Exact Controllability of an Impulsive Semilinear System with Deviated Argument in a Banach Space

Abstract

A functional differential equation with deviated argument coupled with impulsive conditions is studied for the existence and uniqueness of the mild solution and exact controllability of the system. The results are obtained by using Banach contraction principle and 𝐶₀ semigroup theory without imposing additional assumptions such as analyticity and compactness conditions on the generated semigroup and the nonlinear term. An example is provided to illustrate the presented theory.

1. Introduction

It is possible to steer various dynamical systems from an arbitrary initial state to final state using the set of admissible controls. Such systems are called controllable. The theory for abstract linear control systems in finite-dimensions is well established. Many authors have extended the theory to infinite dimensional systems represented by nonlinear evolution equations. References of which are available in various articles such as [1–4].

As the change of state occurs abruptly in several physical phenomena, many authors have discussed impulsive differential equations in their articles. For instance, Chang [5] and Li et al. [6] discussed the controllability of impulsive functional differential system in Banach spaces using Schaefer’s fixed point theorem. Jeong et al. [7] investigated the controllability of semilinear retarded control systems in Hilbert spaces. Tai and Wang [8] proposed sufficient conditions for the controllability of fractional impulsive neutral functional integrodifferential systems in a Banach space. Sakthivel and Anandhi [9] discussed approximate controllability of impulsive differential equations with state dependent delay.

Controllability results are available in overwhelming majority for abstract impulsive differential systems rather than for impulsive functional differential equation with deviated arguments.

As demand for efficient dynamic performances increase, mathematical models are required to behave more like real processes. Models are designed to deal with aftereffect, hereditary systems, equations with deviated arguments, and so forth. Delay differential systems are still resistant to quite a number of “classical controllers.” Many partial differential systems can be reduced to functional differential equations with deviated arguments. Thus controllability of functional differential equation with deviated argument has to be extensively studied. However, Gal [10] studied the existence and uniqueness of local and global solutions for initial value problem with deviated argument 𝑢^′(𝑡)=𝐴𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡),𝑢[𝛼(𝑢(𝑡),𝑡)]),𝑡∈𝑅₊,𝑢(0)=𝑢₀. (1) Pandey et al. [11] used analytic semigroup theory and fixed point arguments to study the existence and uniqueness of mild solutions of a neutral differential equation with a deviated argument in a Banach space.

Chang and Liu [12] discussed the existence of mild and strong solutions for some neutral partial differential equations with nonlocal conditions using Sadovskii fixed point theorem and compact analytic semigroup of uniformly bounded linear operators.

Fractional operators, analyticity, and compactness are mostly used to establish these results which impose more restriction on the semigroup and the nonlinear part of the semilinear system. Interestingly controllability results for functional differential equation with deviated argument coupled with impulsive conditions are not widely available so far.

In this work the exact controllability of a semilinear system governed by an impulsive functional differential equation is studied. The semilinear system is illustrated in Section 2 as an abstract differential equation with deviated argument and impulsive conditions. Impulsive conditions reflect abrupt changes in real physical phenomenon that are more interesting than initial conditions. In Section 3 simple Lipschitz conditions and 𝐶₀ semigroup theory is required to study the differential equation with deviated argument removing the more strict assumptions like analyticity and compactness of the semigroup or the nonlinear part.

2. Preliminaries

In this section some basic definitions and lemmas are stated. Naito [13] and Pazy [14] illustrated the basic concepts of controllability and semigroup theory, respectively.

Let 𝐽=[0,𝑇], 0<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑇 and 𝐽^′=[0,𝑇]⧵{𝑡₁,𝑡₂,…,𝑡_𝑚}⊂𝐽. Let 𝑋 and 𝑈 be Banach spaces. The complete controllability of the following impulsive differential equation is studied. Consider 𝑑𝑥(𝑡)(𝑡)((𝑡)(𝑎(𝑥(𝑡)(𝑡),𝑑𝑡=𝐴𝑥+𝑓𝑡,𝑥,𝑥,𝑡)))+𝐵𝑢𝑡∈𝐽^′,𝑥(0)=𝑥₀,Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡_𝑘,𝑘=1,2,…,𝑚,0<𝑡₁<⋯<𝑡_𝑚<𝑇. (2) The state function 𝑥(⋅) takes values in Banach space 𝑋. The control function 𝑢(⋅)∈𝐿²(𝐽,𝑈), where 𝑈 is a Banach space. 𝐵∶𝑈→𝑋 is a bounded linear operator. 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a the infinitesimal generator of a strongly continuous semigroup 𝑆(𝑡), 𝑡>0 on 𝑋. 𝑓 and 𝑎 are suitably defined functions satisfying certain conditions to be specified in Section 3. 𝐼_𝑘∈𝐶(𝑋,𝑋), 𝑘=1,2,…,𝑚 and Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘).

We define 𝑃𝐶(𝐽,𝑋)={𝑥∶[0,𝑇]→𝑋∣𝑥∈𝐶(𝐽^′,𝑋), 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) exists, and 𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘), 1≤𝑘≤𝑚}.

𝑃𝐶(𝐽,𝑋) is a Banach space with norm ‖𝑥‖=sup_𝑡∈𝐽‖𝑥(𝑡)‖.

Let =𝐶_𝐿(𝐽,𝑋)={𝑢∈𝑃𝐶(𝐽,𝑋)∶‖𝑢(𝑡)−𝑢(𝑠)‖≤𝐿|𝑡−𝑠|,∀𝑡,𝑠∈𝐽^′}.

Definition 1 (see [13]). System (2) is said to be controllable on the interval 𝐽 if for every 𝑥₀, 𝑥_𝑇∈𝑋, there exists a control 𝑢∈𝐿²(𝐽,𝑈) such that the mild solution of (2) satisfies 𝑥(0)=𝑥₀, 𝑥(𝑇)=𝑥_𝑇.

Definition 2. A mild solution of problem (2) is defined as function 𝑥∶𝐽→𝐷 such that it satisfies the following:(i)𝑥(0)=𝑥₀,(ii)Δ𝑥(𝑡_𝑘)=𝐼_𝑘(𝑥(𝑡_𝑘)), 𝑘=1,2,…,𝑚,(iii)the restriction of 𝑥(⋅) to the interval 𝐽^′ is continuous and the following integral equation is satisfied: 𝑥(𝑡)=𝑆(𝑡)𝑥₀+^𝑡₀𝑆(𝑓((𝑠)(𝑎(𝑥(𝑠)+(𝑠𝑡−𝑠)𝑠,𝑥,𝑥,𝑠)))𝐵𝑢)()]𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘, (3)
where 𝑆(𝑡), for all 𝑡∈𝐽, denotes the 𝐶₀ semigroup generated by 𝐴.

Throughout this paper, the following assumptions are made. Let 𝑍_𝑘={𝑥(𝑡)∈𝐷∶‖𝑥(𝑡)‖≤𝑘}. (4)(H1)𝐴 generates a strongly continuous semigroup 𝑆(𝑡) in the Banach space 𝑋 and there exists a constant 𝑀 such that ‖𝑆(𝑡)‖≤𝑀.(H2)The nonlinear operator 𝑓∶𝐽×𝑋×𝑋→𝑋 and satisfy the Lipschitz condition such that ∃ constants 𝑀₁,𝑀₂>0 such that for all 𝑥_𝑖∈𝑍_𝑘, 𝑦_𝑖∈𝑍_𝑘, 𝑖=1,2, we have 𝑓𝑡,𝑥₁,𝑦₁−𝑓𝑡,𝑥₂,𝑦₂≤𝑀₁𝑥₁−𝑥₂+𝑦₁−𝑦₂ (5) and max_𝑡∈𝐽‖𝑓(𝑡,0,𝑥(0))‖=𝑀₂.(H3)Moreover assume 𝑎∶𝐷×ℝ₊→ℝ₊ is globally continuous on 𝑋×ℝ₊ and satisfies 𝑎(⋅,0)=0 and for each 𝛿>0, there is a 𝐿_𝑎=𝐿_𝑎(𝛿)>0 such that |𝑎(𝑢,𝑡)−𝑎(𝑣,𝑡)|≤𝐿_𝑎‖𝑢−𝑣‖, ∀𝑢, 𝑣∈𝑋, whenever 0≤𝑡≤𝛿.(H4)The linear operator 𝑊∶𝐿₂[𝐽,𝑈]→𝑋, defined by 𝑊𝑢=^𝑇₀𝑆(𝑇−𝑠)𝐵𝑢(𝑠)𝑑𝑠, (6) has an invertible operator 𝑊⁻¹ (induced by 𝑊) defined on 𝐿₂[𝐽,𝑈]/Ker𝑊 and there exist positive constants 𝐾₁>0, 𝐾₂>0 such that ‖𝑊⁻¹‖≤𝐾₁ and ‖𝐵‖≤𝐾₂.(H5)There exist constants 𝑑, 𝐿_𝑘, 𝐼, 𝑑_𝑘>0, 𝑘=1,…,𝑚, such that ‖𝐼_𝑘(⋅)‖<𝑑_𝑘, ∑^𝑚_𝑘=1𝑑_𝑘=𝑑, ‖𝐼_𝑘(𝑥)−𝐼_𝑘(𝑦)‖≤𝐿_𝑘‖𝑥−𝑦‖∀𝑥, 𝑦∈𝑋, and ∑^𝑚_𝑘=1𝐿_𝑘=𝐼.

3. Main Result

In this section we discuss the existence and uniqueness of mild solution and exact controllability of the semilinear differential equation with deviated argument and impulsive conditions described in Section 2.

Now, 𝑀𝑥₀+𝑇𝐾₁𝐾₂𝑥_𝑇+𝑀𝑀₂𝐾₁𝐾₂𝑇²+𝑇𝑀𝐾₁𝐾₂𝑥₀+𝑀𝑀₁𝐾₁𝐾₂𝑇²𝑘+𝐿²𝐿_𝑎𝑇+2𝑀₁𝑇𝑘+𝐿²𝐿_𝑎𝑇+𝑀₂𝑇+1+𝑀𝐾₁𝐾₂𝑇𝑑≤𝑘, (7) where 𝑘>0 is a constant. Let 𝑝=2+𝐿𝐿_𝑎𝑀𝐾₁𝐾₂𝑇+1𝑀𝑀₁𝑇+𝑀𝐼+𝑀²𝐾₁𝐾₂𝑇𝐼 (8) be such that 0≤𝑝<1.

Theorem 3. If hypotheses (H1)–(H5) and (7) and (8) are satisfied then system (2) is controllable on 𝐽.

Proof. Let 𝑥∈𝑍_𝑘. Using hypothesis (H4) for an arbitrary function 𝑥(⋅),we define the control 𝑢(𝑡)=𝑊⁻¹𝑥_𝑇−𝑆(𝑇)𝑥₀−^𝑇₀𝑆()𝑓((𝑠)(𝑎(𝑥(𝑠)−𝑇−𝑠𝑠,𝑥,𝑥,𝑠)))𝑑𝑠_𝑚𝑘=1𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘(𝑡). (9) Using this control and Banach contraction principle, the operator defined by (𝑡)(𝑡)𝑥Φ𝑥)(=𝑆₀+^𝑡₀𝑆(𝑡−𝑠)(𝐵𝑢)(𝑠)+𝑓(𝑠,𝑥(𝑠),𝑥(𝑎(𝑥(𝑠),𝑠)))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘 (10) is proved to have a fixed point; that is, (Φ𝑥)(𝑇)=𝑥_𝑇. This fixed point is then a solution of (2). This implies that ∃ the control 𝑢 which steers the semilinear system from the initial state 𝑥₀ to 𝑥_𝑇 in time 𝑇 provided we can obtain a fixed point of the nonlinear operator Φ. First we prove Φ maps 𝑍_𝑘 into itself. For this, using conditions (H1)–(H5), (4), (9), (7), and triangle inequality, we have Φ(𝑥𝑡≤𝑆(𝑡)𝑥‖())()‖₀+∫^𝑡₀𝑆(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠+∫^𝑡₀𝑆()𝑓((𝑠)(𝑎(𝑥(𝑠)+∑𝑡−𝑠𝑠,𝑥,𝑥,𝑠)))𝑑𝑠_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘=𝑆(𝑡)𝑥₀+∫^𝑡₀𝑆(𝑡−𝑠)𝐵𝑊⁻¹×𝑥_𝑇−𝑆(𝑇)𝑥₀−∫^𝑇₀𝑆𝑓𝜂𝑎𝑥𝜂+𝑇−𝜂𝜂,𝑥,𝑥,𝜂𝑑𝜂_𝑚𝑘∑₌₁𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘(𝑠)𝑑𝑠+∫^𝑡₀𝑆()𝑓((𝑠)(𝑎(𝑥(𝑠)+∑𝑡−𝑠𝑠,𝑥,𝑥,𝑠)))𝑑𝑠_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘<𝑆(𝑡)𝑥₀+∫^𝑡₀‖𝑆(𝑡−𝑠)‖‖𝐵‖𝑊⁻¹×𝑥_𝑇−𝑆(𝑇)𝑥₀−∫^𝑇₀𝑆𝑓𝜂𝑎𝑥𝜂+𝑇−𝜂𝜂,𝑥,𝑥,𝜂𝑑𝜂_𝑚𝑘∑₌₁𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘𝑠)+∫(𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),𝑥(𝑎(𝑥(𝑠),𝑠)))𝑑𝑠+𝑀∑_{𝑘=1,2,…,𝑚}𝑑_𝑘≤𝑀𝑥₀+𝑀𝐾₁𝐾₂×∫^𝑡₀𝑥_𝑇+𝑀𝑥₀+𝑀∑_{𝑘=1,2,…𝑚}𝑑_𝑘+𝑀∫^𝑇₀𝑓𝜂𝑎𝑥𝜂(0)+𝑓(0)𝜂,𝑥,𝑥,𝜂−𝑓𝜂,0,𝑥𝜂,0,𝑥𝑑𝜂𝑑𝑠+𝑀∫^𝑡₀𝑓((𝑠)(𝑎(𝑥(𝑠)((0+𝑓((0𝑥𝑠,𝑥,𝑥,𝑠)))−𝑓𝑠,0,𝑥))𝑠,0,𝑥))𝑑𝑠+𝑀𝑑≤𝑀₀+𝑀𝐾₁𝐾₂×∫^𝑡₀𝑥_𝑇+𝑀𝑥₀+𝑀𝑑+𝑀𝑀₁∫^𝑇₀𝑥𝜂+𝑥𝑎𝑥𝜂,𝜂−𝑥(𝑎(𝑥(0),0))𝑑𝜂+𝑀𝑀₂𝑇𝑑𝑠+𝑀𝑀₁∫^𝑡₀{‖𝑥(𝑠)‖+‖𝑥(𝑎(𝑥(𝑠),𝑠))−𝑥(𝑎(𝑥(0),0))‖}𝑑𝑠+𝑀𝑀₂𝑥𝑇+𝑀𝑑≤𝑀₀+𝑀𝐾₁𝐾₂𝑇𝑥_𝑇+𝑀²𝐾₁𝐾₂𝑇𝑥₀+𝑀²𝑀₁𝐾₁𝐾₂𝑇²𝑘+𝐿²𝐿_𝑎𝑇+𝑀²𝑀₂𝐾₁𝐾₂𝑇²+𝑀²𝐾₁𝐾₂𝑇𝑑+𝑀𝑀₁𝑇𝑘+𝐿²𝐿_𝑎𝑇+𝑀𝑀₂𝑥𝑇+𝑀𝑑≤𝑀₀+𝑇𝐾₁𝐾₂𝑥_𝑇+𝑇𝑀𝐾₁𝐾₂𝑥₀+𝑀𝑀₁𝐾₁𝐾₂𝑇²𝑘+𝐿²𝐿_𝑎𝑇+𝑀𝑀₂𝐾₁𝐾₂𝑇²+2𝑀₁𝑇𝑘+𝐿²𝐿_𝑎𝑇+𝑀₂𝑇+1+𝑀𝐾₁𝐾₂𝑇𝑑≤𝑘. (11) Therefore by hypothesis (H5), ‖(Φ𝑥)(𝑡)‖≤𝑘, which proves Φ(𝑍_𝑘)⊆𝑍_𝑘. Thus Φ maps 𝑍_𝑘 into itself. Next Φ is proved to be a contraction map. Let 𝑥₁, 𝑥₂∈𝑍_𝑘. Then Φ𝑥₁(𝑡)−Φ𝑥₂(𝑡=∫_𝑡0)𝑆(𝑡−𝑠)𝐵𝑊⁻¹−∫^𝑇₀𝑆×𝑓𝑇−𝜂𝜂,𝑥₂𝜂,𝑥₂𝑎𝑥₂𝜂,𝜂−𝑓𝜂,𝑥₁𝜂,𝑥₁𝑎𝑥₁𝜂+,𝜂𝑑𝜂_𝑚𝑘∑₌₁𝑆𝑇−𝑡_𝑘𝐼_𝑘𝑥₂𝑡_𝑘−𝐼_𝑘𝑥₁𝑡_𝑘(𝑠)𝑑𝑠+∫^𝑡₀𝑆()×𝑓𝑡−𝑠𝑠,𝑥₂(𝑠),𝑥₁𝑎𝑥₂(𝑠),𝑠−𝑓𝑠,𝑥₁(𝑠),𝑥₂𝑎𝑥₁(𝑠)+∑,𝑠𝑑𝑠_{0<𝑡𝑘<𝑡}𝑆𝑡−𝑡_𝑘𝐼_𝑘𝑥₂𝑡_𝑘−𝐼_𝑘𝑥₁𝑡_𝑘≤𝑀𝐾₁𝐾₂×∫^𝑡₀𝑀𝑀₁×∫^𝑇₀𝑥₂𝜂−𝑥₁𝜂+𝑥₂𝑎𝑥₂𝜂,𝜂−𝑥₁𝑎𝑥₁𝜂,𝜂𝑑𝜂+𝑀∑^𝑚_𝑘=1𝐿_𝑘𝑥₁−𝑥₂𝑑𝑠+𝑀𝑀₁∫^𝑡₀𝑥₁(𝑠)−𝑥₂(𝑠+𝑥₁)𝑎𝑥₁(𝑠),𝑠−𝑥₂𝑎𝑥₂(𝑠),𝑠𝑑𝑠+𝑀∑_{𝑘=1,2…,𝑚}𝐿_𝑘𝑥₂−𝑥₁≤𝑀²𝑀₁𝐾₁𝐾₂𝑇²×𝑥₂−𝑥₁+𝑥₂𝑎𝑥₂𝜂,𝜂−𝑥₁𝑎𝑥₂𝜂+𝑥,𝜂₁𝑎𝑥₂𝜂,𝜂−𝑥₁𝑎𝑥₁𝜂,𝜂+𝑀²𝐾₁𝐾₂𝑇𝐼𝑥₁−𝑥₂+𝑀𝑀₁𝑇𝑥₂−𝑥₁+𝑥₂𝑎𝑥₂(𝑠),𝑠−𝑥₁𝑎𝑥₂(𝑠)+𝑥,𝑠₁𝑎𝑥₂(𝑠),𝑠−𝑥₁𝑎𝑥₁(𝑠)𝑥,𝑠+𝑀𝐼₂−𝑥₁≤𝑀²𝑀₁𝐾₁𝐾₂𝑇²2+𝐿𝐿_𝑎𝑥₂−𝑥₁+𝑀²𝐾₁𝐾₂𝑇𝐼𝑥₁−𝑥₂+𝑀𝑀₁𝑇2+𝐿𝐿_𝑎𝑥₂−𝑥₁+𝑀𝐼𝑥₂−𝑥₁≤2+𝐿𝐿_𝑎𝑀𝐾₁𝐾₂𝑇+1𝑀𝑀₁𝑇+𝑀𝐼+𝑀²𝐾₂𝐾₁𝑇𝐼𝑥₂−𝑥₁≤𝑝𝑥₁−𝑥₂. (12) By applying conditions (H2), (H3), (H5), and (8) we find Φ is a contraction mapping on 𝑍_𝑘. Hence there exists a unique fixed point 𝑥∈𝑍_𝑘 such that (Φ𝑥)(𝑡)=𝑥(𝑡). The unique fixed point of Φ is a mild solution of (2) on 𝐽, which satisfies 𝑥(𝑇)=𝑥_𝑇. Hence the system is controllable on 𝐽.

4. Example

Consider the following third order dispersion equation: 𝑧_𝑡()=𝜕𝑡,𝑥³𝜕𝑥³𝑧()(𝑡)(()(𝑎(𝑧()𝑡𝑥,𝑡+𝐵𝑢+𝑝𝑡,𝑧𝑥,𝑡,𝑧𝑥,𝑡,𝑡)))Δ𝑧_𝑖,𝑥=𝐼_𝑘𝑧𝑡_𝑖,𝑥,𝑡_𝑖∈𝐽^′,𝑖=1,…,𝑚 (13) and the given initial and boundary conditions 𝜕^𝑘𝜕𝑥^𝑘𝑧()=𝜕0,𝑡^𝑘𝜕𝑥^𝑘𝑧(2𝜋,𝑡)=0,𝑘=0,1,2; (14)𝑥∈𝐼=(0,2𝜋), 𝑡∈𝐽=[0,𝑇].

Here 𝐵∶𝑈→𝑋 is a linear operator such that there exists an inverse operator 𝑊⁻¹ on 𝐿²(𝐽;𝑈)/Ker𝑊, where 𝑊 is defined by 𝑊𝑢=^𝑇₀𝑆(𝑇−𝑠)𝐵𝑢(𝑠)𝑑𝑠. (15) The operator 𝑆(𝑡) is uniformly bounded and 𝑝∶𝐽×𝑋×𝑋→𝑋 is continuous and uniformly bounded.

This problem can be brought to the form (2) by making suitable choices of 𝐴, 𝐵, and 𝑓.

Let 𝑈=𝐿²[0,2𝜋], 𝑋={𝑥∈𝐿²[0,2𝜋]∶[𝑥]=∫₀^2𝜋𝑥(𝑠)𝑑𝑠=0}, 𝐴𝑤=−𝑤_𝑥𝑥𝑥, and 𝐷(𝐴)={𝑤∈𝐻³(0,2𝜋)∶(𝜕^𝑘/𝜕𝑥^𝑘)𝑤(0)=(𝜕^𝑘/𝜕𝑥^𝑘)𝑤(1),𝑘=0,1,2}.

Let 𝑓(𝑡,𝑤,𝑤(𝑎(𝑤,𝑡)))=𝑝(𝑡,𝑤(𝑥),𝑤(𝑎(𝑤(𝑥),𝑡))) for all (𝑡,𝑤,𝑤(𝑎(𝑤,𝑡)))∈𝐽×𝑋×𝑋, 𝑥∈𝐼 be such that condition (H2), (H3), and (H5) are satisfied.

Then this system becomes an abstract formulation of the type (2). The solutions are all bounded.

Further, all the conditions stated in the above theorem are satisfied. Hence system is exactly controllable on 𝐽.

5. Conclusion

Impulsive semilinear system of type (2) with deviated argument is studied, subsequently proving the sufficient conditions for exact controllability. Only 𝐶₀ semigroup is used throughout the paper. Our approach eliminates the need to assume the conditions such as analyticity and compactness of the semigroup and the function 𝑓. Banach contraction mapping principle is used to prove that the unique fixed point exists which is the mild solution of (2). Hence the system (2) is proved to be controllable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere gratitude to the reviewers for their valuable suggestions. The first author would like to thank Ministry of Human Resource and Development with Grant no. MHR-02-23-200-429/304 for their funding.