An analysis on the controllability and stability to some fractional delay dynamical systems on time scales with impulsive effects

In this article, we establish a new class of mixed integral fractional delay dynamic systems with impulsive effects on time scales. We investigate the qualitative properties of the considered systems. In fact, the article contains three segments, and the first segment is devoted to investigating the existence and uniqueness results. In the second segment, we study the stability analysis, while the third segment is devoted to investigating the controllability criterion. We use the Leray–Schauder and Banach fixed point theorems to prove our results. Moreover, the obtained results are examined with the help of an example.


Introduction
The notion of fractional differential equations (FDEs) has been a field of intense research for the last few decades. In 1695, the notion of FDEs was initiated with a coincidence between Leibniz and L'Hospital. Nowadays, FDEs play an important role in establishing mathematical modeling of many problems occurring in control theory, bioengineering, mathematical networks, aerodynamics, blood flows, engineering, physics, signal processing, etc. [1,2].
In the past few decades, because of these applications in various fields of interest, impulsive differential equations got considerable attention. In order to unify the difference and differential calculus, Hilger [44] provided the idea of time scales at the end of the twentieth century, which is now a well-known subject. For more details, see [45][46][47][48][49][50][51]. Lupulescu and Zada [49] provided the basics and fundamental notions of linear impulsive systems on time scales in 2010.
In 1960, Kalman presented the notion of controllability, which is the principal notion in mathematical control theory. In general, controllability provides steering the state of a control dynamical equation to the desired terminal state from an arbitrary initial state by utilizing a suitable control function. Numerous researchers examined the controllability results of dynamical systems [52,53]. Moreover, controllability results on time scales is a new area, and few results have been achieved [54,55]. Especially, there are a few articles that examined the existence, controllability, and Ulam type stability regarding a mixed structure of the impulsive fractional dynamical system on time scales.
Definition 2.1 ([57]) At a point ς ∈ T k , the delta derivative g (ς) of a mapping g : T → R is a number (provided it exists) if, for > 0, a neighborhood U of ς exists provided that Theorem 2.2 ([57]) Let c, d ∈ T and f ∈ C rd (T, R), then where n = [σ ] + 1 and the delta nth derivative of f is denoted by f n .
where E σ (Aς σ ) is the matrix representation of the aforesaid Mittag-Leffler function given by To achieve our results, we consider the following: (A): The mappings G, G : T 0 × R n × R n × R n → R n are continuous, and there exist L i , L i , i = 1, 2, 3, as the positive constants such that (i = 1, 2, 3) (B): The mappings G, G : T 0 × R n × R n × R n → R n are continuous, and there exist l i , m i , i = 1, 2, 3, positive constants such that possesses a bounded invertible operator ( σ W T ς 0 ) -1 , and these operators admit values in L 2 (I, R)\ ker( σ W T ς 0 ). Also, there exists a positive constant provided that • When T = {q m : q > 1, m ∈ Z} ∪ Z, then Throughout the manuscript, we set

Existence of solution
Existence criteria are investigated here.
Now, regarding the mixed impulsive system (2), we have the following result.
Next, for both mixed impulsive systems (1) and (2), we investigate the existence of at least one solution via the weaker condition (B) and the Leray-Schauder alternative fixed point method.
Theorem 3.3 The mixed impulsive system (1) has at least one solution provided assumption (B) holds and K > 0 exists so that Proof Firstly, we prove that σ defined by (10) is a completely continuous operator. We see that the continuity of the mappings , , F , and G provides that σ is a continuous operator. Also, assume that 1 ⊆PS along with the fact that the operators , , F , and G are bounded. Then there exist L 1 , L 2 , M 1 , and M 2 (positive constants) such Note that we take L = L 4 + L 5 + a 1 , M = M 1 + M 2 , (ςs) σ -1 ≤ L 1 , and L + L 1 a 2 M(ς fς 0 ) = G.

It implies
Thus, from (18), we conclude that is uniformly bounded. Now, we prove that σ is completely continuous. For this, we discuss the following possibilities.
Case 1: Assume that all points on T are isolated, i.e., time scales consist of discrete points. Using Theorem 2.2, σ becomes Clearly, on a discrete finite set, (19) is a collection of summation operators. Further, the continuity of j , j , F , and G implies that σ is completely continuous. Case 2: Assume that all the points of T are dense, i.e., T is continuous.
Clearly, we observe from the above that it approaches 0 as ς f 2 → ς f 1 . Hence, the operator σ is equicontinuous. Finally, using the Arzela-Ascoli theorem, we conclude that σ is completely continuous.
Case 3: Assume that T involves isolated points along with dense ones, i.e., continuous and discrete. Now, utilizing Theorem 2.2 for the isolated points, we can write σ as the summation operator which is completely continuous (discussed in case 1). For the dense points, one can prove that σ is a completely continuous operator (discussed in case 2). Consequently, σ can be written as a sum of two operators for isolated and dense points. As a result, we know that the sum of two operators which are completely continuous is also completely continuous. Thus, the operator σ is a completely continuous operator. Hence, by summarizing the above three possibilities, we arrive at the conclusion that σ is a completely continuous operator.
We have a similar conclusion for the mixed impulsive system (2).

Theorem 3.4 The mixed impulsive system (2) admits at least one solution if assumption (B) is satisfied and K * > 0 exists such that
Proof It is similar to the previous argument for σ in Theorem 3.3.

Stability analysis
Now, to start this section, we first consider the following inequalities: and for each > 0. (1) is said to be UH stable on T if, for any ω ∈ PC 1 (T, R n ) fulfilling (21), there exists ω ∈ PC 1 (T, R n ) as a solution of (1) such that ω(s)ω(s) ≤ C for C > 0, s ∈ T. (2) is termed as UH stable if, for any > 0 and ω ∈ PC 1 (D, R n ) that fulfills (22), there exists, ω ∈ PC 1 (D, R n ) as a solution of (2) provided

Lemma 4.3
Each function ω ∈ PC 1 (T, R n ) that fulfills (21) also satisfies the following inequality: Proof If ω ∈ PC 1 (T, R n ) satisfies (21), then via Remark 4.1 and the argument is finished.

Lemma 4.4
Each map ω ∈ PC 1 (D, R n ) that fulfills (22) also satisfies the inequalities given below: Proof If ω ∈ PC 1 (D, R n ) satisfies (22), in this case, by virtue of Remark 4.2, Clearly, equation (23) implies that Using a similar method, we get and this ends the argument.
Now, we provide a sufficient condition for the UH stability of mixed impulsive systems (1) and (2).

Theorem 4.5 The mixed impulsive system (1) is UH stable provided assumption (A) and
inequality (9) are satisfied.
Proof Let ω be the solution of the mixed impulsive system (1) and ω be the solution of inequality (21). Therefore, from Theorem 3.1, we have where E σ (Aς σ ) stands for the matrix representation of the Mittag-Leffler function. Using a similar approach as that in Theorem 3.1, we get Hence . Hence, the mixed impulsive system H (a 3 ,a 2 ,L F ,L G 1 ,L G 2 L F 1 ,L G 3 ,L F 2 ,L ,L ) (0) = 0, then our impulsive system (1) becomes generalized UH stable.

Theorem 4.6 The mixed impulsive system (2) is UH stable provided that assumption (A)
and inequality (13) are satisfied.

Definition 5.4
The function ω ∈ T is said to be the solution of the mixed impulsive system (4) if ω satisfies ω(0) = ω 0 and ω is the solution of the following integral equations: where E σ (Aς σ ) stands for the matrix representation of the Mittag-Leffler function.
One can indicate a similar theorem for the mixed impulsive system (4). (4) is controllable on T such that hypotheses (A) and (W) are satisfied and the following inequality holds:
Therefore, from (34), * * σ ( * * ) ⊆ * * . Also, for ς ∈ (s i , ς i+1 ] ∩ T, i = 1, . . . , m, with ω 0 = ω 0 , we have * * σ ω(ς) - * * σ ω(ς) Therefore, the operator * * σ is strictly contractive. Thus, using the Banach fixed point theorem method, * * σ has a unique fixed point, which is the unique solution of the mixed impulsive system (4). Also, using Lemma 5.6, we conclude that ω(ς) fulfills ω(T) = ω T . Consequently, the mixed impulsive system (4) is controllable. regard. In the next step, Ulam-Hyers stability and a generalized version of it were proved for the mentioned mixed impulsive systems. After that, we investigated the controllability property for the aforesaid systems. Lastly, an illustrative example was proposed to examine the results established in the previous sections. For future projects, the main aim of the authors is that these qualitative specifications can be checked and established on some real-world impulsive systems arising in mathematical models of brain.