On Partial Exact Controllability of Fractional Control Systems in Conformable Sense

. In this work, we investigate the partial exact controllability of fractional semilinear control systems in the sense of conformable derivatives. Initially, we establish the existence and uniqueness of the mild solution for this type of fractional control systems. Ten, by employing a contraction mapping principle, we obtain sufcient conditions for the conformable fractional semilinear system to be partially exactly controllable, assuming that its associated linear part is partially exactly controllable. To demonstrate the efcacy of the theoretical fndings, a typical example is provided at the end.


Introduction
Controllability concepts have been essential in a variety of disciplines, such as engineering, control theory, and applied mathematics.For instance, controllability is essential for the design of feedback controllers, which are used to regulate and control the behavior of systems.Following Kalman [1]'s defnition, controllability is characterized by the capability to transition a control system's solution from a given initial state to a desirable state at a fnal time.Later, controllability has been split into two concepts: exact and approximate controllability.Exact controllability aligns with Kalman's defnition, where the system can be directed from any initial state to any desired state within a fnite time.On the other hand, approximate controllability implies that the system can be moved from any starting state to any desired state arbitrarily closely in a fnite time.Te distinction between these two concepts is crucial because some dynamical systems exhibit approximate controllability without achieving exact controllability (see Fattorini [2]).Several researchers have developed suitable controllability conditions for deterministic and stochastic control systems.For example, see [3][4][5][6][7][8][9][10].
Fractional diferential equation (FDE) has emerged as an important attractive area of applied mathematics because of its powerful uses in the felds of engineering and physical sciences [11][12][13].Te utilization of FDEs has been showcased as a highly efective method for enhancing the modeling of various real-world problems and phenomena, such as heat transfer processes and dielectric polarization [14].Tey provide a more accurate model of physical systems than traditional diferential equations, enabling the resolution of problems that prove challenging under traditional modeling.Notably, in the domain of electrical circuits, fractional-order models have proven valuable for simulating electrical components and circuits, including resistors, domino ladders, capacitors, tree structures, and inductors [15].
In electrical circuit, there are some dissipative efects stemming from electrical resistance, ohmic friction, or temperature that standard theoretical calculations fail to consider.Te ordinary derivative is insufcient to take into account these nonconservative features.Consequently, in order to place these dissipation efects on to a relevant theoretical basis, the fractional calculus emerges as a valuable mathematical tool in addressing this kind of electrical problems.For instance, the fractional derivatives allow capturing the nonlocal and hereditary properties neglected in integer-order models [16].
Tis novel type of calculus has attracted the attention of mathematicians, who have been working to develop new results and extend existing concepts to these fractional systems [17][18][19].Researchers in this feld have focused efforts on fndings new theoretical results and expanding controllability notions to apply to fractional control systems.For example, Mahmudov [20] derived a collection of approximate controllability conditions for Sobolev-type equations with fractional derivatives.Sakthivel et al. [21] applied a fxed point theorem to obtain controllability conditions for nonlinear systems with fractional order.Jneid [22] used a compact semigroup operator and Schauder fxed point technique to derive a set of sufcient conditions for the approximate controllability of integrodiferential control systems of non-integer order.Dineshkumar et al. [23] utilized Bohnenblust-Karlin's fxed point theorem, cosine and sine functions of operators to acquire sufcient conditions for the approximate controllability of fractional stochastic diferential inclusions with order 1 < r < 2. Sivasankar et al. [24] studied the nonlocal controllability of stochastic control systems involving Hilfer fractional derivative by using almost sectorial operators with the help of the fxed point technique and measures of noncompactness.
Te majority of previous researches concerning controllability issues of fractional control systems have employed the Riemann-Liouville, Caputo, and Hilfer fractional derivatives.Nevertheless, there has been limited investigation into the controllability problems associated with fractional systems using the conformable fractional derivative [25][26][27][28].Tis represents a notable gap in the existing body of literature, considering that the conformable fractional derivative ofers several advantages compared to the Riemann-Liouville, Caputo and Hilfer fractional derivatives, including its greater naturalness and geometric intuitiveness.Motivated by this observation, this current work focuses on addressing the controllability problems of semilinear control systems with conformable fractional derivatives in Hilbert spaces.Furthermore, we introduce and expand the partial controllability concepts to fractional diferential systems.Roughly speaking, the study of partial controllability is an important part of controllability research overall.Tis signifcance arises from the fact that controllability theorems are often formulated for frst-order diferential equation systems.Nevertheless, many real-world systems, such as higher-order fractional diferential equations and fractional wave equations, can only be written in frst-order form by enlarging the state space dimension.As a consequence, the standard controllability conditions for these systems are too strong since they consider the expanded state space, while controllability notions need to focus on the original state space.To address this, we introduce an additional projection operator P that maps the enlarged state space back to the original state space.To illustrate the workings of partial controllability, we provide two typical examples in Section 2. To our knowledge, the specifc research problem under study has not been previously investigated.We carried out a thorough investigation and literature review and failed to uncover any studies addressing partial exact controllability of fractional control systems.Tis underlines a gap in the existing literature that our research attempts to fll.
Te rest of the paper is organized as follows: In Section 2, useful notations and defnitions, a mathematical model of partial controllability notions, and benefcial preliminary results concerning the partial controllability of linear systems in conformable fractional sense are obtained.In Section 3, we obtain a set of sufcient conditions for the partial exact controllability of fractional semilinear systems, assuming partial exact controllability of its associated linear systems.In Section 4, we give an illustrative example to prove the applicability of the theoretical fndings.In Section 5, we provide a brief discussion of the results that are shown in the illustrative example.Finally, in Section 6, a short conclusion is given to recap the obtained results.

Preliminaries
Troughout this paper, we will utilize the following: (i) (X, ‖.‖) and (U, ‖.‖) are Hilbert spaces with the norms generated by convenient inner products as , where C(0, τ; U) is defned as the vector space of all U− valued continuous functions on [0, τ] endowed with the sup-norm as follows: (iii) C(0, τ; X) × U ad is the product space of two Banach spaces which is also a Banach space equipped with the following norm: (viii) x(t) and u(t) are denoted as x t u t , respectively.Now, let us review some important concepts and fndings about the conformable fractional derivative and the controllability of linear systems.We also establish the necessary assumptions, which will be required in the upcoming sections.
Defnition 1 (see [29]).Te conformable fractional derivative (CFD) of h: on condition that the expression on the right side is exists as a fnite value.Consequently, For every constant c and r ∈ R, the following properties hold.
e x q /q ) � e x q /q Defnition 2 (see [29]).Te CF integral of h: [0, ∞) ⟶ R, is defned by on condition that the improper integral on the right side is a fnite value.
Defnition 3 (see [30]).Te CF Laplace transform of h is defned by It is easy to show that Let the CF− linear system be given as follows: where and A is as defned above.By using CF Laplace transform, we obtain which clearly gives that where I is the identity operator.Now, applying CF inverse Laplace transform and relevant properties from [30], we can derive the mild solution of the system (8) as follows: where Θ (t q /q) is called CF− semigroup generated by A, and (sI − A) − 1 is the inverse operator of (sI − A).Partial controllability is a useful concept for control systems that can be modeled as frst-order diferential equations by augmenting the original state space.Tis is because the partial controllability concepts are more suited for such systems than the traditional notions of controllability.Te projection operator P can be used to map the expanded state space to the original state space, which makes it easier to analyze and design controllers for the system.Partial controllability has several advantages, which will be illustrated in the following examples.
Example 1.Let the n-times conformable fractional diferential system be Consider R as the state space for the system (12).By defnition, controllability concepts for this system revolve around whether the relevant reachable set is either equal to or densely spread across the real space R. Expressing this system in the form of a frst-order diferential equation is straightforward as follows: Journal of Mathematics Te fractional control system ( 13) is defned within an ndimensional Euclidean space, denoted by R n , and as a result, its reachable set shall be a subset in R n .Consequently, the controllability criteria for the system (13) are more stringent compared to those of the system (12).Nonetheless, these criteria can be facilitated through the use of the projection operator P, which can be defned as follows: Tis operator makes the conditions for partial controllability of system (13) the same as the conditions for controllability of system (12).
Example 2. Given a time CF-control system of wave equation as follows: where 0 < q ≤ 1, y is a real valued function defned on [0, ∞) × (0, 1) and z q c is a partial conformable fractional operator of order q .Te state space of this system is L 2 (0, 1).By enlarging the state space, we can rewrite this system in the frst-order CF-control system as follows: if where x ∈ L 2 (0, 1) × L 2 (0, 1).Te ordinary controllability concept for the system (20) are too strong comparable with the same for the system (17).However, we can reduce this difculty by defning the projection operator P as which makes the studying the partial CF controllability of the system (20) the same as the studying ordinary CF controllability of the system (17).
Consider the abstract fractional semilinear system with conformable derivatives as follows: where x and u are state and control values, respectively.Now, let impose the following assumptions (A2) P is a projection operator from X to H.
Under the above assumptions, the ( 22) has a unique mild solution x ∈ C(0, τ; X) for every u ∈ U ad and x 0 ∈ X (see, Jaiswal and Bahuguna [31]), and this solution can be written as follows: Defne the set that stands for the attainable set of control system (22) at a fnite time τ.Defnition 4. Te fractional system ( 22) is said to be partially exactly controllable on U ad if R(τ, ψ) � H for every ψ ∈ X.
Let the fractional controllability operator with a conformable fractional derivative Π t r be given as: where * indicates the adjoint operator.
where x, u, X, U, A, and B are as defned above.Partial controllability of linear control systems is similar to ordinary controllability in many ways.In particular, if we replace the controllability operator with its partial version, 4 Journal of Mathematics most of the results on ordinary controllability can be applied to partial controllability as well.Tis is possible by imposing certain conditions on the partial controllability operator  Π T 0 .In the following theorem, we give necessary and sufcient conditions for partial controllability of conformable fractional linear systems.Theorem 5.Under the above assumptions, the following assertions are equivalent:a (i) Te linear system (32) is partially exactly controllable Proof.Te proof of this theorem closely follows the proofs of similar theorems presented in many papers, for example, see the works of Mahmudov [32] and Jneid [33].So we are not going to repeat the proof here.

Partial Exact Controllability
In this section, we provide a sufcient condition set of partial exact controllability for the semi-linear fractional control system in conformable sense by using a contraction mapping principle.
Lemma .Assume that the assumptions (A1)-(A3) hold true.Ten, for every 0 ≤ t ≤ τ, the following inequalities valid Proof.It is clear that for every 0 Hence, Terefore, where Ten, where, for every 0 ≤ t ≤ τ, Te following estimate is true: where Starting with the frst norm ‖Y − X‖ C(0,τ;X) , we obtain In a similar way, for the second norm ‖V − W‖ U ad , one can obtain: 6

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By combining the inequalities ( 25) and ( 32), the proof is done.
Proof.Due to Lemma 7 and the inequality (40), it is clear that Q is a contraction mapping on C(0, τ; X) × U ad .Terefore, Tank to the well-known Banach fxed point theorem Q admits a single fxed point.
Proof.Let h ∈ H.We would prove that there is a control state u ∈ U ad such that h � Px τ .To do this, defne u by By inserting (40) into (22), one can obtain: Now, setting t � τ in (41), we acquire

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Px T � P Θ Terefore, we observe that the control u ∈ U ad steers the control system (20) from ψ at initial time to x τ at terminal time τ, such that the partial state Px τ � h is accomplished.Terefore, the given control system (20) in conformable sense is partially exactly controllable on U ad for the terminal time τ.

Illustrative Example
Example 3. Given a control system of fractional equations in conformable sense where u ∈ C(0, τ; R), (y, While, the concept of partial controllability is understood in R as Te control system (43) can be interpreted in R 2 as where Since the operator A is a matrix, the CF− semigroup Θ (t q /q) can be simply calculated as follows:

Θ
t q q   � e A t q /q ( ) � Hence, we can compute M as follows: (49) By carrying out elementary calculations, the CF− controllability operator (24) of the system(46) can be computed as follows: s q− 1 e As q /q BB * e A * s q /q ds � Clearly, Π 1 0 is not coercive.Terefore, the control system (43) is not exactly controllable.However, the system (43) can be exactly controllable under the appropriate conditions of the function G if we analyze the partial exact controllability considering only the frst component y t of the state vector x t .Tis can be done by defning the projector operator P as P � 1 0  .Terefore, Tis shows that the corresponding linear part of the semilinear system (46) is partially exactly controllable.
We now compute the Lipschitz constant N for the function G � (G 1 , G 2 ) T as follows: First, we fnd the Lipschitz constant for G 1 (t, y, z, u) � t 2 /40 + 3t 2 cos (y t + z t + u t ).To fnd the Lipschitz constants in y denoted by N 1 y , in z denoted by N 1 z and in u denoted by N 1 u , we take the partial derivatives in terms of y, z and u respectively.Ten calculate the supremum of the absolute values of the partial, we obtain For N, we have Now, evaluating the expression L q , defned previously by (39) we acquire Let q � 2/3.Substitute this value into (53), we obtain as follows: which guaranties that the condition (39) holds.Let q � 5/7.Substitute this value into (53), we obtain as follows: which guaranties that the condition (39) holds.Let q � 9/10.Substitute this value into (53), we obtain as follows: L q � 3 + which guaranties that the condition (39) holds.Let q � 1. Substitute this value into (53), we obtain as follows: which guaranties that the condition (39) holds.
Hence, regarding the above four values of q, all the assumptions of Teorem 9 are fulflled.Consequently, the CF control system described in (43) is partially exactly controllability over the interval [0, 1].Given a control system of fractional equations in conformable sense

Discussion
As observed in the variation of the parameter q, it becomes clear that each choice of the fractional-order q leads to a distinct fractional control system that can be partially exactly controllable.Tis emphasizes the idea that fractional Journal of Mathematics calculus introduces new opportunities and dimensions to control theory, while still preserving known fndings from ordinary calculus when q � 1.Furthermore, the order q becomes a powerful modeling parameter that can be optimized and precisely calibrated to suit the control requirements.Tis underscores the potential of fractional calculus to open new unexplored boundaries in control theory.

Conclusion
In this paper, the notion of partial exact controllability for conformable fractional control systems is introduced and a sufcient condition set for them is obtained.Tese conditions are obtained for semilinear control system, given that its associated linear part is also partially exactly controllable.Te method employed for this type of system closely resembles the one used for nonpartial systems, with a small adjustment.Te efectiveness of this approach has been demonstrated through an illustrative example.
Future research studies will focus on the partial controllability of stochastic fractional control systems with infnite and fnite delay, fractional systems with noninstantaneous impulses, and partially observable stochastic control systems with non-integer orders.