1. Introduction
State-dependent delays are ubiquitous in applications, such as 3D printing and oil drilling. The formulation of the problem working with the control of nonlinear systems with state-dependent delays on the input can be studied by designing a “nonlinear predictor feedback” law that compensates the input delay. In [
1], the authors introduced the concept of nonlinear predictor feedback starting from nonlinear systems with constant delays all the way through to predictor feedback for nonlinear systems with state-dependent delays. In [
2], Bekiaris-Liberis considered nonlinear control systems with long, unknown input delays that depend on either time or the plant state and studied the robustness of nominal constant-delay predictor feedbacks. He showed that when the delay perturbation and its rate have sufficiently small magnitude, the local asymptotic stability of the closed-loop system, under the nominal predictor-based design, is preserved. For the special case of linear systems, and under only time-varying delay perturbations, he proved robustness of global exponential stability of the predictor feedback when the delay perturbation and its rate are small in any one of four different metrics. In addition, he presented two examples, one that is concerned with the control of a Direct Current (DC) motor through a network and another of a bilateral teleoperation between two robotic systems.
Very recently, Xuetao and Quanxin [
3] studied a class of stochastic partial differential equations with Poisson jumps, which is more realistic for establishing mathematical models and has been widely applied in many fields. Under reasonable conditions, they not only established the existence and uniqueness of the mild solution for the investigated system but proved that the
pth moment was exponentially stable by using fixed point theory. They also proved that the mild solution is almost surely the
pth moment, and, therefore, exponentially stable using the well-known Borel-Cantelli lemma. In another publication, Xuetao, Quanxin, and Zhangsong [
4] discussed the exponential stability problem of a class of nonlinear hybrid stochastic heat equations (with Markovian switching) in an infinite state space. Here, the fixed point theory was utilized to discuss the existence, uniqueness, and
pth moment’s exponential stability of the mild solution. Moreover, they acquired the Lyapunov exponents by combining fixed point theory and the Gronwall inequality. In [
5], the authors investigated the stability problem for this class of new systems since Poisson jumps are considered to fill the mathematical gap. By using fixed point theory, they first studied the existence and uniqueness of the solution as well as the
pth moment’s exponential stability for the considered system. Then, based on the well-known Borel-Cantelli lemma, they proved that the solution was almost the
pth moment and exponentially stable.
It is shown in [
6], as another application, that the queuing delay involved in the congestion control algorithm is state-dependent and does not depend on the current time. Then, using an accurate formulation for buffers, networks with arbitrary topologies are built. At equilibrium, their model reduces to the widely used set-up. Using this model, the delay derivative is analyzed, and it is proven that the delay time derivative does not exceed one for the considered topologies. It is then shown that the considered congestion control algorithm globally stabilizes a delay-free single buffer network. Finally, using the specific linearization result for systems with state-dependent delays from Cooke and Huang [
7], they showed the local stability of the single bottleneck network. Hartung, Herdman, and Turi [
8] discussed the existence, uniqueness and numerical approximation for neutral equations with state-dependent delays.
There have been two main foci of mathematical control theory, which, at times, have seemed to work in entirely different directions. One of these is based on the idea that a good model of the object to be controlled is available, and one wants to minimize its behavior. For instance, physical principles and engineering specifications are used in order to calculate the optimal trajectory of a spacecraft that minimizes total travel time or fuel consumption. The techniques being used here are closely related to the classical calculus of variations and some areas of optimization; the end result is typically a preprogrammed flight plan. The other main focus is based on the constraints imposed by uncertainty about the model or about the environment in which the object operates. The central tool here is the use of feedback (state-dependent delay) in order to correct for deviations from the desired behavior. Thus, state-dependent delay systems are very important applicable systems.
The theory of semigroups of bounded linear operators is closely related to the solution of differential and integro-differential equations in Banach spaces. Using the method of semigroups, various types of solutions of semilinear evolution equations have been discussed by Pazy [
9]. The theory of neutral differential equations in Banach spaces has been studied by several authors [
10,
11,
12,
13,
14,
15].
The notion of controllability is of great importance in mathematical control theory. It makes it possible to steer from any initial state of the system to any final state in some finite time using an admissible control. The concept of controllability plays a major role in finite-dimensional control theory; thus, it is natural to try to generalize it to infinite dimensions. The controllability of nonlinear systems, represented by ordinary differential equations in a finite dimensional space, is studied by means of fixed point principles [
16]. This concept has been extended to infinite-dimensional spaces by applying semigroup theory [
9]. Controllability of nonlinear systems, with different types of nonlinearity, has been studied with the help of fixed point principles [
17]. Several authors have studied the problem of controllability of semilinear and nonlinear systems represented by differential and integro-differential equations in finite or infinite dimensional Banach spaces [
18,
19,
20,
21].
The impulsive differential systems can be used to model processes that are subjected to abrupt changes at certain moments. Examples include population biology, the diffusion of chemicals, the spread of heat, the radiation of electromagnetic waves, etc. [
22,
23,
24]. The study of dynamical systems with impulsive effects has been an object of investigations [
25,
26,
27,
28]. It has been extensively studied under various conditions on the operator
A and the nonlinearity
f by several authors [
29,
30,
31]. Chalishajar and Acharya studied controllability of neutral impulsive differential inclusion with nonlocal conditions [
32], (see also [
33,
34,
35,
36]), Anguraj and Karthikeyan [
37] discussed the existence of solutions for impulsive neutral functional differential equations with nonlocal conditions. Ahmad, Malar, and Karthikeyan [
38] studied nonlocal problems of impulsive integro-differential equations with a measure of noncompactness. Very recently, Klamka, Babiarz, and Niezabitowaski [
39] did the survey based on Banach fixed point theorem in semilinear controllability problems and studied Schauder’s fixed-point theorem in approximate controllability problems [
40]. In addition, Balasubramaniam and Tamilalagan [
41] studied approximate controllability of fractional neutral stochastic integro-differential inclusion with infinite delay by using Mainardi’s function and Bohnenblust-Karlin’s fixed point theorem. They also discussed approximate controllability of fractional stochastic equations driven by mixed fractional Brownian motion in a Hilber space with Hurst papramenter
(see [
42]). They proved the solvability and optimal controls for fractional stochastic integro differential equations (see [
43]).
Motivated by the above-mentioned works, we show that a particular class of impulsive neutral evolution integro-differential systems with state-dependent delay in Banach spaces is controllable provided that some conditions are satisfied, using the theory of resolvent operators. The system considered here is untreated in the literature, which is a main motivation of the current work. Here, we have defined a new phase space for state-dependent infinite delay.
2. Preliminaries
In this section, we recall some relevant definitions, notations, and results that we need in the sequel. Throughout this paper,
is a Banach space and
generates the evolution operator in
X. In addition,
are closed linear operators defined on a common domain
), which is dense in
X. The notation
represents the domain of
endowed with the graph norm. Let
and
be Banach spaces. The notation
represents the Banach space of bounded linear operators from
Z onto
W endowed with the uniform operator topology, and we abbreviate this notation to
when
In this paper, we establish the controllability of impulsive neutral evolution integro-differential equations with state-dependent delay described by
where the unknown
takes values in a Banach space
X and the control function
, a Banach space of admissible control functions with
U as a Banach space. Furthermore,
B is a bounded linear operator from
U to
X.
, for
is a bounded linear operator on
X,
I is an interval of the form
are prefixed numbers. The history
given by
belongs to some abstract phase space
defined axiomatically with
,
and
are appropriate functions. The symbol
represents the jump of the function
ξ at
t, which is defined by
To obtain our results, we assume that the abstract impulsive integro-differential system. System (1)–(3) has an associated resolvent operator of bounded linear operators on X.
Definition 1. A resolvent operator of the problems (1)–(3) is a bounded operator-valued function on X, the space of bounded linear operators on X, having the following properties:- (a)
, for every and , for every and . for some constants M and β.
- (b)
For .
- (c)
For each , is continuously differentiable in and - (d)
For each and is continuously differentiable in andwith and strongly continuous on .
Here,
can be extracted from the evolution operator of the generator
. We need to use Sadovskii’s fixed point theorem as stated below (see [
44]).
Lemma 1 (Sadovskii’s Fixed Point Theorem)
. Let N be the condensing operator on a Banach space X. If for a convex, closed and bounded set S of X, then N has a fixed point in S.
Remark 1. Sadovskii’s fixed point theorem is important for studying the stability of the given system. However, we have not discussed the stability, as it is a separate problem to study.
In this paper, we present the abstract phase space
Assume that
be a continuous function with
Define,
Here,
is endowed with the norm
Then, it is easy to show that is a Banach space.
Lemma 2. Suppose then, for each Moreover,where Proof. For any
, it is easy to see that
is bounded and measurable on
for
, and
Since
, then
. Moreover,
The proof is complete. ☐
In this work, we employ an axiomatic definition for the phase space
, which is similar to those introduced in [
45]. More precisely,
will be a linear space of functions mapping
into
X endowed with the seminorm
and satisfying the following axioms:
If
, is continuous on
and
, then, for every
, the following conditions hold:
- (a)
is in .
- (b)
.
- (c)
sup where is constant; is continuous, is locally bonded and are independent of
The space is complete.
Here, we consider some examples of phase spaces.
Example 1 (The phase space
)
. A function is said to be normalized piecewise continuous if ψ is left continuous and restriction of ψ to any interval is piecewise continuous. Let be a continuous nondecreasing function that satisfies the conditions in the terminology of [45]. Next, we slightly modify the definition of phase spaces and in [45]. We denote by the space formed by normalized piecewiese continuous functions ψ such that is bounded on and by the subspace of consisting of function ψ such that as It is easy to see that and endowed with the norm := sup are phase spaces in the sense defined above. Example 2 (The phase space
)
. Let and let be a nonneagative measurable function that satisfies the condition in the terminology of [45]. Briefly, this means that ρ is locally integrable and there exists a non-negative, locally bounded function γ on such that and where is a set with Lebesgue measure zero. The space consists of all classes of Lebesgue-measurable functions such that and is Lebesgue integrable on The seminorm in this space is defined by Proceeding as in the proof of [[45], Theorem 1.3.8], it follows that is a space that satisfies axioms –. Moreover, when this space coincides with and if, in addition, , we can take and for Remark 2. Let and The notation represents the function defined by Consequently, if the function in axiom is such that , then We observe that is well-defined for since the domain of ψ is We also note that, in general, consider, for example, functions of the type where is the characteristic function of and in the space
3. A Controllability Result for a Neutral System
In this section, we study the controllability results for the system (1)–(3). Throughout this section,
is a positive constant such that
, for every
. In the rest of this work,
φ is a fixed function in
and
will be the functions defined by
and
. We adopt the notation of mild solutions for (1)–(3) from the one given in [
46].
Definition 2. A function is called a mild solution of the impulsive neutral evolution integro-differential system (1)–(3) on [0,b] iff is differentiable on [0,b], and satisfies the following integral equation: Definition 3. The system (1)–(3) is said to be controllable on the interval I iff, for every , there exists a control such that the mild solution of (1)–(3) satisfies and [17]. To establish our controllability result, we introduce the following assumptions:
- (H1)
The function
satisfies the following conditions:
- (i)
For each , the function is continuous.
- (ii)
For every the function is strongly measurable.
- (iii)
There exists
and a continuous non-decreasing function
such that
- (H2)
- (i)
The function
g is continuous, and there exists a function
such that
- (ii)
The function is compact.
- (H3)
The linear operator defined by has an induced inverse operator that takes values in ker W, and there exists a constant such that
- (H4)
- (i)
satisfies the Caratheodory condition, which is is measurable with respect to t and continuous with respect to .
- (ii)
The function is well-defined and continuous from the set
into , and there exists a continuous and bounded function such that for every
- (H5)
For
there exist constants
such that
In addition, there exists
such that
for all
and
- (H6)
For
there exists a function
such that
- (H7)
There exists a positive constant Λ defined by
.
Remark 3. In the remainder of this section, and are constants such that and
Lemma 3. Let be continuous on [0,b] and If holds, then sup where [47]. Theorem 1. If the assumptions – are satisfied, and , then the system (1)–(3) is controllable on I provided Proof. Using assumption (H3) for an arbitrary function
, we define the control
Consider the space
endowed with the uniform convergence topology. For any
. Thus,
is a Banach space. For each positive number
r, set
It is clear that is a bounded, closed, convex set in
Consider the map
by
for
and for all
where
is such that
and
on
I. From
, the strong continuity of
, and our assumption on
ϕ; we infer that
is well-defined and continuous.
It is easy to see that
. We prove that there exists
such that
. If this property fails, then for every
,
and
such that
Then, from Lemma 3, we find that
which contradicts our assumption.
Let be such that is the number defined by and
To prove that Γ is a condensing operator, we introduce the decomposition
, where
On the other hand, for
and
we see that
where
, which implies that
is a contraction to
.
Now, we prove that is completely continuous from into .
First, we prove that the set is relatively compact on X, for every .
The case
is trivial. Let
. From the assumptions, we can fix the numbers
such that
if
for some
. Let
. From the mean value theorem for the Bochner integral (see [
48]), we see that
where dia
when
. This proves that
is totally bounded and hence relatively compact in
X, for every
.
Second, we prove that the set is equicontinuous on
Let
and
be such that
, for every
with
. Under these conditions,
and
with
We get
where
which shows that the set of functions
is right-equicontinuous at
A similar procedure proves the right-equicontinuity at zero and left-equicontinuity at
Thus,
is equicontinuous on
I.
Finally, we show that the map is continuous on .
Let be a sequence in and such that in . At first, we study the convergence of sequences
If
is such that
, we can fix
such that
, for every
(by assumption
. In this case, for
, we see that
which proves that
and
in
as
, for every
such that
. Similarly, if
and
such that
, for every
, we get
and
which also shows that
and
in
as
, for every
such that
. Combining the previous arguments, we can prove that
, for every
such that
= 0.
Now, assumption (H1) and the Lebesgue Dominated Convergence Theorem permit us to assert that in . Thus, is continuous, which completes the proof that is completely continuous. Those arguments enable us to conclude that is a condensing map on . By Lemma 1, there exists a fixed point for Γ on . Obviously, is a mild solution of the system (1)–(3) satisfying ☐
4. Example
Consider the partial integro-differential equation:
where
is continuous on
and the constant
is small,
. In this system,
are positive numbers,
are continuous, and the function
is positive. Moreover, we have identified
Put
and
, where
is continuous and
where
The system (6)–(9) is the abstract forms of (1)–(3). We choose the space and (see Example 1 for details). We also consider the operators , given by , for and , for . It is well-known that A is the infinitesimal generator of an analytic semigroup on X. Furthermore, A has a discrete spectrum with eigenvalues , and corresponding normalized eigenfunctions are given by sin .
In addition,
is an orthonormal basis of
and
for
and
. In addition, for
the fractional power
of
A is given by
where
Now, we define operator
, where
By assuming that
is continuous in
t, and there exists
such that
for all
,
, it follows that the system
has an (associated) evolution family
with
for
and
, for every
.
Under the above conditions, we can represent the system
in the abstract form (1)–(3).
Lemma 4 below is a consequence of [
27].
Lemma 4. There exists an operator resolvent for the system (6)–(9). Consider the problem of controllability for the system (6)–(9). For this, the following conditions are assumed:- (i)
The function is continuous, and
- (ii)
The functions i = 1,2 are continuous.
- (iii)
The functions φ, belong to , and the expressions
sup and are finite.
Define the operators
and
, given by
which are well-defined, then the system (6)–(9) are represented in the abstract forms (1)–(3). Moreover,
g is a bounded linear operator
With these choices of
and
, the identity operator, assume that the linear operator
W from
ker
W into
X, defined by
has an invertible operator and satisfies the condition (H3).
Furthermore, all of the conditions stated in Theorem 1 are satisfied, and it is possible to choose and check (5). Hence, by Theorem 1, the system (6)–(9) controllable on I.