On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions

Abstract The aim of this study is to investigate the finite approximate controllability of certain Hilfer fractional evolution systems with nonlocal conditions. To achieve this, we first transform the mild solution of the Hilfer fractional evolution system into a fixed point problem for a condensing map. Then, by using the topological degree approach, we present sufficient conditions for the existence and uniqueness of the solution of the Hilfer fractional evolution systems. Using the variational approach, we obtain sufficient conditions for the finite approximate controllability of semilinear controlled systems. Finally, an example is provided to illustrate main results.


Introduction
In this study, we investigate the following Hilfer fractional evolution system: is a given function that will be specified later. The control function u is taken in ( ′ ) L J U , 2 and the admissible control set U is a Hilbert space, B is a bounded linear operator from U into H. Finally, x 0 is an element of H [1][2][3][4][5][6][7][8][9][10].
Hilfer [11] presented the generalized Riemann-Liouville fractional derivative, which is called the Hilfer fractional derivative. In [12], by using the fixed point theory, semigroup theory, the authors investigated the approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. More results on this topic can be found in [13,14].
It is known that finite approximate controllability is a consequence of approximate controllability. For a finite dimensional subspace of bounded domains, there exists an orthogonal projection over it of the solution at the end of control time, such that the final state satisfies simultaneously a finite number of exact constraints. Due to the strong practicality and applicability, in recent years, some researchers have studied the finite approximate controllability of some differential systems. In [15], semilinear variational inequalities with distributed controls have been studied. Zuazua [16] presented finite dimensional version of null controllability for the semilinear heat equation in bounded domains with Dirichlet boundary conditions. By applying the fixed point result of Leray-Schauder, Menezes [17] investigated the finite approximate controllability for a nonlocal parabolic problem. Mahmudov [18] replaced the uniform boundedness of the nonlinear function by some weaker natural conditions and examined the simultaneous approximate controllability and finite approximate controllability of some semilinear abstract equations.
The topological degree method is a powerful tool to certify the existence of solutions to fractional systems with the weaker conditions. In [19], by using the coincidence degree theory approach, Mawhin studied the existence of solutions to nonlinear boundary value problems. Dinca et al. [20] used the topological degree method to prove the existence of solutions of the Dirichlet problems with p-Laplacian. Isaia [21] applied the topological degree method along with condensing maps and proved the existence of solutions of a nonlinear integral equation. Wang et al. [22] used the topological degree method to solve a class of fractional equations. In [23], Iqbal et al. studied the solutions of coupled systems of multipoint boundary value problems of fractional order hybrid differential equations.
Motivated by the aforementioned works, specially [18] and [22], we offer to study the finite approximate controllability of certain Hilfer fractional evolution systems with nonlocal conditions. Our aim is to obtain some suitable growth conditions for the existence of solutions of system (1.1) by using the topological degree approach under the assumption that the operators are bounded and to apply the variational approach to prove the finite approximate controllability.

Preliminaries
In this section, we recall some definitions, notations, and preliminary results concerning fractional calculus and finite approximate controllability, which will be used later, and furthermore, we consider the mild solution of (1.1) using two classes of operators which will be specified later. Let = [ ] J b 0, and E be a Banach space with norm ∥⋅∥ E . Now E ⁎ denotes its dual and 〈⋅ ⋅〉 , E denotes the duality pairing between E ⁎ and E. We use ( ) to denote the space of bounded linear operators with the norm ∥⋅∥ ( ) be the Banach space of all continuous functions from J to E. Set = + − γ ν μ νμ, < < γ 0 1, and then − = ( − )( − ) , : lim exists and is finite We collect some definitions and remarks of fractional calculus of Hilfer type, finite approximate controllability, non-compactness of Kuratowski type, and the topological degree theories. For more details, we refer to [13][14][15][16]21,[24][25][26][27][28][29][30][31][32][33].
of system (1.1) satisfies the following conditions: where E is a finite dimensional subspace of H and Π E is the orthogonal projection from H to E.
Definition 2.1. It states that the approximate control u ε can be chosen such that condition 2.1 holds and simultaneously a finite number of exact constraints that condition 2.2 also holds.
In this study, B denotes the family of all bounded sets for the Banach space X, ∈ Q B.
Definition 2.2. The Kuratowski measure v of non-compactness of Q, denoted by ( ) v Q , is the infimum of the set of all numbers > k 0 such that Q admits a finite cover by sets with diameters > k 0, that is The Kuratowski measure v of non-compactness has the following properties: Obviously, v-condensing mapping is v-Lipschitz as = L 1.
Denote the collection of all strict v-contractions is Lipschitz with constant L 1 , then F is v-Lipschitz with the same constant L 1 .
are Lipschitz with constants L 1 and L 2 , respectively, then + → Ω X : The following definition is based on Definition 2.3 in [14] and Definition 5 in [34].
and there exists ∈ f L H 2 a.e. on ∈ ′ t J as follows: In Section 1, we assumed that ( )( ≥ ) T t t 0 is uniformly bounded, so there exists an

Existence of mild solutions
Before proceeding to the proof of the existence of mild solutions for (1.1), we propose the following assumptions: is a function such that: . ( ) H B : the following linear fractional control system is approximately controllable on ′ J . Under the aforementioned assumptions of ( ) ( ) H f H ψ , , we will prove that the Hilfer fractional evolution system (1.1) has at least one solution ∈ x Y. Define operators , .
is Lipschitz with constant L g . Consequently, is v-Lipschitz with the same constant L g . Furthermore, satisfies the following growth condition: . In view of Proposition 2.8, is v-Lipschitz with the same constant L g . □ Lemma 3.2. The operator is continuous and satisfies the following growth condition: Mb a x m Γ μ 1 . Proof. To prove the compactness of , consider the bounded set ⊂ B k D , and we need to show that ( ) D is relatively compact in Y.
In fact, from Lemma 3.2, we know ( ) D is bounded. For ∈ x n D and < < ≤   x Y and the set of the solution is bounded in Y.
Proof. In this section, we have defined three operators , , and they are continuous and bounded. Furthermore, the operator → Y Y : is Lipschitz with constant L g , is v-Lipschitz with zero constant. Define In this section, we study the finite approximate controllability of (1.1). Take into account two relevant operators: where denotes the identity operator, B ⁎ denotes the adjoint of B, and (⋅) μ ⁎ T is the adjoint of (⋅) μ T . We can choose the functional defined by