Total controllability of non-autonomous second-order measure evolution systems with state-dependent delay and non-instantaneous impulses

: This paper investigates a new class of non-autonomous second-order measure evolution systems involving state-dependent delay and non-instantaneous impulses. We introduce a stronger concept of exact controllability called total controllability. The existence of mild solutions and controllability for the considered system are obtained by applying strongly continuous cosine family and the M¨onch fixed point theorem. Finally, an example is used to verify the practical application of the conclusion.


Introduction
Impulsive differential equations (IDEs) are often used to simulate some evolutionary processes in the real world, such as shocks, population dynamics, harvesting. These evolutionary processes tend to have some sudden changes that are negligible compared to the whole duration of the process. The theory of IDEs has been widely used in mathematical modeling and has become an important research field in recent years. However, some dynamical behaviors of physical phenomena cannot be modeled by instantaneous IDEs, see [1][2][3][4][5].
Wang and Fěckan [7] investigates the existence of mild solutions to nonlinear operator equations with non-instantaneous impulses. Malik and Kumar [8] consider a control system represented by a second-order IDEs and obtained the existence, uniqueness and stability of mild solution. Since noninstantaneous IDEs can model some physical phenomena that remain active in fixed time intervals, this has attracted extensive attention of many researchers. For detailed results, see [9][10][11][12][13][14][15][16].
Furthermore, the complex case with infinitely many perturbation points in a finite interval, called Zeno behavior [17], cannot be modeled by IDEs. Consequently, Pandit et al. [18] and Sharma et al. [19] established an important class of differential equations, called measure differential equations (MDEs), to deal with this complex phenomenon. Thenceforth, in [20], the existence of regulated solutions for MDEs is obtained by utilizing fixed point theorem. Federson and Mesquita [21] investigated the Lyapunov stability for abstract MDEs on time scales. For detailed results, see [18,[22][23][24][25][26].
As we know, since controllability can be described as a qualitative property of a control system, it is of great interest to researchers [1,[27][28][29][30]. A common way to study controllability problems is to transform it into fixed point problems for continuous operators in Banach spaces. For instance, Ankit and Rames [31] considered the approximate controllability results in the infinite dimensional space for non-autonomous evolution system with time delay from fixed point theorem. Zhao and Liu [32] investigated the existence and controllability of solutions for semilinear fractional control systems with finite delay via differentiable resolvent operators and Banach's fixed point theorem. In [1], by utilizing the fixed point theorem and fundamental solution, the approximate controllable result for nonlinear measure evolution system with time delay and non-instantaneous impulses was obtained.
More generally, by fixed point theorem, Cao and Sun [33] studies that the control system is exactly controllable, and its system form is Subsequently, Liu et al. [34] considered fractional evolution equations with state-dependent delay as follows where c D α is Caputo derivative. By utilizing the Kuratowski measure of noncompactness and Mönch fixed point theorem, the exact controllability was investigated. Wang et al. [35] studied the following non-instantaneous impulsive evolution system.
Using the Banach fixed point theorem, exact controllability was obtained.
But, Wang et al. achieved controllability by using control only at the last time subinterval rather than at each impulses point. In many engineering applications, it is necessary to control the system within each impulse time interval. So, we introduce a stronger concept of exact controllability, namely total controllability. This control will be applied to each subinterval. Moveover, many physical phenomena, such as the lateral motion of stretchable beams and the vibration of hinged rods, can be modeled with partial differential equations (PDEs). These PDEs can be formulated as abstract second-order IDEs in the infinite dimensional spaces. So, it is necessary to study such systems. For detailed results, see [8-10, 31, 36].
Inspired by the above discussion, this manuscript will focus on the controllability of secondorder non-instantaneous impulsive measure evolution systems with state-dependent delay and nonautonomous conditions in a Banach space (X, ∥ · ∥), which is of form H(η, θ)v ρ(θ,v θ ) dθ is a Volterra integral operator with an integral kernel H ∈ C(Ω, R + ), The main contributions, significance and novelties of this article are as follows: • The existence and total controllability of solution for (1.1) are obtained by employing the Mönch fixed point theorem and the theory of strongly continuous cosine family.
• It is well known that systems with Zeno trajectories often appear in game theory, non-smooth mechanics, etc (see [17,37]). MDEs are often used to model such systems. It may exhibit an infinite number of discontinuities within a finite interval, µ is a function of bounded variation.
• We point out that if µ is the sum of an absolutely continuous and a step function, then system (1.1) implies a complex situation that exhibits both non-instantaneous and instantaneous impulses. Few existing article have investigated such cases, which is an approach for solving hybrid systems separately.
• It is important to note that the obtained controllability results reach the desired state not only at the end of the time interval but also at each impulse point η i+1 .
• As [1,36] said, MDEs are more extensive than ODEs. In fact, when a system described by is acted upon by perturbation, the perturbed system is generally given by where the perturbation term G(t, x) is assumed to be continuous or integrable and as such the state of the system changes continuously with respect to time. But in physical systems one cannot expect the perturbations to be well-behaved and it is therefore important to consider the case when the perturbations are impulsive. This will give rise to equations of the form where dx and dµ denote the distributional derivatives of the solution x and the function µ, respectively. If µ is a function of bounded variation, dµ can be identified with a Stieltjes measure and will have the effect of suddenly changing the state of the system at the points of discontinuity of µ. Equations (1.2) is so called measure differential equations.
• The derived results are involving non-instantaneous impulse and state-dependent delay; therefore, they improved and generalized the results of the paper of Cao and Sun [33]. Our findings are also generalizing the results of Liu et al. [34] for the case of control system with state-dependent delay.
• Finally, an example is used to verify the practical application of the conclusion.
The structure of this paper is as follows. In Section 2, the fundamental concepts, lemma, and preliminary knowledge are given which will be required for the later sections. We obtain the controllability results for system (1.1) by using the theory of cosine families and fixed point theorem in Section 3. Finally, in Section 4, an example to verify the usefulness of our abstract results.

Preliminaries
In present section, we review some of the basic concepts, definitions and lemmas required.
is regulated.
Let HLS p µ ([ξ, ζ]; X) be a p-order Henstock-Lebesgue-Stieltjes integrable regulated function space with respect to µ, with norm defined as Next, we define The Kuratowski measure of noncompactness α(·) on bounded subset S of X is defined as The Kuratowski measure of noncompactness of a bounded set in G(J; X) and X are denoted by α G (·), and α(·), respectively. Lemma 2.7. [41] Let S ⊂ HLS 1 µ (J; X) be a countable set and function ν ∈ HLS 1 µ (J; R + ) such that ∥r(η)∥ ≤ ν(η) holds for all r ∈ S and η ∈ J. Then, Lemma 2.8. [34] Let A ⊂ G(J; X) be equiregulated and bounded on J. Then, Theorem 2.9. [34] (Mönch fixed point theorem) Let E ⊂ X be a closed convex set where X is a Banach space. Assume that T : E → E is a continuous operator and satisfies the following properties: Therefore, T has a fixed point in E. Now let us consider a second-order non-autonomous control system: is a closed dense operator and g : [0, b] → X is an appropriate function. It can be known from [31,42] that before considering the existence of the solution of (2.1), we first discuss the existence of the evolution operator of the following homogeneous equation For further study, we first review the definition of evolution operator.
Definition 2.10. The bounded linear operator Φ(η, θ) : J×J → L(X) is said to be the evolution operator of (2.2) if the following properties are satisfied: Let the evolution operator Φ(η, θ) be associated with the operator {A(η) : η ∈ J}. For simplicity, one can also consider that There exist constants M and N such that sup 0≤η,θ≤b If function g : J → X is integrable, then the solution v of the system (2.1) is given by Obviously, for v ∈ D(A), it is a Banach space endowed with the norm ∥v∥ A = ∥v∥ + ∥Av∥.
Based on the works in [36], we consider the following system where A(·): R → L(D, X) such that for each v ∈ D, the map η → A(η)v is a continuously differentiable function in X. Similar to Eq (2.3), we can also obtain that the mild solution of system (2.4) defined as For more theoretical results on second-order differential equations, see [31,36,42,43] and its cited references. Now, we are ready to introduce the concept of a mild solution of the system (1.1) as follows. (2.5)

Main results
This section mainly studies the controllability of (1.1).
Definition 3.1. System (1.1) is said to be exactly controllable on J if for the initial state v(0) ∈ B h and any final state v b ∈ X, there is a control z ∈ HLS 2 µ (J; U) such that the mild solution v(η) of (1.1) Remark 3.3. If the total controllability of (1.1) is established, obviously it is exactly controllable. On the contrary, it does not hold.
The following assumptions are introduced to demonstrate the main results of this paper. (H1) A(η) is the infinitesimal generator of a strongly continuous cosine family Ψ(η, θ). (H2) The function F : J × B h × B h → X satisfies that: (i) For each η ∈ J, F(η, ·, ·) : B h × B h → X is a continuous function.
Remark 3.4. It can be seen from the discussion in Section 2 that A produces a strongly continuous cosine family and corresponding strongly continuous sine family. This can represent a mild solution of system (1.1). Then, we can define an operator T . (H2) requires F to be continuously measurable, ensuring that the operator T is continuous. Assumptions (H3), we defined an invertible linear operator L i+1 , which will be used to define the control function z v (η). Assumptions (H4),(H5) ensures that the operator T is continuously bounded, which helps us to discuss the controllability of the system (1.1) in a closed convex set.

3)
and Proof. From (3.1), the operator T is defined as follows: (3.5) Next, we divide the proof into four steps.
Step 1. We show that there is a constant r > 0 such that T (E r ) ⊂ E r . If it is not true, for any r > 0, there exists v r ∈ E r such that ∥T v r ∥ ∞ > r.
Step 2. To prove that T is continuous on E r . For this, we take a sequence {v n } ∞ n=1 ⊂ E r such that v n → v as n → ∞.
Step 4. Claim that if C ⊂ E r is countable and there exists w 0 ∈ E r such that C ⊂ co({w 0 } ∪ T (C)), (3.12) then C is relatively compact. Without loss of generality, suppose C = {v n } ∞ n=1 ⊂ E r .
Conclusion: System (4.1) is totally controllable on J.
Therefore, it is easy to know from Theorem 3.6 that system (4.1) is totally controllable. □

Conclusions
This paper is mainly concerned with a class of non-autonomous second-order measure evolution systems involving state-dependent delay and non-instantaneous impulses. We first give the definition of Henstock-Lebesgue-Stieltjes integral and phase space. Then, the existence of the mild solution of system (1.1) is obtained by using the integral equation given by the strongly continuous family of cosines. By using the Kuratowski measure of noncompactness and Mönch fixed point theorem, we obtain some sufficient conditions to ensure the total controllability of system (1.1). Finally, an illustrative example is given to show the practical usefulness of the analytical results.