# Local exact controllability of the 1D nonlinear Schrödinger equation in the
case of Dirichlet boundary conditions

Alessandro Duca 111Université Paris-Saclay, UVSQ, CNRS, Laboratoire de
Mathématiques de Versailles, 78000, Versailles, France; e-mail:
<EMAIL_ADDRESS>Vahagn Nersesyan 222NYU-ECNU Institute of Mathematical
Sciences, NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China,
e-mail<EMAIL_ADDRESS>333Université Paris-Saclay, UVSQ, CNRS,
Laboratoire de Mathématiques de Versailles, 78000, Versailles, France

###### Abstract

We consider the 1D nonlinear Schrödinger equation with bilinear control. In
the case of Neumann boundary conditions, local exact controllability of this
equation near the ground state has been proved by Beauchard and Laurent
[BL10]. In this paper, we study the case of Dirichlet boundary conditions. To
establish the controllability of the linearised equation, we use a bilinear
control acting through four directions: three Fourier modes and one generic
direction. The Fourier modes are appropriately chosen so that they satisfy a
saturation property. These modes allow to control approximately the linearised
Schrödinger equation. We show that the reachable set for the linearised
equation is closed. This is achieved by representing the resolving operator as
a sum of two linear continuous mappings: one is surjective (here the control
in generic direction is used) and the other is compact. A mapping with dense
and closed image is surjective, so the linearised Schrödinger equation is
exactly controllable. Then local exact controllability of the nonlinear
equation is derived using the inverse mapping theorem.

AMS subject classifications: 35Q55, 81Q93, 93B05

Keywords: Nonlinear Schrödinger equation, local exact controllability,
linearisation, approximate controllability, saturation property

###### Contents

  1. 0 Introduction
  2. 1 Controllability of the linearised equation
    1. 1.1 Approximate controllability
    2. 1.2 Exact controllability
  3. 2 Proof of the Main Theorem
  4. 3 Appendix
    1. 3.1 Well-posedness of the linear Schrödinger equation
    2. 3.2 Local well-posedness of the NLS equation
    3. 3.3 Proof of Lemma 1.4
    4. 3.4 Saturation property
    5. 3.5 A closed image theorem

##  0 Introduction

We study the controllability of the one-dimensional nonlinear Schrödinger
(NLS) equation with bilinear control and Dirichlet boundary conditions. To
simplify the presentation, we consider in this introduction the case of the
cubic NLS equation

$\displaystyle
i\partial_{t}\psi=-\partial_{xx}^{2}\psi+\kappa|\psi|^{2}\psi+\langle
u(t),Q(x)\rangle\psi,\quad x\in I=(0,1),$ (0.1)
$\displaystyle\psi(t,0)=\psi(t,1)=0,$ (0.2)

where $Q:I\to{\mathbb{R}}^{q}$ is a given external field and $\kappa$ is a
real number. We fix any $T>0$ and consider the amplitude
$u:[0,T]\to{\mathbb{R}}^{q}$ as a control term and the solution at time $T$,
i.e., $\psi(T)$, as a state.

To formulate the main result of this paper, let us introduce some notation. We
consider $L^{2}(I;{\mathbb{C}})$ as a real Hilbert space endowed with the
scalar product

$\langle f,g\rangle_{L^{2}}=\mathop{\rm
Re}\nolimits\int_{0}^{1}f(x)\overline{g(x)}{\textup{d}}x$

and the corresponding norm $\|\cdot\|_{L^{2}}$. Let $A$ be the Dirichlet
Laplacian operator

$A=-\partial_{xx}^{2},\quad\quad{\mathbb{D}}(A)=H^{2}\cap
H^{1}_{0}(I;{\mathbb{C}}),$

and let $\phi_{k}(x)=\sqrt{2}\sin(k\pi x)$, $k\geq 1$ be its eigenfunctions
associated with the eigenvalues $\lambda_{k}=k^{2}\pi^{2}$. We use the spaces
$H^{s}_{(0)}={\mathbb{D}}(A^{\frac{s}{2}})$, $s\geq 0$ endowed with the scalar
products $\langle f,g\rangle_{(s)}=\langle
A^{\frac{s}{2}}f,A^{\frac{s}{2}}g\rangle_{L^{2}}$ and the corresponding norms
$\|\cdot\|_{(s)}$. The system (0.1), (0.2) is supplemented with the initial
condition

$\psi(0,x)=\psi_{0}(x),$ (0.3)

which is assumed to belong to the unit sphere ${\cal S}$ in
$L^{2}(I;{\mathbb{C}})$. The following is the main result of this paper.

######  Main Theorem.

Assume that $Q=(Q_{1},\ldots,Q_{q})$ is a smooth field such that the vector
space

${\mathbb{Q}}=\mathop{\rm span}\nolimits_{\mathbb{R}}\\{Q_{k}:k=1,\ldots,q\\}$
(0.4)

contains the functions ${\mathbf{1}}$, $\cos(\pi x)$, $\cos(2\pi x)$, and a
function $\mu$ verifying the inequality

$|\langle\mu\phi_{1},\phi_{k}\rangle_{L^{2}}|\geq\frac{c}{k^{3}},\quad k\geq
1$ (0.5)

for some number $c>0$. Then there is an at most countable set ${\cal
K}\subset(-\infty,0)$ such that, for any $\kappa\in{\mathbb{R}}\setminus{\cal
K}$, the NLS equation is locally exactly controllable near the ground state
$\phi_{1}$. More precisely, for any $T>0$, there is a number $\delta>0$ such
that, for any $\psi_{0},\psi_{1}\in H^{3}_{(0)}(I;{\mathbb{C}})\cap{\cal S}$
with

$\|\psi_{j}-\phi_{1}\|_{(3)}<\delta,\quad j=0,1,$

there is a control $u\in L^{2}([0,T];{\mathbb{R}}^{q})$ and a solution
$\psi\in C([0,T];H^{3}_{(0)}(I;{\mathbb{C}}))$ of the problem (0.1)-(0.3)
satisfying $\psi(T)=\psi_{1}$.

A more general version of this theorem is stated in Section 2 (see Theorem
2.2). In that version, the nonlinear term has the form $|\psi|^{2p}\psi$ with
any integer $p\geq 1$, and the conditions on the field $Q$ and the number
$\kappa$ are formulated in terms of a general saturation property.

The controllability of the Schrödinger equation (0.1) has been extensively
studied in the literature in the case $\kappa=0$. Note that in that case, even
if the equation is linear in $\psi$, the associated control problem is still
nonlinear. Ball, Marsden, and Slemrod [BMS82] proved that the reachable set
for this equation from any initial condition in
$H^{2}_{(0)}(I;{\mathbb{C}})\cap{\cal S}$ with controls in $L^{2}$ has empty
interior in $H^{2}_{(0)}(I;{\mathbb{C}})\cap{\cal S}$. In particular, this
means that the problem is not locally exactly controllable in that phase
space. Beauchard [Bea05] obtained the first positive controllability result:
in the case $Q(x)=x$, she proved local exact controllability in some
$H^{7}_{(0)}$-neighborhood of any eigenstate by using a Nash–Moser theorem.
Beauchard and Coron [BC06] obtained exact controllability between
neighborhoods of different eigenstates. Later, in the paper [BL10], Beauchard
and Laurent found a way to use the classical inverse mapping theorem to prove
local exact controllability of the Schrödinger equation; more precisely, they
proved exact controllability in some $H^{3}_{(0)}$-neighborhood of any
eigenstate in the case when $Q=\mu$ satisfies condition (0.5). The methods of
[BL10] have been further developed by Morancey and the authors of this paper
in [Mor14, MN15, Duc20] to study simultaneous exact controllability of several
Schrödinger equations.

All the above papers deal with the one-dimensional Schrödinger equation. In
the multidimensional case, exact controllability remains an open problem. In
that situation, approximate controllability property has been studied by many
authors; for the first results we refer the reader to the works by Boscain et
al. [CMSB09, BCCS12], Mirrahimi [Mir09], and the second author [Ner10].

In the case of the NLS equation (0.1) with $\kappa\neq 0$ and Neumann boundary
conditions, local exact controllability is established by Beauchard and
Laurent [BL10]. They use the inverse mapping theorem and exact controllability
of the linearised equation. The latter is proved by using a convenient change
of the unknown that reduces the original problem to another linear system with
explicit spectrum. The controllability of the reduced system is proved by
using a moment problem approach and the Ingham inequality. In the case of
Dirichlet boundary conditions, it is not clear whether such reduction is
possible, so we proceed in a different way.

We note that when ${\mathbf{1}}$ and $\cos(2\pi x)$ belong to the vector space
${\mathbb{Q}}$ defined by (0.4), the ground state $\phi_{1}$ is a stationary
solution of the NLS equation (0.1) corresponding to some constant control $u$.
The linearisation of the equation around the couple $(\phi_{1},u)$ is given by

$i\partial_{t}\xi=-\partial_{xx}^{2}\xi-\pi^{2}\xi+2\kappa\phi_{1}^{2}\mathop{\rm
Re}\nolimits(\xi)+\langle v(t),Q(x)\rangle\phi_{1}.$ (0.6)

We prove exact controllability of this equation in two steps. First, we show
that if the number $\kappa$ is in the complement of some at most countable set
${\cal K}$, then the directions ${\mathbf{1}}$ and $\cos(\pi x)$ are
saturating. As a consequence, we obtain approximate controllability of Eq.
(0.6). The saturation argument employed here is inspired by the papers of
Agrachev, Sarychev [AS05, AS06] and Shirikyan [Shi06], which study approximate
controllability of the nonlinear Navier–Stokes and Euler systems.

Next we show that approximate controllability of Eq. (0.6) implies its exact
controllability. To this end, we decompose the solution as follows
$\xi=\xi_{1}+\xi_{2}$, where $\xi_{1}$ and $\xi_{2}$ are solutions of
equations

$\displaystyle i\partial_{t}\xi_{1}$
$\displaystyle=-\partial_{xx}^{2}\xi_{1}-\pi^{2}\xi_{1}+\langle
v(t),Q(x)\rangle\phi_{1},$ (0.7) $\displaystyle i\partial_{t}\xi_{2}$
$\displaystyle=-\partial_{xx}^{2}\xi_{2}-\pi^{2}\xi_{2}+2\kappa\phi_{1}^{2}\mathop{\rm
Re}\nolimits(\xi_{1}+\xi_{2}).$ (0.8)

Eq. (0.7) is exactly controllable. Indeed, as in [BL10], this can be seen by
rewriting the control system as a moment problem and then by solving it with
the help of the Ingham inequality and the assumption (0.5). On the other hand,
we show that the resolving operator of Eq. (0.8) is compact. According to a
functional analysis result, in a Banach space, the sum of compact and
surjective linear continuous mappings has closed image. On the other hand,
this image is dense, by approximate controllability of Eq. (0.6). An operator
with closed and dense image is obviously surjective, so the linearised
Schrödinger equation (0.6) is exactly controllable. Applying the inverse
mapping theorem, we derive local exact controllability of the nonlinear
equation.

In the case $\kappa\in{\cal K}$, the linearised control system may possibly
miss one direction. However, we expect that using the nonlinear term one can
prove that the result of the Main Theorem still remains true. It would be
natural to study this case by applying a power series expansion in the spirit
of the paper [CC04] by Coron and Crépeau (see also [Cor07, BC06]). This
question will be considered elsewhere.

The use of saturation property to prove approximate controllability of the
linearised Schrödinger equation and the argument allowing to derive exact
controllability from approximate controllability are among the novelties of
this paper. We believe these arguments can be employed in other useful
situations.

Global controllability of the NLS equation (0.1) with $\kappa\neq 0$ is a
challenging open problem. First results in this direction have been obtained
recently by the authors [DN21] and by Coron et al. [CXZ21], who consider
approximate controllability between some particular states (in a semiclassical
sense in the second reference). From the Main Theorem and the time
reversibility of the Schrödinger equation it follows that global exact
controllability will be established if one shows approximate controllability
to the ground state $\phi_{1}$ in the $H^{3}_{(0)}$-norm (see Theorem 3.2 in
[Ner10] for the case $\kappa=0$).

#### Acknowledgement

This paper was finalised when the second author was visiting the School of
Mathematical Sciences of Shanghai Jiao Tong University. He thanks the
institute for hospitality.

#### Notation

In this paper, we use the following notation.

$\langle\cdot,\cdot\rangle$ and $|\cdot|$ denote the Euclidian scalar product
and norm in ${\mathbb{R}}^{q}$.

${\mathbf{1}}$ is the function identically equal to $1$ on the interval
$I=(0,1)$.

$L^{2}=L^{2}(I;{\mathbb{C}})$ and $H^{s}=H^{s}(I;{\mathbb{C}})$, $s>0$ are the
usual Lebesgue and Sobolev spaces of functions $g:I\to{\mathbb{C}}$ with the
norms $\|\cdot\|_{L^{2}}$ and $\|\cdot\|_{H^{s}}$. In the case of the spaces
of real-valued functions, we write $L^{2}(I;{\mathbb{R}})$ and
$H^{s}(I;{\mathbb{R}})$.

For any $V\in H^{3}(I;{\mathbb{R}})$, we denote by $A_{V}$ the Schrödinger
operator

$A_{V}=-\partial_{xx}^{2}+V,\quad\quad{\mathbb{D}}(A_{V})=H^{2}\cap
H^{1}_{0}(I;{\mathbb{C}}).$ (0.9)

$\\{\phi_{k,V}\\}$ is an orthonormal basis in $L^{2}(I;{\mathbb{R}})$ formed
by eigenfunctions of $A_{V}$, and $\lambda_{1,V}<\ldots<\lambda_{k,V}<\ldots$
are the corresponding eigenvalues.

$H^{s}_{(V)}={\mathbb{D}}(A_{V}^{\frac{s}{2}})$, $s\geq 0$. Note that

$H^{3}_{(V)}=H^{3}_{(0)}=\left\\{\psi\in
H^{3}(I,{\mathbb{C}}):\psi|_{x=0,1}=\psi^{\prime\prime}|_{x=0,1}=0\right\\}.$

$\ell^{2}=\left\\{\\{a_{k}\\}_{k\geq
1}\in{\mathbb{C}}^{\mathbb{N}}:\sum_{k=1}^{+\infty}|a_{k}|^{2}<+\infty\right\\}$,
$\ell^{2}_{r}=\left\\{\\{a_{k}\\}_{k\geq
1}\in\ell^{2}:a_{1}\in{\mathbb{R}}\right\\}$.

We write $J_{T}$ instead of $[0,T]$.

Let $X$ be a Banach space endowed with a norm $\|\cdot\|$.

$B_{X}(a,r)$ denotes the closed ball in $X$ of radius $r>0$ centred at $a\in
X$.

$L^{2}(J_{T};X)$ is the space of Borel-measurable functions $g:J_{T}\to X$
with the norm

$\|g\|_{L^{2}(J_{T};X)}=\left(\int_{0}^{T}\|g(t)\|_{X}^{2}{\textup{d}}t\right)^{\frac{1}{2}}.$

$C(J_{T};X)$ is the space of continuous functions $g:J_{T}\to X$ with the norm
$\|g\|_{C(J_{T};X)}=\max_{t\in J_{T}}\|g(t)\|_{X}.$

##  1 Controllability of the linearised equation

In this section, we study the controllability of the following linear
Schrödinger equation (cf. Eq. (0.6)):

$\displaystyle
i\partial_{t}\xi=-\partial_{xx}^{2}\xi+V(x)\xi-\lambda\xi+W(x)\mathop{\rm
Re}\nolimits(\xi)+\langle v(t),Q(x)\rangle\phi(x),$ (1.1)
$\displaystyle\xi(t,0)=\xi(t,1)=0,$ (1.2)

where $Q:I\to{\mathbb{R}}^{q}$ is a given field, $V,W:I\to{\mathbb{R}}$ are
given potentials, $\phi=\phi_{1,V}$ is the ground state of the Schrödinger
operator $A_{V}$ (see (0.9)), and $\lambda=\lambda_{1,V}$ is the associated
eigenvalue. The following well-posedness result is a consequence of
Proposition 3.1.

######  Proposition 1.1.

For any $T>0,Q\in H^{3}(I,{\mathbb{R}}^{q}),V,W\in H^{3}(I;{\mathbb{R}}),v\in
L^{2}(J_{T};{\mathbb{R}}^{q})$, and $\xi_{0}\in H^{3}_{(0)},$ there is a
unique solution $\xi\in C(J_{T};H^{3}_{(0)})$ of the problem (1.1), (1.2)
satisfying $\xi(0)=\xi_{0}$. Moreover, there is a constant $C_{T}>0$ such that

$\|\xi\|_{C(J_{T};H^{3}_{(0)})}\leq
C_{T}\left(\|\xi_{0}\|_{(3)}+\|v\|_{L^{2}(J_{T};{\mathbb{R}}^{q})}\right).$

Let

${\cal T}_{\phi}=\\{\psi\in L^{2}(I,{\mathbb{C}}):\mathop{\rm
Re}\nolimits\langle\psi,\phi\rangle_{L^{2}}=0\\}$

be the tangent space to the unit sphere ${\cal S}$ at $\phi$. As the functions
$Q,V,W,$ and $v$ are real-valued, we have $\xi(t)\in{\cal T}_{\phi}$ for any
$t\in J_{T}$, provided that $\xi_{0}\in{\cal T}_{\phi}$. Let

${\cal R}:H^{3}_{(0)}\times L^{2}(J_{T};{\mathbb{R}}^{q})\to
C(J_{T};H^{3}_{(0)}),\quad(\xi_{0},v)\mapsto\xi$

be the resolving operator of the problem (1.1), (1.2), and let ${\cal R}_{T}$
be its restriction at time $T$. In Section 1.1, we show that this problem is
approximately controllable under some saturation condition. Then, in Section
1.2, assuming additionally that the vector space ${\mathbb{Q}}$ spanned by the
components of $Q$ contains a function $\mu$ satisfying an inequality similar
to (0.5), we prove exact controllability of the problem.

###  1.1 Approximate controllability

Assume that the field $Q$ and the potentials $V,W$ are smooth. Let us define
finite-dimensional vector spaces by

${\cal H}=\mathop{\rm
span}\nolimits_{\mathbb{R}}\\{Q_{j}\phi:\,\,j=1,\ldots,q\\}$

and

${\cal F}({\cal H})=\mathop{\rm
span}\nolimits_{\mathbb{R}}\left\\{f+i(-\partial_{xx}^{2}g+Vg-\lambda
g+W\mathop{\rm Re}\nolimits(g)):\,\,f,g\in{\cal H}\right\\}.$ (1.3)

In the spirit of the papers [AS05, AS06, Shi06], we define a non-decreasing
sequence of finite-dimensional spaces $\\{{\cal H}_{j}\\}$ in the following
way

${\cal H}_{0}={\cal H},\quad{\cal H}_{j}={\cal F}({\cal
H}_{j-1})\quad\textrm{for}\,\,\,j\geq 1,\quad{\cal
H}_{\infty}=\bigcup_{j=0}^{\infty}{\cal H}_{j}.$ (1.4)

Let ${\mathsf{P}}_{1}$ be the orthogonal projection onto the closed subspace
$H^{3}_{(0)}\cap{\cal T}_{\phi}$ in $H^{3}_{(0)}$.

######  Definition 1.2.

We say that a field $Q$ is saturating for the problem (1.1), (1.2) if ${\cal
H}_{\infty}\subset H^{3}_{(0)}$ and the projection ${\mathsf{P}}_{1}{\cal
H}_{\infty}$ is dense in $H^{3}_{(0)}\cap{\cal T}_{\phi}$.

######  Proposition 1.3.

Assume that $Q$ is saturating. Then the problem (1.1), (1.2) is approximately
controllable in the sense that the image of the linear mapping

${\cal R}_{T}(0,\cdot):L^{2}(J_{T};{\mathbb{R}}^{q})\to H^{3}_{(0)}\cap{\cal
T}_{\phi},\quad v\mapsto\xi(T)$

is dense in $H^{3}_{(0)}\cap{\cal T}_{\phi}$ for $T>0$.

###### Proof.

Step 1. Reduction. For any $0\leq\tau\leq t\leq T$, let

$R(t,\tau):H^{3}_{(0)}\to H^{3}_{(0)},\quad\xi_{0}\mapsto\xi(T)$

be the resolving operator of the problem

$\displaystyle
i\partial_{t}\xi=-\partial_{xx}^{2}\xi+V(x)\xi-\lambda\xi+W(x)\mathop{\rm
Re}\nolimits(\xi),$ $\displaystyle\xi(t,0)=\xi(t,1)=0,$
$\displaystyle\xi(\tau,x)=\xi_{0}.$

Let the operator ${\cal A}:L^{2}(J_{T};H^{3}_{(0)})\to H^{3}_{(0)}$ be defined
by

${\cal A}(v)=\int_{0}^{T}R(T,\tau)v(\tau)\,{\textup{d}}\tau,\quad v\in
L^{2}(J_{T};H^{3}_{(0)}),$

and let ${\mathsf{P}}_{\cal H}$ be the orthogonal projection onto ${\cal H}$
in $H^{3}_{(0)}$. The proposition will be proved if we show that the image of
the operator

${\cal A}_{1}:L^{2}(J_{T},H_{(0)}^{3})\to H^{3}_{(0)}\cap{\cal
T}_{\phi},\quad{\cal A}_{1}={\cal A}{\mathsf{P}}_{\cal H}$

is dense in $H^{3}_{(0)}\cap{\cal T}_{\phi}$. The latter will be achieved by
showing that the kernel of the adjoint ${\cal A}_{1}^{*}$ of ${\cal A}_{1}$ is
trivial. Note that ${\cal A}_{1}^{*}$ is given by

${\cal A}_{1}^{*}:H^{3}_{(0)}\cap{\cal T}_{\phi}\to L^{2}(J_{T},{\cal
H}),\quad z\mapsto{\mathsf{P}}_{\cal H}R(T,\cdot)^{*}z,$

where $R(T,\tau)^{*}:H^{3}_{(0)}\to H^{3}_{(0)}$ is the $H^{3}_{(0)}$-adjoint
of $R(T,\tau)$, $\tau\in J_{T}.$

Step 2. Triviality of the kernel of ${\cal A}_{1}^{*}$. Let $z$ be an
arbitrary element of the kernel of ${\cal A}_{1}^{*}$. Our goal is to show
that $z=0$. To this end, we take any $g\in{\cal H}$ and note that

$\langle g,R(T,\tau)^{*}z\rangle_{(3)}=0\quad\text{for almost any $\tau\in
J_{T}$}.$

By continuity in $\tau$ of $R(T,\tau)g$, this is equivalent to

$\langle R(T,\tau)g,z\rangle_{(3)}=0\quad\text{for any $\tau\in J_{T}$}.$
(1.5)

Taking $\tau=T$ in this equality, we see that $z$ is orthogonal to ${\cal H}$
in $H^{3}_{(0)}$. In what follows, we show that $z$ is orthogonal to ${\cal
H}_{j}$ for any $j\geq 1$. This, together with the saturation assumption, will
imply that $z=0.$

Let us fix any $T_{1}\in(0,T)$ and rewrite (1.5) as follows:

$\langle R(T_{1},\tau)g,R(T,T_{1})^{*}z\rangle_{(3)}=0\quad\text{for any
$\tau\in J_{T_{1}}$}.$ (1.6)

Note that $\zeta(\tau)=R(T_{1},\tau)g$ is the solution of the problem

$\displaystyle
i\partial_{\tau}\zeta=-\partial_{xx}^{2}\zeta+V(x)\zeta-\lambda\zeta+W(x)\mathop{\rm
Re}\nolimits(\zeta),$ $\displaystyle\zeta(\tau,0)=\zeta(\tau,1)=0,$
$\displaystyle\zeta(T_{1},x)=g(x).$

Taking the derivative of (1.6) in $\tau$ and choosing $\tau=T_{1}$, we get

$\langle i(-\partial_{xx}^{2}g+Vg-\lambda g+W\mathop{\rm
Re}\nolimits(g)),R(T,T_{1})^{*}z\rangle_{(3)}=0\quad\text{for any $g\in{\cal
H}$}.$

Thus

$\langle g,R(T,T_{1})^{*}z\rangle_{(3)}=0\quad\text{for any $g\in{\cal
H}_{1}$}.$

As $T_{1}\in(0,T)$ is arbitrary, we see that $z$ is orthogonal to ${\cal
H}_{1}$ in $H^{3}_{(0)}$. Iterating this argument, we derive orthogonality of
$z$ to ${\cal H}_{j}$ for any $j\geq 1$. Since ${\mathsf{P}}_{1}{\cal
H}_{\infty}$ is dense in $H^{3}_{(0)}\cap{\cal T}_{\phi}$, we conclude that
$z=0$. ∎

Let us close this section with an example of saturating field $Q$. This
example will be used in the proof of the Main Theorem formulated in the
Introduction. Let us introduce the operator

$A_{\kappa}g=-\partial_{xx}^{2}g-\pi^{2}g+2\kappa\phi_{1}^{2}g,\quad\quad{\mathbb{D}}(A_{\kappa})=H^{2}\cap
H^{1}_{0}(I;{\mathbb{R}}),$ (1.7)

and let $\lambda_{1,\kappa}<\ldots<\lambda_{k,\kappa}<\ldots$ be the sequence
of its eigenvalues. The following lemma is proved in Section 3.3.

######  Lemma 1.4.

There is an at most countable set ${\mathbb{K}}\subset(-\infty,0]$ such that,
for any $\kappa\in{\mathbb{R}}\setminus{\mathbb{K}}$ and any $k\geq 1$, we
have $\lambda_{k,\kappa}\neq 0$.

Recall that ${\mathbb{Q}}$ denotes the vector space spanned by the components
of $Q$ (see (0.4)). The following proposition is proved in Section 3.4.

######  Proposition 1.5.

Let $V(x)=0$ and $W(x)=2\kappa\phi_{1}^{2}(x)$ for any $x\in I$, let
$\kappa\in{\mathbb{R}}\setminus{\mathbb{K}}$, where ${\mathbb{K}}$ is the set
in Lemma 1.4, and let $Q$ be a smooth field such that ${\mathbf{1}}$,
$\cos(\pi x)\in{\mathbb{Q}}$. Then $Q$ is saturating in the sense of
Definition 1.2.

###  1.2 Exact controllability

The aim of this section is to prove the following proposition.

######  Proposition 1.6.

Let $V$ and $W$ be smooth functions such that the following boundary
conditions are verified:

$W(0)=W(1)=W^{\prime}(0)=W^{\prime}(1)=0.$ (1.8)

Moreover, assume that $Q$ is saturating, and there is a function
$\mu\in{\mathbb{Q}}$ satisfying the inequality

$|\langle\mu\phi,\phi_{k,V}\rangle_{L^{2}}|\geq\frac{c}{k^{3}},\quad k\geq 1$
(1.9)

for some $c>0.$ Then the problem (1.1), (1.2) is exactly controllable in the
sense that the mapping ${\cal R}_{T}(0,\cdot):L^{2}(J_{T};{\mathbb{R}}^{q})\to
H^{3}_{(0)}\cap{\cal T}_{\phi}$ is surjective for any $T>0$.

###### Proof.

To prove this proposition, we represent the solution $\xi$ of the problem
(1.1), (1.2) as follows $\xi=\xi_{1}+\xi_{2}$, where $\xi_{1}$ and $\xi_{2}$
are the solutions of problems

$\displaystyle
i\partial_{t}\xi_{1}=-\partial_{xx}^{2}\xi_{1}+V(x)\xi_{1}-\lambda\xi_{1}+\langle
v(t),Q(x)\rangle\phi,$ $\displaystyle\xi_{1}(t,0)=\xi_{1}(t,1)=0,$
$\displaystyle\xi_{1}(0,x)=0$

and

$\displaystyle
i\partial_{t}\xi_{2}=-\partial_{xx}^{2}\xi_{2}+V(x)\xi_{2}-\lambda\xi_{2}+W(x)\mathop{\rm
Re}\nolimits(\xi_{1}+\xi_{2}),$ $\displaystyle\xi_{2}(t,0)=\xi_{2}(t,1)=0$
$\displaystyle\xi_{2}(0,x)=0.$

Let

${\cal R}^{j}_{T}(0,\cdot):L^{2}(J_{T};{\mathbb{R}}^{q})\to
H^{3}_{(0)}\cap{\cal T}_{\phi},\quad v\mapsto\xi_{j}(T),\quad j=1,2$

be the resolving operators of these problems.

######  Lemma 1.7.

The mapping ${\cal R}_{T}^{1}(0,\cdot):L^{2}(J_{T};{\mathbb{R}}^{q})\to
H^{3}_{(0)}\cap{\cal T}_{\phi}$ is surjective for any $T>0$.

######  Lemma 1.8.

The mapping ${\cal R}_{T}^{2}(0,\cdot):L^{2}(J_{T};{\mathbb{R}}^{q})\to
H^{3}_{(0)}\cap{\cal T}_{\phi}$ is compact for any $T>0$.

Taking these lemmas for granted, let us complete the proof of the proposition.
We have

${\cal R}_{T}(0,\cdot)={\cal R}_{T}^{1}(0,\cdot)+{\cal R}_{T}^{2}(0,\cdot),$

where the linear bounded operators ${\cal R}_{T}^{1}(0,\cdot)$ and ${\cal
R}_{T}^{2}(0,\cdot)$ are, respectively, surjective and compact from
$L^{2}(J_{T};{\mathbb{R}}^{q})$ to $H^{3}_{(0)}\cap{\cal T}_{\phi}$. Lemma 3.5
implies that the image of the mapping ${\cal R}_{T}(0,\cdot)$ is closed in
$H^{3}_{(0)}\cap{\cal T}_{\phi}$. On the other hand, by Proposition 1.3, the
image of ${\cal R}_{T}(0,\cdot)$ is dense in $H^{3}_{(0)}\cap{\cal T}_{\phi}$.
We conclude that ${\cal R}_{T}(0,\cdot)$ is surjective. ∎

###### Proof of Lemma 1.7.

Let us consider the problem

$\displaystyle
i\partial_{t}\xi_{3}=-\partial_{xx}^{2}\xi_{3}+V(x)\xi_{3}-\lambda\xi_{3}+v(t)\mu(x)\phi,$
$\displaystyle\xi_{3}(t,0)=\xi_{3}(t,1)=0,$ $\displaystyle\xi_{3}(0,x)=0,$

and denote by ${\cal R}^{3}_{T}(0,\cdot):L^{2}(J_{T};{\mathbb{R}})\to
H^{3}_{(0)}\cap{\cal T}_{\phi}$ its resolving operator. From the assumption
that $\mu\in{\mathbb{Q}}$ it follows that the image of ${\cal
R}^{3}_{T}(0,\cdot)$ is contained in that of ${\cal R}^{1}_{T}(0,\cdot)$, so
it suffices to prove the surjectivity of ${\cal R}^{3}_{T}(0,\cdot)$. The
latter is proved by rewriting the system as a moment problem which is then
solved using the Ingham inequality (see Proposition 4 in [BL10]). Indeed,
$\xi_{3}$ satisfies the equality

$\xi_{3}(t)=-i\int_{0}^{t}e^{-i(A_{V}-\lambda)(t-s)}\left(v(s)\mu(x)\phi\right){\textup{d}}s,\quad
t\in J_{T}.$

We write $\xi_{3}(T)$ in the form

$\xi_{3}(T)=-i\sum_{k=1}^{+\infty}e^{-i(\lambda_{k,V}-\lambda)T}\langle\mu\phi,\phi_{k,V}\rangle_{L^{2}}\phi_{k,V}\int_{0}^{T}e^{i(\lambda_{k,V}-\lambda)s}v(s){\textup{d}}s,$

which is equivalent to

$\int_{0}^{T}e^{i(\lambda_{k,V}-\lambda)s}v(s){\textup{d}}s=\frac{ie^{i(\lambda_{k,V}-\lambda)T}}{\langle\mu\phi,\phi_{k,V}\rangle_{L^{2}}}\int_{0}^{1}\xi_{3}(T,x)\phi_{k,V}(x){\textup{d}}x,\quad
k\geq 1.$

In view of assumption (1.9), for any $\tilde{\xi}\in H^{3}_{(0)}\cap{\cal
T}_{\phi}$, we have

$\left\\{\frac{ie^{i(\lambda_{k,V}-\lambda)T}}{\langle\mu\phi,\phi_{k,V}\rangle_{L^{2}}}\int_{0}^{1}\tilde{\xi}(x)\phi_{k,V}(x){\textup{d}}x\right\\}_{k\geq
1}\in\ell^{2}_{r}.$

By the asymptotic formula for the eigenvalues (e.g., see Theorem 4 in [PT87]),

$\lambda_{k,V}=k^{2}\pi^{2}+\int_{0}^{1}V(x){\textup{d}}x+r_{k},\quad\text{where
}\\{r_{k}\\}_{k\geq 1}\in\ell^{2}.$ (1.10)

Hence, $\lambda_{k,V}-\lambda_{k-1,V}\to+\infty$ as $k\to+\infty$. Applying
Corollary 1 in [BL10], we find that there is $v\in L^{2}(J_{T};{\mathbb{R}})$
such that

$\int_{0}^{T}e^{i\lambda_{k,V}s}v(s){\textup{d}}s=\frac{ie^{i\lambda_{k,V}T}}{\langle\mu\phi,\phi_{k,V}\rangle_{L^{2}}}\int_{0}^{1}\tilde{\xi}(x)\phi_{k,V}(x){\textup{d}}x,\quad
k\geq 1.$

This shows that ${\cal R}^{3}_{T}(0,\cdot)$ is surjective and completes the
proof of Lemma 1.7. ∎

###### Proof of Lemma 1.8.

The proof follows immediately from Corollary 3.3. Indeed, the operator ${\cal
R}_{T}^{2}(0,\cdot)$ is compact since it can be represented as a composition
of the compact operator $\Phi$ in Corollary 3.3 with linear continuous
mappings. ∎

##  2 Proof of the Main Theorem

Let us consider the NLS equation

$\displaystyle
i\partial_{t}\psi=-\partial_{xx}^{2}\psi+V(x)\psi+\kappa|\psi|^{2p}\psi+\langle
u(t),Q(x)\rangle\psi,\quad x\in I,$ (2.1)
$\displaystyle\psi(t,0)=\psi(t,1)=0,$ (2.2)

where the field $Q$ and the potential $V$ are smooth, $p\geq 1$ is an integer,
and $\kappa$ is a real number. The following proposition establishes local
well-posedness of this equation. It is proved in Section 3.2.

######  Proposition 2.1.

Assume that, for some $T>0$, $\hat{\psi}_{0}\in H^{3}_{(0)}$, and $\hat{u}\in
L^{2}(J_{T};{\mathbb{R}}^{q})$, there is a solution $\hat{\psi}\in
C(J_{T};H^{3}_{(0)})$ of the problem (2.1), (2.2) satisfying the initial
condition $\hat{\psi}(0)=\psi_{0}$. Then there are positive numbers
$\delta=\delta(T,\Lambda)$ and $C=C(T,\Lambda)$, where

$\Lambda=\|\hat{\psi}\|_{C(J_{T};H^{3}_{(0)})}+\|\hat{u}\|_{L^{2}(J_{T};{\mathbb{R}}^{q})},$
(2.3)

such that the following properties hold.

  1. (i)

For any $\psi_{0}\in H^{3}_{(0)}$ and $u\in L^{2}(J_{T};{\mathbb{R}}^{q})$
satisfying

$\|\psi_{0}-\hat{\psi}_{0}\|_{(3)}+\|u-\hat{u}\|_{L^{2}(J_{T};{\mathbb{R}}^{q})}<\delta,$
(2.4)

the problem (2.1), (2.2) has a unique solution $\psi\in C(J_{T};H^{3}_{(0)})$
satisfying the initial condition $\psi(0)=\psi_{0}$.

  2. (ii)

Let $\Psi_{T}$ be the mapping taking a couple $(\psi_{0},u)$ satisfying (2.4)
to $\psi(T)$. Then $\Psi_{T}$ is $C^{1}$, and for any $v\in
L^{2}(J_{T};{\mathbb{R}}^{q})$, we have
$\partial_{u}\Psi_{T}(\hat{\psi}_{0},\hat{u})v=\xi(T)$, where $\xi$ is the
solution of linearised system

$\displaystyle
i\partial_{t}\xi=-\partial_{xx}^{2}\xi+V(x)\psi+(p+1)\kappa|\psi|^{2p}\xi+p\kappa\psi^{2}|\psi|^{2(p-1)}\overline{\xi}$
$\displaystyle+\langle\hat{u}(t),Q(x)\rangle\xi+\langle
v(t),Q(x)\rangle\hat{\psi},$ (2.5) $\displaystyle\xi(t,0)=\xi(t,1)=0,$ (2.6)
$\displaystyle\xi(0,x)=0.$ (2.7)

The following theorem is a generalisation of the Main Theorem.

######  Theorem 2.2.

Assume that the field $Q$ is saturating in the sense of Definition 1.2 with
potentials $V(x)$ and $W(x)=2p\kappa\phi^{2p}(x)$, and the vector space
${\mathbb{Q}}$ contains the functions ${\mathbf{1}},\phi^{2p}$, and a function
$\mu$ verifying inequality (1.9) for some $c>0$. Then, for any $T>0$, there is
a number $\delta>0$ such that, for any $\psi_{0},\psi_{1}\in
H^{3}_{(0)}\cap{\cal S}$ with

$\|\psi_{j}-\phi_{1}\|_{(3)}<\delta,\quad j=0,1,$ (2.8)

there is a control $u\in L^{2}([0,T];{\mathbb{R}}^{q})$ and a solution
$\psi\in C([0,T];H^{3}_{(0)})$ of the problem (2.1), (2.2) satisfying
$\psi(0)=\psi_{0}$ and $\psi(T)=\psi_{1}$.

###### Proof.

From the assumption that the functions ${\mathbf{1}}$ and $\phi^{2p}$ belong
to ${\mathbb{Q}}$ it follows that $\hat{\psi}(t)=\phi$ is a stationary
solution of the problem (2.1), (2.2) corresponding to some constant control
$\hat{u}(t)$ such that

$\langle\hat{u}(t),Q(x)\rangle=-\kappa\phi(x)^{2p}-\lambda_{1,V}.$ (2.9)

Proposition 2.1 implies that the operator
$\Psi_{T}:(\psi_{0},u)\mapsto\psi(T)$ is well-defined and $C^{1}$-smooth in
some neighbourhood of $(\phi,\hat{u})$. Furthermore, for any $v\in
L^{2}(J_{T};{\mathbb{R}}^{q})$, we have
$\partial_{u}\Psi_{T}(\phi,\hat{u})v=\xi(T)$, where, in view of (2.9), $\xi$
is the solution of equation

$i\partial_{t}\xi=-\partial_{xx}^{2}\xi+V(x)\xi-\lambda_{1,V}\xi+2p\kappa\phi^{2p}\mathop{\rm
Re}\nolimits(\xi)+\langle v(t),Q(x)\rangle\phi.$ (2.10)

The conditions of Proposition 1.6 are satisfied for the linearised system
(2.10), (2.6), (2.7), so it is exactly controllable. Applying the inverse
mapping theorem to the mapping $u\mapsto\Psi_{T}(\phi,u)$, we find that there
is a number $\delta>0$ such that, for any $\psi_{0},\psi_{1}\in
H^{3}_{(0)}\cap{\cal S}$ verifying (2.8), there are controls $u_{0},u_{1}\in
L^{2}([0,T];{\mathbb{R}}^{q})$ such that
$\Psi_{T}(\phi_{1},u_{0})=\overline{\psi_{0}}$ and
$\Psi_{T}(\phi_{1},u)=\psi_{1}$. By the time-reversibility property of the
Schrödinger equation, $\Psi_{T}(\psi_{0},w)=\phi$ with $w(t)=u_{0}(T-t)$.
Setting $u(t)=w(t)$ for $t\in[0,T]$ and $u(t)=u_{1}(t-T)$ for $t\in[T,2T]$, we
derive $\Psi_{2T}(\psi_{0},u)=\psi_{1}$. As the time $T>0$ is arbitrary, we
complete the proof of Theorem 2.2. ∎

###### Proof of the Main Theorem..

Main Theorem is proved by applying Theorem 2.2 with $p=1$ and $V=0$. Indeed,
the assumption that ${\mathbb{Q}}$ contains the functions ${\mathbf{1}}$ and
$\cos(2\pi x)$ implies that $\phi_{1}^{2}$ is also in ${\mathbb{Q}}$.
Proposition 1.5 implies the saturation assumption with
$W(x)=2\kappa\phi_{1}^{2}(x)$ and
$\kappa\in{\mathbb{R}}\setminus{\mathbb{K}}$.

When444Note that zero belongs to the set ${\mathbb{K}}$ in Lemma 1.4, since
$\lambda_{1,0}=0$. $\kappa=0$, the linearised problem is still exactly
controllable by Lemma 1.7, so the conclusions of the theorem hold in that case
too. Thus we proved the Main Theorem with ${\cal
K}={\mathbb{K}}\setminus\\{0\\}\subset(-\infty,0)$. ∎

######  Remark 2.3.

Let us emphasise that, under the conditions of the Main Theorem, the
linearised problem (1.1), (1.2) with $V(x)=0$, $W(x)=2\kappa\phi_{1}^{2}(x)$,
and $\kappa\in{\cal K}$ may still be exactly controllable. Indeed, in that
situation we only know that the linearised system can miss at most one
direction (see Corollary 3.4). As mentioned in the Introduction, even if the
linearised system misses one direction, the nonlinear system can still be
locally exactly controllable near the ground state.

######  Remark 2.4.

It is not difficult to construct examples of saturating fields in the case of
any integer $p\geq 1$. Indeed, the results of Lemma 1.4 and Proposition 1.5
generalise without difficulties to the case when $V(x)=0$,
$W(x)=2p\kappa\phi^{2p}$, and ${\mathbf{1}},\cos(\pi x),\ldots,\cos(N_{p}\pi
x)\in{\mathbb{Q}}$ with sufficiently large integer $N_{p}\geq 1$.

##  3 Appendix

###  3.1 Well-posedness of the linear Schrödinger equation

Let us consider the following linear Schrödinger equation with a source term:

$\displaystyle i\partial_{t}\xi=-\partial_{xx}^{2}\xi+V(x)\xi+W(x)\mathop{\rm
Re}\nolimits(\xi)+f(t,x),$ (3.1) $\displaystyle\xi(t,0)=\xi(t,1)=0.$ (3.2)

Well-posedness of this equation is essentially established in [BL10].

######  Proposition 3.1.

For any $T>0$, $V,W\in H^{3}(I;{\mathbb{R}})$, $f\in L^{2}(J_{T};H^{3}\cap
H^{1}_{0}(I;{\mathbb{C}}))$, and $\xi_{0}\in H^{3}_{(0)},$ there is a unique
solution $\xi\in C(J_{T};H^{3}_{(0)})$ of this problem with the initial
condition $\xi(0)=\xi_{0}$ in the sense that

$\xi(t)=e^{-iA_{V}t}\xi_{0}-i\int_{0}^{t}e^{-iA_{V}(t-s)}\left(W\mathop{\rm
Re}\nolimits(\xi(s))+f(s)\right){\textup{d}}s,\quad t\in J_{T}.$

Moreover, there is a constant $C_{T}>0$ such that

$\|\xi\|_{C(J_{T};H^{3}_{(0)})}\leq
C_{T}\left(\|\xi_{0}\|_{(3)}+\|f\|_{L^{2}(J_{T};H^{3}\cap
H^{1}_{0}(I;{\mathbb{C}}))}\right).$

By Lemma 1 in [BL10], the function

$G_{f}:t\mapsto\int_{0}^{t}e^{-iA_{V}(t-s)}f(s){\textup{d}}s\quad\textup{belongs
to $C(J_{T};H^{3}_{(0)})$}$ (3.3)

and satisfies the inequality 555More precisely, in the paper [BL10], the case
$V=0$ is considered. The general case is proved in a similar way by using the
asymptotics (1.10) for the eigenvalues of $A_{V}$.

$\|G_{f}\|_{C(J_{T};H^{3}_{(0)})}\leq C_{T}\|f\|_{L^{2}(J_{T};H^{3}\cap
H^{1}_{0}(I;{\mathbb{C}}))}.$ (3.4)

Proposition 3.1 is derived from this by applying a usual fixed point argument
to the mapping $L:C(J_{T};H^{3}_{(0)})\to C(J_{T};H^{3}_{(0)})$ defined by

$\displaystyle
L(\xi)(t)=e^{-iA_{V}t}\xi_{0}-i\int_{0}^{t}e^{-iA_{V}(t-s)}\left(W\mathop{\rm
Re}\nolimits(\xi(s))+f(s)\right)$ $\displaystyle{\textup{d}}s,$
$\displaystyle\textup{for }\xi\in$ $\displaystyle
C(J_{T};H^{3}_{(0)}),\,\,\,t\in J_{T}.$

We shall not dwell on the details of the proof.

The next lemma follows from the Arzelà–Ascoli theorem in a standard way; again
we skip the details.

######  Lemma 3.2.

Under the conditions of Proposition 3.1, the mapping

$G_{\cdot}:L^{2}(J_{T};H^{3}_{(0)})\to C(J_{T};H^{3}_{(0)}),\quad f\mapsto
G_{f}$

is compact.

Let us consider a particular case of the problem (3.1), (3.2) given by

$\displaystyle i\partial_{t}\xi=-\partial_{xx}^{2}\xi+V(x)\xi+W(x)\mathop{\rm
Re}\nolimits(\xi+\eta),$ (3.5) $\displaystyle\xi(t,0)=\xi(t,1)=0,$ (3.6)
$\displaystyle\xi(0,x)=0,$ (3.7)

and assume that $W$ satisfies the boundary conditions (1.8). Then the mapping
$\eta\mapsto W(x)\mathop{\rm Re}\nolimits(\eta)$ is continuous from
$L^{2}(J_{T};H_{(0)}^{3})$ to itself, and the following is an immediate
consequence of Lemma 3.2.

######  Corollary 3.3.

Under the conditions of Proposition 3.1, the mapping

$\Phi:L^{2}(J_{T};H_{(0)}^{3})\to C(J_{T};H^{3}_{(0)}),\quad\eta\mapsto\xi$

is compact, where $\xi$ is the solution of the problem (3.5)-(3.7).

###  3.2 Local well-posedness of the NLS equation

Here we prove Proposition 2.1.

Step 1. Uniqueness. Let $\psi_{1},\psi_{2}\in C(J_{T};H^{3}_{(0)})$ be
solutions of (2.1), (2.2) with the same control $u\in
L^{2}(J_{T};{\mathbb{R}}^{q})$ and the same initial condition
$\psi_{1}(0)=\psi_{2}(0)$. Then the difference $\varphi=\psi_{1}-\psi_{2}$
satisfies

$\displaystyle
i\partial_{t}\varphi=-\partial_{xx}^{2}\varphi+V(x)\varphi+\kappa|\psi_{1}|^{2p}\psi_{1}-\kappa|\psi_{2}|^{2p}\psi_{2}+\langle
u(t),Q(x)\rangle\varphi,$ $\displaystyle\varphi(t,0)=\varphi(t,1)=0,$
$\displaystyle\varphi(0)=0.$

From (3.4) it follows that

$\displaystyle\|\varphi(t)\|_{(3)}^{2}$ $\displaystyle\leq
C_{T}\Big{(}\||\psi_{1}|^{2p}\psi_{1}-|\psi_{2}|^{2p}\psi_{2}\|^{2}_{L^{2}(J_{t};H^{3}\cap
H^{1}_{0}(I;{\mathbb{C}}))}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\|\langle
u,Q\rangle\varphi\|^{2}_{L^{2}(J_{t};H^{3}\cap
H^{1}_{0}(I;{\mathbb{C}}))}\Big{)}$ $\displaystyle\leq
C_{1}\int_{0}^{t}(1+|u(s)|^{2})\|\varphi(s)\|_{(3)}^{2}{\textup{d}}s,\quad
t\in J_{T},$

where $C_{1}>0$ is a constant depending on
$\|\psi_{j}\|_{C(J_{T};H^{3}_{(0)})}$, $j=1,2$. Applying the Gronwall
inequality, we infer that $\varphi(t)=0$ for any $t\in J_{T}$.

Step 2. Local-in-time existence. Let us take any $\psi_{0}\in H^{3}_{(0)}$ and
$u\in L^{2}(J_{T},{\mathbb{R}}^{q})$ satisfying (2.4), any $T_{1}\in J_{T}$,
any $\psi\in B_{C(J_{T_{1}};H^{3}_{(0)})}(\hat{\psi},1)$, and define

$M(\psi)(t)=e^{-iA_{V}t}\psi_{0}-i\int_{0}^{t}e^{-iA_{V}(t-s)}\left(\kappa|\psi(s)|^{2p}\psi(s)+\langle
u(s),Q(x)\rangle\psi(s)\right){\textup{d}}s$

for $t\in J_{T_{1}}$. Then (3.3) implies that $M(\psi)\in
C(J_{T_{1}};H^{3}_{(0)})$. Moreover, using (3.4), we get

$\displaystyle\|M(\psi)(t)-\hat{\psi}(t)\|_{(3)}\leq$ $\displaystyle
C_{2}\Big{(}\|\psi_{0}-\hat{\psi}_{0}\|_{(3)}+\|u-\hat{u}\|_{L^{2}(J_{T};{\mathbb{R}}^{q})}$
$\displaystyle+(\|\hat{u}\|_{L^{2}(J_{T_{1}};{\mathbb{R}}^{q})}+T_{1}^{\frac{1}{2}})\|\psi-\hat{\psi}\|_{C(J_{t};H^{3}_{(0)})}\Big{)},\quad
t\in J_{T_{1}},$

where 666All the constants $C_{i}$ below depend on $T$ and $\Lambda$.
$C_{2}(T,\Lambda)>0$ (see (2.3)). This implies that $M$ maps
$B_{C(J_{T_{1}};H^{3}_{(0)})}(\hat{\psi},1)$ into itself for sufficiently
small $T_{1}$ and $\delta$. In a similar way, for any $\psi_{1},\psi_{2}\in
B_{C(J_{T_{1}};H^{3}_{(0)})}(\hat{\psi},1)$ and $t\in J_{T_{1}}$, we have

$\displaystyle\|M(\psi_{1})(t)-M(\psi_{2})(t)\|_{(3)}\leq$ $\displaystyle
C_{3}\Big{(}(\|u\|_{L^{2}(J_{T_{1}};{\mathbb{R}}^{q})}+T_{1}^{\frac{1}{2}})\|\psi_{1}-\psi_{2}\|_{C(J_{t};H^{3}_{(0)})}\Big{)}.$

Thus, $M$ is a contraction in $B_{C(J_{T_{1}};H^{3}_{(0)})}(\hat{\psi},1)$ for
sufficiently small $T_{1}$ and $\delta$. Hence, there is $\psi\in
B_{C(J_{T_{1}};H^{3}_{(0)})}(\hat{\psi},1)$ such that $M(\psi)=\psi$.

Step 3. Existence up to $T$. Let $\psi$ be a maximal solution of (2.1), (2.2)
corresponding to $\psi_{0}\in H^{3}_{(0)}$ and $u\in
L^{2}(J_{T},{\mathbb{R}}^{q})$ satisfying (2.4). Then there is a maximal time
$T_{*}\in J_{T}$ such that $\psi$ is defined on $[0,T_{*})$ and

$\|\psi(t)\|_{(3)}\to+\infty\quad\text{ as $t\to{T_{*}}^{-}$ when $T_{*}<T.$}$

The difference $\varphi=\psi-\hat{\psi}$ satisfies the inequality

$\displaystyle\|\varphi(t)\|_{(3)}$ $\displaystyle\leq
C_{4}\delta+C_{4}\int_{0}^{t}\left(\|\varphi(s)\|_{(3)}+\|\varphi(s)\|_{(3)}^{2p+1}\right){\textup{d}}s$
$\displaystyle\quad+C_{4}\left(\int_{0}^{t}|u(s)|^{2}\|\varphi(s)\|_{(3)}^{2}{\textup{d}}s\right)^{\frac{1}{2}},\quad
t\in[0,T_{*}).$ (3.8)

Let us denote

$\tau=\inf\\{t\in J_{T}:\|\varphi(t)\|_{(3)}=1\\},$

where the infimum over an empty set is equal to $T$. Let us show that, for
sufficiently small $\delta$, we have $\tau=T$. Arguing by contradiction, let
us assume that $\tau<T$. From (3.8) we derive

$\|\varphi(t)\|_{(3)}^{2}\leq
C_{5}\delta^{2}+C_{5}\int_{0}^{t}\|\varphi(s)\|_{(3)}^{2}(1+|u(s)|^{2}){\textup{d}}s,\quad
t\in[0,\tau).$

The Gronwall inequality implies

$\|\varphi(t)\|_{(3)}^{2}\leq
C_{5}\delta^{2}\exp\left(C_{5}\int_{0}^{t}(1+|u(s)|^{2}){\textup{d}}s\right)<1,\quad
t\in[0,\tau)$

for small $\delta$ and any $u\in L^{2}(J_{T},{\mathbb{R}}^{q})$ satisfying
(2.4). This contradicts the definition of $\tau$. Thus $\tau=T$ for small
$\delta$.

Step 4. Differentiability. The proof of $C^{1}$ regularity of the resolving
operator is similar to the case considered in Sections 2.2 and 3.2 in [BL10],
so we do not provide the details.

###  3.3 Proof of Lemma 1.4

Let us fix any $k\geq 1$. Let $\phi_{k,\kappa}$ be the eigenfunction of the
operator $A_{\kappa}$ (see (1.7)) associated with the eigenvalue
$\lambda_{k,\kappa}$. By Theorem 3 in Chapter 2 in [PT87], both
$\phi_{k,\kappa}$ and $\lambda_{k,\kappa}$ are real-analytic functions in
$\kappa$. By differentiating in $\kappa$ the identity

$\left(-\partial_{xx}^{2}-\pi^{2}+2\kappa\phi_{1}^{2}-\lambda_{k,\kappa}\right)\phi_{k,\kappa}=0,$

we obtain

$\left(-\partial_{xx}^{2}-\pi^{2}+2\kappa\phi_{1}^{2}-\lambda_{k,\kappa}\right)\frac{{\textup{d}}\phi_{k,\kappa}}{{\textup{d}}\kappa}+\left(2\phi_{1}^{2}-\frac{{\textup{d}}\lambda_{k,\kappa}}{{\textup{d}}\kappa}\right)\phi_{k,\kappa}=0.$

Taking the scalar product in $L^{2}(I;{\mathbb{R}})$ of this identity with
$\phi_{k,\kappa}$, we get

$\frac{{\textup{d}}\lambda_{k,\kappa}}{{\textup{d}}\kappa}=\langle
2\phi_{1}^{2},\phi_{k,\kappa}^{2}\rangle_{L^{2}}>0$

for any $\kappa\in{\mathbb{R}}$. We conclude that $\lambda_{k,\kappa}$ is
strictly increasing in $\kappa$, so it can vanish for at most one value of
$\kappa\in{\mathbb{R}}$. Moreover, as $\lambda_{1,0}=0$, strict monotonicity
of $\lambda_{k,\kappa}$ implies that
$0<\lambda_{1,\kappa}\leq\lambda_{k,\kappa}$ for any $k\geq 1$ and $\kappa>0$.
This completes the proof of the lemma.

###  3.4 Saturation property

This section is devoted to the proof of Proposition 1.5. It suffices to
consider the case $q=2$ and $Q(x)=({\mathbf{1}},\cos(\pi x))$. It is easy to
see that

${\cal H}=\mathop{\rm span}\nolimits_{\mathbb{R}}\\{\phi_{1},\phi_{2}\\}.$
(3.9)

Let us denote

$F(g)=i(-\partial_{xx}^{2}g-\pi^{2}g+2\kappa\phi_{1}^{2}\mathop{\rm
Re}\nolimits(g))\quad\textup{ for }g\in H^{2}(I;{\mathbb{C}}).$

From (1.3) and (1.4) it follows that $F(g)\in{\cal H}_{j}$ for $g\in{\cal
H}_{j-1}$ and $j\geq 1$. In particular, $F(g)\in{\cal H}_{\infty}$ for
$g\in{\cal H}_{\infty}$. Furthermore, from (3.9) we derive

${\cal H}_{\infty}\subset\mathop{\rm
span}\nolimits_{\mathbb{C}}\\{\phi_{k}:k\geq 1\\}\subset H^{3}_{(0)}.$

The proposition will be established if we show that ${\cal H}_{\infty}$ is
dense in $H^{3}_{(0)}$. The proof of this is divided into three steps.

Step 1. First let us show that the eigenfunctions $\phi_{2k+1}$, $k\geq 0$
belong to ${\cal H}_{\infty}$. We proceed by recurrence. By (3.9), we have
$\phi_{1}\in{\cal H}\subset{\cal H}_{\infty}$. Let us take any $N\geq 1$,
assume that $\phi_{2k+1}\in{\cal H}_{\infty}$ for any $0\leq k\leq N-1$, and
prove that $\phi_{2N+1}\in{\cal H}_{\infty}$. We have

$F(\phi_{2N-1})=ic_{2N-1}\phi_{2N-1}+i2\kappa\phi_{1}^{2}\phi_{2N-1}\in{\cal
H}_{\infty},$

where $c_{2N-1}=\pi^{2}\left((2N-1)^{2}-1\right)$. Combining this with the
identity

$\phi_{1}^{2}\phi_{2N-1}=\frac{1}{2}\left(2\phi_{2N-1}-\phi_{2N-3}-\phi_{2N+1}\right),$

we obtain

$F(F(\phi_{2N-1}))=\left(c_{2N-1}^{2}-c_{2N-1}\right)\phi_{2N-1}+\frac{1}{2}\left(c_{2N-3}\phi_{2N-3}+c_{2N+1}\phi_{2N+1}\right)\in{\cal
H}_{\infty}.$

As $\phi_{2N-3},\phi_{2N-1}\in{\cal H}_{\infty}$, we conclude that
$\phi_{2N+1}\in{\cal H}_{\infty}$.

In a similar way, as $\phi_{2}\in{\cal H}\subset{\cal H}_{\infty}$, one shows
that $\phi_{2k}\in{\cal H}_{\infty}$ for any $k\geq 1$.

Step 2. Let us prove that the vector space

${\cal G}=\mathop{\rm span}\nolimits_{{\mathbb{R}}}\\{iF(\phi_{k}):k\geq 1\\}$

is dense in $H^{3}_{(0)}(I;{\mathbb{R}})$. Indeed, let $A_{\kappa}$ be the
operator defined by (1.7) with $\kappa\in{\mathbb{R}}\setminus{\mathbb{K}}$.
Then $\lambda_{k,\kappa}\neq 0$ for any $k\geq 1$, so the image of
$A_{\kappa}$ is dense in $H^{3}_{(0)}(I;{\mathbb{R}})$. It remains to note
that $A_{\kappa}g=-iF(g)$ for $g\in H^{2}$, so ${\cal G}$ is dense in
$H^{3}_{(0)}(I;{\mathbb{R}})$.

Step 3. Combining the results of steps 1 and 2, and using the fact that

$\mathop{\rm span}\nolimits_{{\mathbb{R}}}\\{\phi_{k},F(\phi_{k}):k\geq
1\\}\subset{\cal H}_{\infty},$

we see that ${\cal H}_{\infty}$ is dense in $H^{3}_{(0)}$. This completes the
proof of the proposition.

######  Corollary 3.4.

Let ${\mathbb{K}}\subset(-\infty,0]$ be the set in Lemma 1.4,
$Q(x)=({\mathbf{1}},\cos(\pi x))$, $V(x)=0$, and
$W(x)=2\kappa\phi_{1}^{2}(x)$. Then, for any $\kappa\in{\mathbb{K}}$, the
codimension of ${\cal H}_{\infty}$ in $H^{3}_{(0)}$ is one.

###### Proof.

As $\kappa\in{\mathbb{K}}$, there is a unique $k\geq 1$ such that
$\lambda_{k,\kappa}=0$. From the proof of Proposition 1.5 it follows that the
vector space spanned by $i\phi_{k,\kappa}$ is the orthogonal complement of
${\cal H}_{\infty}$ in $H^{3}_{(0)}$. ∎

###  3.5 A closed image theorem

In this section, we formulate a simple functional analysis result used in the
proof of exact controllability of the linear Schrödinger equation (see Section
1.2). Being unable to find a proper reference, we give a complete proof.

Let $X$ and $Y$ be Banach spaces, and let $X^{*}$ and $Y^{*}$ be their duals.

######  Lemma 3.5.

Assume that $A:X\to Y$ and $B:X\to Y$ are linear continuous operators such
that $A$ is surjective and $B$ is compact. Then the image of $A+B$ is closed
in $Y$.

###### Proof.

Let $A^{*}:Y^{*}\to X^{*}$ and $B^{*}:Y^{*}\to X^{*}$ be the adjoint operators
of $A$ and $B$. By Theorem 2.19 in [Bre11], the set $(A+B)(X)$ is closed in
$Y$ if and only if $(A^{*}+B^{*})(Y^{*})$ is closed in $X^{*}$. We will prove
that $(A^{*}+B^{*})(Y^{*})$ is closed in $X^{*}$ in three steps.

Step 1. Let us first show that the kernel of the operator $A^{*}+B^{*}$ is
finite-dimensional. To this end, we prove that any bounded sequence
$\\{y_{n}\\}$ in the kernel of $A^{*}+B^{*}$ has a convergent subsequence.
Indeed, by Theorem 6.4 in [Bre11], $B^{*}$ is compact. So there is a
subsequence $\\{y_{n_{k}}\\}$ such that $\\{B^{*}(y_{n_{k}})\\}$ converges.
The equality $A^{*}(y_{n_{k}})+B^{*}(y_{n_{k}})=0$ implies that
$\\{A^{*}(y_{n_{k}})\\}$ also converges. As $A^{*}$ is injective and
$A^{*}(Y^{*})$ is closed, the open mapping theorem implies that
$\\{y_{n_{k}}\\}$ converges.

Thus the kernel of $A^{*}+B^{*}$ is finite-dimensional, so it is complemented.
Without loss of generality, we can assume that $A^{*}+B^{*}$ is injective.

Step 2. Let $\\{y_{n}\\}\subset Y^{*}$ be a sequence such that
$\\{(A^{*}+B^{*})(y_{n})\\}$ converges. Let us show that $\\{y_{n}\\}$ is
bounded. Arguing by contradiction, let us assume that there is a subsequence
such that $\|y_{n_{k}}\|_{Y^{*}}\to+\infty$. Then, for the sequence
$\tilde{y}_{n}=y_{n_{k}}/\|y_{n_{k}}\|_{Y^{*}}$ we have
$(A^{*}+B^{*})(\tilde{y}_{n})\to 0$ as $n\to+\infty$. By the fact that
$\|\tilde{y}_{n}\|_{Y^{*}}=1$ and the compactness of $B^{*}$, there is a
subsequence $\\{\tilde{y}_{\tilde{n}_{k}}\\}$ such that
$\\{B^{*}(\tilde{y}_{\tilde{n}_{k}})\\}$ converges. From the equality

$A^{*}(\tilde{y}_{\tilde{n}_{k}})=(A^{*}+B^{*})(\tilde{y}_{\tilde{n}_{k}})-B^{*}(\tilde{y}_{\tilde{n}_{k}})$

it follows that $\\{A^{*}(\tilde{y}_{\tilde{n}_{k}})\\}$ converges too. As in
step 1, this implies that $\\{\tilde{y}_{\tilde{n}_{k}}\\}$ converges to some
limit $\tilde{y}$. Since $\|\tilde{y}_{\tilde{n}_{k}}\|_{Y^{*}}=1$, we have
$\|\tilde{y}\|_{Y^{*}}=1$. By continuity,

$(A^{*}+B^{*})(\tilde{y}_{\tilde{n}_{k}})\to(A^{*}+B^{*})(\tilde{y})\quad\textup{as
}k\to+\infty.$ (3.10)

On the other hand, from the construction of $\\{\tilde{y}_{n}\\}$ it follows
that

$(A^{*}+B^{*})(\tilde{y}_{\tilde{n}})\to 0\quad\textup{as }n\to+\infty.$
(3.11)

From (3.10) and (3.11) it follows that $\tilde{y}$ is a non-zero element of
the kernel of $A^{*}+B^{*}$. This contradicts the injectivity of
$A^{*}+B^{*}$.

Step 3. Let $\\{y_{n}\\}\subset Y^{*}$ be a bounded sequence such that

$(A^{*}+B^{*})(y_{n})\to x\quad\textup{as }n\to+\infty.$ (3.12)

By compactness of $B^{*}$, there is a subsequence $\\{y_{n_{k}}\\}$ such that
$\\{B^{*}(y_{n_{k}})\\}$ converges. From (3.12) it follows that
$\\{A^{*}(y_{n_{k}})\\}$ converges too. As above, this implies that
$\\{y_{n_{k}}\\}$ converges to some limit $y$. By continuity and (3.12), we
have $(A^{*}+B^{*})(y)=x$. We conclude that the set $(A^{*}+B^{*})(Y^{*})$ is
closed in $X^{*}$. ∎

## References

  * [AS05] A. A. Agrachev and A. V. Sarychev.  Navier–Stokes equations: controllability by means of low modes forcing.  J. Math. Fluid Mech., 7(1):108–152, 2005.
  * [AS06] A. A. Agrachev and A. V. Sarychev.  Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing.  Comm. Math. Phys., 265(3):673–697, 2006.
  * [BMS82] J. M. Ball, J. E. Marsden, and M. Slemrod.  Controllability for distributed bilinear systems.  SIAM J. Control Optim., 20(4):575–597, 1982.
  * [Bea05] K. Beauchard.  Local controllability of a 1-D Schrödinger equation.  J. Math. Pures Appl. (9), 84(7):851–956, 2005.
  * [BC06] K. Beauchard and J.-M. Coron.  Controllability of a quantum particle in a moving potential well.  J. Funct. Anal., 232(2):328–389, 2006.
  * [BL10] K. Beauchard and C. Laurent.  Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control.  J. Math. Pures Appl. (9), 94(5):520–554, 2010.
  * [BCCS12] U. Boscain, M. Caponigro, T. Chambrion, and M. Sigalotti.  A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule.  Comm. Math. Phys., 311(2):423–455, 2012.
  * [Bre11] H. Brezis.  Functional analysis, Sobolev spaces and partial differential equations.  Universitext. Springer, New York, 2011.
  * [CMSB09] T. Chambrion, P. Mason, M. Sigalotti, and U. Boscain.  Controllability of the discrete-spectrum Schrödinger equation driven by an external field.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 26(1):329–349, 2009.
  * [Cor07] J.-M. Coron.  Control and Nonlinearity, volume 136.  AMS, Providence, RI, 2007.
  * [CC04] J.-M. Coron and E. Crépeau.  Exact boundary controllability of a nonlinear KdV equation with critical lengths.  J. Eur. Math. Soc., 6(3):367–398, 2004.
  * [CXZ21] J.-M. Coron, S. Xiang, and P. Zhang.  On the global approximate controllability in small time of semiclassical 1-d schrödinger equations between two states with positive quantum densities.  Preprint, arXiv:2110.15299, 2021.
  * [Duc20] A. Duca.  Simultaneous global exact controllability in projection of infinite 1D bilinear Schrödinger equations.  Dyn. Partial Differ. Equ., 17(3):275–306, 2020.
  * [DN21] A. Duca and V. Nersesyan.  Bilinear control and growth of Sobolev norms for the nonlinear Schrödinger equation.  Preprint, arXiv:2101.12103, 2021.
  * [Mir09] M. Mirrahimi.  Lyapunov control of a quantum particle in a decaying potential.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 26(5):1743–1765, 2009.
  * [Mor14] M. Morancey.  Simultaneous local exact controllability of 1D bilinear Schrödinger equations.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 31(3):501–529, 2014.
  * [MN15] M. Morancey and V. Nersesyan.  Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations.  J. Math. Pures Appl. (9), 103(1):228–254, 2015.
  * [Ner10] V. Nersesyan.  Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 27(3), 2010.
  * [PT87] J. Pöschel and E. Trubowitz.  Inverse spectral theory, volume 130 of Pure and Applied Mathematics.  Academic Press, Inc., Boston, MA, 1987.
  * [Shi06] A. Shirikyan.  Approximate controllability of three-dimensional Navier–Stokes equations.  Comm. Math. Phys., 266(1):123–151, 2006.


$F\simeq(\mathbb{P}^{1}\times\mathbb{P}^{2})\sqcup\mathbb{P}^{2}$. It follows
that $X$ has to dominate the component isomorphic to the projective plane. But
this component is linearly embedded via the Plücker embedding. As above, if
$X$ were to dominate this component, then $\deg\gamma=45$ which contradicts
(7.2). ∎

###  7.5. The Fano surface: Proof of proposition 7.5

According to proposition 6.9 we have $\dim F\leqslant 3$ with equality if and
only if $Y$ is either the twisted plane in example 6.8 or a cone over a
nonnormal cubic surface. These cases are excluded by looking at the Fano
variety:

  * •

If $Y$ is a cone over a nonnormal cubic curve, then $F$ is irreducible and has
dimension $3$; see example 6.4.

  * •

If $Y$ is a cone over a nonnormal cubic surface that is not itself a cone over
a nonnormal cubic curve, then the $2$-dimensional irreducible components of
$F$ are linearly embedded via the Plücker embedding; see example 6.6. This
would imply $\deg\gamma=45$ contradicting (7.2).

  * •

If $Y$ is the twisted plane, then by example 6.8 the $2$-dimensional
irreducible components of $F$ have degree $1$ and $4$ with respect the Plücker
embedding. Again, this contradicts (7.2).

The above shows that we can assume $\dim F=2$. It follows from proposition 6.7
that the Hilbert polynomial of $F$ with respect to the Plücker embedding is

$\chi(F,\mathcal{L}^{\otimes i})=\textstyle 45\binom{i+1}{2}-45i+6,$

where $\mathcal{L}$ is the line bundle on
$\operatorname{Gr}_{2}(\operatorname{Lie}A)$ obtained as restriction via the
Plücker embedding of the line bundle $\mathcal{O}(1)$ on
$\mathbb{P}(\operatorname{Alt}^{2}\operatorname{Lie}A)$. On the other hand,
for all $i\geqslant 0$,

$\chi(X,\gamma^{\ast}\mathcal{L}^{\otimes i})=\textstyle
45\binom{i+1}{2}-45i+6.$

To see this, first notice that $\gamma^{\ast}\mathcal{L}\simeq\Omega^{2}_{X}$
by definition of the morphism $\gamma$, thus by Serre’s duality theorem, we
have

$\chi(X,\gamma^{\ast}\mathcal{L})=\chi(X,\mathcal{O}_{X})=6.$

On the other hand, the self-intersection of $\gamma^{\ast}\mathcal{L}$ is
$c_{1}(X)^{2}=45$, hence the Hirzebruch-Riemann-Roch theorem implies the
wanted identity. Now, consider the short exact sequence of
$\mathcal{O}_{F}$-modules

$0\longrightarrow\mathcal{O}_{F}\longrightarrow\gamma_{\ast}\mathcal{O}_{X}\longrightarrow
Q\longrightarrow 0.$

Since the morphism $\gamma$ is finite, the $\mathcal{O}_{F}$-module
$\gamma_{\ast}\mathcal{O}_{X}$ is coherent, thus so is $Q$. Taking the tensor
product with $\mathcal{L}^{\otimes i}$, we have the short exact sequence

$0\longrightarrow\mathcal{L}^{\otimes
i}\longrightarrow\gamma_{\ast}\gamma^{\ast}\mathcal{L}^{\otimes
i}\longrightarrow Q\otimes\mathcal{L}^{\otimes i}\longrightarrow 0$

for all integer $i\geqslant 0$. The finiteness of the morphism $\gamma$ also
implies the vanishing of the higher direct images
$\textup{R}^{q}\gamma_{\ast}\gamma^{\ast}\mathcal{L}^{\otimes i}$ for
$q\geqslant 1$, which yields the identity

$\chi(F,\gamma_{\ast}\gamma^{\ast}\mathcal{L}^{\otimes
i})=\chi(X,\gamma^{\ast}\mathcal{L}^{\otimes i})$

Since Euler characteristics are additive in short exact sequences, this
implies that the Hilbert polynomial of $Q$ vanishes altogether, hence $Q=0$.
The morphism $\gamma$ being finite, this implies that $\gamma\colon X\to F$ is
an isomorphism. By hypothesis $X$ is smooth, thus so is $F$. Since $F$ has
dimension $2$ we can apply [AK77, th. 4.2] to derive that $Y$ is smooth, too.
∎

##  Appendix A Hodge number estimates

Let $X$ be a smooth complex projective variety of dimension $d$. The arguments
of Lazarsfeld-Popa [LP10, th. C(iii)] and Lombardi [Lom13, cor. 2.3] can be
applied in a slightly more general context to get bounds on the Euler
characteristics $\chi(X,\Omega^{p}_{X})$ with the Albanese variety replaced by
an arbitrary ambient abelian variety. For a linear subspace
$V\subset\textup{H}^{0}(X,\Omega^{1}_{X})$ put

$m(V)\;:=\;\min\bigl{\\{}\,\operatorname{codim}(X,Z(\omega))\mid
0\neq\omega\in V\,\bigr{\\}},$

where $Z(\omega)\subset X$ denotes the zero locus of
$\omega\in\textup{H}^{0}(X,\Omega^{1}_{X})$ as in loc. cit. We will be
interested in the case $m(V)=d$, which is to say that every nonzero
differential form $\omega\in V$ has at most isolated zeroes. In this case, we
have:

######  Theorem A.1.

If $V\subset\textup{H}^{0}(X,\Omega^{1}_{X})$ satisfies $m(V)=d$ and $\dim
V>d+1$, then

$(-1)^{d-p}\chi(X,\Omega^{p}_{X})\;\geqslant\;\begin{cases}\dim
V-d+1&\text{for\leavevmode\nobreak\ $p\,\in\,\\{0,d\\}$},\\\\[1.00006pt]
2&\text{for\leavevmode\nobreak\ $p\,\in\,\\{1,d-1\\}$},\\\\[1.00006pt]
1&\text{for\leavevmode\nobreak\ $2\leqslant p\leqslant d-2$}.\end{cases}$

###### Proof.

For $0\leqslant p\leqslant d$ and any $v\in V\smallsetminus\\{0\\}$, the cup
product with $v$ gives rise to a complex

$\mathbb{L}^{p}_{X,v}\colon\;\bigl{[}0\;\longrightarrow\;\textup{H}^{0}(X,\Omega^{p}_{X})\;\stackrel{{\scriptstyle
v\smile}}{{\longrightarrow}}\;\textup{H}^{1}(X,\Omega^{p}_{X})\;\stackrel{{\scriptstyle
v\smile}}{{\longrightarrow}}\;\cdots\;\stackrel{{\scriptstyle
v\smile}}{{\longrightarrow}}\;\textup{H}^{d}(X,\Omega^{p}_{X})\;\longrightarrow\;0\bigr{]}.$

For $m(V)=d$, this complex is exact in all degrees $i\neq d-p$ by [GL87, prop.
3.4], Hodge symmetry and Serre duality, see [Lom13, prop. 2.1]. By varying the
vector $v$ we can view these complexes as the fibers of a complex of vector
bundles on the projective space $\mathbb{P}=\mathbb{P}(V)$, the Green-
Lazarsfeld derivative complex

$\mathbb{L}^{p}_{X,V}\colon\;\bigl{[}\;\cdots\rightarrow\mathcal{O}_{\mathbb{P}}(-d+i)\otimes\textup{H}^{i}(X,\Omega^{p}_{X})\stackrel{{\scriptstyle\delta_{i}}}{{\longrightarrow}}\mathcal{O}_{\mathbb{P}}(-d+i+1)\otimes\textup{H}^{i+1}(X,\Omega^{p}_{X})\rightarrow\cdots\;\bigr{]}$

sitting in degrees $i=0,1,\dots,d$. By the above, we have

  1. (1)

$\mathcal{H}^{i}(\mathbb{L}^{p}_{X,V})=0$ for all $i\neq d-p$, and

  2. (2)

the morphisms $\delta_{i}$ have constant rank for all $i$.

Property (2) implies that the cokernel
$F=\operatorname{Coker}(\delta_{d-p-1})$ is locally free and then also that

$E\;:=\;\operatorname{Ker}\left(F\;\rightarrow\;\mathcal{O}_{\mathbb{P}}(-p+1)\otimes\textup{H}^{d-p+1}(X,\Omega^{p}_{X})\right)$

is locally free. Property (1) then shows that the rank of the vector bundle
$E$ is given by

$\operatorname{rk}(E)\;=\;(-1)^{d-p}\chi(X,\Omega^{p}_{X}).$

This shows $(-1)^{d-p}\chi(X,\Omega^{p}_{X})\geqslant 1$, since necessarily
$E\neq 0$: Else the complex $\mathbb{L}^{p}_{X,V}$ would be exact in all
degrees, which is impossible because any exact sequence of direct sums of line
bundles on $\mathbb{P}=\mathbb{P}(V)$ with differentials of degree one has
length at least $\dim V+1$ by [Lom13, lemma 2.2], and we assumed $\dim V>d$.

When $p=d-1$, let $G:=\operatorname{Ker}(\delta_{1})$ so the complex
$\mathbb{L}^{p}_{X,V}$ becomes an exact sequence

$0\;\longrightarrow\;G\;\longrightarrow\;\mathcal{O}_{\mathbb{P}}(-d+1)\otimes\textup{H}^{1}(X,\Omega^{d-1}_{X})\;\longrightarrow\;\cdots\longrightarrow\;\mathcal{O}_{\mathbb{P}}\otimes\textup{H}^{d}(X,\Omega^{d-1}_{X})\;\longrightarrow\;0.$

The differential $\delta_{0}$ factors through an injective map
$h\colon\mathcal{O}_{\mathbb{P}}(-d)\otimes\textup{H}^{0}(X,\Omega^{1}_{X})\hookrightarrow
G$. The cokernel of $h$ is locally free and it is nonzero by [Lom13, lemma
2.2]. If it were of rank $1$, it would be of the form
$\mathcal{O}_{\mathbb{P}}(j)$ for some integer $j$, hence we would have a
short exact sequence

$0\longrightarrow\mathcal{O}_{\mathbb{P}}(-d)\otimes\textup{H}^{0}(X,\Omega^{1}_{X})\longrightarrow
G\longrightarrow\mathcal{O}_{\mathbb{P}}(j)\longrightarrow 0.$

The vanishing of $\textup{H}^{1}(\mathbb{P},\mathcal{O}_{\mathbb{P}}(d+j))$
implies that the previous short exact sequence is split, hence $G$ is a sum of
line bundles on $\mathbb{P}$. By loc. cit. this is not possible, thus

$-\chi(X,\Omega^{d-1}_{X})\;=\;\operatorname{rk}G\;\geqslant 2.$

It remains to prove the estimate $\chi(X,\Omega^{d}_{X})\geqslant\dim V-d+1$.
For this take $p=d$, and let $G:=\operatorname{Ker}(\delta_{0})$. We then have
an exact sequence of vector bundles

$0\;\longrightarrow\;G\;\longrightarrow\;\mathcal{O}_{\mathbb{P}}(-d)\otimes\textup{H}^{0}(X,\Omega^{d}_{X})\;\longrightarrow\;\mathcal{O}_{\mathbb{P}}(-d+1)\otimes\textup{H}^{1}(X,\Omega^{d}_{X})\;\longrightarrow\;\cdots$

Taking the dual of the above sequence one sees that

$\textup{H}^{i}(\mathbb{P},G^{\vee}(j))\;=\;0\quad\text{for
all\leavevmode\nobreak\ $j\in\mathbb{Z}$ and all\leavevmode\nobreak\
$i\in\\{1,2,\dots,\dim V-d-1\\}$}.$

If $\operatorname{rk}G\leqslant\dim V-d$, then by the Evans-Griffiths theorem
[Laz04, ex. 7.3.10] it would follow that $G$ is a direct sum of line bundles.
This is seen to be impossible by applying [Lom13, lemma 2.2 (ii)] with $e=d$,
$q=\dim V$ and $a=-j$. Therefore, we have

$\operatorname{rk}G\;\geqslant\;\dim V-d+1$

and the claim follows since from the above exact sequence
$\operatorname{rk}G=\chi(X,\Omega_{X}^{d})$. ∎

The above in particular applies when $X\subset A$ is a subvariety of an
abelian variety and $V=\textup{H}^{0}(A,\Omega_{A}^{1})$. For a smooth ample
divisor $D\subset A$ the Euler characteristics can be computed exactly:

$(-1)^{g-1-i}\chi(D,\Omega^{i}_{D})\;=\;nA(g,i),$

where $n=D^{g}/g!$ and $A(g,i)$ is the Eulerian number; see [LS20, sect. 3.2].
For subvarieties of codimension $\geqslant 2$, we obtain:

######  Corollary A.2.

Let $A$ be a complex abelian variety of dimension $g$, and let $X\subset A$ be
a smooth subvariety of dimension $d\leqslant g-2$ with ample normal bundle.
Then we have

$(-1)^{d-p}\chi(X,\Omega^{p}_{X})\;\geqslant\;\begin{cases}g-d+1&\text{for\leavevmode\nobreak\
$p\,\in\,\\{0,d\\}$},\\\\[1.00006pt] 2&\text{for\leavevmode\nobreak\
$p\,\in\,\\{1,d-1\\}$},\\\\[1.00006pt] 1&\text{for\leavevmode\nobreak\
$2\leqslant p\leqslant d-2$}.\end{cases}$

###### Proof.

Since $X\subset A$ has ample normal bundle, we know by [Laz04, prop.
6.3.10(i)] that for every

$0\;\neq\;\omega\;\in\;V\;:=\;\textup{H}^{0}(A,\Omega^{1}_{A})\;\subset\;\textup{H}^{0}(X,\Omega^{1}_{X}),$

the zero locus $Z(\omega)\subset X$ is finite. Hence, $m(V)=d$, and theorem
A.1 applies. ∎

## References

  * [AK77] A. B. Altman and S. L. Kleiman, _Foundations of the theory of Fano schemes_ , Compositio Math. 34 (1977), no. 1, 3–47.
  * [BBDG18] A. Beilinson, J. Bernstein, P. Deligne, and O. Gabber, _Faisceaux pervers. Actes du colloque “Analyse et Topologie sur les Espaces Singuliers”. Partie I_ , 2nd edition ed., Astérisque, vol. 100, Paris: Société Mathématique de France (SMF), 2018.
  * [Bea82] A. Beauville, _Les singularités du diviseur theta de la jacobienne intermédiaire de l’hypersurface cubique dans $\mathbb{P}^{4}$._, Algebraic threefolds, Proc. 2nd 1981 Sess. C.I.M.E., Varenna/Italy 1981, Lect. Notes Math. 947, 190-208 (1982)., 1982.
  * [Bou02] N. Bourbaki, _Lie groups and Lie algebras. Chapters 4–6_ , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, Translated from the 1968 French original by Andrew Pressley.
  * [BW79] J. W. Bruce and C. T. C. Wall, _On the classification of cubic surfaces_ , J. London Math. Soc. (2) 19 (1979), no. 2, 245–256.
  * [CG72] C. H. Clemens and P. A. Griffiths, _The intermediate Jacobian of the cubic threefold_ , Ann. of Math. (2) 95 (1972), 281–356.
  * [Deb95] O. Debarre, _Fulton-Hansen and Barth-Lefschetz theorems for subvarieties of abelian varieties_ , J. Reine Angew. Math. 467 (1995), 187–197.
  * [DMOS82] P. Deligne, J. S. Milne, A. Ogus, and K. Shih, _Hodge cycles, motives, and Shimura varieties_ , Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982.
  * [Dol12] I. V. Dolgachev, _Classical algebraic geometry: A modern view_ , Cambridge University Press, Cambridge, 2012.
  * [EH87] D. Eisenbud and J. Harris, _On varieties of minimal degree (a centennial account)_ , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 3–13.
  * [FK00] J. Franecki and M. Kapranov, _The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties_ , Duke Math. J. 104 (2000), no. 1, 171–180.
  * [FP01] G. Fischer and J. Piontkowski, _Ruled varieties_ , Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 2001, An introduction to algebraic differential geometry.
  * [GL87] M. Green and R. Lazarsfeld, _Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville_ , Invent. Math. 90 (1987), no. 2, 389–407.
  * [Har92] J. Harris, _Algebraic geometry_ , Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992, A first course.
  * [JKLM23] A. Javanpeykar, T. Krämer, C. Lehn, and M. Maculan, _The monodromy of families of subvarieties on abelian varieties_ , 2023, arXiv:2210.05166.
  * [KM23] T. Krämer and M. Maculan, _Arithmetic finiteness of irregular varieties_ , 2023, arXiv:2310.08485.
  * [Krä16] T. Krämer, _Cubic threefolds, Fano surfaces and the monodromy of the Gauss map_ , Manuscr. Math. 149 (2016), no. 3-4, 303–314.
  * [Krä20] by same author, _Summands of theta divisors on Jacobians_ , Compos. Math. 156 (2020), no. 7, 1457–1475.
  * [Krä21] by same author, _Characteristic cycles and the microlocal geometry of the Gauss map, II_ , J. Reine Angew. Math. 774 (2021), 53–92.
  * [Krä22] by same author, _Characteristic cycles and the microlocal geometry of the Gauss map, I_ , Ann. Sci. Éc. Norm. Supér. (4) 55 (2022), 1475–1527.
  * [KW15] T. Krämer and R. Weissauer, _Vanishing theorems for constructible sheaves on abelian varieties_ , J. Algebraic Geom. 24 (2015), no. 3, 531–568.
  * [Laz04] R. Lazarsfeld, _Positivity in algebraic geometry. II_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 49, Springer-Verlag, Berlin, 2004, Positivity for vector bundles, and multiplier ideals.
  * [Lom13] L. Lombardi, _Inequalities for the Hodge numbers of irregular compact Kähler manifolds_ , Int. Math. Res. Not. 2013 (2013), no. 1, 63–83.
  * [LP10] R. Lazarsfeld and M. Popa, _Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds_ , Invent. Math. 182 (2010), no. 3, 605–633.
  * [LS20] B. Lawrence and W. Sawin, _The Shafarevich conjecture for hypersurfaces in abelian varieties_ , arXiv, 2020, arXiv:2004.09046.
  * [MP12] V. O. Marcucci and G. P. Pirola, _Points of order two on theta divisors_ , Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 23 (2012), no. 3, 319–323.
  * [Ran81] Z. Ran, _On subvarieties of abelian varieties_ , Invent. Math. 62 (1981), no. 3, 459–479.
  * [Sak10] Y. Sakamaki, _Automorphism groups on normal singular cubic surfaces with no parameters_ , Trans. Amer. Math. Soc. 362 (2010), no. 5, 2641–2666.
  * [Sch15] C. Schnell, _Holonomic $\mathcal{D}$-modules on abelian varieties_, Publ. Math., Inst. Hautes Étud. Sci. 121 (2015), 1–55.
  * [Sch17] S. Schreieder, _Decomposable theta divisors and generic vanishing_ , Int. Math. Res. Not. IMRN (2017), no. 16, 4984–5009.
  * [SS18] C. Sabbah and C. Schnell, _The MHM project_ , 2018, https://perso.pages.math.cnrs.fr/users/claude.sabbah/MHMProject/mhm.pdf.
  * [SY23] C. Schnell and R. Yang, _Hodge modules and singular Hermitian metrics_ , Math. Z. 303 (2023), no. 2, 20.


# NUMERICAL APPROXIMATION OF NONLINEAR STOCHASTIC VOLTERRA INTEGRAL EQUATION
USING WALSH FUNCTION 111Numerical Approximation of NLSVIE using Walsh
function.

Prit Pritam Paikaray<EMAIL_ADDRESS>Nigam Chandra
Parida22footnotemark: 2<EMAIL_ADDRESS>Sanghamitra Beuria33footnotemark: 3
<EMAIL_ADDRESS>
###### Abstract

This article proposes an efficient numerical method for solving nonlinear
stochastic Volterra integral equations using the operational matrices of the
Walsh function and the collocation method. In this method, a nonlinear
stochastic Volterra integral equation is reduced to a system of algebraic
equations, which are then solved to obtain an approximation of the solution.
Error analysis has been performed, confirming the effectiveness of the
methodology, which results in a linear order of convergence. Examples were
computed to demonstrate the efficacy and precision of the method.

###### keywords:

Nonlinear stochastic Volterra integral equation, Brownian motion, It$\hat{o}$
integral, Walsh approximation, Collocation method, Lipschitz condition.

###### MSC:

[2010] 60H05, 65C30

[label1]organization=Department of Mathematics, College of Basic Science and
Humanities, addressline=OUAT, city=Bhubaneswar, postcode=751003, state=Odisha,
country=India

##  1 Introduction

Non-linear stochastic Volterra integral equations (NLSVIEs) are extensively
used in numerous fields, including the physical sciences, biological sciences,
agricultural sciences, telecommunication networks, and financial mathematics,
among others [1, 2]. However, it is not possible to determine the exact
solutions to all NLSVIEs; consequently, a variety of numerical methods are
used to determine the approximate solutions to the problems in which
orthogonal function plays a vital role.

Orthogonal functions such as block pulse function (BPF) [9], Haar wavelet
[10], and Legendre polynomials [11] have been implemented to approximate the
solution of NLSVIEs. Walsh function has also been utilised to solve the
stochastic Volterra integral equation [12]. In this paper, we apply Walsh
function [3] to approximate solution $x(t)$ of the NLSVIE

$x(t)=x_{0}+\int_{0}^{t}k_{1}(s,t)\beta(x(s))ds+\int_{0}^{t}k_{2}(s,t)\sigma(x(s))dB(s)$
(1)

where $x(t)$, $k_{1}(s,t)$, $k_{2}(s,t)$ for $s,t\in[0,T)$, represents the
stochastic processes based on the same probability space $(\Omega,F,P)$ and
$x(t)$ is unknown.Here $B(t)$ is Brownian motion [1, 2] and
$\int_{0}^{t}k_{2}(s,t)x(s)dB(s)$ is the It$\hat{o}$ integral.

Convergence and error analysis of the method are performed to demonstrate the
method’s validity and precision. In the final segment, an illustrative example
is solved using the method and compared to a known exact solution to
demonstrate the method’s efficacy.

##  2 Walsh Function and its Properties

######  Definition 2.1 (Rademacher Function).

Rademacher function $r_{i}(t)$, $i=1,2,\ldots$, for $t\in[0,1)$ is defined by
[3]

$r_{i}(t)=\Biggl{\\{}\begin{aligned} 1&\quad\text{ $i=0$},\\\ sgn(sin(2^{i}\pi
t))&\quad\text{otherwise}\\\ \end{aligned}$

where,

$sgn(x)=\Biggl{\\{}\begin{aligned} 1&\quad\text{ $x>0$},\\\ 0&\quad\text{
$x=0$},\\\ -1&\quad\text{$x<0$}.\end{aligned}$

######  Definition 2.2 (Walsh Function).

The $n^{th}$ Walsh function for $n=0,1,2,\cdots,$ denoted by $w_{n}(t)$,
$t\in[0,1)$ is defined [3] as

$w_{n}(t)=(r_{q}(t))^{b_{q}}.(r_{q-1}(t))^{b_{q-1}}.(r_{q-2}(t))^{b_{q-2}}\ldots(r_{1}(t))^{b_{1}}$

where $n=b_{q}2^{q-1}+b_{q-1}2^{q-2}+b_{q-2}2^{q-3}+\ldots+b_{1}2^{0}$ is the
binary expression of $n$. Therefore, $q$, the number of digits present in the
binary expression of $n$ is calculated by $q=\big{[}\log_{2}n\big{]}+1$ in
which $\big{[}\cdot\big{]}$ is the greatest integer less than or equal’s to
${}^{\prime}\cdot^{\prime}$.

The first $m$ Walsh functions for $m\in\mathbb{N}$ can be written as an
$m$-vector by $W(t)=\begin{bmatrix}w_{0}(t)&w_{1}(t)&w_{2}(t)\ldots
w_{m-1}(t)\end{bmatrix}^{T}$, $t\in[0,1)$. The Walsh functions satisfy the
following properties.

### Orthonormality

The set of Walsh functions is orthonormal. i.e.,

$\int_{0}^{1}w_{i}(t)w_{j}(t)dt=\Biggl{\\{}\begin{aligned}
1&\quad\text{i=j,}\\\ 0&\quad\text{otherwise}.\end{aligned}$

### Completeness

For every $f\in L^{2}([0,1))$

$\int_{0}^{1}f^{2}(t)dt=\sum_{i=0}^{\infty}f_{i}^{2}\lvert\lvert
w_{i}(t)\lvert\lvert^{2}$

where $f_{i}=\int_{0}^{1}f(t)w_{i}(t)dt$.

### Walsh Function Approximation

Any real-valued function $f(t)\in L^{2}([0,1))$ can be approximated as

$f_{m}(t)\simeq\sum_{i=0}^{m-1}c_{i}w_{i}(t)$

where, $c_{i}=\int_{0}^{1}f(t)w_{i}(t)dt$.
The matrix form is given by

$f(t)\simeq F^{T}T_{W}W(t)$ (2)

where $F=\begin{bmatrix}f_{0}&f_{1}&f_{2}\ldots f_{m-1}\end{bmatrix}^{T}$,
$f_{i}=\int_{ih}^{(i+1)h}f(s)ds$.
Here, $T_{W}=[w_{i}(\eta_{j})]$ is called as the Walsh operational matrix
where $\eta_{j}\in[jh,(j+1)h)$.

Similarly, function $k(s,t)\in L^{2}([0,1)\times[0,1))$ can be approximated by

$k_{m}(s,t)\simeq\sum_{i=0}^{m-1}\sum_{j=0}^{m-1}c_{ij}w_{i}(s)w_{j}(t)$

where, $c_{ij}=\int_{0}^{1}\int_{0}^{1}k(s,t)w_{i}(s)w_{j}(t)dtds$ with the
matrix form represented by

$k(s,t)\simeq W^{T}(s)T_{W}KT_{W}W(t)=W^{T}(t)T_{W}K^{T}T_{W}W(s)$ (3)

where $K=[k_{ij}]_{m\times
m},k_{ij}=\int_{ih}^{(i+1)h}\int_{jh}^{(j+1)h}k(s,t)dtds$.

##  3 Relationship between Walsh Function and Block Pulse Functions (BPFs)

######  Definition 3.1 (Block Pulse Functions).

For a fixed positive integer $m$, an $m$-set of BPFs $\phi_{i}(t),t\in[0,1)$
for $i=0,1,...,m-1$ is defined as

$\phi_{i}(t)=\biggl{\\{}\begin{aligned} 1&\quad\text{if $\frac{i}{m}\leq
t<\frac{(i+1)}{m},\quad$}\\\ 0&\quad\text{ otherwise}\end{aligned}$

$\phi_{i}$ is known as the $i$th BPF.

The set of all $m$ BPFs can be written concisely as an $m$-vector,
$\Phi(t)=\begin{bmatrix}\phi_{0}(t)&\phi_{1}(t)&\phi_{2}(t)\ldots\phi_{m-1}(t)\end{bmatrix}^{T}$,
$t\in[0,1)$.

The BPFs are disjoint, complete, and orthogonal [4].

The BPFs in vector form satisfy

$\Phi(t)\Phi(t)^{T}X=\tilde{X}\Phi(t)\;\textrm{and}\;\Phi^{T}(t)A\Phi(t)=\hat{A}\Phi(t)$

where $X\in\mathbb{R}^{m\times 1},\tilde{X}$ is the $m\times m$ diagonal
matrix with $\tilde{X}(i,i)=X(i)\,\textrm{for}\,i=1,2,3\cdots
m,A\in\mathbb{R}^{m\times m}$ and
$\hat{A}=\begin{bmatrix}a_{11}&a_{22}&\ldots&a_{mm}\end{bmatrix}^{T}$ is the
$m$-vector with elements equal to the diagonal entries of $A$. The integration
of BPF vector $\Phi(t)$, $t\in[0,1)$ can be performed by [5]

$\int_{0}^{t}\Phi(\tau)d\tau=P\Phi(t),t\in[0,1),$ (4)

Hence, the integral of every function $f(t)\in L^{2}[0,1)$ can be approximated
as

$\int_{0}^{t}f(s)ds=F^{T}P\Phi(t)$

The It$\hat{o}$ integral of BPF vector $\Phi(t)$, $t\in[0,1)$ can be performed
by [6]

$\int_{0}^{t}\Phi(\tau)dB(\tau)=P_{S}\Phi(t),t\in[0,1)$ (5)

Hence, the It$\hat{o}$ integral of every function $f(t)\in L^{2}[0,1)$ can be
approximated as

$\int_{0}^{t}f(s)dB(s)=F^{T}P_{S}\Phi(t).$

The following theorem describes a relationship between the Walsh function and
the block pulse function.

######  Theorem 1.

[12] Let the $m$-set of Walsh function and BPF vectors be $W(t)$ and $\Phi(t)$
respectively. Then the BPF vectors $\Phi(t)$ can be used to approximate $W(t)$
as $W(t)=T_{W}\Phi(t)$, $m=2^{k}$, and $k=0,1,\ldots$, where
$T_{W}=\big{[}c_{ij}\big{]}_{m\times m}$, $c_{ij}=w_{i}(\eta_{j})$, for some
$\eta_{j}=\big{(}\frac{j}{m},\frac{j+1}{m}\big{)}$ and $i,j=0,1,2,\ldots m-1$.

One can see that [7]

$T_{W}T_{W}^{T}=mI\,\textrm{and}\,T_{W}^{T}=T_{W}$

which implies that $\Phi(t)=\frac{1}{m}T_{W}W(t).$

######  Lemma 1.

[12][Integration of Walsh function] Suppose that $W(t)$ is a Walsh function
vector, then the integral of $W(t)$ w.r.t. $t$ is given by
$\int_{0}^{t}W(s)ds=\wedge W(t)$, where $\wedge=\frac{1}{m}T_{W}PT_{W}$ and

$P=\frac{1}{h}\begin{bmatrix}1&2&2&\ldots&2\\\ 0&1&2&\ldots&2\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\ldots&1\end{bmatrix}$

######  Lemma 2.

[12][Stochastic integration of Walsh function] Suppose that $W(t)$ is a Walsh
function vector, then the It$\hat{o}$ integral of $W(t)$ is given by
$\int_{0}^{t}W(s)dB(s)=\wedge_{S}W(t)$, where
$\wedge_{S}=\frac{1}{m}T_{W}P_{S}T_{W}$ and

$P_{S}=\begin{bmatrix}B(\frac{h}{2})&B(h)&\ldots&B(h)\\\
0&B(\frac{3h}{2})-B(h)&\ldots&B(2h)-B(h)\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\ldots&B(\frac{(2m-1)h}{2})-B((m-1)h)\end{bmatrix}.$

##  4 Numerical Approximation of Nonlinear Stochastic Volterra Integral
Equation

Consider the following nonlinear stochastic integral equation (NLSDE)

$x(t)=x_{0}+\int_{0}^{t}k_{1}(s,t)\beta(x(s))ds+\int_{0}^{t}k_{2}(s,t)\sigma(x(s))dB(s)$
(6)

where $x(t)$, $k_{1}(s,t)$, $k_{2}(s,t)$ for $s,t\in[0,T)$, represents the
stochastic processes based on the same probability space $(\Omega,F,P)$ and
$x(t)$ is unknown.Here $B(t)$ is Brownian motion [1, 2] and
$\int_{0}^{t}k_{2}(s,t)x(s)dB(s)$ is the It$\hat{o}$ integral.
Let $z_{1}(t)=\beta(x(s))$ and $z_{2}(t)=\sigma(x(s))$ which implies,

$x(t)=x_{0}+\int_{0}^{t}k_{1}(s,t)z_{1}(t)ds+\int_{0}^{t}k_{2}(s,t)z_{2}(t)dB(s).$

We approximate $z_{1}(t)$, $z_{2}(t)$, $k_{1}(s,t)$, $k_{2}(s,t)$ for
$s,t\in[0,T)$ as For $\eta=1,2$,

$z_{\eta}(t)\simeq Z_{\eta}^{T}T_{W}W(t),$ (7)

where $Z_{\eta}=\begin{bmatrix}c_{0}&c_{1}&c_{2}\ldots
c_{m-1}\end{bmatrix}^{T}$ and $c_{i}=\int_{ih}^{(i+1)h}z_{\eta}(s)ds$.
Similarly, for $\gamma=1,2$

$k_{\gamma}(s,t)\simeq
W^{T}(s)T_{W}K_{\gamma}T_{W}W(t)=W^{T}(t)T_{W}K^{T}_{\gamma}T_{W}W(s)$ (8)

where $K_{\gamma}=[k_{ij}]_{m\times
m},k_{ij}=\int_{ih}^{(i+1)h}\int_{jh}^{(j+1)h}k_{\gamma}(s,t)dtds$.

Assume that

$x(t)\simeq X^{T}T_{W}W(t),$ (9)

where $X=\begin{bmatrix}x_{0}&x_{1}&x_{2}\ldots x_{m-1}\end{bmatrix}^{T}$ and
$x_{i}=\int_{ih}^{(i+1)h}x(s)ds$.

Substituting (7), (9) and (8) in (6) we have,

$\displaystyle X^{T}T_{W}W(t)$ $\displaystyle\simeq$ $\displaystyle
x_{0}+\int_{0}^{t}W^{T}(t)T_{W}K^{T}_{1}T_{W}W(s)W^{T}(s)T_{W}Z_{1}ds$ (10)
$\displaystyle+\int_{0}^{t}W^{T}(t)T_{W}K^{T}_{2}T_{W}W(s)W^{T}(s)T_{W}Z_{2}dB(s)$
$\displaystyle=$ $\displaystyle
x_{0}+W^{T}(t)T_{W}K^{T}_{1}T_{W}\int_{0}^{t}W(s)W^{T}(s)T_{W}Z_{1}ds$
$\displaystyle+W^{T}(t)T_{W}K^{T}_{2}T_{W}\int_{0}^{t}W(s)W^{T}(s)T_{W}Z_{2}dB(s)$

Now

$\int_{0}^{t}W(s)W^{T}(s)T_{W}Z_{1}ds=T_{W}\tilde{Z_{1}}PT_{W}W(t)$ (11)

Similarly,

$\int_{0}^{t}W(s)W^{T}(s)T_{W}Z_{2}dB(s)=T_{W}\tilde{Z_{2}}P_{S}T_{W}W(t)$
(12)

Substituting (11) and (12) in (10) we get,

$\displaystyle X^{T}T_{W}W(t)$ $\displaystyle=$ $\displaystyle
x_{0}+mW^{T}(t)T_{W}K^{T}_{1}\tilde{Z_{1}}PT_{W}W(t)$
$\displaystyle+mW^{T}(t)T_{W}K^{T}_{2}\tilde{Z_{2}}P_{S}T_{W}W(t)$
$\displaystyle=$ $\displaystyle
x_{0}+W^{T}(t)T_{W}H_{1}T_{W}W(t)+W^{T}(t)T_{W}H_{2}T_{W}W(t)$
$\displaystyle=$ $\displaystyle
x_{0}+m\hat{H_{1}}^{T}T_{W}W(t)+m\hat{H_{2}}^{T}T_{W}W(t)$

i.e.,

$\Big{(}X^{T}-m\hat{H_{1}}^{T}-m\hat{H_{2}}^{T}\Big{)}T_{W}W(t)=x_{0}$ (13)

and,

$\displaystyle
Z_{1}^{T}T_{W}W(t)=\beta(x_{0}+m\hat{H_{1}}^{T}T_{W}W(t)+m\hat{H_{2}}^{T}T_{W}W(t))$
(14) $\displaystyle
Z_{2}^{T}T_{W}W(t)=\sigma(x_{0}+m\hat{H_{1}}^{T}T_{W}W(t)+m\hat{H_{2}}^{T}T_{W}W(t))$

where, $H_{1}=mK^{T}_{1}\tilde{Z_{1}}P$, $H_{2}=mK^{T}_{2}\tilde{Z_{2}}P_{S}$
and $\hat{H}_{i}$ is the $m$-vector with elements equal to the diagonal
entries of $H_{i}$.
To evaluate $Z_{1}$ and $Z_{2}$, we collocate the above equation (14) at
$t_{j}=\frac{2j+1}{2m}$ for $j=0,1,\ldots,m-1$ and solve the system

$\displaystyle
Z_{1}^{T}T_{W}W(t_{j})=\beta(x_{0}+m\hat{H_{1}}^{T}T_{W}W(t_{j})+m\hat{H_{2}}^{T}T_{W}W(t_{j}))$
(15) $\displaystyle
Z_{2}^{T}T_{W}W(t_{j})=\sigma(x_{0}+m\hat{H_{1}}^{T}T_{W}W(t_{j})+m\hat{H_{2}}^{T}T_{W}W(t_{j}))$

##  5 Error Analysis

In this section, we analyse the error between the approximate solution and the
exact solution of the stochastic Volterra integral equation. Before we start
the analysis let us define, $E(|X|^{2})^{\frac{1}{2}}=\|X\|_{2}$.

######  Theorem 2.

[12] If $f\in L^{2}[0,1)$ satisfies the Lipschitz condition with Lipschitz
constant $C$, then $\|e_{m}(t)\|_{2}=O(h)$, where
$e_{m}(t)=|f(t)-\sum_{i=0}^{m-1}c_{i}w_{i}(t)|$ and
$c_{i}=\int_{0}^{1}f(s)w_{i}(s)ds$.

######  Theorem 3.

[12] Suppose $k\in L^{2}\big{(}[0,1)\times[0,1)\big{)}$ satisfies the
Lipschitz condition with Lipschitz constant $L$. If
$k_{m}(x,y)=\sum_{i=0}^{m-1}\sum_{j=0}^{m-1}c_{ij}w_{i}(x)w_{j}(y)$,
$c_{ij}=\int_{0}^{1}\int_{0}^{1}k(s,t)w_{i}(s)w_{j}(t)dtds$, then
$\|e_{m}(x,y)\|_{2}=O(h)$, where $|e_{m}(x,y)|=|k(x,y)-k_{m}(x,y)|$.

######  Theorem 4.

Suppose $x_{m}(t)$ be the approximate solution of the NLSIE (6). If

  1. a.

$f\in L^{2}[0,1)$, $k_{1}(s,t)$ and $k_{2}(s,t)\in
L^{2}\big{(}[0,1)\times[0,1)\big{)}$ satisfies the Lipschitz condition with
Lipschitz constants with Lipschitz constants $C$, $L_{1}$ and $L_{2}$
respectively

  2. b.

$|k_{1}(s,t)|\leq\rho_{1}$, $|k_{2}(s,t)|\leq\rho_{2}$,
$|\beta(x(t))|\leq\zeta_{1}$ and $|\sigma(x(t))|\leq\zeta_{2}$, and

  3. c.

for $\eta_{1},\eta_{2}\geq 0$,
$|\beta(x(t))-\beta(x_{m}(t))|\leq\eta_{1}|x(t)-x_{m}(t)|$,
$|\sigma(x(t))-\sigma(x_{m}(t))|\leq\eta_{2}|x(t)-x_{m}(t)|$

then $\|x(t)-x_{m}(t)\|_{2}^{2}=O(h^{2})$.

###### Proof.

Let (6) be the given Volterra integral equation and $x_{m}(t)$ be the
approximation to the solution using the Walsh function. Then

$\displaystyle x(t)-x_{m}(t)$ $\displaystyle=$ $\displaystyle f(t)-f_{m}(t)$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\big{(}k_{1}(s,t)\beta(x(s))-k_{1m}(s,t)\beta(x_{m}(s))\big{)}ds$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\big{(}k_{2}(s,t)\sigma(x(s))-k_{2m}(s,t)\sigma(x_{m}(s))\big{)}dB(s)$

or,

$\displaystyle|x(t)-x_{m}(t)|$ $\displaystyle\leq$
$\displaystyle|f(t)-f_{m}(t)|$ $\displaystyle+$
$\displaystyle\biggl{|}\int_{0}^{t}\big{(}k_{1}(s,t)\beta(x(s))-k_{1m}(s,t)\beta(x_{m}(s))\big{)}ds\biggr{|}$
$\displaystyle+$
$\displaystyle\biggl{|}\int_{0}^{t}\big{(}k_{2}(s,t)\sigma(x(s))-k_{2m}(s,t)\sigma(x_{m}(s))\big{)}dB(s)\biggr{|}.$

We know that, $(a+b+c)^{2}\leq 3a^{2}+3b^{2}+3c^{2}$

$\displaystyle|x(t)-x_{m}(t)|^{2}$ $\displaystyle\leq$ $\displaystyle
3|f(t)-f_{m}(t)|^{2}$ $\displaystyle+$ $\displaystyle
3\biggl{|}\int_{0}^{t}\big{(}k_{1}(s,t)\beta(x(s))-k_{1m}(s,t)\beta(x_{m}(s))\big{)}ds\biggr{|}^{2}$
$\displaystyle+$ $\displaystyle
3\biggl{|}\int_{0}^{t}\big{(}k_{2}(s,t)\sigma(x(s))-k_{2m}(s,t)\sigma(x_{m}(s))\big{)}dB(s)\biggr{|}^{2}.$

which implies that

$\displaystyle E\big{(}|x(t)-x_{m}(t)|^{2}\big{)}$ $\displaystyle\leq$
$\displaystyle E\biggl{(}3|f(t)-f_{m}(t)|^{2}\biggr{)}$ $\displaystyle+$
$\displaystyle
E\biggl{(}3\biggl{|}\int_{0}^{t}\big{(}k_{1}(s,t)\beta(x(s))-k_{1m}(s,t)\beta(x_{m}(s))\big{)}ds\biggr{|}^{2}\biggr{)}$
$\displaystyle+$ $\displaystyle
E\biggl{(}3\biggl{|}\int_{0}^{t}\big{(}k_{2}(s,t)\sigma(x(s))-k_{2m}(s,t)\sigma(x_{m}(s))\big{)}dB(s)\biggr{|}^{2}\biggr{)}.$

Suppose,

$I_{1}=E\biggl{(}3\biggl{|}\int_{0}^{t}\big{(}k_{1}(s,t)\beta(x(s))-k_{1m}(s,t)\beta(x_{m}(s))\big{)}ds\biggr{|}^{2}\biggr{)}$

and

$I_{2}=E\biggl{(}3\biggl{|}\int_{0}^{t}\big{(}k_{2}(s,t)\sigma(x(s))-k_{2m}(s,t)\sigma(x_{m}(s))\big{)}dB(s)\biggr{|}^{2}\biggr{)}.$

Using theorem 2 in (5) we have,

$E\big{(}|x(t)-x_{m}(t)|^{2}\big{)}\leq C^{2}h^{2}+I_{1}+I_{2}$ (17)

Now,

$\displaystyle|k_{1}(s,t)\beta(x(s))-k_{1m}(s,t)\beta(x_{m}(s))|=$
$\displaystyle|k_{1}(s,t)\beta(x(s))-k_{1}(s,t)\beta(x_{m}(s))$
$\displaystyle+$ $\displaystyle
k_{1}(s,t)\beta(x_{m}(s))-k_{1m}(s,t)\beta(x_{m}(s))\|$ $\displaystyle\leq$
$\displaystyle|k_{1}(s,t)||\beta(x(s))-\beta(x_{m}(s))|$ $\displaystyle+$
$\displaystyle|k_{1}(s,t)-k_{1m}(s,t)||\beta(x_{m}(s))|$ $\displaystyle=$
$\displaystyle|k_{1}(s,t)||\beta(x(s))-\beta(x_{m}(s))|$ $\displaystyle+$
$\displaystyle|k_{1}(s,t)-\beta
k_{1m}(s,t)||\beta(x(s))-\beta(x(s))+\beta(x_{m}(s))|$ $\displaystyle\leq$
$\displaystyle|k_{1}(s,t)||\beta(x(s))-\beta(x_{m}(s))|$ $\displaystyle+$
$\displaystyle|k_{1}(s,t)-k_{1m}(s,t)||\beta(x(s))|$ $\displaystyle+$
$\displaystyle|k_{1}(s,t)-k_{1m}(s,t)||\beta(x(s))-\beta(x_{m}(s))|$

Let $|k_{1}(s,t)|\leq\rho_{1}$, $|\beta(x(t))|\leq\zeta_{1}$,
$|\beta(x(t))-\beta(x_{m}(t))|\leq\eta_{1}|x(t)-x_{m}(t)|$ and using Theorem
3, we get

$\displaystyle|k_{1}(s,t)\beta(x(s))-k_{1m}(s,t)\beta(x_{m}(s))|$
$\displaystyle\leq$ $\displaystyle\rho_{1}\eta_{1}|x(s)-x_{m}(s)|$
$\displaystyle+$
$\displaystyle\sqrt{2}L_{1}h\zeta_{1}+\sqrt{2}L_{1}h\eta_{1}|x(s)-x_{m}(s)|$

which implies,

$\displaystyle|k_{1}(s,t)\beta(x(s))-k_{1m}(s,t)\beta(x_{m}(s))|\leq\sqrt{2}L_{1}h\zeta_{1}+(\rho_{1}\eta_{1}+\sqrt{2}L_{1}h\eta_{1})|x(s)-x_{m}(s)|$
(18)

which gives,

$\displaystyle I_{1}$ $\displaystyle\leq$ $\displaystyle
E\biggl{(}3\biggl{(}\int_{0}^{t}\biggl{|}k_{1}(s,t)\beta(x(s))-k_{1m}(s,t)\beta(x_{m}(s))\biggr{|}ds\biggr{)}^{2}\biggr{)}$
$\displaystyle\leq$ $\displaystyle
E\biggl{(}3\biggl{(}\int_{0}^{t}\big{(}\sqrt{2}L_{1}h\zeta_{1}$
$\displaystyle+$
$\displaystyle(\rho_{1}\eta_{1}+\sqrt{2}L_{1}h\eta_{1})|x(s)-x_{m}(s)|\big{)}ds\biggr{)}^{2}\biggr{)}$

By Cauchy- Schwarz inequality, for $t>0$ and $f\in L^{2}[0,t)$

$\biggl{|}\int_{0}^{t}f(s)ds\biggr{|}^{2}\leq t\int_{0}^{t}|f|^{2}ds$

this implies,

$\displaystyle I_{1}$ $\displaystyle\leq$ $\displaystyle
E\biggl{(}3\int_{0}^{t}\biggl{(}\sqrt{2}L_{1}h\zeta$ $\displaystyle+$
$\displaystyle(\rho_{1}\eta_{1}+\sqrt{2}L_{1}h\eta_{1})|x(s)-x_{m}(s)|\biggr{)}^{2}ds\biggr{)}$
$\displaystyle\leq$ $\displaystyle
E\biggl{(}6\int_{0}^{t}\biggl{(}\bigl{(}\sqrt{2}L_{1}h\zeta_{1}\bigr{)}^{2}$
$\displaystyle+$
$\displaystyle\bigl{(}(\rho_{1}\eta_{1}+\sqrt{2}L_{1}h\eta_{1})|x(s)-x_{m}(s)|\bigr{)}^{2}\biggr{)}ds\biggr{)}$
$\displaystyle\leq$ $\displaystyle
E\biggl{(}6\bigl{(}\sqrt{2}L_{1}h\zeta_{1}\bigr{)}^{2}$ $\displaystyle+$
$\displaystyle
6(\rho_{1}\eta_{1}+\sqrt{2}L_{1}h\eta_{1})^{2}\int_{0}^{t}|x(s)-x_{m}(s)|^{2}ds\biggr{)}$

Hence,

$\displaystyle I_{2}$ $\displaystyle\leq$ $\displaystyle
6\bigl{(}\sqrt{2}L_{1}h\zeta_{1}\bigr{)}^{2}$ $\displaystyle+$ $\displaystyle
6(\rho_{1}\eta_{1}+\sqrt{2}L_{1}h\eta_{1})^{2}\int_{0}^{t}E\bigl{(}|x(s)-x_{m}(s)|^{2}\bigr{)}ds.$

Now,

$\displaystyle I_{2}\leq$ $\displaystyle
E\biggl{(}3\int_{0}^{t}\biggl{|}k_{2}(s,t)\sigma(x(s))-k_{2m}(s,t)\sigma(x_{m}(s))\biggr{|}^{2}ds\biggr{)}$

Let $|k_{2}(s,t)|\leq\rho_{2}$, $|\sigma(x(t))|\leq\zeta_{2}$,
$|\sigma(x(t))-\sigma(x_{m}(t))|\leq\eta_{2}|x(t)-x_{m}(t)|$ and using Theorem
3, we get

$|k_{2}(s,t)\sigma(x(s))-k_{2m}(s,t)\sigma(x_{m}(s))|\leq\sqrt{2}L_{2}h\zeta_{2}+(\rho_{2}\eta_{2}+\sqrt{2}L_{2}h\eta_{2})|x(s)-x_{m}(s)|$
$\displaystyle I_{2}\leq$ $\displaystyle
E\biggl{(}3\int_{0}^{t}\big{(}\sqrt{2}L_{2}h\zeta_{2}+(\rho_{2}\eta_{2}+\sqrt{2}L_{2}h\eta_{2})|x(s)-x_{m}(s)|\big{)}^{2}ds\biggr{)}$
$\displaystyle\leq$ $\displaystyle
E\biggl{(}6\int_{0}^{t}\big{(}(\sqrt{2}L_{2}h\zeta_{2})^{2}+(\rho_{2}\eta_{2}+\sqrt{2}L_{2}h\eta_{2})^{2}|x(s)-x_{m}(s)|^{2}\big{)}ds\biggr{)}$
$\displaystyle=$ $\displaystyle
E\biggl{(}6(\sqrt{2}L_{2}h\zeta_{2})^{2}+6(\rho_{2}\eta_{2}+\sqrt{2}L_{2}h\eta_{2})^{2}\int_{0}^{t}|x(s)-x_{m}(s)|^{2}ds\biggr{)}$

Hence,

$\displaystyle I_{2}$ $\displaystyle\leq$ $\displaystyle
6(\sqrt{2}L_{2}h\zeta_{2})^{2}$ $\displaystyle+$ $\displaystyle
6(\rho_{2}\eta_{2}+\sqrt{2}L_{2}h\eta_{2})^{2}\int_{0}^{t}E\bigl{(}|x(s)-x_{m}(s)|^{2}\bigr{)}ds$

Using (5) and (5) in (17), we have

$\displaystyle E\big{(}|x(t)-x_{m}(t)|^{2}\big{)}$ $\displaystyle\leq$
$\displaystyle C^{2}h^{2}$ $\displaystyle+$ $\displaystyle
6\bigl{(}\sqrt{2}L_{1}h\zeta_{1}\bigr{)}^{2}$ $\displaystyle+$ $\displaystyle
6(\rho_{1}\eta_{1}+\sqrt{2}L_{1}h\eta_{1})^{2}\int_{0}^{t}E\bigl{(}|x(s)-x_{m}(s)|^{2}\bigr{)}ds$
$\displaystyle+$ $\displaystyle 6(\sqrt{2}L_{2}h\zeta_{2})^{2}$
$\displaystyle+$ $\displaystyle
6(\rho_{2}\eta_{2}+\sqrt{2}L_{2}h\eta_{2})^{2}\int_{0}^{t}E\bigl{(}|x(s)-x_{m}(s)|^{2}\bigr{)}ds$

which implies that

$\displaystyle E\big{(}|x(t)-x_{m}(t)|^{2}\big{)}$ $\displaystyle\leq$
$\displaystyle\bigl{(}C^{2}h^{2}+6\bigl{(}\sqrt{2}L_{1}h\zeta_{1}\bigr{)}^{2}+6(\sqrt{2}L_{2}h\zeta_{2})^{2}\bigr{)}$
$\displaystyle+$
$\displaystyle\bigl{(}6(\rho_{1}\eta_{1}+\sqrt{2}L_{1}h\eta_{1})^{2}+6(\rho_{2}\eta_{2}+\sqrt{2}L_{2}h\eta_{2})^{2}\bigr{)}\int_{0}^{t}E\bigl{(}|x(s)-x_{m}(s)|^{2}\bigr{)}ds.$

Let

$\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle\bigl{(}C^{2}h^{2}+6\bigl{(}\sqrt{2}L_{1}h\zeta_{1}\bigr{)}^{2}+6(\sqrt{2}L_{2}h\zeta_{2})^{2}\bigr{)}$
$\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle\bigl{(}6(\rho_{1}\eta_{1}+\sqrt{2}L_{1}h\eta_{1})^{2}+6(\rho_{2}\eta_{2}+\sqrt{2}L_{2}h\eta_{2})^{2}\bigr{)}$

By using Gronwall’s inequality, we have

$\displaystyle E\big{(}|x(t)-x_{m}(t)|^{2}\big{)}$ $\displaystyle\leq$
$\displaystyle R_{1}\exp\biggl{(}\int_{0}^{t}R_{2}ds\biggr{)}.$ (22)

which implies that,

$\|x(t)-x_{m}(t)\|_{2}^{2}=E\big{(}|x(t)-x_{m}(t)|^{2}\big{)}\leq
R_{1}e^{R_{2}}=O(h^{2})$ (23)

∎

##  6 Numerical Example

In this section, we use the proposed method to solve a variety of SVIEs. The
first three examples compare approximate and analytical results to demonstrate
the method’s convergence. Define error $E$ as $\lVert
E\rVert_{\infty}=\underset{1\leq i\leq m}{max}\lvert X_{i}-Y_{i}\rvert$, where
$X_{i}$, $Y_{i}$ are the Walsh coefficient of exact solution and approximate
solution respectively. The number of iterations in the following instances is
$n$, the mean of error $E$ is $\bar{x}_{E}$, and the standard deviation for
error E is $s_{E}$. The computations are carried out using Matlab 2013(a).

######  Example 1.

[1] Consider the NLSVIE

$x(t)=x_{0}-\frac{a^{2}}{2}\int_{0}^{t}tanh(x(s))sech^{2}(x(s))ds+a\int_{0}^{t}sech(x(s))dB(s)$

where, $x_{0}=0$ and $a=\frac{1}{30}$ and $x(t)$ represents the unknown
stochastic process based on the same probability space $(\Omega,F,P)$, and
$B(t)$ is a Brownian motion process. The exact solution is given by
$x(t)=arcsinh(aB(t)+sinh(x_{0}))$.

Table 1: Mean error, standard deviation of error, and interval of confidence for mean error in Example 1 with m=16 and 50 iterations. t | $\bar{x}_{E}$ | $s_{E}$ | 95% interval of confidence for error mean.
---|---|---|---
|  |  | Lower | Upper
0.1 | $0.0073\times 10^{-5}$ | $0.0137\times 10^{-5}$ | $0.0348\times 10^{-6}$ | $0.0111\times 10^{-5}$
0.3 | $0.0330\times 10^{-5}$ | $0.1151\times 10^{-5}$ | $0.0109\times 10^{-6}$ | $0.0649\times 10^{-5}$
0.5 | $0.0900\times 10^{-5}$ | $0.3352\times 10^{-5}$ | $0.0290\times 10^{-6}$ | $0.1829\times 10^{-5}$
0.7 | $0.1402\times 10^{-5}$ | $0.4698\times 10^{-5}$ | $0.0995\times 10^{-6}$ | $0.2704\times 10^{-5}$
0.9 | $0.1939\times 10^{-5}$ | $0.6806\times 10^{-5}$ | $0.0519\times 10^{-6}$ | $0.3825\times 10^{-5}$
Figure 1: Example 1’s approximate and exact solutions for m=32 and m=64  Figure 2: Example 1’s error trend for m=16,m=32, and n=50 Table 2: Mean error, standard deviation of error, and interval of confidence for mean error in Example 1 with m=32 and 50 iterations. t | $\bar{x}_{E}$ | $s_{E}$ | 95% interval of confidence for error mean.
---|---|---|---
|  |  | Lower | Upper
0.1 | $0.0192\times 10^{-5}$ | $0.0223\times 10^{-5}$ | $0.0130\times 10^{-5}$ | $0.0254\times 10^{-5}$
0.3 | $0.0809\times 10^{-5}$ | $0.1477\times 10^{-5}$ | $0.0400\times 10^{-5}$ | $0.1219\times 10^{-5}$
0.5 | $0.1577\times 10^{-5}$ | $0.3446\times 10^{-5}$ | $0.0621\times 10^{-5}$ | $0.2532\times 10^{-5}$
0.7 | $0.2262\times 10^{-5}$ | $0.4336\times 10^{-5}$ | $0.1061\times 10^{-5}$ | $0.3464\times 10^{-5}$
0.9 | $0.2992\times 10^{-5}$ | $0.5677\times 10^{-5}$ | $0.1418\times 10^{-5}$ | $0.4565\times 10^{-5}$

##  7 Conclusion

The proposed numerical method solves the integral equation using the Walsh
function, which converts the problem into a system of algebraic equations, and
then solves the system to acquire an approximation of the solution to the
problem. The order of convergence of the method is determined to be $O(h)$
through convergence and error analysis of the method. In the final section,
the suggested technique is used to solve a numerical examples with known exact
solutions, and the numerical result are presented in tables and figures. These
numerical result indicate that the current method is highly precise and
effective. As one can see, high-precision results can be achieved by engaging
a limited number of basis functions. An increase in the number of basis
functions also reduces absolute errors. The method can further be modified to
obtain solutions to nonlinear stochastic integral equations with fractional
Brownian motion.

## References

  * [1] P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, in: Applications of Mathematics, Springer-Verlag, Berlin, 1999.
  * [2] B. Oksendal, Stochastic Differential Equations, fifth ed., in: An Introduction with Applications, Springer-Verlag, New York, 1998.
  * [3] J. L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math. vol. 55 (1923) pp. 5-24.
  * [4] G. Prasada Rao, Piecewise Constant Orthogonal Functions and Their Application to Systems and Control, Springer, Berlin, 1983.
  * [5] Saeed Hatamzadeh-Varmazyar, Zahra Masouri, Esmail Babolian, Numerical method for solving arbitrary linear differential equations using a set of orthogonal basis functions and operational matrix, Applied Mathematical Modelling 40 (2016) 233–253.
  * [6] K. Maleknejad, M. Khodabin, M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions,Mathematical and Computer Modelling, Mathematical and Computer Modelling 55 (2012) 791–800.
  * [7] C. F. Cheng, Y. Tsay, T. T. Wu, Walsh operational matrices for fractional calculus and their application to distributed systems,Journal of The Franklin Institute-engineering and Applied Mathematics(1997).
  * [8] B. Golubov, A. Efimov, V. Skvortsov, Walsh Series and Transforms Theory and Applications.
  * [9] M. Asgari, F. H. Shekarabi, Numerical solution of nonlinear stochastic differential equations using the block pulse operational matrices. Math Sci 7, 47 (2013).
  * [10] J. Wu, G. Jiang, X. Sang, Numerical solution of nonlinear stochastic Itô–Volterra integral equations based on Haar wavelets. Adv Differ Equ 2019, 503 (2019).
  * [11] R. Zeghdane, Numerical approach for solving nonlinear stochastic Itô-Volterra integral equations using shifted Legendre polynomials, International Journal of Dynamical Systems and Differential Equations 11.1 (2021): 69-88.
  * [12] Paikaray P. P., Beuria S., Parida N. Ch., Numerical approximation of p-dimensional stochastic Volterra integral equation using Walsh function. J Math Comput SCI-JM.(2023); 31(4):448–460