Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2s (2025) 

 

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Controllability for Volterra Integro-Dynamic Sylvester Matrix Systems 

with Impulse on Time Scales 

 

A Sreenivasulu1*, B V Appa Rao1, Durga Prasad Ravutla2 and Gudala Balaji Prakash3 
1Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur-522502, 

Andhra Pradesh, India. 

2Department of Human and Science, KG Reddy College of Engineering and Technology, Moinabadh Mandal, R.R 

District -501504, Telangana State, India. 

3Department of Mathematics, Aditya University, Surampalem, Andhra Pradesh, Pin code 533437, India 

*Corresponding author. E-mail: asreenivasulu@kluniversity.in 

Contributing author: bvardr2010@kluniversity.in, durgaprasadravutla@kgr.ac.in, and balajiprakashgudala@gmail.com 

 

Article History: 

Received: 16-09-2024 

Revised: 23-10-2024 

Accepted: 04-11-2024 

Abstract: 

This article presents the complete controllability for a Volterra integro-dynamic Sylvester 

matrix system with time scale impulses in a finite-dimensional space Rn. We utilized the 

Banach fixed point theorem and nonlinear functional analysis to determine the existence of a 

unique solution for the system, and conducted an analysis of complete controllability using 

the Gramian matrix and various parameter changes. We provided a numerical example using 

simulation to demonstrate the application of these conclusions for two different time scales, 

T=R and T=P_1,1. 

Keywords: Controllability, impulses, Volterra integro-dynamic system, time scale. 

AMS Subject Classification: 34A37, 93B05, 34H05, 34N05. 

 

1. Introduction 

There are numerous health issues that exhibit abrupt shifts in their states. We refer to these abrupt 

alterations as impulsive impacts within the system. Impulsive differential equations are those that 

incorporate the impact of impulses. These have substantial applications in several real-world situations, 

specifically in mechanical systems involving impact, biological systems like heartbeats and population 

dynamics, blood flow, ecology, medicine, control theory, and more. Within the current body of 

knowledge, there are two distinct categories of impulsive systems. There are two types of systems: 

impulsive and non-impulsive. In the impulsive system, the period of abrupt changes is significantly 

shorter compared to the overall duration of an evolutionary process, such as shocks, natural disasters, 

and non-impulsive events. The duration of these modifications persists throughout a limited time span. 

Insulin administration into the bloodstream is an important application of non-impulses is the 

administration of insulin into the bloodstream. This involves a sudden shift followed by a gradual 

absorption process, with the insulin remaining active for a specific period of time. Some references [1, 

10, 14]. 

Kalman presented the idea of controllability and observability in the year 1960, and it quickly became 

a subject of examination for a significant number of researchers immediately after its introduction. In 

a general sense, controllability refers to the ability of a control dynamical system to guide itself from 

mailto:asreenivasulu@kluniversity.in
mailto:bvardr2010@kluniversity.in


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a state initial to the intended final by making use of a control that is accessible within the system. 

Recently, numerous authors have published their research articles [3, 8, 9, 11-13, 15-24]. 

We design with nonlinear time-varying complete controllability Volterra integro-dynamic Sylvester 

matrix system with an impulse control system in Rn 

𝑋∆(𝑡) = 𝐴(𝑡)𝑋(𝑡) + 𝑋(𝑡)𝐵(𝑡) + ∫ (𝐾1(𝑡, 𝑠)𝑋(𝑠) + 𝑋(𝑠)𝐾2(𝑡, 𝑠))∆𝑠
𝑡

𝑡0
+

𝐶(𝑡)𝑈(𝑡) + 𝐹(𝑡, 𝑋(𝑡))  

(1.1) 

𝑋(𝑡) = (1+ 𝐷𝑗)𝑋(𝑡𝑗
−),        𝑗 = 1,2, … .,  (1.2) 

Where A(t), B(t), C(t), 𝐾1(𝑡) and 𝐾2(𝑡) are rd-continuous matrices orders 𝑛 × 𝑛. 𝐹: 𝐼 × ℝ𝑛 → ℝ𝑛 is 

rd-continuous on 𝕋0. 𝐷𝑗 ∈ 𝑀𝑛×𝑛(ℝ),  𝑋(𝑡) ∈ ℝ𝑛 is state variable. 𝑈(𝑡) ∈ ℝ𝑚 is the control input.  

Time scale theory incorporates both discrete and continuous theories, as well as a hybrid of the two. 

Thus, in contrast to previous findings in the literature, our findings are more applicable to a wider 

range of situations. In this paper, the following structure is used: We lay the groundwork, provide some 

definitions, state some key lemmas and theorems in Section 2. Section 3 presents the results for 

complete controllability with Gramian matrix. 

2. Preliminaries 

Stefan Hilger’s 1988 doctoral thesis was the first to present the time scales calculus. He is bringing 

together the system’s discrete and continuous analysis. A time scale 𝕋 is defined as a non-empty closed 

subset of ℝ. If max 𝕋 exists, we define 𝕋𝑘 = 𝕋{𝑚𝑎𝑥𝕋}. But if that is not the case, 𝕋𝑘 = 𝕋. According, 

we define (𝑎, 𝑏)𝕋, [𝑎, 𝑏)𝕋, (𝑎, 𝑏]𝕋 and so on as a time scale interval, where [𝑎, 𝑏]𝕋 =

{𝑡 ∈ 𝕋: 𝑎 ≤ 𝑡 ≤ 𝑏}. With the substitution 𝑠𝑢𝑝𝕋 for inf{∅}, the forward jump operator 𝜎: 𝕋𝑘 → 𝕋 is 

defined as 𝜎(𝑡) = 𝑖𝑛𝑓{ 𝑠 ∈ 𝕋: 𝑠 > 𝑡} ∈ 𝕋. The operator 𝜌: 𝕋𝑘 → 𝕋, which is defined as 𝜌(𝑡) =

sup{ 𝑠 ∈ 𝕋: 𝑠 > 𝑡} ∈ 𝕋, can be expanded with the substitution sup{∅} = inf𝕋. At last, for 𝑡 ∈ 𝕋, the 

graininess function 𝜇(𝑡) follows the equation 𝜎(𝑡) − 𝑡. 

 when 𝑡 = 𝑠𝑢𝑝𝕋, choose 𝜏 such that mapping x from 𝕋 to ℝ is not left scattered. If 휀 > 0, then 

the generalized delta derivative of x(t), denoted as 𝑥∆(t), is of the form that. Given that U(t) is a 

neighbourhood, it follows that  

|[𝑥(𝜎(𝑡) − 𝑥(𝑠)] − 𝑥∆(𝑡)[𝜎(𝑡) − 𝑠]| ≤ 휀|𝜎(𝑡) − 𝑠|, for 𝑠 ∈ 𝑈. 

The process of mapping x from 𝕋 to ℝ is known as the generalized delta derivative on time scales 

calculus, where x is delta derivative for every 𝑡 ∈ 𝕋.  

 The right dense points in 𝕋 are considered to represent the origins of rd-continuous M mapping 

from 𝕋 to ℝ, whereas the left dense points in 𝕋 are the locations of its finite left sided limits. The set 

of rd-continuous functions M is denoted by 𝐶𝑟𝑑 = 𝐶𝑟𝑑(𝕋) = 𝐶𝑟𝑑(𝕋,ℝ). Assuming 𝑀∆(𝜏) = 𝑀(𝜏) for 

every 𝜏 ∈ 𝕋𝑘, the mapping from 𝕋𝑘 to ℝ is referred to as the anti- derivative of M from 𝕋𝑘 to ℝ. We 

continue by creating the integral ∫ 𝑚(𝑡)∆𝑡 = 𝑀(𝑏) − 𝑀(𝑎).
𝑏

𝑎
 

Definition 2.1.[5]: The function M(t) that maps from 𝕋 to ℝ is regressive is defined as1 + 𝜇(𝑡)(𝑡) ≠

0 ∀ 𝑡 ∈ 𝕋. The right dense continuous function ℛ = ℛ(𝑡) = ℛ(𝕋,ℝ) is the sum of all regressive 



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functions.  Likewise,  ℛ+ = ℛ+(𝕋,ℝ) = {𝑀 ∈ ℛ: 1 + 𝜇(𝑡)𝑀(𝑡) > 0, ∀ 𝑡 ∈ 𝕋} denotes all positively 

regressive function. 

Lemma 2.1.[6]: When 𝑀,𝑁 ∈ ℛ matrices on 𝕋, thus 

i. 𝑒𝑀
−1(𝜏, 𝑠) ≡

𝑒⊖𝑀
∗ (𝜏, 𝑠); 

ii. 𝑒0(𝜏, 𝑠) ≡ 𝐼 and 

𝑒0(𝜏, 𝜏) ≡ 𝐼; 

iii. 𝑒𝑀(𝜎(𝜏), 𝑠) ≡

(𝐼 + 𝜇(𝜏)𝑀(𝜏))𝑒𝑀(𝜏, 𝑠); 

iv. 𝑒𝑀(𝜏, 𝑠) =

𝑒𝑀
−1(𝑠, 𝜏) = 𝑒⊖𝑀∗

∗ (𝑠, 𝜏); 

v. 𝑒𝑀(𝜏, 𝑠)𝑒𝑁(𝜏, 𝑠) =

𝑒𝑀⊕𝑁(𝜏, 𝑠); 

vi. 𝑒𝑀(𝜏, 𝑠)𝑒𝑀(𝑠, 𝑟) =

𝑒𝑀(𝜏, 𝑟); 

Lemma 2.2.[4]:  Consider a matrix M of size 𝑛 × 𝑛 on a time scale. Assume that the mapping f from 

𝕋 to ℝ𝑛 is continuous and right dense. Given that  𝑡0 belongs to the set 𝕋  and 𝑝0 belongs to ℝ𝑛, this 

implies the initial value problem (IVP). 

𝑝∆(𝑡) = 𝑀(𝑡)𝑝(𝑡) + 𝑙(𝑡), 𝑝(𝑡0) = 𝑝0, 

having one and only one solution p mapping from 𝕋 to ℝ is developed as 

𝑝(𝑡) = 𝑓𝑀(𝑡, 𝑡0)𝑝0 + ∫𝑓𝑀(𝑡, 𝜎(𝜏))𝑙(𝜏)∆𝜏.

𝑡

𝑡0

 

Theorem 2.1. Let Z(t)=Vec X(t), �̂�(𝑡)=Vec U(t), and  𝑓(𝑡, 𝑧(𝑡)) = 𝑉𝑒𝑐𝐹(𝑡, 𝑋(𝑡)). Then the 

Volterra Integro-dynamic Sylvester matrix with an impulse control system (1.1), (1.2) is equivalent 

the system  

z∆(t) = P(t)z(t) + ∫K(t, s)z(s)∆s

t

0

+Q(t)Û(t) + f(t, z(t)) 
(2.1) 

z(t) = [In ⊗ R𝑗]z(𝑡𝑗
−)                                                                (2.2) 

Where 𝑃(𝑡) = [𝐵∗⊗ In + In ⊗𝐴], 𝑄(𝑡) = [In ⊗𝐶], 𝐾(𝑡, 𝑠) = [𝐾2
∗⊗ 𝐼𝑛) + (𝐼𝑛⊗𝐾1)  and 𝑅𝑗 =

(1+ 𝐷𝑗) and  In is the identity matrix. 

Proof: We apply the Vec operator to the equation (1.1), (1.2) and using the above properties of 

Kronecker product [3], we have  

z∆(t) = P(t)z(t) + ∫K(t, s)z(s)∆s

t

0

+Q(t)Û(t) + f(t, z(t)) 
 



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z(t) = [In ⊗ R𝑗]z(𝑡𝑗
−)                                                                   

Lemma 2.3 [7]. For the system (2.2) with 𝑃 ∈ 𝑀𝑛2(ℝ) is a constant, there exists a scalar function 

𝛾0(𝑡, 𝑠), … , 𝛾𝑛2−1(𝑡, 𝑠) ∈ (𝕋
+,ℝ) such that the only one solution has representation. 

          𝑒𝑃(𝑡, 𝑠) = ∑ 𝛾𝑘(𝑡, 𝑠)𝑃
𝑘.𝑛2−1

𝑘=0                                                                           

Theorem 2.2. Each ∀ 𝑡 ∈ (𝑠𝑗 , 𝑡𝑗+1]𝕋, 𝑗 = 1,2, . ., implies the satisfying function is known as the solution 

of a system (2.1) represented by 

𝑧(𝑡) = Ψ(𝑡, 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−) + ∫Ψ(𝑡, 𝜎(𝑠))K(t, s)z(s)∆s

𝑡

𝑠𝑗

+ ∫Ψ(𝑡, 𝜎(𝜏))

𝑡

𝑠𝑗

[𝑄(𝜏)�̂�(𝜏) + 𝑓(𝜏, 𝑧(𝜏))]∆𝜏, 

(2.3) 

Proof: if 𝑡 ∈ [𝑡0, 𝑡1]𝕋, then there exists an only one solution of (2.1), we have 

𝑧(𝑡) = Ψ(𝑡, 𝑡0)𝑧0 + ∫Ψ(𝑡, 𝜎(𝑠))K(t, s)z(s)∆s

𝑡

𝑡0

+ ∫Ψ(𝑡, 𝜎(𝜏)

𝑡

𝑡0

𝑄(𝜏)�̂�(𝜏)∆𝜏

+ ∫Ψ(𝑡, 𝜎(𝜏)

𝑡

𝑡0

𝑓(𝜏, 𝑧(𝜏))∆𝜏 

Next, j=1 then 𝑡 ∈ (𝑠1, 𝑡2]𝕋 we have 

𝑧(𝑡) = Ψ(𝑡, 𝑡0)𝑧(𝑠1) + ∫Ψ(𝑡, 𝜎(𝑠))K(t, s)z(s)∆s

𝑡

𝑠1

 

                                             +∫ Ψ(𝑡, 𝜎(𝜏)
𝑡

𝑠1
𝑄(𝜏)�̂�(𝜏)∆𝜏 + ∫ Ψ(𝑡, 𝜎(𝜏)

𝑡

𝑠1
𝑓(𝜏, 𝑧(𝜏))∆𝜏.      

Also, for 𝑧(𝑠1) = [𝐼𝑛⨂𝑅1]𝑧(𝑡1) substitute above equation, we get 

𝑧(𝑡) = Ψ(𝑡, 𝑡0)[𝐼𝑛⨂𝑅1]𝑧(𝑡1)  + ∫Ψ(𝑡, 𝜎(𝑠))K(t, s)z(s)∆s

𝑡

𝑠1

 

                                            +∫ Ψ(𝑡, 𝜎(𝜏)
𝑡

𝑠1
𝑄(𝜏)�̂�(𝜏)∆𝜏 + ∫ Ψ(𝑡, 𝜎(𝜏)

𝑡

𝑠1
𝑓(𝜏, 𝑧(𝜏))∆𝜏.      

Similarly, we are repeating the above same process for  𝑡 ∈ (𝑠𝑗, 𝑡𝑗+1]𝕋, 𝑗 = 1,2, … ,𝑚, we get 

𝑧(𝑡) = Ψ(𝑡, 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−) + ∫Ψ(𝑡, 𝜎(𝑠))K(t, s)z(s)∆s

𝑡

𝑠𝑗

 



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+ ∫Ψ(𝑡, 𝜎(𝜏))

𝑡

𝑠𝑗

[𝑄(𝜏)�̂�(𝑡) + 𝑓(𝜏, 𝑧(𝜏))] ∆𝜏  

Therefore, the equation (2.3) was derived. 

3. CONTROLLABILITY 

In this section, we provide necessary and sufficient conditions for complete controllability in the 

following system.  

{
 
 

 
 𝑧∆(𝑡) = 𝑃(𝑡)𝑧(𝑡) + ∫ 𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠

𝑡

0

+ 𝑄(𝑡)�̂�(𝑡) + 𝑓(𝑡, 𝑧(𝑡)),       𝑡 ∈ (𝑠𝑗, 𝑡𝑗+1]𝕋, 𝑗 = 1, 2, …

𝑧(𝑡) = [𝐼𝑛⊗𝑅𝑗]𝑧(𝑡𝑗
−),                                                    𝑡 ∈ (𝑠𝑗 , 𝑡𝑗+1]𝕋, 𝑗 = 0,1, …                            

𝑧(𝑡0) = 𝑧0,   𝑡0 ∈ 𝕋                                                                                                                                             

(3.1) 

Definition 3.1: For any 𝑧0 and 𝑧𝑇 ∈ ℝ𝑛
2

, there must be a piece-wise rd-continuous control function 

Û(𝑡): [𝑡0, 𝑇]𝕋  →  ℝ
𝑛2 , so that the solution of the system (3.1) satisfies 𝑧(𝑡0) = 𝑧0 and 𝑧(𝑇) = 𝑧𝑇 . This 

system is called controllability on [𝑡0, 𝑇]𝕋  with 𝑡0 < 𝑇.  

Definition 3.2: For 𝑗 = 1,2, … ,𝑚, it is controllable on both [𝑡0, 𝑡1]𝕋   and [𝑠𝑗 , 𝑡𝑗+1]𝕋, then system (3.1) 

is known as complete controllable in [𝑡0, 𝑇]𝕋 with 𝑡0 < 𝑇. 

The corresponding Gramian matrices are defined by. 

𝒩0(𝑡0, 𝑡1) = ∫Ψ(𝑡0,σ(τ))𝑄(τ)𝑄
∗(τ)Ψ∗(𝑡0, σ(τ))Δτ 

t

t0

 (3.2) 

𝒩𝑗(𝑠𝑗 , 𝑡𝑗+1) = ∫ Ψ (𝑠𝑗 , σ(τ))𝑄(τ)𝑄
∗(τ)Ψ∗ (𝑠𝑗 , σ(τ))Δτ , 𝑗 = 1,2, … ,𝑚,

𝑡𝑗+1

s𝑗

 (3.3) 

If 𝑃(𝑡) = 𝑃 and 𝑄(𝑡) = 𝑄 are constant matrices, then 

𝒩0(𝑡0, 𝑡1) = ∫ 𝑒𝑃(𝑡0,σ(τ))𝑄𝑄
∗𝑒𝑃
∗(𝑡0,σ(τ))Δτ 

t

t0

 (3.4) 

𝒩𝑗(𝑠𝑗 , 𝑡𝑗+1) = ∫ 𝑒𝑃 (𝑠𝑗 , σ(τ))𝑄𝑄
∗𝑒𝑃
∗ (𝑠𝑗 , σ(τ))Δτ, 𝑗 = 1,2, … ,𝑚

𝑡𝑗+1

s𝑗

 (3.5) 

Here (. )∗is represented as transpose of a matrix (. ). 

We define Û(𝑡) as 

Û(𝑡) = {
−𝑄∗(𝑡)𝛹∗(𝑡0, 𝜎(𝜏))Φ0 ,                                   𝑡 ∈ [𝑡0, 𝑡1]𝕋

−𝑄∗(t)Ψ∗(𝑠𝑘,σ(τ))Φ𝑘, 𝑡 ∈ (𝑠𝑗 , 𝑡𝑗+1]𝕋, 𝑗 = 1,2, … ,𝑚.
 (3.6) 

Where  



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Φ0 = 𝒩0
−1(𝑡0, 𝑡1) [𝑧0 − Ψ(𝑡0, 𝑡1)𝑧𝑡1 + ∫ Ψ(𝑡0, 𝜎(𝑠))K(t, s)z(s)∆s

𝑡1

t0

+∫ Ψ(𝑡0,σ(τ))f(τ, z(τ))Δτ 
𝑡1

t0

], 

and  

Φ𝑗 = 𝒩𝑗
−1(𝑠𝑗, 𝑡𝑗+1) [[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗

−) −Ψ(𝑠𝑗 , 𝑡𝑗+1)𝑧𝑡𝑗+1 +∫ Ψ (𝑡𝑗 , σ(𝑠))K(t, s)z(s)∆s 
𝑡𝑗+1

s𝑗

+∫ Ψ (𝑠𝑗 , σ(τ)) f(τ, z(τ))Δτ 
𝑡𝑗+1

s𝑗

]. 

 We need following the conditions: 

(H1): The nonlinear function  𝑓: 𝐽1 × ℝ𝑛
2

→ ℝ𝑛
2

, 𝐽1 = ⋃ [𝑠𝑗 , 𝑡𝑗+1]𝕋
𝑚
𝑗=0  is rd-continuous and  there is 

exists 𝑀𝑓 > 0 such that ‖𝑓(𝑡, 𝑧) − 𝑓(𝑡, 𝑥)‖ ≤ 𝑀𝑓‖𝑧 − 𝑥‖, ∀ 𝑧, 𝑥 ∈ ℝ𝑛
2

, 𝑡 ∈ 𝐽1. Also, there is exists 

𝐿𝑓 > 0 such that ‖𝑓(𝑡, 𝑧)‖ ≤ 𝐿𝑓 , ∀𝑡 ∈ 𝐽1 𝑎𝑛𝑑 𝑧 ∈ ℝ𝑛
2

. 

(H2): The function  [𝐼𝑛⨂𝑅𝑗]:= [𝑡𝑗 , 𝑠𝑗]𝕋 × ℝ𝑛
2

 →  ℝ𝑛
2

 are  rd-continuous there is exists  𝑀[𝐼𝑛⨂𝑅𝑗]
> 0 

such that ‖[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−) − [𝐼𝑛⨂𝑅𝑗]𝑥(𝑡𝑗

−)‖ ≤ 𝑀[𝐼𝑛⨂𝑅𝑗]
‖𝑧 − 𝑥‖, ∀ 𝑧, 𝑥 ∈ ℝ𝑛

2

, 𝑡 ∈ 𝐼𝑗 . Also, there is 

exists 𝐿[𝐼𝑛⨂𝑅𝑗] > 0 such that ‖[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗)‖ ≤ 𝐿[𝐼𝑛⨂𝑅𝑗], ∀𝑡 ∈ 𝐼𝑗  𝑎𝑛𝑑 𝑧 ∈ ℝ𝑛
2

. 

(H3): 𝑀𝛼 = max
1≤𝑘≤𝑚

{𝑀𝛼1
0 , 𝑀𝛼1

𝑗
, 𝑀[𝐼𝑛⨂𝑅𝑗]

} < 1, were  

                      𝑀𝛼1
0 = 𝐿𝑀𝑘𝑀𝑓𝑡1(1+ 𝐿

2𝐿𝑄
2 𝑡1𝛿0), 

   𝑀𝛼1

𝑗
= 𝐿𝑀[𝐼𝑛⨂𝑅𝑗]

+ 𝐿2𝐿𝑄
2 𝑀[𝐼𝑛⨂𝑅𝑗]

𝑇𝛿𝑗 + 𝐿𝑀𝑘𝑀𝑓𝑇(1+ 𝐿
2𝐿𝑄

2 𝑇𝛿𝑗), 𝑗 = 1,2, … ,𝑚 

For notational accommodation, we get 

𝛿0 = ‖𝒩0
−1(𝑡0, 𝑡1)‖, 𝛿𝑗 = ‖𝒩𝑗

−1(𝑠𝑗, 𝑡𝑗+1)‖. 

𝐿 = Ψ(𝑡, 𝑠)(𝑡,𝑠)∈𝐼
𝑚𝑎𝑥

, 𝐿𝑄 = ‖𝑄(t)‖𝑡∈𝐼
𝑚𝑎𝑥 , 𝐿𝐾 = ‖𝐾(t, s)‖𝑡∈𝐼

𝑚𝑎𝑥  

ℋ0 = 𝐿‖𝑧0‖ + 𝐿𝐿𝐾𝑡1 + 𝐿𝐿𝑓𝑡1 + 𝐿𝐿𝑄𝐿𝑈
0 𝑡1. 

𝐿𝑈
0 = 𝐿𝐿𝑄𝛿0(‖𝑧0‖ + 𝐿‖𝑧𝑡1‖ + 𝐿𝐿𝐾𝑡1 + 𝐿𝐿𝑓𝑡1). 

ℋ1
𝑗
= 𝐿𝐿[𝐼𝑛⨂𝑅𝑗] + 𝐿𝐿𝐾𝑇 + 𝐿𝐿𝑓𝑇 +  𝐿𝐿𝑄𝐿𝑈

𝑇 𝑇.  

𝐿
𝑈
𝑗
= 𝐿𝐿𝑄𝛿0 (𝐿[𝐼𝑛⨂𝑅𝑗] + 𝐿‖𝑧𝑡𝑗+1‖ + 𝐿𝐿𝐾𝑡𝑗+1 + 𝐿𝐿𝑓𝑡𝑗+1). 

                   𝛾 ≥ max
1≤𝑘≤𝑚

{ℋ0,ℋ1
𝑗
, 𝐿[𝐼𝑛⨂𝑅𝑗]} . 

 



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Theorem 3.1: Assuming that requirements (H1) - (H3) are satisfied, then there is only one solution to 

system (3.1). 

Proof: The subset 𝒟 ⊆ 𝑃𝐶 is defined as the set  

𝒟 = {𝑧 ∈ 𝑃𝐶: ‖𝑧‖𝑃𝐶 ≤ 𝛾}. 

Currently, we are defining the function 𝒢:𝒟 ⟶ 𝒟, which means that  

For, 𝑡 ∈ [0, 𝑡1]𝕋 

(𝒢𝑧)(𝑡) = Ψ(𝑡, 𝑡0)𝑧0 +∫ Ψ(𝑡, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠
𝑡

𝑡0

+∫ Ψ(𝑡, 𝜎(𝜏))(𝑓(𝜏, 𝑧(𝜏)) + 𝑄(𝜏))∆𝜏.
𝑡

𝑡0

 

(3.8) 

For 𝑡 ∈ (𝑡𝑗 , 𝑠𝑗]𝕋, 𝑗 = 1,2, … ,𝑚  

(𝒢𝑧)(𝑡) = [𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−),  (3.8) 

For ∀𝑡 ∈ (𝑠𝑗, 𝑡𝑗+1]𝕋, 𝑗 = 1,2, … ,𝑚  

(𝒢𝑧)(𝑡) = Ψ(𝑡, 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−) + ∫ Ψ(𝑡, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠

𝑡

𝑠𝑗

+∫ Ψ(𝑡, 𝜎(𝜏)) (𝑓(𝜏, 𝑧(𝜏)) + 𝑄(𝜏)�̂�(𝜏)) ∆𝜏.
𝑡

𝑠𝑗

 

(3.9) 

It is evident that the solution is the Banach fixed point for 𝒢. Let us now consider 𝑡 ∈ (𝑠𝑗 , 𝑡𝑗+1]𝕋, 𝑗 =

1,2, … ,𝑚, and  𝑧 ∈ 𝒟, we obtain  

‖(𝒢𝑧)(𝑡)‖ ≤ ‖Ψ(𝑡, 𝑠𝑗)‖‖[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−)‖ +∫ ‖Ψ(𝑡, 𝜎(𝑠))‖‖𝐾(𝑡, 𝑠)‖‖𝑧(𝑠)‖∆𝑠

𝑡

𝑠𝑗

+∫ ‖Ψ(𝑡,𝑡(𝑡))‖‖𝑡(𝑡,𝑡(𝑡))‖∆𝑡
𝑡

𝑡𝑡

+∫ ‖Ψ(𝑡, 𝜎(𝜏))‖‖Û(𝜏)‖‖𝑄(𝜏)‖
𝑡

𝑠𝑗

∆𝜏 

(3.10) 

≤ 𝐿𝐿[𝐼𝑛⨂𝑅𝑗] + 𝐿𝐿𝐾𝑡𝑗+1 + 𝐿𝐿𝑓𝑡𝑗+1 +  𝐿𝐿𝑄𝐿𝑈
𝑗
𝑡𝑗+1 

≤ ℋ1
𝑗
≤ 𝛾. 

Similarly, for 𝑡 ∈ [0, 𝑡1]𝕋 and  𝑧 ∈ 𝒟, then 



Communications on Applied Nonlinear Analysis 

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‖(𝒢𝑧)(𝑡)‖ ≤ ‖Ψ(𝑡, 𝑡0)‖‖𝑧0‖ + ∫ ‖Ψ(𝑡, 𝜎(𝑠))‖‖𝐾(𝑡, 𝑠)‖‖𝑧(𝑠)‖∆𝑠
𝑡

𝑡0

+∫ ‖Ψ(𝑡, 𝜎(𝜏))‖‖𝑓(𝜏, 𝑧(𝜏))‖∆𝜏
𝑡

𝑡0

+∫ ‖Ψ(𝑡, 𝜎(𝜏))‖‖Û(𝜏)‖‖𝑄(𝜏)‖
𝑡

𝑡0

∆𝜏 

(3.11) 

≤ 𝐿‖𝑧0‖ + 𝐿𝐿𝐾𝑡 + 𝐿𝐿𝑓𝑡 +  𝐿𝐿𝑄𝐿𝑈
0 𝑡 

≤ ℋ0 ≤ 𝛾. 

Similarly, for 𝑡 ∈ (𝑠𝑗, 𝑡𝑗]𝕋, and  𝑧 ∈ 𝒟, we get 

‖(𝒢𝑧)‖𝑃𝐶 ≤ 𝐿[𝐼𝑛⨂𝑅𝑗] ≤ 𝛾.    (3.12) 

After succinct the above inequalities (3.10) - (3.12), we have  

‖(𝒢𝑧)‖𝑃𝐶 ≤ 𝛾. 

Since, 𝒢: 𝒟 ⟶ 𝒟, For any 𝑧, 𝑥 ∈ 𝒟, 𝑡 ∈ (𝑠𝑗 , 𝑡𝑗+1]𝕋, 𝑗 = 1,2, … ,𝑚,  we get 

‖(𝒢𝑧)(𝑡) − (𝒢𝑥)(𝑡)‖ ≤ ‖Ψ(𝑡, 𝑠𝑗)‖‖[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−) − [𝐼𝑛⨂𝑅𝑗]𝑥(𝑡𝑗

−)‖ 

+∫ ‖Ψ(𝑡, 𝜎(𝑠))‖‖𝐾(𝑡, 𝑠)‖‖𝑧(𝑠) − 𝑥(𝑠)‖∆𝑠
𝑡

𝑠𝑗

 

+∫ ‖Ψ(𝑡, 𝜎(𝜏))‖‖𝑓(𝜏, 𝑧(𝜏)) − 𝑓(𝜏, 𝑥(𝜏))‖∆𝜏
𝑡

𝑠𝑗

 

+∫ [‖Ψ(𝑡, 𝜎(𝜏))‖‖𝑄(τ)‖‖𝑄∗(𝜏)‖‖Ψ∗(𝑡, 𝜎(𝜏))‖
𝑡

𝑠𝑗

× ‖𝒩𝑗
−1(𝑠𝑗, 𝑡𝑗+1)‖‖[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗

−) − [𝐼𝑛⨂𝑅𝑗]𝑥(𝑡𝑗
−)‖

+ ∫ ‖Ψ(𝑡,𝑡(𝑡))‖‖𝑡(𝑡,𝑡)‖‖𝑡(𝑡) −𝑡(𝑡)‖∆𝑡
𝑡𝑡+1

𝑡𝑡

+∫ ‖Ψ (𝑠𝑗 , 𝜎(𝑠))‖ ‖𝑓(𝑠, 𝑧(𝑠)) − 𝑓(𝑠, 𝑥(𝑠))‖∆𝑠
𝑡𝑗+1

𝑠𝑗

 ] ∆𝜏 

≤ 𝐿𝑀[𝐼𝑛⨂𝑅𝑗]
‖𝑧(𝑡𝑗

−) − 𝑥(𝑡𝑗
−)‖ + 𝐿𝑀𝐾∫ ‖𝑧(𝑠) − 𝑥(𝑠)‖∆𝑠

𝑡

𝑠𝑗

+ 𝐿𝑀𝑓∫ ‖𝑧(𝜏) − 𝑥(𝜏)‖∆𝜏 + 𝐿2𝐿𝑄
2 𝛿𝑗

𝑡

𝑠𝑗

 

×∫ [𝑀[𝐼𝑛⨂𝑅𝑗]
‖𝑧(𝑡𝑗

−) − 𝑥(𝑡𝑗
−)‖

𝑡

𝑠𝑗

+𝑀𝐾∫ ‖𝑧(𝑠) − 𝑥(𝑠)‖∆𝑠
𝑡𝑗+1

𝑠𝑗

+𝑀𝑓∫ ‖𝑧(𝑠) − 𝑥(𝑠)‖∆𝑠
𝑡𝑗+1

𝑠𝑗

]∆𝜏 

≤  𝐿𝑀[𝐼𝑛⨂𝑅𝑗]
‖𝑧 − 𝑥‖𝑃𝐶 + 𝐿𝑀𝐾‖𝑧 − 𝑥‖𝑃𝐶(𝑡 − 𝑠𝑗) +  𝐿𝑀𝑓‖𝑧 − 𝑥‖𝑃𝐶(𝑡 − 𝑠𝑗) 

+𝐿2𝐿𝑄
2 (𝑡 − 𝑠𝑗)𝛿𝑗[𝑀[𝐼𝑛⨂𝑅𝑗]

+ 𝐿𝑀𝐾(𝑡𝑗+1 − 𝑠𝑗)] ‖𝑧 − 𝑥‖𝑃𝐶 + 𝐿𝑀𝑓(𝑡𝑗+1 − 𝑠𝑗)] ‖𝑧 − 𝑥‖𝑃𝐶 



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≤ 𝑀𝛼1

𝑗 ‖𝑧 − 𝑥‖𝑃𝐶 ≤ 𝑀𝛼‖𝑧 − 𝑥‖𝑃𝐶 (3.13) 

For any 𝑧, 𝑥 ∈ 𝒟, 𝑡 ∈ [0, 𝑡1]𝕋, we get   

‖(𝑡𝑡)(𝑡) − (𝑡𝑡)(𝑡)‖

≤ ∫ ‖Ψ(𝑡,𝑡(𝑡))‖‖𝑡(𝑡,𝑡)‖‖𝑡(𝑡) −𝑡(𝑡)‖∆𝑡
𝑡

0

+∫ ‖Ψ(𝑡, 𝜎(𝜏))‖‖𝑓(𝜏, 𝑧(𝜏)) − 𝑓(𝜏, 𝑥(𝜏))‖∆𝜏
𝑡

0

 

+∫ [‖Ψ(𝑡, 𝜎(𝜏))‖‖𝑄(τ)‖‖𝑄∗(𝜏)‖‖Ψ∗(𝑡, 𝜎(𝜏))‖
𝑡

0

 

× ‖𝒩0
−1(𝑡0, 𝑡1)‖[∫ ‖Ψ(𝑡, 𝜎(𝑠))‖‖𝐾(𝑡, 𝑠)‖‖𝑧(𝑠) − 𝑥(𝑠)‖∆𝑠

𝑡1

0

+∫ ‖Ψ(𝑠𝑘, 𝜎(𝑠))‖‖𝑓(𝑠, 𝑧(𝑠)) − 𝑓(𝑠, 𝑥(𝑠))‖∆𝑠]∆𝜏
𝑡1

𝑡0

 

≤ 𝑀𝛼1
0 ‖𝑧 − 𝑥‖𝑃𝐶 ≤ 𝑀𝛼‖𝑧 − 𝑥‖𝑃𝐶 . (3.14) 

Similarly, for 𝑡 ∈ (𝑠𝑗, 𝑡𝑗]𝕋, we have 

‖(𝒢𝑧)(𝑡) − (𝒢𝑥)(𝑡)‖ ≤ 𝑀[𝐼𝑛⨂𝑅𝑗]
‖𝑧 − 𝑥‖𝑃𝐶 ≤ 𝑀𝛼‖𝑧 − 𝑥‖𝑃𝐶 (3.15) 

After succinct the inequalities (3.13) - (3.15), for 𝑡 ∈ 𝐼, we have 

‖(𝒢𝑧) − (𝒢𝑥)‖𝑃𝐶 ≤ 𝑀𝛼‖𝑧 − 𝑥‖𝑃𝐶 . 

Thus, according to Banach's fixed point theorem, there is only one solution to system (3.1). Because 

of this, 𝒢 is a mapping that strictly contracts. 

Theorem 3.2: Assuming that requirements (H1) - (H3) are satisfied; the system (3.1) is complete 

controllable in [𝑡0, 𝑇]𝕋 if and only if the matrices 𝒩0(𝑡0, 𝑡1) and 𝒩𝑗(𝑠𝑗, 𝑡𝑗+1) are invertible. 

Proof: Let 𝒩0(𝑡0, 𝑡1) and 𝒩𝑗(𝑠𝑗 , 𝑡𝑗+1) are invertible. Then, for the given 𝑧𝑡1and 𝑧𝑡𝑗+1 , and the input 

control Û (t) given by (3.6). Now, put 𝑡 = 𝑡1, in the system (3.1), we have  

𝑧(𝑡1) = Ψ(𝑡1, 𝑡0)𝑧0 +∫ Ψ(𝑡1, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠
𝑡1

𝑡0

+∫ Ψ(𝑡1, 𝜎(𝜏))𝑓(𝜏, 𝑧(𝜏))∆𝜏
𝑡1

𝑡0

−∫ ‖Ψ(𝑡1, 𝜎(𝜏))‖‖𝑄(τ)‖‖𝑄
∗(𝜏)‖‖Ψ∗(𝑡0, 𝜎(𝜏))‖

𝑡1

𝑡0

𝑧0∆𝜏 



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= Ψ(𝑡1, 𝑡0)𝑧0 +∫ Ψ(𝑡1, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠
𝑡1

𝑡0

+∫ Ψ(𝑡1, 𝜎(𝜏))𝑓(𝜏, 𝑧(𝜏))∆𝜏
𝑡1

𝑡0

− Ψ(𝑡1, 𝑡0)𝒩0(𝑡0, 𝑡1)𝒩0
−1(𝑡0, 𝑡1) [𝑧0 − Ψ(𝑡0, 𝑡1)𝑧𝑡1 +∫ Ψ(𝑡1, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠

𝑡1

𝑡0

+∫ Ψ(𝑡0, 𝜎(𝜏))𝑓(𝜏, 𝑧(𝜏))∆𝜏
𝑡1

𝑡0

] 

= 𝑧𝑡1  

Similarly, for 𝑡 ∈ (𝑠𝑗, 𝑡𝑗+1]𝕋, , 𝑗 = 1,2, … ,𝑚,. we replace 𝑡 = 𝑡𝑗+1, in the solution of (3.1), we have 

𝑧(𝑡𝑗+1) = Ψ(𝑡𝑗+1, 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−) + ∫ Ψ (𝑡𝑗+1, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠

𝑡𝑗+1

𝑠𝑗

+∫ Ψ (𝑡𝑗+1, 𝜎(𝜏)) [𝑓(𝜏, 𝑧(𝜏)) − 𝑄(𝜏)𝑄
∗(𝜏)Ψ∗ (𝑠𝑗 , 𝜎(𝜏)) 𝑧𝑗]

𝑡𝑗+1

𝑠𝑗

∆𝜏 

= Ψ(𝑡𝑗+1, 𝑠𝑗)[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−) + ∫ Ψ (𝑡𝑗+1, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠

𝑡𝑗+1

𝑠𝑗

+∫ Ψ(𝑡𝑡+1,𝑡(𝑡))𝑡(𝑡,𝑡(𝑡))∆𝑡
𝑡𝑡+1

𝑡𝑡

− Ψ(𝑡𝑗+1, 𝑠𝑗)𝒩𝑗(𝑠𝑗 , 𝑡𝑗+1)𝒩𝑗
−1(𝑠𝑗, 𝑡𝑗+1)

× [[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−) − Ψ(𝑠𝑗, 𝑡𝑗+1)𝑧𝑡𝑗+1 +∫ Ψ (𝑡𝑗+1, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠

𝑡𝑗+1

𝑠𝑗

+∫ Ψ (𝑡𝑗+1, 𝜎(𝜏)) 𝑓(𝜏, 𝑧(𝜏))∆𝜏
𝑡𝑗+1

𝑠𝑗

] 

= 𝑧𝑡𝑗+1 . 

Hence, for in [𝑡0, 𝑇]𝕋 the system (3.1) is complete controllable  

Conversely, on the interval [𝑡0, 𝑇]𝕋, we presume that system (3.1) is complete controllable. Therefore, 

the matrices 𝒩0(𝑡0, 𝑡1)   and 𝒩𝑗(𝑠𝑗 , 𝑡𝑗+1) are not invertible.  

Then, there exists a non-zero vector 𝑧𝛼, 𝑧𝛼𝑗 ∈ ℝ𝑛
2

 such that  

𝑧𝛼
∗𝒩0(𝑡0, 𝑡1)𝑧𝛼 = 0 𝑎𝑛𝑑     𝑧𝛼𝑗

∗ 𝒩𝑗(𝑠𝑗, 𝑡𝑗+1)𝑧𝛼𝑗 = 0.    (3.16) 

From the equations (3.2), (3.3) and (3.16), we get  

∫𝑧𝛼
∗Ψ(𝑡0, σ(τ))𝑄(𝜏)𝑄

∗(𝜏)Ψ∗(𝑡0,σ(τ))𝑧𝛼Δτ = 0

t

t0

. (3.17) 



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∫ 𝑧𝛼𝑗
∗ Ψ(𝑡0,σ(τ))𝑄(τ)𝑄

∗(𝜏)Ψ∗(𝑡0,σ(τ))𝑧𝛼𝑗Δτ = 0

𝑡𝑗+1

s𝑗

. (3.18) 

On solving the above equations (3.17) and (3.18), we have 

𝑧𝛼
∗Ψ(𝑡0, σ(τ))𝑄(𝜏) = 0, 𝜏 ∈ [𝑡0, 𝑡1]𝕋 

𝑧𝛼𝑗
∗ Ψ (𝑠𝑗 , σ(τ))𝑄(𝜏) = 0, 𝜏 ∈ (𝑠𝑗 , 𝑡𝑗+1]𝕋, 𝑗 = 1,2, … ,𝑚. 

Therefore, the system (3.1) is complete controllable on[𝑡0, 𝑡1]𝕋 ,  

So, if we choose  

𝑧0 = 𝑧𝛼 +  Ψ(𝑡0, 𝑡1)𝑧𝛼𝑗 −∫ Ψ(𝑡0, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠
𝑡1

𝑡0

−∫ Ψ(𝑡0, σ(τ))𝑓(𝜏, 𝑧(𝜏))∆𝜏,
𝑡1

𝑡0

 

in [𝑡0, 𝑡1]𝕋. In that case, there exist a piece-wise rd-continuous control Û(t) that  

𝑧𝛼1 =  Ψ(𝑡1, 𝑡0) (𝑧𝛼 + Ψ(𝑡0, 𝑡1)𝑧𝛼1 −∫ Ψ(𝑡0, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠
𝑡1

𝑡0

−∫ Ψ(𝑡0,σ(τ))𝑓(𝜏, 𝑧(𝜏))∆𝜏
𝑡1

𝑡0

)

+ ∫ Ψ(𝑡0, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠
𝑡1

𝑡0

+∫ Ψ(𝑡0,σ(τ)) (𝑓(𝜏, 𝑧(𝜏)) + 𝑄(τ)Û(𝜏))∆𝜏
𝑡1

𝑡0

, 

Which gives 𝑧𝛼
∗𝑧𝛼 = 0. Similarly, we have  

[𝐼𝑛⨂𝑅𝑗]𝑧(𝑡𝑗
−) = 𝑧𝛼𝑗 +  Ψ(𝑠𝑗 , 𝑡𝑗+1)𝑧𝑡𝑗+1 −∫ Ψ(𝑡0, 𝜎(𝑠))𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠

𝑡𝑗+1

𝑠𝑗

−∫ Ψ(𝑡0,σ(τ))𝑓(𝜏, 𝑧(𝜏))∆𝜏
𝑡𝑗+1

𝑠𝑗

. 

It can be shown that 𝑧𝛼𝑗
∗ 𝑧𝛼𝑗 = 0,  which contradicts the fact that 𝑧𝛼

∗𝑧𝛼 ≠ 0,Therefore, the matrices 

 𝒩0(𝑡0, 𝑡1)   and 𝒩𝑗(𝑠𝑗 , 𝑡𝑗+1) are invertible. 

Theorem 3.3: Assuming that requirements (H1) - (H3) are satisfied; the time-invariant case of system 

(3.1) is said to be complete controllable in interval [𝑡0, 𝑇]𝕋  if and only if the rank of the matrix   

[𝑄  𝑃𝑄  𝑃2𝑄… 𝑃𝑛−1𝑄]   = 𝑛2 (3.19) 

Proof: Assume that system (3.1) is to be complete controllable in [𝑡0, 𝑇]𝕋. But the rank of 𝐶 ≠ 𝑛2(∵

 [𝑄  𝑃𝑄  𝑃2𝑄… 𝑃𝑛
2−1𝑄] = 𝐶), then there exists non-zero vector 𝑧𝛼 ∈ ℝ𝑛

2

such that  

𝑧𝛼
∗𝑃𝑖𝐵 = 0, 𝑖 = 0,1, … , 𝑛2 − 1. (3.20) 

Furthermore, based on equations (3.4) and (3.5), we can deduce  

𝑧𝛼
∗𝒩0(𝑡0, 𝑡1)𝑧𝛼 = ∫ 𝑧𝛼

∗𝑒𝑃(𝑡0,σ(τ))𝑄𝑄
∗𝑒𝑃
∗(𝑡0,σ(τ))𝑧𝛼Δτ 

t

t0

 (3.21) 



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𝑧𝛼
∗𝒩𝑘(𝑠𝑗 , 𝑡𝑗+1)𝑧𝛼 = ∫ 𝑧𝛼

∗𝑒𝑃 (𝑠𝑗 , σ(τ))𝑄𝑄
∗𝑒𝑃
∗ (𝑠𝑗 , σ(τ)) 𝑧𝛼Δτ 

𝑡𝑗+1

s𝑗

. (3.22) 

Now, we are using Theorem 2.4. and from equation (3.20) in the above equation (3.21) and (3.22), we 

have 

𝑧𝛼
∗𝒩0(𝑡0, 𝑡1)𝑧𝛼 = ∫ [∑ 𝛾𝑗

𝑛2−1

𝑗=0

(𝑡0, σ(τ))𝑧𝛼
∗𝑃𝑖𝑄]𝑄∗𝑒𝑃

∗(𝑡0, σ(τ))𝑧𝛼Δτ = 0 
t

t0

             

𝑧𝛼
∗𝒩𝑗(𝑠𝑗 , 𝑡𝑗+1)𝑧𝛼 = ∫ [∑ 𝛾𝑗

𝑛2−1

𝑗=0

(𝑠𝑗 , σ(τ)) 𝑧𝛼
∗𝑃𝑖𝑄]𝑄∗𝑒𝑃

∗ (𝑠𝑗 , σ(τ)) 𝑧𝛼Δτ = 0.
𝑡𝑗+1

s𝑗

 

Thus, 𝒩0(𝑡0, 𝑡1) and 𝒩𝑗(𝑠𝑗 , 𝑡𝑗+1), are not invertible. Theorem 3.1states that system (3.1) is not 

completely controllable. Therefore, it contradicts. The rank of 𝐶 = 𝑛2.  

Conversely, the matrices 𝒩0(𝑡0, 𝑡1) and 𝒩𝑗(𝑠𝑗 , 𝑡𝑗+1), are not invertible and we assume that the rank of 

𝐶 = 𝑛2. the system (3.1) is not to be complete controllable. This means that there exists non-zero 

vectors  𝑧𝛼, 𝑧𝛼𝑗 ∈ ℝ𝑛
2

, such that  

𝑧𝛼
∗𝒩0(𝑡0, 𝑡1)𝑧𝛼 = 0. (3.23) 

and  

𝑧𝛼𝑗
∗ 𝒩𝑗(𝑠𝑗, 𝑡𝑗+1)𝑧𝛼𝑗 = 0, 𝑗 = 1,2… ,𝑚, (3.24) 

Now, from the equations (3.4), (3.5), (3.23) and (3.24), we have 

𝑧𝛼
∗𝑒𝑃(𝑡0, 𝑡1)𝑄 = 0, ∀ 𝑡 ∈ [𝑡0, 𝑡1]𝕋 (3.25) 

and  

𝑧𝛼
∗𝑒𝑃(𝑠𝑗, 𝑡𝑗+1)𝑄 = 0, ∀ 𝑡 ∈ (𝑠𝑗 , 𝑡𝑗+1]𝕋, , 𝑗 = 1,2, … ,𝑚, (3.26) 

Now, for 𝑗 = 1,2, … ,𝑚, the 𝑒𝑃(𝑡0, . ), 𝑒𝑃(𝑠𝑗, . ) are rd-continuous and 𝜎([𝑡0, 𝑡1]𝕋), 𝜎((𝑠𝑗 , 𝑡𝑗+1]𝕋) are 

density argument [𝜎(𝑡0), 𝜎(𝑡1)]𝕋 = [𝑡0, 𝑡1]𝕋, (𝜎(𝑠𝑗), 𝜎(𝑡𝑗+1)]𝕋 = (𝑠𝑗, 𝑡𝑗+1]𝕋, Hence, from the above 

equations (3.25) and (3.26), we have  

𝑧𝛼
∗𝑒𝑃(𝑡0, 𝑡)𝑄 = 0, ∀ 𝑡 ∈ [𝑡0, 𝑡1]𝕋. (3.27) 

𝑧𝛼
∗𝑒𝑃(𝑠𝑗, 𝑡)𝑄 = 0, ∀ 𝑡 ∈ (𝑠𝑗 , 𝑡𝑗+1]𝕋, 𝑗 = 1,2, … ,𝑚,      (3.28) 

At 𝑡 = 𝑡0, an equation (3.27) becomes 𝑧𝛼
∗𝑄 = 0. Also, 𝑒𝑃(𝑡0, . ) is delta differentiable, we get 

𝑒𝑃
∆𝑡(𝑡0, 𝑡) = −𝑒𝑃(𝑡0, 𝜎(𝑡))𝑃. 

Then subsequent derivatives and the density equations of (3.27) give 

(−1)𝑖𝑧𝛼
∗𝑒𝑃(𝑡0, 𝑡)𝑃

𝑖−1𝑄 = 0, 𝑖 = 0,1,2, … , 𝑛2 − 1, 𝑡 ∈ [𝑡0, 𝑡1]𝕋. (3.29) 

Put 𝑡 = 𝑡0 in the above equation (3.29), we have 



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𝑧𝛼
∗𝑃𝑖−1𝑄 = 0, 𝑖 = 0,1,2, … , 𝑛2 − 1. 

Therefore,  𝑧𝛼
∗ [𝑄  𝑃𝑄  𝑃2𝑄… 𝑃𝑛

2−1𝑄] = 0, hence, our assumption is wrong: Therefore, it is a 

contradictory that the rank of  𝐶 = 𝑛2.Similarly, we iterate the procedure on equation (3.28), yielding  

𝑧𝛼𝑘
∗ [𝑄  𝑃𝑄  𝑃2𝑄… 𝑃𝑛

2−1𝑄] = 0 

Once again, the contradiction demonstrate that system (3.1) is completely controllable throughout the 

time interval [𝑡0, 𝑇]𝕋 . 

Example 3.1: The following non-linear Kreneker product of Volterra integro-dynamic with an impulse 

control system  

 

{
 
 

 
 𝑧∆(𝑡) = 𝑃(𝑡)𝑧(𝑡) + ∫ 𝐾(𝑡, 𝑠)𝑧(𝑠)∆𝑠

𝑡

0

+ 𝑄(𝑡)�̂�(𝑡) + 𝑓(𝑡, 𝑧(𝑡)),       𝑡 ∈ (𝑠𝑗 , 𝑡𝑗+1]𝕋, 𝑗 = 0,1, 2,

𝑧(𝑡) = [𝐼𝑛⊗𝑅𝑘]𝑧(𝑡𝑗
−),                                                    𝑡 ∈ (𝑠𝑗, 𝑡𝑗+1]𝕋,   𝑗 = 1,2, …                            

𝑧(𝑡0) = 𝑧0,   𝑧0 ∈ ℝ2                                                                                                                                             

(3.30) 

Where 𝑧(𝑡) = [

𝑧11(𝑡)
𝑧12(𝑡)
𝑧21(𝑡)
𝑧22(𝑡)

] , 𝑡0 = 𝑠0 = 0, 𝑡1 = 0.8, 𝑠1 = 0.9, 𝑡2 = 2.1, 𝑠2 = 2.2, 𝑡3 = 𝑇 = 3, 𝑃(𝑡) = [𝐵∗⊗

In + In ⊗𝐴] = [

−2 0 0 0
0 −2 0 0
0

0

0

0

−3
0

0

−3

], 

𝐾(𝑡, 𝑠) = [𝐾2
∗⊗ 𝐼𝑛) + (𝐼𝑛⊗𝐾1)  = [

𝑠𝑖𝑛𝑡 0 0 0
0 𝑐𝑜𝑠𝑡 0 0
0

0

0

0

𝑠𝑖𝑛𝑡
0

0

𝑐𝑜𝑠𝑡

], 

 𝑄(𝑡) = [In ⊗𝐶] =

[
 
 
 
 
 

1 0
2

25
𝑒1(𝜎(𝑡), 0) 0

0

0

1
2

25
𝑒1(𝜎(𝑡), 0)]

 
 
 
 
 

 , 𝑓(𝑡, 𝑧(𝑡) =
1

35

[
 
 
 
 
 
sin(𝑧22(𝑡))

𝑒𝑡
2+2

0
0

cos(𝑧11(𝑡))

𝑒𝑡
2+2 ]

 
 
 
 
 

,  

[𝐼𝑛⊗𝑅𝑗]𝑧(𝑡𝑗
−) =

1

20

[
 
 
 
 
 
 
 
 
 

𝑧2(𝑡𝑘
−)

𝑒𝑡
2+2(1+ 𝑖𝑡)

0

𝑧1(𝑡𝑘
−)

𝑒𝑡
2+3(1+ 𝑖𝑡2)

0

0

0

𝑧2(𝑡𝑘
−)

𝑒𝑡
2+2(1+ 𝑖𝑡)

𝑧1(𝑡𝑘
−)

𝑒𝑡
2+3(1+ 𝑖𝑡2)]

 
 
 
 
 
 
 
 
 

, 𝑗 = 1,2. 



Communications on Applied Nonlinear Analysis 

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The matrix that provides the fundamental solution to the system (3.30) is 

𝑒𝑃(𝑡, 0) =

[
 
 
 
𝑒−2(𝑡, 0) 0 0 0

0 𝑒−2(𝑡, 0) 0 0

0

0

0

0
𝑒−3(𝑡, 0)

0

0

𝑒−3(𝑡, 0)]
 
 
 
. 

Therefore 𝑒𝑃(0, 𝑡) =

[
 
 
 
𝑒2(0, 𝑡) 0 0 0

0 𝑒2(0, 𝑡) 0 0

0

0

0

0
𝑒3(0, 𝑡)

0

0

𝑒3(0, 𝑡)]
 
 
 
. Also we can easily compute  

𝒩(0, σ(τ)) = Ψ(0, σ(τ))𝑄(τ)𝑄∗(τ)Ψ∗(0,σ(τ)) 

=

[
 
 
 
 
 
 
 (𝑒−2(0,σ(τ)) +

8

625
)
2

0 0 0

0 (𝑒−2(0,σ(τ)) +
8

625
)
2

0 0

0

0

0

0
(𝑒−3(0, σ(τ)) +

8

625
)
2

0

0

(𝑒−3(0,σ(τ)) +
8

625
)
2

]
 
 
 
 
 
 
 

. 

𝒩(𝑠1,σ(τ)) = Ψ(𝑠1, σ(τ))𝑄(τ)𝑄
∗(τ)Ψ∗(𝑠1,σ(τ)) 

=

[
 
 
 
 
 
 
 (𝑒−2(𝑠1,σ(τ)) +

8

625
)
2

0 0 0

0 (𝑒−2(𝑠1,σ(τ)) +
8

625
)
2

0 0

0

0

0

0
(𝑒−3(𝑠1, σ(τ)) +

8

625
)
2

0

0

(𝑒−3(𝑠1, σ(τ)) +
8

625
)
2

]
 
 
 
 
 
 
 

. 

𝒩(𝑠2,σ(τ)) = Ψ(𝑠2, σ(τ))𝑄(τ)𝑄
∗(τ)Ψ∗(𝑠2,σ(τ)) 

=

[
 
 
 
 
 
 
 (𝑒−2(𝑠2,σ(τ)) +

8

625
)
2

0 0 0

0 (𝑒−2(𝑠2,σ(τ)) +
8

625
)
2

0 0

0

0

0

0
(𝑒−3(𝑠2, σ(τ)) +

8

625
)
2

0

0

(𝑒−3(𝑠2, σ(τ)) +
8

625
)
2

]
 
 
 
 
 
 
 

. 

Now consider the following two cases: 

Case (1): If 𝕋 = ℝ, then 𝑒𝑎(𝑡, 0) = 𝑒𝑎𝑡. Therefore,  

𝒩0(0, 𝑡1) = ∫𝒩(0, σ(τ))dτ 
t

0

= [

7.2665 0 0 0
0 7.2665 0 0
0

0

0

0

24.156
0

0

24.156

]. 



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𝒩1(𝑠1, 𝑡2) = ∫ 𝒩(𝑠1,σ(τ))dτ 
𝑡2

𝑠1

= [

16.513 0 0 0
0 16.513 0 0
0

0

0

0

74.52
0

0

74.52

]. 

𝒩2(𝑠2, 𝑇) = ∫ 𝒩(𝑠2,σ(τ))dτ 
𝑇

𝑠2

= [

53.762 0 0 0
0 53.762 0 0
0

0

0

0

137.24
0

0

137.24

]. 

It follows that the matrices 𝒩0(0, 𝑡1), 𝒩1(𝑠1, 𝑡2), and 𝒩2(𝑠2, 𝑇) are all invertible.  In addition, all three 

assumptions (H1) − (H3) hold. with M𝛼 = max {0.9104, 0.2328, 0.7531, 0.01718} <1. The system 

(3.30) is complete controllable, since all the criteria of Theorem 3.2, are satisfied. 

Case (2): If 𝕋 = ℙ1,1 =∪𝑗=0
∞ [2𝑗. 2𝑗 + 1], then 𝑒𝑎(𝑡, 0) = (1+ 𝑎)𝑗𝑒𝑎(𝑡−𝑗). Therefore,  

𝒩0(0, 𝑡1) = ∫𝒩(0, σ(τ))dτ 
t

0

= [

7.2665 0 0 0
0 7.2665 0 0
0

0

0

0

24.156
0

0

24.156

]. 

𝒩1(𝑠1, 𝑡2) = ∫ 𝒩(𝑠1,σ(τ))dτ 
𝑡2

𝑠1

= [

6.781 0 0 0
0 6.781 0 0
0

0

0

0

15.167
0

0

15.167

]. 

𝒩2(𝑠2, 𝑇) = ∫ 𝒩(𝑠2,σ(τ))dτ 
𝑇

𝑠2

= [

73.2665 0 0 0
0 73.2665 0 0
0

0

0

0

827.24
0

0

827.24

]. 

It follows that the matrices 𝒩0(0, 𝑡1), 𝒩1(𝑠1, 𝑡2), and 𝒩2(𝑠2, 𝑇) are all invertible.  In addition, all three 

assumptions (H1) − (H3) hold. with M𝛼 = max {0.9480, 0.2107, 0.9513, 0.00367} <1. The system 

(3.30) is complete controllable, since all the criteria of Theorem 3.1, are satisfied. 

Declarations 

Conflict of interest: The authors declare that they have no competing interests. 

Author Contributions: All authors contributed equally to this article. All authors read and approved 

the final manuscript. 

Data availability: Not applicable. 

Funding: Not applicable. 

References: 

[1] Agarwal, R. P., Bohner, M., Regan, D.O., and Peterson, A.: Dynamic equations on Time      

[2]       scales. A survey, J comput. Appl. Math, No.4, 1-26, (2002). 

[3] Alexander, G.: Kronecker products and matrix calculus; with applications, Ellis Hordwood   Ltd., England, (1981). 

[4] Appa Rao, B. V., and Prasad, KASNV.: Study of controllability of matrix integrodifferential equations on time scales. 

International Journal of Chemical Sciences, 13 (3), pp. 1324 – 1332, (2015). 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 2s (2025) 

 

365 https://internationalpubls.com 

[5] Appa Rao, B. V., and Prasad, KASNV.: Controllability and Observability of Sylvester Matrix Dynamical Systems on 

Time Scales Kyungpook Math.J. 56, 529-539, (2016). 

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https://doi.org/10.1016/j.nahs.2020.100986
https://doi.org/10.1186/s13662-021-03665-6
https://doi.org/10.1007/s12190-021-01688-6
https://doi.org/10.1007/s12190-021-01688-6