Novasinergia 2022 5(1), 06-16. https://doi.org/10.37135/ns.01.09.01 http://novasinergia.unach.edu.ec
Research article
Approximate controllability of non-instantaneous impulsive semilinear time-
dependent control systems with unbounded delay and non-local condition
Controlabilidad aproximada de sistemas de control semilineales no autónomos con
impulsos no instantáneos, retardo no acotado y condiciones no locales
Katherine García , Hugo Leiva *
School of Mathematical and Computational Sciences, Department of Mathematics, University Yachay Tech, Imbabura, Ecuador, 00119;
katherine.garciap@yachaytech.edu.ec
*Correspondencia: hleiva@yachaytech.edu.ec
Citación: Garcia, K., & Leiva, H.,
(2022). Approximate Controllability
of Non-Instantaneous Impulsive
Semilinear Time-Dependent Control
Systems with Unbounded Delay and
Non-Local Condition. Novasinergia.
4(5). 06-16.
https://doi.org/10.37135/ns.01.09.01
Recibido: 07 diciembre 2021
Aceptado: 29 enero 2022
Publicado: 31 enero 2022
Novasinergia
ISSN: 2631-2654
Abstract: In this work, we study the approximate controllability of a
control system with unbounded delay, non-instantaneous impulse, and
non-local conditions. These results prove once again that the
controllability of a linear system is preserved if we consider the impulses,
the non-local conditions and the delays as disturbances of it, which is very
natural in real life problems, never the critical points of a differential
equation is exactly the critical point of the model that it represents, the
same happens with the impulses, the delay and non-local conditions; they
are intrinsic phenomena to the real problem, that many times they are not
taken into account at the moment of carrying out the mathematical
modeling. To achieve our result, we will use a technique developed by A.
Bashirov et al., which does not use fixed point theorems. On the other
hand, as the delay is infinite, we consider a phase space that satisfies the
axiomatic theory propose by Hale-Kato to study retarded differential
equations with unbounded delay.
Keywords: Approximate controllability, Bashirov et. at technique, Hale-
Kato axiomatic theory, non-instantaneous impulses, non-local conditions,
semilinear retarded equations with infinite delay
2020 Mathematics Subject Classification: Primary 34K35. Secondary
37L05
Copyright: 2022 derechos otorgados
por los autores a Novasinergia.
Este es un artículo de acceso abierto
distribuido bajo los términos y
condiciones de una licencia de
Creative Commons Attribution (CC
BY NC).
(http://creativommons.org/licenses/
by/4.0/).
Resumen: En este trabajo estudiamos la controlabilidad aproximada de un
sistema de control con retardo no acotado, impulso no instantáneo y condiciones
no locales. Estos resultados prueban una vez más que la controlabilidad de un
sistema lineal se preserva si consideramos los impulsos, las condiciones no locales
y los retardos como perturbaciones del mismo, lo cual es muy natural en los
problemas de la vida real, nunca los puntos críticos de una ecuación diferencial
corresponden exactamente el punto crítico del modelo que representa, lo mismo
ocurre con los impulsos, el retardo y las condiciones no locales; son fenómenos
intrínsecos al problema real, que muchas veces no son tomados en cuenta al
momento de realizar la modelación matemática. Para lograr nuestro resultado,
utilizaremos una técnica desarrollada por A. Bashirov et al., que no utiliza
teoremas de punto fijo. Por otro lado, como el retardo es infinito, consideramos
un espacio de fase que satisface la teoría axiomática propuesta por Hale-Kato para
estudiar ecuaciones diferenciales retardadas con retardo no acotado.
Palabras claves: Condiciones no locales, controlabilidad aproximada,
ecuaciones semilineales retardadas con retardo infinito, impulsos no
instantáneos, técnica de Bashirov et. at, teoría axiomática de Hale-Kato
Novasinergia 2022, 5(1), 06-16 7
1. Introduction
In this work, we study the approximate controllability of the following system with
unbounded delay, non-instantaneous impulse, and non-local condition. To achieve our result
we will use a technique developed by A. Bashirov et al. (Bashirov & Ghahramanlou, 2014;
Bashirov & Jneid, 2013; Bashirov, Mahmudov, Şemi, & Etikan, 2007), which does not use fixed
point theorems as many researchers do. On the other hand, as the delay is infinite, we consider
a phase space that satisfies the axiomatic theory propose by Hale-Kato to study retarded
differential equations with unbounded delay (Hale & Kato, 1978; Hino, Murakami, & Naito,
2013). These results prove once again that the controllability of a linear system is preserved if
we consider the impulses, the non-local conditions and the delays as disturbances of it, which
is very natural in real life problems, never the critical points of a differential equation is exactly
the critical point of the model that it represents, the same happens with the impulses, the delay
and non-local conditions; they are intrinsic phenomena to the real problem, that many times
they are not taken into account at the moment of carrying out the mathematical modeling:
(1.1)
where , , , .
There exists a fixed number such that , where is the maximal interval
of local existence of solutions of problem (1.1); and , , selected
under specific rules marked by the real-life problem that the mathematical model could
represent, such as:
. The advantage of using non-local conditions is that
measurements at more places can be incorporated to get better models. For more details and
physical interpretations see Byszewski & Lakshmikantham (1991), Byszewski (1990),
Byszewski (1991), Chabrowski (1984), Vrabie (2015), and references therein. ,
, is the phase space to be specified later. , is a
smooth enough function, , are continuous and represents the
impulsive effect in the system (1.1), i.e., we are considering that the system can have abrupt
changes that stay there for an interval of time. These alterations in state might be due to certain
external factors, which cannot be well described by pure ordinary differential equations, (see,
for instance, Lakshmikantham, Bainov, & Simeonov (1989) and Selvi & Arjunan (2012) and
reference therein). , and , where is the phase space that will
be defined later (see section 2). For this type of problems the phase space for initial functions
plays an important role in the study of both qualitative and quantitative theory, for more
details, in case without impulses and non-local conditions, we refer to Hale & Kato (1978), Hino
et al. (2013) and Shin (1987, 1987). The function for illustrate the
Novasinergia 2022, 5(1), 06-16 8
history of the state up to the time , and also remembers much of the historical past of ,
carrying part of the present to the past.
Additionally, we assume the following conditions on the nonlinear term
(1.2)
where is a continuous function. In particular, , with .
Associated with the semilinear system (1.1), we also consider the linear system
(1.3)
Also, we shall assume the following hypothesis:
H1) The linear control system (1.3) is exactly controllable on any interval , for all with
.
The hypothesis H1) is satisfied in the case that and are constant matrices
since the algebraic Kalman's condition (Lee & Markus, 1986) for exact controllability of linear
autonomous systems do not depend on the time interval.
Other examples of time-dependent systems satisfying the hypothesis H1) can be found in Leiva
& Zambrano (1999). In addition, there are several papers on the existence of solutions of
semilinear evolution equations with impulses, with impulses and bounded delay, with
bounded delay and non-local condition, and with non-local conditions and impulses. To
mention, one can see Selvi & Arjunan, (2012). Recently, in Abbas, Arifi, Benchohra, & Graef
(2020), the existence of periodic mild solution of infinite delay evolution equations with non-
instantaneous impulses has been studied by using Koratowski's measure of non-compactness
and Sadowski's fixed point theorem. In recently work Ayala-Bolagay, Leiva, & Tallana-
Chimarro (2020), using some ideas from the preceding paper and Hale & Kato (1978), Liu
(2000), Liu, Naito, & Van Minh (2003), to define a particular phase space satisfying Hale-Kato
axiomatic theory, the existence of solutions for this type of systems has been proved applying
Karakosta's fixed point theorem, which is an extension of Krasnosel'skii's Fixed Point Theorem
for contraction and compact mappings, as in Karakostas (2003), Leiva & Sundar (2017). But, as
far as we know, this system's controllability has not been studied before.
2. Preliminaries
This section is dedicated mainly to select the appropriate phase space to set this
problem, which must satisfy the axiomatic theory proposed by Hale and Kato to study
differential equations with infinite delay; that on the one hand, on the other hand, we will give
a formula for the solutions of the problem posed through the evolution operator or transition
Novasinergia 2022, 5(1), 06-16 9
matrix corresponding to the associated linear system. To this end, we denote by the
fundamental matrix of the linear system
(2.4)
i.e.,
then the evolution operator is defined by . For , we
consider the following bound for the evolution operator
Now, we shall define the space of normalized piecewise continuous function, denoted by
, as the set of functions such that their restriction to any interval of the
form is a piecewise continuous function. i.e.,
is a piecewise continuous function
Using some ideas from Liu (2000), we consider a function such that
1. ,
2. ,
3. is decreasing.
Remark 2.1. A particular function is with .
Now, we define the following functions space
In Abbas et al. (2020), Hale & Kato (1978), Liu (2000), Liu et al. (2003), and other references it is
mentioned that this space is a Banach space, which certainly it is true because we did the proof.
Lemma 2.2. The space endowed with the norm
is a Banach space.
Our phase space will be
equipped with the norm
Novasinergia 2022, 5(1), 06-16 10
Now, we shall consider the following larger space
for a fixed
and is a continuous except at
where side limits exist and
From Lemma 2.2, we have the following,
Lemma 2.3. is a Banach space endowed with the norm
where
.
For more details about it, one can see Abbas et al. (2020), Hale & Kato (1978), Liu (2000), Liu et
al. (2003).
Thus, will be a linear space of functions mapping into endowed with a norm .
Now, let us denote by
i.e.,
and the norm in the space is given by
Definition 2.4. (Exact Controllability) The system (1.1) is said to be exactly controllable on
if for every ,
1n
z
R
there exists such that the solution of (1.1)
corresponding to verifies:
and
Definition 2.5. (Approximate Controllability) The system (1.1) is said to be approximately
controllable on if for every , and , there exists such
that the solution of (1.1) corresponding to verifies:
and
Novasinergia 2022, 5(1), 06-16 11
3. Controllability of Linear System
In this section, we shall present some known characterization of the controllability of the
linear system (3.5) without impulses, delays, and non-local conditions. To this end, we note
that for all and the initial value problem
(3.5)
admits only one solution given by
(3.6)
Definition 3.1. Corresponding with (3.5), we define the following matrix: The Gramian
controllability matrix by:
(3.7)
Proposition 3.1. (See Leiva, Cabada, & Gallo (2020)) The system (3.5) is controllable on
if, and only if, the matrix is invertible.
Moreover, a control steering the system (3.5) from initial state to a final state on the
interval is given by
(3.8)
i.e.,
The corresponding solution of the linear system (3.5) satisfies the boundary condition:
and
4. Main Result
This section is devoted to the main result of the present work, i.e., the approximate
controllability of the semilinear system in (1.1) with infinite delay, nonlocal conditions and non-
instantaneous impulses. According to Abbas et al. (2020), Ayala-Bolagay et al. (2020) for all
and , the problem (1.1) admits only one solution , which is given,
for , by
(4.9)
Novasinergia 2022, 5(1), 06-16 12
Theorem 4.1. If the functions are smooth enough, condition (1.2) holds, and the linear system
(3.5) is exactly controllable on any interval , , then system (1.1) is approximately
controllable on .
Proof. Given , a final state and , we want to find a control steering
the system to a ball of center and radius on . Indeed, we consider any fixed control
and the corresponding solution of the problem (1.1).
For
0 min ,
p
sMK
−
, we define the control as follows
if
if
where and
Since , then ; and using the cocycle property , the
associated solution of the time-dependent impulsive semilinear retarded
differential equation with infinite delay and non-local (1.1), at time , can be expressed as
follows:
Therefore,
The corresponding solution of the initial value problem (3.5) at
time , for the control and the initial condition , is given by:
and from Proposition (3.1), we get that
Thus,
Novasinergia 2022, 5(1), 06-16 13
(4.10)
Now, since and , then and
Hence, since satisfies
, from (1.3), we get:
which completes the proof.
5. Final Remark
In this work, we have proved the approximate controllability of a control system
governed by a retarded differential equation with unbounded delay, non-local conditions and
non-instantaneous impulses, without the need to use fixed point theorems, only applying a
technique used by Bashirov & Ghahramanlou (2014), Bashirov & Jneid (2013), and Bashirov et
al. (2007). However, to prove exact controllability we can use Rothe's Fixed Point Theorem or a
new technique that appears in assuming certain conditions in nonlinear terms on the one
hand,or assuming certain conditions on the other hand in the Gramian matrix; this is part of
ongoing research. Also, the ideas presented here can be used to study the controllability of
infinite-dimensional systems in Hilbert spaces where the dynamical is given by the
infinitesimal generator of a compact semigroup , in this case we only get
approximate controllability of the system.
Interest conflict
The funders had no role in the study design; in the collection, analysis or interpretation
of data; in the writing of the manuscript or in the decision to publish the results.
Authors’ contributions
Following the internationally established taxonomy for assigning credits to authors
of scientific articles (https://casrai.org/credit/). The authors declare their contributions in the
following matrix:
Novasinergia 2022, 5(1), 06-16 14
García, K.
Leiva, H.
Conceptualization
Formal Analysis
Investigation
Methodology
Resources
Validation
Writing-review & editing
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