A Necessary Optimality Condition for Extended Weighted Generalized Fractional Optimal Control Problems

Using the recent weighted generalized fractional order operators of Hattaf, a general fractional optimal control problem without constraints on the values of the control functions is formulated and a corresponding (weak) version of Pontryagin's maximum principle is proved. As corollaries, necessary optimality conditions for Caputo-Fabrizio, Atangana-Baleanu and weighted Atangana-Baleanu fractional dynamic optimization problems are trivially obtained. As an application, the weighted generalized fractional problem of the calculus of variations is investigated and a new more general fractional Euler-Lagrange equation is given.


Introduction
In 1744, Euler published his "Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti" [1].In this work, Euler seeks a method to find a curve that minimizes or maximizes any quantity expressed by an integral, generalizing the problems studied before by the Bernoulli brothers', but retaining the geometrical approach developed by Johann Bernoulli to solve them.Euler found what is now known as the Euler-Lagrange differential equation, which is a necessary condition for a function to maximize or minimize a given integral functional that depends on the unknown function and its derivative.In 1760, Lagrange published an "Essay on a new method of determining the maxima and minima of indefinite integral formulas" [2], giving the analytic method that allows to attack all types of variational problems [3,4].
At the end of the fifties of the 20th century, Pontryagin and his collaborators formulated one of the most important tools to characterize the optimal trajectories of a controllable system [15].In the particular case when the controls are continuous functions without constraints on its values, the necessary optimality conditions given by Pontryagin's maximum principle (PMP) reduce to the classical Euler-Lagrange equations.This is known as the weak PMP [16], in contrast with the general (strong) PMP [17].
Fractional optimal control problems (FOCPs) give an extension of the classical optimal control problems (see, e.g., [18]), where not only the control system is described by some fractional differential equation [19], but the cost functional may be also provided by a fractional integral operator [20].
Motivated by the new generalized fractional operators explored by Hattaf [21] and the weighted generalized fractional integration by parts formula proved by Zine et al. [14] (see Section 2), together with the urgent need to use advanced techniques to quantify the optimal controls for the considered optimization problems, we prove in Section 3 a weak version of a weighted generalized fractional Pontryagin's maximum principle, supported by an adequate application to the calculus of variations (Section 4).We end with Section 5 of conclusions and possible future work.

Preliminaries
In this section, we recall the necessary mathematical prerequisites to formulate and prove our main theorem, which gives necessary optimality conditions of Pontryagin type in the weighted generalized fractional setting.We adopt here the following notations: where 0 ≤ α < 1 and B(α) is a normalization function obeying B(0 Let H 1 (a, b) be the Sobolev space of order one defined as follows: Definition 1 (The left-weighted generalized fractional derivative and integral [21]).Let 0 ≤ α < 1, β > 0 and f ∈ H 1 (a, b).The left-weighted generalized fractional derivative of order α of function f , in the Riemann-Liouville sense, is defined by where E β denotes the Mittag-Leffler function of parameter β defined by where RL a,w I β is the standard left weighted Riemann-Liouville fractional integral of order β given by RL a,w Definition 2 (The right-weighted generalized fractional derivative and integral [14]).Let 0 ≤ α < 1, β > 0 and f ∈ H 1 (a, b).The right-weighted generalized fractional derivative of order α of function f , in the Riemann-Liouville sense, is defined by where w ∈ C 1 ([a, b]) with w > 0. The corresponding fractional integral is defined by where RL I β b,w is the standard right weighted Riemann-Liouville fractional integral of order β: Theorem 1 (The weighted generalized formula of integration by parts [14]).
Remark 1.It should be noted that the fractional integration by parts formulas (8) and (9) are not always true: they only hold under certain conditions.In general, f must be in L p while the second function g must be in L q in such a way 1/p + 1/q = 1.In our work these conditions are verified.Indeed, as we shall see, the state variable of our control system is in H 1 , which means that it is in L 2 , and the costate variable λ is an absolutely continues function on a bounded and closed interval [0, T ], which means that it is also a L 2 function.

Main Result
In this section we formulate an extended weighted generalized fractional optimal control problem (EWGFOCP) and derive necessary conditions for the optimality of the EWGFOCP.
The set of all admissible controls is denoted by U ad .The problem consists to determine the optimal control u(•) and corresponding state trajectory x(•) that minimizes a given cost functional performance J, subject to a given dynamic control system, assumed to be controllable, and given boundary conditions where L, f : [a, b] × R × R → R are differentiable functions with respect to all their arguments.
Remark 2. As direct corollaries of Theorem 2, one can obtain weak versions of Pontryagin's maximum principle for several different fractional operators: 1. if w(t) ≡ 1 and β = 1, then we obtain a weak Caputo-Fabrizio fractional Pontryagin's maximum principle with Hamiltonian system 2. if w(t) ≡ 1 and β = α, then we obtain a weak Atangana-Baleanu fractional Pontryagin's maximum principle with Hamiltonian system 3. if β = α, then we obtain a weak weighted Atangana-Baleanu fractional Pontryagin's maximum principle with Hamiltonian system 4 Application: A Fractional Euler-Lagrange Equation Let L : [a, b]×R×R → R be a differentiable function with respect to all its arguments and consider the following fundamental problem of the calculus of variations: subject to the boundary conditions Theorem 3 (The weighted generalized fractional Euler-Lagrange equation).If x is a minimizer of problem (23)-(24), then x satisfies the following weighted generalized fractional Euler-Lagrange equation: Proof.Set u(t) := R a,w D α,β x(t).The performance index of (23) becomes equivalent to the objective functional subject to a control system (11) with f (t, x, u) = u.Applying our main Theorem 2 to the current problem, that is, using the Hamiltonian system (13) and the stationary condition ( 14), we have Since which is the new Euler-Lagrange equation (25).Remark 3.According to the cases listed in Remark 2, we can obtain several interesting Euler-Lagrange equations as corollaries of Theorem 3, namely, the Caputo-Fabrizio, Atangana-Baleanu, and weighted Atangana-Baleanu fractional Euler-Lagrange equations.

Conclusion
Using as mathematical tools some recent results of the literature related to the fractional calculus theory with non singular kernels, we formulated a general fractional optimal control problem without constraints on the values of the control functions, and proved a necessary optimality condition of Pontryagin type.As an application, a general fractional Euler-Lagrange equation was obtained, showing the usefulness and interest of our main result.Our results open many future possibilities for future research: to extend our scalar problems and results to the vectorial case, to generalize the integral cost functional to be given as a fractional integral operator, etc.In principle, all the developments done to the fractional calculus of variations after the works of Riewe, see [22] and references therein, can now be done for extended weighted generalized fractional variational problems.The objective of our current study was to setup a general theoretical result related to a new formulation of the generalized weighted fractional optimal control problem.The idea was to contribute with theoretical tools to construct a generalized fractional theory of optimal control.As future work, it would be interesting to develop numerical schemes for such kind of problems.
) Using the weighted generalized fractional integration by parts formula (9) of Theorem 1, we get b a λ(t) R a,w D α,β δx(t) dt = b a w(t) 2 δx(t) R D α,β b,w