Pontryagin maximum principle for fractional delay differential equations and controlled weakly singular Volterra delay integral equations

In this article, we explore two distinct issues. Initially, we examine the utilization of the Pontriagin maximum principle in relation to fractional delay differential equations. Additionally, we discuss the optimal approach for solving the control problem for equation (1.1) and its associated payoff function (1.2). Following that, we investigate the application of the Pontryagin Maximum principle in the context of Volterra delay integral equations (1.3). We strengthen the results of our study by providing illustrative examples at the end of the article.


INTRODUCTION
The Pontryagin Maximum Principle is a powerful tool in optimal control theory that is used to derive necessary conditions for finding optimal control trajectories.This principle was first introduced by Lev S. Pontryagin in 1956, and it is widely used in various mathematical and applied fields.
In recent years, the Pontryagin Maximum Principle has been extended to fractional delay differential equations (FDDEs) and delayed Volterra integral equations (DVIEs).These equations are widely used in various areas of science and engineering, including control systems, ecology, finance, biology, physics, and chemistry.
The Caputo FDDE is a type of fractional differential equation that contains a time delay term.It describes the dynamics of systems that exhibit memory effects, and it has been used to model various phenomena such as heat conduction, viscoelasticity, and fractional order control systems.The delayed Volterra integral equation, on the other hand, is a type of integral equation that contains a time delay term.It describes the behavior of systems that exhibit history-dependent dynamics, and it has been used to model various phenomena such as population dynamics, chemical reaction kinetics, and electrical networks.
The Pontryagin Maximum Principle for Caputo FDDEs and DVIEs is a set of necessary conditions that must be satisfied by optimal control trajectories for these equations.It provides a method to solve optimization problems involving these equations, such as finding the control inputs that minimize the energy consumption or maximize the system performance.
The necessary conditions derived from the Pontryagin Maximum Principle for Caputo FDDEs and DVIEs involve the optimal control trajectory, the corresponding adjoint function, and other variables that depend on the specific problem being considered.These conditions are often expressed in terms of differential or integral equations, and they can be used to derive optimal control strategies for a wide range of systems.
The article deals with two issues: Firstly, let us consider the following Pontriagin Maximum Principle for delayed differential equation.
C D α t y(t) = f (t, y(t), y(t − h), u(t)), a.e.t ∈ [0, T ], y(t) = 0, −h ≤ t ≤ 0, h > 0. (1.1) In the overhead, η(•) and f (•, •, •, •, •) are given functions, called the free term and the generator of the state equation, respectively, y(•) is called the state trajectory taking values in the Euclidean space R n , and u(•) is called the control taking values in some separable metric space U .In order to evaluate the effectiveness of the control, we establish a payoff functional ( with a term on the right hand expressing the running cost.
In article [ [1]], Lin and Yong addressed the issue of a state equation without delay, focusing on a cost functional that encompasses both running costs and pre-specified instant costs.However, in our paper, we extend this framework by considering a state equation generator that incorporates delay variables.Furthermore, our proposed cost functional not only includes running costs but also incorporates delay variables into these running costs.

Preliminaries
In the following passage, we will share initial discoveries that will be helpful for future investigations.Firstly, we will focus on a specific period of time labeled as T > 0.Then, we will establish certain spaces: Also, we define In the subsequent analysis, we utilize the notation ∆ We call a continuous and strictly increasing function ω(•) : R + → R + a modulus of continuity if ω(0) = 0.
Definition 2.1.( [19]) The fractional integral of order α > 0 for a function φ : [0, T ] → R is defined by where Γ : (0, ∞) → R is the well-known Euler gamma function defined as Definition 2.2.( [19]) The Riemann-Liouville fractional derivative of the order 0 < α ≤ 1 for a function The relationship Caputo and Riemann-Liouville fractional differentiation operators for the function φ(•) ∈ L ∞ [0, T ] is as follows: First, we will denote the following necessary theorem, which we use it for proof of the lemma.

Main results
Suppose that U , a separable metric space with the metric ρ.U can be either a nonempty bounded or unbounded set in R m with the metric induced by the usual Euclidean norm.We can view U as a measurable space by considering the Borel σ-field.Let u 0 be a fixed element in U .For any p ≥ 1, we define

Pontryagin's maximum principle for the fractional differential equation
Let us consider the following problem (3.1) In this paragraph, we negotiate the optimal control problem for (3.1) with payoff functional (3.2) Let us acquaint the following hypothesises.The assumed conditions are more than sufficient.Without loss generality, we state to utilize these conditions.( for some p > 1 α and α ∈ (0, 1), u 0 ∈ U . ) for some modulus of continuity ω(•).Moreover, it is evident that L is included in a smaller space, compared to the space to which L 0 belongs.(H2): Let g : [0, T ] × R n × R n × U → R be a transformation with t → g(t, y, y h , u) being measurable, (y, y h ) → g(t, y, y h , u) being continuously differentiable, and (y, y h , u) → (g(t, y, y h , u), g y (t, y, y h , u), g y h (t, y, y h , u)) being continuous.There is a constant L > 0 such that Obviously, the payoff functional (3.2) is well-defined under (H1) and (H2).Thus, we are able to consider the following optimal control problem (OCP).
Arbitrary u * (•) satisfying (3.8) is called an optimal control of Problem (P), the appropriate state y * (•) is called an optimal state, and (y * , u * ) is called optimal pair.Now, Pontryagin's maximum principle can be given for the Problem (P).
Theorem 3.1.Let (H1) and (H2) satisfy.Assume (y such that the following maximum condition satisfies: Proof.Proof of theorem consists of two steps. step1.A variational inequality.Let (y * (•), u * (•)) be an optimal pair of Problem(P).Fix an arbitrary and by using lemma 2.2 , accordingly, for arbitrary δ > 0, there exists an where f (•) is denoted the system of (3.11), and By using (H1), we obtain Obviously, ϕ(•) ∈ L q (0, T ) for some q ∈ ( 1 α , p).We get By applying the Gronwall's inequality (Lemma 2.1), selecting q ′ ∈ ( 1 α , q), and using Hölder inequality, we have It follows Y δ (•) be a solution of the following : Let be a solution of the following variational equation We get By applying the Gronwall's inequality (Lemma 2.1), mention q > 1 α and L(•) ∈ L By using the dominated convergence theorem, we get In addition, by (3.12), Applying the Gronwall's inequality, As a result, using the dominated convergence theorem, we get With optimality of (y * (•), u * (•)), we have where g(•) is denoted in (3.11), and Therefore, by utilizing (3.12) and the convergence where and Y (•) is the solution of the following differential equation: Let u ∈ U be fixed and let s 0 ∈ (0, T ) be a Lebesgue point of Ψ(s, u * ) − Ψ(s, u).Then for any δ > 0, let Therefore, from the above, we get Consequently, by the Lebesgue point theorem, we get the maximum condition (3.10).

Maximum principle for the delayed Volterra equation with the weak singular kernel
Let us research the following issue where η(•) ∈ L p (0, T ; R).
Now, we discuss the optimal control problem for (3.15) with payoff functional (3.16).We introduce certain assumptions that are considered to be more than enough for our requirements.Therefore, we will make use of these conditions without any loss of generality.(H1 ′ ): Let f : ∆ × R n × R n × U → R n be a transformation with (t, s) → f (t, s, y, y h , u) being measurable, (y, y h ) → f (t, s, y, y h , u) being continuously differentiable, (y, y h , u) → f (t, s, y, y h , u) , (y, y h , u) → f y (t, s, y, y h , u) and (y, y h , u) → f y h (t, s, y, y h , u)being continuous.Furthermore, there are non-negative functions involved in these conditions.L 0 (•), L(•) with for some p > 1 α and α ∈ (0, 1), u 0 ∈ U .
for some modulus of continuity ω(•).Moreover, it is evident that L is included in a smaller space, compared to the space to which L 0 belongs.(H2 ′ ): Let g : [0, T ] × R n × R n × U → R be a transformation with t → g(t, y, y h , u) being measurable, (y, y h ) → g(t, y, y h , u) being continuously differentiable, and (y, y h , u) → (g(t, y, y h , u), g y (t, y, y h , u), g y h (t, y, y h , u)) being continuous.There is a constant L > 0 such that Given that conditions (H1 ′ ) and (H2 ′ ) hold, we may proceed to analyze the optimal control problem (OCP) defined by the payoff functional (3.16).
An optimal control for Problem (P ′ ) is any arbitrary u(•) that satisfies equation (3.22).The corresponding state y(•) is known as an optimal state, and when combined with the optimal control u, they form the optimal pair (y, u * ).Now, Pontryagin's maximum principle can be applied to Problem (P ′ ) and can be restated as follows.
Lemma 3.1.Let (H1 ′ ) and (H2 ′ ) satisfy.Then the following result holds: where Y is the solution to the first variational equation related to the optimal pair (y * , u given by such that the following maximum condition satisfies: Let u ∈ U be fixed and let s 0 ∈ (0, T ) be a Lebesgue point of Ψ(s, u * ) − Ψ(s, u).Then for any δ > 0, let Therefore, from the above, we get Consequently, by the Lebesgue point theorem, we get the maximum condition (3.27).

Example
Let us consider the following fractional delay optimal control problem (FDOCP).
where x ∈ R n , u ∈ R m , A and B are n × n matrices, C is an n × m matrix with n > m.The solution of (4.8) is given as x(s) = W s α+1 − Ae −sh s by using inverse Laplace transform A k e −skh s α(k+1)+1 L −1 1 s α+1 − Ae −sh s (t) =   For our problem, it is easy to check that, X α (t) and X α,α (t − s) are given the following form A k (t − kh) αk Γ(αk + 1) , and X α,α (t − s) = It means that the pair (y * (t), u * (t)) is the only pair, which can be a locally optimal solution of problem (4.1)-(4.4).