Controllability of Impulsive Non–Linear Delay Dynamic Systems on Time Scale

In this paper, we obtain the results of controllability for first order impulsive non–linear time varying delay dynamic systems and Hammerstein impulsive system on time scale, using the theory of fixed points such as Banach fixed point theorem combined with Lipchitz conditions and non linear functional analysis. We also provide examples to support our theoretical results.


I. INTRODUCTION
The theory of differential equations with impulses has been well utilized in mathematical modeling. In real life problems, there are numerous procedures and phenomena that are characterized by the fact that at certain occasions they experienced sudden changes in their states. These procedures are exposed to short-term perturbations and is known as impulsive effects in the system. Impulsive differential equations can be used to represent various real world problems containing variation in its state. Recently, the theory of differential equations with impulses has influenced many author's attention. For more details we refer the work of Samoilenko and Perestyuk [1], Lakshmikantham et al. [2] and Rogovchenko [3].
On the other hand, in control theory, controllability is a mathematical problem which consists of determining a specific control parameter to steers the solutions of the problem throughout the process. Due to many applications of control in practical problems, controllability received an increasing interest [4]- [8]. In the last few decades, impulsive control has attracted the interest of many researchers [9]. Such control arises naturally in a wide variety of phenomena, such as orbital transfer of satellite [10], [11], ecosystems The associate editor coordinating the review of this manuscript and approving it for publication was Guangdeng Zong . management [12], synchronization in chaotic secure communication systems [13] and control of money supply in a financial market [14]. For more results on the controllability of impulsive systems one can see the work of Shubov et al. [15] and Park et al. [16]. Impulsive systems with time-delay describe the models of practical processes where both dependence on the past and instantaneous disturbances are observed. The interaction between the impulsive effects and the time-delay makes it rather difficult to analyze controllability of such systems [17], [18].
The idea of time scale was firstly introduced by Hilger [19]. He introduced this theory in 1988, as a unification of discrete and continuous calculus. Due to its importance, this theory has been well utilized in differential and difference equations. For details of mentioned topic, see [20]- [24] and the books [25], [26]. More recently, many researchers discussed the existence, uniqueness, stability and controllability of abstract equations on time scale. In [27], Lupulescu and Zada discussed the basic concepts of impulsive linear systems and studied the solutions of linear impulsive dynamic systems on time scale. For the controllability results of dynamical systems on time scale we recommend [28]- [30], [35].
Recently, Shah et al. [31] studied the existence of solutions and Hyers-Ulam stability of first order delay dynamic systems on time scale. The concept of Hyers-Ulam stability [32]- [34], is related to the difference between the exact and approximate solutions of the considered problem, and thus this concept can be used in approximation theory and numerical analysis.
In this paper, we investigate the models presented in [31] for controllability. In fact, we study the controllability results for delay dynamic systems of the form: and of impulsive non-linear delay dynamic system of the form: where T is a time scale with 0, b, θ k , ∈ T. A(θ) and M (θ ) is a family of linear bounded operators which is continuous and piecewise continuous on I , respectively. H : be the right and left side limits of χ (θ ) at θ k , where θ k are not isolated points and satisfies Moreover, ν : T → T ∪ [−λ, 0] I will be a continuous and delay function such that ν(θ ) ≤ θ . B and N are bounded linear operators from a Banach space U to R n and u(·) is control function given in L 2 (I , U ). This paper is organized as follows. In first and second sections, we provide introduction, basic notations and definitions which are required for the main results. In third and fourth sections, we give the main results of controllability for systems (1) and (2), respectively. In last section, an example is provided to utilize the applicability of obtained results.

II. PRELIMINARIES
Here, we recall fundamental definitions, lemmas and some notations, which will be utilized throughout the paper.
Let (Y , · ) be a Banach space and B(Y ) be the space of all linear and bounded operators on Y . Furthermore, PC(I , Y ) is a Banach space of all piecewise continuous functions from I to Y , induced with the norm, y PC = sup t∈I y(t) .
The time scale T is a non-empty subset of real numbers. The backward and forward jump operators ρ : T → T and σ : T → T are respectively defined as: ρ(s) = sup{t ∈ T : t < s} and σ (s) = inf{t ∈ T : t > s}. Also, t is called right or left dense if t < sup{T} and σ (t) = t or t > inf{T} and ρ(t) = t, respectively. The point t is called the dense point if at the same time it is right as well as left dense. A function is said to be regulated if its right hand limit exists at all right dense points and left hand limit exists at all left dense points in T. A function f : T → Y is said to be rd-continuous, if it is regulated and continuous at all rightdense points. The derived form of T is denoted by T z , and is defined as: T z = T\{max T} if max T exists, otherwise T z = T. Remark 1: Throughout this paper, we consider that T Z, where T is a time scale and Z is the set of integers. Also, the impulses χ(θ k ) on the isolated points are assumed to be zero.
The −derivative and −integral of f : T → R are respectively defined as where F = f on T z . Definition 1: A function χ ∈ PC(I , Y ) is called a mild solution of (1) if it satisfies χ (0) = χ 0 and the following equation Definition 2: A function χ ∈ PC(I , Y ) is called a mild solution of (2) if it satisfies χ (0) = χ 0 and the following equation

A. CONTROLLABILITY OF EQUATION (1)
To analyze the controllability results for equation (1). The following presumptions will be needed: (A 1 ) The function H is continuous and satisfying the following Proof: Define the operator ϒ : Using the Picard operator on C(T ∪ [−λ, 0] T , R n ) and the presumption (A 3 ), ϒ is strictly contractive. So, the operator ϒ has a unique fixed point (FP). Hence (1) has a unique solution steers the state function χ(θ ) from initial state to final state at θ = b. Also, the estimate for the control function u(θ ) is Proof: By substituting θ = b in the mild solution (3) of the system (1), we get Hence, the control function (5) steers the state function For θ ∈ I and χ ∈ , we have Also, for µ 1 , µ 2 ∈ and θ ∈ I , Thus, T is a contraction operator. So, by a Banach FP theorem, T has a unique FP on I . Therefore, system (1) has a solution on I and hence we conclude that the system (1) is controllable on I . (2) Before proving the controllability result for equation (2), let us assume the following presumptions:

B. CONTROLLABILITY OF EQUATION
Proof: Define an operator ϒ : We see that for any z 1 , steers the state function χ(t) from initial state to final state at θ = b. Also, the estimate for the control function u(θ ) is M u where Proof: By substituting θ = b in the mild solution (4) of the system (2), we get ) N (s)u(s)+H(s, χ(s), χ(ν(s))) s VOLUME 8, 2020 Hence, the control function (6) steers the state function χ(θ ) from initial state χ 0 to final state χ b at time θ = b. Also, × H(s, χ(s), χ(ν(s))) s Proof: Define an operator : → by For θ ∈ I and χ ∈ , we have × H(s, µ 1 (s), µ 1 (ν(s))) − H(s, µ 2 (s), µ 2 (ν(s))) s Thus, is a contraction operator. So, by a Banach FP theorem, has a unique FP on I . Therefore, system (2) has a solution on I and hence we conclude that the system (2) is controllable on I .

C. CONTROLLABILITY OF HAMMERSTEIN IMPULSIVE SYSTEM
In this section, we establish the controllability results for integro-differential equation with impulses which is known as Hammerstein impulsive system.

IV. CONCLUSION
In this paper, we established the controllability of systems (1) and (2). Also, we studied the controllability of Hammerstein impulsive system (7) on time scale. We used the theory of FP combined with non linear functional analysis to establish the main results. The considered problems are new for controllability on time scale. We believe that the obtained results will be valuable and give remarkable contribution to the existing literature on the topic.

CONFLICTS OF INTEREST
The author declares that there is no competing interest.

ACKNOWLEDGMENT
The author acknowledges the support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Group. His research interests include fixed-point theory, variational analysis, random operator theory, optimization theory, and approximation theory. He has provided and developed many mathematical tools in his fields productively over the past years. He has over 600 scientific articles and projects either presented or published. Moreover, he delivers many invited talks on different international conferences every year all around the world.
WIYADA KUMAM received the Ph.D. degree in applied mathematics from the King Mongkut's University of Technology Thonburi (KMUTT). She is currently an Assistant Professor with the Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT). Her research interests include fuzzy optimization, fuzzy regression, least-squares method, minimization problems, optimization problems, and fuzzy nonlinear mappings.
GOHAR ALI received the Ph.D. degree in mathematics from the Abdus Salam School of Mathematical Sciences (ASSMS), GC University, Lahore, Pakistan. He was a Postdoctoral Fellow with the University of Liverpool, U.K., from 2012 to 2013. He has published research articles in reputed international journals of mathematics. His research interests include graph theory, combinatorial mathematics, image processing via graph cut, analysis, mathematical inequalities, and fluid dynamics.
JEHAD ALZABUT received the Ph.D. degree from Middle East Technical University, Turkey. His areas of interest are qualitative properties of delay, difference, fractional, and impulsive differential equations. He has particular interest in mathematical models describing biological and medical phenomena. VOLUME 8, 2020