The exact boundary controllability of the higher order nonlinear Schr¨odinger equation with constant coefficients on a bounded domain with various boundary conditions is studied

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EXACT BOUNDARY CONTROLLABILITY FOR HIGHER ORDER NONLINEAR SCHR ¨ODINGER EQUATIONS WITH

CONSTANT COEFFICIENTS

JUAN CARLOS CEBALLOS V., RICARDO PAVEZ F., OCTAVIO PAULO VERA VILLAGR ´AN

Abstract. The exact boundary controllability of the higher order nonlinear Schr¨odinger equation with constant coefficients on a bounded domain with various boundary conditions is studied. We derive the exact boundary con- trollability for this equation for sufficiently small initial and final states.

1. Introduction We consider the initial-value problem

iut+αuxx+iβuxxx+|u|²u= 0, x, t∈R

u(x,0) =u0(x) (1.1)

whereα, β ∈R, β6= 0 anduis a complex valued function. The above equation is a particular case of the equation

iu_t+αu_xx+iβu_xxx+γ|u|²u+iδ|u|²u_x+iu²u_x= 0, x, t∈R

u(x,0) =u₀(x) (1.2)

where α, β, γ, δ, with β 6= 0 and uis a complex valued function. This equation was first proposed by Hasegawa and Kodama [10] as a model for the propagation of a signal in a fiber optic (see also [13]). The equation (1.2) can be reduced to other well known equations. For instance, settingα= 1,β ==γ= 0 in (1.2) we have the semi linear Schr¨odinger equation, i. e.,

ut−iuxx−iγ|u|²u= 0. (1.3) If we letβ=γ= 0 andα= 1 in (1.2) we obtain the derivative nonlinear Schr¨odinger equation

u_t−iu_xx−δ|u|²u_x−u²u_x= 0. (1.4)

2000Mathematics Subject Classification. 35K60, 93C20.

Key words and phrases. KdVK equation; boundary control; Hilbert uniqueness method;

Ingham’s inequality; smoothing properties.

c

2005 Texas State University - San Marcos.

Submitted January 20, 2005. Published October 31, 2005.

J. C. Ceballos was supported by grant 0528081/R from Proyectos de Investigacion Internos, Universidad del B´ıo-B´ıo. Concepci´on. Chile.

1

Lettingα=γ = = 0 in (1.2), the equation that arises is the complex modified Korteweg-de Vries equation,

u_t+βu_xxx+δ|u|²u_x= 0. (1.5) The initial-value problem for the equations (1.3), (1.4) and (1.5) has been exten- sively studied, see for instance [1, 8, 14, 18, 20, 21, 24, 26] and references therein.

In 1992, Laurey [17] considered the equation (1.2) and proved local well-posedness of the initial-value problem associated for data inH^s(R) withs >3/4, and global well-posedness in H^s(R) where s ≥ 1. In 1997, Staffilani [30] established local well-posedness for data inH^s(R) withs≥1/4, improving Laurey’s result. Similar results were given in [6, 7] for (1.2) wherew(t),β(t) are real functions.

For the case of the (1.1) if we consider the Gauge transformation u(x, t) =e^iα3x+i2α

3

27v(x−α²

3 t, t)≡e^θv(η, ξ) whereθ=i^α₃x+i2^α₂₇³, η=x−^α₃²tandξ=t,then

u_t=i2α³

27e^θv−α²

3 e^θv_η+e^θv_ξ uxx=−α²

9 e^θv+i2

3αe^θvη+e^θvηη

u_xxx=−iα³

27e^θv−1

3α²e^θv_η+iαe^θv_ηη+e^θv_ηηη.

Replacing in (1.1) and consideringβ = 1(rescaling the equation) we obtain iv_ξ+iv_ηηη+|v|²v− 4

27α³v= 0, x, t∈R v(x,0) =v0(x)≡u0(x)e^−iα3

(1.6) Thus (1.1) is reduced to a complex modified Korteweg-de Vries type equation. In this paper, we consider the boundary control of the Schr¨odinger equation

iu_t+αu_xx+iβu_xxx+|u|²u+iδu_x= 0 (1.7) whereα, β, δ∈R,β6= 0 and uis a complex valued function on the domain (a, b), t >0,and with the boundary condition

u(a, t) =h₀, u(b, t) =h₁, u_x(a, t) =h₂, u_x(b, t) =h₃. (1.8) In this paper we want to study directly the exact boundary controllability problem for the higher order Schr¨odinger equation by adapting the method of [21] which combines the Hilbert Uniqueness Method (HUM) and multiplier techniques. This method has been successfully applied to study controllability of wave and plate equations, Schr¨odinger and KdV equations (see for instance [1, 8, 9, 11, 14, 15, 18, 20, 22, 24] and references therein). The first result of this paper concerns boundary controllability of the higher order linear Schr¨odinger equation.

Theorem 1.1. Let H_p² ={w ∈H²(0,2π) : w(0) =w(2π), w⁰(0) =w⁰(2π)} and T > 0. Then, for any y0, yT ∈ (H_p²)⁰ (the dual space of H_p²), there exist hk ∈ L²(0, T)(k= 0,1,2) such that the solution y ∈C([0, T] : (H_p²)⁰) of the boundary initial-value higher order Schr¨odinger equation

iy_t+iβy_xxx+αy_xx= 0, (x, t)∈(0,2π)×(0, T); (1.9)

∂_x^ky(2π, t)−∂_x^ky(0, t) =hk(t), k= 0,1,2; (1.10)

y(.,0) =y0 (1.11)

satisfiesy(., T) =yT.

We see that explicit controls may be given. Unfortunately, the state y is only known to belong to C([0, T] : (H_p²)⁰) so it seems quite difficult to deduce from Theorem 1.1 controllability results for higher order nonlinear Schr¨odinger equation (1.7).

The second result relates exact boundary controllability for the linear higher order Schr¨odinger equation with boundary control on yx at x = L. In this part a condition on the coefficients α and β given by the second and the third order derivatives that appear in (HSCHROD) is needed. A condition on the length L of the domain appears.

Theorem 1.2. Let |α|<3β,δ >0and N =n

2πβ s

k²+kl+l²

3βδ+α² :k, l∈N^∗ o.

Then for anyT >0andL∈(0,+∞)\N, and for anyy₀, y_T ∈L²(0, L), there exists h∈L²(0, T)such that the mild solutiony∈C([0, T] :L²(0, L))∩L²(0, T :H¹(0, L)) of the system

iy_t+iβy_xxx+αy_xx+iδy_x= 0 (1.12)

y(0, t) =y(L, t) = 0 (1.13)

yx(L, t) =h(t) (1.14)

y(x,0) =y₀(x) (1.15)

satisfiesy(., T) =yT.

To prove this we use the Hilbert uniqueness method and the multiplier method.

It turns out that the study of (1.12)-(1.15) as a boundary initial-value problem is more delicate than the study of (1.9)-(1.11), and -because of the extra termy_x in (1.14)- the observability result holds true if and only ifL /∈ N. On the other hand, the solutionybelongs this time to a functional space in which we may give a sense to the nonlinear term |y|²y in (1.1). By means of the Banach Contraction Fixed Point Theorem and Theorem 1.2 we get the main result of the paper, that is the exact boundary controllability of the higher order nonlinear Schr¨odinger equation on a bounded domain.

Theorem 1.3. Let |α|<3β,δ >0, T >0 andL >0. Then, there exists r0>0 such that for any y0,yT ∈L²(0, L)with ky0k_L2(0,L)< r0,kyTk_L2(0,L) < r0, there is functiony in

C([0, T] :L²(0, L))∩L²([0, T] :H¹(0, L))∩W^1,1([0, T] :H⁻²(0, L)) (1.16) which is a solution of

iy_t=−(iβyxxx+αy_xx+|y|²y+iδy_x) inD^,(0, T :H⁻²(0, L)) (1.17)

y(0, .) = 0 inL²(0, L) (1.18)

and such that y(.,0) = y0,y(., T) =yT. If moreover L /∈ N, then in addition, it is possible to assume thaty(L, .) = 0 inL²(0, T)and take y_x(L, .)in L²(0, T)as a control function.

In a forthcoming paper we study the case |α| ≥ 3β for Theorems 1.2 and 1.3 using the Gauge transformation (KdVm described above) and following the same idea shown here.

This paper is organized as follows: Section 2 outlines briefly the notation and terminology to be used subsequently and some previous result. Section 3 we derive from the Hilbert uniqueness method a direct proof of the exact controllability result for the higher order linear Schr¨odinger equation. In section 4, we consider another boundary controllability problem for the higher order linear Schr¨odinger equation, in which only the value of the first spatial derivative (at x = L) of the state function is assumed to be controlled: this boundary initial-value problem is first shown to admit solutions, later on, an observability result is given and used to show using the Hilbert uniqueness method the exact boundary controllability for higher order linear Schr¨odinger equation with these boundary conditions. Finally, in section 5, we prove the main result of this paper, that is, the exact local boundary controllability of the higher order nonlinear Schr¨odinger equation on a bounded domain.

2. Preliminaries

For an arbitrary Banach spaceX, the associated norm will be denoted byk · kX. If Ω = (a, b) is a bounded open interval andka non-negative integer, we denote by C^k(Ω) =C^k(a, b) the functions that, along with their firstk ones, are continuous on [a, b] with the norm

kfk_Ck(Ω)= sup

x∈Ω,0≤j≤k

|f^(j)(x)|. (2.1)

As usual, D(Ω) is the subspace of C^∞(Ω) consisting of functions with compact support in Ω. Its dual space D⁰ is the space of Schwartz distributions on Ω. For 1≤p <∞,L^p(Ω) denotes those functionsfwhich arep-power absolutely integrable on Ω with the usual modification n case p = ∞. If s ≥ 0 is an integer and 1 ≤ p ≤ ∞, W^s,p(Ω) is the Sobolev space consisting of those L^p(Ω)-functions whose firstsgeneralized derivatives lie inL^p(Ω), with the usual norm

kfk^p_Ws,p(Ω)=

s

X

k=0

kf^(k)k^p_Lp(Ω). (2.2) Ifp= 2 we writeH²(Ω) forW^s,2(Ω). The notationH^s(Ω) is frequent wheresis a positive integer.

k · ks=k · k_Hs(a,b). (2.3) Fors≥1,H₀^s((a, b)) is the closed linear subspace ofH^s((a, b)) of functionsf such that f(a) =f⁰(a) =· · ·=f^s−1(a) = 0. H_loc^s (Ω) is the set of real-valued functions f defined on Ω such that, for eachϕ∈ D(Ω),ϕf ∈H^s(Ω). This space is equipped with the weakest topology such that all of the mapping f 7→ ϕf, for ϕ ∈ D(Ω), are continuous fromH^s(Ω) intoH_loc^s (Ω). With this topology,H_loc^s (Ω) is a Fr´echet space. IfX is a Banach space,T a positive real number and 1≤p≤+∞, we will denote byL^p(0, T;X) the Banach space of all measurable functionsu: (0, T)7→X, such thatt7→ ku(t)kX is inL^p(0, T),with the norm

kuk_Lp(0,T;X)=Z T 0

ku(t)k^p_Xdx^1/p

if 1≤p <+∞,

and ifp=∞, then

kuk_L^∞_(0,T;X)= sup

0<t<T

kukX.

Similarly, if k is a positive integer, then C^k(0, T : X) denote the space of all continuous functionsu: [0, T]7→X, such that their derivatives up to thek order exist and are continuous.

For notation, we write∂=∂/∂x,∂t=∂/∂t anduj =∂^j_xu=∂^ju/∂x^j. Definition. Fork={2,3}, we define the space

H_p^k =n

u∈H^k(0,2π) : d^ju

dx^j(0) = d^ju

dx^j(2π) for 0≤j≤k−1o

We remark thatH^k(0,2π) denotes the classical Sobolev space on the interval (0,2π).

Definition. Forn∈Z, let then-th Fourier coefficient ofu∈L²(0,2π),

bu(n) = 1 2π

Z 2π

0

e^−inxu(x)dx (2.4)

Lemma 2.1. Forn∈Z, we have X

n∈Z

|u(n)|b ²= 1 2π

Z 2π

0

|u(x)|²dx (2.5)

The proof of the above lemma is straightforward. We remark that for k = 2 (similarly fork= 3) we have

u(n) =b 1 2π

Z 2π

0

e^−inxu(x)dx=−1

n²∂²bu(x)

then −n²u(n) =b ∂²u(n). Applyingb | · | and squaring we obtain [n²|bu(n)|²]² =

|∂²bu(n)|² where by applyingP

n∈Z and using (2.2) it follows that X

n∈Z

[n²|u(n)|b ²]²=X

n∈Z

|∂²u(n)|b ²= 1 2π

Z 2π

0

|∂²u(x)|²dx <∞.

Hence, we have that for allu∈L²(0,2π),k∈ {2,3}

u∈H_p^k if and only if X

n∈Z

[n^k|bu(n)|²]²<∞, (2.6) and the Sobolev norm

kuk_Hk(0,2π)=hX^k

j=0

Z 2π

0

|∂^ju(x)|²dxi^1/2

=hX^k

j=0

k∂^juk²_L2(0,2π)

i^1/2

reduces to

kuk_Hk(0,2π)=h X

n∈Z

(1 +n²+. . .+n^2k)|bu(n)|²i^1/2

foru∈H_p^k. (2.7) In what follows, the Hilbert spaceH_p^k is endowed with the norm kuk_Hk(0,2π).

Lemma 2.2 (Ingham’s Inequality [12]). Assume the strictly increasing sequence {λk}k∈Zof real numbers satisfies the “gap” conditionλk+1−λk≥γ, for allk∈Z, for some γ > 0. Then, for all T > 2π/γ there are two positive constants C1, C2

depending only onγ andT such that C1(T, γ)

∞

X

k=−∞

|ak|²≤ Z T

0

∞

X

k=−∞

ake^itλk

dx≤C2(T, γ)

∞

X

k=−∞

|ak|² (2.8) for every complex sequence (ak)_k∈Z∈l², where

C1(T, γ) =2T

π 1− 4π² T²γ²

>0, C2(T, γ) = 8T

π 1 + 4π² T²γ²

>0

(2.9)

andl²is the Hilbert space of square summable sequences, sequences {ak} such that P

k∈N|ak|²<∞.

Finally, we denote by c, a generic constant, not necessarily the same at each occasion, which depends in an increasing way on the indicated quantities.

3. Exact boundary controllability of the higher order linear Schr¨odinger equation by means of control on data [∂^ky(., t)]^2π₀ for

k= 0,1,2

For simplicity, in this section, we restrict ourselves to the case where the space domain [0, L] is [0,2π]; although Theorem 1.1 holds for arbitraryL >0.

Lemma 3.1. Let A denote the operator Au = (−β∂³ +iα∂²)u on the domain D(A) = H_p³ ⊆L²(0,2π). Then A generates a strongly continuous unitary group (S(t))_t∈R onL²(0,2π).

Proof. LetA:D(A)⊆L²(0,2π)7→L²(0,2π) such thatu7→Au=−β∂³u+iα∂²u.

We have

hAu, vi=h−β∂³u+iα∂²u, vi

=−βh∂³u, vi+iαh∂²u, vi

=βhu, ∂³vi+iαhu, ∂²vi

=hu, β∂³vi+hu,−iα∂²vi

=hu,−(−β∂³v+iα∂²v)i

=hu,−Avi

thenA^∗ =−A. Hence, by the Stone theorem [25],A is the infinitesimal generator of a unitary group of class C0 (all groups of class C0 are strongly continuous) on

L²(0,2π).

Definition. LetT > 0. For uT =P

n∈Zcne^int∈L²(0,2π), the mild solution of the uncontrolled problem

∂_tu+β∂³u−iα∂²u= 0, x∈(0,2π), t∈R;

∂^ku(0, t) =∂^ku(2π, t), k= 0,1,2;

u(., T) =u_T(.)

(3.1)

is given by

u(x, t) =X

n∈Z

c_ne^{i(βn3−αn2)(t−T)+inx} (3.2) Remark 3.2. Letu(x, t) =P

n∈Zu(n, t)eb ^inx, then u(x, t) =X

n∈Z

cne^{i[(βn3−αn2)(t−T)+nx]}. In fact,

∂tu(x, t) =X

n∈Z

∂tu(n, t)eb ^inx

∂²u(x, t) =X

n∈Z

(in)²bu(n, t)e^inx=−X

n∈Z

n²bu(n, t)e^inx

∂³u(x, t) =X

n∈Z

(in)³u(n, t)eb ^inx =−iX

n∈Z

n³u(n, t)eb ^inx, hence, ifuis the solution of (3.1), we obtain

X

n∈Z

∂_tu(n, t)eb ^inx−iβX

n∈Z

n³u(n, t)eb ^inx+iαX

n∈Z

n²bu(n, t)e^inx= 0.

Multiplying bye^−imx(m∈Z) and integrating overx∈(0,2π) we obtain X

n∈Z

∂tu(n, t)b −i(βn³−αn²)u(n, t)b Z 2π

0

e^i(n−m)xdx= 0.

Using that

Z 2π

0

e^i(n−m)xdx=

(0, ifn6=m 2π, ifn=m we have that P

n∈Z∂_tbu(n, t)−i(βn³−αn²)u(n, t) = 0, thenb ∂_tbu(n, t)−i(βn³− αn²)u(n, t) = 0 whereb

∂_th

e^{−i(βn3−αn2)t}u(n, t)b i

= 0.

Integrating overt∈[0, T] yields

u(n, t) =b bu(n,0)e^{i(βn3−αn2)t} multiplying bye^inx and applyingP

n∈Z we obtain u(x, t) =X

n∈Z

bu(n, t)e^inx

=X

n∈Z

bu(n,0)e^{i[(βn3−αn2)t+nx]}

=X

n∈Z

bu(n,0)e^{i(βn3−αn2)T}e^{i[(βn3−αn2)(t−T)+nx]}

=X

n∈Z

cne^{i(βn3−αn2)(t−T)+inx}. wherec_n=bu(n,0)e^{i(βn3−αn2)T} andu(x, T) =u_T =P

n∈Zc_ne^inx.

For the rest of this article,uwill denote the solution of (3.1) associated withu_T. We show the following result for the non-homogeneous problem.

Theorem 3.3. Let H_p² = {w ∈H²(0,2π) : w(0) = w(2π), w⁰(0) = w⁰(2π)} and T > 0. Then for any y0, yT ∈ (H_p²)⁰ (the dual space of H_p²), there exist hk ∈ L²(0, T)(k= 0,1,2) such that the solution y ∈C([0, T] : (H_p²)⁰) of the boundary initial-value higher order Schr¨odinger equation

∂ty+β∂³y−iα∂²y= 0, (x, t)∈(0,2π)×(0, T);

∂^ky(2π, t)−∂^ky(0, t) =hk(t), k= 0,1,2;

y(.,0) =y0

(3.3)

satisfiesy(., T) =y_T.

Remark 3.4. Given y₀ ∈ (H_p²)⁰, h_k ∈ L²(0, T) (k = 0,1,2), we want to find y such that it satisfies (3.3). We first prove that (3.3) admits a unique solution y ∈ C([0, T] : (H_p²)⁰) in a certain sense, and this solution is the classical one whenevery∈D(A), andhk(k= 0,1,2) are smooth enough and vanish at 0.

Lemma 3.5. (1) Assume that hk ∈C₀²([0, T]) ={h∈C²([0, T] :C) :h(0) = 0}

and y0 ∈ H_p³. Then there exists a unique solution y ∈ C([0, T] : H³(0,2π))∩ C¹([0, T] :L²(0,2π)) of (3.3). Moreover, for any u_T ∈H_p³ and any t ∈[0, T] we have

Z 2π

0

u(x, t)y(x, t)dx

= Z 2π

0

u(x,0)y₀(x)dx−(β−iα) Z t

0

∂²u(0, s)h₀(s)ds +β

Z t

0

∂u(0, s)h1(s)ds+ Z t

0

u(0, s)(βh2(s) +iαh1(s))ds.

(3.4)

(2)ForuT ∈H_p²,u∈C([0, T] :H_p²)and∂²u(0, .) makes sense inL²(0, T).

(3)Assume now thaty0∈(H_p²)⁰ andhk∈L²(0, T)(k= 0,1,2). Then, there exists a uniquey∈C([0, T] : (H_p²)⁰)such that for all u_T ∈H_p² and for allt∈[0, T],

hu(., t), y(t)i_H2 p×(H_p²)⁰

=hu(.,0), y₀iH_p²×(H_p²)⁰−(β−iα) Z t

0

∂²u(0, s)h₀(s)ds +β

Z t

0

∂u(0, s)h1(s)ds+ Z t

0

u(0, s)(βh2(s) +iαh1(s))ds

(3.5)

Proof. (1)Letφ_i ∈C^∞([0,2π])(i= 0,1,2) be such that φ^(k)_i (0) = 0 and φ^(k)_i (2π) =

(−1, i=k 0, i6=k.

We consider the change of functionz(x, t) =P2

i=0[h_i(t)φ_i(x)+(S(t)y₀)(x)+y(x, t)], then

z(2π, t)−z(0, t) =

2

X

i=0

hi(t)φi(2π) + (S(t)y0)(2π) +y(2π, t)

−

2

X

i=0

hi(t)φi(0) + (S(t)y0)(0) +y(0, t)

=−h₀(t) + (S(t)y₀)(2π)−(S(t)y₀)(0) +y(2π, t)−y(0, t)

=−h0(t) + (S(t)y0)(2π)−(S(t)y0)(0) +h0(t)

= (S(t)y0)(2π)−(S(t)y0)(0)

using thaty0 ∈ H_p³ we obtain z(2π, t) = z(0, t). The other initial conditions are calculated in a similar way. Hence, this change of the function yields an equivalent problem to (3.3): Findz such that

∂tz+β∂³z−iα∂²z=f(x, t)

=

2

X

i=0

h

h⁰_i(t)φi(x) +βhi(t)φ⁽³⁾_i (x)−iαhi(t)φ⁽²⁾_i (x)i

∂^kz(2π, t) =∂^kz(0, t), k= 0,1,2 z(.,0) = 0

(3.6)

Since f ∈ C¹([0, T] : L²(0,2π)), this non-homogeneous problem admits a unique solution (see [25]),z∈C([0, T] :H_p³)∩C¹([0, T] :L²(0,2π)). This proves the first assertion in (1).

LetuT ∈ H_p³, thenu∈C([0, T] :H_p³)∩C²([0, T] : L²(0,2π)). Multiplying the equation (3.1) byy and integrating inx∈[0,2π] and t∈[0, T] we have

Z t

0

Z 2π

0

y[∂su]dx ds+β Z t

0

Z 2π

0

y[∂³u]dx ds−iα Z t

0

Z 2π

0

y[∂²u]dx ds= 0.

Each term is treated separately. Integrating by parts, Z t

0

Z 2π

0

y[∂su]dx ds

= Z 2π

0

y(x, t)u(x, t)dx− Z 2π

0

y(x,0)u(x,0)dx− Z t

0

Z 2π

0

[∂sy]u dx ds ,

β Z t

0

Z 2π

0

y[∂⁵u]dx ds=β Z t

0

h₀(s)[∂²u(0, s)]ds−β Z t

0

h₁(s)[∂u(0, s)]ds +β

Z t

0

h₂(s)u(0, s)ds− Z t

0

Z 2π

0

[∂³y]u dx ds ,

−iα Z t

0

Z 2π

0

y[∂²u]dx ds

=−iα Z t

0

h0(s)[∂²u(0, s)]ds+iα Z t

0

h1(s)u(0, s)ds−iα Z t

0

Z 2π

0

[∂²y]u dx ds . Therefore,

Z 2π

0

y(x, t)u(x, t)dx− Z 2π

0

y(x,0)u(x,0)dx− Z t

0

Z 2π

0

[∂_sy]u dx ds +β

Z t

0

h0(s)[∂²u(0, s)]ds−β Z t

0

h1(s)[∂u(0, s)]ds+β Z t

0

h2(s)u(0, s)ds

− Z t

0

Z 2π

0

[∂³y]u dx ds−iα Z t

0

h0(s)[∂²u(0, s)]ds+iα Z t

0

h1(s)u(0, s)ds

−iα Z t

0

Z 2π

0

[∂²y]u dx ds= 0,

where

Z 2π

0

u(x, t)y(x, t)dx

= Z 2π

0

u(x,0)y₀(x)dx−(β−iα) Z t

0

[∂²u(0, s)]h₀(s)ds +β

Z t

0

[∂u(0, s)]h1(s)ds+ Z t

0

u(0, s)(βh2(s) +iαh1(s))ds.

Result (1) follows.

Now, we proof (2). By (3.2), fort1, t2∈[0, T] u(x, t1) =X

n∈Z

cne^{i(βn3−αn2)(t1−T)+inx}, u(x, t2) =X

n∈Z

cne^{i(βn3−αn2)(t2−T)+inx}; hence

u(x, t1)−u(x, t2)

=X

n∈Z

cne^{i(βn3−αn2)T} e^{i(βn3−αn2)t1}−e^{i(βn3−αn2)t2} e^inx. From (2.3), if u_T ∈ H_p² then P

n∈Z|n²c_n|²<∞ and P

n∈Z|ncn|²<∞. Using Lebesgue’s Theorem [27],

|u(x, t1)−u(x, t2)|=X

n∈Z

(n²+n)cn e^{i(βn3−αn2)t1}−e^{i(βn3−αn2)t2}

2

which approaches 0 ast₁→t₂. We conclude thatu∈C([0, T] :H_p²). Henceu(0, .),

∂u(0, .) exist in C([0, T])⊆L²(0, T). The same argument shows that ifuT ∈H_p³, u∈C([0, T] :H_p³) and

∂²u(0, t) =X

n∈Z

−n²cne^{−i(βn3−αn2)T}

e^{i(βn3−αn2)t}. (3.7) The sum in (3.7) makes sense in L²(0, T) wherever P

n∈Z(n²|cn|)²<∞, that is, u_T ∈H_p². From now on,∂²u(0, .) denotes foru_T ∈H_p², the sum in (3.7).

Remark 3.6. The linear mapu_T 7→∂²u(0, .) is continuous since

X

n∈Z

n²cne^{−i(βn3−αn2)T}

e^{i(βn3−αn2)t} ≤ [T

2π] + 1 X

n∈Z

[n²|cn|]² (3.8) where [x] denotes the integral part of a real numberx. IdentifyingL²(0,2π) with its dual by means of the conjugate linear mapy7→ h., yi_L2(0,2π), we have the following dense and compact embedding (see [23])

H_p²,→L²(0,2π),→(L²(0,2π))⁰ ,→(H_p²)⁰. (3.9) Moreover,

hu, yi_H2

p×(H_p²)⁰ =hu, yi_L2(0,2π)= Z 2π

0

uy dx (3.10)

foru∈H_p² andy∈L²(0,2π). Then hu(., t), y(t)i_H2

p×(H²_p)⁰

=hu(.,0), y0i_H2

p×(H_p²)⁰−(β−iα) Z t

0

[∂²u(0, s)]h0(s)ds +β

Z t

0

∂u(0, s)h₁(s)ds+ Z t

0

u(0, s)(βh₂(s) +iαh₁(s))ds

forhk ∈C₀²([0, T]) (k = 0,1,2) andy0, uT ∈H_p³. SinceH_p³ is dense in H_p², using (2), we see that (3.5) also is true foru_T ∈H_p².

Definition.For y0 ∈ (H_p²)⁰ and hk ∈ L²(0, T) (k = 0,1,2), we define a weak solution of (3.3) as a function y ∈ C([0, T] : (H_p²)⁰) such that (3.5) holds for all uT ∈H_p² and allt∈[0, T].

Claim. Fortfixed in [0, T], (3.5) definesy(t)∈(H_p²)⁰ in a unique manner.

In fact, from the proof of (2) the map Ξ :H_p²→C,uT 7→Ξ(uT), given by Ξ(u_T) =−(β−iα)

Z t

0

h₀(s)[∂²u(0, s)]ds+β Z t

0

h₁(s)[∂u(0, s)]ds +

Z t

0

(βh2(s)−iαh1(s))u(0, s)ds

is a continuous linear form. On the other hand, the map Φ : H_p² 7→ H_p² with uT → Φ(uT) = u(., t) is an automorphism of the Hilbert space, hence, for each t∈[0, T], y(t) is uniquely defined in (H_p²)⁰. Moreover, fort∈[0, T],

ky(t)k_(H2

p)⁰= sup

ku(.,t)k_H2 p≤1

|hu(., t), y(t)i|

= sup

ku(.,t)k_H2 p≤1

|hu(.,0), y₀iH_p⁴×(H_p²)⁰−(β−iα) Z t

0

[∂²u(0, s)]h₀(s)ds

+β Z t

0

[∂u(0, s)]h1(s)ds+ Z t

0

u(0, s)(βh2(s) +iαh1(s))ds|

≤ sup

ku(.,t)k_H2 p≤1

|hu(.,0), y0i_H2 p×(H_p²)⁰|

+ (|β|+|α|) sup

ku(.,t)k_H2 p≤1

Z t

0

|h0(s)[∂²u(0, s)]|ds

+|β| sup

ku(.,t)k_H2 p≤1

Z t

0

|h1(s)[∂u(0, s)]|ds

+ sup

ku(.,t)k_H2 p≤1

Z t

0

|(βh2(s)−iαh1(s))u(0, s)|ds

≤ sup

ku(.,t)k_H2 p≤1

ku(.,0)k(H²_p)⁰ky0kH_p²

+ (|β|+|α|) sup

ku(.,t)k_H2 p≤1

kh₀(t)k_L2(0,T)k∂²u(0, t)k_L2(0,T)

+|β| sup

ku(.,t)k_H2 p≤1

kh1(t)k_L2(0,T)k∂u(0, t)k_L2(0,T)

+ sup

ku(.,t)k_H2 p≤1

k(βh2(s)−iαh1(s))k_L2(0,T)ku(0, t)k_L2(0,T)

≤c ky0k(H_p²)⁰+kh0kL²(0,T)+kh1kL²(0,T)+kh2kL²(0,T)

where c is a positive constant which does not depend on t or on y0, h0, h1, h2. Since

y∈C([0, T] :L²(0,2π))⊆C([0, T] : (H_p²)⁰) (3.11) for y ∈H_p³ and (h0, h1, h2)∈ [C₀²([0, T])]³, and since H_p³ is dense inL²(0, T) and C₀²([0, T]) is dense in L²(0, L), it follows from (3.11) thaty∈C([0, T] : (H_p²)⁰).

Lemma 3.7 (Observability result). Let T >0. There exist positive numbersC₁^T, C₂^T such that for everyuT ∈H_p²

C₁^TkuTk²_H2

p(0,2π)≤ ku(0, .)k²_L2(0,T)+k∂u(0, .)k²_L2(0,T)+k∂²u(0, .)k²_L2(0,T)

≤C₂^TkuTk²_H2 p(0,2π)

(3.12) Proof. InL²(0, T) we have that

u(0, t) =X

n∈Z

c_ne^{i(βn3−αn2)(t−T)}

∂u(0, t) =X

n∈Z

incne^{i(βn3−αn2)(t−T)}

∂²u(0, t) =X

n∈Z

−n²cne^{i(βn3−αn2)(t−T)}. Hence

ku(0, t)k²_L2(0,T)+k∂u(0, t)k²_L2(0,T)+k∂²u(0, t)k²_L2(0,T)

≤ [T

2π] + 1 X

n∈Z

(1 +n²+n⁴)|cn|²

≤C₂^T ku_Tk²_H2

p(0,2π) foru_T ∈H_p²

(3.13)

where C₂^T = ([_2π^T ] + 1). To prove the left inequality we first take T⁰ ∈(0, T) and γ >2π/T⁰. LetN ∈N^∗ be such that

n∈Z, |n| ≥N ⇒[β(n+ 1)⁵−α(n+ 1)³]−[βn⁵−αn³]≥γ.

By Ingham’s inequality [12] there existsc^T0 >0 such that for all sequences (an)_n∈Z inl²(Z),

X

|n|≥N

|an|²≤c^T0 Z T⁰

0

X

|n|≥N

a_ne^{i(βn3−αn2)(t−T)}

2

dt. (3.14)

LetZ_n= Span(e^inx) forn∈ZandZ =⊕_n∈ZZ_n ⊆H_p². We define a semi-normp inZ by: ∀u∈ Z,

p(u) = (|u(0)|²+|∂u(0)|²+|∂²u(0)|²)^1/2

=

X

n∈Z

u(n)b

2+

X

n∈Z

inbu(n)

2+

X

n∈Z

−n²bu(n)

2^1/2 (3.15) (Foru∈ Z, bu(n) = 0 for|n|large enough).

LetuT ∈ Z ∩(⊕_|n|<NZn)^⊥, that is,cn= 0 for|n|< N or for|n| large enough.

Using (3.2) and (3.14) we have kuTk²_H2

p((0,2π))= X

n≥N

(1 +n²+n⁴)|cn|²≤c^T0 Z T

0

[p(u(., t))]²dt. (3.16)

SinceT > T⁰, it follows from (3.13), (3.16) and a result by Komornik (see [14]) that there exists a constantC₁^T >0 such that for alluT in Z,

C₁^TkuTk²_H2 p(0,2π)≤

Z T

0

[p(u(., t))]²dt

=ku(0, .)k²_L2(0,T)+k∂u(0, .)k²_L2(0,T)+k∂²u(0, .)k²_L2(0,T)

(3.17)

and the result follows.

We remark that by a density argument we obtain the left inequality in (3.12) in the general case (uT ∈H_p²).

Proof of Theorem 3.3. Without loss of generality we may assume thaty0 = 0. In fact, if y0, yT ∈ (H_p²)⁰, if there exist hk ∈ L²(0, T) (k = 0,1,2) such that the weak solution ey of (3.3) and y(.,e 0) = 0 satisfies ey(., T) = yT −S(T)y0, then yT =S(T)y0+ey(., t) is the weak solution of (3.3) with the same control functions and its such thaty(., T) =yT. In what follows we assume thaty0= 0. ForuT ∈H_p² we let Λ :H_p²7→(H_p²)⁰,

u_T 7→Λ(u_T) =y_T.

where y is the weak solution of (3.3) and hk(k= 0,1,2) are chosen the following way:

h0(t) = −1

(β+iα)∂²u(0, t), h1(t) = 1

β∂u(0, t), h2(t) =i1

βu(0, t) +iα

β²∂u(0, t)

As aboveustands for the solutions of (3.1) associated withu_T. Clearly Λ :H_p²7→

(H_p²)⁰ is a conjugate linear continuous map. Moreover huT,Λ(uT)iH²_p×(H_p²)⁰ =

Z T

0

(|u(0, t)|²+|∂u(0, t)|²+|∂²u(0, t)|²)dt

≥C₁^TkuTk²_H2 p(0,2π).

By Lemmas 3.5 and 3.7 it follows from Lax-Milgram’s Theorem (see [34]) that Λ is

invertible. Then the theorem follows.

Remark 3.8. IfT = 2π, Lemma 3.7 is trivial. Indeed, for anyu_T ∈H_p², ku_Tk²_H4

p(0,2π)=ku(0, .)k²_L2(0,2π)+k∂u(0, .)k²_L2(0,2π)+k∂²u(0, .)k²_L2(0,2π). 4. Exact boundary controllability of the higher order linear

Schr¨odinger equation by means of the control ∂y(L, t) We consider now, the scalar spaceR. In this section,Lstands for some positive number. We shall prove the controllability inL²(0, L) of

∂_ty+β∂³y−iα∂²y+δ∂y= 0 y(0, t) =y(L, t) = 0

∂y(L, t) =h(t) y(.,0) =y₀

(4.1)

where h∈L²(0, T) stands for the control function. More precisely we shall prove that, for anyL >0,T >0,y0,yT ∈L²(0, L) there exists h∈L²(0, T) such that a mild solution

y∈C([0, T] :L²(0, L))∩L²(0, T :H¹(0, L))∩H¹(0, T :H⁻²(0, L)) (4.2) of (4.1) which verifies the equation (4.1) inD⁰(0, T :H⁻²(0, L)) andy0inL²(0, L) may be found such thaty(., T) =yT.

We begin by showing the well-posedness of the initial-value homogeneous prob- lem with|α|<3β

∂ty+β∂³y−iα∂²y+δ∂y= 0 y(0, t) =y(L, t) = 0

∂y(L, t) = 0 y(.,0) =y0.

(4.3)

Let A denote the operator Aw = −βw⁰⁰⁰+iαw⁰⁰−δw⁰ on the (dense) domain D(A)⊆L²(0, L), defined by

D(A) ={w∈H³(0, L) :w(0) =w(L) =w⁰(L) = 0}

Lemma 4.1. OperatorAgenerates a strongly continuous semigroup of contractions onL²(0, L).

Proof. Ais closed. Letw∈D(A). Then Rehw, Awi_L2(0,L)

= Re Z L

0

[−βw⁰⁰⁰+iαw⁰⁰−δw⁰]w(x)dx

= Reh

−β Z L

0

w⁰⁰⁰(x)w(x)dx+iα Z L

0

w⁰⁰w(x)dx−δ Z L

0

w⁰(x)w(x)dxi . Each term is treated separately. Integrating by parts,

Z L

0

w⁰⁰⁰(x)w(x)dx= 1

2[w⁰(0)]², Z L

0

w⁰⁰(x)w(x)dx=− Z L

0

[w⁰(x)]²dx Then

Rehw, Awi_L2(0,L)=−β

2[w⁰(0)]²≤0 if β > 1 3|α|

hence,Ais dissipative. It can be seen thatA^∗(w) =βw⁰⁰⁰−iαw⁰⁰+δw⁰with domain D(A^∗) ={w∈H⁵(0, L) :w(0) =w(L) =w⁰(0) = 0}, so that

Rehw, A^∗wiL²(0,L)=−β

2[w⁰(L)]²≤0, ifβ >1 3|α|

andA^∗is dissipative. Hence, by the Lumer-Phillips Theorem,Ais the infinitesimal generator of aC0semigroup of contractions onL²(0, L). The result follows.

We denote by (S(t))_t≥0the semi-group of contractions associated withA, and we letHdenote the Banach spaceC([0, T] :L²(0, L))∩L²([0, T] :H¹(0, L)) endowed

with the norm

kyk_H= sup

t∈[0,T]

ky(., t)kL²(0,L)+Z T 0

ky(., t)k²_H1(0,L)dt^1/2

= sup

t∈[0,T]

ky(., t)k_L2(0,L)+ky(., t)k_L2(0,T:H¹(0,L)).

(4.4)

Using the multiplier method, we get useful estimates for the mild solutions of (4.3).

Lemma 4.2. Let |α|<3β. Then

(1) The mapy0∈L²(0, L)7→S(·)y0∈His continuous.

(2) For y0 ∈ L²(0, L), ∂y(0, .) makes sense in L²(0, L), and for all y0 ∈ L²(0, L),

k∂y(., t)k_L2(0,T)≤ ky0k_L2(0,L) (4.5) ky0k²_L2(0,L)≤ 1

TkS(·)y0k²_L2((0,T)×(0,L))+k∂y(0, .)k²_L2(0,T) (4.6) Proof. (1) Fory₀∈L²(0, L) we writeythe mild solutionS(·)y0of (R₂). By Lemma 4.1,y∈C([0, T] :L²(0, L)) and

kykC([0,T]:L²(0,L)) ≤ ky0kL²(0,L) (4.7) To see thaty∈L²(0, T :H²(0, L)) we first assume thaty∈D(A). Letξ=ξ(x, t)∈ C^∞([0, T]×[0, L]). Then, multiplying the equation (4.3) byiξywe have

iξy∂ty+iξy∂³y+αξy∂²y+iδξy∂y= 0

−iξy∂ty−iξy∂³y+αξy∂²y−iδξy∂y= 0

(applying conjugates). Subtracting, integrating overx∈(0, L) and using straight- forward calculus, we obtain

i∂t

Z L

0

ξ|y|²dx−i Z L

0

∂tξ|y|²dx+iβ Z L

0

ξy∂³y dx+iβ Z L

0

ξy∂³y dx +α

Z L

0

ξy∂²y dx−α Z L

0

ξy∂²y dx−iδ Z L

0

∂ξ|y|²dx= 0.

Each term is treated separately. Integrating by parts Z L

0

ξy∂³y dx= Z L

0

∂²ξy∂y dx+ 2 Z L

0

∂ξ|∂y|²dx−ξ(0, t)|∂y(0, t)|² +

Z L

0

ξ∂y∂²y dx Z L

0

ξy∂³y dx= Z L

0

∂²ξy∂y dx+ Z L

0

∂ξ|∂y|²dx− Z L

0

ξ∂y∂²y dx, Z L

0

ξy∂²y dx=− Z L

0

∂ξy∂y dx− Z L

0

ξ|∂y|²dx, Z L

0

ξy∂²y dx=− Z L

0

∂ξy∂y dx− Z L

0

ξ|∂y|²dx . Then

i∂t

Z L

0

ξ|y|²dx−i Z L

0

∂tξ|y|²dx+iβ Z L

0

∂²ξy∂y dx+ 2iβ Z L

0

∂ξ|∂y|²dx

−iβξ(0, t)|∂y(0, t)|²+iβ Z L

0

ξ∂y∂²y dx+iβ Z L

0

∂²ξy∂y dx+iβ Z L

0

∂ξ|∂y|²dx

−iβ Z L

0

ξ∂y∂²y dx−α Z L

0

∂ξy∂y dx−α Z L

0

ξ|∂y|²dx+ Z L

0

∂ξy∂y dx +

Z L

0

ξ|∂y|²dx−iδ Z L

0

∂ξ|y|²dx= 0. Hence,

i∂_t Z L

0

ξ|y|²dx−i Z L

0

∂_tξ|y|²dx+iβ Z L

0

∂²ξ∂(|y|²)dx+ 3iβ Z L

0

∂ξ|∂y|²dx

−iβξ(0, t)|∂y(0, t)|²−2iαIm Z L

0

∂ξy∂y dx−iδ Z L

0

∂ξ|y|²dx= 0. Thus

∂t

Z L

0

ξ|y|²dx− Z L

0

∂tξ|y|²dx−β Z L

0

∂³ξ|y|²dx+ 3β Z L

0

∂ξ|∂y|²dx

−βξ(0, t)|∂y(0, t)|²−δ Z L

0

∂ξ|y|²dx

= 2αIm Z L

0

∂ξy∂y dx

≤ |α|

Z L

0

∂ξ|y|²dx+|α|

Z L

0

∂ξ|∂y|²dx, where

∂t

Z L

0

ξ|y|²dx+ Z L

0

[3β− |α|]∂ξ|∂y|²dx− Z L

0

∂tξ|y|²dx−β Z L

0

∂³ξ|y|²dx

−βξ(0, t)|∂y(0, t)|²−δ Z L

0

∂ξ|y|²dx− |α|

Z L

0

∂ξ|y|²dx≤0.

(4.8)

Choosingξ(x, t) =xleads to

∂t

Z L

0

x|y|²dx+ Z L

0

[3β− |α|]|∂y|²dx−(δ+|α|) Z L

0

|y|²dx≤0.

Integrating overt∈[0, T] we obtain Z L

0

x|y|²dx+ [3β− |α|]

Z T

0

Z L

0

|∂y|²dx dt

≤(δ+|α|) Z T

0

Z L

0

|y|²dx dt+ Z L

0

x|y₀|²dx

≤(δ+|α|) Z T

0

Z L

0

|y|²dx dt+L Z L

0

|y0|²dx.

Using that |α|<3β, the second and the third terms in the left hand on the above equation are positive, thus we obtain

[3β− |α|]k∂yk²_L2(0,T:L²(0,L))

≤

(|δ|+|α|)kyk²_L2(0,T:L²(0,L))+Lky0k²_L2(0,L)

,