On boundary controllability for the higher order nonlinear Schr\"odinger equation

A control problem with terminal overdetermination is considered for the higher order nonlinear Schr\"odinger equation on a bounded interval. The boundary condition on the space derivative is chosen as the control. Results on global existence of solutions under small input date are established.


Introduction
In this paper the higher order nonlinear Schrödinger equation (HNLS) x u = f (t, x), (1.1) posed on an interval I = (0, R), is considered.Here a, b, λ, β, γ are real constants, p 0 , p 1 ≥ 1, u = u(t, x) and f are complex-valued functions (as well as all other functions below, unless otherwise stated).
For an arbitrary T > 0 in a rectangle Q T = (0, T )×I consider an initial-boundary value problem for equation (1.1) with an initial condition u(0, x) = u 0 (x), x ∈ [0, R], (1.2) and boundary conditions u(t, 0) = µ(t), u(t, R) = ν(t), u x (t, R) = h(t), t ∈ [0, T ], (1.3) where the function h is unknown and must be chosen such, that the corresponding solution of problem (1.1)-(1.3)satisfies the condition of terminal overdetermination for given function u T .Equation (1.1) is a generalized combination of the nonlinear Schrödinger equation (NLS) iu t + au xx + λ|u| p u = 0 and the Korteweg-de Vries equation (KdV) It has various physical applications, in particular, it models propagation of femtosecond optical pulses in a monomode optical fiber, accounting for additional effects such as third order dispersion, self-steeping of the pulse, and self-frequency shift (see [5,6,8,9] and the references therein).
The first result on boundary controllability for the KdV equation on a boundary interval appeared in the pioneer paper by L. Rosier [12].In the case b = 1, initial condition (1.2) and boundary conditions (1.3) for µ = ν ≡ 0 it was proved that under small u 0 , u T ∈ L 2 (0, R) there existed a solution under the restriction on the length of the interval , ∀k, l ∈ N.
In paper [2] this result was extended to the truncated HNLS equation with cubic nonlinearity iu t + au xx + ibu x + iu xxx + |u| 2 u = 0 again under homogeneous boundary conditions (1.3), under restriction on the length of the interval and under the conditions b > 0, |a| < 3 (in fact, the equation considered in [2] there was a positive coefficient before the third derivative, but it could be easily eliminated by the scaling with respect to t, which is possible since the time interval was arbitrary).The argument repeated the one from [12].
In the present paper the same result is established for the general HNLS equation (1.1) under non-homogeneous boundary conditions (1.3) and without any conditions on the coefficients a and b.
Note that in the recent paper [4] the inverse initial-boundary value problem (1.1)-(1.3)was considered with an integral overdetermination for given functions ω and ϕ.Either boundary function h or the function F in the right-hand side f (t, x) = F (t)g(t, x) for given function g were chosen as controls.Results on well-posedness under either small input data or small time interval were established.
In [1] a direct initial-boundary value problem on a bounded interval with homogeneous boundary conditions (1.3) for equation (1.1) in the case p 0 = p 1 = p was studied.For p ∈ [1,2] and the initial function u 0 ∈ H s (I), 0 ≤ s ≤ 3, results on global existence and uniqueness of mild solutions were obtained.For u 0 ∈ L 2 (I) the result on global existence was extended either to p ∈ (2, 3) or p ∈ (2, 4), γ = 0. Non-homogeneous boundary conditions were considered in [3] in the real case and nonlinearity uu x .Note also that in [4] there is a brief survey of other results concerning the direct initial-boundary value problems for equation (1.1).
Solutions of the considered problems are constructed in a special functional space Introduce the notion of a weak solution of problem (1.1)-(1.3) , and for all test functions φ(t, x), , and the following integral identity is verified: To describe the properties of the boundary data µ and ν introduce the fractionalorder Sobolev spaces.Let f (ξ) ≡ F [f ](ξ) and F −1 [f ](ξ) be the direct and inverse Fourier transforms of a function f respectively.In particular, for f ∈ S(R) For s ∈ R define the fractional-order Sobolev space and for certain T > 0 let H s (0, T ) be a space of restrictions on (0, T ) of functions from H s (R).Now we can pass to the main result of the paper.
Then there exists δ > 0 such that under the assumption c 0 ≤ δ there exists a function h ∈ L 2 (0, T ) and the corresponding unique solution of problem (1.1)-(1.3)u ∈ X(Q T ) verifying condition (1.4).
Remark 1.4.The smoothness assumption µ, ν ∈ H 1/3 (0, T ) on the boundary data is natural, since if one considers the initial value problem then, by [7], its solution v ∈ C(R; L 2 (R) (which can be constructed via the Fourier transform) satisfies the following relations for any Further we use the following simple interpolating inequality: there exists a constant c = c(R, q) such that for any ϕ ∈ H 1 (I) where the second term in the right-hand side is absent if ϕ ∈ H 1 0 (I).The paper is organized as follows.In Section 2 results on the corresponding linear problem are presented, Section 3 contains the proof of nonlinear results.

Auxiliary linear problem
Besides the nonlinear problem consider its linear analogue and start with the following one with homogeneous boundary conditions Define an operator Lemma 2.1.The operator A generates a continuous semi-group of contractions e tA , t ≥ 0 in L 2 (I).
Proof.This assertion is proved in [2, Lemma 4.1] but under the restriction |a| < 3.However, the slight correction of that proof provides the desired result.In fact, the operator A is obviously closed.Next, for y ∈ D(A) Here, and so the operator A is dissipative.Next, the operator A * y = y ′′′ − iay ′′ + by ′ with the domain Therefore, the operator A * is also dissipative.Application of the Lumer-Phillips theorem (see [11]) finishes the proof.
Remark 2.2.Note that the weak solution of problem (2.1), (2.2) can be considered in the space L 1 (0, T ; L 2 (I)) in the sense of an integral identity valid for any test function from Definition 1.1.Then the general theory of semigroups (see [11]) provides that for Then there exist a unique weak solution to problem (2.1), (2.2) u ∈ X(Q T ) and a function θ ∈ L 2 (0, T ), such that for certain constant c = c(T ), non-decreasing with respect to T , and for a.e.t ∈ (0, T ) ) Proof.First, consider regular solutions in the case Then multiplying equality (2.1) by 2ū(t, x)ρ(x), extracting the imaginary part and integrating one obtains an equality and equality (2.6) provides estimate (2.4) in the regular case.This estimate gives an opportunity to establish existence of a weak solution with property (2.4) in the general case via closure.Moreover, equality (2.6) is also verified.In particular, this equality implies that the function u(t, •)ρ 1/2 2 L2(I) is absolutely continuous on [0, T ] and then (2.5) follows.
Three following lemmas are proved in [2] in the case |a| < 3, b > 0. The proof in the general case is similar, however, we present it here, moreover, in a more transparent way.The first auxiliary lemma is concerned with the properties of the operator A. Since the function y has the compact support, the function y can be extended to the entire function on C. Note that (κ, σ) = (0, 0), otherwise y ≡ 0. The roots of the function κ − σe −iRξ are simple and have the form ξ 0 + 2πn/R for certain complex number ξ 0 and integer number n.Then the roots of the function ξ 3 − aξ 2 − bξ + p must also be simple and coincide with the roots of the numerator.As a result, for certain complex number ξ 0 and natural k, l the roots of the denominator can be written in such a form: Exploiting the Vieta formulas we express ξ 0 from the first one, substitute it into the second one and derive an equality Remark 2.6.It can be shown, that the restriction on the size of the interval is also sufficient for existence of such eigenfunctions, but this is not used further.
Lemma 2.7.For T > 0 let F T denote the space of initial functions u 0 ∈ L 2 (I) Proof.It is obvious that For any T > 0 the set F T is a finite-dimensional vector space, In fact, if u 0n is a sequence in a unit ball {y ∈ F T : y L2(I) ≤ 1} it follows from (2.4) that the corresponding sequence of weak solutions {u n } is bounded in L 2 (0, T ; H 1 (I))) and, therefore, the set . With the use of the continuous embeddings , where the first one is compact, by the standard argument (see [10]) we obtain that the set u n is relatively compact in L 2 (Q T ).Extracting the subsequence, we derive that it is convergent in L 2 (Q T ), whence it follows from (2.8) that the corresponding subsequence of u 0n is convergent in L 2 (I).It means that the considered unit ball is compact and the Riesz theorem (see [13]) implies that the space F T has a finite dimension.Let T ′ > 0 is given.To prove that F T ′ = {0}, it is sufficient to find T ∈ (0, T ′ ) such that F T = {0}.Since the map T → dim(F T ) is non-increasing and step-like, there exist T, ǫ > 0 such that T < T + ǫ < T ′ and dim F T = dim F T +ǫ .Let u 0 ∈ F T and t ∈ (0, ǫ).Since e tA e τ A u 0 = e (t+τ )A u 0 for τ ≥ 0 and u 0 ∈ F T +ǫ , then Let M T = {u = e τ A u 0 : τ ∈ [0, T ], u 0 ∈ F T } ⊂ C([0, T ]; L 2 (I)).Since u ∈ H 1 (0, T + ǫ; H −2 (I)), there exists On the other hand, by (2.10) Therefore, (the last property holds since u ∈ C([0, T ]; H 3 (I))).Hence, has at least one nontrivial eigenfunction, which contradicts Lemma 2.5.
As in the proof of the previous lemma the corresponding sequence of weak solutions {u n } is bounded in L 2 (0, T ; H 1 (I)) and according to (2.9) the sequence u nt is bounded in L 2 (0, T ; H −2 (I)).Again as in the proof of the previous lemma extract a subsequence of {u n }, for simplicity also denoted as {u n }, such that it is convergent in L 2 (Q T ).Then by (2.8) {u 0n } converges in L 2 (I) to certain function u 0 .Inequality (2.7) implies that P u 0n → P u 0 in L 2 (0, T ).Then u 0 L2(I) = 1 and P u 0 L2(0,T ) = 0, which contradicts Lemma 2.7.
Now consider the non-homogeneous linear equation (2.12) The notion of a weak solution to the corresponding initial-boundary value problem with initial and boundary conditions (1.2), (1.3) is similar to Definition 1.1.In particular, the corresponding integral identity (for the same test functions as in Definition 1.1) is written as follows: The following result is established in [4].
Establish a result on boundary controllability in the linear case.
(2.22) Remark 2.13.Note that the function h can not be defined in a unique way.Indeed, choose h = 0 in L 2 (0, T /2).Move the time origin to the point T /2 and for u 0 ≡ S T /2 (0, 0, 0, h, 0, 0) and u T ≡ 0 construct the solution of the corresponding boundary controllability problem, which is, of course, nontrivial.However, h ≡ 0 and u ≡ 0 solve the same problem.
The contraction principle used in the previous proof ensures uniqueness of the solution u only in the ball X r (Q T ).The next theorem provides uniqueness in the whole space X(Q T ), which finishes the proof of Theorem 1.3.Proof.Let u, u ∈ X(Q T ) be two weak solutions of the same problem (1.1)-(1.3).Denote w ≡ u− u, then the function w ∈ X(Q T ) is the weak solution of the problem of (2.1), (2.2) type for f ≡ f 0 − f 1x , where f 0 (t, x) ≡ f 00 (t, x; u) − f 00 (t, x; u) + f 01 (t, x; u) − f 01 (t, x; u), f 1 (t, x) ≡ f 1 (t, x; u) − f 1 (t, x; u), given by formulas (3.5), (3.6).Similarly to the previous proof f 0 ∈ L 1 (0, T ; L 2 (I)), f 1 ∈ L 2 (Q T ).Then the corresponding equality (2.5) in the case ρ(x) ≡ 1 + x yields that