A new approach on approximate controllability of Sobolev-type Hilfer fractional differential equations

The approximate controllability of Sobolev-type Hilfer fractional control differential systems is the main emphasis of this paper. We use fractional calculus, Gronwall’s inequality, semigroup theory, and the Cauchy sequence to examine the main results for the proposed system. The application of well-known fixed point theorem methodologies is avoided in this paper. Finally, a fractional heat equation is discussed as an example


Introduction
Differential systems of fractional order are found to be useful models for a variety of physical, biological, and engineering challenges. As a result, they have gotten greater attention from researchers in the last two decades. Fractional derivatives are a stronger tool for illustrating memory and hereditary features. As a result, they've found widespread use in physics, electrodynamics, economics, aerodynamics, control theory, viscoelasticity, and heat conduction. In recent years, significant advances in the theory and applications of fractional systems have been made, one can review the books [1][2][3][4]. The notation of exact and approximate controllability is useful in analysis and design control frameworks. In [5] authors studied the existence and controllability of fractional integro-differential system of order 1 < r < 2 via measure of noncompactness using fixed point theory approach. In [6][7][8][9][10][11][12][13] Anurag et al. studied the controllability of semilinear deterministic and stochastic systems of integral and fractional order with several important extensions using different approaches. The numerical model of numerous physical phenomena, such as the movement of liquid through split rocks, thermodynamics, and so on, is usually Sobolev-type. (see [14][15][16][17]).
Another type of fractional order derivative introduced by Hilfer [18] is the Caputo fractional and Riemann-Liouville derivative. Several authors have focused on the Hilfer fractional derivative including [19][20][21][22][23][24][25][26][27] for the existence and controllability of deterministic and stochastic fractional order systems. Many academics have recently considered the exact and approximate controllability of *Corresponding Author systems characterized by impulsive functional inclusions, integro-differential equations, semilinear functional equations, neutral functional differential equations, and evolution inclusions, to name a few examples, see [23,24,27] and references in that. In [28][29][30][31][32][33][34] Ravi et. al. studied the existence, uniqueness, controllability, and optimal control of fractional differential control systems and their real-life mathematical applications using various types of approaches. Consider the following Sobolev-type Hilfer fractional control system as below.
The non densely defined closed linear operator The function F : J × X × U → X is a purely nonlinear function and B : X → U is a bounded linear operator.
This article makes the following major contributions: • Using two separate situations, investigate the sufficient conditions for the approximate controllability of the suggested systems (1)-(2). • In the first case, we assume that B = I (where I is an identity operator) and in the second case, we assume that B ̸ = I. • controllability results are achieved using Gronwall's inequality and the Cauchy sequence. • Results are obtained with weaker conditions (Lipschitz) on nonlinearity and can be extended for the delay differential equations. • The suggested method is simple in terms of hefty estimations as compared to standard ways such as the fixed point theory approach. • The results are advanced and weighted enough as contribution in control differential equations.
We have divided this paper into the following sections: Section (1) provides a review of some essential concepts and preparatory outcomes Section (2). The main discussion of our manuscript is presented in Section (3). Finally, in Section (4), an application for drawing the theory of the primary outcomes is discussed.

Preliminaries
Let the spaces of all continuous functions is denoted by C(J, X). Take η = ℘ + ϖ − ℘ϖ, The linear operators A : (P 1 ) A and L are closed linear operators.
Additionally, because (P 1 ) and (P 2 ), L −1 is closed, by (P 3 ) and from closed graph theorem, we have the boundedness of AL −1 : Introducing acquaint essential facts relevant to fractional theory. (The readers can check [18,35]).
The left sided Riemann-Liouville fractional integral of order ϖ having lower limit c for F : [c, +∞) → R is presented as The left-sided Hilfer fractional derivative of order 0 ≤ ℘ ≤ 1 and 0 < ϖ < 1 function of F (ϱ) is given by
In case F ≡ 0, then the system (1)-(2) reduces to the corresponding linear system. The reachable set in this case is denoted by K c (0)." Definition 6.
[6] "If K c (F ) = X, then the semilinear control system is approximately controllable on [0, c]. Here K c (F ) represents the closure of K c (F ). It is clear that, if K c (0) = X, then linear system is approximately controllable."

Controllability results
3.1. Controllability of semilinear system: when B = I The linear system has an approximate controllability is proven to reach from the semilinear system under specified nonlinear term constraints in this study. Clearly, X = U . Let us consider the subsequent linear system and the semilinear system We need to present the following assumptions to prove the primary aim of this section, which is the approximate controllability of (6)- (7):
We consider for the given u(σ) and w(σ), there exists v(ϱ) fulfilling (10) (We need to verify the existence and uniqueness of v).
The mild solution of (4)-(5) is given by and for the system (8)-(9) is given by From (11) and (12), we get Applying norm on both sides, one can get Using assumption (2), we get By referring the Gronwall's inequality, w(σ) = z(σ), ∀ σ ∈ [0, c]. As a result, the linear system's solution w along the control u is a semilinear system's solution z along the control v, i.e., K c (F ) ⊃ K c (0). Because K c (0) is dense in X (according to assumption 1), K c (F ) is dense in X as well, implying that system (6)- (7) is approximate controllable. The proof is finished.
We need to verify that there exists a Assume that v 0 ∈ X and v n+1 = u − F (σ, w(σ), v n ) : n = 0, 1, 2, .... Thus, one can get Hence, by referring assumption (2), When n → ∞ (since l < 1), the R.H.S of (15)goes to zero. As a result, {v n } is a Cauchy sequence in X that converges to v ∈ X. Next, Because, R.H.S of (16) approaches to zero when n → ∞, one can obtain Now, we will show that v is unique. For proving it let v 1 = u − F (σ, w(σ), v 1 ) and v 2 = u − F (σ, w(σ), v 2 ). Then using assumption (2), we get Hence v is unique. □

Controllability of semilinear system: when B ̸ = I
The approximate controllability of the semilinear system under simple conditions B and F as indicated by assumptions (3)- (6). Let us consider the subsequent linear system and the semilinear system We must make the following assumptions to prove the fundamental aim of this section, namely, the approximate controllability of (19)- (20):  Proof. Assume that w(σ) and the control u are the mild solution of (17)- (18). Assume that the semilinear system that follows is In the above, the control function v in (21)- (22) fulfills Bv(σ) = Bu(σ) − F (σ, w(σ), v(σ)), and assumption (5), concludes that the considered equation is well defined. By employing assumption (6) and the way of approached followed in Theorem 2, we can easily prove that provided that l < ξ, ∃ v(σ) ∈ U such that Bv(σ) = Bu(σ) − F (σ, w(σ), v(σ)).

Conclusion
The focus of this study is on the Sobolev-type approximate controllability of Hilfer fractional semilinear control systems. The results were obtained using Gronwall's inequality, the Cauchy sequence, and the fixed point technique was avoided. With appropriate changes, these conclusions may be extended to include many types of delay for both deterministic and stochastic systems.
Remark 3. One can replace the Lipschitz condition on the nonlinearity by monotonic nonlinearity or integral contractor type nonlinearity and obtained a different set of sufficient conditions for the approximate controllability of the proposed system.