Hyers–Ulam Stability, Exponential Stability, and Relative Controllability of Non-Singular Delay Difference Equations
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.
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 , 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  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 . Ulam in a seminar, in his presentation he pointed out some problems associated with the stability of group homomorphism. After a year in , 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 . He also used this idea in the Cauchy difference system. Obloza  used this idea in differential equations, and later Jung  and Khan et al.  used it in the difference equations. This stability was also discussed in fractional differential equation by Gao et al. , and some results on Ulam-type stability of a first-order non-linear delay dynamic system were discussed by Shah et al. in . 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 . Also in , 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  on controllability assumes that controls are functions on time having values on some non-empty subset of . 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 .
The delay difference system can be used in the characterization of automatic engine, control theory, and physiology system. Khusainov et al  solved the linear autonomous delay-time system with commutable matrices. Diblik and Khusainov  gave the description about the solutions of discrete delayed system using the idea . Then, Wang et al.  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. . Kuruklis  and Yu  studied the asymptotic behavior of the variable type delay difference equation. Kosmala and Teixeira  provided a good insight and discussed the behavior of solution of the difference equation of the type . Liu et al  designed the exponential behavior of switch discrete-time delay system. Marwen and Sakly  discussed the stability techniques about the switched non-linear time-delay difference equations. Yuanyuan  described the stability techniques of high-order difference systems. The stability of higher-order rational difference systems was studied by Khaliq .
Our present study is focused on the Hyers–Ulam stability and exponential stability of non-singular delay difference system of the formandwhere the commutable constant matrices are and is non-singular. , the space of bounded sequences, and , the space of convergent sequences, where , , and . Also, our focus is on relative controllability of the systemwhere , and the control function takes values from . The continuous form of this work is given in . The Hyers–Ulam stability of (3) was recently presented in .
Here, we discuss some notations and definitions, which will be needed for our main work. By and , we will denote the -dimensional Euclidean space with vector norm and matrices with real-valued entries. The vector infinite-norm is defined as and the matrix infinite-norm is given as where and ; also, and are the elements of the vector and the matrix . will be the space of all bounded sequences from to with norm . We will use and for the set of real, integer, and non-native integer numbers, respectively. Also, we define .
Definition 1. The solution of system (1) is said to be exponentially stable if there exist positive real numbers and , such that
Definition 4. System (4) is said to be relatively controllable, if for initial vector function and final state of the vector function , there exists a control such that (4) has a solution which satisfies the boundary condition .
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: : the linear system is well modelled. : .
Theorem 1. If assumptions and hold, then system (1) has a unique solution .
Proof. Define byNow, for any , we haveThis implies thatThus, is contraction if , 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 .
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: : the linear system is well posed. : there exists a constant such that
Proof. The solution of difference system (1) isFrom Remark 1, the solution ofisNow, we havewhere . 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: : the map satisfies the Carathéodory condition for some and for all .
Proof. The solution of delay difference system (2) isAlso, from Remark 1, the solution ofisNow, we haveThus, system (2) is Hyers–Ulam stable.
The Hyers–Ulam stability of system (3) over bounded discrete interval is discussed in .
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: : the operator family , , , . : the linear system is well posed. : also, assume that for each , and for .
Proof. The exact solution of non-autonomous difference system (1) isLet be the approximate solution of system (1); then, clearly, for a sequence , with , we haveandNow, we havewhere . 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: : the continuous function satisfies the Carathéodory condition for every .
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 and such that for all . Before going to the result, we will consider the following assumptions:(1)Let be the eigenvalues of with(2) for some positive number and for all .(3).(4)There exists , such that .(5)There exists , such that (6).(7).
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 ; then, (4) reduces to the linear system
We define a delay Gramian matrix
Theorem 10. The linear system (6) is relatively controllable, if and only if is non-singular.
Proof. Sufficiency: since is non-singular, then its inverse is well defined. So, we select a control function as follows: