and the norms of the coefficients. From Lemma 17 we know that for any r > 0 and all s, t ∈ [0, T ], Z k (s, t, ·) L ∞ (Br(0)) < C(r) and ∂ z Z k (s, t, ·) L ∞ (Br(0)) < C(r) where C(r) > 0 depends only on r, A L 2 (0,T ;C 1 b ) and B L 2 (0,T ;C 1 b ) . Then we can conclude from the implicit description (40) that which yields f k (t) L ∞ ≤ c by Gronwall's lemma. By differentiating (40) and using (44) the z-derivative can be bounded similarly by Finally one can easily show that ∂ t f k L 2 (0,T ;L 2 ) ≤ c by expressing ∂ t f k by the Vlasov equation. Since all f k (t) are compactly supported in B (0) this yields Then, according to the Banach-Alaoglu theorem, there exists f ∈ H 1 (]0, T [×R 6 ) such that f k f after extraction of a subsequence. Moreover there exists some function f * ∈ L ∞ (]0, T [×R 6 ) such that f k * f * up to a subsequence, i.e., a subsequence of (f k ) converges to f * with respect to the weak-*-topology on We will now show that f is a strong solution of (24) by verifying the conditions of Definition 20.
Condition (iv) is also obvious because supp f k ⊂ B (0) for all k ∈ N, t ∈ [0, T ]. The radius does not depend on k and satisfies < ζ(3 + r).
This implies that ψ f k → ψ f and Φ a,f k → Φ a,f in L 2 ([0, T ] × R 3 ) and the assertion easily follows.
Condition (iii): Finally, according to Mazur's lemma, there exists some sequence where for all k ∈ N,f k is a convex combination of f 1 , ..., f k . This meansf k (0) =f and hence Consequently f is a strong solution but we still have to prove uniqueness. We assume that there exists another strong solutionf and define d := f −f . Then, by the fundamental theorem of calculus, T ] by Gronwall's lemma. This proves (a).
To prove (b) we only have to approximate B. Therefore we choose some sequence (B k ) ⊂ C([0, T ]; C 1,γ ) such that Then for any k ∈ N there exists a unique classical solution f k of the system (24) to the coefficients a,f , A, B k and χ according to Proposition 18. Recall that for all t ∈ [0, T ], supp f k (t) ⊂ B (0) where := ζ(r + 1) with r = max{r 0 , r 2 }. Again, let Z k denote the solution of the characteristic system to A and B k satisfying Z k (t, t, z) = z and in the following the letter c denotes some generic positive constant depending only on T , r 0 , r 2 and the norms of the coefficients. Now for all s, t ∈ [0, T ] (where s ≤ t without loss of generality) and z ∈ B (0), Similarly, for any i ∈ {1, ..., 6} the difference of the i-th derivative can be bounded by for all s, t ∈ [0, T ] and z ∈ B (0). Thus By expressing ∂ t f k and ∂ t f j by their corresponding Vlasov equation we can easily verify the estimate This means that (f k ) is a Cauchy sequence in W 1,2 (0, T ; C b ) ∩ C([0, T ]; C 1 b ) and thus it converges to some function f ∈ W 1,2 (0, . From the strong convergence one can easily conclude that f satisfies the system (24) almost everywhere and thus f is a strong solution according to Definition 20.
Moreover, by the definition of convergence, we can find k ∈ N such that f −f k W 1,2 (0,T ; ) according to Proposition 18 and B k is bounded by B k L 2 (0,T ;C 1,γ ) ≤ 2 B L 2 (0,T ;C 1,γ ) .
We will now assume that r 1 = ζ(r 0 ). As it has already been discussed in the comment to Proposition 18 the values of f k | Br 0 (0) do not depend on the choice of χ as long as (31) and (33) hold. As f k | Br 0 (0) converges to f | Br 0 (0) uniformely on [0, T ] × B r 0 (0) this result holds true for f | Br 0 (0) .

Fréchet differentiability of the field-state operator
Again, let K > 0 be arbitrary. We can now use the results of Section 5 to establish Fréchet differentiability of the control state operator onB K (that is the interior of B K ).
and some radius > 0 depending only on T, K,f and β. Then the following holds: (a) The field-state operator f. is Fréchet differentiable onB K with respect to the norm on C([0, T ]; L 2 (R 6 )), i.e., for any B ∈B K there exists a unique linear and bounded operator f B : V → C([0, T ]; L 2 (R 6 )) such that The Fréchet derivative is given by (b) For all B, H ∈B K , the solution f H B depends Hölder-continuously on B in such a way that there exists some constant C > 0 depending only onf , T, K and β such that for all A, B ∈B K , Remark 23. As K > 0 was arbitrary the obove results hold true onB 2K instead ofB K . Hence they are especially true for B ∈ B K .
Proof Let C denote some generic positive constant depending only onf , K, T and β. First note that the system (45) is of the type (24) whereby the coefficients of (45) satisfy the regularity and support conditions of Corollary 21(a). Hence (45) has a strong solution To prove Fréchet differentiability of the field-state operator we must consider the difference f B+H − f B with B ∈B K and H ∈ V such that B + H ∈B K . Therefore we will assume that H V < δ for some sufficiently small δ > 0. Now we expand the nonlinear terms in the Vlasov equation (1) to pick out the linear parts. We have are nonlinear remainders. Then R := R 1 − R 2 lies in L 2 (0, T ; H 1 ∩ C b ) and from Lemma 2 and Corollary 15 we can conclude that R L 2 (0,T ; almost everywhere on [0, T ]×R 6 . From Corollary 21 (a) we know that this solution is unique. Also according to Corollary 21 (a) the system has a unique strong solution f R . Then f H B + f R is a solution of (47) due to linearity and thus f B+H − f B = f H B + f R because of uniqueness. One can easily show that Applying first the standard version and then the quadratic version of Gronwall's lemma (cf. Dragomir [2, p. 4]) yields Let now ε > 0 be arbitrary. Then for all t ∈ [0, T ], if δ is sufficiently small. Hence assertion (a) is proved and the Fréchet derivative is determined by the system (45).
if k tends to infinity. From Corollary 21 (and its proof) we can conclude that Since the (x, v)-supports of all occurring functions are contained in some ball B (0) whose radius r depends only onf , K, T and β but not on k, we can apply the Rellich-Kondrachov theorem to obtain up to a subsequence. As A k , B k and H k satisfy the regularity condition (29), f H k A k and f H k B k are classical solutions and can be described implicitely by the representation formula (40). Note that Lemma 9 holds true for instead of R. Hence for all s, t ∈ [0, T ], Also recall that we know from Lemma 10 (with instead of R) that for all s, t ∈ [0, T ], Using the implicit description (40) it follows by a simple computation and the application of Gronwall's lemma that 7 An optimal control problem with a tracking type cost functional Letf ∈ C 2 c (R 6 ) be any given initial datum and let T > 0 denote some fixed final time. The aim is to control the time evolution of the distribution function in such a way that its value at time T matches a desired distribution function f d ∈ C 2 c (R 6 ) as closely as possible. More precisely we want to find a magnetic field B such that the L 2 -difference f B (T ) − f d L 2 becomes as small as possible. Therefore we intend to minimize the quadratic cost functional: where λ is a nonnegative parameter. The field B is the control in this model. As the state f B (t) preserves the p-norm, i.e., f B (t) p = f p for all 1 ≤ p ≤ ∞, t ∈ [0, T ], it makes sense to assume that f d p = f p for all 1 ≤ p ≤ ∞ because otherwise the exact matching f (T ) = f d would be foredoomed to fail.
At first appearance the term λ 2 D x B 2 L 2 seems to be useless or even counterproductive as we actually want to minimize the expression f (T ) − f d L 2 . However, in optimal control theory, such a term is usually added because of its smoothing effect on the control. If λ > 0 a magnetic field is punished by high values of the cost functional if its derivatives become large. Of course the weight of punishment depends on the size of λ. For that reason the additional term is referred to as the regularization term. Note that the regularity B ∈ H is now necessary to avoid infinite values of the cost functional.
Of course such an optimization problem does only make sense if there actually exists at least one globally optimal solution. This fact will be established in the next Theorem. The proof is quite short as most of the work has already been done in the previous sections.
Proof Suppose that λ > 0 (if λ = 0 the proof is similar but even easier). The cost functional J is bounded from below since J(B) ≥ 0 for all B ∈ B K . Hence M := inf B∈B K J(B) exists and we can choose a minimizing sequence (B k ) k∈N such that J(B k ) → M if k → ∞. Without loss of generality we can assume that J(B k ) ≤ M + 1 for all k ∈ N. As B K ⊂ V is weakly compact according to Lemma 5 it holds that B k B in L 2 (0, T ; W 2,β )∩L 2 (0, T ; H 1 ) for some weak limitB ∈ B K after extraction of a subsequence. Then we know from Proposition 16 that f B k fB in W 1,2 (0, T ; L 2 ) after subsequence extraction. By the fundamental theorem of calculus this implies that f B k (T ) fB(T ) in L 2 (R 6 ). Together with the weak lower semicontinuity of the L 2 -norm this yields By the definition of infimum this proves J(B) = M .
Of course this theorem does not provide uniqueness of a globally optimal solution. In general, the optimization problem may have more than one globally optimal solution and, of course, it may have more than one locally optimal solution. Therefore we will characterize the local minimizers in the following subsections by necessary and sufficient conditions.

Necessary conditions for local optimality
A locally optimal solution is defined as follows: Definition 25. A controlB ∈ B K is called a locally optimal solution of the optimization problem (49) iff there exists δ > 0 such that is the open ball in V with radius δ and centerB.
To establish necessary optimality conditions of first order we need Fréchet differentiability of the cost functional J.
for all H ∈ V. LetB ∈ B K be a locally optimal solution of the optimization problem (49). Then Proof As the control-state operator is Fréchet differentiable on B K so is the cost functional J by chain rule. Thus, the function [0, 1] t → J(B + tH) ∈ R is differentiable with respect to t and sinceB is also a local minimizer of this function, we have for any H ∈ B K with B + H ∈ B K . IfB is an inner point of B K this line even holds with "=" instead of "≤".
If we consider B K as a subset of L 2 ([0, T ] × R 3 ; R 3 ) it might be possible to find an adjoint operator f B (T ) . Then, by integration by parts, for all H ∈ V. This means that the derivative J would have the explicit description If nowB ∈ int B K were a locally optimal solution it would satisfy the semilinear Poisson equation In general such an adjoint operator is not uniquely determined. This means that we cannot deduce uniqueness of our optimal solution. A common technique to find an adjoint operator is the Lagrangian technique. For B ∈ B K and f, g L is called the Lagrangian. Obviously, by integration by parts, In the definition of the Lagrangian f , B and g are independent functions. However, inserting It is important that this equality does not depend on the choice of g. Since L is Fréchet differentiable with respect to f in the H 1 (]0, T [×R 6 )-sense and with respect to B in the V-sense we can use this fact to compute the derivative of J alternatively. By chain rule, for all B ∈ B K , H ∈ V and any g ∈ H 1 (]0, T [×R 6 ). Here ∂ f L and ∂ B L denote the partial Fréchet derivative of L with respect to f and B. We will now fix f, g and B. Then where Φ f,g is given by (34). Moreover, for all H ∈ V. Apparently, the derivative with respect to B looks pretty nice while the derivative with respect to f is rather complicated. However if we insert those terms in (51) we can still choose g. Now the idea of the Lagrangian technique is to choose g in such a way that the term We consider the following final value problem which is referred to as the costate equation or the adjoint equation: where χ ∈ C 2 c (R 6 ; [0, 1]) with χ = 1 on B R Z (0) and supp χ ∈ B 2R Z (0) denotes an arbitrary but fixed cut-off function. Here R Z is the constant from Lemma 9, i.e., for all s, t ∈ [0, T ], Z B (s, t, B R (0)) ⊂ B R Z (0). Existence and uniqueness of a strong solution to this system will be established in the following theorem: Theorem 27. Let B ∈ B K be arbitrary. The costate equation (54) has a unique strong solution g B ∈ W 1,2 0, with compact support supp g B (t) ⊂ B R * (0) for all t ∈ [0, T ] and some radius R * > 0 depending only onf , f d , T, K and β.
In this case g B B R (0) does not depend on the choice of χ.
Moreover g B depends Lipschitz/Hölder-continuously on B in such a way that there exists some constant C ≥ 0 depending only onf , f d , T, K, β and χ C 1 b such that for all B, H ∈ B K , Remark 28. Note that only the values of g B on the ball B R (0) will matter in the following approach. Therefore it is essential that those values are not influenced by the cut-off function χ.

Proof
Step 1 : Obviously the system (54) has a unique strong solution g B in the sense of Corollary 21 (a). Unfortunately the coefficients do not satisfy the stronger regularity conditions of Corollary 21 (b) as the final value f B (T ) − f d is not in C 2 c (R 6 ). However, because of linearity, it holds that g B =g B − h B whereg B is a solution of and h B is a solution of h t=T = f d Now the first system has a unique strong solution in the sense of Corollary 21 (a) and the second one possesses a strong solution in the sense of Corollary 21 (b) since f d ∈ C 2 c (R 6 ). Indeed the solutiong B is much more regular. As Φ f B ,f B = 0 one can easily see that f B is a solution of the first system and thus, because of uniqueness,g B = f B . Consequently Step 2 : We will now prove the Hölder estimate. It suffices to establish the result for h. as the result has already been proved for f. in Corollary 15. Therefore let B, H ∈ B K be arbitrary and let C > 0 denote some generic constant depending only onf , f d , T , K, β and if k → ∞. By Corollary 21 (b) (and its proof) the induced strong solutions h B k and h H k satisfy The constant C does not depend on k since B k V and H k V are bounded by 2K. Also note that there exists some constant > 0 depending only onf , f d , T, K and β (but not on k) such that supp h B k ⊂ B (0) and also supp h H k ⊂ B (0). As h B k and h H k are classical solutions they satisfy the implicit representation formula (41). We also know from Lemma 10 (with instead of R) that for all t ∈ [0, T ]. Together with Lemma 9 we can show that In summary, we have established that For k → ∞ this directly implies that and hence Step 3 : We must still prove that g B ∈ L ∞ (0, T ; H 2 ). As f B ∈ L ∞ (0, T ; H 2 ) has already been established in Theorem 13 it suffices to show that h B is twice weakly differentiable with respect to z and D 2 z h B ∈ L ∞ (0, T ; L 2 ). Recall that for any k ∈ N, f B k ∈ C([0, T ]; C 2 b ) according to Theorem 8 and h B k ∈ C([0, T ]; C 1 b ) according to Theorem 18. Thus for all i ∈ {1, 2, 3}, ; H 2 (B r (0)) , r > 0. The third line follows from Lemma 2. Consequently, since χ is compactly supported. We also know from Lemma 9 (with instead of R) that Z B k is twice continuously differentiable with respect to z and

Now recall the implicit representation formula (41) for h B k that is
for all (t, z) ∈ [0, T ]×R 6 . As f d ∈ C 2 c (R 6 ) and Z B k (T, t, ·) ∈ C 2 (R 6 ), the term f d (Z B k (T, t, z)) is twice continuously differentiable with respect to z by chain rule. By approximating Φ f B k ,h B k χ by sufficiently smooth functions one can easily show that the integral term of (58) is twice weakly differentiable and the derivatives can be computed by chain rule (with weak instead of classical derivatives if necessary). Now, one can show that the weak derivative ∂ z i ∂ z j h B k can be bounded by

By (57) this finally yields
is converging with respect to the weak-*-topology on [L 1 (0, T ; L 2 )] * = L ∞ (0, T ; L 2 ) up to a subsequence. Because of uniqueness, the weak-*-limit of the sequence (∂ z i ∂ z j h B k ) must be ∂ z i ∂ z j h B and especially h B ∈ L ∞ (0, T ; H 2 ). This completes the proof.
Now inserting the state f B and its costate g B in (51) yields This provides a necessary optimality condition: Theorem 29.
(a) The Fréchet derivative of J at the point B ∈ B K is given by (b) Let us assume thatB ∈ B K is a locally optimal solution of the optimization problem (49). Then for all B ∈ B K , (c) If we additionally assume thatB ∈B K thenB satisfies the semilinear Poisson equation In this caseB ∈ C([0, T ]; for all t ∈ [0, T ] and x ∈ R 3 . ThusB does not depend on the choice of χ as long as χ = 1 on B R Z (0) as it only depends on gB B R (0) .
Proof (a) follows immediately from (53)  HenceB is uniquely determined by (61). We must still prove thatB lies in C([0, T ]; C 2 b (R 3 )). Recall that fB and gB are in W 1,2 (0, T ; C b (R 6 )) ∩ C([0, T ]; C 1 b (R 6 )) asB ∈ B K . Thus is continuous with supp p(t) ⊂ B R (0) for all t ∈ [0, T ]. By approximating fB by C([0, T ]; C 2 b )functions and using integration by parts one can easily show that p is continuously differentiable whereby the partial derivatives are given by Since gB does not depend on χ as long as χ = 1 on B R Z (0) the same holds forB.
Note that Theorem 29 provides only a necessary but not a sufficient condition for local optimality. If a control B satisfies the above condition it could still be a saddle point or even a local maximum point. Theorem 29 does also not provide uniqueness of the locally optimal solution. However the globally optimal solution that is predicted by Theorem 24 is also locally optimal. Thus we have at least one control to satisfy the necessary optimality condition of Theorem 29.
Assuming that there exists a locally optimal solutionB ∈B K we can easily deduce from Theorem 29 that the triple (fB, gB,B) is a classical solution of some certain system of equations.
Corollary 30. Suppose thatB ∈B K is a locally optimal solution of the optimization problem (49). Let fB and gB be its induced state and costate. Then fB, gB ∈ C 1 ([0, T ] × R 6 ) and the triple (fB, gB,B) is a classical solution of the optimality system 1 |x−y| w × ∂ v f (t, y, w) g(t, y, w) d(y, w) .

A sufficient condition for local optimality
To prove that our cost functional is twice continuously Fréchet differentiable we will need Fréchet differentiability of first order of the costate. 37 The proof proceeds analogously to the proof of Theorem 22.
Remark 32. As K was arbitrary the above results hold true ifB K is replaced byB 2K . Hence they are especially true on B K .
Continuous differentiability of the cost functional then follows: Corollary 33. The cost functional J of the optimization problem (49) is twice Fréchet differentiable onB K . The Fréchet derivative of second order at the point B ∈B K can be described as a bilinear operator J (B) : V 2 → R that is given by The following theorem provides a sufficient condition for local optimality: Theorem 34. Suppose thatB ∈ B K and let fB and gB be its induced state and costate. Let 0 < α < 2 + γ be any real number. We assume that the variation inequality

Uniqueness of the optimal solution on small time intervals
We know from Corollary 30 that for any locally optimal solutionB ∈B K the triple (fB, gB,B) is a classical solution of the optimality system 1 |x−y| w × ∂ v f (t, y, w) g(t, y, w) d(y, w) .

(68)
The following theorem states that the solution of this system of equations is unique if the final time T is small compared to λ. As we will have to adjust T λ it is necessary to assume that 0 < λ ≤ λ 0 for some constant λ 0 > 0. Of course large regularaization parameters λ do not make sense in our model, so we will just assume that λ 0 = 1.
Proof Suppose that the triple (f ,g,B) is another classical solution that is satisfying the support condition with radiusr. Without loss of generality we assume that r =r. Let C = C(T ) ≥ 0 denote some generic constant that may depend on T ,f , f d , r, χ C 1 b and the C([0, T ]; C 1 b )-norm of f ,f , g andg. We can assume that C = C(T ) is monotonically increasing in T . First of all, by integration by parts, Let now Z andZ denote the solutions of the characteristic system of the Vlasov equation to the fields B andB satisfying Z(t, t, z) = z andZ(t, t, z) = z for any t ∈ [0, T ] and z ∈ R 6 . Then for any s, t ∈ [0, T ] (where s ≤ t without loss of generality) and z ∈ R 6 ,