Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

 

493 
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Controllability and Observability of Matrix Sylvester Adjoint Dynamic 

Impulsive Systems on Time Scales  
 

 A. Sreenivasulu1*, B. V. Appa Rao1, P. Lakshmi Pallavi2, Gudala Balaji Prakash3,  

M. Srinivasa Reddy4, J Peter Praveen5 and D. Ramesh1  
1Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation, Green fields, Vaddeswaram, 

Guntur-522302, Andhra Pradesh, India. 
2Department of Mathematics, B V Raju Institute of Technology, Narsapur, 502313 Telangana, India. 

3Department of Mathematics, Aditya University, Surampalem, 533437 Andhra Pradesh, India. 
4Freshman Engineering department, Lakireddy Bali Reddy College of Engineering, Mylavaram-521230, NTR-district, 

India. 
5Vignan Institute of Information Technology, Duvvada, Visakhapatnam,530049, Inida. 

Corresponding author Email: asreenivasulu@kluniversity.in; 

Author: bvardr2010@kluniversity.in; lakshmipallavi.p@bvrit.ac.in; balajiprakashgudala@gmail.com; 

maths4444@gmail.com ; jppraveen17@gmail.com and ram.fuzzy@gmail.com 

 

Article History: 

Received: 12-11-2024 

Revised: 17-12-2024 

Accepted: 06-01-2025 

Abstract:  

This paper investigates the controllability and observability of matrix Sylvester adjoint 

dynamic impulsive systems within the framework of time scales. By applying the 

vectorization operator, the system is reformulated into an equivalent Kronecker product-

based dynamic impulsive system, enabling more efficient analysis. The study derives 

necessary and sufficient conditions for controllability and observability through the adjoint 

matrix approach, highlighting its analytical significance. Additionally, the research 

incorporates the Gramian matrix to establish results for impulsive dynamic systems on time 

scales, providing a comprehensive criterion for evaluating system properties. This unified 

framework bridges discrete and continuous-time dynamics, enabling the modeling and 

analysis of hybrid systems with abrupt state transitions. The findings advance the 

mathematical understanding of Sylvester matrix systems and offer practical insights into 

the design and control of complex impulsive systems across varied applications. This work 

contributes to the growing field of time-scale calculus and its applications in dynamic 

system analysis. 

Keywords: Controllability, observability, Kronecker product, Time scales. 

Mathematics Subject Classification: 93B05, 93B07,39A12, 18A40, 34N05.   

1. Introduction 

The study of adjoint matrix Sylvester dynamic impulsive systems on time scales combines the 

strengths of continuous and discrete systems, offering a unified framework to analyze complex real-

world phenomena. This approach effectively models systems that exhibit both abrupt changes and 

continuous evolution, making it particularly relevant in fields such as engineering, biology, and 

economics. The use of adjoint matrix techniques brings significant advantages, such as simplifying the 

process of solving linear dynamic equations, reducing computational complexity, and enhancing 

analytical clarity. Matrices also provide a compact representation of system dynamics, facilitating 

efficient manipulation, scalability, and the application of powerful algebraic methods. By 

incorporating the time scales framework, this method seamlessly integrates hybrid systems, enabling 

comprehensive analyses of controllability, stability, and optimization under impulsive effects. 

Consequently, this framework bridges the gap between discrete and continuous dynamics, advancing 

both theoretical mathematics and practical applications. 

mailto:asreenivasulu@kluniversity.in;
mailto:bvardr2010@kluniversity.in
mailto:lakshmipallavi.p@bvrit.ac.in
mailto:balajiprakashgudala@gmail.com
mailto:maths4444@gmail.com
mailto:jppraveen17@gmail.com


Communications on Applied Nonlinear Analysis 

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Impulsive differential equations play a crucial role in describing systems with sudden changes in state, 

governed by continuous dynamics interspersed with jump criteria. These equations provide a logical 

and robust framework for modeling abrupt transitions commonly observed in real-world processes. 

Their versatility has led to extensive research and development in this area. For instance, in [4], 

solutions to fractional equations such as the Fitzhugh-Nagumo equation, the Newell-Whitehead-Segel 

equation, and the Zeldovich equation were explored, demonstrating their practical significance. 

Similarly, [5] addressed an inverse coefficient problem for the conformable time-diffusion equation, 

retrieving time-dependent diffusion coefficients with precision. Furthermore, [6] proposed a novel 

mathematical formation model using the fractional Atangana-Baleanu-Caputo derivative, showcasing 

the reliability and computational efficiency of this method. Applications of impulsive systems extend 

to biological models, as demonstrated in [8], where insect population dynamics were analyzed using 

exponential, hyperbolic, and trigonometric functions to solve second-order linear dynamic equations. 

Additionally, [9] discussed stability conditions that ensure input-to-state stability for hybrid systems, 

even under instability. The concept of controllability further enhances system stabilization by 

constraining behavior through the analysis of linear and nonlinear operators [10].  

Matrix-based approaches further bolster the study of impulsive systems. They allow for efficient 

representation of multi-dimensional systems and enable the use of spectral analysis, eigenvalue 

computation, and matrix decompositions to study system properties like stability and controllability. 

The study of impulsive systems on time scales has also proven the existence and uniqueness of 

solutions for nonlinear impulsive dynamic equations [11]. In [13], the properties of impulsive Dirac 

systems on Sturmian time scales were examined, including the construction of self-adjoint operators. 

Additionally, [14] introduced a new transition matrix to analyze the controllability and observability 

of impulsive systems on time scales. This research highlights the flexibility and adaptability of the 

impulsive framework, enabling its application to nonuniform time domains [15].  

In this paper, we address the sufficient and necessary controllability and observability conditions for 

matrix Sylvester adjoint dynamic impulsive systems over various time scales. 

{
 
 

 
 
𝑋𝛥(𝑡) = 𝑃(𝑡)𝑋(𝑡) + 𝑋(𝑡)𝑄(𝑡) + 𝜇(𝑡)𝑃(𝑡)𝑋(𝑡)𝑄(𝑡) + 𝑇1(𝑡)𝑈(𝑡)𝑇2

∗(𝑡)

𝑋(𝑡𝑘
+) = (𝐼 + 𝐿𝑘)𝑋(𝑡𝑘),    𝑡 = 𝑡𝑘  k = 1,2,3....

𝑌(𝑡) = 𝐶(𝑡)𝑋(𝑡) + 𝐷(𝑡)𝑈(𝑡)                                                                           
𝑋(𝑡0) = 𝑋0.

 (1.1) 

 Where 𝑋(𝑡) is an  𝑛 × 𝑛  matrix, 𝑈(𝑡) is m× 𝑛 input pricewise rd-continuous matrix called control 

input and 𝑌(𝑡) is  𝑝 × 𝑛 output rd-continuous. Here 𝑃 (𝑡), 𝑄(𝑡),  𝑇1(𝑡), 𝑇2(𝑡) and 𝐿𝑘 are  

𝑛 × 𝑛, 𝑛 × 𝑛, 𝑛 × 𝑛, 𝑛 × 𝑚 and 𝑛 × 𝑛  rd-continuous matrices respectively. C(t), D(t) are rd-

continuous matrices of order 𝑝 × 𝑛 and 𝑝 × 𝑚 respectively. 𝑋∆(𝑡) is the generalized Delta derivative 

of X, and t is from a time scales 𝕋, which is a non-empty closed subset of ℝ and 𝜇 is a graininess 

function. when 𝑄 = 𝑃∗(* denotes the transpose of matrix) equation (1.1) is called matrix Lyapunov 

dynamical system on time scale.  

2. Preliminaries 

         We provide some preliminary information to help you understand the notation used in this paper. 

A summary of the time scales can be found in [6, 7]. A nonempty closed subset of the R real line is 

called a time scale 𝕋. We usually write 𝕋𝑘 = 𝕋{𝑚𝑎𝑥𝕋} if 𝑚𝑎𝑥𝕋 < ∞, otherwise 𝕋𝑘 = 𝕋. 

Definition 2.1[6] Let 𝑓: 𝕋 → ℝ and 𝑡 ∈ 𝕋𝑘 the delta derivative of 𝑓𝛥(𝑡) is the number (when it exists), 

with the property that, for any 𝜀 > 0, there is a neighbourhood 𝑈 of  𝜏 such that 



Communications on Applied Nonlinear Analysis 

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         |[𝑓(𝜎(𝜏)) − 𝑓(𝑠)] − 𝑓𝛥(𝜏)[𝜎(𝜏) − 𝑠]| ≤ 𝜀|𝜎(𝜏) − 𝑠|, for all 𝑠 ∈ 𝑈 

Definition 2.2.[7]: The regressive function y(t) mapping from 𝕋 to ℝ is defined as1+ 𝜇(𝑡)𝑦(𝑡) ≠
0 ∀ 𝑡 ∈ 𝕋. The combination of all regressive and right dense continuous function is represented as ℛ =
ℛ(𝑡) = ℛ(𝕋,ℝ). Similarly all positively regressive function is denoted by  ℛ+ = ℛ+(𝕋,ℝ) = {𝑦 ∈
ℛ: 1 + 𝜇(𝑡)𝑦(𝑡) > 0, ∀ 𝑡 ∈ 𝕋} 

Definition 2.3[6] If 𝐹: 𝕋𝑘 → ℝ is said to be anti-derivative of   𝑓: 𝕋𝑘 → ℝ provided  𝐹𝛥(𝑡) = 𝑓(𝑡) 
fulfilled, for all  𝑡 ∈ 𝕋𝑘, then  

                ∫ 𝑓(𝑠)𝛥𝑠 = 𝐹(𝑡) − 𝐹(𝑎)
𝑡

𝑎
 

Definition 2.4 Let the matrices are 𝐴 ∈ 𝐶𝑚×𝑛(ℝ𝑚×𝑛) and 𝐵 ∈ 𝐶𝑝×𝑞(ℝ𝑝×𝑞) the the Kronecker product 

of A and B. we have defined to be the partitioned matrix written (𝐴 ⊗ 𝐵) is 

𝐴⊗ 𝐵 = [

𝑎11𝐵 𝑎12𝐵 ⋯ 𝑎1𝑛𝐵
𝑎21𝐵 𝑎22𝐵 ⋯ 𝑎2𝑛2𝐵
⋯ ⋯ ⋯ ⋯

𝑎11𝐵 𝑎12𝐵    ⋯     𝑎1𝑛𝐵

] 

is an 𝑚𝑝 × 𝑛𝑞 matrix is in  𝐶𝑚×𝑛(ℝ𝑚×𝑛). 

Definition 2.5. Let 𝐴 = [𝑎𝑖𝑗] ∈ ℝ𝑚×𝑛, we denote 

   �̂� = 𝑉𝑒𝑐𝐴 = [

𝐴1

𝐴2

⋮
𝐴𝑛

], where 𝐴.𝑗 = [

𝑎1𝑗
𝑎2𝑗
⋮
𝑎𝑚𝑗

] (1 ≤ 𝑗 ≤ 𝑛) 

        Here we converted the linear matrix Sylvester dynamic impulsive system on time scales to an 

equivalent KP dynamic impulsive system on time scales using vectorization operator. The dynamical 

system is  

{
 
 

 
 𝑧

𝛥(𝑡) = 𝐺(𝑡)𝑧(𝑡) + 𝐴(𝑡)𝑈(𝑡),  t ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋
𝑧(𝑡𝑘

+) = [𝐼𝑛⊗𝑅𝑘]𝑧(𝑡𝑘), 𝑡 = 𝑡𝑘,  k = 1,2,3…

�̂�(𝑡) = (I⊗𝐶)(𝑡)𝑧(𝑡) + (𝐼 ⊗ 𝐷)�̂�(𝑡)         
𝑧(𝑡0) = 𝑧0.

 (2.1) 

Where z(t) = Vec X(t), �̂�(𝑡) = Vec U(t), �̂�(𝑡) = Vec Y(t), 𝑅𝑘 = (𝐼𝑛 + 𝐿𝑘)  and G(t)= [𝑄∗⊗ 𝐼 + 𝐼 ⊗
𝑃 + 𝜇(𝑡)(𝑄∗⊗𝑃)] A(t)= [𝑇2

∗⊗𝑇1].  

 Now rearranged the linear adjoint dynamic system (2.1) as follows 

{
 
 

 
 𝑧

𝛥(𝑡) = −𝐺𝑘
𝑇(𝑡)𝑧𝜎(𝑡) + 𝐴𝑘(𝑡)�̂�(𝑡),  t ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋

𝑧(𝑡𝑘
+) = [𝐼𝑛⊗𝑅𝑘]𝑧(𝑡𝑘), 𝑡 = 𝑡𝑘,  k = 1,2,3…

�̂�(𝑡) = (I⊗𝐶𝑘)(𝑡)𝑧
𝜎(𝑡) + (𝐼 ⊗ 𝐷𝑘)�̂�(𝑡)                   

𝑧(𝑡0) = 𝑧0.

 (2.2) 

Remark 2.1.[3]. Clearly observe that, the matrix valued function X(t) is a solution (2.1) on 𝕋 if and 

only if the vector valued function 𝑧(𝑡) = 𝑉𝑒𝑐𝑋(𝑡) is a solution of the system (2.1) on 𝕋.  

Theorem 2.1.[7]. If 𝐺 ∈ 𝐶𝑟𝑑ℛ (𝕋+, 𝑀𝑛2×𝑛2(ℝ)) and 𝑙 ∈ 𝐶𝑟𝑑 (𝕋+, 𝑀𝑛2×1(ℝ)), then for each (𝜏, 𝜂) ∈

𝕋+ × ℝ𝑛
2

 the initial value problem 



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    𝑧𝛥(𝑡) = 𝐺(𝑡)𝑧(𝑡) + 𝑙(𝑡),  z(τ) = 𝜂,  

has a unique solution 𝑧: 𝕋(𝜏) → ℝ𝑛
2

. 

Lemma 2.1.[7]. If 𝐺 ∈ 𝐶𝑟𝑑ℛ (𝕋+, 𝑀𝑛2(ℝ)) and 𝑙 ∈ 𝐶𝑟𝑑 (𝕋+, 𝑀𝑛2×1(ℝ)), then for each (𝜏, 𝜂) ∈

𝕋+ × ℝ𝑛
2

 the initial value problem 

                              𝑧𝛥(𝑡) = 𝐺(𝑡)𝑧(𝑡) + 𝑙(𝑡),  z(τ) = 𝜂, 

has one and only one solution 𝑧: 𝕋(𝜏) → ℝ𝑛
2

 is given by 

       𝑧(𝑡) = 𝜓𝐺(𝑡, 𝜏)𝜂 + ∫ 𝜓𝐺(𝑡, 𝜎(𝑠))𝑙(𝑠)𝛥𝑠
𝑡

𝜏
, 𝑡 ≥ 𝜏. 

Lemma 2.2.[7]. If 𝐺 ∈ 𝐶𝑟𝑑ℛ (𝕋+, 𝑀𝑛2(ℝ)) and 𝑙 ∈ 𝐶𝑟𝑑 (𝕋+, 𝑀𝑛2×1(ℝ)), then for each (𝜏, 𝜂) ∈

𝕋+ × ℝ𝑛
2

 the initial value problem 

                              𝑧𝛥(𝑡) = −𝐺𝑇(𝑡)𝑧𝜎(𝑡) + 𝑙(𝑡),  z(τ) = 𝜂, 

has one and only one solution  𝑧: 𝕋(𝜏) → ℝ𝑛
2

 is given by 

      𝑧(𝑡) = 𝜓⊝𝐺𝑇(𝑡, 𝜏)𝜂 + ∫ 𝜓⊝𝐺𝑇(𝑡, 𝜎(𝑠))𝑙(𝑠)𝛥𝑠
𝑡

𝑡0
, 𝑡 ∈ 𝕋(𝜏).       

Proposition 2.1. [8] The system (2.2) with  𝐺𝑘 ∈ 𝑀𝑛2(𝑅) constant, there exist scalar functions 

𝜒0(𝑡, 𝜏), 𝜒1(𝑡, 𝜏), . . . . . 𝜒𝑛2−1(𝑡, 𝜏) ∈ 𝐶𝑟𝑑
∞ (𝕋+,ℝ) such that the one and only one solution is given by 

                                             𝑒𝐺𝑘
𝑇(𝑡, 𝜏) = ∑ 𝜒𝑖

𝑛2−1
𝑖=0 (𝑡, 𝜏)𝐺𝑖.                                                        

3. Complete Controllability 

In this section, we present the controllability in time variant and time invariant adjoint dynamic system 

(3) on time scales.  

Lemma 3.1. for any ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋, 𝑙 = 1,2, . . . , 𝑘 the solution of initial value problem (2.2) is given by 

𝑧(𝑡) =

{
 
 
 
 
 
 
 

 
 
 
 
 
 
 𝜓𝐺1

𝑇 (𝑡0, 𝑡)𝑧0 +∫ 𝜓𝐺1
𝑇 (𝜏, 𝑡)𝐴1(𝜏)�̂�(𝜏)∆𝜏, 𝑙 = 1

𝑡

𝑡0

𝜓𝐺𝑙
𝑇 (𝑡𝑙−1, 𝑡){∏ [𝐼𝑛⊗𝑅𝑗] ∏ 𝜓𝐺𝑗

𝑇 (𝑡𝑗−1, 𝑡𝑗)𝑧0 +

1

𝑗=𝑙−1

1

𝑗=𝑙−1

∑(∏[𝐼𝑛⊗𝑅𝑗] ∏ 𝜓𝐺𝑖
𝑇 (𝑡𝑖−1, 𝑡𝑖)∫ 𝜓𝐺𝑗

𝑇 (𝜏, 𝑡𝑗)𝐴𝑗(𝜏)�̂�(𝜏)∆𝜏 
𝑡𝑗

𝑡𝑗−1

𝑗+1

𝑖=𝑙−1

𝑗

𝑖=𝑙−1

)

𝑙−2

𝑗=1

+[𝐼𝑛⊗𝑅𝑙−1]∫ 𝜓𝐺𝑙−1
𝑇 (𝜏, 𝑡𝑙−1)𝐴𝑙−1(𝜏)�̂�(𝜏)∆𝜏 

𝑡𝑙−1

𝑡𝑙−2

}

+∫ 𝜓𝐺𝑙
𝑇 (𝜏, 𝑡)𝐴𝑙(𝜏)�̂�(𝜏)∆𝜏 , 𝑙 = 2,3, ⋯ , 𝑘.

𝑡

𝑡𝑙−1

 (3.1) 

  Proof: Form lemma 2.2 for 𝑡 ∈ [𝑡0, 𝑡1]𝕋, we have 

𝑧(𝑡) = 𝜓𝐺1
𝑇 (𝑡0, 𝑡)𝑧0 + ∫ 𝜓𝐺1

𝑇 (𝜏, 𝑡)𝐴1(𝜏)�̂�(𝜏)∆𝜏
𝑡

𝑡0

, 



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similarly, for 𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋, we have  

𝑧(𝑡) = 𝜓𝐺𝑙
𝑇 (𝑡𝑙−1, 𝑡){𝑧(𝑡𝑙)} + ∫ 𝜓𝐺𝑙

𝑇 (𝜏, 𝑡)𝐴𝑙(𝜏)�̂�(𝜏)∆𝜏 ,
𝑡

𝑡𝑙−1

 

Also, form the system (2.2), we have  

𝑧(𝑡𝑙) = ∏ [𝐼𝑛⊗𝑅𝑗] ∏ 𝜓𝐺𝑗
𝑇 (𝑡𝑗−1, 𝑡𝑗)𝑧0

1

𝑗=𝑙−1

1

𝑗=𝑙−1

+∑( ∏ [𝐼𝐼⊗𝐼𝐼] ∏ 𝐼𝐼𝐼

𝐼 (𝐼𝐼−1,𝐼𝐼)∫ 𝐼𝐼𝐼

𝐼 (𝐼,𝐼𝐼)𝐼𝐼(𝐼)�̂�(𝐼)∆𝐼 
𝐼𝐼

𝐼𝐼−1

𝐼+1

𝐼=𝐼−1

𝐼

𝐼=𝐼−1

)

𝐼−2

𝐼=1

+ [𝐼𝑛⊗𝑅𝑙−1]∫ 𝜓𝐺𝑙−1
𝑇 (𝜏, 𝑡𝑙−1)𝐴𝑙−1(𝜏)�̂�(𝜏)∆𝜏

𝑡𝑙−1

𝑡𝑙−2

, 𝑙 = 2,3, ⋯ , 𝑘, 

therefore for 𝑡 ∈ (𝑡𝑙−1, 𝑡𝑙)𝕋, we have 

𝑧(𝑡) =  𝜓𝐺𝑙
𝑇 (𝑡𝑙−1, 𝑡) {∏ [𝐼𝑛⊗𝑅𝑗] ∏ 𝜓𝐺𝑗

𝑇 (𝑡𝑗−1, 𝑡𝑗)𝑧0

1

𝑗=𝑙−1

1

𝑗=𝑙−1

+∑( ∏ [𝐼𝐼⊗𝐼𝐼] ∏ 𝐼𝐼𝐼

𝐼 (𝐼𝐼−1,𝐼𝐼)∫ 𝐼𝐼𝐼

𝐼 (𝐼,𝐼𝐼)𝐼𝐼(𝐼)�̂�(𝐼)∆𝐼 
𝐼𝐼

𝐼𝐼−1

𝐼+1

𝐼=𝐼−1

𝐼

𝐼=𝐼−1

)

𝐼−2

𝐼=1

+ [𝐼𝑛⊗𝑅𝑙−1]∫ 𝜓𝐺𝑙−1
𝑇 (𝜏, 𝑡𝑙−1)𝐴𝑙−1(𝜏)�̂�(𝜏)∆𝜏 

𝑡𝑙−1

𝑡𝑙−2

} +∫ 𝜓𝐺𝑙
𝑇 (𝜏, 𝑡)𝐴𝑙(𝜏)�̂�(𝜏)∆𝜏.

𝑡

𝑡𝑙−1

 

After repeating the above same process, we get the desire results. 

Theorem 3.1.  

i. If there exist at least k∈ {1,2, … , 𝑙} such that 

     𝑟𝑎𝑛𝑘 {𝐻𝑘(𝑡𝑘−1, 𝑡𝑘, 𝑡𝑓)} = 𝑛
2 

then the impulsive system (2.2) is controllable on [𝑡0, 𝑡1]𝕋(𝑡𝑓 ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋). 

ii. Suppose that (𝐼𝑛⊗𝑅𝑗) ≠ −1, 𝑗 = 1,2, … , 𝑘. If impulsive system (2.2) is controllable on 

[𝑡0, 𝑡1]𝕋(𝑡𝑓 ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋), then        

 𝑟𝑎𝑛𝑘 {𝐻1, … , 𝐻𝑙} = 𝑛2. 

Proof: (i). Let k∈ {1,2, … , 𝑙} such that the 𝑟𝑎𝑛𝑘 {𝐻𝑘(𝑡𝑘−1, 𝑡𝑘 , 𝑡𝑓)} = 𝑛2 i.e., the matrix  

𝐻𝑘(𝑡𝑘−1, 𝑡𝑘 , 𝑡𝑓) is invertible then for a given  𝑧0 ∈ ℝ𝑛
2

, we choose a control function given as   



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�̂�(𝑡)

=

{
 
 
 
 
 

 
 
 
 
 

−𝐴1
𝑇(𝑡)𝜓𝐺1(𝑡, 𝑡𝑓)𝐻1

−1𝜓𝐺1
𝑇 (𝑡0, 𝑡𝑓)𝑧0,    𝑓𝑜𝑟 𝑡 ∈ [𝑡0, 𝑡1]𝕋, 2 ≤ 𝑘 ≤ 𝑙 − 1,

−𝐴𝑘
𝑇(𝑡)𝜓𝐺𝑘(𝑡, 𝑡𝑓)𝐻𝑘

−1𝜓𝐺𝑘
𝑇 (𝑡𝑘−1, 𝑡𝑓) ∏ [𝐼𝑛⊗𝑅𝑗]

1

𝑗=𝑘−1

∏ 𝜓𝐺𝑘
𝑇 (𝑡𝑗−1, 𝑡𝑗)

1

𝑗=𝑘−1

𝑧0, 𝑓𝑜𝑟 𝑡 ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋  

0        𝑖𝑓 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋\
[𝑡𝑘−1, 𝑡𝑘)𝕋,

−𝐴𝑙
𝑇(𝑡)𝜓𝐺𝑙(𝑡, 𝑡𝑓)𝐻𝑙

−1𝜓𝐺𝑙
𝑇 (𝑡𝑙−1, 𝑡𝑓) ∏ [𝐼𝑛⊗𝑅𝑗]

1

𝑗=𝑙−1

∏ 𝜓𝐺𝑗
𝑇 (𝑡𝑗−1, 𝑡𝑗)

1

𝑗=𝑙−1

𝑧0, 𝑓𝑜𝑟 𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋

0        𝑖𝑓 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋\
[𝑡𝑙−1, 𝑡𝑙)𝕋,

 

(3.5

) 

obviously, the control function �̂�(𝑡) is a piecewise rd-continuous on [𝑡0, 𝑡1]𝕋. By Lemma 3.1. we 

obtain  

𝑧(𝑡𝑓) = 𝜓𝐺1
𝑇 (𝑡0, 𝑡𝑓)𝑧0 − ∫ 𝜓𝐺1

𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡0

𝐴1(𝜏)𝐴1
𝑇(𝜏)𝜓𝐺1(𝜏, 𝑡𝑓)𝐻1

−1𝜓𝐺1
𝑇 (𝑡0, 𝑡𝑓)𝑧0∆𝜏, 

by using equation (3.2), we have   

𝑧(𝑡𝑓) = 𝜓𝐺1
𝑇 (𝑡0, 𝑡𝑓)𝑧0 −𝐻1𝐻1

−1𝜓𝐺1
𝑇 (𝑡0, 𝑡𝑓)𝑧0 = 0 𝑓𝑜𝑟 𝑡 ∈ [𝑡0, 𝑡1]𝕋, 

then the system (2.2) is a controllable on [𝑡0, 𝑡1]𝕋. 

Next, for 2 ≤ 𝑘 ≤ 𝑙 − 1, and 𝑡 ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋, 

𝑧(𝑡𝑓) = 𝜓𝐺𝑘
𝑇 (𝑡𝑘−1, 𝑡𝑓) ∏ [𝐼𝑛⊗𝑅𝑗]

1

𝑗=𝑘−1

∏ 𝜓𝐺𝑗
𝑇 (𝑡𝑗−1, 𝑡𝑗)

1

𝑗=𝑘−1

𝑧0

− ∫ 𝜓𝐺𝑘
𝑇 (𝜏, 𝑡𝑓)

𝑡

𝑡𝑘−1

𝐴𝑘(𝜏)𝐴𝑘
𝑇(𝜏)𝜓𝐺𝑘(𝜏, 𝑡𝑓)𝐻𝑘

−1𝜓𝐺𝑘
𝑇 (𝑡𝑘−1, 𝑡𝑓) ∏ [𝐼𝑛

1

𝑗=𝑘−1

⊗𝑅𝑗] ∏ 𝜓𝐺𝑗
𝑇 (𝑡𝑗−1, 𝑡𝑗)

1

𝑗=𝑘−1

𝑧0∆𝜏, 

it follows that from equation (3.3) 

𝑧(𝑡𝑓) = 0 𝑓𝑜𝑟 𝑡 ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋, 

and similarly, we have 

𝑧(𝑡𝑓) = 0 𝑓𝑜𝑟 𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋, 

thus, the system (2.2) is a controllable on [𝑡0, 𝑡𝑓]𝕋. So (i) holds. 

(ii). Suppose that (2.2) is controllable on [𝑡0, 𝑡𝑓]𝕋. We have to show that  

    𝑟𝑎𝑛𝑘 {𝐻1, … , 𝐻𝑙} = 𝑛2 

Assume that  

𝑟𝑎𝑛𝑘 {𝐻1, … , 𝐻𝑙} < 𝑛
2 



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Then, there exists a non-zero 𝑧𝛼 ≠ 0 ∈ ℝ𝑛
2

such that  

𝑧𝛼
𝑇𝐻𝑗(𝑡𝑗−1, 𝑡𝑗 , 𝑡𝑓)𝑧𝛼 = 0, 𝑗 = 1,2, … , 𝑙. 

For j=1 

𝑧𝛼
𝑇𝐻1𝑧𝛼 = ∫ 𝑧𝛼

𝑇𝜓𝐺1
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡0

𝐴1(𝜏)𝐴1
𝑇(𝜏)𝜓𝐺1(𝜏, 𝑡𝑓)𝑧𝛼∆𝜏, 

as  𝑧𝛼
𝑇𝜓𝐺1

𝑇 (𝑡, 𝑡𝑓)𝐴1(𝑡) is rd-continuous functions so 

‖𝑧𝛼
𝑇𝜓𝐺1

𝑇 (𝜏, 𝑡𝑓)𝐴1(𝑡)‖
2
= 0. 

Which according to   

𝐴1
𝑇(𝜏)𝜓𝐺1

𝑇 (𝑡, 𝑡𝑓)𝑧𝛼 = 0, 𝑡 ∈ [𝑡0, 𝑡1]𝕋. (3.6) 

For 𝑘 = 2,3, ⋯ , 𝑙 − 1. 

𝑧𝛼
𝑇𝐻𝑘𝑧𝛼 = ∫ 𝑧𝛼

𝑇𝜓𝐺𝑘
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡𝑘−1

𝐴𝑘(𝜏)𝐴𝑘
𝑇(𝜏)𝜓𝐺𝑘(𝜏, 𝑡𝑓)𝑧𝛼∆𝜏 = 0, 

𝑧𝛼
𝑇𝜓𝐺𝑘

𝑇 (𝑡, 𝑡𝑓)𝐴𝑘(𝑡)𝐴𝑘
𝑇(𝑡)𝜓𝐺𝑘(𝑡, 𝑡𝑓)𝑧𝛼 = ‖𝑧𝛼

𝑇𝜓𝐺𝑘
𝑇 (𝑡, 𝑡𝑓)𝐴𝑘(𝑡)‖

2
 

𝐴𝑘
𝑇(𝑡)𝜓𝐺𝑘

𝑇 (𝑡, 𝑡𝑓)𝑧𝛼 = 0, 𝑡 ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋,     (3.7) 

Similarly,  

𝐴𝑙
𝑇(𝑡)𝜓𝐺𝑙

𝑇 (𝑡, 𝑡𝑓)𝑧𝛼 = 0, 𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋, (3.8) 

However, the impulsive system (2.2) is controllability on [𝑡0, 𝑡1]𝕋, and so choosing 𝑧0 = 𝑧𝛼, there 

exists a piecewise rd-continuous control function  �̂�(𝑡) such that  

0 = 𝑧(𝑡𝑓) = 𝜓𝐺1
𝑇 (𝑡0, 𝑡𝑓)𝑧𝛼 + ∫ 𝜓𝐺1

𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡0

𝐴1(𝜏)�̂�(𝜏)∆𝜏; 𝑙 = 1 (3.9) 

Multiply through by 𝑧𝛼
𝑇 in (3.9) and by using the transpose of the equations (3.6), we have  

𝜓𝐺1
𝑇 (𝑡0, 𝑡𝑓)𝑧𝛼

𝑇𝑧𝛼 = 0.      (3.10) 

Similarly,  



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𝑧(𝑡𝑓) = 𝜓𝐺𝑙
𝑇 (𝑡𝑙−1, 𝑡𝑓){∏ [𝐼𝑛⊗𝑅𝑗]

1

𝑗=𝑙−1

∏ 𝜓𝐺𝑗
𝑇 (𝑡𝑗−1, 𝑡𝑗)

1

𝑗=𝑙−1

𝑧𝛼

+∑( ∏ [𝐼𝐼

1

𝐼=𝐼−1

𝐼−2

𝐼=1

⊗𝑅𝑗] ∏ 𝜓𝐺𝑖
𝑇 (𝑡𝑖−1, 𝑡𝑖)

𝑗+1

𝑖=𝑙−1

∫ 𝜓𝐺𝑗
𝑇 (𝜏, 𝑡𝑗)

𝑡𝑗

𝑡𝑗−1

𝐴𝑗(𝜏)𝑈(𝜏)∆𝜏)

+ [𝐼𝑛⊗𝑅𝑙−1] ∫ 𝜓𝐺𝑙−1
𝑇 (𝜏, 𝑡𝑙−1)

𝑡𝑙−1

𝑡𝑙−2

𝐴𝑙−1(𝜏)�̂�(𝜏)∆𝜏} 

+ ∫ 𝜓𝐺𝑙
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡𝑙−1

𝐴𝑙(𝜏)�̂�(𝜏)∆𝜏, 𝑙 = 2,3, …                                      

(3.11) 

Multiply by 𝜓𝐺1
𝑇 (𝑡1, 𝑡2)𝜓𝐺2

𝑇 (𝑡2, 𝑡3)…𝜓𝐺𝑘
𝑇 (𝑡𝑙−1, 𝑡𝑓) and 𝑧𝛼

𝑇 in the equation (3.11), using equations (3.7), 

and (3.8), we have 

∏[𝐼𝑛⊗𝑅𝑗]𝑧𝛼
𝑇𝑧𝛼 = 0,

𝑙

𝑖=2

 (3.12) 

From equations (3.10) and (3.12), according to that 𝑧𝛼
𝑇𝑧𝛼 = 0. This contradicts 𝑧𝛼 ≠ 0 and so, we 

conclude that  

𝑟𝑎𝑛𝑘 {𝐻1, … , 𝐻𝑙} = 𝑛
2 

Theorem 3.2. Suppose that (𝐼𝑛⊗𝑅𝑗) ≠ −1, 𝑗 = 1,2, … , 𝑘 and 𝐺𝑘(𝑡) = 𝐺𝑘, 𝐴𝑘(𝑡) = 𝐴𝑘 are constant 

matrices. Then, the system (2.2) is a controllable on [𝑡0, 𝑡𝑓]𝕋
(𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋), if and only if  

𝑟𝑎𝑛𝑘 {𝑀1, 𝑀2, … , 𝑀𝑙} = 𝑛2 (3.13) 

Since, 𝑀𝑗 = [𝐴𝑗
𝑇 𝐴𝑗

𝑇𝐺𝑗  … 𝐴𝑗
𝑇𝐺𝑗

𝑛2−1] , 𝑗 = 1,2, … , 𝑙. 

Proof: Suppose that the system (2.2) is a controllability on [𝑡0, 𝑡𝑓]𝕋. If the rank condition (3.13) does 

not hold, if there exist 𝑧𝛼 ∈ ℝ𝑛
2

 with  𝑧𝛼 ≠ 0, such that  

𝐴𝑗𝐺𝑗
𝑖𝑧𝛼 = 0. (3.14) 

For,  j = 1, … , k, i = 0,1, … , 𝑛2 − 1 

We Consider  

𝐻1(𝑡0, 𝑡𝑓, 𝑡𝑓)𝑧𝛼 = ∫ 𝑒𝐺1
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡0

𝐴1𝐴1
𝑇𝑒𝐺1(𝜏, 𝑡𝑓)𝑧𝛼∆𝜏. 

From equation (3.14) and using Proposition 2.1, we have  



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𝐻1(𝑡0, 𝑡𝑓, 𝑡𝑓)𝑧𝛼 = ∫ 𝑒𝐺1
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡0

𝐴1𝐴1
𝑇 ∑ 𝜒1𝑖(𝜏, 𝑡𝑓)

 𝑛2−1

𝑖=0

𝐺1
𝑖𝑧𝛼∆𝜏, 

= ∫ 𝑒𝐺1
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡0

𝐴1 ∑ 𝜒1𝑖(𝜏, 𝑡𝑓)

 𝑛2−1

𝑖=0

𝐴1
𝑇𝐺1

𝑖
𝑧𝛼∆𝜏 = 0. 

By again equation (3.14) and using Proposition 2.1, according to   

𝐻𝑘(𝑡𝑘−1, 𝑡𝑘, 𝑡𝑓)𝑧𝛼 = ∫ 𝑒𝐺𝑘
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡𝑘−1

𝐴𝑘𝐴𝑘
𝑇𝑒𝐺𝑘(𝜏, 𝑡𝑓)𝑧𝛼∆𝜏

= ∫ 𝑒𝐺𝑘
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡𝑘−1

𝐴𝑘 ∑ 𝜒1𝑖(𝜏, 𝑡𝑓)

 𝑛2−1

𝑖=0

𝐴𝑘
𝑇𝐺𝑘

𝑖
𝑧𝛼∆𝜏 = 0. 

For 2 ≤ 𝑘 ≤ 𝑙 − 1, similarly, 𝐻𝑙(𝑡𝑙−1, 𝑡𝑙 , 𝑡𝑓)𝑧𝛼 = 0, according to  

𝑟𝑎𝑛𝑘 {𝑀1, … ,𝑀𝑙} < 𝑛
2  

Hence, it is contradicting the conclusion (ii) of Theorem (3.1) and thus, we can conclude that the 

condition (3.13) is true. 

Conversely, assume that the condition (3.13) is satisfied. If the impulsive system (2.2) is not 

controllable on 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋
(𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋), then it follows that from the conclusion (i) of Theorem 

3.1., that the matrices 𝐻1(𝑡0, 𝑡𝑓 , 𝑡𝑓), 𝐻𝑘(𝑡𝑘−1, 𝑡𝑘, 𝑡𝑓) and 𝐻𝑙(𝑡𝑙−1, 𝑡𝑙 , 𝑡𝑓) are not invertible. If there exist 

𝑧𝛼 ∈ ℝ𝑛
2

 with  𝑧𝛼 ≠ 0, such that  

𝑧𝛼
𝑇𝐻1(𝑡0, 𝑡𝑓, 𝑡𝑓)𝑧𝛼 = ∫ 𝑧𝛼

𝑇𝑒𝐺1
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡0

𝐴1𝐴1
𝑇𝑒𝐺1(𝜏, 𝑡𝑓)𝑧𝛼∆𝜏 = 0, 

𝑧𝛼
𝑇𝐻𝑘(𝑡𝑘−1, 𝑡𝑘, 𝑡𝑓)𝑧𝛼 = ∫ 𝑧𝛼

𝑇𝑒𝐺𝑘
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡𝑘−1

𝐴𝑘𝐴𝑘
𝑇𝑒𝐺𝑘(𝜏, 𝑡𝑓)𝑧𝛼∆𝜏 = 0, 2 ≤ 𝑘 ≤ 𝑙 − 1 

𝑧𝛼
𝑇𝐻𝑙(𝑡𝑙−1, 𝑡𝑙, 𝑡𝑓)𝑧𝛼 = ∫ 𝑧𝛼

𝑇𝑒𝐺𝑙
𝑇 (𝜏, 𝑡𝑓)

𝑡𝑓

𝑡𝑙−1

𝐴𝑙𝐴𝑙
𝑇𝑒𝐺𝑙(𝜏, 𝑡𝑓)𝑧𝛼∆𝜏 = 0, 

Exactly same as in proof of Theorem 3.1., according to   

𝐴1
𝑇𝑒𝐺1(𝑡, 𝑡𝑓)𝑧𝛼 = 0. 𝑓𝑜𝑟 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋 (3.15) 

𝐴𝑘
𝑇𝑒𝐺𝑘(𝑡, 𝑡𝑓)𝑧𝛼 = 0. 𝑓𝑜𝑟 𝑡 ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋. (3.16) 

Where 2 ≤ 𝑘 ≤ 𝑙 − 1, and  

𝐴𝑙
𝑇𝑒𝐺𝑙(𝑡, 𝑡𝑓)𝑧𝛼 = 0. 𝑓𝑜𝑟 𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋 (3.17) 

Differentiating equations (3.13), (3.14) and (3.15)  𝑖𝑡ℎ times, where (0 ≤ 𝑖 ≤ 𝑛2 − 1), we obtain 



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𝐴1
𝑇𝐺1

𝑖𝑒𝐺1(𝜏, 𝑡𝑓)𝑧𝛼 = 0. 𝑓𝑜𝑟 𝑡 ∈ [𝑡0, 𝑡1]𝕋. (3.18) 

𝐴𝑘
𝑇𝐺𝑘

𝑖𝑒𝐺𝑘(𝜏, 𝑡𝑓)𝑧𝛼 = 0. 𝑓𝑜𝑟 𝑡 ∈ [𝑡𝑘−1, 𝑡𝑘)𝕋.    (3.19) 

Where 2 ≤ 𝑘 ≤ 𝑙 − 1, and  

𝐴𝑙
𝑇𝐺𝑙

𝑖𝑒𝐺𝑙(𝜏, 𝑡𝑓)𝑧𝛼 = 0. 𝑓𝑜𝑟 𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋.    (3.20) 

If we take 𝑡 = 𝑡𝑓 in equations (3.18), (3.19) and (3.20), then it follows that 𝐴𝑗
𝑇𝐺𝑗

𝑖𝑧𝛼 = 0, 𝑓𝑜𝑟 𝑗 =

1, … , 𝑘  and 𝑖 = 0,1, … , 𝑛2 − 1. Which implies that the rank condition (3.11) fails, which gives 

contradiction. So, the impulsive system (2.1) is controllable on 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋
(𝑡𝑓 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋). So that 

the system (2.2) is controllable by theorem 3.2. 

  4. Complete Observability  

In this section, we present the observability in time variant and time invariant adjoint dynamic system 

(2.3) on time scales.  

Definition 4.1. The  system (2.2) is said to be completely observability on [𝑡0, 𝑡𝑓]𝕋
(𝑡𝑓 > 𝑡0) if any 

initial state 𝑧(𝑡0) = 𝑧0 ∈ ℝ𝑛
2

is uniquely determined by the corresponding system input �̂�(𝑡) and the 

system output y(t) for [𝑡0, 𝑡𝑓]𝕋. 

Theorem 4.1. Suppose that [𝐼𝑛⊗𝑅𝑗] ≥ 0, 𝑗 = 1,2, … , 𝑙. Then, the impulsive system (2.2) is 

observable on 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋
(𝑡𝑓 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋) if and only if the matrix 

𝑊(𝑡0, 𝑡𝑓) ≔ 𝑊(𝑡0, 𝑡0, 𝑡1) +∑∏[𝐼𝑛⊗𝑅𝑖]𝑊(𝑡0, 𝑡𝑗−1, 𝑡𝑗) +

𝑗

𝑖=1

𝑙−1

𝑗=2

∏[𝐼𝑛⊗𝑅𝑖]𝑊(𝑡0, 𝑡𝑙−1, 𝑡𝑓)

𝑙

𝑖=1

 

is invertible, where  

𝑊(𝑡0, 𝑡0, 𝑡1) ≔ ∫ 𝜓𝐺1(𝑡0, 𝜏)(I⊗𝐶1)
𝑇(𝜏)(I⊗𝐶1)(𝜏)𝜓𝐺1

𝑇 (𝑡0, 𝜏)∆𝜏

𝑡1

𝑡0

, 

𝑊(𝑡0, 𝑡𝑗−1, 𝑡𝑗) ≔ ∫ Ω𝑗(𝑡0, 𝜏)(I⊗𝐶𝑗)
𝑇
(𝜏)(I⊗𝐶𝑗)(𝜏)Ω𝑗

𝑇(𝑡0, 𝜏)∆𝜏

𝑡𝑗

𝑡𝑗−1

, 𝑗 = 2, … , 𝑙 − 1, 

and  

𝑊(𝑡0, 𝑡𝑙−1, 𝑡𝑓) ≔ ∫ Ω𝑙(𝑡0, 𝜏, )(I⊗𝐶𝑙)
𝑇(𝜏)(I⊗𝐶𝑙)(𝜏)Ω𝑙

𝑇(𝑡0, 𝜏)∆𝜏

𝑡𝑓

𝑡𝑙−1

, 

with  

Ω𝑗
𝑇(𝑡0, 𝜏) = 𝜓𝐺𝑗

𝑇 (𝑡𝑗−1, 𝜏)𝜓𝐺𝑗−1
𝑇 (𝑡𝑗−2, 𝑡𝑗−1)…𝜓𝐺𝑗

𝑇 (𝑡0, 𝑡1), 𝑗 = 1, … , 𝑘 

Proof: Assume that the matrix  𝑊(𝑡0, 𝑡𝑓)𝕋 is invertible. From the system (2.2) and the equation (3.1), 

we have  

For 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋 



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𝑦(𝑡) = (I⊗𝐶1)(𝑡)𝜓𝐺1
𝑇 (𝑡0, 𝜏)𝑧0 + (I⊗𝐶1)(𝑡) ∫ 𝜓𝐺1

𝑇 (𝜏, 𝑡)𝐴1(𝜏)𝑈(𝑡)∆𝜏

𝑡1

𝑡0

+ (I⊗𝐷1)�̂�(𝑡), 

(4.1) 

and  

for 𝑡 ∈ (𝑡𝑘−1, 𝑡𝑘]𝕋, 𝑘 = 2,3, … 𝑙. 

𝑦(𝑡) = (I⊗𝐶𝑘)(𝑡)𝜓𝐺𝑘
𝑇 (𝑡𝑘−1, 𝑡){ ∏ [𝐼𝑛⊗𝑅𝑗]

1

𝑗=𝑘−1

∏ 𝜓𝐺𝑗
𝑇 (𝑡𝑗−1, 𝑡𝑗)

1

𝑗=𝑘−1

𝑧0(I⊗𝐶𝑘)(𝑡) 

+∑(∏ [𝐼𝑛⊗𝑅𝑖]

𝑘

𝑖=𝑘−1

∏ 𝜓𝐺𝑖
𝑇 (𝑡𝑖−1, 𝑡𝑖)

𝑘+1

𝑖=𝑘−1

∫ 𝜓𝐺𝑗
𝑇 (𝜏, 𝑡𝑗)

𝑡𝑗

𝑡𝑗−1

𝐴𝑗(𝜏)�̂�(𝜏)∆𝜏)

𝑘−2

𝑗=1

(I⊗𝐶𝑘)(𝑡) 

+[𝐼𝑛⊗𝑅𝑙−1] ∫ 𝜓𝐺𝑘−1
𝑇 (𝜏, 𝑡𝑘−1)

𝑡𝑘−1

𝑡𝑘−2

𝐴𝑘−1(𝜏)�̂�(𝜏)∆𝜏}(I⊗𝐶𝑘)(𝑡) 

+ ∫ 𝜓𝐺𝑘
𝑇 (𝜏, 𝑡)

𝑡

𝑡𝑘−1

𝐴𝑘(𝜏)�̂�(𝜏)∆𝜏 + (I⊗𝐷𝑘)�̂�(𝑡) (4.2) 

From the Definition 4.1., that the observability of the system (2.2) is  

𝑦(𝑡) =

{
 

 
(I⊗𝐶𝑘)(𝑡)𝜓𝐺𝑘

𝑇 (𝑡0, 𝑡)𝑧0, 𝑡 ∈ [𝑡0, 𝑡1]𝕋

∏ [𝐼𝑛⊗𝑅𝑗](I⊗𝐶𝑘)𝜓𝐺𝑘
𝑇 (𝑡0, 𝑡)

1

𝑗=𝑘−1

𝑧0, 𝑡 ∈ (𝑡𝑘−1, 𝑡𝑘]𝕋, 𝑘 = 2,3, … , 𝑙
 (4.3) 

as �̂�(𝑡) = 0. Now multiply by Ω𝑘(𝑡0, 𝑡)(I⊗𝐶𝑘)
𝑇(𝑡) to both sides of the equation (4.3) and integrating 

with respect to 𝑡0 𝑡𝑜 𝑡𝑓 , we get 

∫ Ω𝐼(𝐼0,𝐼, )(I⊗ 𝐼𝐼)
𝐼(𝐼)𝐼(𝐼)∆𝐼

𝐼𝐼

𝐼0

= [ ∫ 𝐼𝐼1
(𝐼0,𝐼, )(I⊗𝐼1)

𝐼(𝐼)(I⊗𝐼1)(𝐼)𝐼𝐼1

𝐼 (𝐼0,𝐼)∆𝐼

𝐼1

𝐼0

+∑∏[𝐼𝐼⊗𝐼𝐼]

𝐼

𝐼=𝐼

∫ Ω𝐼(𝐼0,𝐼, )(I⊗𝐼𝐼)
𝐼(𝐼)(I⊗𝐼𝐼)(𝐼)Ω𝐼

𝐼(𝐼0,𝐼)∆𝐼

𝐼𝐼

𝐼𝐼−1

𝐼−1

𝐼=2

+∏[𝐼𝑛⊗𝑅𝑖]

𝑘

𝑖=1

∫ Ω𝑙(𝑡0, 𝜏, )(I⊗𝐶𝑙)
𝑇(𝜏)(I⊗𝐶𝑙)(𝜏)Ω𝑙

𝑇(𝑡0, 𝜏)∆𝜏

𝑡𝑓

𝑡𝑙−1

] 𝑧0 

and so, 



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∫ Ω𝑘(𝑡0, 𝜏, )(I⊗𝐶𝑘)
𝑇(𝜏)𝑦(𝜏)∆𝜏

𝑡𝑓

𝑡0

= 𝑊(𝑡0, 𝑡𝑓)𝑧0 .  (4.4) 

Obviously, the left-hand side of equation (4.4) depends on 𝑦(𝑡), 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋
. Since the matrix 

𝑊(𝑡0, 𝑡𝑓) is invertible, then from linear algebraic equations (4.4) we deduce that 𝑧(𝑡0) = 𝑧0 is a 

uniquely determined by the corresponding system output 𝑦(𝑡), 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋. 

Conversely, assume that the matrix 𝑊(𝑡0, 𝑡𝑓) is not invertible, then there exists a nonzero 𝑧𝛼 ∈ ℝ𝑛
2

, 

such that 

𝑧𝛼
𝑇𝑊(𝑡0, 𝑡𝑓)𝑧𝛼 = 0.    

Since, [𝐼𝑛⊗𝑅𝑗] ≥ 0, 𝑗 = 1,2, … , 𝑙, 𝑊(𝑡0, 𝑡0, 𝑡1), 𝑊(𝑡0, 𝑡𝑗−1, 𝑡𝑗) 𝑓𝑜𝑟 𝑗 = 2,3, … 𝑙 − 1 and 

𝑊(𝑡0, 𝑡𝑙−1, 𝑡𝑓) are positive semidefinite matrices, we get 

𝑧𝛼
𝑇𝑊(𝑡0, 𝑡0, 𝑡1)𝑧𝛼 = 0.    

𝑧𝛼
𝑇𝑊(𝑡0, 𝑡𝑗−1, 𝑡𝑗)𝑧𝛼 = 0.   𝑓𝑜𝑟 𝑗 = 2, … 𝑙 − 1   (4.5) 

𝑧𝛼
𝑇𝑊(𝑡0, 𝑡𝑙−1, 𝑡𝑓)𝑧𝛼 = 0.    

We choose 𝑧0 = 𝑧𝛼. Thus, from equations (4.3) and (4.5), according to  

∫ 𝑦𝑇(𝜏)𝑦(𝜏)∆𝜏

𝑡𝑓

𝑡0

= ∫ 𝑧𝛼
𝑇𝜓𝐺1(𝑡0, 𝜏, )(I⊗𝐶1)

𝑇(𝜏)(I⊗𝐶1)(𝜏)𝜓𝐺1
𝑇 (𝑡0, 𝜏)𝑧𝛼∆𝜏

𝑡1

𝑡0

 

+∑[∏[𝐼𝑛⊗𝑅𝑖]

𝑗

𝑖=1

]

2
𝑙−1

𝑗=2

∫ 𝑧𝛼
𝑇Ω

𝑗
(𝑡0, 𝜏, )(I⊗𝐶𝑗)

𝑇
(𝜏)(I⊗𝐶𝑗)(𝜏)Ω𝑗

𝑇(𝑡0, 𝜏)𝑧𝛼∆𝜏

𝑡𝑗

𝑡𝑗−1

 

+[∏[𝐼𝑛⊗𝑅𝑖]

𝑙

𝑖=1

]

2

∫ 𝑧𝛼
𝑇Ω𝑙(𝑡0, 𝜏, )(I⊗𝐶𝑙)

𝑇(𝜏)(I⊗𝐶𝑙)(𝜏)Ω𝑙
𝑇(𝑡0, 𝜏)𝑧𝛼∆𝜏

𝑡𝑓

𝑡𝑙−1

. 

implies  

∫‖𝑦(𝜏)‖2∆𝜏

𝑡𝑓

𝑡0

= 0. 

According to  

0 = 𝑦(𝑡) =

{
 
 
 

 
 
 

(I⊗𝐶1)(𝑡)𝜓𝐺1
𝑇 (𝑡0, 𝑡)𝑧0, 𝑡 ∈ [𝑡0, 𝑡1]𝕋

∏[𝐼𝑛⊗𝑅𝑖](I⊗𝐶𝑘)Ω𝑘
𝑇(𝑡0, 𝑡)

𝑘

𝑖=1

𝑧0, 𝑡 ∈ (𝑡𝑘−1, 𝑡𝑘]𝕋, 𝑘 = 2, … , 𝑙 − 1,

∏[𝐼𝑛⊗𝑅𝑖](I⊗𝐶𝑙)Ω𝑙
𝑇(𝑡0, 𝑡)

𝑙

𝑖=1

𝑧0, 𝑡 ∈ (𝑡𝑙−1, 𝑡𝑙]𝕋.

 



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The last equality implies, by Definition 4.1., that the system (2.2) is not observable on 

𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋
(𝑡𝑓 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋).  

Theorem 4.2. Assume that [𝐼𝑛⊗𝑅𝑗] ≥ 0, 𝑗 = 1,2, … , 𝑙 and 𝐺𝑘(𝑡) = 𝐺𝑘, (I⊗𝐶𝑘)(𝑡) = (I⊗𝐶𝑘) are 

constant matrices. Then, the system (2.2) is observable on 𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋
(𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋), if and only if 

𝑟𝑎𝑛𝑘 (𝑆) = 𝑛2. 

Let us define the following matrix  

𝑆 =

[
 
 
 
 

(I⊗𝐺1𝑗)

(I⊗𝐺1𝑘)𝑃𝑗
𝑇

⋮

(I⊗𝐺𝑘)(𝑃𝑗
𝑇)
𝑛2−1

]
 
 
 
 

 (4.6) 

Proof: Assume that r𝑎𝑛𝑘 (𝑆) = 𝑛2. And we aim to show that the system (2.2) is observability on 

𝑡 ∈ [𝑡0, 𝑡𝑓]𝕋
(𝑡 ∈ [𝑡𝑙−1, 𝑡𝑙)𝕋). If otherwise, namely the system (2.2) is not observability then by 

Theorem 4.1., according to the matrix  𝑊(𝑡0, 𝑡𝑓) is not invertible, which leads to that there exists a 

nonzero vector  𝑧𝛼 ≠ 0. Then by using Theorem 4.1., we have 

𝑧𝛼
𝑇𝑊(𝑡0, 𝑡0, 𝑡1)𝑧𝛼 = ∫ 𝑧𝛼

𝑇𝑒𝐺1(𝑡0, 𝜏)(I⊗𝐶1)(I⊗𝐶1)
𝑇𝑒𝐺1

𝑇 (𝑡0, 𝜏)𝑧𝛼∆𝜏

𝑡1

𝑡0

 

= ∫[(I ⊗𝐶1)𝑒𝐺1
𝑇 (𝑡0, 𝜏)𝑧𝛼]

𝑇
[(I⊗𝐶1)𝑒𝐺1

𝑇 (𝑡0, 𝜏)𝑧𝛼]
𝑇
∆𝜏

𝑡1

𝑡0

, 

Similarly,  

(I⊗𝐶𝑗)Ω𝑗
𝑇(𝑡0, 𝑡)𝑧𝛼 = 0, 𝑗 = 1, … , 𝑙 − 1, (4.8) 

and 

(I⊗𝐶𝑙)Ω𝑙
𝑇(𝑡0, 𝑡)𝑧𝛼 = 0,     (4.9) 

Where Ω𝑗
𝑇(𝑡0, 𝑡) = 𝑒𝐺𝑗

𝑇 (𝑡𝑗−1, 𝑡)𝑒𝐺𝑗−1
𝑇 (𝑡𝑗−2, 𝑡𝑗−1)… 𝑒𝐺𝑗

𝑇 (𝑡0, 𝑡1). 

Obviously, at 𝑡 = 𝑡0, we obtain (I⊗𝐶𝑗)𝑧𝛼 = 0, 𝑓𝑜𝑟  𝑗 = 1, … , 𝑙 − 1, and differentiating the equations 

(4.7), (4.8) and (4.9) 𝑛2 − 1 times and evaluating the results at 𝑡 = 𝑡0 gives 

(I⊗𝐶𝑗)𝐺𝑗
𝑖𝑧𝛼 = 0, 𝑖 = 0,1, … , 𝑛2 − 1, 𝑗 = 1,2, … , 𝑙 (4.10) 

Therefore, by the equations (4.6) and (4.9) we have 𝑆𝑧𝛼 = 0, And furthermore,  𝑧𝛼 ≠ 0 implies that 

𝑟𝑎𝑛𝑘 (𝑆) < 𝑛2 which leads to a contradiction with the assumptions that 𝑟𝑎𝑛𝑘 (𝑆) = 𝑛2.  

Conversely, we assume that r𝑎𝑛𝑘 (𝑆) < 𝑛2. Thus, there exists 𝑧𝛼 ≠ 0 such that 𝑆𝑧𝛼 = 0, which leads 

to the equation (4.10). 

From equation (4.10) and using Proposition 2.1. we obtain 

𝑊(𝑡0, 𝑡0, 𝑡1)𝑧𝛼 = ∫ ∑ 𝜒1𝑖

𝑛2−1

𝑖=0

(𝑡0, 𝜏)𝑒𝐺1(𝑡0, 𝜏)(I⊗𝐶1)
𝑇(I⊗𝐶1)𝑒𝐺1

𝑇 (𝑡0, 𝜏)𝑧𝛼∆𝜏

𝑡1

𝑡0

 



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= ∫ ∑ 𝜒1𝑖

𝑛2−1

𝑖=0

(𝑡0, 𝜏)𝑒𝐺1(𝑡0, 𝜏)(I⊗𝐶1)
𝑇(I⊗𝐶1)𝑒𝐺1

𝑇 (𝑡0, 𝜏)𝑧𝛼∆𝜏

𝑡1

𝑡0

= 0, 

Similarly, for 𝑗 = 1,2, … , 𝑙 − 1 

𝑊(𝑡0, 𝑡𝑗−1, 𝑡𝑗)𝑧𝛼 = 0, 

and  

𝑊(𝑡0, 𝑡𝑙−1, 𝑡𝑙)𝑧𝛼 = 0. 

The equation (4.10) yields 𝑊(𝑡0, 𝑡𝑓)𝑧𝛼 = 0. Since 𝑧𝛼 ≠ 0, the matrix 𝑊(𝑡0, 𝑡𝑓) is not invertible. Hence 

the system (2.2) is not observable, and it is contradicting with the assumption of observability.  

References.  

[1] Agarwal R. P, Bohner M., Regan D.O and Peterson A. Dynamic equations on Time scales. A survey, J comput. Appl. 

Math, No.4, (2002), 1-26. 

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

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

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

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