# New scattered linearized quadrinomials

Valentino Smaldore, Corrado Zanella and Ferdinando Zullo

###### Abstract

Let $1<t<n$ be integers, where $t$ is a divisor of $n$. An
$\operatorname{R-}q^{t}\operatorname{-}$partially scattered polynomial is a
${{\mathbb{F}}_{q}}$-linearized polynomial $f$ in $\mathbb{F}_{q^{n}}[X]$ that
satisfies the condition that for all $x,y\in\mathbb{F}_{q^{n}}^{*}$ such that
$x/y\in\mathbb{F}_{q^{t}}$, if $f(x)/x=f(y)/y$, then
$x/y\in{{\mathbb{F}}_{q}}$; $f$ is called scattered if this implication holds
for all $x,y\in\mathbb{F}_{q^{n}}^{*}$. Two polynomials in
$\mathbb{F}_{q^{n}}[X]$ are said to be equivalent if their graphs are in the
same orbit under the action of the group $\operatorname{\Gamma L}(2,q^{n})$.
For $n>8$ only three families of scattered polynomials in
$\mathbb{F}_{q^{n}}[X]$ are known: $(i)$ monomials of pseudoregulus type,
$(ii)$ binomials of Lunardon-Polverino type, and $(iii)$ a family of
quadrinomials defined in [1, 10] and extended in [8, 13]. In this paper we
prove that the polynomial
$\varphi_{m,{q^{J}}}=X^{q^{J(t-1)}}+X^{q^{J(2t-1)}}+m(X^{q^{J}}-X^{q^{J(t+1)}})\in\mathbb{F}_{q^{2t}}[X]$,
$q$ odd, $t\geq 3$ is $\operatorname{R-}q^{t}\operatorname{-}$partially
scattered for every value of $m\in\mathbb{F}_{q^{t}}^{*}$ and $J$ coprime with
$2t$. Moreover, for every $t>4$ and $q>5$ there exist values of $m$ for which
$\varphi_{m,q}$ is scattered and new with respect to the polynomials mentioned
in $(i)$, $(ii)$ and $(iii)$ above. The related linear sets are of
$\operatorname{\Gamma L}$-class at least two.

2020 MSC: 11T71; 11T06; 94B05
Keywords: Scattered polynomials; linearized polynomials; MRD codes

##  1 Introduction

In this paper $q=p^{\varepsilon}$ denotes a power of a prime $p$. Let $n$ be a
positive integer. An _${{\mathbb{F}}_{q}}$ -linearized polynomial_, or _$q$
-polynomial_, in ${{\mathbb{F}}_{q^{n}}}[X]$ is of type

$f=\sum_{i=0}^{d}a_{i}X^{q^{i}},\quad a_{i}\in{{\mathbb{F}}_{q^{n}}},\
i=0,1,\ldots,d.$ (1)

If $a_{d}\neq 0$, then the $q$-polynomial $f$ has _$q$ -degree_ $d$. The set
of linearized polynomials over ${{\mathbb{F}}_{q^{n}}}$ is denoted by
$L_{n,q}$. Such a set, equipped with the operations of sum, multiplication by
elements of ${{\mathbb{F}}_{q}}$, and the composition, results to be an
${{\mathbb{F}}_{q}}$-algebra. The quotient algebra
$\mathcal{L}_{n,q}=L_{n,q}/(X^{q^{n}}-X)$ is isomorphic to the
${{\mathbb{F}}_{q}}$-algebra of the ${{\mathbb{F}}_{q}}$-linear endomorphisms
of ${{\mathbb{F}}_{q^{n}}}$. Hence, for every ${{\mathbb{F}}_{q}}$-linear
endomorphism $\Phi$ of ${{\mathbb{F}}_{q^{n}}}$ there exists a unique
$q$-polynomial $f$ of $q$-degree less than $n$ such that $f(x)=\Phi(x)$ for
any $x\in{{\mathbb{F}}_{q^{n}}}$. By abuse of notation, any $f\in L_{n,q}$ is
identified with the class $f+(X^{q^{n}}-X)\in\mathcal{L}_{n,q}$.

Linearized polynomials have found many applications in various areas of
combinatorics, coding theory and cryptography. Special attention has been paid
to _scattered polynomials_ , introduced by Sheekey in [16], in the context of
optimal codes in the rank metric, as we will see later. A _scattered
polynomial_ is an $f\in\mathcal{L}_{n,q}$ such that for any
$x,y\in\mathbb{F}_{q^{n}}^{*}$,

$\frac{f(x)}{x}=\frac{f(y)}{y}\ \Longrightarrow\
\frac{x}{y}\in\mathbb{F}_{q}.$ (2)

A generalization of the notion of scattered polynomials was recently
introduced in [9]. Let $f$ be an ${{\mathbb{F}}_{q}}$-linearized polynomial in
$\mathcal{L}_{n,q}$, and $t$ a nontrivial divisor of $n$; so, $n=tt^{\prime}$,
and $1<t,t^{\prime}<n$. We say that $f$ is
_$\operatorname{L-}q^{t}\operatorname{-}$ partially scattered_ if for any
$x,y\in\mathbb{F}_{q^{n}}^{*}$,

$\frac{f(x)}{x}=\frac{f(y)}{y}\ \Longrightarrow\
\frac{x}{y}\in\mathbb{F}_{q^{t}},$ (3)

and that $f$ is _$\operatorname{R-}q^{t}\operatorname{-}$ partially scattered_
if for any $x,y\in\mathbb{F}_{q^{n}}^{*}$,

$\frac{f(x)}{x}=\frac{f(y)}{y}\,\mbox{ and
}\,\frac{x}{y}\in\mathbb{F}_{q^{t}}\ \Longrightarrow\
\frac{x}{y}\in{{\mathbb{F}}_{q}}.$ (4)

A $q$-polynomial is scattered if and only if it is both
$\operatorname{L-}q^{t}\operatorname{-}$ and
$\operatorname{R-}q^{t}\operatorname{-}$partially scattered.

The _graph_ of $f\in{{\mathbb{F}}_{q^{n}}}[X]$ is $U_{f}=\\{(x,f(x))\colon
x\in{{\mathbb{F}}_{q^{n}}}\\}$. If $f\in\mathcal{L}_{n,q}$, then $U_{f}$ is an
${{\mathbb{F}}_{q}}$-subspace of $\mathbb{F}_{q^{n}}^{2}$. We say that two
polynomials $f$ and $g$ in $\mathcal{L}_{n,q}$ are _$\operatorname{\Gamma L}$
-equivalent_, or simply _equivalent_ , if their graphs are in the same orbit
under the action of the group $\operatorname{\Gamma L}(2,q^{n})$. Up to such a
notion of equivalence, only three families of scattered polynomials in
$\mathcal{L}_{n,q}$ are known for $n>8$:

  * $(i)$

$X^{q^{s}}$, with $\gcd(s,n)=1$, known as _pseudoregulus type_ ;

  * $(ii)$

$X^{q^{n-s}}+\delta X^{q^{s}}$, with $\gcd(s,n)=1$ and
$\operatorname{N}_{q^{n}/q}(\delta)=\delta^{\frac{q^{n}-1}{q-1}}\neq 0,1$,
known as _Lunardon-Polverino_ type [12, 16];

  * $(iii)$

$\psi_{s,h}=X^{q^{s}}+X^{q^{s(t-1)}}+h^{1+q^{s}}X^{q^{s(t+1)}}+h^{1-q^{s(2t-1)}}X^{q^{s(2t-1)}}\in\mathcal{L}_{2t,q}$,
where $q$ is odd, $\gcd(s,2t)=1$, $h\in\mathbb{F}_{q^{2t}}$ and
$h^{q^{t}+1}=-1$ [1, 8, 10, 13].

It is convenient to split the family $(iii)$ into two separate families
$(iii\text{-}a)$ of all polynomials $\psi_{s,h}$ such that
$h\in{{\mathbb{F}}_{q^{t}}}$, and $(iii\text{-}b)$, where
$h\notin{{\mathbb{F}}_{q^{t}}}$.

One of the reasons that scattered polynomials and their generalizations have
attracted so much attention is their connection to optimal codes in the theory
of rank-metric codes. See [15, 16] for a general overview of this topic. Rank-
metric codes are sets of $n\times m$ matrices over a finite field
$\mathbb{F}_{q}$, endowed with the rank metric. When $n=m$, the rank-metric
codes can be represented in terms of linearized polynomials, since the
${{\mathbb{F}}_{q}}$-algebra $\mathbb{F}_{q}^{n\times n}$ is isomorphic to
$\mathcal{L}_{n,q}$. Of particular interest is the family of _maximum rank
distance (MRD)_ codes, which are of maximum size for given $n$ and given
minimum rank distance. The first construction of a family of MRD codes was due
to Delsarte [3] and independently to Gabidulin [4]. The codes of this family
are now known as the _Gabidulin codes_. Sheekey in [16] pointed out a way to
construct special classes of MRD codes: if $f\in\mathcal{L}_{n,q}$ is a
scattered polynomial then $\operatorname{\mathcal{C}}_{f}=\langle
X,f\rangle_{{{\mathbb{F}}_{q^{n}}}}$ is an MRD code of size $q^{2n}$ and
minimum distance $n-1$. In the same paper, he also proved that the equivalence
of linearized polynomials corresponds to the equivalence of rank-metric codes.
Therefore, the scattered polynomials $(i)$, $(ii)$ and $(iii)$ give rise to
three disjoint families of MRD codes.

The _adjoint_ of a linearized polynomial
$f=\sum_{i=0}^{n-1}a_{i}X^{\sigma^{i}}$ with respect to the trace form is the
polynomial $f^{\top}$ which satisfies
$\operatorname{Tr}_{q^{n}/q}(f(x)y)=\operatorname{Tr}_{q^{n}/q}(xf^{\top}(y))$
for every $x,y\in{{\mathbb{F}}_{q^{n}}}$
(111$\operatorname{Tr}_{q^{n}/q}(x)=x+x^{q}+\cdots+x^{q^{n-1}}$ for
$q\in{{\mathbb{F}}_{q^{n}}}$.). It holds

$f^{\top}=\sum_{i=0}^{n-1}a_{i}^{\sigma^{n-i}}X^{\sigma^{n-i}}.$

Such $f^{\top}$ is scattered if and only if $f$ is. Even, $f$ and $f^{\top}$
determine the same linear set $L_{f}=L_{f^{\top}}$ [2]. However, the
polynomials $f$ and $f^{\top}$ need not to be equivalent. The classes $(i)$,
$(ii)$, and $(iii\text{-}b)$ described above contain their own adjoints up to
equivalence. Furthermore, every polynomial in the collection $(iii\text{-}b)$
is equivalent to its adjoint [8, Theorem 4.6], [13, Proposition 4.17].

The scattered polynomials can also be used to construct scattered linear sets.
We refer the reader to [6, 14, 15] for generalities on this topic. Any linear
set in the projective line $\operatorname{PG}(1,q^{n})$ can be defined, up to
the action of GL$(2,q^{n})$, by a linearized polynomial $f$ as follows

$L_{f}=\\{\langle(x,f(x))\rangle_{{{\mathbb{F}}_{q^{n}}}}\colon
x\in\mathbb{F}_{q^{n}}^{*}\\},$

and we say that it is _scattered_ if $|L_{f}|=(q^{n}-1)/(q-1)$. It is a
straightforward check that $f$ is scattered if and only if $L_{f}$ is
scattered. Scattered linear sets on the projective line have been related to
various objects in Galois geometries, such as linear minimal blocking sets of
largest order, translation planes and many more; see for example [6, 10, 14].

In this paper, we study the linearized polynomials

$\varphi_{m,\sigma}=X^{{\sigma}^{t-1}}+X^{{\sigma}^{2t-1}}+m(X^{\sigma}-X^{{\sigma}^{t+1}})\in\mathcal{L}_{2t,q},$

where $t\geq 3$, $m\in\mathbb{F}_{q^{t}}^{*}$, $x\mapsto x^{\sigma}$ is a
generator of $\operatorname{Gal}({{\mathbb{F}}_{q^{n}}}/{{\mathbb{F}}_{q}})$,
i.e. $\sigma={q^{J}}$, $J\in\\{1,2,\ldots,2t-1\\}$, $\gcd(J,2t)=1$, and $q$ is
odd. In Section 2, we will first prove that any such $\varphi_{m,\sigma}$ is
R-$q^{t}$-partially scattered, and then we will show some conditions on $m$
that ensure that $\varphi_{m,\sigma}$ is scattered. In Section 3, we find the
stabilizer of the graph of $\varphi_{m,\sigma}$ under the action of the group
$\mathrm{GL}(2,q^{n})$, which turns out to be an invariant for equivalence. In
the last section we consider the question of equivalence between
$\varphi_{m,q}$ and the known families of scattered polynomials. The adjoint
$\varphi_{m,q}^{\top}$ of a scattered polynomial $\varphi_{m,q}$ is not
equivalent to $\varphi_{k,q}$ for any $k\in{{\mathbb{F}}_{q^{t}}}$; hence,
$\varphi_{m,q}$ does not belong to the family $(iii\text{-}b)$ (Proposition
4.4). The main result of this paper is Theorem 4.7. It states that for $t>4$,
if $t$ is even and $q>3$ or $t$ is odd and $q>5$, there exists at least one
scattered polynomial of type $\varphi_{m,q}$ that is not equivalent to any
known scattered polynomial.

In [2] the notion of a _$\operatorname{\Gamma L}$ -class_ of a linear set $L$
has been introduced, which is the number of nonequivalent polynomials $f$ such
that $L_{f}=L$. As a consequence of Proposition 4.4, the $\operatorname{\Gamma
L}$-class of any scattered linear set of type $L_{\varphi_{m,q}}$ is of at
least two.

##  2 A family of $\operatorname{R-}q^{t}\operatorname{-}$partially scattered
polynomials

From now on we will assume that $q$ is an odd prime power, $t\geq 3$ is an
integer, and $n=2t$. We will show a family of
$\operatorname{R-}q^{t}\operatorname{-}$partially scattered polynomials in
$\mathcal{L}_{n,q}$.

For $m\in{{\mathbb{F}}_{q^{n}}}$ and $\sigma=q^{J}$, $J\in\\{1,\ldots,n-1\\}$,
$\gcd(J,n)=1$, consider the following $q$-polynomials in $\mathcal{L}_{n,q}$:

$\alpha=\alpha_{\sigma}=X^{{\sigma}^{t-1}}+X^{{\sigma}^{2t-1}}\,\,\,\text{and}\,\,\,\beta=\beta_{m,\sigma}=m(X^{\sigma}-X^{{\sigma}^{t+1}}).$

Define

$W=\\{x\in\mathbb{F}_{q^{n}}\colon x^{q^{t}}+x=0\\},$

which is a one-dimensional $\mathbb{F}_{q^{t}}$-subspace of
$\mathbb{F}_{q^{n}}$. Note that $x^{-1},x^{q},x^{\sigma}\in W$ for any $x\in
W$, $x\neq 0$. Some results in the sequel, such as the next ones, are based on
the fact that the kernel of a non-trivial ${{\mathbb{F}}_{q}}$-linear map of
type $\Phi(x)=\sum_{i=0}^{d}a_{i}x^{\sigma^{i}}$ has dimension at most $d$
over ${{\mathbb{F}}_{q}}$ (see e.g. [5, Theorem 5]).

######  Lemma 2.1.

The following hold:

  1. 1.

if $A\in\mathbb{F}_{q^{t}}$ and $B\in W$ then $AB\in W$;

  2. 2.

if $A,B\in W$ then $AB\in\mathbb{F}_{q^{t}}$;

  3. 3.

$\mathbb{F}_{q^{2t}}=\mathbb{F}_{q^{t}}\oplus W$;

  4. 4.

$\ker(\alpha)=W$;

  5. 5.

$\mathrm{Im}(\alpha)=\mathbb{F}_{q^{t}}$;

  6. 6.

$\ker(\beta)=\mathbb{F}_{q^{t}}$;

  7. 7.

$\mathrm{Im}(\beta)=W$

###### Proof.

We only prove 4. and 5. Raising to the $\sigma$ one obtains

$\ker(\alpha)=\\{x^{\sigma^{t}}+x=0\\}.$

Furthermore, if $x\in W$, then
$x^{\sigma^{t}}=(((x^{q^{t}})^{q^{t}})\cdots)^{q^{t}}$, an odd number of
powers, hence $x^{\sigma^{t}}=-x$, that is $x\in\ker(\alpha)$. This implies
$W\subseteq\ker(\alpha)$, and 4. follows from
$\dim_{{{\mathbb{F}}_{q}}}\ker(\alpha)\leq t$.

Next, ${{\mathbb{F}}_{q^{t}}}=\\{y\in{{\mathbb{F}}_{q^{n}}}\colon
y^{\sigma^{t}}-y=0\\}$; if $y=\alpha(x)$, then

$y^{\sigma^{t}}-y=(x^{{\sigma}^{t-1}}+x^{{\sigma}^{2t-1}})^{\sigma^{t}}-(x^{{\sigma}^{t-1}}+x^{{\sigma}^{2t-1}})=0.$

∎

######  Theorem 2.2.

Let $t\geq 3$ be an integer. Assume $m\in\mathbb{F}_{q^{t}}^{*}$ and
$\sigma=q^{J}$, $J\in\\{1,\ldots,2t-1\\}$, $\gcd(J,2t)=1$. Then the polynomial

$\varphi_{m,\sigma}=X^{{\sigma}^{t-1}}+X^{{\sigma}^{2t-1}}+m(X^{\sigma}-X^{{\sigma}^{t+1}})\in\mathcal{L}_{2t,q}$
(5)

is $\operatorname{R-}q^{t}\operatorname{-}$partially scattered.

###### Proof.

The polynomial $\varphi_{m,\sigma}=\alpha+\beta$ is
$\operatorname{R-}q^{t}\operatorname{-}$partially scattered if and only if
$\varphi_{m,\sigma}$ satisfies the condition that for any
$\rho\in\mathbb{F}_{q^{t}}$ and $x\in\mathbb{F}_{q^{n}}$ such that $x\neq 0$,
if

$\varphi_{m,\sigma}(\rho x)=\rho\varphi_{m,\sigma}(x),$ (6)

then $\rho\in\mathbb{F}_{q}$. So, suppose that (6) holds and, because of 3. of
Lemma 2.1, we can write $x=x_{1}+x_{2}$, where $x_{1}\in\mathbb{F}_{q^{t}}$
and $x_{2}\in W$. Using 1., 2., 4. and 5. of Lemma 2.1 we obtain

$\varphi_{m,\sigma}(\rho x)=\alpha(\rho x_{1})+\beta(\rho x_{2})$

and

$\rho\varphi_{m,\sigma}(x)=\rho\alpha(x_{1})+\rho\beta(x_{2}).$

Since $\mathbb{F}_{q^{2t}}=\mathbb{F}_{q^{t}}\oplus W$, by 5. and 7. of Lemma
2.1 we have

$\left\\{\begin{array}[]{ll}\alpha(\rho x_{1})=\rho\alpha(x_{1}),\\\
\beta(\rho x_{2})=\rho\beta(x_{2}),\end{array}\right.$ (7)

which can be rewritten as

$\left\\{\begin{array}[]{ll}(\rho^{{\sigma}^{t-1}}-\rho)x_{1}^{{\sigma}^{t-1}}=0,\\\
m(\rho^{\sigma}-\rho)x_{2}^{\sigma}=0,\end{array}\right.$

and since $m\neq 0$ and at least one among $x_{1}$ and $x_{2}$ is nonzero,
then $\rho^{{\sigma}^{t-1}}-\rho=0$ or $\rho^{\sigma}-\rho=0$. In each case we
get $\rho\in{{\mathbb{F}}_{q}}$. ∎

In the following we show that for certain values of $m$ the polynomial
$\varphi_{m,\sigma}$ is also $\operatorname{L-}q^{t}\operatorname{-}$partially
scattered.

######  Theorem 2.3.

Let $t\geq 3$ be an integer, and $W=\\{x\colon
x\in\mathbb{F}_{q^{2t}},\,x^{q^{t}}+x=0\\}$. Assume $\sigma=q^{J}$,
$J\in\\{1,\ldots,2t-1\\}$, $\gcd(J,2t)=1$. If $m\in{{\mathbb{F}}_{q^{t}}}$ is
neither a $(q-1)$-th power nor a $(q+1)$-th power of any element of $W$ then
the polynomial
$\varphi_{m,\sigma}=X^{{\sigma}^{t-1}}+X^{{\sigma}^{2t-1}}+m(X^{\sigma}-X^{{\sigma}^{t+1}})\in\mathcal{L}_{2t,q}$
is scattered.

###### Proof.

By Theorem 2.2, it is enough to prove that $\varphi_{m,\sigma}$ is
$\operatorname{L-}q^{t}\operatorname{-}$partially scattered; that is, it
satisfies the condition that for any $\rho,x\in\mathbb{F}_{q^{n}}$ such that
$x\neq 0$, if

$\varphi_{m,\sigma}(\rho x)=\rho\varphi_{m,\sigma}(x),$ (8)

then $\rho\in{{\mathbb{F}}_{q^{t}}}$. So, suppose that (8) holds. Because of
3. of Lemma 2.1, we can write

$\rho=h+r\,\,\,\text{and}\,\,\,x=x_{1}+x_{2},$

where $h,x_{1}\in\mathbb{F}_{q^{t}}$ and $r,x_{2}\in W$. Using 1., 2., 4. and
5. of Lemma 2.1 we obtain

$\varphi_{m,\sigma}(\rho
x)=\alpha(hx_{1})+\alpha(rx_{2})+\beta(hx_{2})+\beta(rx_{1})$

and

$\rho\varphi_{m,\sigma}(x)=h\alpha(x_{1})+h\beta(x_{2})+r\alpha(x_{1})+r\beta(x_{2}).$

Since $\mathbb{F}_{q^{2t}}=\mathbb{F}_{q^{t}}\oplus W$, by 5. and 7. of Lemma
2.1 we have

$\left\\{\begin{array}[]{ll}\alpha(hx_{1})+\alpha(rx_{2})=h\alpha(x_{1})+r\beta(x_{2}),\\\
\beta(hx_{2})+\beta(rx_{1})=h\beta(x_{2})+r\alpha(x_{1}),\end{array}\right.$
(9)

which can be rewritten as

$\left\\{\begin{array}[]{ll}mx_{2}^{\sigma}r-x_{2}^{{\sigma}^{t-1}}r^{{\sigma}^{t-1}}=(h^{{\sigma}^{t-1}}-h)x_{1}^{{\sigma}^{t-1}},\\\
x_{1}^{{\sigma}^{t-1}}r-mx_{1}^{\sigma}r^{\sigma}=-m(h-h^{\sigma})x_{2}^{\sigma},\end{array}\right.$

and raising the first equation to the ${\sigma}$ we obtain

$\left\\{\begin{array}[]{ll}-x_{2}r+m^{\sigma}x_{2}^{{\sigma}^{2}}r^{\sigma}=(h-h^{\sigma})x_{1},\\\
-x_{1}^{{\sigma}^{t-1}}r+mx_{1}^{\sigma}r^{\sigma}=m(h-h^{\sigma})x_{2}^{\sigma}.\end{array}\right.$
(10)

Suppose by contradiction that $r\neq 0$. We divide the proof in four cases.
Since $J$ is odd, every $(\sigma-1)$-th power (resp. $(\sigma+1)$-th power) of
an element of $W$ is also a $(q-1)$-th power (resp. $(q+1)$-th power) of an
element of $W$.
Case 1: $x_{1}=0$.
In this case $x_{2}\neq 0$, and from the first equation of (10) we obtain

$m^{\sigma}=x_{2}^{1-\sigma^{2}}r^{1-\sigma}=\left(x_{2}^{-1-\sigma}r^{-1}\right)^{\sigma-1},$

that is $m=\delta^{q-1}$ for some $\delta\in W$, a contradiction to our
assumptions.
Case 2: $x_{2}=0$.
We have $x_{1}\neq 0$. From the second equation of (10) we obtain

$m=x_{1}^{\sigma^{t-1}-\sigma}r^{1-\sigma}=\left(x_{1}^{\sigma(1+\sigma+\cdots+\sigma^{t-3})}/r\right)^{\sigma-1},$

again a contradiction.
Case 3: $h-h^{\sigma}=0$.
Argue as in Case 1. or 2, depending on whether $x_{2}\neq 0$ or $x_{1}\neq 0$.
Case 4: $x_{1}x_{2}(h-h^{\sigma})\neq 0$.
We start by proving that

$D=\det\left(\begin{array}[]{cc}-x_{2}&m^{\sigma}x_{2}^{{\sigma}^{2}}\\\
-x_{1}^{{\sigma}^{t-1}}&mx_{1}^{\sigma}\end{array}\right)$

is non zero. Indeed, if $D=0$ then, since (10) admits solutions for $r$ and
$r^{\sigma}$, we have

$\mathrm{rk}\left(\begin{array}[]{ccc}-x_{2}&m^{\sigma}x_{2}^{{\sigma}^{2}}&(h-h^{\sigma})x_{1}\\\
-x_{1}^{{\sigma}^{t-1}}&mx_{1}^{\sigma}&m(h-h^{\sigma})x_{2}^{\sigma}\end{array}\right)=1,$

and in particular

$\det\left(\begin{array}[]{cc}m^{\sigma}x_{2}^{{\sigma}^{2}}&(h-h^{\sigma})x_{1}\\\
mx_{1}^{\sigma}&m(h-h^{\sigma})x_{2}^{\sigma}\end{array}\right)=0,$

that is

$m^{{\sigma}}x_{2}^{{\sigma}^{2}+{\sigma}}-x_{1}^{{\sigma}+1}=0\ \Rightarrow\
m=u^{\sigma+1},$

where

$u=\frac{x_{1}^{\sigma^{t-1}}}{x_{2}}\in W,$

a contradiction. Therefore $D\neq 0$. From (10) we obtain

$r=\frac{\det\left(\begin{array}[]{cc}(h-h^{\sigma})x_{1}&m^{\sigma}x_{2}^{{\sigma}^{2}}\\\
m(h-h^{\sigma})x_{2}^{\sigma}&mx_{1}^{\sigma}\end{array}\right)}{D}=\frac{mx_{1}^{{\sigma}+1}-m^{{\sigma}+1}x_{2}^{{\sigma}^{2}+{\sigma}}}{D}(h-h^{\sigma})$

and

$r^{\sigma}=\frac{\det\left(\begin{array}[]{cc}-x_{2}&(h-h^{\sigma})x_{1}\\\
-x_{1}^{{\sigma}^{t-1}}&m(h-h^{\sigma})x_{2}^{\sigma}\end{array}\right)}{D}=\frac{-mx_{2}^{{\sigma}+1}+x_{1}^{{\sigma}^{t-1}+1}}{D}(h-h^{\sigma}).$

Therefore,

$r^{{\sigma}-1}=\frac{-mx_{2}^{{\sigma}+1}+x_{1}^{{\sigma}^{t-1}+1}}{mx_{1}^{{\sigma}+1}-m^{{\sigma}+1}x_{2}^{{\sigma}^{2}+{\sigma}}}=\frac{1}{m}(-mx_{2}^{{\sigma}+1}+x_{1}^{{\sigma}^{t-1}+1})^{1-{\sigma}},$

that is $m$ is a $(\sigma-1)$-th power of an element in $W$, again a
contradiction.

In each of the cases analyzed, the condition $r\neq 0$ leads to a
contradiction. It follows that $\rho\in\mathbb{F}_{q^{t}}$. ∎

By the following result at least one of the assumptions above cannot be
removed.

######  Proposition 2.4.

If $m$ is a $(\sigma+1)$-th power of an element of $W$, then
$\varphi_{m,\sigma}$ is not scattered.

###### Proof.

By assumption $m=w^{\sigma+1}$ where $w\in W$. Define $x_{1}=1$,
$x_{2}=w^{-1}$. Under these assumptions the equations in (10) coincide up to a
factor with

$-r+w^{\sigma+1}r^{\sigma}=w(h-h^{\sigma}).$

The images of the ${{\mathbb{F}}_{q}}$-linear maps $r\in
W\mapsto-r+w^{\sigma+1}r^{\sigma}\in W$ and $h\in{{\mathbb{F}}_{q^{t}}}\mapsto
w(h-h^{\sigma})\in W$ both are of ${{\mathbb{F}}_{q}}$-dimension at least
$t-1$; this implies that their intersection is not trivial, and $r\in W$,
$h\in{{\mathbb{F}}_{q^{t}}}$ exist such that $r\neq 0$ and (10) is satisfied.
∎

######  Proposition 2.5.

  1. $(i)$

For any $t\geq 3$ there are precisely $(q^{t}-1)/(q-1)$ elements of
${{\mathbb{F}}_{q^{t}}^{*}}$ which are $(q-1)$-th powers of elements in $W$;
more precisely, they are the solutions of

$x^{\frac{q^{t}-1}{q-1}}=-1,\quad x\in{{\mathbb{F}}_{q^{n}}}.$

  2. $(ii)$

If $t$ is even, then there are precisely $(q^{t}-1)/(q+1)$ elements of
${{\mathbb{F}}_{q^{t}}^{*}}$ which are $(q+1)$-th powers of elements in $W$;
more precisely, they are the solutions of

$x^{\frac{q^{t}-1}{q+1}}=-1,\quad x\in{{\mathbb{F}}_{q^{n}}}.$

  3. $(iii)$

If $t$ is odd, then there are precisely $(q^{t}-1)/2$ elements of
${{\mathbb{F}}_{q^{t}}^{*}}$ which are $(q+1)$-th powers of elements in $W$;
more precisely, they are the solutions of

$x^{\frac{q^{t}-1}{2}}=(-1)^{\frac{q+1}{2}},\quad x\in{{\mathbb{F}}_{q^{n}}}.$

###### Proof.

Let $W^{*}=W\setminus\\{0\\}$. For any positive even integer $D$ define the
set $S_{D}$ of all $D$-powers of elements of $W^{*}$. Note that, by Lemma 2.1,
$S_{D}\subseteq{{\mathbb{F}}_{q^{t}}^{*}}$. Let $\delta=\gcd(D,q^{t}-1)$. The
$D$-powers of elements of ${{\mathbb{F}}_{q^{t}}^{*}}$ are precisely the
solutions of the equation

$x^{\frac{q^{t}-1}{\delta}}=1,\quad x\in{{\mathbb{F}}_{q^{n}}}.$

Let $w_{0}\in W^{*}$. It holds
$(w_{0}^{D})^{\frac{q^{t}-1}{\delta}}=(w_{0}^{q^{t}-1})^{D/\delta}=(-1)^{D/\delta}$.
We have

$S_{D}=\\{w_{0}^{D}y^{D}\colon
y\in{{\mathbb{F}}_{q^{t}}^{*}}\\}=\\{w_{0}^{D}x\colon
x^{\frac{q^{t}-1}{\delta}}=1,\,x\in{{\mathbb{F}}_{q^{n}}}\\}.$

Therefore, $S_{D}$ has equation

$x^{\frac{q^{t}-1}{\delta}}=(-1)^{D/\delta}.$ (11)

Taking into account that

$\gcd(q+1,q^{t}-1)=\left\\{\begin{array}[]{cc}q+1&\mbox{ for $t$ even,}\\\
2&\mbox{ for $t$ odd,}\end{array}\right.$

the statements $(i)$, $(ii)$, and $(iii)$ follow from (11). ∎

######  Corollary 2.6.

There is at least one scattered polynomial of type $\varphi_{m,\sigma}$ for
any $\sigma$ and $t\geq 3$ such that $t$ is even or $q>3$.

###### Proof.

The sum of the sizes of $S_{q-1}$ and $S_{q+1}$ is less than $q^{t}-1$, hence
there exist in ${{\mathbb{F}}_{q^{t}}^{*}}$ elements which are neither
$(q-1)$\- nor $(q+1)$-powers of elements of $W$. Therefore, Theorem 2.3 can be
applied for at least one value of $m$. ∎

######  Remark 2.7.

By Theorem 2.3 and Proposition 2.5, if $t$ is even, or $t$ is odd and $q\equiv
1\pmod{4}$, then $\varphi_{1,q}$ is scattered. On the other hand, by
Proposition 2.4, if $t$ is odd and $q\equiv 3\pmod{4}$, then $\varphi_{1,q}$
is not scattered. This is consistent with [10, Theorem 2.4] for $k=1$. It
follows that that the family we are studying contains examples of
R-$q^{t}$-partially scattered polynomials that are not scattered.

##  3 Matrices stabilizing the graph of $\varphi_{m,\sigma}$

Here we investigate the matrices in $\mathbb{F}_{q^{n}}^{2\times 2}$ that
stabilize the graph $U_{m,\sigma}$ of $\varphi_{m,\sigma}$. More precisely, we
will compute the set $\mathbb{S}_{m,\sigma}$ of matrices $A$ such that
$AU_{m,\sigma}\subseteq U_{m,\sigma}$. Such $\mathbb{S}_{m,\sigma}$ has been
studied in [17], where in particular it has been proved that it is isomorphic
to the right idealizer of $\mathcal{C}_{\varphi_{m,\sigma}}$. As a
consequence, $\mathbb{S}_{m,\sigma}$ is invariant up to equivalence of
polynomials.

######  Proposition 3.1.

Suppose that $t>4$, or that $t=4$ and $\varphi_{m,\sigma}$ is scattered. Then
the set $\mathbb{S}_{m,\sigma}$ of matrices $A\in\mathbb{F}_{q^{n}}^{2\times
2}$ such that $AU_{m,\sigma}\subseteq U_{m,\sigma}$ is equal to

$\displaystyle K_{1}$ $\displaystyle=\left\\{\begin{pmatrix}a&0\\\
0&a^{q}\end{pmatrix}\ \colon\ a\in\mathbb{F}_{q^{2}}\right\\},$
$\displaystyle\mbox{ if $t$ is even},$ $\displaystyle K_{2}$
$\displaystyle=\left\\{\begin{pmatrix}a&0\\\ 0&a\end{pmatrix}\ \colon\
a\in\mathbb{F}_{q}\right\\},$ $\displaystyle\mbox{ if $t$ is odd and $m\neq
1$},$ $\displaystyle K_{3}$ $\displaystyle=\left\\{\begin{pmatrix}a&b\\\
-4b&a\end{pmatrix}\ \colon\ a\in\mathbb{F}_{q},\ b\in\mathbb{F}_{q^{2}},\
b^{q}+b=0\right\\},$ $\displaystyle\mbox{ if $t$ is odd and $m=1$}.$

The set $K_{1}$ is a subring of $\mathbb{F}_{q^{n}}^{2\times 2}$ isomorphic to
$\mathbb{F}_{q^{2}}$, $K_{2}$ is isomorphic to ${{\mathbb{F}}_{q}}$, $K_{3}$
is isomorphic to $\mathbb{F}_{q^{2}}$ if $q\equiv 1\pmod{4}$, whereas it is
not a field for $q\equiv 3\pmod{4}$.

###### Proof.

If $t=4$ and $\varphi_{m,\sigma}$ is scattered, then $\varphi_{m,\sigma}$ is
in standard form, and the set of matrices stabilizing $U_{m,\sigma}$ is
$K_{2}$ [11, 17].

From now on assume $t>4$. Let $A=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathbb{F}_{q^{n}}^{2\times 2}$, and

$A\begin{pmatrix}x\\\ \varphi_{m,\sigma}(x)\end{pmatrix}=\begin{pmatrix}y\\\
\varphi_{m,\sigma}(y)\end{pmatrix};$

that is,
$cx+d\varphi_{m,\sigma}(x)=\varphi_{m,\sigma}(ax+b\varphi_{m,\sigma}(x))$ for
all $x\in{{\mathbb{F}}_{q^{n}}}$. This leads to the following polynomial
identity:

$\displaystyle cX+$ $\displaystyle
d(X^{{\sigma}^{t-1}}+X^{{\sigma}^{2t-1}}+m(X^{\sigma}-X^{{\sigma}^{t+1}}))=$
$\displaystyle
a^{{\sigma}^{t-1}}X^{{\sigma}^{t-1}}+b^{{\sigma}^{t-1}}(X^{{\sigma}^{2t-1}}+X^{{\sigma}^{t-2}}+m^{{\sigma}^{t-1}}(X^{{\sigma}^{t}}-X))$
$\displaystyle+a^{{\sigma}^{2t-1}}X^{{\sigma}^{2t-1}}+b^{{\sigma}^{2t-1}}(X^{{\sigma}^{t-2}}+X^{{\sigma}^{2t-2}}+m^{{\sigma}^{2t-1}}(X-X^{{\sigma}^{t}}))$
$\displaystyle+ma^{{\sigma}}X^{{\sigma}}+mb^{{\sigma}}(X^{{\sigma}^{t}}+X+m^{{\sigma}}(X^{{\sigma}^{2}}-X^{{\sigma}^{t+2}}))$
$\displaystyle-
ma^{{\sigma}^{t+1}}X^{{\sigma}^{t+1}}-mb^{{\sigma}^{t+1}}(X+X^{{\sigma}^{t}}+m^{{\sigma}^{t+1}}(X^{{\sigma}^{t+2}}-X^{{\sigma}^{2}})).$
(12)

Taking into account the coefficients of $X^{{\sigma}^{t-1}}$,
$X^{{\sigma}^{t+1}}$ and $X^{{\sigma}^{2t-1}}$,

$d=a^{{\sigma}^{t-1}},\ md=ma^{{\sigma}^{t+1}},\ d=a^{{\sigma}^{2t-1}},$

respectively; hence $a\in\mathbb{F}_{q^{\gcd(t,2)}}$ and $d=a^{q}$. Next,
since the coefficient of $X^{{\sigma}^{t-2}}$ is zero,
$-(b^{{\sigma}^{t-1}}+b^{{\sigma}^{2t-1}})=0$, that is, $b\in W$. This in turn
implies (looking at $X^{{\sigma}^{2}}$)
$0=-m(m^{\sigma}b^{\sigma}+b^{{\sigma}^{t+1}})=-mb^{\sigma}(m-1)^{\sigma}$.
Therefore, $b=0$ or $m=1$. If $m\neq 1$, taking into account the coefficient
of $X$,

$c=-b^{{\sigma}^{t-1}}m^{{\sigma}^{t-1}}+b^{{\sigma}^{2t-1}}m^{{\sigma}^{2t-1}}+b^{\sigma}m-b^{{\sigma}^{t+1}}m,$
(13)

hence $b=c=0$. This leads to $K_{2}$.

Now assume that $t$ is even and $m=1$. The identity related to
$X^{{\sigma}^{t}}$ is

$0=-b^{{\sigma}^{t-1}}+b^{{\sigma}^{2t-1}}-b^{\sigma}+b^{{\sigma}^{t+1}}=-2(b^{{\sigma}^{t-1}}+b^{\sigma}).$
(14)

One obtains $b^{{\sigma}^{t}}+b^{{\sigma}^{2}}=0$, that is, since $b\in W$,
$b^{{\sigma}^{2}}=b$, hence $b^{{\sigma}^{t}}=b$ and finally $b=c=0$.
Therefore, we have the expression $K_{1}$; one can directly check that the
matrices in $K_{1}$ stabilize $U_{m,\sigma}$.

The last case is $t$ odd and $m=1$. From (14), $b^{q}+b=0$, and by (13),
$c=-4b$. This gives $K_{3}$.

As regards the last assertion, note that the equation $x^{2}+4y^{2}=0$ has
non-trivial solutions $x\in{{\mathbb{F}}_{q}}$ and $y^{q}+y=0$ if and only if
$q\equiv 3\pmod{4}$. ∎

######  Remark 3.2.

For the case $t\geq 3$ is odd, $m=1$ and $q\equiv 3\pmod{4}$, it can be shown
that in this case the kernel and the image of any matrix of rank one in
$\mathbb{S}_{1,\sigma}$ are points of $\operatorname{PG}(1,q^{n})$ of weight
$n/2$, i.e., such that their intersection with the graph is an
${{\mathbb{F}}_{q}}$-subspace of dimension $n/2$. Polynomials of this type
have been studied in [17].

##  4 Nonequivalence with previously known scattered polynomials

Our main goal is to show that there are new scattered polynomials in the
family we introduce. For this purpose it will be sufficient to consider the
case where $\sigma=q$, $n=2t$, $t\geq 3$. We compare our construction with the
known examples of scattered polynomials.

The first nonequivalence is a simple consequence of the fact that the right
idealizer of the MRD code associated with a polynomial of pseudoregulus type
in $\mathcal{L}_{n,q}$ is isomorphic to ${{\mathbb{F}}_{q^{n}}}$, combined
with Proposition 3.1. Analogously we proceed with the second nonequivalence if
$t$ is odd, since the right idealizer of the MRD code associated with a
polynomial of Lunardon-Polverino type in $\mathcal{L}_{n,q}$ is isomorphic to
$\mathbb{F}_{q^{2}}$.

######  Proposition 4.1.

Let $f=X^{q^{s}}\in\mathcal{L}_{n,q}$ be a scattered polynomial of
pseudoregulus type. Then $\varphi_{m,q}$ and ${f}$ are not equivalent.

######  Proposition 4.2.

Let $g=X^{q^{2t-s}}+\delta X^{q^{s}}$ be a Lunardon-Polverino scattered
polynomial, $t>4$. Then ${\varphi_{m,q}}$ and ${g}$ are not equivalent.

###### Proof.

The case $t$ odd belongs from the different order of the right idealizers. Let
$t$ be even. The subspaces $U_{\varphi_{m,q}}$ and $U_{g}$ are in the same
orbit under the action of $\mathrm{\Gamma L}(2,q^{n})$ if and only if an
integer $k$ and a matrix

$M=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\operatorname{GL}(2,q^{2t})$

exist such that for any $x\in\mathbb{F}_{q^{2t}}$ there is
$y\in\mathbb{F}_{q^{2t}}$ satisfying

$M\left(\begin{array}[]{c}x^{p^{k}}\\\
\varphi_{m,q}(x)^{p^{k}}\end{array}\right)=\left(\begin{array}[]{c}y\\\
g(y)\end{array}\right).$

Let $e=m^{p^{k}}$ and $z=x^{p^{k}}$. The condition above is equivalent to

$az+b(z^{q^{t-1}}+z^{q^{2t-1}})+be(z^{q}-z^{q^{t+1}})=y,$
$cz+d(z^{q^{t-1}}+z^{q^{2t-1}})+de(z^{q}-z^{q^{t+1}})=y^{q^{2t-s}}+\delta
y^{q^{s}},$

from which we derive the polynomial identity

$a^{q^{2t-s}}Z^{q^{2t-s}}+b^{q^{2t-s}}(Z^{q^{t-s-1}}+Z^{q^{2t-s-1}})+b^{q^{2t-s}}e^{q^{2t-s}}(Z^{q^{2t-s+1}}-Z^{q^{t-s+1}})+$
$+\delta a^{q^{s}}Z^{q^{s}}+\delta b^{q^{s}}(Z^{q^{t+s-1}}+Z^{q^{s-1}})+\delta
b^{q^{s}}e^{q^{s}}(Z^{q^{s+1}}-Z^{q^{t+s+1}})=$
$=cZ+d(Z^{q^{t-1}}+Z^{q^{2t-1}})+de(Z^{q}-Z^{q^{t+1}}).$

We consider the cases $s\in\\{1,t-1,t+1,2t-1\\}$.
Let $s=1$. When $t>4$, we have the following system

$\left\\{\begin{array}[]{lllllll}b^{q^{2t-1}}e^{q^{2t-1}}+\delta b^{q}=c\\\
\delta a^{q}=de\\\ \delta b^{q}e^{q}=0\\\ b^{q^{2t-1}}=0\\\ 0=d\\\
-b^{q^{2t-1}}e^{q^{2t-1}}+\delta b^{q}=0\\\ 0=-de\\\ -\delta b^{q}e^{q}=0\\\
b^{q^{2t-1}}=0\\\ a^{q^{2t-1}}=d.\end{array}\right.$ (15)

Which implies $a=b=c=d=0$. For $s\in\\{t-1,t+1,2t-1\\}$ we get analogous
conditions that yield to $a=b=c=d=0$.

Finally, for $s\notin\\{1,t-1,t+1,2t-1\\}$ the analogous system as in (15)
leads to the same conclusion. In fact, the exponents of the indeterminate $Z$
depending on $s$ equal some of the exponents of $Z$ in the right side of the
polynomial identity $(1,q,q^{t+1},q^{t-1},q^{2t-1})$ if and only if
$s\in\\{1,t+1,t-1,2t-1,t+2,t-2\\}$. Since $t$ is even we exclude $t-2$ and
$t+2$, and then, apart from the already considered cases, we get the condition
$c=d=0$. ∎

The fact that $\varphi_{1,\sigma}$ belongs to the family $(iii\text{-}a)$
motivates our interest in the next result.

######  Proposition 4.3.

Let $t\in\mathbb{N}$, $t>4$. Then ${\varphi_{m,q}}$ and ${\varphi_{1,\sigma}}$
are equivalent only if $\operatorname{N}_{q^{t}/q}(m)=1$.

###### Proof.

Assume that ${\varphi_{m,q}}$ and ${\varphi_{1,\sigma}}$ are equivalent. Then
there exist an integer $k$ and a matrix

$M=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\operatorname{GL}(2,q^{n})$

such that for any $x\in\mathbb{F}_{q^{2t}}$ there is $y\in\mathbb{F}_{q^{2t}}$
satisfying

$M\left(\begin{array}[]{c}x^{p^{k}}\\\
\varphi_{m,q}(x)^{p^{k}}\end{array}\right)=\left(\begin{array}[]{c}y\\\
\varphi_{1,\sigma}(y)\end{array}\right).$

Let $e=m^{p^{k}}$ and $z=x^{p^{k}}$. The condition above is equivalent to

$az+b(z^{q^{t-1}}+z^{q^{2t-1}})+be(z^{q}-z^{q^{t+1}})=y,$
$cz+d(z^{q^{t-1}}+z^{q^{2t-1}})+de(z^{q}-z^{q^{t+1}})=y^{\sigma}-y^{{\sigma}^{t+1}}+y^{{\sigma}^{t-1}}+y^{{\sigma}^{2t-1}},$

and, taking into account $e^{{\sigma}^{t}}=e$ and that $J$ is odd, we get the
following identity in $\mathcal{L}_{n,q}$:

$cZ+d(Z^{q^{t-1}}+Z^{q^{2t-1}})+de(Z^{q}-Z^{q^{t+1}})=$
$=a^{\sigma}Z^{q^{J}}+b^{\sigma}(Z^{t+J-1}+Z^{q^{J-1}})+b^{\sigma}e^{\sigma}(Z^{q^{J+1}}-Z^{q^{t+J+1}})$
$-a^{{\sigma}^{t+1}}Z^{q^{t+J}}-b^{{\sigma}^{t+1}}(Z^{q^{J-1}}+Z^{q^{t+J-1}})-b^{{\sigma}^{t+1}}e^{\sigma}(Z^{q^{t+J+1}}-Z^{q^{J+1}})$
$+a^{{\sigma}^{t-1}}Z^{q^{t-J}}+b^{{\sigma}^{t-1}}(Z^{q^{2t-J-1}}+Z^{q^{t-J-1}})+b^{{\sigma}^{t-1}}e^{{\sigma}^{t-1}}(Z^{q^{t-J+1}}-Z^{q^{2t-J+1}})$
$+a^{{\sigma}^{2t-1}}Z^{q^{2t-J}}+b^{{\sigma}^{2t-1}}(Z^{q^{t-J-1}}+Z^{q^{2t-J-1}})+b^{{\sigma}^{2t-1}}e^{{\sigma}^{2t-1}}(Z^{q^{2t-J+1}}-Z^{q^{t-J+1}}).$
(16)

By comparing the monomials having the same degree, if $J\notin\\{\pm 1,t\pm
1\\}$, one obtains without any assumption on $\operatorname{N}_{q^{t}/q}(m)$
that $M$ has a zero row.

If $J=1$, by (16),

$\left\\{\begin{array}[]{lllllll}c=b^{q}-b^{q^{t+1}}-b^{q^{t-1}}e^{q^{t-1}}+b^{q^{2t-1}}e^{q^{t-1}}\\\
de=a^{q}\\\ 0=b^{q}+b^{q^{t+1}}\\\ 0=b^{q^{t-1}}+b^{q^{2t-1}}\\\
d=a^{q^{t-1}}\\\
0=b^{q}-b^{q^{t+1}}+b^{q^{t-1}}e^{q^{t-1}}-b^{q^{2t-1}}e^{q^{t-1}}\\\
de=a^{q^{t+1}}\\\ 0=-b^{q}e^{q}-b^{q^{t+1}}e^{q}\\\
0=b^{q^{t-1}}+b^{q^{2t-1}}\\\ d=a^{q^{2t-1}}.\end{array}\right.$ (17)

If $a\neq 0$, then $d\neq 0$. By comparing the seventh and the second
equation, one obtains $a\in{{\mathbb{F}}_{q^{t}}}$. From the second and the
fifth equation, $a^{q-q^{t-1}}=e$ that implies
$\operatorname{N}_{q^{t}/q}(m)=1$. If $a=0$, then $cb\neq 0$, $b^{q^{t}}+b=0$
by the third equation and $b^{q^{2}}=eb$ by the sixth. This implies
$\operatorname{N}_{q^{t}/q}(m)=1$.

The cases $J=2t-1$ or $J=t\pm 1$ for $t$ even can be dealt with in a similar
way leading in each case to $\operatorname{N}_{q^{t}/q}(m)=1$. ∎

We now investigate the equivalence between polynomials of type $\varphi_{m,q}$
and their adjoints. Define
$\phi_{\mu}=X^{q}+X^{q^{t+1}}+\mu(X^{q^{2t-1}}-X^{q^{t-1}})$,
$\mu\in{{\mathbb{F}}_{q^{t}}}$; it holds

$\varphi_{m,q}^{\top}=\phi_{m^{q^{t-1}}}$ (18)

for any $m\in{{\mathbb{F}}_{q^{t}}}$.

######  Proposition 4.4.

Let $t>4$. For any $m,\mu\in{{\mathbb{F}}_{q^{t}}}$ such that $\varphi_{m,q}$
is scattered, $\varphi_{m,q}$ and $\phi_{\mu}$ are nonequivalent.

###### Proof.

Starting from

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\begin{pmatrix}x\\\
x^{q^{t-1}}+x^{q^{2t-1}}+m(x^{q}-x^{q^{t+1}})\end{pmatrix}=\begin{pmatrix}y\\\
y^{q}+y^{q^{t+1}}+\mu(y^{q^{2t-1}}-y^{q^{t-1}})\end{pmatrix}$

one obtains the polynomial identity

$\displaystyle cX+$ $\displaystyle
d[X^{q^{t-1}}+X^{q^{2t-1}}+m(X^{q}-X^{q^{t+1}})]=a^{q}X^{q}+b^{q}[X^{q^{t}}+X+m^{q}(X^{q^{2}}-X^{q^{t+2}})]$
$\displaystyle+a^{q^{t+1}}X^{q^{t+1}}+b^{q^{t+1}}[X+X^{q^{t}}+m^{q^{t+1}}(X^{q^{t+2}}-X^{q^{2}})]$
$\displaystyle+\mu a^{q^{2t-1}}X^{q^{2t-1}}+\mu
b^{q^{2t-1}}[X^{q^{t-2}}+X^{q^{2t-2}}+m^{2^{2t-1}}(X-X^{q^{t}})]$
$\displaystyle-\mu a^{q^{t-1}}X^{q^{t-1}}-\mu
b^{q^{t-1}}[X^{q^{2t-2}}+X^{q^{t-2}}+m^{q^{t-1}}(X^{q^{t}}-X)].$

This gives ten equations, four equations equivalent to
$b\in{{\mathbb{F}}_{q^{t}}}$ and four equations equivalent to
$d=m^{-1}a^{q}=-\mu a^{q^{t-1}}=-m^{-1}a^{q^{t+1}}$, that is

$a\in W,\ d=m^{-1}a^{q},\ \mu=-m^{-1}a^{q-q^{t-1}}.$

By comparing the coefficients of $X^{q^{t}}$ one obtains
$\mu=m^{-q^{t-1}}b^{q-q^{t-1}}$, and finally

$m^{q-1}=\left(\frac{a}{b}\right)^{q^{2}-1}.$

As a consequence, $m^{(q^{t}-1)/(q+1)}=(a/b)^{q^{t}-1}=-1$ for $t$ even end
$m^{(q^{t}-1)/2}=(a/b)^{(q^{t}-1)(q+1)/2}=(-1)^{(q+1)/2}$ for $t$ odd and this
implies by Propositions 2.4 and 2.5 that $\varphi_{m,q}$ is not scattered. ∎

In [2] the notion of a _$\operatorname{\Gamma L}$ -class_ of a linear set $L$
has been introduced, which is the number of nonequivalent polynomials $f$ such
that $L_{f}=L$. As a corollary of Proposition 4.4 it results:

######  Theorem 4.5.

For $t>4$ the $\operatorname{\Gamma L}$-class of any scattered linear set of
type $L_{\varphi_{m,q}}$ is at least two.

It is possible to prove the nonequivalence of the polynomials of type
$\varphi_{m,q}$ with the polynomials in the class $(iii\text{-}b)$ with a
case-by-case analysis. However, the following result makes it possible to
shorten the proof.

######  Proposition 4.6.

[7] Let $f,g\in\mathcal{L}_{n,q}$ be equivalent. Then $f^{\top}$ and
$g^{\top}$ are equivalent.

The main result of this paper is a summary of the propositions of this
section.

######  Theorem 4.7.

Let $q$ be odd and $t>4$. If $t$ is even and $q>3$ or $t$ is odd and $q>5$,
then there exists $m\in{{\mathbb{F}}_{q^{t}}}$ such that

$\varphi_{m,q}=X^{{q}^{t-1}}+X^{{q}^{2t-1}}+m(X^{q}-X^{{q}^{t+1}})$

is a scattered $q$-polynomial that is not equivalent to any previously known
scattered $q$-polynomial in $\mathbb{F}_{q^{2t}}[X]$.

###### Proof.

Take into account a scattered $\varphi_{m,q}$. This $\varphi_{m,q}$ does not
belong to the families $(i)$ and $(ii)$ by Propositions 4.1 and 4.2. Since any
element of the family $(iii\text{-}b)$ is equivalent to its adjoint, by
Propositions 4.4 and 4.6 such family does not contain $\varphi_{m,q}$. The
family $(iii\text{-}a)$ contains elements of type $\varphi_{1,\sigma}$ and if
$\operatorname{N}_{q^{t}/q^{\gcd(2,t)}}(m)\neq 1$, $\varphi_{m,q}$ is
nonequivalent to these by Proposition 4.3. So it remains to prove that at
least one scattered $\varphi_{m,q}$ exists satisfying that condition. Taking
into account Theorem 2.3, it is enough to show that the sum of the
cardinalities of the following three sets is less than $q^{t}-1$.

  1. 1.

$S_{q-1}$, that is, the set of elements in $\mathbb{F}_{q^{t}}^{*}$ which are
$(q-1)$-powers of elements of $W$;

  2. 2.

$S_{q+1}$, that is, the set of elements in $\mathbb{F}_{q^{t}}^{*}$ which are
$(q+1)$-powers of elements of $W$;

  3. 3.

$T$, the set of elements $m\in{{\mathbb{F}}_{q^{t}}}$ such that
$\operatorname{N}_{q^{t}/q}(m)=1$.

The equation $\operatorname{N}_{q^{t}/q}(x)=1$ has $(q^{t}-1)/(q-1)$
solutions.

Assume that $t$ is even. Combining with Proposition 2.5,

$|S_{q-1}\cup S_{q+1}\cup
T|\leq\frac{q^{t}-1}{q-1}+\frac{q^{t}-1}{q+1}+\frac{q^{t}-1}{q-1}=\frac{(q^{t}-1)(3q+1)}{q^{2}-1},$
(19)

that for $q>3$ is less than $q^{t}-1$.

In the case that $t$ is odd one obtains similarly

$|S_{q-1}\cup S_{q+1}\cup
T|\leq\frac{q^{t}-1}{q-1}+\frac{q^{t}-1}{2}+\frac{q^{t}-1}{q-1}=\frac{q^{t}-1}{2(q-1)}(q+3).$

For $q>5$ this is less than $q^{t}-1$. ∎

## Acknowledgements

The authors are very thankful to Giovanni Longobardi for pointing out
Proposition 4.6. The authors were partially supported by the Italian National
Group for Algebraic and Geometric Structures and their Applications (GNSAGA -
INdAM). The research of the third author was supported by the project “VALERE:
VAnviteLli pEr la RicErca” of the University of Campania “Luigi Vanvitelli”
and by the project COMBINE from “VALERE: VAnviteLli pEr la RicErca” of the
University of Campania “Luigi Vanvitelli”.

## References

  * [1] D. Bartoli, C. Zanella and F. Zullo: A new family of maximum scattered linear sets in $PG(1,q^{6})$, _Ars Math. Contemp._ 19(1) (2020), 125–145.
  * [2] B. Csajbók, G. Marino and O. Polverino: Classes and equivalence of linear sets in $\operatorname{PG}(1,q^{n})$. _J. Combin. Theory Ser. A_ 157 (2018), 402–426.
  * [3] P. Delsarte: Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A 25 (1978), 226–241.
  * [4] E. Gabidulin: Theory of codes with maximum rank distance, _Problems of information transmission_ , 21(3) (1985), 3–16.
  * [5] R. Gow and R. Quinlan: Galois extensions and subspaces of alterning bilinear forms with special rank properties, Linear Algebra Appl. 430 (2009), 2212–2224.
  * [6] M. Lavrauw: Scattered spaces in Galois Geometry, Contemporary Developments in Finite Fields and Applications (2016), 195–216.
  * [7] G. Longobardi: Private communication, (2024).
  * [8] G. Longobardi, G. Marino, R. Trombetti and Y. Zhou: A large family of maximum scattered linear sets of $\mathrm{PG}(1,q^{n})$ and their associated MRD codes, Combinatorica 43 (2023), 681–716.
  * [9] G. Longobardi, C. Zanella: Partially scattered linearized polynomials and rank metric codes, _Finite Fields Appl._ 76 (2021), 101914.
  * [10] G. Longobardi, C. Zanella: Linear sets and MRD-codes arising from a class of scattered linearized polynomials, _J. Algebraic Combin._ 53 (2021), 639–661.
  * [11] G. Longobardi, C. Zanella: A standard form for scattered linearized polynomials and properties of the related translation planes, https://arxiv.org/abs/2205.15429.
  * [12] G. Lunardon, O. Polverino: Blocking sets and derivable partial spreads, _J. Algebraic Combin._ 14 (2001), 44–56.
  * [13] A. Neri, P. Santonastaso, F. Zullo: Extending two families of maximum rank distance codes Finite Fields Appl. 81 (2022), Paper No. 102045.
  * [14] O. Polverino: Linear sets in finite projective spaces, _Discrete Math._ 310(22) (2010), 3096–3107.
  * [15] O. Polverino, F. Zullo: Connections between scattered linear sets and MRD-codes, _Bull. Inst. Combin. Appl._ 89 (2020), 46–74.
  * [16] J. Sheekey: A new family of linear maximum rank distance codes, Adv. Math. Commun. 10(3) (2016), 475–488.
  * [17] V. Smaldore, C. Zanella, F. Zullo: On the stabilizer of the graph of linear functions over finite fields, https://arxiv.org/abs/2401.06085v1.

Valentino Smaldore (ORCID: 0000-0003-3433-3164) and Corrado Zanella (ORCID:
0000-0002-5031-1961)
Dipartimento di Tecnica e Gestione dei Sistemi Industriali
Università degli Studi di Padova
Stradella S. Nicola, 3
36100 Vicenza VI - Italy
<EMAIL_ADDRESS>
Ferdinando Zullo (ORCID: 0000-0002-5087-2363)
Dipartimento di Matematica e Fisica,
Università degli Studi della Campania “Luigi Vanvitelli”,
Viale Lincoln 5,
81100 Caserta CE - Italy
<EMAIL_ADDRESS>


††∗ Corresponding author<EMAIL_ADDRESS>
# Controllability of a second-order impulsive neutral differential equation
via resolvent operator technique

A. Afreen∗, A. Raheem & A. Khatoon  A. Afreen
Department of Mathematics, Aligarh Muslim University,
Aligarh - 202002, India<EMAIL_ADDRESS>A. Raheem
Department of Mathematics, Aligarh Muslim University,
Aligarh - 202002, India<EMAIL_ADDRESS>A. Khatoon
Department of Mathematics, Aligarh Muslim University,
Aligarh - 202002, India<EMAIL_ADDRESS>
###### Abstract.

This paper uses the resolvent operator technique to investigate second-order
non-autonomous neutral integrodifferential equations with impulsive conditions
in a Banach space. We study the existence of a mild solution and the system’s
approximate controllability. The semigroup and resolvent operator theory,
graph norm, and Krasnoselskii’s fixed point theorem are used to demonstrate
the results. Finally, we present our findings with an example.

###### Key words and phrases:

non-autonomous, neutral system, controllability, resolvent operator theory,
semigroup theory

###### 2010 Mathematics Subject Classification:

93B05, 34G20, 34K30, 34K40, 34K45

##  1\. Introduction

Let $\big{(}\mathfrak{B},\|\cdot\|\big{)}$ be a Banach space. Consider the
control systems governed by the following neutral integrodifferential
equations with impulses:

$\displaystyle\left\\{\begin{array}[]{lll}\dfrac{d^{2}}{dt^{2}}E(t,\vartheta_{t})=\mathcal{A}(t)E(t,\vartheta_{t})+\displaystyle\int_{0}^{t}\zeta(t,s)E(s,\vartheta_{s})ds+\pounds_{1}\big{(}t,\vartheta(t)\big{)}+\beta
u(t),\\\ \hskip 221.93158ptt\in T=[0,\ell],~{}t\neq t_{q},&\\\
\Delta\vartheta(t_{q})=\mathrm{I}_{q}\big{(}\vartheta(t_{q})\big{)},\quad
q=1,2,\ldots,N,&\\\
\Delta\vartheta^{\prime}(t_{q})=\mathrm{J}_{q}\big{(}\vartheta(t_{q})\big{)},\quad
q=1,2,\ldots,N,&\\\
\vartheta_{0}=\varPhi\in\wp,\quad\vartheta^{\prime}(0)=x^{1}\in\mathfrak{B}.\par\end{array}\right.$
(1.6)

$\mathcal{A}(t):\mathfrak{D}\big{(}\mathcal{A}(t)\big{)}\subseteq\mathfrak{B}\rightarrow\mathfrak{B}$
is a densely defined closed linear operator and
$\vartheta_{t}:(-\infty,0]\rightarrow\mathfrak{B},~{}\vartheta_{t}(s)=\vartheta(t+s)$
belongs to an abstract phase space $\wp$.
$\pounds_{1}:[0,\ell]\times\mathfrak{B}\rightarrow\mathfrak{B};~{}\pounds_{2},E:[0,\ell]\times\wp\rightarrow\mathfrak{B},$
with $E(t,\varPsi)=\varPsi(0)+\pounds_{2}(t,\varPsi)$ are appropriate
functions.
$\zeta(t,s):\mathfrak{D}(\zeta)\subseteq\mathfrak{B}\rightarrow\mathfrak{B}$
is a closed linear operator whose domain does not depend on $(t,s).$ Also,
assume that
$\frac{d}{dt}\pounds_{2}(t,\vartheta_{t})\big{\rvert}_{t=0}=y^{1}.$ Let
$0=t_{0}<t_{1}<t_{2}<\ldots<t_{N}<t_{N+1}=\ell;~{}\mathrm{I}_{q},\mathrm{J}_{q}:\mathfrak{B}\rightarrow\mathfrak{B},~{}q=1,2,\ldots,N$
are impulsive functions, $\Delta\vartheta(t_{q})$ and
$\Delta\vartheta^{\prime}(t_{q})$ indicate the jump of $\vartheta$ and
$\vartheta^{\prime}$ at $t_{q}.$ Let
$PC(T,\mathfrak{B})=\big{\\{}\vartheta:T\rightarrow\mathfrak{B}~{}|~{}\vartheta(t)~{}\mbox{is
continuously differentiable at}~{}t\neq t_{q};\mbox{
and}~{}\vartheta(t_{q})=\vartheta(t_{q}^{-}),\vartheta^{\prime}(t_{q})=\vartheta^{\prime}(t_{q}^{-})\big{\\}}.$
$u(\cdot)$ is the control function in a Banach space $L^{2}(T,U),$ where $U$
is a Banach space; the operator $\beta:U\rightarrow\mathfrak{B}$ is linear and
continuous.

To investigate the above problem, the existence of a resolvent operator
associated with the homogeneous system

$\displaystyle\vartheta^{\prime\prime}(t)=\mathcal{A}(t)\vartheta(t)+\int_{0}^{t}\zeta(t,s)\vartheta(s)ds,\quad
t\in T=[0,\ell],$

is used. Researchers have expressed an interest in studying various problems
using the resolvent operator technique in recent years. The resolvent operator
takes the place of the $C_{0}$-semigroup in evolution equations and is crucial
in solving differential equations in both the weak and strict sense [15, 21,
22, 23, 25, 28]. Using the resolvent operators, Grimmer [7] investigated the
existence of mild solutions to evolution equations. Rezapour et al. [24] noted
the existence of mild solutions via the resolvent operator technique.

It is well known that many real-life phenomena are affected by the sudden
change in their state at certain moments, such as heartbeats and blood flow in
the human body. These phenomena are discussed in the form of impulses whose
duration is negligible compared to the whole process. Such processes are
modeled using impulsive differential equations. Analyzing mathematical models
requires an understanding of impulsive systems. It has a wide range of
applications, including drug diffusion in the human body, population dynamics,
theoretical physics, mathematical economy, chemical technology, engineering,
control theory, medicine, and so on. More information can be found in the
references [1, 15, 16, 17, 19, 26, 30, 31].

The goal of controllability theory is the ability to control a specific system
to the desired state. One of the fundamental ideas for investigating and
analyzing various dynamical control processes is controllability theory. Many
research papers on the controllability of second-order linear and nonlinear
differential systems were presented, for example, see [3, 4, 5, 12, 29]. Most
researchers have been found to study autonomous and non-autonomous systems
using various techniques, see [6, 8, 9, 10, 11, 12, 13, 20, 27].

Many real-world phenomena can be represented mathematically by a dynamical
system governed by neutral differential equations with nonlocal conditions
[2]. Nonlocal problems are more commonly used in applications than classical
problems. Many researchers are now paying close attention to this theory and
its applications. For more details, refer [8, 14, 15, 18].

There are five sections to this paper. The first two sections include an
introduction, notations, some required definitions, assumptions, as well as
some lemmas. The third section discusses the existence of a mild solution and
the system’s controllability. To demonstrate the results, an example is
provided in the fourth section, and in the last section, a brief conclusion is
given.

##  2\. Preliminaries and Assumptions

The space $\mathfrak{D}(\mathcal{A})$ provided with the graph norm induced by
$\mathcal{A}(t)$ is a Banach space. We will assume that all of these norms are
equivalent. A simple condition for obtaining this property is that there
exists $\lambda\in\rho(\mathcal{A}(t)),$ the resolvent set of
$\mathcal{A}(t)$, so that $(\lambda
I-\mathcal{A}(t))^{-1}:\mathfrak{B}\rightarrow\mathfrak{D}(\mathcal{A})$ is a
bounded linear operator. In what follows, by $[\mathfrak{D}(\mathcal{A})]$ we
represent the vector space $\mathfrak{D}(\mathcal{A})$ provided with any of
these equivalent norms, and we denote

$\|\vartheta\|_{[\mathfrak{D}(\mathcal{A})]}=\|\vartheta\|+\|\mathcal{A}(t)\vartheta\|,~{}\vartheta\in\mathfrak{D}(\mathcal{A}).$

Let us assume that for every
$\vartheta\in\mathfrak{D}(\mathcal{A}),~{}t\longmapsto\mathcal{A}(t)\vartheta$
is continuous. Therefore, we consider that $\mathcal{A}(\cdot)$ generates
$\\{S(t,s)\\}_{0\leq s\leq t\leq\ell},$ refer to [12]. We regard that
$S(\cdot)\vartheta$ is continuously differentiable for all
$\vartheta\in\mathfrak{B}$ with derivative uniformly bounded on bounded
intervals. We introduce another operator ${C(t,s)}$ associated with the
evolution operator $S(t,s)$ as

$C(t,s)=-\frac{\partial S(t,s)}{\partial s}.$

Let $\varOmega=\big{\\{}(t,s):0\leq s\leq t\leq\ell\big{\\}}.$ The following
assumptions hold throughout the paper.

  * (A1)

The operator
$\zeta(\cdot,\cdot):[\mathfrak{D}(\mathcal{A})]\rightarrow\mathfrak{B}$ is
bounded and continuous, i.e.,

$\|\zeta(t,s)\vartheta\|\leq\hbar_{1}\|\vartheta\|_{[\mathfrak{D}(\mathcal{A})]},$

where $(t,s)\in\varOmega$ and $\hbar_{1}>0.$

  * (A2)

There exists $L_{\zeta}>0$ such that

$\|\zeta(t_{2},s)\vartheta-\zeta(t_{1},s)\vartheta\|\leq
L_{\zeta}|t_{2}-t_{1}|\|\vartheta\|_{[\mathfrak{D}(\mathcal{A})]},$

for all $\vartheta\in\mathfrak{D}(\mathcal{A}),~{}0\leq s\leq t_{1}\leq
t_{2}\leq\ell.$

  * (A3)

There exists $\hbar_{2}>0$ such that

$\bigg{\|}\int_{\eta}^{t}S(t,s)\zeta(s,\eta)\vartheta
ds\bigg{\|}\leq\hbar_{2}\|\vartheta\|,$

for all $\vartheta\in\mathfrak{D}(\mathcal{A}).$

######  Definition 2.1.

A two-parameter family $\\{\Re(t,s)\\}_{t\geq s}$ on $\mathfrak{B}$ is said to
be a resolvent operator for the system

$\displaystyle\left\\{\begin{array}[]{ll}\vartheta^{\prime\prime}(t)=\mathcal{A}(t)\vartheta(t)+\displaystyle\int_{s}^{t}\zeta(t,\tau)\vartheta(\tau)d\tau,\quad
s\leq t\leq\ell,&\\\
\vartheta(s)=0,\quad\vartheta^{\prime}(s)=x\in\mathfrak{B},\end{array}\right.$
(2.3)

if

  * (i)

the map $\Re:\varOmega\rightarrow L(\mathfrak{B})$ is strongly continuous,
$\Re(t,\cdot)\vartheta$ is continuously differentiable $\forall$
$\vartheta\in\mathfrak{B},\Re(s,s)=0,\dfrac{\partial}{\partial
t}\Re(t,s)\Big{\rvert}_{t=s}=\mathcal{I}$ and $\dfrac{\partial}{\partial
s}\Re(t,s)\Big{\rvert}_{s=t}=-\mathcal{I};$

  * (ii)

$\Re(t,\cdot)x$ is a solution for (2.3), i.e.,

$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}\Re(t,s)x=\mathcal{A}(t)\Re(t,s)x+\int_{s}^{t}\zeta(t,\tau)\Re(\tau,s)xd\tau,$

for all $0\leq s\leq t\leq\ell,$ $x\in\mathfrak{D}(\mathcal{A}).$

We may assume that

$\|\Re(t,s)\|\leq M_{1},~{}\Big{\|}\frac{\partial}{\partial
s}\Re(t,s)\Big{\|}\leq M_{2},\quad(t,s)\in\varOmega,$

where $M_{1}>0$ and $M_{2}>0.$

Next, the linear operator

$\displaystyle\tilde{F}(t,\eta)x=\int_{\eta}^{t}\zeta(t,s)\Re(s,\eta)xds,\quad
x\in\mathfrak{D}(\mathcal{A}),~{}0\leq\eta\leq t\leq\ell,$

can be extended to $\mathfrak{B}$ and still denoted by itself
$\tilde{F}(t,\eta),~{}\tilde{F}:\varOmega\rightarrow L(\mathfrak{B})$ is
strongly continuous, and

$\displaystyle\Re(t,\eta)x=S(t,\eta)+\int_{\eta}^{t}S(t,s)\tilde{F}(s,\eta)xds,\quad\mbox{for
all}~{}x\in\mathfrak{B}.$ (2.4)

Clearly, $\Re(\cdot)$ is uniformly Lipschitz continuous, i.e., there exists a
constant $L_{\Re}>0$ such that

$\displaystyle\|\Re(t+h,\eta)-\Re(t,\eta)\|\leq L_{\Re}|h|,\quad\mbox{for
all}~{}t,t+h,\eta\in[0,\ell],$ (2.5)

and

$\displaystyle\bigg{\|}\frac{\partial\Re(t+h,\eta)}{\partial
s}-\frac{\partial\Re(t,\eta)}{\partial s}\bigg{\|}\leq M_{\Re}|h|,$ (2.6)

for $M_{\Re}>0.$

The phase space $\wp$ has the following properties:

  * (B1)

If
$\vartheta:(-\infty,\mu+\ell)\rightarrow\mathfrak{B},\ell>0,\mu\in\mathbb{R}$
is continuous on $[\mu,\mu+\ell)$ and $\vartheta_{\mu}\in\wp,$ then for every
$t\in[\mu,\mu+\ell),$ we have
(i) $\vartheta_{t}$ is in $\wp;$
(ii) $\|\vartheta(t)\|\leq K_{1}\|\vartheta_{t}\|_{\wp};$
(iii) $\|\vartheta_{t}\|_{\wp}\leq K_{2}(t-\mu)\sup\\{\|\vartheta(s)\|:\mu\leq
s\leq t\\}+K_{3}(t-\mu)\|\vartheta_{\mu}\|_{\wp},$
where $K_{1}>0$ is a constant; $K_{2},K_{3}:[0,\infty)\rightarrow[1,\infty),$
$K_{2}(\cdot)$ is continuous, $K_{3}(\cdot)$ is locally bounded, and
$K_{1},K_{2},K_{3}$ are independent of $\vartheta(\cdot).$

  * (B2)

For $\vartheta(\cdot)$ in (B1), $t\rightarrow\vartheta_{t}$ is continuous from
$[\mu,\mu+\ell)$ into $\wp.$

  * (B3)

$\wp$ is complete.

For each $r>0,$ define the set
$\Theta_{r}=\big{\\{}\vartheta\in\mathcal{K}(\ell):\|\vartheta(t)\|\leq
r~{}\mbox{for all}~{}t\in T\big{\\}},$ where
$\mathcal{K}(\ell)=\big{\\{}\vartheta\in
PC(T,\mathfrak{B}):\vartheta(0)=\varPhi(0)\big{\\}}$ is a convex closed subset
of $PC(T,\mathfrak{B}).$ To establish the results, we will introduce the
required assumptions as follows:

  * (C1)

The function $\pounds_{1}:T\times\mathfrak{B}\rightarrow\mathfrak{B}$ is
continuous for all $t\in T$ and strongly measurable for each
$\vartheta\in\mathfrak{B}.$ In addition, there is $\nu_{r}(\cdot)\in
L^{2}(T,\mathbb{R^{+}})$ for each $r>0$ such that

$\displaystyle\sup\big{\\{}\|\pounds_{1}(t,\vartheta(t))\|:\|\vartheta(t)\|\leq
r\big{\\}}\leq\nu_{r}(t),\quad\mbox{for a.e.}~{}t\in T,$

and

$\displaystyle\lim_{r\rightarrow\infty}\inf\frac{1}{r}\|\nu_{r}\|_{L^{2}}=\sigma<\infty.$

  * (C2)

The closure of $\big{\\{}\Re(t,s)\big{[}\pounds_{1}(s,\vartheta(s))+\beta
u(s)\big{]}:(t,s)\in\Omega,\|\vartheta(s)\|\leq r\big{\\}}$ is compact in
$\mathfrak{B}.$

  * (C3)

The function $\pounds_{2}:T\times\wp\rightarrow\mathfrak{B}$ is continuous and
satisfying:
(i) The set
$\big{\\{}\pounds_{2}(t,\varPsi):\|\varPsi\|_{\wp}\leq\delta\big{\\}}$ is
equicontinuous on $T$ and is relatively compact in $\mathfrak{B},$ where
$\delta>0.$
(ii) There exist $r_{1},r_{2}>0$ such that

$\|\pounds_{2}(t,\varPsi)\|\leq r_{1}+r_{2}\|\varPsi\|_{\wp},$

$\forall$ $t\in T$ and $\varPsi\in\wp.$
(iii) There exists $L_{2}>0$ such that

$\big{\|}\pounds_{2}(t,\varPsi^{1})-\pounds_{2}(s,\varPsi^{2})\big{\|}\leq
L_{2}\big{\|}\varPsi^{1}-\varPsi^{2}\big{\|}_{\wp},$

$\forall$ $t,s\in T;$ $\varPsi^{1},\varPsi^{2}\in\wp.$

  * (C4)

The function
$\mathrm{I}_{q},\mathrm{J}_{q}:\mathfrak{B}\rightarrow\mathfrak{B},\quad
q=1,2,\ldots,N,$ are continuous operators and there are $d_{q},e_{q}>0$ such
that

$\big{\|}\mathrm{I}_{q}(\vartheta)\big{\|}\leq
d_{q}\big{(}\|\vartheta\|+1\big{)},$
$\big{\|}\mathrm{J}_{q}(\vartheta)\big{\|}\leq
e_{q}\big{(}\|\vartheta\|+1\big{)},$

where $q=1,2,\ldots,N,~{}\vartheta\in\mathfrak{B}.$

  * (C5)

The control $u(\cdot)=u(t,\vartheta)\in L^{2}(T,U)$ is continuous on $T$ and
bounded for every $\vartheta\in\mathfrak{B},$ i.e., $\exists$ $\lambda>0$ such
that $\|u(t)\|=\|u(t,\vartheta)\|\leq\lambda\|\vartheta\|.$

######  Lemma 2.2.

[15] $\Re(t,s)$ is compact and continuous in the uniform operator topology of
$L(\mathfrak{B})$ for $t,s>0.$

######  Definition 2.3.

A function $\vartheta:(-\infty,\ell]\rightarrow\mathfrak{B}$ is called a mild
solution for (1.6) if $\vartheta$ is continuous on
$[0,\ell],~{}\vartheta_{0}=\varPhi\in\wp,~{}\frac{d}{dt}E(t,\vartheta_{t})\big{\rvert}_{t=0}=x^{1}+y^{1}$
and

$\displaystyle\vartheta(t)$ $\displaystyle=$
$\displaystyle-{\frac{\partial\Re(t,s)\big{[}\varPhi(0)+\pounds_{2}(0,\varPhi)\big{]}}{\partial
s}}\bigg{\rvert}_{s=0}+\Re(t,0)\big{[}x^{1}+y^{1}\big{]}-\pounds_{2}(t,\vartheta_{t})$
$\displaystyle\quad+\int_{0}^{t}\Re(t,s)\big{[}\pounds_{1}\big{(}s,\vartheta(s)\big{)}+\beta
u(s)\big{]}ds$
$\displaystyle\quad-\sum_{0<t_{q}<t}\frac{\partial\Re(t,s)}{\partial
s}\bigg{\rvert}_{s=t_{q}}\mathrm{I}_{q}\big{(}\vartheta(t_{q})\big{)}+\sum_{0<t_{q}<t}\Re(t,t_{q})\mathrm{J}_{q}\big{(}z(t_{q})\big{)}$

is satisfied for $t\in T.$

######  Definition 2.4.

The system (1.6) is approximately controllable on $T,$ if for each final state
$x_{\ell}\in\mathfrak{B}$ and each $\varepsilon>0,$ there exists a control
$u(\cdot)\in L^{2}(T,U)$ such that the mild solution $\vartheta(\cdot,u)$ of
(1.6) satisfies $\|\vartheta(\ell,u)-x_{\ell}\|<\varepsilon.$

To show the controllability of (1.6), we consider the linear system:

$\displaystyle\left\\{\begin{array}[]{ll}\dfrac{d^{2}}{dt^{2}}E(t,\vartheta_{t})=\mathcal{A}(t)E(t,\vartheta_{t})+\displaystyle\int_{0}^{t}\zeta(t,s)E(s,\vartheta_{s})ds+\beta
u(t),\\\ \hskip 199.16928ptt\in T=[0,\ell],&\\\
\vartheta_{0}=\varPhi\in\wp,\quad\vartheta^{\prime}(0)=x^{1}\in\mathfrak{B}.\par\end{array}\right.$
(2.10)

Define the controllability operator
$\Gamma_{0}^{\ell}:\mathfrak{B}\rightarrow\mathfrak{B}$ as

$\displaystyle\Gamma_{0}^{\ell}=\int_{0}^{\ell}\Re(\ell,s)\beta\beta^{*}\Re^{*}(\ell,s)ds,$

where $*$ denotes the adjoint and the resolvent of $\Gamma_{0}^{\ell}$ is
given by

$\displaystyle\mathcal{V}\big{(}\varepsilon,\Gamma_{0}^{\ell}\big{)}=\big{(}\varepsilon
I+\Gamma_{0}^{\ell}\big{)}^{-1},\quad\varepsilon>0.$

######  Lemma 2.5.

[14] The linear control system (2.10) is approximately controllable on $T$ if
and only if
$\varepsilon\mathcal{V}\big{(}\varepsilon,\Gamma_{0}^{\ell}\big{)}\rightarrow
0$ as $\varepsilon\rightarrow 0^{+}$ in the strong operator topology.

##  3\. Main results

######  Lemma 3.1.

If the conditions (C1),(C2),(C3) and (C5) hold and $u(\cdot)\in L^{2}(T,U)$ is
bounded, then the operator $P_{1}:\Theta_{r}\rightarrow\Theta_{r},$ defined by

$\displaystyle(P_{1}\vartheta)(t)$ $\displaystyle=$
$\displaystyle-{\frac{\partial\Re(t,s)\big{[}\varPhi(0)+\pounds_{2}(0,\varPhi)\big{]}}{\partial
s}}\bigg{\rvert}_{s=0}-\pounds_{2}(t,\vartheta_{t})$ (3.1)
$\displaystyle\quad+\int_{0}^{t}\Re(t,s)\big{[}\pounds_{1}\big{(}s,\vartheta(s)\big{)}+\beta
u(s)\big{]}ds$

is compact.

###### Proof.

First, we show that $P_{1}(\Theta_{r})$ is equicontinuous on $[0,\ell].$ For
any $s_{1},s_{2}\in[0,\ell]$ with $s_{1}<s_{2}$ and $\vartheta\in\Theta_{r},$
we have
$\big{\|}(P_{1}\vartheta)(s_{2})-(P_{1}\vartheta)(s_{1})\big{\|}$

$\displaystyle\leq$
$\displaystyle\bigg{\|}\bigg{(}{\frac{\partial\Re(s_{1},s)}{\partial
s}}-{\frac{\partial\Re(s_{2},s)}{\partial
s}}\bigg{)}\bigg{\rvert}_{s=0}\big{[}\varPhi(0)+\pounds_{2}(0,\varPhi)\big{]}\bigg{\|}$
$\displaystyle\quad+\Big{\|}\pounds_{2}\big{(}s_{1},\vartheta_{s_{1}}\big{)}-\pounds_{2}\big{(}s_{2},\vartheta_{s_{2}}\big{)}\Big{\|}$
$\displaystyle\quad+\bigg{\|}\int_{0}^{s_{1}}\big{[}\Re(s_{2},s)-\Re(s_{1},s)\big{]}\big{[}\pounds_{1}\big{(}s,\vartheta(s)\big{)}+\beta
u(s)\big{]}ds\bigg{\|}$
$\displaystyle\quad+\bigg{\|}\int_{s_{1}}^{s_{2}}\Re(s_{2},s)\big{[}\pounds_{1}\big{(}s,\vartheta(s)\big{)}+\beta
u(s)\big{]}ds\bigg{\|}$ $\displaystyle\leq$ $\displaystyle
M_{\Re}|s_{2}-s_{1}|\big{[}r+\big{(}r_{1}+r_{2}\|\varPhi\|_{\wp}\big{)}\big{]}+L_{2}\|\vartheta_{s_{1}}-\vartheta_{s_{2}}\|_{\wp}$
$\displaystyle\quad+\int_{0}^{s_{1}}L_{\Re}|s_{2}-s_{1}|\big{[}\|\pounds_{1}\big{(}s,\vartheta(s)\big{)}\|+\|\beta
u(s)\|\big{]}ds$
$\displaystyle\quad+M_{1}\int_{s_{1}}^{s_{2}}\big{[}\|\pounds_{1}\big{(}s,\vartheta(s)\big{)}\|+\|\beta
u(s)\|\big{]}ds.$

Therefore,
$\big{\|}(P_{1}\vartheta)(s_{2})-(P_{1}\vartheta)(s_{1})\big{\|}\rightarrow 0$
as $s_{2}\rightarrow s_{1}.$ Hence, $P_{1}(\Theta_{r})$ is equicontinuous on
$[0,\ell]$. Following the idea used in [24], we can easily show that
$P_{1}(\Theta_{r})$ is relatively compact. Hence, by Arzela-Ascoli theorem,
the operator $P_{1}$ is compact. ∎

######  Theorem 3.2.

If the conditions (B1)-(B3) and (C1)-(C5) hold, then the system (1.6) has a
mild solution in some $\Theta_{r},$ if

$\displaystyle
2M_{1}+M_{2}+r_{2}K_{2}(t-\mu)+M_{1}\ell\sigma+M_{1}\|\beta\|\lambda\ell+M_{1}\sum\limits_{q=0}^{N}e_{q}+M_{2}\sum\limits_{q=0}^{N}d_{q}<1.$
(3.2)

###### Proof.

Take the control $u(\cdot)$ as
$u(t)=u\big{(}t,\vartheta(\ell)\big{)},~{}\vartheta\in\Theta_{r}.$ Define
$\tilde{Q}:\Theta_{r}\rightarrow PC(T,\mathfrak{B})$ by

$\displaystyle\big{(}\tilde{Q}\vartheta\big{)}(t)=\Re(t,0)\big{[}x^{1}+y^{1}\big{]}+(P_{1}\vartheta)(t)+(P_{2}\vartheta)(t),$
(3.3)

where the map $P_{1}$ is defined by (3.1) and

$\displaystyle(P_{2}\vartheta)(t)=\sum_{0<t_{q}<t}\Re(t,t_{q})\mathrm{J}_{q}\big{(}\vartheta(t_{q})\big{)}-\sum_{0<t_{q}<t}\frac{\partial\Re(t,s)}{\partial
s}\bigg{\rvert}_{s=t_{q}}\mathrm{I}_{q}\big{(}\vartheta(t_{q})\big{)}.$ (3.4)

We show that the map $\tilde{Q}$ maps from $\Theta_{r}$ to $\Theta_{r}$ for
some $r>0.$ Let us assume contrarily that for each $r>0,$ there exist
$\vartheta_{r}\in\Theta_{r}$ such that
$\big{\|}\tilde{Q}\vartheta_{r}(t)\big{\|}>r$ for some $t\in T$. Therefore,

$\displaystyle r\leq\big{\|}(\tilde{Q}\vartheta_{r})(t)\big{\|}$
$\displaystyle\leq$ $\displaystyle
M_{1}\big{\|}x^{1}+y^{1}\big{\|}+M_{2}\big{[}\|\varPhi(0)\|+\|\pounds_{2}(0,\varPhi)\|\big{]}+\big{\|}\pounds_{2}(t,\vartheta_{t})\big{\|}$
(3.5)
$\displaystyle\quad+M_{1}\int_{0}^{t}\big{[}\|\pounds_{1}\big{(}s,\vartheta(s)\big{)}\|+\|\beta
u(s)\|\big{]}ds+M_{1}\sum\limits_{q=0}^{N}\big{\|}\mathrm{J}_{q}\big{(}\vartheta(t_{q})\big{)}\big{\|}$
$\displaystyle\quad+M_{2}\sum\limits_{q=0}^{N}\big{\|}\mathrm{I}_{q}\big{(}\vartheta(t_{q})\big{)}\big{\|}$
$\displaystyle\leq$ $\displaystyle
2rM_{1}+M_{2}\big{[}r+\big{(}r_{1}+r_{2}\|\varPhi\|_{\wp}\big{)}\big{]}+\big{(}r_{1}+r_{2}\|\vartheta_{t}\|_{\wp}\big{)}+M_{1}\ell\|\nu_{r}\|_{L^{2}}$
$\displaystyle\quad+M_{1}\|\beta\|\lambda\ell
r+M_{1}\sum_{q=0}^{N}e_{q}(r+1)+M_{2}\sum_{q=0}^{N}d_{q}(r+1)$
$\displaystyle\leq$ $\displaystyle
2rM_{1}+M_{2}\big{[}r+\big{(}r_{1}+r_{2}\|\varPhi\|_{\wp}\big{)}\big{]}+r_{1}+r_{2}rK_{2}(t-\mu)$
$\displaystyle\quad+r_{2}K_{3}(t-\mu)\|\vartheta_{\mu}\|_{\wp}+M_{1}\ell\|\nu_{r}\|_{L^{2}}+M_{1}\|\beta\|\lambda\ell
r$
$\displaystyle\quad+M_{1}\sum\limits_{q=0}^{N}e_{q}(r+1)+M_{2}\sum\limits_{q=0}^{N}d_{q}(r+1).$

Dividing both sides of (3.5) by $r$ and then taking $r\rightarrow\infty,$ we
get

$\displaystyle
1<2M_{1}+M_{2}+r_{2}K_{2}(t-\mu)+M_{1}\ell\sigma+M_{1}\|\beta\|\lambda\ell+M_{1}\sum\limits_{q=0}^{N}e_{q}+M_{2}\sum\limits_{q=0}^{N}d_{q}.$

This is a contradiction. So, there exists $r>0$ such that
$\tilde{Q}(\Theta_{r})\subseteq\Theta_{r}.$ Further, we prove that $\tilde{Q}$
is continuous on $\Theta_{r}.$ Take any sequence
$\\{\vartheta_{n}\\}\subset\Theta_{r}$ with
$\vartheta_{n}\rightarrow\vartheta\in\Theta_{r}$ as $n\rightarrow\infty.$
Then, for each $t\in T,$ we have

  * (a)

$\big{\|}\pounds_{1}\big{(}t,\vartheta_{n}(t)\big{)}-\pounds_{1}\big{(}t,\vartheta(t)\big{)}\big{\|}\rightarrow
0$ as $n\rightarrow\infty.$

  * (b)

$\big{\|}\pounds_{1}\big{(}t,\vartheta_{n}(t)\big{)}-\pounds_{1}\big{(}t,\vartheta(t)\big{)}\big{\|}\leq
2\nu_{r}(t).$

  * (c)

$\big{\|}\beta u\big{(}t,\vartheta_{n}(t)\big{)}-\beta
u\big{(}t,\vartheta(t)\big{)}\big{\|}<\infty.$

Using Lebesgue Dominated Convergence theorem, we have

$\big{\|}(\tilde{Q}\vartheta_{n})(t)-(\tilde{Q}\vartheta)(t)\big{\|}$

$\displaystyle\leq$
$\displaystyle\big{\|}\pounds_{2}\big{(}t,(\vartheta_{n})_{t}\big{)}-\pounds_{2}\big{(}t,\vartheta_{t}\big{)}\big{\|}+\bigg{\|}\int_{0}^{t}\Re(t,s)\big{[}\pounds_{1}\big{(}s,\vartheta_{n}(s)\big{)}-\pounds_{1}\big{(}s,\vartheta(s)\big{)}\big{]}ds\bigg{\|}$
$\displaystyle\quad+\bigg{\|}\int_{0}^{t}\Re(t,s)\big{[}\beta
u(s,\vartheta_{n}(s))-\beta u(s,\vartheta(s))\big{]}ds\bigg{\|}$
$\displaystyle\quad+\sum\limits_{0<t_{q}<t}\Big{\|}\Re(t,t_{q})\big{[}\mathrm{J}_{q}\big{(}\vartheta_{n}(t_{q})\big{)}-\mathrm{J}_{q}\big{(}\vartheta(t_{q})\big{)}\big{]}\Big{\|}$
$\displaystyle\quad+\sum\limits_{0<t_{q}<t}\bigg{\|}\frac{\partial\Re(t,s)}{\partial
s}\bigg{\rvert}_{s=t_{q}}\big{[}\mathrm{I}_{q}\big{(}\vartheta_{n}(t_{q})\big{)}-\mathrm{I}_{q}\big{(}\vartheta(t_{q})\big{)}\big{]}\bigg{\|}$
$\displaystyle\leq$ $\displaystyle
L_{2}\big{\|}(\vartheta_{n})_{t}-\vartheta_{t}\big{\|}_{\wp}+M_{1}\int_{0}^{\ell}\big{\|}\pounds_{1}\big{(}s,\vartheta_{n}(s)\big{)}-\pounds_{1}\big{(}s,\vartheta(s)\big{)}\big{\|}ds$
$\displaystyle\quad+M_{1}\int_{0}^{\ell}\big{\|}\beta
u\big{(}s,\vartheta_{n}(s)\big{)}-\beta
u\big{(}s,\vartheta(s)\big{)}\big{\|}ds+M_{1}\sum\limits_{q=0}^{N}\Big{\|}\mathrm{J}_{q}\big{(}\vartheta_{n}(t_{q})\big{)}-\mathrm{J}_{q}\big{(}\vartheta(t_{q})\big{)}\Big{\|}$
$\displaystyle\quad+M_{2}\sum\limits_{q=0}^{N}\Big{\|}\mathrm{I}_{q}\big{(}\vartheta_{n}(t_{q})\big{)}-\mathrm{I}_{q}\big{(}\vartheta(t_{q})\big{)}\Big{\|},$
$\displaystyle\rightarrow 0~{}\mbox{as}~{}n\rightarrow\infty.$

Thus, $\tilde{Q}$ is continuous on $\Theta_{r}.$ Let
$T_{0}=[0,t_{1}],T_{1}=(t_{1},t_{2}],\ldots,T_{N}=(t_{N},\ell],$ then the map
$P_{2}$ can be rewritten as

$\displaystyle(P_{2}\vartheta)(t)=\left\\{\begin{array}[]{lll}0,\quad t\in
T_{0},&\\\
\Re(t,t_{1})\mathrm{J}_{1}\big{(}\vartheta(t_{1})\big{)}-\dfrac{\partial\Re(t,s)}{\partial
s}\bigg{\rvert}_{s=t_{1}}\mathrm{I}_{1}\big{(}\vartheta(t_{1})\big{)},\quad
t\in T_{1},&\\\ \ldots,&\\\
\sum\limits_{q=1}^{N}\Re(t,t_{q})\mathrm{J}_{q}\big{(}\vartheta(t_{q})\big{)}-\sum\limits_{q=1}^{N}\dfrac{\partial\Re(t,s)}{\partial
s}\bigg{\rvert}_{s=t_{q}}\mathrm{I}_{q}\big{(}\vartheta(t_{q})\big{)},\quad
t\in T_{N}.\end{array}\right.$

Since $\mathrm{I_{1}},\mathrm{J_{1}}$ are continuous and $\Re$ is compact. As
a result, the set $\\{P_{2}\vartheta:\vartheta\in\Theta_{r}\\}$ is relatively
compact in $\mathfrak{B}$ for every $t\in\overline{T_{1}}$ (closure of
$T_{1}$). For any $s_{1},s_{2}\in\overline{T_{1}}$ with $s_{1}<s_{2}$ and
$\vartheta\in\Theta_{r},$ it follows from Lemma 2.2 that

$\displaystyle\bigg{\|}\mathrm{J}_{1}\big{(}\vartheta(t_{1})\big{)}\big{[}\Re(s_{2},t_{1})-\Re(s_{1},t_{1})\big{]}\bigg{\|}+\bigg{\|}\mathrm{I}_{1}\big{(}\vartheta(t_{1})\big{)}\bigg{(}\frac{\partial\Re(s_{2},s)}{\partial
s}\bigg{\rvert}_{s=t_{1}}-\frac{\partial\Re(s_{1},s)}{\partial
s}\bigg{\rvert}_{s=t_{1}}\bigg{)}\bigg{\|}$ $\displaystyle\leq
e_{1}(r+1)L_{\Re}|s_{2}-s_{1}|+d_{1}(r+1)M_{\Re}|s_{2}-s_{1}|,$
$\displaystyle\rightarrow 0~{}\mbox{as}~{}s_{1}\rightarrow
s_{2}~{}\mbox{independent of }~{}\vartheta\in\Theta_{r}.$

Thus, the set $\big{\\{}P_{2}\vartheta:\vartheta\in\Theta_{r}\big{\\}}$ is
equicontinuous on $\overline{T_{1}}.$ Therefore, by Arzela-Ascoli theorem, the
set $\big{\\{}P_{2}\vartheta:\vartheta\in\Theta_{r}\big{\\}}$ is relatively
compact in $C(\overline{T_{1}},\mathfrak{B}).$ In the same manner, we can show
for $T_{q},q=2,3,\ldots,N.$ It follows from [15] that
$\big{\\{}P_{2}\vartheta:\vartheta\in\Theta_{r}\big{\\}}$ is relatively
compact in $PC(T,\mathfrak{B}).$ Moreover, the map
$\Re(t,0)\big{[}x^{1}+y^{1}\big{]}:\Theta_{r}\rightarrow\Theta_{r}$ is
contraction as $\Re$ is Lipschitz continuous. But the map
$P_{1}+P_{2}:\Theta_{r}\rightarrow\Theta_{r}$ is completely continuous.
Therefore, by Krasnoselskii’s fixed point theorem, $\tilde{Q}$ has a fixed
point in $\Theta_{r},$ which is the mild solution of (1.6). ∎

######  Theorem 3.3.

Suppose that hypotheses (B1)-(B3), (C1)-(C4) hold, and the functions
$\pounds_{1},\pounds_{2},\mathrm{I}_{q},\mathrm{J}_{q},(q=1,2,\ldots,N)$ are
uniformly bounded. If the associated linear system (2.10) is approximately
controllable on $T,$ then (1.6) is approximately controllable on $T.$

###### Proof.

Let $b\in\mathfrak{B}$ and $\varepsilon>0$ be constant. Define the control
$u(t)$ as

$\displaystyle
u(t):=u_{\varepsilon}(t,\vartheta)=\beta^{*}\Re^{*}(\ell,t)\mathcal{V}\big{(}\varepsilon,\Gamma_{0}^{\ell}\big{)}p(\vartheta),$
(3.7)

where

$\displaystyle p(\vartheta)$ $\displaystyle=$ $\displaystyle
b+{\frac{\partial\Re(\ell,s)\big{[}\varPhi(0)+\pounds_{2}(0,\varPhi)\big{]}}{\partial
s}}\bigg{\rvert}_{s=0}-\Re(\ell,0)\big{[}x^{1}+y^{1}\big{]}+\pounds_{2}(\ell,\vartheta_{\ell})$
$\displaystyle\quad-\int_{0}^{\ell}\Re(\ell,s)\pounds_{1}\big{(}s,\vartheta(s)\big{)}ds+\sum\limits_{q=1}^{N}\frac{\partial\Re(\ell,s)}{\partial
s}\bigg{\rvert}_{s=t_{q}}\mathrm{I}_{q}\big{(}\vartheta(t_{q})\big{)}$
$\displaystyle\quad-\sum\limits_{q=1}^{N}\Re(\ell,t_{q})\mathrm{J}_{q}\big{(}\vartheta(t_{q})\big{)}.$

It follows from Lemma 3.1 and Theorem 3.2 that the control $u(t)$ satisfies
(C5). Define a map $\tilde{Q}_{\varepsilon}:PC(T,\mathfrak{B})\rightarrow
PC(T,\mathfrak{B})$ as defined in Theorem 3.2

$\displaystyle(\tilde{Q}_{\varepsilon}\vartheta)(t)$ $\displaystyle=$
$\displaystyle\Re(t,0)\big{[}x^{1}+y^{1}\big{]}+(P_{1}\vartheta)(t)+\sum\limits_{q=1}^{N}\Re(t,t_{q})\mathrm{J}_{q}\big{(}\vartheta(t_{q})\big{)}$
(3.8)
$\displaystyle\quad-\sum\limits_{q=1}^{N}\frac{\partial\Re(t,s)}{\partial
s}\bigg{\rvert}_{s=t_{q}}\mathrm{I}_{q}\big{(}\vartheta(t_{q})\big{)},$

where $P_{1}$ is given by (3.1). From Theorem 3.2, it is clear that (3.8) has
a fixed point $\vartheta_{\varepsilon},$ which is a mild solution of (1.6).
Let $\vartheta_{\varepsilon}\in\tilde{Q}_{\varepsilon}$ be a fixed point.
Easily, we see that

$\displaystyle\vartheta_{\varepsilon}(\ell)$ $\displaystyle=$
$\displaystyle\Re(\ell,0)\big{[}x^{1}+y^{1}\big{]}+(P_{1}\vartheta_{\varepsilon})(\ell)+\sum\limits_{q=1}^{N}\Re(\ell,t_{q})\mathrm{J}_{q}\big{(}\vartheta_{\varepsilon}(t_{q})\big{)}$
(3.9)
$\displaystyle\quad-\sum\limits_{q=1}^{N}\frac{\partial\Re(\ell,s)}{\partial
s}\bigg{\rvert}_{s=t_{q}}\mathrm{I}_{q}\big{(}\vartheta_{\varepsilon}(t_{q})\big{)}$
$\displaystyle=$ $\displaystyle
b-\varepsilon\mathcal{V}\big{(}\varepsilon,\Gamma_{0}^{\ell}\big{)}p(\vartheta_{\varepsilon}).$

The uniform boundedness of $\pounds_{1},\pounds_{2},\mathrm{I}_{q},$
$\mathrm{J}_{q}$ and compactness of $\Re(t,s)$ imply that there are
subsequences of
$\int_{0}^{\ell}\Re(\ell,s)\pounds_{1}(s,\vartheta_{\varepsilon}(s))ds,$
$\pounds_{2}(\ell,(\vartheta_{\varepsilon})_{\ell}),$
$\sum\limits_{q=1}^{N}\Re(\ell,t_{q})\mathrm{J}_{q}\big{(}\vartheta_{\varepsilon}(t_{q})\big{)}$
and $\sum\limits_{q=1}^{N}\frac{\partial\Re(\ell,s)}{\partial
s}\Big{\rvert}_{s=t_{q}}\mathrm{I}_{q}\big{(}\vartheta_{\varepsilon}(t_{q})\big{)}$
denoted by themselves, respectively; that converge to
$\hat{\pounds_{1}},\hat{\pounds_{2}},\hat{J}$ and $\hat{I},$ respectively.
Let
$\chi=b+{\dfrac{\partial\Re(\ell,s)\big{[}\varPhi(0)+\pounds_{2}(0,\varPhi)\big{]}}{\partial
s}}\bigg{\rvert}_{s=0}-\Re(\ell,0)\big{[}x^{1}+y^{1}\big{]}+\hat{\pounds}_{2}-\hat{\pounds}_{1}+\hat{I}-\hat{J},$
then

$\displaystyle\big{\|}p(\vartheta_{\varepsilon})-\chi\big{\|}$
$\displaystyle\leq$
$\displaystyle\big{\|}\pounds_{2}\big{(}\ell,(\vartheta_{\varepsilon})_{\ell}\big{)}-\hat{\pounds_{2}}\big{\|}+\bigg{\|}\int_{0}^{\ell}\Re(\ell,s)\pounds_{1}\big{(}s,\vartheta_{\varepsilon}(s)\big{)}ds-\hat{\pounds_{1}}\bigg{\|}$
(3.10)
$\displaystyle\quad+\Bigg{\|}\sum\limits_{q=1}^{N}\frac{\partial\Re(\ell,s)}{\partial
s}\bigg{\rvert}_{s=t_{q}}\mathrm{I}_{q}\big{(}\vartheta_{\varepsilon}(t_{q})\big{)}-\hat{I}\Bigg{\|}$
$\displaystyle\quad+\Bigg{\|}\sum\limits_{q=1}^{N}\Re(\ell,t_{q})\mathrm{J}_{q}\big{(}\vartheta_{\varepsilon}(t_{q})\big{)}-\hat{J}\Bigg{\|},$
$\displaystyle\rightarrow 0~{}\mbox{as}~{}\varepsilon\rightarrow 0.$

Using (3.9), (3.10) and Lemma 2.5, we obtain

$\displaystyle\big{\|}\vartheta_{\varepsilon}(\ell)-b\big{\|}$
$\displaystyle\leq$
$\displaystyle\big{\|}\varepsilon\mathcal{V}\big{(}\varepsilon,\Gamma_{0}^{\ell}\big{)}(\chi)\big{\|}+\big{\|}\varepsilon\mathcal{V}\big{(}\varepsilon,\Gamma_{0}^{\ell}\big{)}\big{\|}\big{\|}p(\vartheta_{\varepsilon})-\chi\big{\|}$
$\displaystyle\rightarrow 0~{}\mbox{as}~{}\varepsilon\rightarrow 0.$

∎

##  4\. Application

Example 4.1. Consider the following neutral integro-differential equations:

$\displaystyle\left\\{\begin{array}[]{lll}\dfrac{\partial^{2}}{\partial
t^{2}}\big{[}\varrho(t,y)+\pounds_{2}\big{(}t,\varrho(t-\delta_{1},y)\big{)}\big{]}\\\
\hskip 17.07182pt=\dfrac{\partial^{2}}{\partial
y^{2}}\big{[}\varrho(t,y)+\pounds_{2}\big{(}t,\varrho(t-\delta_{1},y)\big{)}\big{]}+\mathcal{F}(t)\big{[}\varrho(t,y)+\pounds_{2}\big{(}t,\varrho(t-\delta_{1},y)\big{)}\big{]}\\\
\hskip
22.76228pt+\displaystyle\int_{0}^{t}\hbar(t-s)\frac{\partial^{2}}{\partial
y^{2}}\big{[}\varrho(t,y)+\pounds_{2}\big{(}t,\varrho(t-\delta_{1},y)\big{)}\big{]}ds\\\
\hskip 28.45274pt+\pounds_{1}\big{(}t,\varrho(t,y)\big{)}+u(t,y),\quad
y\in[0,2\pi],~{}t\in[0,\ell],~{}t\neq t_{q},&\\\
\varrho(t,0)=\varrho(t,2\pi)=0,\quad t\in[0,\ell],&\\\
\varrho(\theta,y)=\varPhi(\theta,y),{\dfrac{\partial}{\partial
t}}\varrho(t,y)\Big{\rvert}_{t=0}=b_{1}(y),\quad\theta\in(-\infty,0],~{}y\in[0,2\pi],&\\\
\Delta\varrho(t,y)\big{\rvert}_{t=t_{q}}=\displaystyle\int_{0}^{2\pi}\varphi_{q}(\xi,y)\frac{(\varrho(t_{q},\xi))^{2}}{\pi\big{(}1+(\varrho(t_{q},\xi))^{2}\big{)}}d\xi,\quad
q=1,2,3,\ldots,N,&\\\ \Delta\frac{\partial}{\partial
t}\varrho(t,y)\big{\rvert}_{t=t_{q}}=\displaystyle\int_{0}^{2\pi}\varsigma_{q}(\xi,y)\frac{(\varrho(t_{q},\xi))^{4}}{2e^{2}\big{(}1+(\varrho(t_{q},\xi))^{4}\big{)}}d\xi,\quad
q=1,2,3,\ldots,N,\end{array}\right.$ (4.9)

where $t_{q}$ are the impulse points on $[0,\ell]$ such that
$0=t_{0}<t_{1}<t_{2}<\ldots<t_{N}<t_{N+1}=\ell$ ($q=0,1,2,\ldots,N$),
$\mathcal{F},\hbar:[0,\ell]\rightarrow\mathbb{R},$
$\varPhi:(-\infty,0]\times[0,2\pi]\rightarrow\mathbb{R}$ and
$b_{1}:[0,2\pi]\rightarrow\mathbb{R}$ satisfy suitable constraints.

Let $\mathfrak{B}=L^{2}(0,2\pi)$ be the Banach space endowed with norm
$\|\cdot\|$ such that $\varPhi(\theta,\cdot),b_{1}(\cdot)\in\mathfrak{B}.$
Here, $\wp$ be a phase space and $\varPhi(\theta)(y)=\varPhi(\theta,y)$ for
$\theta\in(-\infty,0],y\in[0,2\pi].$ We will assume that the map $\pounds_{1}$
satisfies the conditions (C1)-(C2).

We define the operator $\mathcal{F}_{0}$ as

$\mathcal{F}_{0}\big{[}\varrho(t,y)+\pounds_{2}\big{(}t,\varrho(t-\delta_{1},y)\big{)}\big{]}=\frac{\partial^{2}}{\partial
y^{2}}\big{[}\varrho(t,y)+\pounds_{2}\big{(}t,\varrho(t-\delta_{1},y)\big{)}\big{]}$

with domain

$\mathfrak{D}(A)=\big{\\{}\varrho\in\mathfrak{B}|\varrho,\varrho^{\prime}~{}\mbox{are
absolutely
continuous}~{}\varrho^{\prime\prime}\in\mathfrak{B},\mbox{and}~{}\varrho(0)=\varrho(2\pi)=0\big{\\}}.$

Clearly, $\mathcal{F}_{0}$ generates a cosine family $\\{C(t)\\}_{t\geq 0}$ on
$\mathfrak{B}$ associated with sine family $\\{S(t)\\}_{t\geq 0}.$ Moreover,
it has discrete spectrum with eigen values $-m^{2}$ for $m\in\mathbb{N},$
whose corresponding eigen vectors are

$\vartheta_{m}(x)=\frac{1}{\sqrt{2\pi}}e^{imx},~{}m\in\mathbb{N}.$

The set $\\{\vartheta_{m}:m\in\mathbb{N}\\}$ is an orthonormal basis of
$\mathfrak{B}.$ Therefore, we have

$\mathcal{F}_{0}\varrho=\sum\limits_{m\in\mathbb{N}}{-m^{2}}\langle\varrho,\vartheta_{m}\rangle\vartheta_{m},~{}\varrho\in\mathfrak{D}(\mathcal{F}_{0}),$
$C(t)\varrho=\sum\limits_{m\in\mathbb{N}}\cos(mt)\langle\varrho,\vartheta_{m}\rangle\vartheta_{m},~{}t\in\mathbb{R},$

and

$S(t)\varrho=\sum\limits_{m\in\mathbb{N}}\frac{\sin(mt)}{m}\langle\varrho,\vartheta_{m}\rangle\vartheta_{m},~{}t\in\mathbb{R}.$

Define $\mathcal{A}(t)\varrho=\mathcal{F}_{0}\varrho+\mathcal{F}(t)\varrho$ on
$\mathfrak{D}(\mathcal{A}).$ By defining
$\zeta(t,s)=\hbar(t-s)\mathcal{F}_{0}$ for $0\leq s\leq t\leq\ell$ on
$\mathfrak{D}(\mathcal{A})$ and following the above, we can rewrite (4.9) into
an abstract form (1.6). Furthermore, there exists a resolvent operator
$\\{\Re(t,s)\\}_{0\leq s\leq t\leq\ell}$ associated with the homogeneous
system of (4.9). Also, we can easily show that all the assumptions are
satisfied.

Remark(i) Under the above conditions, Theorem 3.2 guarantees that $\exists$ a
mild solution for (4.9).
(ii) In addition, if the functions
$\pounds_{1},\pounds_{2},\varphi_{q},\varsigma_{q}(q=1,2,\ldots,N)$ are
uniformly bounded and the associated linear system of (4.9) is approximately
controllable on $[0,\ell],$ then by Theorem 3.3, the system (4.9) is
approximately controllable on $[0,\ell].$

##  5\.  Conclusion

In this work, we established some sufficient conditions for the existence of
mild solutions of the second-order non-autonomous neutral integrodifferential
equations with impulsive conditions. And then, we study the approximate
controllability of the system. The resolvent operator technique was used to
obtain the results. We gave an example to show the effectuality of the
results. By applying the same technique to fractional-order neutral
differential equations, we will make some remarks on the nature of the system.

Acknowledgment The authors would like to thank the referees for their valuable
suggestions. The first and third authors acknowledge UGC, India, for providing
financial support through MANF F.82-27/2019 (SA-III)/ 4453 and F.82-27/2019
(SA-III)/191620066959, respectively.

## References

  * [1] Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. New York, Hindawi Publishing Corporation (2006)
  * [2] Balasubramaniam, P., Ntouyas, S. K.: Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space. J. Math. Anal. Appl. 324, 161-176 (2006)
  * [3] Das, S., Pandey, D.N., Sukavanam, N.: Approximate controllability of a second-order neutral stochastic differential equation with state-dependent delay. Nonlinear Anal. Model. Control. 21(6), 751-769 (2016)
  * [4] Das, S., Pandey, D.N., Sukavanam, N.: Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state dependent delay. Acta Math. Sci. Ser. B Engl. Ed. 36(5), 1509-1523 (2016)
  * [5] Dauer, J. P.: Nonlinear perturbations of quasi-linear control systems. J. Math. Anal. Appl. 54, 717-725 (1976)
  * [6] Fu, X., Haung, R.: Existence of solutions for neutral integro-differential equations with state-dependent delay. Appl. Math. Comput. 224, 743-759 (2013)
  * [7] Grimmer, R.: Resolvent operators for integral equations in a Banach spaces. Trans. Am. Math. Soc. 273, 333-349 (1982)
  * [8] Haloi, R.: Approximate controllability of non-autonomous nonlocal delay differential equations with deviating arguments. Electron. J. Differential Equations. 111, 1-12 (2017)
  * [9] Hernández, E., Azevedo, K.A.G., O’Regan, D.: On second order differential equations with state-dependent delay. Appl. Anal. 97, 2610-2617 (2018)
  * [10] Hernández, E., O’Regan, D.: Existence of solutions for abstract non-autonomous neutral differential equations. Canad. Math. Bull. 55(4), 736-751 (2011)
  * [11] Hernández, E., Wu, J., Fernandes, D.: Existence and uniqueness of solutions for abstract neutral differential equations with state-dependent delay. Appl. Math. Optim. 81, 89-111 (2020)
  * [12] Henríquez, H. R.: Existence of solutions of non-autonomous second order functional differential equations with infinite delay. Nonlinear Analysis TMA 74, 3333-3352 (2011)
  * [13] Henríquez, H. R., Pozo, J.C.: Existence of solutions of abstract non-autonomous second order integro-differential equations. Bound. value Probl. 168, 1-24 (2016)
  * [14] Kamaljeet, Bahuguna, D.: Approximate controllability of nonlocal neutral fractional integro-differential equations with finite delay. J. Dyn. Control Syst. 22, 485-504 (2016)
  * [15] Kamaljeet, Sukavanam, N.: Approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory and an approximating technique. Appl. Math. Comput. 364, (2019)
  * [16] Kumar, A., Muslim, M., Sakthivel, R.: Controllability of the second-order nonlinear differential equations with non-instantaneous impulses. J. Dyn. Control Syst. 24, 325–342 (2018)
  * [17] Kumar, P.: On the new concepts of solutions and existence results for impulsive differential equations with a deviating argument. Nonlinear Dyn. Syst. Theory. 14(1), 58-75 (2014)
  * [18] Kumar, R.: Nonlocal cauchy problem for analytic resolvent integro-differential equations in Banach spaces. Appl. Math. Comput. 47, 352-362 (2008)
  * [19] Leiva, H.: Controllability of semilinear impulsive nonautonomous systems. Int. J. Control 88(3), 585-592 (2015)
  * [20] Mahmudov, N. I, Vijayakumar, V., Murugesu,R.: Approximate controllability of second-order evolution differential inclusions in Hilbert spaces, Mediterr. J. Math. 13, 3433-3454 (2016)
  * [21] Mokkedem, F., Fu, X.: Approximate controllability of semi-linear neutral integro-differential systems with finite delay. Appl. Math. Comput. 242, 202-215 (2014)
  * [22] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983)
  * [23] Ravichandran, C., Valliammal, N., Nieto, J.J.: New results on exact controllability of a class of fractional neutral integro-differential systems with state-dependent delay in Banach spaces. J. Frankl. Inst. 356, 1535-1565 (2019)
  * [24] Rezapour, S., Henríquez, H. R., Vijayakumar, V., Nisar, K.S., Shukla, A.: A note on existence of mild solutions for second order neutral integro-differential evolution equations with state-dependent delay. Fractal Fract. 5, 126 (2021)
  * [25] Santos, J., Henríuez, H., Hernández, E.: Existence results for neutral integro-differential equations with unbounded delay. J. Integral Equ. Appl. 23(2), 289-330 (2011)
  * [26] Singh, V., Chaudhary, R., Pandey, D.N.: Approximate controllability of second-order non-autonomous stochastic impulsive differential systems. Stoch. Anal. Appl. 39(2), 339-356 (2021)
  * [27] Vijayakumar, V.: Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces. IMA J. Math. Control Inf. 35, 297-314 (2018)
  * [28] Vijayakumar, V.: Approximate controllability results for analytic resolvent integro-differential inclusions in Hilbert spaces. Int. J. Control 91, 204-214, (2018)
  * [29] Vijayakumar, V., Murugesu, R.: Controllability for a class of second-order evolution differential inclusions without compactness. Appl. Anal. 98(7), 1367-1385 (2019)
  * [30] Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of impulsive differential equations. Series in Modern Applied Mathematics, World Scientific Publishing (1989)
  * [31] Yang, T.: Impulsive Systems and Control: Theory and Applications. Berlin Germany: Springer-Verlag (2001)