# Hyers–Ulam Stability, Exponential Stability, and Relative Controllability of Non-Singular Delay Difference Equations

Authors: Sawitree Moonsuwan; Gul Rahmat; Atta Ullah; Muhammad Yasin Khan; Kamran; Kamal Shah
Journal: Complexity
 (2022)

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## Abstract

In this paper, we study the uniqueness and existence of the solutions of four types of non-singular delay difference equations by using the Banach contraction principles, fixed point theory, and Gronwall’s inequality. Furthermore, we discussed the Hyers–Ulam stability of all the given systems over bounded and unbounded discrete intervals. The exponential stability and controllability of some of the given systems are also characterized in terms of spectrum of a matrix concerning the system. The spectrum of a matrix can be easily obtained and can help us to characterize different types of stabilities of the given systems. At the end, few examples are provided to illustrate the theoretical results.


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## Body

## 1. Introduction

In mathematics, we usually observed that many of the biological systems and models can be resolved by using differential equations. Differential equations have a lot of applications in various fields of natural sciences, economics, statistics, and engineering (see [1–4] and the references therein). Although differential equations are too useful, when we discuss a real-life problem, we need to take the sample in discrete form and show the model in a form of difference equations (for details, see [5, 6]). The applications of difference equation have appeared recently in many fields of sciences and technology, mathematical physics, and biological systems. The theory of difference equations will continue its role in mathematics as a whole because during the period of development of mathematics together with information revolution, there are many difference equations to describe the real problem such as the monographs and wind flow. Similarly, many models were described by fractional-order differential equation (FODE), in which the order of derivative is in fraction form rather than an integer form. These types of differential equations have a lot of applications in real life [7, 8]. In [7], the theoretical study of the Caputo–Fabrizio fractional modelling for hearing loss due to mumps virus with optimal control was discussed which is useful contribution in natural science. Also in [8] some novel mathematical analysis of fractal-fractional model of the AH1N1/09 virus and its generalized Caputo-type version was explained.Any type of system has some properties (qualitative properties), in which the stability is more important. Every differential system has some qualitative properties, in which the stability plays a vital role. Using this, the system performance can be checked. A differential have various types of stabilities, but here we are interested in Hyers–Ulam stability, because nowadays many researchers wants to know about this stability. The idea of Hyers–Ulam stability started in 1940 [9]. Ulam in a seminar, in his presentation he pointed out some problems associated with the stability of group homomorphism. After a year in [10], Hyers gave a positive answer to the Ulam's question by considering Banach Space in place of that group. The general approach of this stability was given in 1978, by Rassias [11]. He also used this idea in the Cauchy difference system. Obloza [12] used this idea in differential equations, and later Jung [13] and Khan et al. [14] used it in the difference equations. This stability was also discussed in fractional differential equation by Gao et al. [15], and some results on Ulam-type stability of a first-order non-linear delay dynamic system were discussed by Shah et al. in [16]. Recently, the Hyers–Ulam stability of second order differential equations by using Mahgoub transform and generalized Hyers–Ulam stability of a coupled hybrid system of integro-differential equations involving ϕ-caputo fractional operator was studied in [17,18]. The existence and Hyers–Ulam stability of solution for almost periodical fractional stochastic differential equation was discussed in [19]. Also in [20], the existence and Hyers–Ulam stability of random impulsive stochastic functional differential equations with finite delays was discussed, which showed that the Hyers–Ulam stability have a lot of contribution in fractional calculus.Controllability is one of the fundamental concepts in modern mathematical control theory. Kalman’s result [21] on controllability assumes that controls are functions on time having values on some non-empty subset of Rn. This is a qualitative property of control systems and is of particular importance in control theory. Systematic study of controllability was started at the beginning of 1960s and theory of controllability is based on the mathematical description of the dynamical system. Many dynamical systems are such that the control does not affect the complete state of the dynamical system but only a part of it. On the other hand, very often in real industrial processes, it is possible to observe only a certain part of the complete state of the dynamical system. Therefore, it is very important to determine whether or not control of the complete state of the dynamical system is possible. Roughly speaking, controllability generally means that it is possible to steer dynamical system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. Controllability plays an essential role in the development of the modern mathematical control theory. There are important relationships between controllability, stability, and stabilizability of linear control systems [22, 23]. Controllability is also strongly connected with the theory of minimal realization of linear time-invariant control systems. Moreover, it should be pointed out that there exists a formal duality between the concepts of controllability and observability [24].The delay difference system can be used in the characterization of automatic engine, control theory, and physiology system. Khusainov et al [25] solved the linear autonomous delay-time system with commutable matrices. Diblik and Khusainov [26] gave the description about the solutions of discrete delayed system using the idea [25]. Then, Wang et al. [27] studied relative controllability and exponential stability of non-singular systems. Recently, the generalized Hyers–Ulam–Rassias stability of impulsive difference equations was demonstrated by Almalki et al. [28]. Kuruklis [29] and Yu [30] studied the asymptotic behavior of the variable type delay difference equation. Kosmala and Teixeira [31] provided a good insight and discussed the behavior of solution of the difference equation of the type Uk+1=A+Uk−1/BUk+Uk−1. Liu et al [32] designed the exponential behavior of switch discrete-time delay system. Marwen and Sakly [33] discussed the stability techniques about the switched non-linear time-delay difference equations. Yuanyuan [34] described the stability techniques of high-order difference systems. The stability of higher-order rational difference systems was studied by Khaliq [35].Our present study is focused on the Hyers–Ulam stability and exponential stability of non-singular delay difference system of the form(1)EVn+1=AVn+BVn−k,n≥0,k≥0,Vn=Φn,−k≤n≤0,(2)EVn+1=AVn+BVn−k+fn,Vn,n≥0,k≥0,Vn=ϕn,−k≤n≤0,and(3)EVn+1=AVn+BVn−k+Fn,Vn−k,n≥0,k≥0,Vn=ψn,−k≤n≤0,where the commutable constant matrices are E,A,B∈ℝn×n and E is non-singular. ϕ∈BZ+,X, the space of bounded sequences, and F∈CSZ+×X,X, the space of convergent sequences, where J=−k,−k+1,…,0, Z+=0,1,2,…, and X=ℝn. Also, our focus is on relative controllability of the system(4)EVn+1=AVn+BVn−k+yn,Vn+CUn,n∈I,k≥0,Vn=Ψn,−k≤n≤0,where I=0,1,2,…,n,n>0,C∈ℝn×n,y∈CSZ+×X,X, and the control function U⋅ takes values from L2I,ℝn. The continuous form of this work is given in [27]. The Hyers–Ulam stability of (3) was recently presented in [36].


## 2. Preliminaries

Here, we discuss some notations and definitions, which will be needed for our main work. Byℝn and ℝn×n, we will denote the n-dimensional Euclidean space with vector norm ⋅ and n×n matrices with real-valued entries. The vector infinite-norm is defined as v=max1≤i≤nvi and the matrix infinite-norm is given as A=max1≤i≤n∑j=1naij where v∈ℝn and A∈ℝn×n; also, vi and aij are the elements of the vector v and the matrix A. BI,X will be the space of all bounded sequences from I to X with norm vC=supn∈Ivn. We will use ℝ,Z and Z+ for the set of real, integer, and non-native integer numbers, respectively. Also, we define B′I,X=v∈BI,X;v′∈B′I,X.Lemma 1.
The non-singular delay difference systems (1)–(4) have the solutions:(5)Vn=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiVi−k,Vn=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,Vi+An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi,Vn=AnE−nψ0+An−1E−n∑i=0kA−iEiBψi−k+Fi,ψi−k+An−1E−n∑i=k+1nA−iEiBVi−k+Fi,Vi−k,and(6)Vn=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+yi,Vi+CUi+An−1E−n∑i=k+1nA−iEiBVi−k+yi,Vi+CUi,respectively, where AE=EA,AB=BA, and EB=BE. The proofs can easily be obtained by successively putting the values of n∈−k,−k+1,….Definition 1.
The solution of system (1) is said to be exponentially stable if there exist positive real numbers λ1 and λ2, such that(7)Vn≤λ1e−λ2n,∀n≥0.Definition 2.
For a positive numberϵ, the sequence ψn is said to be an ϵ-approximate solution of (1)–(3) if the following holds:(8)Eψn+1−Aψn−Bψn−k≤ϵ,n≥0,k≥0ψn−Φn≤ϵ,−k≤n≤0,Eψn+1−Aψn−Bψn−k−fn,ψn≤ϵ,n≥0,k≥0ψn−ϕn≤ϵ,−k≤n≤0,Eψn+1−Aψn−Bψn−k−Fn,ψn−k≤ϵ,n≥0,k≥0ψn−Ψn≤ϵ,−k≤n≤0..Definition 3.
Systems (1)–(3) are said to be Hyers–Ulam stable if for every ϵ-approximate solutions ψn of systems (1)–(3) there are exact solutions Yn of (1)–(3) and a non-negative real number K such that(9)Yn−ψn≤Kϵ,n∈I.Definition 4.
System (4) is said to be relatively controllable, if for initial vector function Ψ∈B′J,X and final state of the vector function v1∈X, there exists a control u∈ℒ2I,X such that (4) has a solution v∈B−v,…,n1,X which satisfies the boundary condition vn1=v1.Remark 1.
It is clear from (5) that Y∈B′I,X satisfied (5) if and only if there exists f∈BI,X satisfying(10)fn≤ϵ,n∈I,Eyn+1=Ayn+Byn−k+fn,n∈Z+,yn=Φn,−k≤n≤0.fn≤ϵ,n∈I,Eyn+1=Ayn+Byn+fn,yn−k+fn,n∈Z+,yn=ϕn,−k≤n≤0.fn≤ϵ,n∈I,Eyn+1=Ayn+Byn−k+Fn,yn−k+fn,n∈Z+,yn=Ψn,−k≤n≤0.


## 3. Existence and Uniqueness of Solutions

Here, we will discuss the existence and uniqueness of the solution of system (1). For this, we need the following assumptions:Λ1: the linear system AGn+1=MGn+NGn−k is well modelled.Λ2: An−1E−nL<1.Theorem 1.
If assumptionsΛ1 and Λ2 hold, then system (1) has a unique solution V∈BI,X.Proof.
DefineT:BI,X⟶BI,X by(11)TVn=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiVi−k.
Now, for anyV,V′∈BI,X, we have(12)TVn−TV′n=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiVi−k−An−1E−nΦ0−BAn−1E−n∑i=k+1nA−iEiΦi−k+−BAn−1E−n∑i=k+1nA−iEiVi−k′..
This implies that(13)TVn−TV′n≤BAn−1E−n∑i=k+1nA−iEiVi−k−Vi−k′≤BAn−1E−nLV−V′.
Thus,T is contraction if AnE−nL<1, so (by BCP) it has a unique fixed point and will be the solution of system (1). Similarly, we can show the existence and uniqueness of solutions of systems (2)–(4). For (3), we also refer to [36].


## 4. Hyers–Ulam Stability over Bounded Discrete Interval

In this part of the paper, we will discuss the Hyers–Ulam stability over bounded discrete interval. Before the result, we will put the following assumptions:Λ1: the linear system EVn+1=AVn+BVn−k is well posed.Λ2: there exists a constant η such that(14)∑r=1n−kϕr≤ηforeachn∈I.Theorem 2.
IfΛ1 and Λ2 and Remark 1 are satisfied, then system (1) is Hyers–Ulam stable over bounded interval.Proof.
The solution of difference system (1) is(15)Vn=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiVi−k.
From Remark1, the solution of(16)EUn+1=AUn+BUn−k+fn,n≥0,k≥0,Un=Φn,−k≤n≤0,is(17)Un=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+An−1E−n∑i=k+1nA−iEiBUi−k+fi−k.
Now, we have(18)Un−Vn=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−kAn−1E−nΦ0−BAn−1E−n∑i=k+1nA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiUi−k=An−1E−n∑i=k+1nA−iEiBYi−k+fi−k−BAn−1E−n∑i=k+1nA−iEiVi−k≤An−1E−n∑i=k+1nA−iE−iBUi−k−BVi−k+An−1E−n∑i=k+1nA−iEifi−k,Un−Vn=An−1E−n∑i=k+1nA−iEifi−k≤An−1E−n∑i=k+1nA−iEiϵϕi−k=ϵAn−1E−n∑r=1n−kA−k−rEk+rϕr=ϵL4∑r=1n−kϕr≤ϵL4η=lϵ,where l=L4η. Hence, system (1) is Hyers–Ulam stable over bounded discrete interval.
Next, we will show that system (2) is Hyers–Ulam stable. Again, we need one more assumption:Λ3: the map F:I×X⟶X satisfies the Carathéodory condition(19)Fn,ϑ−Fn,ϑ′≤Kϑ−ϑ′,for someK≥0 and for all ϑ,ϑ′∈BI,X.Theorem 3.
IfΛ1–Λ3 along with (2.6) and Remark 1 are satisfied, then system (2) is Hyers–Ulam stable over bounded interval.Proof.
The solution of delay difference system (2) is(20)Vn=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,Vi+An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi.
Also, from Remark1, the solution of(21)EUn+1=AUn+BUn−k+fn,Un+fn,n≥0,k≥0,Un=ϕn,−k≤n≤0,is(22)Un=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,Ui+An−1E−n∑i=k+1nA−iEiBUi−k+fi,Ui+fi−k.
Now, we have(23)Un−Vn=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,UiAn−1E−n∑i=k+1nA−iEiBUi−k+fi,Ui+fi−kAn−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi=An−1E−n∑i=k+1nA−iEiBUi−k+fi,Ui+fi−k=An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi=An−1E−n∑i=k+1nA−iEiBUi−k+fi,Ui+An−1E−n∑i=k+1nA−iEifi−k−An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi≤An−1E−n∑i=k+1nA−iEiBUi−k−BVi−k+fi,Ui−fi,Vi+fi−k,Un−Vn≤An−1E−n∑i=k+1nA−iEiBUi−k−BVi−k+LUi−Vi+fi−k=An−1E−n∑i=k+1nA−iEifi−k≤An−1E−n∑i=k+1nA−iEiϵϕi−k=ϵAn−1E−n∑r=1n−kA−k−rEk+rϕr=L4∑r=1n−kϕr≤L4η=Kϵ.
Thus, system (2) is Hyers–Ulam stable.
The Hyers–Ulam stability of system (3) over bounded discrete interval is discussed in [36].


## 5. Hyers–Ulam Stability over an Unbounded Discrete Interval

Here, we discuss the Hyers–Ulam stability of systems (1)–(3) over an unbounded discrete interval; we have some assumptions:A1: the operator family L4≤Ne−νn, n≥0, ν≥0, N≥1.A2: the linear system AGn+1=MGn+NGn−k is well posed.A3: also, assume that(24)∑r=1n−1ϕr≤η,for eachn∈ℤ+, and for η≥0 .Theorem 4.
IfA1–A3 along with (2.6) and Remark 1 are satisfied, then system (1) is Hyers–Ulam stable over an unbounded interval.Proof.
The exact solution of non-autonomous difference system (1) is(25)Vn=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiVi−k.
LetY be the approximate solution of system (1); then, clearly, for a sequence fn, with fn≤ϵ, we have(26)EYn+1=AYn+BYn−k+fn,n≥0,k≥0,Yn=Φn,−k≤n≤0,and(27)Yn=AnE−nΦ0+An−1E−n∑i=0kA−iEiΦi−k+An−1E−n∑i=k+1nA−iEiBYi−k+fi−k.
Now, we have(28)Yn−Un=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+An−1E−n∑i=k+1nA−iEiBYi−k+fi−k−An−1E−nΦ0−BAn−1E−n∑i=k+1nA−iEiΦi−k+−BAn−1E−n∑i=k+1nA−iEiUi−k=An−1E−n∑i=k+1nA−iEiBYi−k+fi−k−BAn−1E−n∑i=k+1nA−iEiUi−k≤An−1E−n∑i=k+1nA−iEiBYi−k−BYi−k+fi−k,≤An−1E−n∑i=k+1nA−iEiBYi−k−BYi−k+An−1E−n∑i=k+1nA−iEifi−k≤An−1E−n∑i=k+1nA−iEifi−k≤An−1E−n∑i=k+1nA−iEiϵϕi−k=ϵAn−1E−n∑r=1n−kA−k−rEk+rϕr=ϵL4∑r=1n−kϕr≤ϵL4η≤Ne−νnηϵ=L,where L=Me−νkη. Thus, system (1) is Hyers–Ulam stable over an unbounded interval.
To prove the Hyers–Ulam stability of system (2), we have to add one more assumption:A4: the continuous function ℍ:ℤ+×X⟶X satisfies the Carathéodory condition(29)ℍn,ω−ℍn,ω′≤Kω−ω′,K≥0,for everyn∈ℤ+ω,ω′∈X.Theorem 5.
IfA1–A4 along with (2.6) and Remark 1 are satisfied, then system (2) is Hyers–Ulam stable over an unbounded interval.Proof.
The solution of delay difference system (2) is(30)Vn=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,Vi+An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi.
Also, from Remark1, the solution of(31)EUn+1=AUn+BUn−k+fn,Un+fn,n≥0,k≥0,Un=ϕn,−k≤n≤0,is(32)Un=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,Ui+An−1E−n∑i=k+1nA−iEiBUi−k+fi,Ui+fi−k.
Now, we have(33)Un−Vn=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,Ui+An−1E−n∑i=k+1nA−iEiBUi−k+fi,Ui+fi−k−AnE−nϕ0−An−1E−n∑i=0kA−iEiBϕi−k+fi,Vi−An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi=An−1E−n∑i=k+1nA−iEiBUi−k+fi,Ui+fi−k−An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi=An−1E−n∑i=k+1nA−iEiBUi−k+fi,Ui+An−1E−n∑i=k+1nA−iEifi−k−An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi≤An−1E−n∑i=k+1nA−iEiBUi−k−BVi−k+fi,Ui−fi,Vi+fi−k,≤An−1E−n∑i=k+1nA−iEiBUi−k−BVi−k+LUi−Vi+fi−k=An−1E−n∑i=k+1nA−iEifi−k≤An−1E−n∑i=k+1nA−iEiϕi−k=An−1E−n∑r=1n−kA−k−rEk+rϕr=L4∑r=1n−kϕr≤Ne−νnη=K,where K=Ne−νnη. Thus, system (2) is Hyers–Ulam stable.Theorem 6.
System (3) is Hyers–Ulam stable over an unbounded interval.
For the proof, see [36].


## 6. Exponential Stability

In this part of the paper, we will present the exponential stability of system (1). First, we recall that a discrete system is said to be exponentially stable if there exist two positive constants M and α such that Vn≤Me−αn for all n∈Z+. Before going to the result, we will consider the following assumptions:(1)
LetσAE−1=λ1,λ2,…,λn be the eigenvalues of AE−1 with(34)λ1≤λ2≤⋯λn≤r,r∈0,1.(2)
AnE−n≤Ne−αn for some positive number α and for all n∈Z+.(3)
N2B−α<0.(4)
There existsL>0, such that fi,Vi≤LVifori≥0andVi∈ℝn.(5)
There existsM>0, such that Fi,Vi≤MVifori≥0andVi∈ℝn.(6)
N2BL2−α<0.(7)
N2MB−α<0.Theorem 7.
Assume that (1)–(3) are satisfied. Then, system (1) is exponentially stable.Proof.
The solution of system (1) is(35)Vn=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiVi−k.
Now,(36)Vn=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiV≤AnE−nΦ0+BAn−1E−n∑i=0kA−iEiVi−k+BAn−1E−n∑i=k+1nA−iEi≤Ne−αnΦ0+BNe−αn∑i=0kNe−αnVi−k,Vn≤Ne−αnΦ0+B∑i=0kNe−αnΦi−k+B∑i=k+1nNe−αnVi−k,eαnVn≤NΦ0+B∑i=0kN2e−αnΦi−k+B∑i=k+1nN2e−αnVi−k=Mϕ,ϕi+B∑i=k+1nαe−αnVi−k,where Mϕ,ϕi=NΦ0+B∑i=0kN2e−αnΦi−k; now, using the Gronwall inequality, we have(37)eαnVn≤Mϕ,ϕieN2Bn.
From this, we have(38)Vn≤Mϕ,ϕieN2B−αn.
Using definition of stability and assumption (3), the result follows.Theorem 8.
Assume that (1), (2), (4), and (6) are satisfied. Then, system (2) is exponentially stable.Proof.
The solution of (2) is in the form of(39)Vn=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,Vi+An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi.
Now,(40)Vn=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,ViAn−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi,Vn≤AnE−nϕ0+An−1E−n∑i=0kA−iE1Bϕi−k+fi,Vi+An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi≤Ne−αnϕ0+Ne−αn∑i=0kNe−αnBϕi−k+Ne−αn∑i=0kNe−αnfi,Vi+Ne−αn∑i=k+1nNe−αnBVi−k+Ne−αn∑i=k+1nNe−αnfi,Vi,Vn≤e−αnNϕ0+∑i=0kN2e−αnBϕi−k+∑i=0kN2e−αnfi,Vi+∑i=k+1nN2e−αn‖Vi−k+∑i=k+1nN2e−αnfi,Vi.
Using (4), we have(41)eαnVn≤Nϕ0+∑i=0kN2e−αnBϕi−k+∑i=0kN2e−αnLVi+∑i=k+1nN2e−αnBVi−k+∑i=k+1nN2e−αnLVi.
Using the Gronwall inequality, we have(42)eαnVn≤Mϕ,ϕ1eN2BL2n,where Mϕ,ϕ1=Nϕ0+∑i=0kN2e−αnBϕi−k>0. From this, we have(43)Vn≤Mϕ,ϕ1eN2BL2−αn.
From (50), the desired result holds.Theorem 9.
Assume that (1), (2), (5), and (7) are satisfied. Then, system (3) is exponentially stable.Proof.
The solution of (3) is in the form of(44)Vn=AnE−nψ0+An−1E−n∑i=0kA−iEiBψi−k+Fi,ψi−k+An−1E−n∑i=k+1nA−iEiBVi−k+Fi,Vi−k.
Now consider(45)Vn=AnE−nψ0+An−1E−n∑i=0kA−iEiBψi−k+Fi,ψi−k+An−1E−n∑i=k+1nA−iEiBVi−k+Fi,Vi−k≤AnE−nψ0+An−1E−n∑i=0kA−iEiBψi−k+Fi,ψi−k+An−1E−n∑i=k+1nA−iEiBVi−k+Fi,Vi−k≤Ne−αnψ0+Ne−αn∑i=0kNe−αnBψi−k+Fi,ψi−k+Ne−αn∑i=k+1nNe−αnBVi−k+Fi,Vi−k=e−αnNψ0+∑i=0kN2e−αnBψi−k+Fi,ψi−k+∑i=k+1nN2e−αnBVi−k+∑i=k+1nN2e−αnFi,Vi−k.
This implies that(46)eαnVn≤Nψ0+∑i=0kN2e−αnBψi−k+Fi,ψi−k+∑i=k+1nN2e−αnBVi−k+∑i=k+1nN2e−αn=Mψ,ψ1+∑i=k+1nN2e−αnBVi−k+∑i=k+1nN2e−αnFi,Vi−k.
Using (5), we have(47)eαnVn≤Mψ,ψ1+∑i=k+1nN2e−αnBVi−k+∑i=k+1nN2e−αnMVi−k,where Mψ,ψ1=Nψ0+∑i=0kN2e−αnBψi−k+Fi,ψi−k.
Using again the Gronwall inequality, we have(48)eαnVn≤Mψ,ψ1eN2MBn.
That is,(49)Vn≤Mψ,ψ1eN2MB−αn.
From (51), the desired result holds.


## 7. Controllability

In this portion, we will discuss the controllability of system (4). First, we will discuss the linear problem and then the non-linear problem.Linear Problem. We assume that y=0; then, (4) reduces to the linear system(50)EVn+1=AVn+BVn−k+CUn,n∈I,k≥0,Vn=ψn,−k≤n≤0.We define a delay Gramian matrix(51)Wc0,n1=∑i=k+1nA−1EiAn−1E−nCCTAn−1E−nTA−1EiT.Theorem 10.
The linear system (6) is relatively controllable, if and only if Wc0,n1 is non-singular.Proof.
Sufficiency: sinceWc0,n1 is non-singular, then its inverse is well defined. So, we select a control function as follows:(52)Un=CTAn−1E−nTA−1EiTWc−10,n1η,where(53)η=v1−AnE−nΨ0−An−1E−n∑i=0kA−iEiBΨi−k+CUi.
Then,(54)vn1=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+CUi+An−1E−n∑i=k+1nA−iEiCUi=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+CUi+An−1E−n∑i=k+1nA−1EiCCTAn−1E−nTA−1EiTWc−10,n1η=v1.
Clearly, from Definition4, we have that (6) is relatively controllable.
Necessity. We will prove by contradiction; assume that Wc0,n1 is singular, i.e., there exists at least one non-zero state ṽ ∈X such that(55)v∘Wc0,n1v∘=0.
So, we obtained(56)0=vWc0,n1v=∑i=0kvTA−1EiAn−1E−nCCTAn−1E−nTA−1EiTv=∑i=0kvTA−1EiAn−1E−nC2,which implies that(57)v∘TA−1EiAn−1E−nC=0,…,0︸n=0T∀n∈I.
Since (6) is relatively controllable, from Definition 4, there exists U1n that drives the initial state to zero at n1, that is,(58)vn1=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+CUi+An−1E−n∑i=k+1nA−iEiCU1n=0.
Similarly, there also exists a controlU2n that drives the initial state to the state \mathringv at n1:(59)vn1=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+CUi+An−1E−n∑i=k+1nA−iEiCU2n=v∘.
From the above, we have(60)v=An−1E−n∑i=k+1nA−iEiCU2n−U2n.
Multiplying both sides of (60) by v and via (8), we have(61)vTv=An−1E−n∑i=k+1nvT∘A−iEiCU2n−U2n=0.
This implies thatv∘=0, which contradicts the fact that v∘ is non-zero. So, the delay Gramian matrix WC0,n1 is non-singular, which completes the proof.
Non-Linear Problem. To discuss the controllability of a non-linear system (4), consider the following conditions:(1)
The operatorW:L2I,X⟶X defined by(62)Wu=∑i=k+1nA−1EiAn−1EnCUnhas inverse operatorW−1, which takes values from L2I,X/kerW and the set M1=W−1Ln1X,L2I,X/kerW. For the next result, we put another assumption.(2)
The mapu:I×X⟶X is continuous and there exist a constant p>1 and Ly⋅∈LpI,X such that(63)un,b−un,b≤Lynb−a,b,a∈X.Theorem 11.
Let us suppose that (1)–(3), (8), and (9) are satisfied. Then, system (4) is relatively controllable if(64)b1+NM1CAn1−1E−n11−e−αn11−e−α<1,whereb=An1−1E−n1N1−e−αqn1/1−e−αq1/qLyLpI,Xand1/q+1/p=1,q,p>1.Proof.
Using (8), for an arbitrary v⋅∈BI,X, we define a control function uvn by(65)uvn=W−1v1−AnE−nΨ0−An−1E−n∑i=0kA−iEiBΨi−k+yi,ViAn−1E−n∑i=k+1nA−iEiBVi−k+yi,Vin,n∈I.
We show that the operatorP:BI,X⟶BI,X defined by(66)Pvn=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+yi,Vi+An−1E−n∑i=k+1nA−iEiBVi−k+yi,Vi+An−1E−n∑i=0nA−iEiCUihas a fixed point, which is the solution of (4), by using the above control function.
We need to check thatPvn1=v1 and Pv0=v0, which means that uv steers system (4) from v0 to v1 in finite n1 and this implies that system (4) is relatively controllable on I.
Foe every real numberr, let Br=v∈BI,X:vc≤r. Set F=supn∈Iyn,0. We will prove this theorem in following three steps.Step 1.
We claim that there exists a positive real numberr such that PBr⊆Br.
Note that(67)∑i=0ne−αiLyi≤∑i=0ne−αiq1/q∑i=0nLypi1/p≤1−e−αnq1−e−αq1/qLyLpI,X,and ∑i=0ne−αiyi,0≤F∑i=0ne−αi=F1−e−αn/1−e−α.
Now using (1), (2), (8), and (60), we have(68)uvn=W−1v1−An1E−n1Ψ0−An1−1E−n1∑i=0kA−iEiBΨi−k+yi,Vi−An1−1E−n1∑i=k+1nA−iEiBVi−k+yi,Vi≤W−1ℒn1X,ℒ2I,X/kerWv1+An1E−n1Ψ0+An1−1E−n1∑i=0kA−iEiBΨi−k+yi,Vi+An1−1E−n1∑i=k+1nA−iEiBVi−k+yi,Vi≤M1v1+Ne−αn1Ψ0+An1−1E−n1∑i=0kNe−αiBΨi−k+yi,Vi+An1−1E−n1∑i=k+1nNe−αiBVi−k+yi,Vi≤M1v1+Ne−αn1Ψ0+An1−1E−n1∑i=0kNe−αiBΨi−k+An1−1E−n1∑i=0nNe−αiyi,Vi+An1−1E−n1∑i=k+1nNe−αiBVi−k=M1v1+Ne−αn1Ψ0+An1−1E−n1∑i=0kNe−αiBΨi−k+An1−1E−n1∑i=0nNe−αiyi,Vi−yi,0+yi,0+An1−1E−n1∑i=k+1nNe−αiBVi−k≤M1v1+Ne−αn1Ψ0+An1−1E−n1∑i=0kNe−αiBΨi−k+An1−1E−n1∑i=0nNe−αiLyiVi+yi,0+An1−1E−n1∑i=k+1nNe−αiBVi−k,=M1v1+Ne−αn1Ψ0+An1−1E−n1∑i=0kNe−αiBΨi−k+An1−1E−n1∑i=0nNe−αiLyiVi+An1−1E−n1∑i=0nNe−αiyi,0+An1−1E−n1∑i=k+1nNe−αiBVi−k≤M1v1+Ne−αn1Ψ0+An1−1E−n1∑i=0kNe−αiBΨi−k+An1−1E−n1N1−e−αn1q1−e−αq1/qLyLpI,Xvc+An1−1E−n1NF1−e−αn11−e−α+An1−1E−n1∑i=k+1nNe−αiBVi−k=M1v1+M1a+M1bvc,where(69)a=v1++Ne−αn1Ψ0+An1−1E−n1∑i=0kNe−αiBΨi−k+An1−1E−n1NF1−e−αn11−e−α+An1−1E−n1∑i=k+1nNe−αiBVi−k,and(70)b=An1−1E−n1N1−e−αn1q1−e−αq1/qLyLpI,X.
From (57) and (60), we have(71)Pvn=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+yi,Vi+An−1E−n∑i=k+1nA−iEiBVi−k+yi,Vi+An−1E−n∑i=0nA−iEiCUi+1+An−1E−nCM1N1−e−αn1−e−αV1≤r,where(72)r=a1+An−1E−nCM1N1−e−αn1−e−α+b1+An−1E−nCM1N1−e−αn1−e−αVc++1+An−1E−nCM1N1−e−αn1−e−αV1.
So, we obtainPBr⊆Br.Step 2.
Now, we define a mapP1 on Br and we will show that it is a contraction mapping.(73)P1vn=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+An−1E−n∑i=0nA−iEiCUvi.
Letρ,ϱ∈Br. Using (57) and (60) for each n∈I, we have(74)uρn−uϱn=W−1v1−A1nE−n1Ψ0−An1−1E−n1∑i=0kA−iEiBΨi−k+yi,ρi−An1−1E−n1∑i=k+1nA−iEiBVi−k+yi,ρi−W−1v1−An1E−n1Ψ0−An1−1E−n1∑i=0kA−iEiBΨi−k+yi,ϱi−An1−1E−n1∑i=k+1nA−iEiBVi−k+yi,ϱi≤W−1An1−1E−n1∑i=0nA−iEiyi,ρi−yi,ϱi≤M1An1−1E−n1∑i=0nNe−αiLyiρ−ϱcuρn−uϱn≤M1An1−1E−n1N1−e−αn1q1−e−αq1/qLyLpI,Xρ−ϱc≤M1bρ−ϱc.
Thus,(75)P1ρn−P1ϱn=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+An−1E−n∑i=0nA−iEiCUρi−AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−k+An−1E−n∑i=0nA−iEiCUϱi=An−1E−n∑i=0nA−iEiCUρi−Uϱi≤An−1E−n∑i=0nA−iEiCUρi−Uϱi≤An−1E−nN∑i=0ne−αiCM1bρ−ϱc≤An−1E−nNCM1bρ−ϱc1−e−αn1−e−α=Mρ−ϱc,where(76)M=An−1E−nNCM1b1−e−αn1−e−α.
From (10), M<1, so P1 is contraction.Step 3.
Here we define a mapP2:Br⟶BI,X and will show that it is a compact and continuous operator.(77)P2vn=An−1E−n∑i=0nA−iEiyi,Vi,for n∈I. Let νn∈Br with νn⟶ν in Br as n⟶∞. Using (57), we have y⋅,νn⟶y⋅,ν in BI,X as n⟶∞, and thus(78)P2νnn−P2νn=An−1E−n∑i=0nA−iEiyi,Vni−An−1E−n∑i=0nA−iEiyi,Vi≤An−1E−nN∑i=0ne−αiyi,Vni−yi,Vi⟶0asn⟶∞,which implies that P2 is continuous on Br.
To show thatP2 is compact on Br, we have to prove that P2Br is equicontinuous and bounded. For any ν∈Br,n1≥n+h≥0, note that(79)P2νn+h−P2νn=An+h−1E−n+h∑i=0n+hA−iEiyi,νi−An−1E−n∑i=0nA−iEiyi,νi,P2νn+h−P2νn=An+h−1E−n+h∑i=0n+hA−iEiyi,νiAn−1E−n∑i=0nA−iEiyi,νi,P2νn+h−P2νn≤An+h−1E−n+h∑i=0n+hA−iEiyi,νi−An−1E−n∑i=0nA−iEiyi,νi≤S1−S2.
Let(80)S1=An+h−1E−n+h∑i=0n+hA−iEiyi,νi,and(81)S2=An−1E−n∑i=0nA−iEiyi,νi.
Now, we have to checkS1⟶S2 as h⟶0. Now,(82)S1=An+h−1E−n+h∑i=0n+hA−iEiyi,Vi,≤An+h−1E−n+hN∑i=0n+he−αiLyiνi+yi,0≤An+h−1E−n+hN1−e−αn+hq1−e−αq1/qLyLpI,X+An+h−1E−n+hNF1−e−αn+h1−e−α⟶S2ash⟶0,which implies that(83)P2νn+h−P2νn⟶0ash⟶0.
Therefore,P2Br is equicontinuous.
Next, we show thatP2Br is bounded, and we have(84)P2ν=An−1E−n∑i=0nA−iEiyi,Vi≤An−1E−nN∑i=0ne−αiLyi+yi,0≤An−1E−nN1−e−αnq1−e−αq1/qLyLpI,XAn−1E−nNF1−e−αn1−e−α.
Hence,P2Br is bounded. From the Arzelà–Ascoli theorem, P2Br is compact in BI,X. Thus, P2Br is a compact and continuous operator.
Now, Krasnoselskii’s fixed point theorem guarantees thatP has a fixed point ν in Br. Clearly, ν is a solution of (4) satisfying νn1=ν1, and the boundary condition νn=Ψn,−k≤n≤0 holds from the solution of system (4), which completes the proof.


## 8. Numerical Examples

In this section, we give some examples on Hyers–Ulam stability and controllability for the theoretical results.Example 1.
Consider the following non-singular delay difference equation:(85)EVn+1=AVn+BVn−3,V0=1,n∈0,1,2,3,Vn=Φn,−3≤n≤0,with inequality(86)EVn+1−AVn−BVn−3≤0.8,V0=1,n∈0,1,2,3,Vn−Φn≤1,−3≤n≤0,where k=3.
If we fixed
A=−5−2−4−3, B=4123, E=2002, and ϕn=cosn+π/2cosn+π/2t, obviously, ϕn=00t, when n=0. Then, we get that
AB=−24−11−22−13=BA, AE=−10−4−8−6=EA, BE=8246=EB, E−1=0.5000.5, AE−1=−2.5−1−2−1.5=E−1A, and BE−1=20.511.5=E−1B.
Moreover, ifV satisfied (86), then there exists fn such that fn≤0.8, and(87)EVn+1=AVn+BVn−3+fn,V0=1,n∈0,1,2,3Vn=Φn,−3≤n≤0.
Also, the solution of (85) is(88)Vn=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiVi−k.
Letϵ=0.8, and f:Z+⟶X be as given below.
fn=0.6cosn+π/20.6sinn+π/2t.
Then, clearly(89)fn=0.6cosn+π22+0.6sinn+π22=0.62cos2n+π2+0.62sin2n+π21/2=0.62=0.6≤=0.8.
Now, the perturbed delay difference systems (11)–(13) have the solution(90)Hn=AnE−nΦ0+BAn−1E−n∑i=0kA−iEiΦi−k+BAn−1E−n∑i=k+1nA−iEiVi−k+fi−k.
The plots of exact and perturbed solutions obtained using Mathematica are shown in Figure1.Figure 1
The plots of exact and perturbed solutions.Example 2.
Consider the following non-singular delay difference equation:(91)EVn+1=AVn+BVn−3+fn,V,V0=1,n∈0,1,2,3,Vn=ϕn,−3≤n≤0,with inequality(92)EVn+1−AVn−BVn−3−fn,V≤0.8,V0=1,n∈0,1,2,3,Vn−ϕn≤1,−3≤n≤0,where k=3.
If again we fixedA=4−323, B=2−321, E=3003, and ϕn=cosn+π/2cosn+π/2t, obviously, ϕn=00t, when n=0. Then, we get AB=2−1510−3=BA, AE=12−969=EA, BE=6−963=EB, E−1=0.333000.333, AE−1=1.332−0.9990.6660.999=E−1A, and BE−1=0.666−0.9990.6660.333=E−1B.
Moreover, ifV satisfied (15), then there exists fn such that fn≤0.8, and(93)EVn+1=AVn+BVn−3+fn,V+fn,V0=1,n∈0,1,2,3,Vn=ϕn,−3≤n≤0.
Also, the solution of (91) is(94)Vn=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,Vi+An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi.
Letϵ=0.8, and f:Z+⟶X be as given below.
fn=0.7cosn+π/20.7sinn+π/2t.
Then, clearly(95)fn=0.7cosn+π22+0.7sinn+π22=0.72cos2n+π2+0.72sin2n+π21/2=0.72=0.7≤=0.8.
Now, the perturbed delay difference systems (91)–(93) have the solution(96)Hn=AnE−nϕ0+An−1E−n∑i=0kA−iEiBϕi−k+fi,Vi+An−1E−n∑i=k+1nA−iEiBVi−k+fi,Vi+fi−k.
The plots of exact and perturbed solutions obtained using Mathematica are shown in Figure2.Figure 2
The plots of exact and perturbed solutions.Example 3.
Setn1=3. Consider the following delay difference controlled system:(97)EVn+1=AVn+BVn−3+yn,Vn+CUn,n∈I=0,1,2,3,Vn=Ψn,−3≤n≤0,which has the solution(98)Vn=AnE−nΨ0+An−1E−n∑i=0kA−iEiBΨi−3+yi,Vi+CUi+An−1E−n∑i=k+1nA−iEiBVi−3+yi,Vi+CUi,where k=3.
LetA=7364, B=1−1−22, E=4004, C=1001, and yn,Vn=0.2nV20.1nV1.
Note thatAB=1−1−22=BA, AE=28122416=EA, BE=4−4−88=EB, E−1=0.25000.25, AE−1=1.750.751.51=E−1A, and BE−1=0.25−0.25−0.50.5=E−1B.
NowC=1. Also, AnE−n≤Ne−αn with N=1,α=2, and n∈0,1,2,3.
Now consider(99)Wc0,n1=∑i=0n1A−1EiAn1−1E−n1CCTAn1−1E−n1TA−1EiT=∑i=03A−1EiA2E−2CCTA2E−2T=A−1EiT=9.31767−14.83364.38789−6.31446.
Then,Wc0,n1−1=−1.009972.37257−0.7018241.49032 and M1=Wc0,n1−1=0.399932. Further for any ν,μ∈X, we have(100)yn,ν−yn,μ=max0.2nν1−μ1,0.1nν2−μ2≤o.2nmaxν1−μ1,ν2−μ2=0.2nν−μ.
Now we setLyn=0.2n∈L2I,X with p=q=2, so LyL2I,X=∑i=0n1Lyi1/2=∑i=030.2i1/2=1.09545.(101)b=An1−1E−n1N1−e−αn1q1−e−αq1/qLyL2I,X=A2E−21−e−121−e−41/21.09545=0.3546,and(102)bAn1−1E−n1CM1N1−e−αn11−e−α=0.3546A2E−2M11−e−61−e−2=0.8573≤1.Example 4.
Consider the following non-singular delay difference equation:(103)EVn+1=AVn+BVn−0.3,n∈0,1,2,3,…,Vn=0.3,0.2T,−0.3≤n≤0,where k=0.3 and we set A=−5−2−4−3, B=4123, E=2002, and ϕn=cosn+π/2cosn+π/2t; obviously, ϕn=00t, when n=0. Then, we get that AB=−24−11−22−13=BA, AE=−10−4−8−6=EA, BE=8246=EB, E−1=0.5000.5, AE−1=−2.5−1−2−1.5=E−1A, and BE−1=20.511.5=E−1B.
Now,
B=0.510.60.340.5000.5=0.47<Ne−0.3α=0.656101,choosingα=1.4048,N=1.
Also,ϕ=0.3; now,(104)Mϕ,ϕi=NΦ0+B∑i=0kN2e−αnΦi−k=0.4324,N2B−α<−0.3725<0,Vn≤Mϕ,ϕieN2B−αn⟶0,asn⟶∞.
Hence, system (1) is exponentially stable.


## 9. Conclusion

In recent years, the qualitative behavior of delay difference equations has a significant contribution in real life. Especially, the discussion regarding the Hyers–Ulam stability, exponential stability, and controllability of delay difference equations has been considered as one of the important topics of the literature, in which different types of conditions have been used in the form of inequalities and mostly results have been obtained through discrete Gronwall inequality. In this paper, we have investigated the existence and uniqueness of the solution through Banach contraction principle, Hyers–Ulam stability over bounded and unbounded discrete interval, exponential stability, and controllability of the delay difference system with the help of Gronwall inequality and Carathéodory condition.



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