Optimal Controls for a Class of Impulsive Katugampola Fractional Differential Equations with Nonlocal Conditions

In this paper, we investigate a class of impulsive Katugampola fractional differential equations with nonlocal conditions in a Banach space. First, by using a fixed-point theorem, we obtain the existence results for a class of impulsive Katugampola fractional differential equations. Secondly, we derive the sufficient conditions for optimal controls by building approximating minimizing sequences of functions twice.


Introduction
In recent years, fractional calculus has received more and more attention in many fields because of the local limit definition of integral-order ordinary differential equation or partial differential equation which is not suitable to describe the history-dependent process and has received more and more attention in many fields. Leibnitz discovered fractional derivatives in 1695. Because fractional-order differential equation can describe objective laws and the essence of things more accurately than integral differential equation, many scholars have devoted themselves to the study of fractional calculus. Now, Riemann-Liouville calculus definition, Caputo differential definition, Grunwald-Letnikov differential definition, etc. are the most commonly used fractional calculus definition in basic mathematics research and engineering application research (see [1]). Fractional differential equations are widely used in practice and greatly enriched the content of mathematical theory and penetrated into many fields of natural science.
In 2011, Katugampola presented a new fractional integral, which generalizes the Riemann-Liouville and Hadamard integral into a single form. When a parameter was fixed at different values, it produces the above integrals as special cases; when ρ ⟶ 1, we can get the Riemann-Liouville operators; when ρ ⟶ 0, we can get the Hadamard operators (see [2]). And he presented two representations of the generalized derivative called Katugampola derivative in [3]. In the same work, he obtained boundedness in an extended Lebesgue measurable space and gave illustrative examples of generalized fractional integral. In [4], he obtained existence and uniqueness results to the solution of initial value problem for a class of generalized fractional differential equations. Since then, many scholars have done a lot of research on Katugampola derivative (see [5][6][7][8][9][10][11][12][13][14][15]).
Since the end of the last century, many scholars have devoted themselves to the study of approximate controllability (see [15][16][17][18][19][20]). Impulsive differential equation, which provides a natural description of observed evolution processes, is an important mathematical tool to solve some practical problems. The theory of impulsive differential equations of integer order has been widely used in practical mathematical modeling and has become an important area of research in recent years, which steadily receives attention of many authors (see [21][22][23][24][25][26][27][28][29][30][31][32][33][34]). Sun et al. [35] considered a class of impulsive fractional differential equations with Riemann-Liouville fractional derivative, the existence of solution was proved by using Darbo-Sadovskii's fixed-point theorem, and the optimal control results were obtained.
On the basis of previous work, in this paper, we initiate to study the existence of the following impulsive Katugampola fractional differential equation with nonlocal condition: where 0 < β < α < 1, ρ > 0, and a ∈ R and ρ I ð1−αÞ ða + Þ and ρ D α ða + Þ are the Katugampola fractional integral and derivative, which will be given in next section. x ∈ X, and X is a real Banach space; f is a nonlinear perturbation; gðxÞ is a given X-valued function; J k ðxðt k ÞÞ is a nonlinear mapping; a < t 1 < t 2 < ⋯<t k ≤ T; and U ad is a control set, the control u ∈ U ad . The rest of the paper is organised as follows. In Section 2, we give some preliminary facts that we need in the subsequent sections. In Section 3, we obtain the existence of solution of system (1) by using Darbo-Sadovskii's fixed-point theorem. In Section 4, we establish the optimal controls for a Lagrange problem. In Section 5, an illustrative example is given to show the practical usefulness of the analytical results.

Preliminaries
In this section, we present some results which will be useful throughout the paper.
Let X be a real Banach space; £ðXÞ is a kind of linear operators in X and not necessarily bounded.
Hausdorff measure of noncompactness σð·Þ defined on each bounded subset Ω of the Banach space X, and σðΩÞ = inf fε > 0g (Ω has a finite ε-net in X).
Lemma 7 (see [37]). If G ⊂ X is a convex bounded and closed set, the continuous mapping Λ : G ⟶ G is a σ-contraction and then has at least one fixed point in G.
The set N which has controls u is another separable reflexive Banach space. Let P f ðNÞ ⊂ N be nonempty closed and convex. We assume that mapping w : ða, T ⟶ P f ðNÞ is 2 Journal of Function Spaces multivalued and measurable, wð·Þ ⊂ O, O ⊂ N is bounded, the control set U ad is defined as U ad = V p w = fu ∈ L p ðOÞ | uðtÞ ∈ wðtÞ, a:e:g, p > 1, and U ad ≠ ∅.

Existence of Solution
In this section, we obtain the existence of solution of system (1) by using Darbo-Sadovskii's fixed-point theorem.

Lemma 8.
A solution x ∈ PC δ ðða, T ; XÞ of problem (1) can be expressed as Proof. Suppose that x ∈ PC δ ðða, T ; XÞ is a solution of problem (1). By (9), we can get By using formula (10), we can get where the second term in the right side is a continuous function on ða, T. In particular, Obviously, the impulsive condition ρ D β a + xðt + k Þ − ρ D β a + xð t − k Þ = J k ðxðt k ÞÞ is a discontinuity condition, which means we can get Since this function satisfies problem (1) in view of (8), for t ∈ ðt 1 , t 2 , we obtain so that Combining (15) and (18) and applying problem (1) yield Repeating the above process, the solution xðtÞ for ðt k−1 , t k can be written as Conversely, by a simple calculation of (12), we can easily get (1). This completes the proof.
(H2). There exists a function φ such that where r is a finite positive constant and r > 0. For a.e., t ∈ ða, T and x ∈ B r , ρ I α a φ r ðtÞ exists and is bounded and continuous.
where q is a constant, q > 0, and σ is the Hausdorff measure of noncompactness.
Proof. Let x 0 ∈ X be fixed. We consider the operator V on P C δ ðða, T ; XÞ such that The proof is given in following several steps.
Step 1. V maps W r into W r .
For any x ∈ W r , by (H1)-(H6) and (12), we can obtain Step 2. V is continuous in W r .

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Let x 1 , x 2 ∈ W r ; we can get Because g, J k , and f are continuous, we can obtain V is continuous.
Step 3. V has at least one fixed point. Let for each x ∈ W r . By condition (H5), we can obtain that the operator I 1 is compact in W r , and σðI 1 W r Þ = 0. From (H1) and (H3), we can get that f ðt, xðtÞÞ is a compact mapping in W r ; we omit the proof of this step as it is similar to that of Theorem 4.34 in [38]. Combining with (H4), we know the operator I 3 is compact in W r , and σ ðI 3 W r Þ = 0. Then, by condition (H6), we can know that fJ k g m k=1 are Lipschitz continuous. For x 1 , x 2 ∈ W r , we have Therefore, By (12), we can obtain By Lemma 3, we can easily obtain that the operator V has a fixed point in x ∈ W r ; i.e., (1) has at least one solution. This completes the proof.

Optimal Controls for Problem
In this section, we establish the optimal controls for a Lagrange problem.
Let x u ∈ W r denote the solution of problem (1) associated with the control u ∈ U ad .
We research the following limited Lagrange problem ðPÞ: For all u ∈ U ad , there exist a x 0 ∈ W r ⊆ PC δ ðða, T ; XÞ and u 0 ∈ U ad such that where x 0 ∈ W r denotes the solution of problem (1) associated with the control u 0 ∈ U ad . We introduce the following hypotheses: The function ψ : ða, T × X × N satisfies the following: (Ψ1). The function ψ : ða, T × X × N ⟶ R ∪ ∞ is Borel measurable.