Novasinergia 2022, 5(2), 06-22. https://doi.org/10.37135/ns.01.10.01 http://novasinergia.unach.edu.ec
Research article
Controllability of impulsive semilinear retarded differential equations
with infinite delay and nonlocal conditions
Controlabilidad de ecuaciones diferenciales retardadas semilineales impulsivas con retardo
infinito y condiciones no locales
Steven Allauca , Hugo Leiva*
School of Mathematical and Computational Sciences, Yachay Tech, Ibarra, Ecuador; 00119;
steven.allauca@yachaytech.edu.ec
*Correspondence: hleiva@yachaytech.edu.ec
Citación: Allauca, S., & Leiva, H.,
(2022). Controllability of
impulsive semilinear retarded
differential equations with
infinite delay and nonlocal
conditions. Novasinergia. 5(2). 06-
22.
https://doi.org/10.37135/ns.01.10.
01
Recibido: 30 mayo 2022
Abstract: In this paper, we prove the exact controllability of
semilinear retarded equations with infinity delay, impulses
and nonlocal conditions; proving the conjecture that the
controllability is preserve under the influence of delay,
impulses and nonlocal conditions if some conditions are
assumed on the nonlinear terms. As an application of our
result, we present an example were all the conditions assumed
are verified.
Aceptado: 22 junio 2022
Publicación: 05 julio 2022
Novasinergia
ISSN: 2631-2654
Keywords: controllability, infinite delay, impulses, nonlocal
conditions, Rothe’s fixed point theorem, semilinear retarded
equations.
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://creativecommons.org/licens
es/by/4.0/).
Resumen: En este artículo, se prueba la controlabilidad exacta de
ecuaciones retardadas semilineales con retardo infinito, impulsos y
condiciones no locales; probando la conjetura que afirma que la
controlabilidad se preserva bajo la influencia de retardo, impulsos y
condiciones no locales si se asumen algunas condiciones sobre los
términos no lineales. Como una aplicación de este resultado, se
presenta un ejemplo donde todas las condiciones asumidas se
satisfacen.
Palabras claves: condiciones no locales, controlabilidad,
ecuaciones retardadas semilineales, impulsos, retardo infinito,
teorema del punto fijo de Rothe.
1. Introduction
This paper is devoted to prove the exact controllability of a semilinear retarded
equations under the influence of infinity delay, impulses and nonlocal conditions; verifying
once again the conjecture that says: The controllability is preserve under the influence of
delay, impulses and nonlocal conditions if some conditions are assumed. Specifically, we
shall prove that the controllability of the associated time dependent linear systems is
preserve by the following semilinear retarded system of differential equations with
impulses, infinite delays, and nonlocal conditions:
Novasinergia 2022, 5(2), 06-22 7
(1.1)
where , , are fixed real numbers,
, , is defined as a function from to by
, , are continuous matrices of dimension and respectively,
the control function belongs to , , with being a particular
phase space satisfying the axiomatic theory defined by Hale and Kato (which will be
specified later), , , , such
that
(1.2)
(1.3)
(1.4)
(1.5)
with , , , and
Next, we will 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.,
Adapting some ideas from Hale & Kato (1978), Liu (2000) and Liu, Naito, & Van Minh
(2003), we confider the function satisfying the following conditions:
1. ,
2. ,
lim ( )
xdx
→−
=
3. is decreasing.
Remark 1.1. For example, we can consider the function as , with .
The following spaces will be defined in order to set our problem
Lemma 1.2. The space equipped with the norm
Novasinergia 2022, 5(2), 06-22 8
turns out to be a Banach space.
So, the phase space for our problem will be
together with the norm
Next, we are going to consider the following bigger space
defined by
As a consequence of Lemma 1.2, we get the following result
Lemma 1.3. is a Banach space endowed with the following norm
where
.
Due to the fact that the function is defined on the entire real line, we can prove the
following lemma satisfying axiom A1)-iii) from Hale and Kato axiomatic theory for the
phase space. This Lemma play an important role in the prove of our main result.
In the same way, the following spaces are defined
is equipped with the following norm
Also, we are going to use the following spaces
with the norm
endowed with norm:
Novasinergia 2022, 5(2), 06-22 9
For a given , we consider the following
expression:
We consider the associated linear system to the semilinear system (1.1):
(1.6)
Next, the most important definition will be given, which is the definition of controllability
for system (1,1):
Definition 1.4 ( Controllability) If for every , , there exists
such that the solution of (1.1) verifies:
the system (1.1) is said to be controllable on figure 1.
figure 1: Controllability definition.
One can find in the literature a large number of works on the controllability of linear systems
(Chukwu, 1991), including books, articles and monographs, see for example, Chukwu
(1992), Lee & Markus (1967), and Sontag (1998). However, the references about the
controllability of non-linear systems is limited, particularly for semilinear systems governed
by differential equations, in this regard, we can refer to the article of Lukes (1972) and the
book by Coron (2007) (see Theorem 3.40 and Corollary 3.41). On the other hand,
Vidyasager (1972) proved the controllability for semilinear systems using Schauder’s Fixed
Point Theorem and assuming that the non-linear term did not depend on the control u. Not
so, Dauer (1976) found condition on semilinear systems that turns out to be weaker than the
previous ones, which also allowed him to prove the controllability of such systems. But, Do
(1990) weakened the previous conditions on the non-linear term and was able to prove the
controllability of the semilinear systems; these new conditions generalize Dauer’s work; it
is good to mention that these conditions strongly depend on the associated linear system
(1.6) through its fundamental matrix; specifically the fundamental matrix of the linear
system , which is in general not available in closed form. We must
emphasize that in all these systems the influence of impulses, non-local conditions and delay
Novasinergia 2022, 5(2), 06-22 10
is not taking into account.
There are many concepts of controllability depending on whether the control variable has
restrictions, such as local controllability, which has been strongly studied by Chukwu (1979,
1980, 1987, 1991, 1992), Mirza & Womack (1972), Sinha & Yokomoto (1980), and Sinha (1985).
But as far as we know, these studies are also not influenced by impulses, non-local
conditions and delay, simultaneously, which is an open problem.
The controllability of differential equations with impulses, nonlocal condition and delay is
at its peak at the moment, many mathematicians and engineers are devoted to the study of
such equations; for example, for infinite dimensional systems governed by evolution
equations, we can look at the works done by Nieto & Tisdell (2010) and Zhu & Lin (2012).
Moreover, the work done in Leiva (2014a, 2014b) can be formulated in infinite dimension
spaces. For infinite dimensional systems one can see Selvi & Mallika (2012), which studied
the controllability of impulsive differential systems with finite delay by using measures of
noncompactness and Monch’s Fixed Point Theorem. In Leiva (2014a, 2014b) the Rothe’s
fixed point Theorem has been applied to prove the controllability of semilinear systems with
impulses, which is the essential motivation for doing this work.
For infinite-dimensional Banach spaces, we are sure that some ideas presented here can be
used to address also the controllability of evolution equations with impulses, delays, and
nonlocal conditions simultaneously, and the nonlinear term involving all the variables, the
time, the state , and the control. On the other hand, some results from Carrasco, Leiva,
Sanchez, & Tineo (2014), Leiva (2014a), and Leiva & Merentes (2015) give us a good ideas
to do this work.
In order to conclude this section, we will set the following result.
Lemma 1.5. For all function the following estimate holds for all :
Theorem 1.6 (Rothe’s Fixed Theorem, (Banas & Goebel, 1980; Isac, 2004; Smart, 1980) Let be
a Banach space, and be a closed convex subset such that the zero of is contained in the
interior of . Let be a continuous mapping with relatively compact in and
. Then, has at least a fixed point in . i.e., There exists a point such that
.
Our main hypotheses will be: The controllability of the linear system (1.6), the continuity of
the fundamental matrix of the uncontrolled linear system and the conditions (1.2)-(1.5)
satisfied by the nonlinear terms , , .
2. Controllability of Linear Systems
Now, we will give characterization for the controllability of linear systems (1.6) in the
Novasinergia 2022, 5(2), 06-22 11
case when impulses, infinite delays and nonlocal conditions are not considered. In doing so,
we shall consider, for all and , the initial value problem
(2.1)
which admits only one solution given by
(2.2)
with given by , where is the fundamental matrix of the
following corresponding differential equation
(2.3)
i.e., the matrix verifies:
(2.4)
where is the identity matrix. Hence, there exist constants and
such that
(2.5)
Definition 2.1. Associated with system (1.6) the following linear operators are defined:
The controllability operator (for ) is defined as follows
(2.6)
The adjoint operators of the operator is given by
(2.7)
and the Controllability Gramian operator is given by
(2.8)
Proposition 2.2. The systems (6) is controllable on if, and only if, .
Also, we will use the following result from Curtain & Pritchard (1978) and Curtain & Zwart
(1995).
Lemma 2.3. Let and be Hilbert space, and the adjoint operator.
Then the following statements holds,
(i)
(ii) .
Lemma 2.4. Then the following claims are equivalent
a) .
b) .
c) / , in .
d) .
e) .
Novasinergia 2022, 5(2), 06-22 12
Hence, the maps given by
(2.9)
is called the steering operator and it is a right inverse of , which means that
(2.10)
Moreover,
(2.11)
and a control steering the system (6) from initial state to a final state at time is
given by
(2.12)
Lemma 2.5. (Leiva, 2014b) Let be any dense subspace of . Then, system (6) is
controllable with control if, and only if, it is controllable with control .
i.e.,
where is the restriction of to .
Remark 2.6. Due to the previous Lemma, if the linear system (1.6) is controllable, it is
controllable with control functions in the following dense subspaces of :
Moreover, the operators , and are well defined in the space of continuous functions:
by
(2.13)
and by
(2.14)
Also, the Controllability Gramian operator still the same
(2.15)
Finally, the operators defined by
(2.16)
is a right inverse of , in the sense that
(2.17)
3. Results
In this part, we will prove the controllability of the nonlinear system (1.1) with
impulses, infinite delays, and nonlocal conditions. To do so, for all and
, due to Leiva (2018), the initial value problem
Novasinergia 2022, 5(2), 06-22 13
(3.1)
has one solution given by
(3.2)
Now, we define the following nonlinear operator
given by the formula:
where and are given as follow:
and
such that:
(3.3)
and (3.4)
with
(3.5)
The following proposition follows trivially from the definition of the operator .
Proposition 3.1. The Semilinear System (1.1) with impulses, infinite delay, and nonlocal
conditions is controllable if, and only if, for all initial state and a final state the
operator given by Leiva & Zambrano (1999), Liu (2000) and Liu et al. (2003) has a fixed
point. i.e.,
Theorem 3.2. Suppose that conditions (1.2)-(1.5) hold and the linear system (1.6) is controllable
Novasinergia 2022, 5(2), 06-22 14
on . If , , , then the nonlinear system
(1.1) is controllable on . Moreover, exists a control such that for a given
the corresponding solution of (1.1) satisfies:
and
with
Proof. We shall prove this theorem by claims.
Statement 1. The operator is continuous. In fact, to prove the continuity of , it is enough
to prove the continuity of the operators and defined above.
The continuity of follows from the continuity of the nonlinear functions ,
, and the following estimate
where,
and
The continuity of the operator follows from the continuity of the operators and
define above.
Statement 2. maps bounded sets of into
equicontinuous sets of .
Consider the following equality
Let be a bounded set. The equicontinuity for
is given by the equicontinuity of each one of its components , , which are
obtained from the continuity of and the following estimates
Since is continuous goes to zero as and so does the
Novasinergia 2022, 5(2), 06-22 15
sum and the integral from to , which implies that is equicontinuous. Moreover,
the equicontinuity of follows from the continuity of the evolution operator .
Hence, maps bounded sets into equicontinuous sets.
Statement 3. The set is relatively compact. Indeed, let be a bounded subset of
. By the continuity of , , and , for it
follows that
where , , , . Therefore,
is uniformly bounded. Now, we consider a sequence a in
. Since is contained in and is an
uniformly bounded and equicontinuous family, by Arzelà-Ascoli Theorem we can assume,
without loss of generality, that converges. On the other hand, since
is contained in , then
, . Taking into account that is bounded and
equicontinuous in , we can apply Arzelà-Ascoli Theorem to ensure the existence of a
subsequence
of , which is uniformly convergent on .
Now, consider the sequence
on the interval . On this interval the
sequence
is uniformly bounded and equicontinuous, and for the same
reason, it has a subsequence
uniformly convergent on . In this way, for the
intervals , , , , we see that the sequence
converges
uniformly on the interval .
Besides, in the interval the function is piecewise continuous, then repeating
the foregoing process we can assume that the subsequence
converges in . This means that is compact, i.e., is relatively
compact.
Statement 4. for , , , the following limit
holds.
where is the norm in the space
.
Using the conditions (1.2)-(1.5), we get that
where
Novasinergia 2022, 5(2), 06-22 16
and
Therefore,
where is given by:
Hence,
Consequently,
Statement 5. The operator has a fixed point. In fact, by Statement 4, we know that for a
fixed there exists big enough such that
Hence, if we denote by the closed ball of center zero and radius , we get that
. Since is a compact operator, is relatively compact in
, and maps the sphere into the interior of the ball
, we can apply Rothe’s fixed point theorem 1.6 to ensure the existence of a fixed point
such that
Hence, applying the Proposition 3.1, we get that the nonlinear system (1.1) is controllable
on . Moreover,
such that for a given , the corresponding solution of (1.1)
satisfies:
Novasinergia 2022, 5(2), 06-22 17
It completes the proof.
Now, we present another version of the previous theorem, which follows from the estimates
considered in Statement 4.
Theorem 3.3. Suppose the linear system (1.6) is controllable on . Then the nonlinear system
(1.1) is controllable if one of the following statements holds:
a) , , and
b) , ,
and
c) , ,
and
d) , ,
and
e) , ,
and
where .
f) , ,
and
where .
g) , ,
and
where
Proof Let us consider any of the conditions . Then, from the estimates obtained in
Statement 4, we get that
Hence, there exists such that
Then, analogously to the previous theorem the proof of Theorem 3.3 immediately follows
by applying Proposition 3.1.
4. Example
In this section, we present an example to illustrate our results. In this regard, we will
apply Theorem 3.2 to the semilinear time dependent control system with impulses, delay
and nonlocal conditions given by
(4.1)
where , with and and constant matrices,
respectively.
Novasinergia 2022, 5(2), 06-22 18
, and
From Leiva & Zambrano (1999), if the Kalman’s rank condition holds true
then the following time dependent linear system
with , , is exactly controllable on (Leiva & Zambrano,
1999). Here, the nonlinear terms and the impulsive functions are given as follows
,
, given by
, , given by
Then
and since and , are bounded, the conditions (1.2)-(1.5) are satisfied.
Hence, the system (29) is exactly controllable on .
Final Remark
In this paper, we have proved that the controllability is robust in the presence of
impulses, infinite delay, and nonlocal conditions. This happens for many real life control
systems where impulses, delays, and nonlocal conditions are intrinsic phenomena of the
system. Moreover, in several papers we have shown that the influence of impulses do not
destroy the controllability of some known systems like the heat equation, the wave equation
and the strongly damped wave equation (Carrasco et al., 2014; Leiva, 2014b, 2015; Leiva &
Merentes, 2015). Therefore, the same ideas presented in this work to prove the exact
controllability can be used to prove the controllability of infinite dimensional systems in
Hilbert spaces where the dynamical is given by the infinitesimal generator of a compact
Novasinergia 2022, 5(2), 06-22 19
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:
Allauca, S.
Leiva, H.
Conceptualization
Formal Analysis
Investigation
Methodology
Resources
Validation
Writing-review & editing
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