OPTIMAL NONLINEARITY CONTROL OF SCHR¨ODINGER EQUATION

. We study the optimal nonlinearity control problem for the nonlinear Schr¨odinger equation iu t = −(cid:52) u + V ( x ) u + h ( t ) | u | α u , which is originated from the Fechbach resonance management in Bose-Einstein condensates and the nonlinearity management in nonlinear optics. Based on the global well- posedness of the equation for 0 < α < 4 N , we show the existence of the optimal control. The Fr´echet diﬀerentiability of the objective functional is proved, and the ﬁrst order optimality system for N ≤ 3 is presented.

In Bose-Einstein condensate, this model describes the dynamics of interacting atoms at zero temperature in an external potential V (x), where h(t) is related to the scattering length which can be tuned through an external magnetic field by the technique of Feshbach resonance to produce robust matter-wave, a typical example is to take h(t) = γ 0 + γ 1 cos(ωt) [9]. In optics, it describes nonlinearity management for transverse beam propagation in layered optical media. In the case of h(t) being periodic, the behavior of the solution of this equation have been investigated in [5,9,10,11,23,29], et al. From the mathematical point of view, the management problem mentioned above is essentially a nonlinearity control problem. In comparison, the coherent manipulation of quantum systems via external potentials corresponding to the linear control where f (u) denotes the nonlinearity term, V (x) is a fixed external potential, and W (x, t) corresponds to the controller. A lot of works have been carried out on this problem, we refer to [1,12,13,14,15,16,17,19,28] for some related studies. However, for the nonlinearity control problem, although it has been studied in physics literature (see [6,7,20,21,22,24] and the references therein), to our knowledge, a mathematical discussion is still lacking.
In this paper, we consider the optimal nonlinearity control problem governed by where T > 0 is the final control time given arbitrarily, h : [0, T ] → R is a real function which denotes the control parameter. It is known that in the case h(t) = constant = 0, equation (1.1) is focusing (h < 0) or defocusing (h > 0), and may have stable solutions of the form u(x, t) = v(x)e −iλt . When h(t) = constant, this is obviously false. Notice that when h(t) is allowed to be sign-changed, equation (1.1) is of mixed type of focusing and defocusing nonlinearity. In such a situation, the global existence and long-time behavior of solutions become rather complex.
Recall that the energy E(t) corresponds to (1.1) reads and its evolution is given by It is easy to see that (1.1) enjoys mass conservation, i.e., u(t) L 2 x = u 0 L 2 x , for all t ∈ [0, T ]. But unlike the case of h(t) =constant, the energy E(t) is not conserved, which will bring some difficulties in our study.
The optimal control problem for Schrödinger equation usually needs to minimize an objective functional and deduce an optimality condition to characterize the minimum of the functional [8]. In general, the objective functional consist of two parts, one is the "distance" between the solution of the state equation and the desired state, the other describes the cost through the control process. There are many possible ways of modeling the cost it takes to reach a certain prescribed expectation value, here we will adopt the framework suggested in [16], and define the objective functional J = J(h) as where h ∈ H 1 (0, T ), γ 1 ≥ 0, γ 2 > 0, u(T ) denotes the final state, and the operator A : L 2 (R N ) → L 2 (R N ) is assumed to be bounded linear and self-adjoint.

Remark 1.
A typical choice of A would be A := I − P ψ , where P ψ denotes the orthogonal projection onto a given target ψ ∈ L 2 x . If we choose ψ L 2 On the other hand, the distance between u(T ) and ψ in L 2 x reads x , which is the same as that in [19]. So, the first term of the right hand side of (1.4) is used to measure the "distance" between the final state u(T ) and the target ψ.
Our aim is to find a control function h * ∈ H 1 (0, T ) such that J(h * ) = inf H 1 (0,T ) J(h). For this purpose, we set where R > 0 is a real number given arbitrarily. We will look for a minimizer h * , and when R → ∞, h * R will tends to an idea control, i.e., J(h * R ) will tends to inf H 1 (0,T ) J(h). In what follows, we will solve the following minimizing problem (1.5) Our main results can be stated as Theorem 1.1. Let 0 < α < 4 N , γ 1 ≥ 0, γ 2 > 0, and V ∈ C ∞ (R N ) be subquadratic, then for every u 0 ∈ Σ, R > 0, the optimal control problem (1.5) has a minimizer h * R ∈ C h (R). where

Remark 2.
As an example, we can take A = id − P ψ , where P ψ : L 2 x → Σ is the orthogonal projection. Then A satisfies the assumption A(Σ) ⊂ Σ. On the other hand, without the assumption A(Σ) ⊂ Σ, the higher regularity estimates of the solution for (1.1) is needed. We refer to [16], where the Gâteaux-differentiability of J(h) is obtained by Σ m -regularity of u with m > N 2 , and in [15], the Fréchetdifferentiability of J(h) is obtained by Σ 2 regularity of u. Theorem 1.2 yields the first order necessary optimality condition for problem (1.5) as following.
be a solution of (1.5), u * = u(h * R ) the solution of (1.1), v * the solution of the adjoint equation (1.8), and ω * the function defined in (1.7) with u replaced by u * . Then for all h ∈ C h (R), , the set of the inner point of C h (R), then for all ν ∈ H 1 (0, T ), J (h * R )ν = 0. The rest of the paper is organized as follows. In section 2 we list some lemmas and show the local and global existence of solution of (1.1). In section 3, we give the proof of Theorem 1.1. In section 4, we first show the the well-posedness of the adjoint equation and the Lipschitz continuity of the solution u(h) with respect to h, and then prove the Fréchet-differentiability of J(h).
Notations and conventions. We use the abbreviations L p x = L p (R N , C), H 1 x = H 1 (R N , C) and W 1,r x = W 1,r (R N , C). The scalar product on L 2 x is defined by where z means to take the real part of the complex number z. We also define Σ 1,r := {u ∈ W 1,r x ; xu ∈ L r x } with the norm u Σ 1,r = u W 1,r x + xu L r x . Σ 1,2 will be denoted by Σ, and Σ * is the dual space of Σ.
The norm of the space L q (I, L r x ) is denoted by · L q t L r The Strichartz space S(I) will also be used in the following, which is defined as the closure of the Schwartz functions under the norm Similarly, we define the space S Σ (I) with the norm We always assume that u 0 ∈ Σ, h ∈ H 1 (0, T ), and the external potential V (x) is assumed to be smooth and subquadratic, that is Throughout this paper, C > 0 will stand for a constant that may be different from line to line when it dose not cause any confusion.

2.
Local and global existence. In this section, we give some preliminary results. We say u is a mild solution of (1.1) if it satisfies The following Strichartz's estimates [3,4] will be invoked throughout this paper.
Let U (t) be defined as above, then there exists an η > 0 such that (1) For any admissible pair (q,r) and every ϕ ∈ L 2 x , there exists a constant then for every admissible pair (q,r), there exists a constant C(q, γ) > 0, such that for all s < t < s + η and f ∈ L γ t L ρ x (s, s + η). In the next, we establish the local existence for system (1.1).
Proof. The proof of the lemma is similar to that of Theorem 4.11.1 in [4]. The main difficulties are to deal with the term including subquadratic potential V (x), and give an estimate for xu. We firstly present the proof for N ≥ 3.
In order to estimate the nonlinearity h|u| α u, we choose an admissible pair (q 0 , r 0 ) which satisfies 1 − 2 r 0 = α 2 * and an easy calculation shows that Since For every u 0 ∈ Σ, from Lemma 2.1, there exist η and K such that Let l ≤ min{T, η} and set is a complete metric space. Denote the right hand side of the (2.1) by Φ(u)(t), in order to prove Lemma 2.2, it suffices to show that Φ is a contraction mapping from E into itself.
Choose l small enough, by Lemma 2.1, Hölder's inequality and Sobolev's inequality, we have On the other hand Similarly, it holds that Hence, back to (2.1), we obtain Since ∇V is sublinear by the assumption of V (x), there exists a C > 0 which depends only on V such that and x (0,l) . (2.6) Collecting (2.2), (2.5) and (2.6), we have (2.7) Since (γ, ρ) in (2.7) is an arbitrarily admissible pair, we infer that Choosing l small enough such that it is easy to see that Φ maps E into itself and is a contraction. And then the contraction mapping theorem gives the existence and uniqueness of the solution for If N = 2, the proof is the same as the case N ≥ 3, except we set θ > 1, (2 − α)θ ≤ 1, and let r 0 = 2θ.
Because we need the global existence of solutions of equation (1.1), hereafter we mainly deal with the case 0 < α < 4 N , because for the case of 4 N ≤ α ≤ 4 N −2 , the solution of (1.1) may blow up in finite time [5,11,22]. Comparing with [4] and [5], our method is slightly different.
The following Gronwall-type estimate is important in our research, for more details we refer to [5,15].
3. Proof of Theorem 1.1. The proof of Theorem 1.1 proceeds in three steps. In step 1, we investigate the convergence of the minimizing sequence {h n } as well as the correspond solution sequence {u n }. In step 2, we show that the limit of u n is the solution for (1.1) with respect to the limit of {h n }. In step 3, by studying the weak lower semi-continuity of the objective functional, we conclude that the limit of {h n } is indeed a minimizer of optimal control problem (1.5).
Step 1. For any given h ∈ C h (R), Lemma 2.4 implies that 0 ≤ J(h) < +∞. Thus J is bounded from below. For a minimizing sequence {h n } +∞ n=1 ⊂ C h (R), the corresponding sequence J(h n ) (n∈N) of the objective function is bounded on R, hence J(h n ) ≤ C < +∞ for all n ∈ N.
By passing to a subsequence, it holds u n (t) → u * (t) in L 2 x and u n (t) → u * (t) in L α+2 x for a.e. t ∈ [0, T ].
Step 2. We show that u * is a mild solution of (1.1) with control h * R . Indeed, for v 1 , v 2 ∈ L α+2 x , we have On the other hand By the aid of Lemma 2.4 in [16], it follows from the Lebesgue dominated convergence theorem that (3.4) so u * ∈ L ∞ ((0, T ), Σ) ∩ W 1,∞ ((0, T ), Σ * ) and satisfies (3.4). Based on Strichartz's estimates, the uniqueness of the weak Σ−solution u * can be deduced by the standard argument. Using the same discussion as in the proof of Theorem 3.3.9 in [4], we conclude that u * is indeed a mild solution of (1.1), i.e., u * = u(h * R ) satisfies u * ∈ C((0, T ), Σ) ∩ C 1 ((0, T ), Σ * ). Furthermore Step 3. In order to prove that h * R ∈ C h (R) is indeed a minimizer of the control problem, we need only to show Since A : Σ → L 2 x is a bounded linear operator by assumption, the sequence (Au n (T )) n∈N converges weakly to Au * (T ) in L 2 To deal with the term T 0 |E (t)| 2 dt, in view of (1.3), we define

KAI WANG, DUN ZHAO AND BINHUA FENG
Notice that 0 ≤ ω n (t) ≤ u n α+2 Σ , then (3.2) implies that ω n , ω * ∈ L ∞ (0, T ). By the convexity of the function |z| α+2 for z ∈ C, it follows from (3.3) that Differentiating with respect to u formally, we have where v ∈ L 2 (R N ) ⊂ Σ * . By a similar argument as in [16], we can get the following adjoint equation where u is the solution of (1.1) with respect to the control h, δJ δu(t) and δJ δu(T ) denote the first variations of J with respect to u(t) and u(T ) respectively. Explicitly  Proof. By Lemma 2.4, we have u ∈ C([0, T ], Σ) ∩ L γ ((0, T ), Σ 1,ρ ). Then, it is easily to infer that δJ(u,h) δu(t) ∈ L 1 ((0, T ), L α+2 α+1 ). According to the assumption on A, we have δJ(u,h) δu(T ) ∈ Σ. Due to the inequality Firstly we study the local Lipschitz continuity of u with respect to the control h. where C only depends on u 0 , T , γ and h H 1 (0,T ) .
To get the Fréchet-differentiability of the unconstrained functional J under Σregularity of u, we recall a fact below.
Let u ∈ Σ * and χ u ∈ Σ be the solution of the equation where u, v Σ * ,Σ is the dual product.
Proof of Theorem 1.2. Assume that h andh = h + φ satisfy the assumption in Proposition 4.2, u andũ are the solutions of (1.1) corresponding to the control h andh respectively. By the definition of Fréchet differentiability, we will show that ). then as φ H 1 (0,T ) → 0, the desired result can be obtained.