# A Necessary Optimality Condition for Extended Weighted Generalized
Fractional Optimal Control Problems††This is a preprint version of the paper
published open access in ’Results in Control and Optimization’
[https://doi.org/10.1016/j.rico.2023.100356].

Houssine Zine1
<EMAIL_ADDRESS>El Mehdi Lotfi2
<EMAIL_ADDRESS>Delfim F. M. Torres1,
<EMAIL_ADDRESS>Corresponding author. Noura Yousfi2
<EMAIL_ADDRESS>
(1Center for Research and Development in Mathematics and Applications (CIDMA),
Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
2Laboratory of Analysis, Modeling and Simulation (LAMS),
Faculty of Sciences Ben M’sik, Hassan II University of Casablanca,
P.O. Box 7955 Sidi Othman, Casablanca, Morocco)

###### Abstract

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.

Keywords: extended weighted generalized fractional optimal control problems;
weak Pontryagin’s maximum principle; fractional calculus of variations;
fractional Euler–Lagrange equation and Hamiltonian approach; weighted
generalized integration by parts.

MSC: 26A33; 49K05.

##  1 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].

In 1996, Riewe introduced the fractional calculus of variations [5, 6], from
which many scholars reintroduced several variants of the fractional calculus
of variations and obtained some wonderful and useful fractional Euler–Lagrange
equations with some additional results: see, e.g., the works of Agrawal [7],
Almeida et al. [8], Atanackovic et al. [9, 10], Frederico and Torres [11],
Klimek [12], Zine et al. [13, 14], just to mention a few relevant names and
works.

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.

##  2 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:

$\displaystyle\phi(\alpha):=\dfrac{1-\alpha}{B(\alpha)},\quad\psi(\alpha):=\dfrac{\alpha}{B(\alpha)},\quad\mu_{\alpha}:=\dfrac{\alpha}{1-\alpha},$

where $0\leq\alpha<1$ and $B(\alpha)$ is a normalization function obeying
$B(0)=B(1)=1$.

Let $H^{1}(a,b)$ be the Sobolev space of order one defined as follows:

$H^{1}(a,b)=\left\\{y\in L^{2}(a,b):y^{\prime}\in L^{2}(a,b)\right\\}.$

######  Definition 1 (The left-weighted generalized fractional derivative and
integral [21]).

Let $0\leq\alpha<1$, $\beta>0$ and $f\in H^{1}(a,b)$. The left-weighted
generalized fractional derivative of order $\alpha$ of function $f$, in the
Riemann–Liouville sense, is defined by

$\,_{a,w}^{R}D^{\alpha,\beta}f(x)=\dfrac{1}{\phi(\alpha)}\dfrac{1}{w(x)}\dfrac{d}{dx}\int^{x}_{a}(wf)(s)E_{\beta}\left[-\mu_{\alpha}(x-s)^{\beta}\right]ds,$
(1)

where $E_{\beta}$ denotes the Mittag–Leffler function of parameter $\beta$
defined by

$E_{\beta}(z)=\sum_{j=0}^{\infty}\dfrac{z^{j}}{\Gamma(\beta j+1)},$ (2)

and $w\in C^{1}([a,b])$ with $w>0$. The corresponding left fractional integral
is defined by

$_{a,w}I^{\alpha,\beta}f(x)=\phi(\alpha)f(x)+\psi(\alpha)\,^{RL}_{a,w}I^{\beta}f(x),$
(3)

where ${}^{RL}_{a,w}I^{\beta}$ is the standard left weighted Riemann–Liouville
fractional integral of order $\beta$ given by

${}^{RL}_{a,w}I^{\beta}f(x)=\dfrac{1}{\Gamma(\beta)}\dfrac{1}{w(x)}\int^{x}_{a}(x-s)^{\beta-1}w(s)f(s)ds,\quad
x>a.$ (4)

######  Definition 2 (The right-weighted generalized fractional derivative and
integral [14]).

Let $0\leq\alpha<1$, $\beta>0$ and $f\in H^{1}(a,b)$. The right-weighted
generalized fractional derivative of order $\alpha$ of function $f$, in the
Riemann–Liouville sense, is defined by

${}^{R}D^{\alpha,\beta}_{b,w}f(x)=\dfrac{-1}{\phi(\alpha)}\dfrac{1}{w(x)}\dfrac{d}{dx}\int^{b}_{x}(wf)(s)E_{\beta}\left[-\mu_{\alpha}(s-x)^{\beta}\right]ds,$
(5)

where $w\in C^{1}([a,b])$ with $w>0$. The corresponding fractional integral is
defined by

$I^{\alpha,\beta}_{b,w}f(x)=\phi(\alpha)f(x)+\psi(\alpha)\,^{RL}I^{\beta}_{b,w}f(x),$
(6)

where ${}^{RL}I_{b,w}^{\beta}$ is the standard right weighted
Riemann–Liouville fractional integral of order $\beta$:

${}^{RL}I_{b,w}^{\beta}f(x)=\dfrac{1}{\Gamma(\beta)}\dfrac{1}{w(x)}\int^{b}_{x}(x-s)^{\beta-1}w(s)f(s)ds,\quad
x<b.$ (7)

######  Theorem 1 (The weighted generalized formula of integration by parts
[14]).

Let $f$, $g$ $\in$ $H^{1}(a,b)$ and $\omega\in C^{1}(a,b)$ with $\omega>0$. If
$0\leq\alpha<1$ and $\beta>0$, then

$\displaystyle\int_{a}^{b}f(x)\left(\,{}^{R}D^{\alpha,\beta}_{b,w}g\right)(x)dx$
$\displaystyle=$
$\displaystyle\int_{a}^{b}w(x)^{2}g(x)\left(\,{}_{a,w}^{R}D^{\alpha,\beta}\left(\dfrac{f}{w^{2}}\right)\right)(x)dx,$
(8) $\displaystyle\int_{a}^{b}f(x)(\,^{R}_{a,w}D^{\alpha,\beta}g)(x)dx$
$\displaystyle=$
$\displaystyle\int_{a}^{b}w(x)^{2}g(x)\left(\,{}^{R}D^{\alpha,\beta}_{b,w}\left(\dfrac{f}{w^{2}}\right)\right)(x)dx.$
(9)

######  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 $\lambda$ is an
absolutely continues function on a bounded and closed interval $[0,T]$, which
means that it is also a $L^{2}$ function.

##  3 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. Let $U\subset\mathbb{R}$ be a given nonempty open
set. A control $u:$ $[0,T]$ $\longrightarrow$ $U$ is called admissible if it
is continuous on $[0,T]$. The set of all admissible controls is denoted by
$U_{ad}$. The problem consists to determine the optimal control $u(\cdot)$ and
corresponding state trajectory $x(\cdot)$ that minimizes a given cost
functional performance $J$,

$J[x,u]=\int_{a}^{b}L\left(t,x(t),u(t)\right)dt\longrightarrow\min,$ (10)

subject to a given dynamic control system,

$\,^{R}_{a,w}D^{\alpha,\beta}x(t)=f\left(t,x(t),u(t)\right),\quad t\in[a,b],$
(11)

assumed to be controllable, and given boundary conditions

$x(a)=x_{a},\quad x(b)=x_{b},$ (12)

where $L,f:[a,b]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ are
differentiable functions with respect to all their arguments.

######  Theorem 2 (Necessary optimality condition for the EWGFOCP (10)–(12)).

Let $0\leq\alpha<1$, $\beta>0$, and $\left(x(\cdot),u(\cdot)\right)$ be a
minimizer of problem (10)–(12). Then, there exists a function $\lambda(\cdot)$
for which the triplet $\left(x(\cdot),\lambda(\cdot),u(\cdot)\right)$
satisfies:

  * (i)

the Hamiltonian system

$\left\\{\begin{array}[]{ll}\,{}^{R}_{a,w}D^{\alpha,\beta}x(t)=\dfrac{\partial\mathcal{H}}{\partial\lambda}\left(t,x(t),\lambda(t),u(t)\right),&t\in[a,b],\\\\[8.5359pt]
w(t)^{2}\,{}^{R}D_{b,w}^{\alpha,\beta}\dfrac{\lambda(t)}{w(t)^{2}}=\dfrac{\partial\mathcal{H}}{\partial
x}\left(t,x(t),\lambda(t),u(t)\right),&t\in[a,b];\end{array}\right.$ (13)

  * (ii)

and the stationary condition

$\dfrac{\partial\mathcal{H}}{\partial
u}\left(t,x(t),\lambda(t),u(t)\right)=0,\quad t\in[a,b],$ (14)

where $\mathcal{H}$ is a scalar function, called the Hamiltonian, defined by

$\mathcal{H}\left(t,x,\lambda,u\right)=L\left(t,x,u\right)+\lambda
f\left(t,x,u\right).$ (15)

###### Proof.

We consider the following augmented functional of the performance index:

$J_{a}[x,\lambda,u]=\int_{a}^{b}\left[\mathcal{H}\left(t,x(t),\lambda(t),u(t)\right)-\lambda(t)f\left(t,x(t),u(t)\right)\right]dt.$
(16)

Regarding the variation of the augmented performance index (16),

$\delta
J_{a}[x,\lambda,u]=\int_{a}^{b}\left\\{\dfrac{\partial\mathcal{H}}{\partial
x}\delta
x(t)+\left[\dfrac{\partial\mathcal{H}}{\partial\lambda}-\,^{R}_{a,w}D^{\alpha,\beta}x(t)\right]\delta\lambda(t)+\dfrac{\partial\mathcal{H}}{\partial
u}\delta u(t)-\lambda(t)\,^{R}_{a,w}D^{\alpha,\beta}\delta x(t)\right\\}dt.$
(17)

Using the weighted generalized fractional integration by parts formula (9) of
Theorem 1, we get

$\int_{a}^{b}\left\\{\lambda(t)\,^{R}_{a,w}D^{\alpha,\beta}\delta
x(t)\right\\}dt=\int_{a}^{b}\left\\{w(t)^{2}\delta
x(t)\,^{R}D_{b,w}^{\alpha,\beta}\frac{\lambda(t)}{w(t)^{2}}\right\\}dt,$ (18)

since $\lambda(t),x(t)\in L^{2}(a,b)$. Then the equality (17) can be written
as

$\delta
J_{a}[x,\lambda,u]=\int_{a}^{b}\left\\{\left[\dfrac{\partial\mathcal{H}}{\partial
x}-w(t)^{2}\,{}^{R}D_{b,w}^{\alpha,\beta}\frac{\lambda(t)}{w(t)^{2}}\right]\delta
x(t)+\left[\dfrac{\partial\mathcal{H}}{\partial\lambda}-\,^{R}_{a,w}D^{\alpha,\beta}x(t)\right]\delta\lambda(t)+\dfrac{\partial\mathcal{H}}{\partial
u}\delta u(t)\right\\}dt.$ (19)

To obtain the necessary optimality conditions for the EWGFOCP, the equality
$\delta J_{a}=0$ should be verified for all independent variations $\delta
x(t)$, $\delta\lambda(t)$ and $\delta u(t)$, which requires that the
corresponding coefficients must vanish. So,

$\displaystyle
w(t)^{2}\,{}^{R}D_{b,w}^{\alpha,\beta}\dfrac{\lambda(t)}{w(t)^{2}}$
$\displaystyle=$ $\displaystyle\dfrac{\partial\mathcal{H}}{\partial
x}\left(t,x(t),\lambda(t),u(t)\right),\quad t\in[a,b],$
$\,{}^{R}_{a,w}D^{\alpha,\beta}x(t)$ $\displaystyle=$
$\displaystyle\dfrac{\partial\mathcal{H}}{\partial\lambda}\left(t,x(t),\lambda(t),u(t)\right),\quad
t\in[a,b],$ $\displaystyle\dfrac{\partial\mathcal{H}}{\partial
u}\left(t,x(t),\lambda(t),u(t)\right)$ $\displaystyle=$ $\displaystyle 0,\quad
t\in[a,b],$

which completes the proof. ∎

######  Remark 2.

As direct corollaries of Theorem 2, one can obtain weak versions of
Pontryagin’s maximum principle for several different fractional operators:

  1. 1.

if $w(t)\equiv 1$ and $\beta=1$, then we obtain a weak Caputo–Fabrizio
fractional Pontryagin’s maximum principle with Hamiltonian system

$\left\\{\begin{array}[]{ll}\,{}^{R}_{a,1}D^{\alpha,1}x(t)=\dfrac{\partial\mathcal{H}}{\partial\lambda}\left(t,x(t),\lambda(t),u(t)\right),&t\in[a,b],\\\\[8.5359pt]
\,{}^{R}D_{b,1}^{\alpha,1}\lambda(t)=\dfrac{\partial\mathcal{H}}{\partial
x}\left(t,x(t),\lambda(t),u(t)\right),&t\in[a,b];\end{array}\right.$ (20)

  2. 2.

if $w(t)\equiv 1$ and $\beta=\alpha$, then we obtain a weak Atangana–Baleanu
fractional Pontryagin’s maximum principle with Hamiltonian system

$\left\\{\begin{array}[]{ll}\,{}^{R}_{a,1}D^{\alpha,\alpha}x(t)=\dfrac{\partial\mathcal{H}}{\partial\lambda}\left(t,x(t),\lambda(t),u(t)\right),&t\in[a,b],\\\\[8.5359pt]
\,{}^{R}D_{b,1}^{\alpha,\alpha}\lambda(t)=\dfrac{\partial\mathcal{H}}{\partial
x}\left(t,x(t),\lambda(t),u(t)\right),&t\in[a,b];\end{array}\right.$ (21)

  3. 3.

if $\beta=\alpha$, then we obtain a weak weighted Atangana–Baleanu fractional
Pontryagin’s maximum principle with Hamiltonian system

$\left\\{\begin{array}[]{ll}\,{}^{R}_{a,w}D^{\alpha,\alpha}x(t)=\dfrac{\partial\mathcal{H}}{\partial\lambda}\left(t,x(t),\lambda(t),u(t)\right),&t\in[a,b],\\\\[8.5359pt]
w(t)^{2}\,{}^{R}D_{b,w}^{\alpha,\alpha}\dfrac{\lambda(t)}{w(t)^{2}}=\dfrac{\partial\mathcal{H}}{\partial
x}\left(t,x(t),\lambda(t),u(t)\right),&t\in[a,b].\end{array}\right.$ (22)

##  4 Application: A Fractional Euler–Lagrange Equation

Let $L:[a,b]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be a
differentiable function with respect to all its arguments and consider the
following fundamental problem of the calculus of variations:

$J[x]=\int_{a}^{b}L\left(t,x(t),^{R}D^{\alpha,\beta}_{a,w}x(t)\right)dt\longrightarrow\min$
(23)

subject to the boundary conditions

$x(a)=x_{a},\quad x(b)=x_{b}.$ (24)

######  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:

$\dfrac{\partial L}{\partial
x}+w(t)^{2}\,{}^{R}D_{b,w}^{\alpha,\beta}\dfrac{\dfrac{\partial L}{\partial
x}}{w(t)^{2}}=0.$ (25)

###### Proof.

Set $u(t):={{}^{R}_{a,w}D}^{\alpha,\beta}x(t)$. The performance index of (23)
becomes equivalent to the objective functional

$J[x,u]=\int_{a}^{b}L\left(t,x(t),u(t)\right)dt$ (26)

subject to a control system (11) with $f\left(t,x,u\right)=u$. Applying our
main Theorem 2 to the current problem, that is, using the Hamiltonian system
(13) and the stationary condition (14), we have

$\begin{split}\dfrac{\partial L}{\partial
x}+w(t)^{2}\,{}^{R}D_{b,w}^{\alpha,\beta}\dfrac{\dfrac{\partial L}{\partial
u}}{w(t)^{2}}&=\dfrac{\partial L}{\partial
x}+w(t)^{2}\,{}^{R}D_{b,w}^{\alpha,\beta}\dfrac{\dfrac{\partial H}{\partial
u}-\lambda}{w(t)^{2}}=\dfrac{\partial L}{\partial
x}+w(t)^{2}\,{}^{R}D_{b,w}^{\alpha,\beta}\dfrac{-\lambda}{w(t)^{2}}\\\
&=\dfrac{\partial L}{\partial x}-\dfrac{\partial H}{\partial x}.\end{split}$
(27)

Since

$\displaystyle\mathcal{H}\left(t,x(t),\lambda(t),u(t)\right)$ $\displaystyle=$
$\displaystyle\lambda(t)f\left(t,x(t),u(t)\right)+L\left(t,x(t),u(t)\right)=\lambda(t)^{R}_{a,w}D^{\alpha,\beta}x(t)+L\left(t,x(t),u(t)\right)$
$\displaystyle=$ $\displaystyle\lambda(t)u(t)+L\left(t,x(t),u(t)\right),$

it follows from (27) that

$\dfrac{\partial L}{\partial
x}+w(t)^{2}\,{}^{R}D_{b,w}^{\alpha,\beta}\dfrac{\dfrac{\partial L}{\partial
x}}{w(t)^{2}}=0,$

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.

##  5 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.

## Acknowledgments

This research was funded by Fundação para a Ciência e a Tecnologia (FCT),
grant number UIDB/04106/2020 (CIDMA). The authors are grateful to two
anonymous reviewers and an editor for valuable comments and suggestions, which
helped them to improve the contents of the submitted version of the article.

## Data availability statement

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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