Closed-form solution of optimal control problem of a fractional order system

Tirumalasetty Chiranjeevi; Raj Kumar Biswas

doi:10.1016/j.jksus.2019.02.010

View/Download PDF

Buy Reprints

PDF

Translate this page into:

31 (

); 1042-1047

doi:

10.1016/j.jksus.2019.02.010

Closed-form solution of optimal control problem of a fractional order system

Tirumalasetty Chiranjeevi^⁎, Raj Kumar Biswas

Department of Electrical Engineering, National Institute of Technology, Silchar 788010, Assam, India

⁎Corresponding author. chirupci479@gmail.com (Tirumalasetty Chiranjeevi)

Received: 2018-7-9, Accepted: 2019-2-18, Published: 2019-10-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

A formulation and an analytical solution technique of fractional optimal control problem (FOCP) is presented in this paper. The performance index (PI) considered is a conformable fractional integral (CFI) function and is a function of both the state and the control variables. Dynamic behaviour of the system is described by conformable fractional differential equation (FDE). The necessary conditions of optimality and the general transversality condition in terms of Hamiltonian are obtained using variational approach. Both the fixed and free final end point conditions have been considered. An analytical solution technique is presented for solving the conformable FOCPs. To validate the formulation and solution scheme numerical examples are presented.

Keywords

Conformable fractional derivative

Conformable fractional integral

Optimal control

Calculus of variations

Show Related Articles from PubMed

1 Introduction

Fractional calculus has been attracted by many researchers from last few decades in different fields because systems described using FDEs give more accurate behaviour (Abuasad et al., 2019; Albzeirat et al., 2018; Dzielinski et al., 2010; Katsikadelis, 2015; Phaochoo et al., 2016; Tarasov, 2016 ). It should be mentioned here that impedance of a real capacitor is $\frac{1}{C s^{α}}, (0 < α < 1)$ , where C is a constant nearly equal to the conventional capacitance (Westeriund and Ekstam, 1994). This leads to a fractional order model and it contains most of the properties of the real capacitor. The capacitor remembers voltages it has been subjected to earlier. As fractional derivative represents nonlocal property or memory, FDE represents better description of the dynamic system. Many applications of fractional derivative could be found in (Bagley and Torvik, 1984; Diethelm, 2013; Ding et al., 2012; He and Li, 2016; Heymans, 2004; Jenson and Jeffreys, 1977; Luo et al., 2018; Mandelbrot, 1982; Rangaig and Convicto, 2018; Suribabu and Chiranjeevi, 2016 ) and references therein. Therefore, this subject is developing extensively.

Several definitions of a fractional derivative (FD) exist in the literature. For example, Riemann-Liouville (RL), Caputo, Riesz, Riesz-Caputo, Grünwald-Letnikov (GL) and Weyl FDs (Oldham and Spanier, 1974; Podlubny, 1999). All these FDs are defined via fractional integrals, so they represent nonlocal properties. As the FD is not a local property and it takes into account the effect of history, perhaps for this reason system description using FDE is more accurate. However, some discrepancies arise with these FDs because these derivatives in general do not follow chain rule, product rule, quotient rule, etc. Due to these discrepancies mathematical analysis becomes difficult. To overcome some of these difficulties, Kolwankar and Gangal (1996) define a type of local fractional derivative (LFD) using Riemann-Liouville formulation of fractional derivative.

It has been observed in the literature (Adda and Cresson, 2001; Babakhani and Daftardar-Gejji, 2002; Chen et al., 2010; Xiaojun and Baleanu, 2013 ), several authors have extended the properties of LFDs and its applications. The advantage of LFD is that it is limit based rather than defined via fractional integral. Recently, Khalil et al. (2014) introduce another kind of limit based FD called conformable fractional derivative (CFD). It is interesting to note that CFD follows most of the properties of integer derivative. Therefore, it is advantageous to use CFD from mathematical analysis point of view. The detailed analysis of CFDs and various solution methods of conformable FDEs are presented in (Abdeljawad, 2015; Abu Hammad and Khalil, 2014a,b,c,d; Atangana, 2015; Atangana et al., 2015; Batarfi et al., 2015; Bayour and Torres, 2017; Benkhettou et al., 2016; Khalil and Abu-Shaab, 2015; Lazo and Torres, 2017; Salahshour et al., 2015; Ünal et al., 2015, 2017; Ünal and Gökdoğan, 2017; Gökdoğan et al., 2016; Yang et al., 2016 ). Moreover, applications of CFD in different fields are found in (Avci et al., 2016,2017a,2017b; Çenesiz and Kurt, 2015, Çenesiz et al., 2017; Chung, 2015; Eslami and Rezazadeh, 2016; Kurt et al., 2015 ).

In Katugampola (2014), CFD is generalized and author introduces a new LFD which is also following most of the classical properties of integer derivative. Anderson and Ulness (2015) follow Katugampola’s derivative and address different applications in quantum mechanics. Zhao and Luo (2017) introduce a new LFD, called “general conformable fractional derivative” (GCFD), and also address the physical interpretation of GCFD. It is demonstrated that the CFD is a special case of GCFD. Also, the CFD has clear physical interpretations (Zhao and Luo, 2017). Therefore, CFD deals better with the dynamics of complex systems than other FDs in the literature. In the present work, we explore the application of CFDs and CFIs in FOCPs.

The problem of finding state and control variables which minimizes a given PI subjected to system constraints is referred as optimal control problem (OCP) (Lewis and Syrmos, 1995; Naidu, 2003). So far most of the FOCPs have been reported in terms of Riemann-Liouville ( Agrawal, 2004; Agrawal and Baleanu, 2007; Baleanu et al., 2009; Biswas and Sen, 2010, 2011b, 2014; Dzielinski and Czyronis, 2012,2013; Heydari et al., 2016; Keshavarz et al., 2016; Lotfi et al., 2011 ), Caputo (Agrawal, 2008; Alizadeh and Effati, 2018; Almeida and Torres, 2015; Biswas and Sen, 2011a; Chiranjeevi and Biswas, 2017; Dehghan et al., 2016; Ezz-Eldien et al., 2017; Javad Sabouri et al., 2017; Lotfi et al., 2013; Nemati and Yousefi, 2016 ) FDs and few works with other FDs like Riesz (Agrawal, 2007), Riesz-Caputo (Almeida, 2012; Frederico and Torres, 2010; Jarad et al., 2012 ) and combined Caputo (Odzijewicz et al., 2012). In this regard, Agrawal (2004), Agrawal and Baleanu (2007), Agrawal (2008) and Baleanu et al. (2009) present formulation and different solution schemes for FOCPs. Biswas and Sen (2010, 2011a) propose FOCP formulation and solution scheme using pseudo-state-space approach. Variational iteration method is used in Alizadeh and Effati (2018) for the solution of FOCPs. Almeida and Torres (2015) propose a discrete method for the solution of FOCPs. Biswas and Sen (2011b, 2014) investigate the formulation and solution of free and fixed final end point conditions of FOCPs. Solution of very large class of FOCPs by means of rational approximation is presented in Tricaud and Chen (2010). Fractional optimal control problem solution method based on Legendre wavelets is presented in Heydari et al. (2016). Javad Sabouri et al. (2017) obtain an approximate solution of FOCP using neural networks. An FOCP of discrete-time system with fixed final time have been investigated in (Dzielinski and Czyronis, 2012, 2013; Chiranjeevi and Biswas, 2017 ). Formulation of both continuous-time and discrete-time FOCPs with control constraints have been presented in Chiranjeevi and Biswas (2018). Dehghan et al. (2016) investigate the solution of FOCPs using the modified Jacobi polynomials. Numerical solution of FOCPs based on Legendre orthonormal polynomials and shifted Legendre orthonormal polynomials are presented in Lotfi et al. (2011, 2013), Ezz-Eldien et al. (2017) and Nemati and Yousefi (2016). The method based upon Bernoulli polynomials is presented in Keshavarz et al. (2016) for solving FOCPs. A Bezier curve method is presented in Ghomanjani (2016) for solving FOCPs and fractional Riccati differential equation.

All the above works consider FOCPs in terms of RL, Caputo, Riesz, Riesz-Caputo FDs. Formulation of FOCP results left and right FD problem. Hence, the analytical solution is difficult. Existing works in the literature present numerical solutions of FOCPs. Analytical solution of FOCP is still an open problem. In this respect this work presents FOCP in terms of CFD. The order of the FD has taken as $0 < α < 1$ . PI is considered in quadratic form. Both the cases of fixed final time OCP have been considered. Hamiltonian approach is used for obtaining the optimal conditions. The necessary conditions are solved using analytical method. With the best of authors’ knowledge this is the first time an analytical solution of FOCP is presented in this paper. However, FOCP in the sense of CFD is already considered in Lazo and Torres (2017). The contribution of the present work is problem formulation for different end point conditions and analytical solution technique.

2 Mathematical preliminaries

The left conformable fractional derivative of order $α \in (0, 1]$ starting from $a \in R$ of a unction $f : [a, b] \to R$ is defined by Lazo and Torres (2017)

(1a)

\frac{d_{a}^{α}}{{dt}_{a}^{α}} f (t) = f_{a}^{(α)} (t) = \lim_{ε \to 0} \frac{f (t + ε {(t - a)}^{1 - α}) - f (t)}{ε}

and the right conformable fractional derivative of order $0 < α \leq 1$ terminating at $b \in R$ of a function $f : [a, b] \to R$ is defined by

(1b)

\frac{_{b} d^{α}}{_{b} d t^{α}} f (t) =_{b} f^{(α)} (t) = - lim_{ε \to 0} \frac{f (t + ε {(b - t)}^{1 - α}) - f (t)}{ε}

The left conformable fractional integral of order $α \in (0, 1]$ starting from $a \in R$ of a function $f \in L^{1} [a, b]$ is defined by Lazo and Torres (2017)

(2a)

I_{a}^{α} f (t) = \int_{a}^{t} f (τ) d_{a}^{α} τ = \int_{a}^{t} \frac{f (τ)}{{(τ - a)}^{1 - α}} d τ

and the right conformable fractional integral of order $0 < α \leq 1$ terminating at $b \in R$ of a function $f \in L^{1} [a, b]$ is defined by

(2b)

_{b} I^{a} f (t) = \int_{t}^{b} f (τ)_{b} d^{a} τ = \int_{t}^{b} \frac{f (τ)}{{(b - τ)}^{1 - α}} d τ

From the definition of CFD, we can observe that at $α = 1$ the CFD reduces to integer derivative, but at $α = 0$ the CFD of a function does not produce the function itself. This is the main drawback of CFD. But CFD is obeying most of the properties of integer derivative and it is having clear physical interpretations. These properties are very useful in mathematical operation and physical applications.

There are some limitations with different types of FDs. For example, Riemann-Liouville FD is not suitable for dynamic systems modelling because the solution of Riemann-Liouville FD problems depend on fractional order initial conditions. However, fractional dynamic systems defined in terms of Caputo FDs accept natural initial conditions. Moreover, these definitions do not satisfy most of the properties of integer derivative. Due to non-local properties of these FDs, the computational effort and storage requirements increase significantly as compared to local fractional derivatives. However, CFD is a local FD and it satisfies most of the properties of integer derivative. Also, it has clear physical interpretation. In this work we consider FOCP where the FD is described as CFD.

3 Formulation of FOCP

Fixed final time FOCP formulation is presented in this section. Here the problem is to find the control $u (t)$ which minimizes the following PI

(3)

J (u) = \frac{1}{2} \int_{0}^{t_{f}} [q (t) x^{2} + r (t) u^{2}] d_{0}^{α} t

subject to

(4)

x_{0}^{(α)} (t) = a (t) x + b (t) u

with initial condition as $x (0) = x_{0}$ ,

where $x \in R$ is the state variable, $u \in R$ is the control variable, $q (t) \geq 0$ and $r (t) > 0$ . Here $x_{0}^{(α)} (t)$ represents conformable fractional derivative of $x (t)$ starting at $t = 0$ . In the rest of this paper we denote $x_{0}^{(α)} (t)$ as $x_{0}^{(α)}$ .

Introducing Lagrange multiplier $λ (t)$ we introduce the augmented performance index as

(5)

J_{a} (u) = \int_{0}^{t_{f}} [\frac{1}{2} (q (t) x^{2} + r (t) u^{2}) + λ (a (t) x + b (t) u - x_{0}^{(α)})] d_{0}^{α} t

The Lagrange multiplier method converts constrained optimization problem to an unconstrained optimization problem. It can be noted that $J_{a} (u) = J (u)$ , if Eq. (4) is satisfied for any $λ (t)$ .

We define Hamiltonian function as

(6)

H = \frac{1}{2} (q (t) x^{2} + r (t) u^{2}) + λ (a (t) x + b (t) u)

Thus, the augmented PI $J_{a} (u)$ in terms of Hamiltonian function is

(7)

J_{a} (u) = \int_{0}^{t_{f}} [H - λ x_{0}^{(α)}] d_{0}^{α} t

The first variation of the augmented PI $J_{a} (u)$ is

(8)

δ J_{a} (u) = \int_{0}^{t_{f}} [\frac{\partial H}{\partial x} δ x + \frac{\partial H}{\partial u} δ u + \frac{\partial H}{\partial λ} δ λ - λ δ x_{0}^{(α)} - x_{0}^{(α)} δ λ] d_{0}^{α} t

Considering the fourth term of above integral and using integration by parts we get

(9)

\int_{0}^{t_{f}} λ δ x_{0}^{(α)} d_{0}^{α} t = λ δ {x|}_{t_{f}} - \int_{0}^{t_{f}} λ_{0}^{(α)} δ x d_{0}^{α} t

Substituting Eq. (9) in Eq. (8), we get

(10)

δ J_{a} (u) = \int_{0}^{t_{f}} [(\frac{\partial H}{\partial x} + λ_{0}^{(α)}) δ x + \frac{\partial H}{\partial u} δ u + (\frac{\partial H}{\partial λ} - x_{0}^{(α)}) δ λ] d_{0}^{α} t - λ δ {x|}_{t_{f}}

The first variation $δ J_{a} = 0$ , for optimum. Which requires the coefficients of $δ x$ , $δ u$ and $δ λ$ in Eq. (10) be zero. Thus, the necessary conditions are

(11)

λ_{0}^{(α)} = - \frac{\partial H}{\partial x} = - q (t) x - a (t) λ

(12)

x_{0}^{(α)} = \frac{\partial H}{\partial λ} = a (t) x + b (t) u

(13)

\frac{\partial H}{\partial u} = 0 \Rightarrow u = - r^{- 1} (t) b (t) λ

Finally, the Eq. (10) becomes

(14)

λ δ {x|}_{t_{f}} = 0

Eqs. (11)–(13) are the necessary conditions for optimality and Eq. (14) represents a general transversality condition for the FOCP in terms of CFD. It should be mentioned here that more general necessary conditions are presented in Lazo and Torres (2017). In contrast, a general transversality condition is obtained in the present work for different end point conditions.

4 Analytical solution scheme for conformable FOCPs

In this section, we present an analytical solution scheme for solving conformable FOCPs. Optimal conditions are

(15)

λ_{0}^{(α)} = - q (t) x - a (t) λ

(16)

x_{0}^{(α)} = a (t) x + b (t) u

(17)

u = - r^{- 1} (t) b (t) λ

Substituting Eq. (17) in Eq. (16), we get

(18)

x_{0}^{(α)} = a (t) x - r^{- 1} (t) b^{2} (t) λ

Eqs. (15) and (18) can be written in vector matrix form as

(19)

[\begin{matrix} x_{0}^{(α)} \\ λ_{0}^{(α)} \end{matrix}] = [\begin{matrix} a (t) & - r^{- 1} (t) b^{2} (t) \\ - q (t) & - a (t) \end{matrix}] [\begin{matrix} x \\ λ \end{matrix}]

Now Eq. (19) can be written as

(20)

z_{0}^{(α)} = A z, z (0) = z_{0}

z_{0}^{(α)} = [\begin{matrix} x_{0}^{(α)} \\ λ_{0}^{(α)} \end{matrix}], A = [\begin{matrix} a (t) & - r^{- 1} (t) b^{2} (t) \\ - q (t) & - a (t) \end{matrix}], z = [\begin{matrix} x \\ λ \end{matrix}] a n d z (0) = [\begin{matrix} x (0) \\ λ (0) \end{matrix}]

Applying the operator $I_{0}^{α}$ to the above equation, we obtain

(21)

z = z_{0} + A (I_{0}^{α} z)

Now we have

(22)

z_{n + 1} = z_{0} + A (I_{0}^{α} z_{n}), n = 0, 1, 2, \dots

where the “conformable fractional integral” is defined by Abdeljawad (2015)

(23)

{(I}_{0}^{α} z) = \int_{0}^{t} z (τ) τ^{α - 1} d τ

For $n = 0$ ,

(24)

z_{1} = z_{0} + A (I_{0}^{α} z_{0}) = [I + \frac{A t^{α}}{α}] z_{0}

For $n = 1$ ,

(25)

z_{2} = z_{0} + A (I_{0}^{α} z_{1}) = [I + \frac{A t^{α}}{α} + A^{2} \frac{t^{2 α}}{α (2 α)}] z_{0}

and finally

(26)

z_{n} = [\sum_{k = 0}^{n} \frac{A^{k} t^{k α}}{α^{k} k!}] z_{0}

Letting

(27)

n \to \infty, z (t) = [\sum_{k = 0}^{\infty} \frac{A^{k} t^{k α}}{α^{k} k!}] z_{0}

Therefore, the solution of Eq. (20) can be given as Abdeljawad (2015)

(28)

z (t) = [\sum_{k = 0}^{\infty} \frac{A^{k} t^{k α}}{α^{k} k!}] z_{0} = e^{(\frac{A t^{α}}{α})} z_{0}

5 Illustrative examples

This section presents two numerical examples to show the validity of the formulation and the solution scheme.

Example 1. Here, we consider the example of fixed final time-fixed final state FOCP. Minimize the PI

(29)

J (u) = \frac{1}{2} \int_{0}^{t_{f}} [x^{2} + u^{2}] d_{0}^{α} t

subject to

(30)

x_{0}^{(α)} = - x + u

with the initial and final conditions

(31)

x (0) = 1 a n d x (1) = 0

The necessary conditions for this problem are

(32)

x_{0}^{(α)} = - x - λ

(33)

λ_{0}^{(α)} = - x + λ

(34)

u = - λ

By solving Eqs. (32)–(34) using the solution technique discussed in Section 4, we obtain the expressions for $x (t)$ and $u (t)$ as

(35)

x (t) = \frac{1}{2 \sqrt{2}} [(\sqrt{2} x_{0} - x_{0} - λ_{0}) e^{\frac{\sqrt{2} t^{α}}{α}} + (\sqrt{2} x_{0} + x_{0} + λ_{0}) e^{\frac{- \sqrt{2} t^{α}}{α}}]

(36)

u (t) = \frac{1}{2 \sqrt{2}} [(x_{0} - {\sqrt{2} λ}_{0} - λ_{0}) e^{\frac{\sqrt{2} t^{α}}{α}} - (x_{0} + \sqrt{2} λ_{0} - λ_{0}) e^{\frac{- \sqrt{2} t^{α}}{α}}]

Using Eqs. (35) and (36) in Eq. (29), we obtain minimum PI. Thus

(37)

J_{\min} = \frac{1}{16} [M (e^{\frac{2 \sqrt{2}}{α}} - 1) + N (1 - e^{\frac{- 2 \sqrt{2}}{α}})]

M = \frac{[{(\sqrt{2} x_{0} - x_{0} - λ_{0})}^{2} + {(x_{0} - \sqrt{2} λ_{0} - λ_{0})}^{2}]}{2 \sqrt{2}} a n d N = \frac{[{(\sqrt{2} x_{0} + x_{0} + λ_{0})}^{2} + {(x_{0} + \sqrt{2} λ_{0} - λ_{0})}^{2}]}{2 \sqrt{2}}

Solving Eqs. (35)–(37) with the given boundary conditions for different values of $α$ we get $x (t), u (t) a n d J_{\min}$ .

Example 2. Here, we consider the example of fixed final time-free final state FOCP. Find the optimal control $u (t)$ which minimizes the PI in Eq. (29), subject to the system dynamics given by Eq. (30) and the initial condition $x (0) = 1$ . Necessary conditions for this problem are given by Eqs. (32)–(34) and the tranversality condition is

(38)

λ (1) = 0

Analytical expressions for $x (t)$ and $u (t)$ are given in Eqs. (35) and (36). By solving Eqs. (35) and (36) using initial and final conditions $x (0) = 1 a n d λ (1) = 0$ for different values of $α$ , we can get optimal state $x (t)$ and optimal control $u (t)$ .

In this section, for different values of $α$ we solve both the free and fixed final state FOCPs. Figs. 1–4 show the $x (t)$ and $u (t)$ for different $α$ . We can observe from these figures that amplitude of $x (t)$ and $u (t)$ increases like in (Agrawal, 2004; Biswas and Sen, 2011a,b ) as $α$ is increased. Moreover, at $α = 1$ conformable FOCP results recover integer order OCP results as expected. Table 1 shows the minimum value of PI $J_{\min}$ for different values of $α$ for fixed final state problem. From this we observe that performance index decreases as $α$ is decreased.

Optimal state x ( t ) for different α for the fixed final time-fixed final state problem ∗ : α = 0.35 , Δ : α = 0.5 , × : α = 0.7 , O : α = 0.85 , - : α = 1.0 .

Optimal control u ( t ) for different α for the fixed final time-fixed final state problem ∗ : α = 0.35 , Δ : α = 0.5 , × : α = 0.7 , O : α = 0.85 , - : α = 1.0 .

Optimal state x ( t ) for different α for the fixed final time-free final state problem ∗ : α = 0.45 , Δ : α = 0.6 , × : α = 0.75 , O : α = 0.9 , - : α = 1.0 .

Optimal control u ( t ) for different α for the fixed final time-free final state problem ∗ : α = 0.45 , Δ : α = 0.6 , × : α = 0.75 , O : α = 0.9 , - : α = 1.0 .

Table 1 Minimum value of PI

J_{\min}

for different

α

for the fixed final time-fixed final state problem.

α	0.5	0.7	0.85	1.0
J_min	0.2117	0.2321	0.2594	0.2956

6 Conclusions

In this paper, a general formulation of fixed final time FOCP defined in terms of CFD, and an analytical solution scheme is presented. Quadratic PI with conformal fractional integral function is considered in this problem. Both free and fixed final state FOCPs are considered. The necessary optimality conditions and general transversality condition are obtained using Hamiltonian approach and conformable fractional calculus of variations. The necessary conditions are solved using analytical method. It can be noted that this is the first time an analytical solution of FOCP in terms of CFD has been presented in this work. The efficacy of the formulation has been demonstrated by illustrative examples. Results are produced for different values of $α$ . It is observed that as $α$ decreases $x (t)$ is also decreased and it demands small control effort. Further, with the decreases in $α$ , PI value also decreases.

It should be mentioned here that existence of optimal solution has not considered in this work. Existence of optimal solution and its stability analysis will be considered in our next work.

References

Abdeljawad T., . On conformable fractional calculus. J. Comput. Appl. Math.. 2015;279:57-66.
[Google Scholar]
Abuasad S., Moaddy K., Hashim I., . Analytical treatment of two-dimensional fractional Helmholtz equations. J. King Saud Univ. – Sci.. 2019;31:659-666.
[CrossRef] [Google Scholar]
Abu Hammad I., Khalil R., . Fractional Fourier series with applications. Am. J. Comput. Appl. Math.. 2014;4(6):187-191.
[Google Scholar]
Abu Hammad M., Khalil R., . Legendre fractional differential equation and Legender fractional polynomials. Int. J. Appl. Math. Res.. 2014;3(3):214-219.
[Google Scholar]
Abu Hammad M., Khalil R., . Abel’s formula and wronskian for conformable fractional differential equations. Int. J. Differ. Equ. Appl.. 2014;13(3):177-183.
[Google Scholar]
Abu Hammad M., Khalil R., . Conformable fractional heat differential equation. Int. J. Pure Appl. Math.. 2014;94(2):215-221.
[Google Scholar]
Adda F.B., Cresson J., . About non-differentiable functions. J. Math. Anal. Appl.. 2001;263:721-737.
[Google Scholar]
Agrawal O.P., . A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn.. 2004;38:323-337.
[Google Scholar]
Agrawal O.P., . Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A: Math. Theor.. 2007;40(24):6287-6303.
[Google Scholar]
Agrawal O.P., Baleanu D., . A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control. 2007;13(9–10):1269-1281.
[Google Scholar]
Agrawal O.P., . A quadratic numerical scheme for fractional optimal control problems. ASME J. Dyn. Sys., Meas. Control. 2008;130:0110101-0110106.
[Google Scholar]
Albzeirat A.K., Ahmad M.Z., Momani S., Ahmad I., . New implementation of reproducing kernel Hilbert space method for solving a fuzzy integro-differential equation of integer and fractional orders. J. King Saud Univ. – Sci.. 2018;30:352-358.
[Google Scholar]
Alizadeh A., Effati S., . An iterative approach for solving fractional optimal control problems. J. Vib. Control. 2018;24(1):18-36.
[Google Scholar]
Almeida R., . Fractional variational problems with the Riesz-Caputo derivative. Appl. Math. Lett.. 2012;25:142-148.
[Google Scholar]
Almeida R., Torres D.F.M., . A discrete method to solve fractional optimal control problems. Nonlinear Dyn.. 2015;80:1811-1816.
[Google Scholar]
Anderson D.R., Ulness D.J., . Properties of the katugampola fractional derivative with potential application in quantum mechanics. J. Math. Phys.. 2015;56:0635021-06350218.
[Google Scholar]
Atangana A., . Derivative with a New Parameter: Theory, Methods and Applications. London, UK: Academic Press; 2015.
Atangana A., Baleanu D., Alsaedi A., . New properties of conformable derivative. Open Math.. 2015;13:889-898.
[Google Scholar]
Avci, D., İskender Eroğlu, B.B., Özdemir, N., 2016. Conformable heat problem in a cylinder. In: Proceedings of the International Conference on Fractional Differentiation and its Applications. Novi Sad, Serbia, July 18-20, pp. 572–581.
Avci D., İskender Eroğlu B.B., Özdemir N., . The Dirichlet problem of a conformable advection-diffusion equation. Therm. Sci.. 2017;21(1):9-18.
[Google Scholar]
Avci D., İskender Eroğlu B.B., Özdemir N., . Conformable heat equation on a radial symmetric plate. Therm. Sci.. 2017;21(2):819-826.
[Google Scholar]
Babakhani A., Daftardar-Gejji V., . On calculus of local fractional derivatives. J. Math. Anal. Appl.. 2002;270:66-79.
[Google Scholar]
Bagley R.L., Torvik P., . On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech.. 1984;51:294-298.
[Google Scholar]
Baleanu B., Defterli O., Agrawal O.P., . A central difference numerical scheme for fractional optimal control problems. J. Vib. Control. 2009;15:583-597.
[Google Scholar]
Batarfi H., Losada J., Nieto J.J., Shammakh W., . Three-point boundary value problems for conformable fractional differential equations. J. Funct. Spaces 2015:1-6.
[Google Scholar]
Bayour B., Torres D.F.M., . Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math.. 2017;312:127-133.
[Google Scholar]
Benkhettou N., Hassani S., Torres D.F.M., . A conformable fractional calculus on arbitrary time scales. J. King Saud Univ. Sci.. 2016;28:93-98.
[Google Scholar]
Biswas R.K., Sen S., . Fractional optimal control problems: a pseudo-state-space approach. J. Vib. Control. 2010;17(7):1034-1041.
[Google Scholar]
Biswas, R.K., Sen, S., 2011a. Fractional optimal control within Caputo’s derivative. In: Proceedings of the ASME International Design Engineering Technical Conference and Computers and Information in Engineering Conference, Washington, DC, USA, August 28-31, pp. 1–8.
Biswas R.K., Sen S., . Fractional optimal control problems with specified final time. ASME J. Comput. Nonlinear Dyn.. 2011;6:0210091-0210096.
[Google Scholar]
Biswas R.K., Sen S., . Free final time fractional optimal control problems. J. Franklin Inst.. 2014;351:941-951.
[Google Scholar]
Çenesiz Y., Kurt A., . The new solution of time fractional wave equation with conformable fractional derivative definition. J. N. Theo.. 2015;7:79-85.
[Google Scholar]
Çenesiz Y., Baleanu D., Kurt A., Tasbozan O., . New exact solutions of Burgers’ type equations with conformable derivative. Wave Random Complex. 2017;21(1):103-116.
[Google Scholar]
Chen Y., Yan Y., Zhang K., . On the local fractional derivative. J. Math. Anal. Appl.. 2010;362:17-33.
[Google Scholar]
Chiranjeevi T., Biswas R.K., . Formulation of optimal control problems of fractional dynamic systems with control constraints. J. Adv. Res. Dynam. Control Syst.. 2018;3:201-212.
[Google Scholar]
Chiranjeevi T., Biswas R.K., . Discrete-time fractional optimal control. Mathematics. 2017;5(2):1-12.
[Google Scholar]
Chung W.S., . Fractional newton mechanics with conformable fractional derivative. J. Comput. Appl. Math.. 2015;290:150-158.
[Google Scholar]
Dehghan M., Hamedi E.A., Khosravian-Arab H., . A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J. Vib. Control. 2016;22(6):1547-1559.
[Google Scholar]
Diethelm K., . A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn.. 2013;71:613-619.
[Google Scholar]
Ding Y., Wang Z., Ye H., . Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Control Syst. Technol.. 2012;20(3):763-769.
[Google Scholar]
Dzielinski A., Sierociuk D., Sarwas G., . Some applications of fractional order calculus. B. Pol. Acad. Sci-Tech.. 2010;58(4):583-592.
[Google Scholar]
Dzielinski A., Czyronis P.M., . Fixed final time optimal control problem for fractional dynamic systems-linear quadratic discrete-time case. In: Mikolaj B., Krzysztof M., eds. Advances in Control Theory and Automation. Poland: Printing House of Bialystok University of Technology; 2012. p. :71-80.
[Google Scholar]
Dzielinski A., Czyronis P.M., . Fixed final time and free final state optimal control problem for fractional dynamic systems-linear quadratic discrete-time case. B. Pol. Acad. Sci-Tech.. 2013;61(3):681-690.
[Google Scholar]
Eslami M., Rezazadeh H., . The first integral method for Wu-Zhang system with conformable time fractional derivative. Calcolo. 2016;53:475-485.
[Google Scholar]
Ezz-Eldien S.S., Doha E.H., Baleanu D., Bhrawy A.H., . A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. J. Vib. Control. 2017;23(1):16-30.
[Google Scholar]
Frederico G.S.F., Torres D.F.M., . Fractional Noether’s theorem in the Riesz-Caputo sense. Appl. Math. Comput.. 2010;217:1023-1033.
[Google Scholar]
Ghomanjani F., . A numerical technique for solving fractional optimal control problems and fractional Riccati differential equations. J. Egypt. Math. Soc.. 2016;24:638-643.
[Google Scholar]
Gökdoğan A., Ünal E., Celik E., . Existence and uniqueness theorems for sequential linear conformable fractional differential equations. Miskolc Math. Notes. 2016;17(1):267-279.
[Google Scholar]
He J.H., Li Z.B., . A fractional model for dye removal. J. King Saud Univ. – Sci.. 2016;28:14-16.
[Google Scholar]
Heydari M.H., Hooshmandasl M.R., Maalek Ghaini F.M., Cattani C., . Wavelets method for solving fractional optimal control problems. Appl. Math. Comput.. 2016;286:139-154.
[Google Scholar]
Heymans N., . Fractional calculus description of non-linear viscoelastic behavior of polymers. Nonlinear Dyn.. 2004;38:221-231.
[Google Scholar]
Jarad F., Abdeljawad T., Baleanu D., . On Riesz-Caputo formulation for sequential fractional variational principles. Abstr. Appl. Anal. 2012:1-15.
[Google Scholar]
Javad Sabouri K., Effati S., Pakdaman M., . A neural network approach for solving a class of fractional optimal control problems. Neural Process Lett.. 2017;45:59-74.
[Google Scholar]
Jenson V.G., Jeffreys G.V., . Mathematical Methods in Chemical Engineering. New York: Academic Press; 1977.
Katsikadelis J.T., . Generalized fractional derivatives and their applications to mechanical systems. Arch. Appl. Mech.. 2015;85:1307-1320.
[Google Scholar]
Katugampola, U.N., 2014. A new fractional derivative with classical properties. J. Amer. Math. Soc. 1-8. arXiv: 1410.6535.
Keshavarz E., Ordokhani Y., Razzaghi M., . A numerical solution for fractional optimal control problems via Bernoulli polynomials. J. Vib. Control. 2016;22(18):3889-3903.
[Google Scholar]
Khalil R., Horani M.A., Yousef A., Sababheh M., . A new definition of fractional derivative. J. Comput. Appl. Math.. 2014;264:65-70.
[Google Scholar]
Khalil R., Abu-Shaab H., . Solution of some conformable fractional differential equations. Int. J. Pure Appl. Math.. 2015;103(4):667-673.
[Google Scholar]
Kolwankar K.M., Gangal A.D., . Fractional differentiability of nowhere differentiable functions and dimensions. Chaos: An Interdiscip. J. Nonlinear Sci.. 1996;6(4):505-513.
[Google Scholar]
Kurt A., Çenesiz Y., Tasbozan O., . On the solution of Burgers’ equation with the new fractional derivative. Open Phys.. 2015;13:355-360.
[Google Scholar]
Lazo M.J., Torres D.F.M., . Variational calculus with conformable fractional derivatives. IEEE/CAA J. Autom. Sin.. 2017;4(2):340-352.
[Google Scholar]
Lewis F.L., Syrmos V.L., . Optimal Control. New York: Wiley; 1995.
Lotfi A., Dehghan M., Yousefi S.A., . A numerical technique for solving fractional optimal control problems. Comput. Math. Appl.. 2011;62:1055-1067.
[Google Scholar]
Lotfi A., Yousefi S.A., Dehghan M., . Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J. Comput. Appl. Math.. 2013;250:143-160.
[Google Scholar]
Luo D., Wang J.R., Feckan M., . Applying fractional calculus to analyze economic growth modeling. J. Appl. Math., Stat. Inform.. 2018;14(1):25-36.
[Google Scholar]
Mandelbrot B., . The Fractal Geometry of Nature. San Francisco: Freeman; 1982.
Naidu D.S., . Optimal Control Systems. Bocaraton, Florida: CRC Press; 2003.
Nemati A., Yousefi S.A., . A numerical method for solving fractional optimal control problems using Ritz method. ASME J. Comput. Nonlinear Dyn.. 2016;11:0510151-0510157.
[Google Scholar]
Odzijewicz T., Malinowska A.B., Torres D.F.M., . Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal.. 2012;75:1507-1515.
[Google Scholar]
Oldham K.B., Spanier J., . The Fractional Calculus. New York: Academic Press; 1974.
Phaochoo P., Luadsong A., Aschariyaphotha N., . The meshless local Petrov-Galerkin based on moving kriging interpolation for solving fractional Black-Scholes model. J. King Saud Univ. – Sci.. 2016;28:111-117.
[Google Scholar]
Podlubny I., . Fractional Differential Equations. New York: Academic Press; 1999.
Rangaig N.A., Convicto V.C., . On fractional modelling of dye removal using fractional derivative with non-singular kernel. J. King Saud Univ. – Sci. 2018
[CrossRef] [Google Scholar]
Salahshour S., Ahmadian A., Ismail F., Baleanu D., Senu N., . A new fractional derivative for differential equation of fractional order under interval uncertainty. Adv. Mech. Eng.. 2015;7(12):1-11.
[Google Scholar]
Suribabu, G., Chiranjeevi, T., 2016. Implementation of fractional order PID controller for an AVR system using GA and ACO optimization techniques. In: Proceedings of the IFAC-ACODS Conference, NIT Trichy, India, Feb 1–5, pp. 456–461.
Tarasov V.E., . Discrete model of dislocations in fractional nonlocal elasticity. J. King Saud Univ. – Sci.. 2016;28:33-36.
[Google Scholar]
Tricaud C., Chen Y.Q., . An approximate method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl.. 2010;59:1644-1655.
[Google Scholar]
Ünal E., Gökdoğan A., Celik E., . Solutions of sequential conformable fractional differential equations around an ordinary point and conformable fractional hermite differential equation. Br. J. Appl. Sci. Technol.. 2015;10(2):1-11.
[Google Scholar]
Ünal E., Gökdoğan A., . Solution of conformable fractional ordinary differential equations via differential transformation method. Optik. 2017;128:264-273.
[Google Scholar]
Ünal E., Gökdoğan A., Cumhur İ., . The operator method for local fractional linear differential equations. Optik. 2017;131:986-993.
[Google Scholar]
Westeriund S., Ekstam L., . Capacitor theory. IEEE Trans. Dielec. El. In.. 1994;1(5):826-839.
[Google Scholar]
Xiaojun Y., Baleanu D., . Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci.. 2013;17(2):625-628.
[Google Scholar]
Yang, X.J., Baleanu, D., Srivastava, H.M., 2016. Local Fractional Integral Transforms and Their Applications. Academic Press, San Diego (CA).
Zhao D., Luo M., . General conformable fractional derivative and its physical interpretation. Calcolo. 2017;54(3):903-917.
[Google Scholar]

Appendix

Sufficient condition

In order to determine the nature of optimization we need to consider the second variation and examine its sign. The second variation of the functional $J_{a} (u)$ is given by $δ^{2} J_{a} (u) = \frac{1}{2} \int_{0}^{t_{f}} [\frac{\partial^{2} H}{\partial x^{2}} δ x^{2} + 2 \frac{\partial^{2} H}{\partial x \partial u} δ x δ u + \frac{\partial^{2} H}{\partial u^{2}} δ u^{2}] d_{0}^{α} t$ $= \frac{1}{2} \int_{0}^{t_{f}} [\begin{matrix} δ x & δ u \end{matrix}] [\begin{matrix} \frac{\partial^{2} H}{\partial x^{2}} & \frac{\partial^{2} H}{\partial x \partial u} \\ \frac{\partial^{2} H}{\partial x \partial u} & \frac{\partial^{2} H}{\partial u^{2}} \end{matrix}] [\begin{matrix} δ x \\ δ u \end{matrix}] d_{0}^{α} t$ $= \frac{1}{2} \int_{0}^{t_{f}} [\begin{matrix} δ x & δ u \end{matrix}] Π [\begin{matrix} δ x \\ δ u \end{matrix}] d_{0}^{α} t$

For the minimum, the second variation $δ^{2} J_{a}$ must be positive. This means that the matrix

$Π = [\begin{matrix} \frac{\partial^{2} H}{\partial x^{2}} & \frac{\partial^{2} H}{\partial x \partial u} \\ \frac{\partial^{2} H}{\partial x \partial u} & \frac{\partial^{2} H}{\partial u^{2}} \end{matrix}]$ must be positive definite.

In most of the cases this reduces to the condition that $\frac{\partial^{2} H}{\partial u^{2}}$ must be positive definite (negative definite) for minimum (maxium). Now using the Hamiltonian in Eq. (6) and calculating the various partials, we have

$Π = [\begin{matrix} q (t) & 0 \\ 0 & r (t) \end{matrix}]$ . Since $q (t) \geq 0 a n d r (t) > 0$ , it follows that $Π$ is only positive semidefinite. However, $\frac{\partial^{2} H}{\partial u^{2}} = r (t)$ is positive definite, thus represents minimum.

Introduction

Mathematical preliminaries

Formulation of FOCP

Analytical solution scheme for conformable FOCPs

Illustrative examples

Conclusions

Fulltext Views
20

PDF downloads
5

View/Download PDF
Download Citations

BibTeX
RIS

Show Sections

[1] Abdeljawad T., . On conformable fractional calculus. J. Comput. Appl. Math.. 2015;279:57-66.
[Google Scholar]

[2] Abuasad S., Moaddy K., Hashim I., . Analytical treatment of two-dimensional fractional Helmholtz equations. J. King Saud Univ. – Sci.. 2019;31:659-666.
[CrossRef] [Google Scholar]

[3] Abu Hammad I., Khalil R., . Fractional Fourier series with applications. Am. J. Comput. Appl. Math.. 2014;4(6):187-191.
[Google Scholar]

[4] Abu Hammad M., Khalil R., . Legendre fractional differential equation and Legender fractional polynomials. Int. J. Appl. Math. Res.. 2014;3(3):214-219.
[Google Scholar]

[5] Abu Hammad M., Khalil R., . Abel’s formula and wronskian for conformable fractional differential equations. Int. J. Differ. Equ. Appl.. 2014;13(3):177-183.
[Google Scholar]

[6] Abu Hammad M., Khalil R., . Conformable fractional heat differential equation. Int. J. Pure Appl. Math.. 2014;94(2):215-221.
[Google Scholar]

[7] Adda F.B., Cresson J., . About non-differentiable functions. J. Math. Anal. Appl.. 2001;263:721-737.
[Google Scholar]

[8] Agrawal O.P., . A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn.. 2004;38:323-337.
[Google Scholar]

[9] Agrawal O.P., . Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A: Math. Theor.. 2007;40(24):6287-6303.
[Google Scholar]

[10] Agrawal O.P., Baleanu D., . A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control. 2007;13(9–10):1269-1281.
[Google Scholar]

[11] Agrawal O.P., . A quadratic numerical scheme for fractional optimal control problems. ASME J. Dyn. Sys., Meas. Control. 2008;130:0110101-0110106.
[Google Scholar]

[12] Albzeirat A.K., Ahmad M.Z., Momani S., Ahmad I., . New implementation of reproducing kernel Hilbert space method for solving a fuzzy integro-differential equation of integer and fractional orders. J. King Saud Univ. – Sci.. 2018;30:352-358.
[Google Scholar]

[13] Alizadeh A., Effati S., . An iterative approach for solving fractional optimal control problems. J. Vib. Control. 2018;24(1):18-36.
[Google Scholar]

[14] Almeida R., . Fractional variational problems with the Riesz-Caputo derivative. Appl. Math. Lett.. 2012;25:142-148.
[Google Scholar]

[15] Almeida R., Torres D.F.M., . A discrete method to solve fractional optimal control problems. Nonlinear Dyn.. 2015;80:1811-1816.
[Google Scholar]

[16] Anderson D.R., Ulness D.J., . Properties of the katugampola fractional derivative with potential application in quantum mechanics. J. Math. Phys.. 2015;56:0635021-06350218.
[Google Scholar]

[17] Atangana A., . Derivative with a New Parameter: Theory, Methods and Applications. London, UK: Academic Press; 2015.

[18] Atangana A., Baleanu D., Alsaedi A., . New properties of conformable derivative. Open Math.. 2015;13:889-898.
[Google Scholar]

[19] Avci, D., İskender Eroğlu, B.B., Özdemir, N., 2016. Conformable heat problem in a cylinder. In: Proceedings of the International Conference on Fractional Differentiation and its Applications. Novi Sad, Serbia, July 18-20, pp. 572–581.

[20] Avci D., İskender Eroğlu B.B., Özdemir N., . The Dirichlet problem of a conformable advection-diffusion equation. Therm. Sci.. 2017;21(1):9-18.
[Google Scholar]

[21] Avci D., İskender Eroğlu B.B., Özdemir N., . Conformable heat equation on a radial symmetric plate. Therm. Sci.. 2017;21(2):819-826.
[Google Scholar]

[22] Babakhani A., Daftardar-Gejji V., . On calculus of local fractional derivatives. J. Math. Anal. Appl.. 2002;270:66-79.
[Google Scholar]

[23] Bagley R.L., Torvik P., . On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech.. 1984;51:294-298.
[Google Scholar]

[24] Baleanu B., Defterli O., Agrawal O.P., . A central difference numerical scheme for fractional optimal control problems. J. Vib. Control. 2009;15:583-597.
[Google Scholar]

[25] Batarfi H., Losada J., Nieto J.J., Shammakh W., . Three-point boundary value problems for conformable fractional differential equations. J. Funct. Spaces 2015:1-6.
[Google Scholar]

[26] Bayour B., Torres D.F.M., . Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math.. 2017;312:127-133.
[Google Scholar]

[27] Benkhettou N., Hassani S., Torres D.F.M., . A conformable fractional calculus on arbitrary time scales. J. King Saud Univ. Sci.. 2016;28:93-98.
[Google Scholar]

[28] Biswas R.K., Sen S., . Fractional optimal control problems: a pseudo-state-space approach. J. Vib. Control. 2010;17(7):1034-1041.
[Google Scholar]

[29] Biswas, R.K., Sen, S., 2011a. Fractional optimal control within Caputo’s derivative. In: Proceedings of the ASME International Design Engineering Technical Conference and Computers and Information in Engineering Conference, Washington, DC, USA, August 28-31, pp. 1–8.

[30] Biswas R.K., Sen S., . Fractional optimal control problems with specified final time. ASME J. Comput. Nonlinear Dyn.. 2011;6:0210091-0210096.
[Google Scholar]

[31] Biswas R.K., Sen S., . Free final time fractional optimal control problems. J. Franklin Inst.. 2014;351:941-951.
[Google Scholar]

[32] Çenesiz Y., Kurt A., . The new solution of time fractional wave equation with conformable fractional derivative definition. J. N. Theo.. 2015;7:79-85.
[Google Scholar]

[33] Çenesiz Y., Baleanu D., Kurt A., Tasbozan O., . New exact solutions of Burgers’ type equations with conformable derivative. Wave Random Complex. 2017;21(1):103-116.
[Google Scholar]

[34] Chen Y., Yan Y., Zhang K., . On the local fractional derivative. J. Math. Anal. Appl.. 2010;362:17-33.
[Google Scholar]

[35] Chiranjeevi T., Biswas R.K., . Formulation of optimal control problems of fractional dynamic systems with control constraints. J. Adv. Res. Dynam. Control Syst.. 2018;3:201-212.
[Google Scholar]

[36] Chiranjeevi T., Biswas R.K., . Discrete-time fractional optimal control. Mathematics. 2017;5(2):1-12.
[Google Scholar]

[37] Chung W.S., . Fractional newton mechanics with conformable fractional derivative. J. Comput. Appl. Math.. 2015;290:150-158.
[Google Scholar]

[38] Dehghan M., Hamedi E.A., Khosravian-Arab H., . A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J. Vib. Control. 2016;22(6):1547-1559.
[Google Scholar]

[39] Diethelm K., . A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn.. 2013;71:613-619.
[Google Scholar]

[40] Ding Y., Wang Z., Ye H., . Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Control Syst. Technol.. 2012;20(3):763-769.
[Google Scholar]

[41] Dzielinski A., Sierociuk D., Sarwas G., . Some applications of fractional order calculus. B. Pol. Acad. Sci-Tech.. 2010;58(4):583-592.
[Google Scholar]

[42] Dzielinski A., Czyronis P.M., . Fixed final time optimal control problem for fractional dynamic systems-linear quadratic discrete-time case. In: Mikolaj B., Krzysztof M., eds. Advances in Control Theory and Automation. Poland: Printing House of Bialystok University of Technology; 2012. p. :71-80.
[Google Scholar]

[43] Dzielinski A., Czyronis P.M., . Fixed final time and free final state optimal control problem for fractional dynamic systems-linear quadratic discrete-time case. B. Pol. Acad. Sci-Tech.. 2013;61(3):681-690.
[Google Scholar]

[44] Eslami M., Rezazadeh H., . The first integral method for Wu-Zhang system with conformable time fractional derivative. Calcolo. 2016;53:475-485.
[Google Scholar]

[45] Ezz-Eldien S.S., Doha E.H., Baleanu D., Bhrawy A.H., . A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. J. Vib. Control. 2017;23(1):16-30.
[Google Scholar]

[46] Frederico G.S.F., Torres D.F.M., . Fractional Noether’s theorem in the Riesz-Caputo sense. Appl. Math. Comput.. 2010;217:1023-1033.
[Google Scholar]

[47] Ghomanjani F., . A numerical technique for solving fractional optimal control problems and fractional Riccati differential equations. J. Egypt. Math. Soc.. 2016;24:638-643.
[Google Scholar]

[48] Gökdoğan A., Ünal E., Celik E., . Existence and uniqueness theorems for sequential linear conformable fractional differential equations. Miskolc Math. Notes. 2016;17(1):267-279.
[Google Scholar]

[49] He J.H., Li Z.B., . A fractional model for dye removal. J. King Saud Univ. – Sci.. 2016;28:14-16.
[Google Scholar]

[50] Heydari M.H., Hooshmandasl M.R., Maalek Ghaini F.M., Cattani C., . Wavelets method for solving fractional optimal control problems. Appl. Math. Comput.. 2016;286:139-154.
[Google Scholar]

[51] Heymans N., . Fractional calculus description of non-linear viscoelastic behavior of polymers. Nonlinear Dyn.. 2004;38:221-231.
[Google Scholar]

[52] Jarad F., Abdeljawad T., Baleanu D., . On Riesz-Caputo formulation for sequential fractional variational principles. Abstr. Appl. Anal. 2012:1-15.
[Google Scholar]

[53] Javad Sabouri K., Effati S., Pakdaman M., . A neural network approach for solving a class of fractional optimal control problems. Neural Process Lett.. 2017;45:59-74.
[Google Scholar]

[54] Jenson V.G., Jeffreys G.V., . Mathematical Methods in Chemical Engineering. New York: Academic Press; 1977.

[55] Katsikadelis J.T., . Generalized fractional derivatives and their applications to mechanical systems. Arch. Appl. Mech.. 2015;85:1307-1320.
[Google Scholar]

[56] Katugampola, U.N., 2014. A new fractional derivative with classical properties. J. Amer. Math. Soc. 1-8. arXiv: 1410.6535.

[57] Keshavarz E., Ordokhani Y., Razzaghi M., . A numerical solution for fractional optimal control problems via Bernoulli polynomials. J. Vib. Control. 2016;22(18):3889-3903.
[Google Scholar]

[58] Khalil R., Horani M.A., Yousef A., Sababheh M., . A new definition of fractional derivative. J. Comput. Appl. Math.. 2014;264:65-70.
[Google Scholar]

[59] Khalil R., Abu-Shaab H., . Solution of some conformable fractional differential equations. Int. J. Pure Appl. Math.. 2015;103(4):667-673.
[Google Scholar]

[60] Kolwankar K.M., Gangal A.D., . Fractional differentiability of nowhere differentiable functions and dimensions. Chaos: An Interdiscip. J. Nonlinear Sci.. 1996;6(4):505-513.
[Google Scholar]

[61] Kurt A., Çenesiz Y., Tasbozan O., . On the solution of Burgers’ equation with the new fractional derivative. Open Phys.. 2015;13:355-360.
[Google Scholar]

[62] Lazo M.J., Torres D.F.M., . Variational calculus with conformable fractional derivatives. IEEE/CAA J. Autom. Sin.. 2017;4(2):340-352.
[Google Scholar]

[63] Lewis F.L., Syrmos V.L., . Optimal Control. New York: Wiley; 1995.

[64] Lotfi A., Dehghan M., Yousefi S.A., . A numerical technique for solving fractional optimal control problems. Comput. Math. Appl.. 2011;62:1055-1067.
[Google Scholar]

[65] Lotfi A., Yousefi S.A., Dehghan M., . Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J. Comput. Appl. Math.. 2013;250:143-160.
[Google Scholar]

[66] Luo D., Wang J.R., Feckan M., . Applying fractional calculus to analyze economic growth modeling. J. Appl. Math., Stat. Inform.. 2018;14(1):25-36.
[Google Scholar]

[67] Mandelbrot B., . The Fractal Geometry of Nature. San Francisco: Freeman; 1982.

[68] Naidu D.S., . Optimal Control Systems. Bocaraton, Florida: CRC Press; 2003.

[69] Nemati A., Yousefi S.A., . A numerical method for solving fractional optimal control problems using Ritz method. ASME J. Comput. Nonlinear Dyn.. 2016;11:0510151-0510157.
[Google Scholar]

[70] Odzijewicz T., Malinowska A.B., Torres D.F.M., . Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal.. 2012;75:1507-1515.
[Google Scholar]

[71] Oldham K.B., Spanier J., . The Fractional Calculus. New York: Academic Press; 1974.

[72] Phaochoo P., Luadsong A., Aschariyaphotha N., . The meshless local Petrov-Galerkin based on moving kriging interpolation for solving fractional Black-Scholes model. J. King Saud Univ. – Sci.. 2016;28:111-117.
[Google Scholar]

[73] Podlubny I., . Fractional Differential Equations. New York: Academic Press; 1999.

[74] Rangaig N.A., Convicto V.C., . On fractional modelling of dye removal using fractional derivative with non-singular kernel. J. King Saud Univ. – Sci. 2018
[CrossRef] [Google Scholar]

[75] Salahshour S., Ahmadian A., Ismail F., Baleanu D., Senu N., . A new fractional derivative for differential equation of fractional order under interval uncertainty. Adv. Mech. Eng.. 2015;7(12):1-11.
[Google Scholar]

[76] Suribabu, G., Chiranjeevi, T., 2016. Implementation of fractional order PID controller for an AVR system using GA and ACO optimization techniques. In: Proceedings of the IFAC-ACODS Conference, NIT Trichy, India, Feb 1–5, pp. 456–461.

[77] Tarasov V.E., . Discrete model of dislocations in fractional nonlocal elasticity. J. King Saud Univ. – Sci.. 2016;28:33-36.
[Google Scholar]

[78] Tricaud C., Chen Y.Q., . An approximate method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl.. 2010;59:1644-1655.
[Google Scholar]

[79] Ünal E., Gökdoğan A., Celik E., . Solutions of sequential conformable fractional differential equations around an ordinary point and conformable fractional hermite differential equation. Br. J. Appl. Sci. Technol.. 2015;10(2):1-11.
[Google Scholar]

[80] Ünal E., Gökdoğan A., . Solution of conformable fractional ordinary differential equations via differential transformation method. Optik. 2017;128:264-273.
[Google Scholar]

[81] Ünal E., Gökdoğan A., Cumhur İ., . The operator method for local fractional linear differential equations. Optik. 2017;131:986-993.
[Google Scholar]

[82] Westeriund S., Ekstam L., . Capacitor theory. IEEE Trans. Dielec. El. In.. 1994;1(5):826-839.
[Google Scholar]

[83] Xiaojun Y., Baleanu D., . Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci.. 2013;17(2):625-628.
[Google Scholar]

[84] Yang, X.J., Baleanu, D., Srivastava, H.M., 2016. Local Fractional Integral Transforms and Their Applications. Academic Press, San Diego (CA).

[85] Zhao D., Luo M., . General conformable fractional derivative and its physical interpretation. Calcolo. 2017;54(3):903-917.
[Google Scholar]

Closed-form solution of optimal control problem of a fractional order system

Abstract

Keywords

Conformable fractional derivative

Conformable fractional integral

Optimal control

Calculus of variations

1 Introduction

2 Mathematical preliminaries

3 Formulation of FOCP

4 Analytical solution scheme for conformable FOCPs

5 Illustrative examples

6 Conclusions

References

Appendix

Sufficient condition

Suggested read for related articles: