Analysis and Optimal Control of ϕ -Hilfer Fractional Semilinear Equations Involving Nonlocal Impulsive Conditions

: The goal of this paper is to consider a new class of ϕ -Hilfer fractional differential equations with impulses and nonlocal conditions. By using fractional calculus, semigroup theory, and with the help of the ﬁxed point theorem, the existence and uniqueness of mild solutions are obtained for the proposed fractional system. Symmetrically, we discuss the existence of optimal controls for the ϕ -Hilfer fractional control system. Our main results are well supported by an illustrative example.


Introduction
In recent years, a lot of research attention has been paid to the study of fractional calculus, which is considered as a generalization of classical derivatives and integrals to non-integer order. Phenomena with memory and hereditary characteristics that arise in ecology, biology, medicine, electrical engineering, and mechanics, etc, may be well modelled by using fractional differential equations (FDEs for short). For more details on FDEs and its applications, see [1][2][3][4][5] and the references therein. In [6], Hilfer derived a new two-parameter fractional derivative D σ 1 ,σ 2 a + of order σ 1 and type σ 2 , which is called Hilfer fractional derivative that combines the Riemann-Liouville and Caputo fractional derivatives. This kind of parameter produces more types of stationary states and gives an extra degree of freedom on the initial conditions. Systems based on Hilfer fractional derivatives are considered by many authors, see [7][8][9][10][11] and the references therein. Recently, Sousa and Oliveira [12] introduced a new fractional derivative with respect to another ϕ-function the so-called ϕ-Hilfer fractional derivative, and discussed their properties as well as important results of the fractional calculus. For more recent works on ϕ-Hilfer fractional derivative and its applications, we refer to [13][14][15][16][17] and the references therein.
Many real-world phenomena and processes which are subjected to external influences for a small time interval during their evolution can be represented as an impulsive differential equations. The impulsive differential equations have become the natural framework for modelling of many evolving processes and phenomena studied in the field of science and engineering such as in mechanical systems, biological systems, population dynamics, physics, economy, and control theory. Recently, based on the theory of semigroup and fixed point approach, many authors studied the qualitative properties of solutions for impulsive differential equations of order one and non-integer [18][19][20][21][22][23][24] and the references therein. The optimal control problem (OCP for short) plays a crucial role in biomedicine, for example, model cancer chemotherapy and recently applied to epidemiological models. When FDEs describe the system dynamics and the cost functional, an OCP reduces to a fractional optimal control problem. The fractional OCP refers to optimize the cost functional subject to dynamical constraints on the control parameter and state variables that where H D σ 1 ,σ 2 ;ϕ t γ + denotes the ϕ-Hilfer fractional derivative of order 1/2 < σ 1 < 1, 0 < σ 2 < 1 and the state z(·) takes values in a Hilbert space As usual z(t + γ ) and z(t − γ ) are the right and left limits of z at the point t γ , respectively. I γ : E → E are impulsive functions that characterize the jump of z at points t γ . The functions ∆ : J 0 × E → E, G : C(J 0 , E) → E are some suitable functions that will be specified later.
The rest of the manuscript is organized as follows. In Section 2, we recall some important concepts and results. In Sections 3 and 4, we derived the mild solution by using semigroup as well as probability density function and proved the existence of mild solutions for the proposed fractional system, receptively. In Section 5, we investigated the existence of optimal controls for the ϕ-Hilfer fractional control system. Moreover, in Section 6, an example is presented to demonstrate the applicability of the obtained symmetry results.

Preliminaries
Let J 1 = [a, b] and ϕ ∈ C m (J 1 , R) an increasing function such that ϕ (t) = 0, ∀ t ∈ J 1 . Definition 1. The ϕ-Riemann fractional integral of order σ 1 > 0 of the function R is given by The ϕ-Hilfer fractional derivative can be written as Lemma 2 ([12]). Let σ 1 > 0 and σ 2 > 0, then I For comprehensive details on ϕ-Hilfer fractional derivative and its properties, we refer to papers [12,14,17]. Consider the weighted space [14] defined as Define the space of piecewise continuous functions as

Representation of Mild Solution
Lemma 3. To reduce the generalized form (1), we consider the linear ϕ-Hilfer fractional differential system: has a mild solution, which is defined as where Proof. Rewrite the problem (2) in the equivalent integral equation provided that the integral in Equation (4) exists. Let β > 0. Applying the generalized Laplace transform It follows that Taking s =t σ 1 , we obtain where We consider the following one-sided stable probability density Using Equation (5), we obtain and Hence, we obtain By using inverse Laplace transform, we obtain Thus, we obtain Hence, we obtain 17,32]). The operators S σ 1 ,σ 2 ϕ and T σ 1 ϕ have the subsequent conditions 1. S σ 1 ,σ 2 ϕ (t, s) and T σ 1 ϕ (t, s) are linear and bounded operators for any fixed t ≥ s ≥ 0, and 2. If T (t) is compact operator for all t > 0, then S σ 1 ,σ 2 ϕ (t, s), T σ 1 ϕ (t, s) are compact for all t, s > 0. Hence, S σ 1 ,σ 2 ϕ (t, s) and T σ 1 ϕ (t, s) are strongly continuous. 3. The operators S σ 1 ,σ 2 ϕ (t, s) and T σ 1 ϕ (t, s) are strongly continuous. For every z ∈ E and Definition 5. A function z ∈ P C(E) is called a mild solution of problem (1) if for every t ∈ J 0 , z(t) fulfills I for every t ∈ [0, t 1 ] and for every t ∈ (t γ , t γ+1 ].

Existence and Uniqueness
In this section, we prove the existence outcomes of the proposed system (1). Let us assume the following hypotheses [X1]: T (t) is compact for every t > 0.
[X2]: The function ∆ : For all z ∈ E, the function t → ∆(t, z) is strongly measurable and the function ∆(t, · ) : E → E is continuous for a.e t ∈ J 0 .
Step 5. We prove that Obviously, δ(0) = {0} is relatively compact. Let t ∈ (t γ , t γ+1 ] be fixed, 0 < < t, and is real number. For z ∈ Ω π , we define Then δ(t) is relatively compact in E. By steps 3-5 and Arzela-Ascoli theorem, Π 2 is completely continuous. Hence, by the fixed point theorem of Krasnoselskii's [33], there exists at least one mild solution on J 0 . Proof. Let z 1 and z 2 be the mild solutions of the ϕ-fractional system (1) in Ω π . Then, for each k ∈ {1, 2}, the mild solutions z k satisfies

Existence of Optimal Controls
Let v takes the value in the separable reflexive Banach space T and V f (T ) is a class of subsets of T , which is nonempty convex and closed. The multifunction g : J → V f (T ) is measurable and g(·) ⊂ , the admissible control set where is a bounded set of T . Then U ad = φ.
We consider the Lagrange problem where the cost functional is where z v be the mild solution of (11) with respect to control v ∈ U ad . Next, we assume For almost all t ∈ J 0 , L(t, ·, ·) is sequentially lower semicontinuous on E × T .

3.
For each z v ∈ E and almost all t ∈ J 0 , L(t, z v , ·) is convex on T .

4.
There exist constants :D is a strongly continuous operator.
in L 2 (J 0 , T ). Since U ad is convex and closed, by using Marzur Lemma, we get v * ∈ U ad . Let z k and z * be the mild solution of system (11) with respect to v k and v * , respectively It follows from the boundedness of {v k }, {v * } and Theorem 2, we obtain there exists a constant Θ > 0 such that z k ∞ , z * ∞ ≤ Θ.
Thus, we have z k − z * P C −→ 0 as k → ∞, this yields that z k −→ z * in P C(E) as k → ∞. Since P C(E) ⊂ L 1 (J 0 , E), by using [X9] and Balder's theorem, we obtain