On Riesz-Caputo Formulation for Sequential Fractional Variational Principles

Abstract

This paper deals with sequential Riesz-Caputo fractional variational problems with and without the presence of delay in the state variables and their derivatives. In both cases the necessary conditions for the optimal control are reported.

1. Introduction

Fractional calculus which is as old as the classical calculus has become a candidate in solving problems of complex systems which appear in various branches of science [1–10].

In the theory of time-delay systems the correct formulation of some initial value problems together with the representation of solutions as well as the asymptotic behavior of solutions are some of the general problems to be discussed. From the point of view of the fractional calculus we have almost the same problems to be discussed. The combination of these two main tools will lead us to a better description of the dynamics of the complex systems. In [10], fractional order modeling of delay dynamics was used to better characterize the delay behavior. Recently, it was shown that the delay parameter modifies the time window for the fractional order time derivative by allowing fading memory of an earlier time and introduces specific information about initial conditions (see [11] and the references therein).

Optimal control problems with time delay in calculus of variations were investigated in [12–14]. Variational optimal control problems within the framework of fractional calculus were considered in [15]. Very recently, the fractional variational problems in the presence of delay were discussed in [16, 17].

The aim of this paper is to find necessary conditions for sequential Riesz-Caputo fractional variational and optimal control problems with or without delay.

The structure of the paper is as follows: in Section 2 the basic definitions in fractional calculus are presented. Section 3 is devoted to the variational and optimal control problem within sequential Riesz-Caputo fractional derivatives in the absence of delay. In Section 4 the Riesz-Caputo variational principles with delay are discussed.

2. Basic Definitions

In this section, some basic definitions related to fractional derivatives are presented.

The left Riemann-Liouville fractional integral and the right Riemann-Liouville fractional integral are defined, respectively, by_𝑎𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑎(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝐼𝑑𝜏,_𝛼𝑏𝑓(𝑡)=1Γ(𝛼)^𝑏_𝑡(𝜏−𝑡)^𝛼−1𝑓(𝜏)𝑑𝜏, (2.1) where 𝑓∈𝐿₁(𝑎,𝑏). Here and in the following Γ(𝛼) represents the gamma function.

The left Riemann-Liouville fractional derivative and the right Riemann-Liouville fractional derivatives are defined, respectively, by_𝑎𝐷^𝛼𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑡_𝑎(𝑡−𝜏)^{𝑛−𝛼−1}𝑓(𝜏)𝐷𝑑𝜏,_𝛼𝑏𝑓(𝑡)=1Γ()−𝑑𝑛−𝛼𝑑𝑡^𝑛𝑏_𝑡(𝜏−𝑡)^{𝑛−𝛼−1}𝑓(𝜏)𝑑𝜏, (2.2) where 𝛼>0,𝑛=[𝛼]+1, and 𝑓∈𝐴𝐶^𝑛[𝑎,𝑏] (the space of complex-valued functions 𝑓(𝑡) having continuous derivatives up to 𝑛−1 on [𝑎,𝑏]) and 𝑓^(𝑛−1)(𝑥)∈𝐴𝐶[𝑎,𝑏] (the space of absolutely continuous functions on [𝑎,𝑏]). In particular,_𝑎𝐷⁰𝑓(𝑡)=𝐷⁰_𝑏𝑓(𝑡)=𝑓_𝑎(𝑡),𝐷^𝑛𝑓(𝑡)=𝑓^(𝑛)(𝑡),𝐷^𝑛_𝑏𝑓(𝑡)=(−1)^𝑛𝑓^(𝑛)(𝑡). (2.3)

The fractional derivative of a constant takes the form_𝑎𝐷^𝛼()𝐶=𝐶𝑡−𝑎^−𝛼Γ(1−𝛼) (2.4) and the fractional derivative of a power of t has the following form:_𝑎𝐷^𝛼(𝑡−𝑎)^𝛽=Γ(𝛼+1)(𝑡−𝑎)^𝛽−𝛼Γ,𝛽−𝛼+1 (2.5) for 𝛽>−1,𝛼≥0.

The left Caputo fractional derivative is^𝐶_𝑎𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎(𝑡−𝜏)^{𝑛−𝛼−1}𝑑𝑑𝜏^𝑛𝑓(𝜏)𝑑𝜏, (2.6) while the right Caputo fractional derivative is^𝐶𝐷^𝛼_𝑏𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑏_𝑡(𝜏−𝑡)^{𝑛−𝛼−1}−𝑑𝑑𝜏^𝑛𝑓(𝜏)𝑑𝜏, (2.7) where 𝑓∈𝐴𝐶^𝑛[𝑎,𝑏] and 𝛼 represents the order of the derivative such that 𝑛=[𝛼]+1. By definition the Caputo fractional derivative of a constant is zero.

The Riemann-Liouville fractional derivatives and Caputo fractional derivatives are connected with each other by the following relations:^𝐶_𝑎𝐷^𝛼𝑓(𝑡)=_𝑎𝐷^𝛼𝑓(𝑡)−^𝑛−1_𝑘=0𝑓^(𝑘)(𝑎)Γ()()𝑘−𝛼+1𝑡−𝑎^𝑘−𝛼,_𝐶𝐷^𝛼_𝑏𝑓(𝑡)=𝐷^𝛼_𝑏𝑓(𝑡)−^𝑛−1_𝑘=0(−1)^𝑘𝑓^(𝑘)(𝑏)Γ()()𝑘−𝛼+1𝑏−𝑡^𝑘−𝛼. (2.8)

In [18, 19], the Riesz Riemann-Liouville fractional derivative or simply the Riesz fractional derivative and the Caputo Riesz fractional derivative are, respectively, represented as^𝑅_𝑎𝐷^𝛼_𝑏𝑓(𝑡)=12^𝑎𝐷^𝛼𝑓(𝑡)+(−1)^𝑛𝐷^𝛼_𝑏𝑓(𝑡), (2.9)_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑓(𝑡)=12^𝐶𝑎𝐷^𝛼𝑓(𝑡)+(−1)^𝑛𝐶𝐷^𝛼_𝑏𝑓(𝑡). (2.10)

In [1], a formula for the fractional integration by parts on the whole interval [𝑎,𝑏] was given by the following lemma.

Lemma 2.1. Let 𝛼>0, 𝑝,𝑞≥1, and 1/𝑝+1/𝑞≤1+𝛼(𝑝≠1𝑎𝑛𝑑𝑞≠1𝑖𝑛𝑡ℎ𝑒𝑐𝑎𝑠𝑒𝑤ℎ𝑒𝑛1/𝑝+1/𝑞=1+𝛼).(a)If 𝜑∈𝐿_𝑝(𝑎,𝑏) and 𝜓∈𝐿_𝑞(𝑎,𝑏), then ^𝑏_𝑎𝜑(𝑡)_𝑎𝐼^𝛼𝜓(𝑡)𝑑𝑡=^𝑏_𝑎𝜓(𝑡)𝐼^𝛼_𝑏𝜑(𝑡)𝑑𝑡. (2.11)(b)If 𝑔∈𝐼^𝛼_𝑏(𝐿_𝑝) and 𝑓∈_𝑎𝐼^𝛼(𝐿_𝑞), then ^𝑏_𝑎𝑔(𝑡)_𝑎𝐷^𝛼𝑓(𝑡)𝑑𝑡=^𝑏_𝑎𝑓(𝑡)𝐷^𝛼_𝑏𝑔(𝑡)𝑑𝑡, (2.12) where _𝑎𝐼^𝛼(𝐿_𝑃)={𝑓∶𝑓=_𝑎𝐼^𝛼𝜑,𝜑∈𝐿_𝑝(𝑎,𝑏)} and 𝐼^𝛼_𝑏(𝐿_𝑃)={𝑓∶𝑓=𝐼^𝛼_𝑏𝜙,𝜙∈𝐿_𝑝(𝑎,𝑏)}.

Lemma 2.1 was generalized in [16, Lemma 2] and [17, Lemma 2], with the same conditions, to be applicable to subintervals [𝑎,𝑟] and [𝑟,𝑏] where 𝑟∈(𝑎,𝑏).

3. The Sequential Riesz-Caputo Fractional Variational Problem

In this section we consider the following variational problem.

Minimize𝐽𝑦=^𝑏_𝑎𝐿𝑡,𝑦(𝑡),_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦(𝑡)𝑑𝑡, (3.1) where 𝑦∈𝐴𝐶^2𝑟[𝑎,𝑏], 0<𝛼≤1, 𝑟−1<𝑘𝛼≤𝑟,𝑟∈ℕ𝑦^(𝑖)(𝑎)=𝑐_𝑖,𝑦^(𝑖)(𝑏)=𝑑_𝑖,𝑖=0,1,…,𝑟−1, 𝑐_𝑖’s and 𝑑_𝑖’s are constant, and 𝐿∶[𝑎,𝑏]×ℝ^𝑘+1→ℝ has first and second partial derivatives with respect to all of its variables.

Theorem 3.1. Let 𝐽(𝑦) be the functional of the form (3.1) defined on the set of functions 𝑦(𝑡) which have continuous Riesz-Caputo derivatives of order 𝑖𝛼,𝑖=1,…,𝑘 and which satisfy the boundary conditions 𝑦^(𝑖)(𝑎)=𝑐_𝑖,𝑦^(𝑖)(𝑏)=𝑑_𝑖. Then a necessary condition for 𝐽(𝑦) to have a minimum for a given 𝑦(𝑡) is that 𝑦(𝑡) satisfy the following Euler-Lagrange equation: 𝜕𝐿+𝜕𝑦^𝑘_𝑖=1(−1)_𝑎^𝑖𝑅𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦=0. (3.2)

Proof. Assume that 𝐽(𝑦) has a minimum for 𝑦^∗(𝑡). Define a family of curves 𝑦(𝑡)=𝑦^∗(𝑡)+𝜖𝜂(𝑡) where 𝜖∈ℝ and 𝜂 is a function in 𝐴𝐶^2𝑟[𝑎,𝑏] satisfying 𝜂^(𝑖)(𝑎)=𝜂^(𝑖)(𝑏)=0,𝑖=0,1,…,𝑟−1. The function 𝛾(𝜖)=𝐽𝑦^∗(𝑡)+𝜖𝜂(𝑡) (3.3) admits a minimum when 𝛾′(0)=0. That is, ^𝑏_𝑎𝜕𝐿𝜂(𝑡)+𝜕𝑦∗^𝑘_𝑖=1𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦∗^𝑎𝑅𝐶𝐷_𝑏^𝑖𝛼𝜂(𝑡)𝑑𝑡=0. (3.4) Using (2.10), (2.8), and [1, Lemma  2.7], (3.4) becomes ^𝑏_𝑎𝜕𝐿+𝜕𝑦∗^𝑘_𝑖=1(−1)_𝑎^𝑖𝑅𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝜂(𝑡)𝑦∗𝑑𝑡=0. (3.5) Since 𝜂 is arbitrary, one has 𝜕𝐿𝜕𝑦^∗+^𝑘_𝑖=1(−1)_𝑎^𝑖𝑅𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦^∗=0. (3.6)

When 𝛼=1, the results in Theorem 3.1 coincide with the results in [20, p. 59].

Example 3.2. Let us consider the following action: ^𝑏_𝑎12𝑦²+𝑓(𝑡)_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦𝑑𝑡, (3.7) where 𝑓(𝑡)∈𝐴𝐶[𝑎,𝑏],0<𝛼≤1,𝑦(𝑎)=𝑐₁,and𝑦(𝑏)=𝑑₁. Then by Theorem 3.1, the necessary condition for optimality is 𝑦=^𝑅_𝑎𝐷^𝛼_𝑏𝑓(𝑡). (3.8)

One can generalize Theorem 3.1 as follows.

Corollary 3.3. Consider the functional of the form 𝐽𝑦₁,𝑦₂,…,𝑦_𝑙=^𝑏_𝑎𝐿𝑡,𝑦₁(𝑡),_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦₁(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦₁(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦₁(𝑡),𝑦₂(𝑡),_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦₂(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦₂(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦₂(𝑡)𝑦,…,_𝑙(𝑡),_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦_𝑙(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦_𝑙(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦_𝑙(𝑡)𝑑𝑡, (3.9) defined on sets of continuous functions 𝑦_𝑗,𝑗=1,2…,𝑙 that have Riesz Caputo fractional derivatives of order 𝑖𝛼,𝑖=1,2,…,𝑘,0<𝛼≤1,(𝑟−1<𝑘𝛼≤𝑟) in the interval [𝑎,𝑏] and satisfy the conditions 𝑦_𝑗^(𝑖)(𝑎)=𝑐_𝑖𝑗,𝑦_𝑗^(𝑖)(𝑏)=𝑑_𝑖𝑗,𝑖=0,1,…,𝑟−1;𝑗=1,2,..𝑙, (3.10) where 𝑎_𝑖𝑗,𝑑_𝑖𝑗 are constant and 𝐿∶[𝑎,𝑏]×ℝ^𝑙(𝑘+1)→ℝ is a function with continuous first and second partial derivatives with respect to all of its arguments. For 𝑦_𝑗,𝑗=1,2,…,𝑙 to be a minimum of (3.9), it is necessary that 𝑦_𝑗(𝑥) satisfy the following Euler-Lagrange equations: 𝜕𝐿𝜕𝑦_𝑗+^𝑘_𝑖=1(−1)_𝑎^𝑖𝑅𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦_𝑗=0,𝑗=1,2,…,𝑙. (3.11)

3.1. The Optimal Control Case

The optimal control problem we consider is to find 𝑢(𝑡) that minimizes the performance index𝐽𝑦,𝑢=^𝑏_𝑎𝐹𝑡,𝑢(𝑡),𝑦(𝑡),_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦(𝑡)𝑑𝑡, (3.12) subject to the constraint𝐺𝑡,𝑢(𝑡),𝑦(𝑡),_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦(𝑡)=0, (3.13) such that𝑦^(𝑖)(𝑎)=𝑐_𝑖,𝑦^(𝑖)(𝑏)=𝑑_𝑖,𝑖=0,1,…,𝑟−1, (3.14) where 𝑐_𝑖,𝑑_𝑖 are constant, 0<𝛼≤1,𝑟−1<𝑘𝛼≤𝑟, and 𝐹,𝐺∶[𝑎,𝑏]×ℝ^𝑘+2→ℝ are functions with continuous first and second partial derivatives with respect to their arguments.

To find the optimal control, one defines the modified performance index as𝐽̂𝑦,𝑢=^𝑏_𝑎[𝐹(⋅)+𝜆(𝑡)𝐺(⋅)]𝑑𝑡, (3.15) where 𝜆(𝑡) is called a Lagrange multiplier or adjoint variable. Using conditions (3.11), the following necessary conditions for optimal control are found: 𝜕𝐹𝜕𝑢+𝜆𝜕𝐺𝜕𝑢=0,𝜕𝐹𝜕𝑦+𝜆𝜕𝐺+𝜕𝑦^𝑘_𝑖=1(−1)_𝑎^𝑖𝑅𝐷_𝑏^𝑖𝛼𝜕𝐹𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦+^𝑅_𝑎𝐷_𝑏^𝑖𝛼𝜆𝜕𝐺𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦=0. (3.16) When 𝛼=1,𝑘=1, (3.16) coincides with the results in [20, Theorem 6.2.2].

4. The Riesz-Caputo Sequential Fractional Variational Principles with Delay

In this section the following problem with delay is considered.

Minimize𝐽𝑦=^𝑏_𝑎𝐿𝑡,_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦(𝑡),𝑦(𝑡),𝑦^′(𝑡),…,𝑦^(𝑟)(𝑡),𝑦(𝑡−𝜏),𝑦^′(𝑡−𝜏),…,𝑦^(𝑟)(𝑡−𝜏)𝑑𝑡 (4.1) such that𝑦(𝑡)=𝑓(𝑡),𝑎−𝜏≤𝑡≤𝑎,𝑦^(𝑖)(𝑏)=𝑑_𝑖,𝑖=0,1,2,…,𝑟−1,𝑟−1<𝑘𝛼≤𝑟, (4.2) where 𝑑_𝑖 are constant, 𝑦∈𝐴𝐶^2𝑟[𝑎,𝑏], 𝑓 is a smooth function, and 𝐿∶[𝑎−𝜏,𝑏]×ℝ^𝑘+2𝑟+2→ℝ is a function with continuous first- and second-order partial derivatives with respect to its arguments.

Theorem 4.1. If y(t) satisfying (4.2) is a minimum of (4.1), then it is necessary that it satisfies the Euler-Lagrange equations 𝜕𝐿()()+𝜕𝑦𝑡−𝜏𝑡+𝜏𝜕𝐿(𝑡)(𝑡)+𝜕𝑦_𝑟𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐿𝜕𝑦^(𝑖)()()+()𝑡−𝜏𝑡+𝜏−1^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡+_𝑘𝑖=1)(−1)_𝑎^𝑖𝑅𝐷^𝑖𝛼_𝑏−𝜏𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)−121Γ()𝐷𝑖𝛼^{𝑖𝛼𝑏−𝜏}×^𝑏_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑠)(𝑠−𝑡)^𝑖𝛼−1𝑑𝑠=0, (4.3) for 𝑎≤𝑡≤𝑏−𝜏𝜕𝐿(𝑡)(𝑡)+𝜕𝑦^𝑟_𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)+^𝑘_𝑖=1(−1)^𝑖𝑅_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)−121Γ(𝑖𝛼)^𝑏−𝜏𝐷^𝑖𝛼×_𝑎𝑎^𝑏−𝜏𝐷^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑠)(𝑡−𝑠)^𝑖𝛼−1𝑑𝑠=0, (4.4) for 𝑏−𝜏≤𝑡≤𝑏, and the transversality condition ^𝑟_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^{(𝑟−𝑙−1)}(𝑡)_𝑎^𝑏−𝜏+^𝑟_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^{(𝑟−𝑙−1)}(𝑡)^𝑏_𝑏−𝜏+^𝑟_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐿𝜕𝑦^(𝑖)(𝑡−𝜏)(𝑡+𝜏)𝜂^{(𝑟−𝑙−1)}(𝑡)_𝑎^𝑏−𝜏=0, (4.5) where 𝜂∈𝐴𝐶^2𝑟[𝑎,𝑏] is any arbitrary function such that 𝜂^(𝑖)(𝑏)=0,𝑖=0,1,…,𝑟−1,𝜂≡0,𝑎−𝜏≤𝑡≤𝑎.

Proof. Assume that 𝐽(𝑦) has a minimum for 𝑦₀(𝑡). Define a family of curves 𝑦(𝑡)=𝑦₀(𝑡)+𝜖𝜂(𝑡) where 𝜖∈ℝ and 𝜂 is a function in 𝐴𝐶^2𝑟[𝑎,𝑏] satisfying 𝜂^(𝑖)(𝑏)=0,𝑖=0,1,…,𝑟−1,𝜂≡0,𝑎−𝜏≤𝑡≤𝑎. The function 𝛾(𝜖)=𝐽𝑦₀(𝑡)+𝜖𝜂(𝑡) (4.6) admits a minimum when 𝛾′(0)=0. That is, at 𝑦₀, one has ^𝑏_𝑎𝜕𝐿(𝑡)(𝑡)𝜂(𝑡)+𝜕𝑦𝜕𝐿()(𝑡)𝜂()+𝜕𝑦𝑡−𝜏𝑡−𝜏^𝑘_𝑖=1𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝜂(𝑡)+^𝑟_𝑖=1𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^(𝑖)(𝑡)+𝜕𝐿𝜕𝑦^(𝑖)()(𝑡)𝜂𝑡−𝜏^(𝑖)(𝑡−𝜏)𝑑𝑡=0. (4.7)
Using (2.8), (2.10), and the fact that 𝜂(𝑡)≡0,𝑎−𝜏≤𝑡≤𝑎 and 𝜂^(𝑖)(𝑏)=0,𝑖=0,1,…,𝑟−1, one gets^𝑏_𝑎𝜕𝐿(𝑡)(𝑡)𝜂(𝑡)+𝜕𝑦𝜕𝐿()(𝑡)𝜂()+𝜕𝑦𝑡−𝜏𝑡−𝜏^𝑘_𝑖=1𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)^𝑅_𝑎𝐷_𝑏^𝑖𝛼𝜂(𝑡)+^𝑟_𝑖=1𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^(𝑖)(𝑡)+𝜕𝐿𝜕𝑦^(𝑖)()(𝑡)𝜂𝑡−𝜏^(𝑖)(𝑡−𝜏)𝑑𝑡=0. (4.8) Splitting the integrals, making the change of variables 𝑡−𝜏↔𝑡 and keeping in mind that 𝜂(𝑡)≡0,𝑎−𝜏≤𝑡≤𝑎, (4.8) becomes _𝑎^𝑏−𝜏𝜕𝐿(𝑡)(𝑡)𝜂(𝑡)+𝜕𝑦𝜕𝐿()()𝜂(𝑡)+𝜕𝑦𝑡−𝜏𝑡+𝜏^𝑘_𝑖=1𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)^𝑅_𝑎𝐷_𝑏^𝑖𝛼𝜂(𝑡)+^𝑟_𝑖=1𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^(𝑖)(𝑡)+𝜕𝐿𝜕𝑦^(𝑖)()()𝜂𝑡−𝜏𝑡+𝜏^(𝑖)(𝑡+)𝑑𝑡_{𝑏𝑏−𝜏}𝜕𝐿(𝑡)(𝑡)𝜂(𝑡)+𝜕𝑦^𝑘_𝑖=1𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)^𝑅_𝑎𝐷_𝑏^𝑖𝛼𝜂(𝑡)+^𝑟_𝑖=1𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^(𝑖)(𝑡)𝑑𝑡=0. (4.9) If one uses the usual integration by parts formula, the integration by parts formulas in [16, Lemma 2] and [17, Lemma 2], and the properties of the function 𝜂, one obtains _𝑎^𝑏−𝜏𝜕𝐿()()+𝜕𝑦𝑡−𝜏𝑡+𝜏𝜕𝐿(𝑡)(𝑡)+𝜕𝑦_𝑟𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐿𝜕𝑦^(𝑖)()()+()𝑡−𝜏𝑡+𝜏−1^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡+_𝑘𝑖=1)(−1)_𝑎^𝑖𝑅𝐷^𝑖𝛼_𝑏−𝜏𝜕𝐿𝜕<_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)−121Γ()𝐷𝑖𝛼^{𝑖𝛼𝑏−𝜏}×^𝑏_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑠)(𝑠−𝑡)^𝑖𝛼−1𝜂(𝑡)+𝑑𝑠𝑑𝑡_{𝑏𝑏−𝜏}𝜕𝐿(𝑡)(𝑡)+𝜕𝑦^𝑟_𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)+^𝑘_𝑖=1(−1)^𝑖𝑅_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)−121Γ(𝑖𝛼)^𝑏−𝜏𝐷^𝑖𝛼×_𝑎𝑎^𝑏−𝜏𝐷^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑠)(𝑡−𝑠)^𝑖𝛼−1𝜂(𝑡)+𝑑𝑠𝑑𝑡_{𝑟𝑖=1𝑖−1𝑙=0}(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^{(𝑟−𝑙−1)}(𝑡)_𝑎^𝑏−𝜏+^𝑟_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐿𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^{(𝑟−𝑙−1)}(𝑡)^𝑏_𝑏−𝜏+^𝑟_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐿𝜕𝑦^(𝑖)()()𝜂𝑡−𝜏𝑡+𝜏^{(𝑟−𝑙−1)}(𝑡)_𝑎^𝑏−𝜏=0. (4.10)
If one chooses 𝜂 such that 𝜂(𝑎)=0 and 𝜂≡0 on [𝑏−𝜏,𝑏], (4.3) holds. If one chooses 𝜂(𝑏)=0 and 𝜂≡0 on [𝑎,𝑏−𝜏], (4.4) holds. Then since the first and second integrals in (4.10) become zero, (4.5) holds.

The result in Theorem 4.1 coincides with the results in [14, Corollaries 2.1 and 2.2].

Example 4.2. Consider ²₀12𝑦²()+1𝑡−12𝑦²(𝑡)+𝑓(𝑡)₀^𝑅𝐶𝐷^𝛼₂𝑦(𝑡)𝑑𝑡, (4.11) where 0<𝛼≤1,𝑦(2)=constant,and𝑦(𝑡)=𝑔(𝑡)for−1≤𝑡≤0. The necessary conditions for optimality according to Theorem 4.1 are 𝑦(𝑡)=12^𝑅0𝐷^𝛼₁𝑓𝑡)+1((𝛼)𝐷4Γ^𝛼12₁𝐷^𝛼₂𝑓(𝑠)(𝑠−𝑡)^𝛼−1(𝑡), (4.12) for 0<𝑡<1 and 𝑦(𝑡)=^𝑅₁𝐷^𝛼₂𝑓𝑡)−1(2Γ(𝛼)¹𝐷^𝛼1₀₀𝐷^𝛼𝑓(𝑠)(𝑡−𝑠)^𝛼−1(𝑡), (4.13) for 1<𝑡<2.

Theorem 4.1 can be generalized as follows

Corollary 4.3. Consider the functional of the form 𝐽𝑦₁,𝑦₂,…,𝑦_𝑚=^𝑏_𝑎𝐿𝑡,_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦₁(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦₁(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦₁(𝑡),𝑦₁(𝑡),𝑦^′₁(𝑡),…,𝑦₁^(𝑟)(𝑡),𝑦₁(𝑡−𝜏),𝑦^′₁(𝑡−𝜏),…,𝑦₁^(𝑟)(𝑡−𝜏_𝑎𝑅𝐶),𝐷^𝛼_𝑏𝑦₂(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦₂(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦₂(𝑡),𝑦₂(𝑡),𝑦^′₂(𝑡)𝑦,…,_2(𝑟)(𝑡),𝑦₂(𝑡−𝜏),𝑦^′₂(𝑡−𝜏),…,𝑦₂^(𝑟)(𝑡−𝜏),…,_𝑎𝑅𝐶𝐷^𝛼_𝑏𝑦_𝑚(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦_𝑚(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦_𝑚(𝑡),𝑦_𝑚(𝑡),𝑦^′_𝑚(𝑡),…,𝑦_𝑚^(𝑟)(𝑡),𝑦_𝑚(𝑡−𝜏),𝑦^′_𝑚(𝑡−𝜏),…,𝑦_𝑚^(𝑟)(𝑡−𝜏)𝑑𝑡, (4.14) defined on sets of functions 𝑦_𝑗(𝑡)∈𝐴𝐶^2𝑟[𝑎,𝑏],𝑗=1,2,…,𝑚 that have Riesz-Caputo fractional derivatives of order 𝑖𝛼,𝑖=1,2,…,𝑘,0<𝛼≤1,𝑟−1<𝑘𝛼≤𝑟, and satisfy the conditions 𝑦_𝑗^(𝑖)=𝑑_𝑖𝑗,𝑦_𝑗(𝑡)=𝑓_𝑗(𝑡),𝑡∈[𝑎−𝜏,𝑎],𝜏<𝑏−𝑎,𝑎<𝑏, (4.15) where 𝑑_𝑖𝑗 are constant, 𝑓_𝑗(𝑡) are smooth functions and 𝐿∶[𝑎−𝜏,𝑏]×ℝ^{𝑚(𝑘+2𝑟+2)}→ℝ is a function having first-and second-order partial derivatives with respect to its arguments. For 𝑦_𝑗,𝑗=1,2,…,𝑚 satisfying (4.15) to be a minimum of (4.14), it is necessary that 𝑦_𝑗 satisfy the Euler-Lagrange equations 𝜕𝐿𝜕𝑦_𝑗()()+𝑡−𝜏𝑡+𝜏𝜕𝐿𝜕𝑦_𝑗(𝑡)(𝑡)+^𝑟_𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐿𝜕𝑦_𝑗^(𝑖)()()+()𝑡−𝜏𝑡+𝜏−1^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐿𝜕𝑦_𝑗^(𝑖)(𝑡)(𝑡)+^𝑘_𝑖=1(−1)_𝑎^𝑖𝑅𝐷^𝑖𝛼_𝑏−𝜏𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦_𝑗(𝑡)𝑡)−1(21Γ()𝐷𝑖𝛼^{𝑖𝛼𝑏−𝜏}×^𝑏_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦_𝑗(𝑡)(𝑠)(𝑠−𝑡)^𝑖𝛼−1𝑑𝑠=0, (4.16) for 𝑎≤𝑡≤𝑏−𝜏,𝑗=1,2…,𝑚𝜕𝐿𝜕𝑦_𝑗(𝑡)(𝑡)+^𝑟_𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐿𝜕𝑦_𝑗^(𝑖)(𝑡)(𝑡)+^𝑘_𝑖=1(−1)^𝑖𝑅_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦_𝑗(𝑡)(𝑡)−121Γ(𝑖𝛼)^𝑏−𝜏𝐷^𝑖𝛼×_𝑎𝑎^𝑏−𝜏𝐷^𝑖𝛼𝜕𝐿𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦_𝑗(𝑡)(𝑠)(𝑡−𝑠)^𝑖𝛼−1𝑑𝑠=0, (4.17) for 𝑏−𝜏≤𝑡≤𝑏,𝑗=1,2…,𝑚, and the transversality condition ^𝑟_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐿𝜕𝑦_𝑗^(𝑖)(𝑡)(𝑡)𝜂_𝑗^{(𝑟−𝑙−1)}(𝑡)_𝑎^𝑏−𝜏+^𝑟_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐿𝜕𝑦_𝑗^(𝑖)(𝑡)(𝑡)𝜂_𝑗^{(𝑟−𝑙−1)}(𝑡)^𝑏_𝑏−𝜏+^𝑟_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐿𝜕𝑦_𝑗^(𝑖)()()𝜂𝑡−𝜏𝑡+𝜏_𝑗^{(𝑟−𝑙−1)}(𝑡)_𝑎^𝑏−𝜏=0, (4.18) where 𝜂=(𝜂₁,𝜂₂,…,𝜂_𝑚) is an an arbitrary vector-valued function satisfying 𝜂(𝑡)=(0,0,…,0),𝑡∈[𝑎−𝜏,𝑎], and 𝜂^(𝑖)(𝑏)=(0,0,…,0),𝑖=0,1,…,𝑟−1.

4.1. The Optimal Control Case with Delay

Find the optimal control variable 𝑢 such that 𝑢 minimizes the performance index 𝐽𝑦,𝑢=^𝑏_𝑎𝐹𝑡,𝑢(𝑡),_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦(𝑡),𝑦(𝑡),𝑦^′(𝑡),…,𝑦^(𝑘)(𝑡),𝑦(𝑡−𝜏),𝑦^′(𝑡−𝜏),…,𝑦^(𝑘)(𝑡−𝜏)𝑑𝑡 (4.19) subject to the constraint 𝐺𝑡,𝑢(𝑡),_𝑎^𝑅𝐶𝐷^𝛼_𝑏𝑦(𝑡),_𝑎^𝑅𝐶𝐷_𝑏^2𝛼𝑦(𝑡),…,_𝑎^𝑅𝐶𝐷_𝑏^𝑘𝛼𝑦(𝑡),𝑦(𝑡),𝑦^′(𝑡),…,𝑦^(𝑘)(𝑡),𝑦(𝑡−𝜏),𝑦^′(𝑡−𝜏),…,𝑦^(𝑘)(𝑡−𝜏)=0 (4.20) such that𝑦(𝑡)=𝑓(𝑡),𝑎−𝜏≤𝑡≤𝑎,𝑦^(𝑖)=𝑑_𝑖,𝑖=0,1,2,…,𝑟−1,0<𝛼≤1, (4.21) where 𝑟−1<𝑘𝛼≤𝑟𝑑_𝑖 are constant, 𝑦∈𝐴𝐶^2𝑟[𝑎,𝑏], 𝑓 is a smooth function and 𝐹,𝐺∶[𝑎−𝜏,𝑏]×ℝ^𝑘+2𝑟+3→ℝ are functions with continuous first and second order partial derivatives with respect to their arguments.

If one defines a modified performance index as𝐽̂𝑦,𝑢=^𝑏_𝑎𝐹(⋅)+𝜆(𝑡)𝐺(⋅)𝑑𝑡, (4.22) applying Corollary 4.3, it is necessary that 𝜕𝐹()()+𝜕𝑦𝑡−𝜏𝑡+𝜏𝜕𝐹(𝑡)(𝑡)+𝜕𝑦_𝑟𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐹𝜕𝑦^(𝑖)()()+()𝑡−𝜏𝑡+𝜏−1^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐹𝜕𝑦^(𝑖)(𝑡)(𝑡+_𝑘𝑖=1)(−1)_𝑎^𝑖𝑅𝐷^𝑖𝛼_𝑏−𝜏𝜕𝐹𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)−121Γ()𝐷𝑖𝛼^{𝑖𝛼𝑏−𝜏}×^𝑏_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜕𝐹𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑠)(𝑠−𝑡)^𝑖𝛼−1𝑑𝑠+𝜆𝜕𝐺()()+𝜆𝜕𝑦𝑡−𝜏𝑡+𝜏𝜕𝐺(𝑡)(𝑡)+𝜕𝑦^𝑟_𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜆𝜕𝐺𝜕𝑦^(𝑖)()()+()𝑡−𝜏𝑡+𝜏−1^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜆𝜕𝐺𝜕𝑦^(𝑖)(𝑡)(𝑡+_𝑘𝑖=1)(−1)_𝑎^𝑖𝑅𝐷^𝑖𝛼_𝑏−𝜏𝜆𝜕𝐺𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)−121Γ()𝐷𝑖𝛼^{𝑖𝛼𝑏−𝜏}×^𝑏_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜆𝜕𝐺𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑠)(𝑠−𝑡)^𝑖𝛼−1𝑑𝑠=0,𝜕𝐹(𝑡)(𝑡)(𝑡)𝜕𝑢+𝜆𝜕𝐺(𝑡)(𝑡)𝜕𝑢=0, (4.23) for 𝑎≤𝑡≤𝑏−𝜏, 𝜕𝐹(𝑡)(𝑡)+𝜕𝑦^𝑟_𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜕𝐹𝜕𝑦^(𝑖)(𝑡)(𝑡)+^𝑘_𝑖=1(−1)^𝑖𝑅_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜕𝐹𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑡)−121Γ(𝑖𝛼)^𝑏−𝜏𝐷^𝑖𝛼×_𝑎𝑎^𝑏−𝜏𝐷^𝑖𝛼𝜕𝐹𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑠)(𝑡−𝑠)^𝑖𝛼−1,𝜆𝑑𝑠𝜕𝐺(𝑡)𝑡)+𝜕𝑦(^𝑟_𝑖=1(−1)^𝑖𝑑^𝑖𝑑𝑡^𝑖𝜆𝜕𝐺𝜕𝑦^(𝑖)(𝑡)𝑡)+_𝑘𝑖=1((−1)^𝑖𝑅_𝑏−𝜏𝐷_𝑏^𝑖𝛼𝜆𝜕𝐺𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)𝑡)−1(21Γ(𝑖𝛼)^𝑏−𝜏𝐷^𝑖𝛼×_𝑎𝑎^𝑏−𝜏𝐷^𝑖𝛼𝜆𝜕𝐹𝜕_𝑎^𝑅𝐶𝐷_𝑏^𝑖𝛼𝑦(𝑡)(𝑠)(𝑡−𝑠)^𝑖𝛼−1𝑑𝑠=0,𝜕𝐹(𝑡)(𝑡)(𝑡)𝜕𝑢+𝜆𝜕𝐺(𝑡)(𝑡)𝜕𝑢=0, (4.24) for 𝑏−𝜏<𝑡<𝑏, and the transversality condition^𝑘_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐹𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^{(𝑟−𝑙−1)}(𝑡)_𝑎^𝑏−𝜏+^𝑘_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐹𝜕𝑦^(𝑖)(𝑡)(𝑡)𝜂^{(𝑟−𝑙−1)}(𝑡)^𝑏_𝑏−𝜏+^𝑘_𝑖=1^𝑖−1_𝑙=0(−1)^𝑙𝑑^𝑙𝑑𝑡^𝑙𝜕𝐹𝜕𝑦^(𝑖)()()𝜂𝑡−𝜏𝑡+𝜏^{(𝑟−𝑙−1)}(𝑡)_𝑎^𝑏−𝜏=0, (4.25) where 𝜂 is any arbitrary function satisfying 𝜂(𝑡)=0,𝑡∈[𝑎−𝜏,𝑡], and 𝜂^(𝑖)(𝑏)=0,𝑖=0,1,2,…,𝑟−1.

5. Conclusion

Fractional variational principles started to be used in several branches of science and engineering. However the delay is present in various phenomena having great impact in science and engineering. One of the main questions is to combine in an optimal way the properties of the fractional calculus and those of the delay having in mind to obtain a more general variational principles. In this paper we have used the fractional Riesz-Caputo derivative and the delay in the state variable. We mention that the definition of the Riesz-Caputo derivative contains both left and right fractional derivative. The necessary conditions for the optimal control were obtained. When 𝛼→1 or the delay is absent the classical results are obtained.

Acknowledgment

This work is partially supported by the Scientific and Technical Research Council of Turkey.