Abstract
The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.
1. Introduction
The idea of fractional derivative dates back to a conversation between two mathematicians: Leibniz and L’Hopital. In 1695, they exchanged about the meaning of a derivative of order 1/2. Their correspondence has been well documented and is stated as the foundation of fractional calculus [1].
A fractional optimal control problem (FOCPs) is an optimal control problem focused on the performance index or the fractional differential equations governing the dynamics of the system or both contain at least one fractional order derivative term.
The formulation and solution of state and variables FOCPs were first established by Agrawal, where the applied fractional variational calculus (FVC) presented a general formulation and solution scheme for FOCPs in the Riemann-Liouville (RL) sense; it was based on variational virtual work coupled with the Lagrange multiplier technique. Since the Caputo fractional derivatives (CFDs) seems more natural and allows incorporating the usual initial conditions, it becomes a popular choice for researchers [2, 3].
The Chebyshev polynomials are used to solve composition FOCPs and Chebyshev polynomials of degree are important in approximation theory, since the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation.
Agrawal and Baleanu [4] obtained necessary conditions for FOCPs with Riemann-Liouville derivative and then were able to solve the problem numerically.
Elbarbary introduced Chebyshev finite difference approximation for the boundary value problems of integer derivatives [5]. In [6], Khader and Hendy studied an efficient numerical scheme for solving fractional optimal control problems. In [7], Akbarian and Keyanpour studied a new approach to the numerical solution of fractional order optimal control Problems.
This paper presents a new approach for computing the approximate optimal control function for optimal control problems with some types of fractional order differential equation such as multi-fractional order differential equation, and composite fractional order differential equations are investigated with first kind Chebyshev approximate polynomial which works very efficiently with fractional orders and optimal control problems. All the discussion and details in addition to the transformations of the steps are given to compute all the useful of the new work. The obtained solutions of the method show that the technique of the approach is very convenient and efficient, and many calculations give high accuracy and may lead to near to exact solution by changing up the order of fractional positive integer numbers.
The multi-fractional differential equations corresponding to optimal control problems and basic theorems have been given with algorithm for multi-composite fractional order optimal control problems. Also, we give illustrative examples for the solution of the approximate systems.
The value of is observed in detail to explain the activity of solution approximation with mixed boundary conditions. It is important to notice that the calculations are written by using the mathematical software MATHCAD by version 14.0.3.332.
This paper consists of the following sections: in Section 2, some basic definitions and properties of fractional order calculus are introduced (R-L and Caputo fractional derivatives). In Section 3, the shifted Chebyshev polynomials and numerical approximations of CFD and RLFD using Chebyshev polynomials are introduced. In Section 4, we derive the necessary optimality conditions of composition order fractional optimal control problems. In Section 5, we give numerical examples to solve composition FOCPs and show the accuracy of the presented method. Finally, conclusions are presented in Section 6.
2. Preliminaries
In this section, we give some definitions and properties of fractional derivatives that will be needed later on.
2.1. Fractional Order Calculus
Definition 1 (see [8]). The left (LRLFD) and right (RRLFD) Riemann-Liouville fractional derivatives of a function are defined aswhere the order of the derivative satisfies , and is the gamma function.
Definition 2 (see [8]). The left (LCFD) and right (RCFD) Caputo fractional derivatives of a function are defined as where .
2.2. Some Properties of the Fractional Calculus Are Presented in Detail Which Will Be Needed Later on
(i) The fractional operators are linear [9]:where is one of or and , are real numbers.
(ii) Let is a positive integer, , and if , ) [9], then
(iii) If , we obtain the following [2]:
Moreover, if is a function such that , we have simpler formulas:
(iv) The relation between the RLFD and the CFD is as follows [2].
For and , the Riemann-Liouville and Caputo fractional derivatives are related by the following formulas:
(v) The constant function and power function of Caputo’s derivative are as follows:
In particular, , and, also from right integral, . where denotes the smallest integer greater than or equal to and . Recall that for , [10].
3. The Shifted Chebyshev Polynomials
The well-known Chebyshev polynomials are defined on the interval and can be determined by the following recurrence formula [11]:The Chebyshev polynomials can be expanded in power series as follows: , and denotes the biggest integral less than or equal to
The Chebyshev polynomials are orthogonal under integration over with the weighting function , with orthogonal condition:where .
In order to use these polynomials on the interval we define the so-called shifted pseudo-spectral Chebyshev polynomials by introducing the change of variable
The shifted Chebyshev polynomials are defined as follows [12]:The analytic form of the shifted pseudo-spectral Chebyshev polynomial of degree n is given by the following [10]:where , and
The orthogonality relation is as follows:with the weight function for
The function can be expressed in terms of shifted Chebyshev polynomials as follows:where the coefficients are given by
3.1. The Chebyshev-Gauss-Lobatto Points
We choose the Chebyshev-Gauss-Lobatto points associated with the interval , as follows:These grids can be written as
Clenshaw and Curtis [13] introduced an approximation of the function , as follows:where (′′) on the summation means that the first and last terms are to be taken with a factor
Theorem 3 (see [2]). The fractional derivative of order in the Caputo sense for the function at the point is given bySuch that where
Proof. The fractional derivative of the approximate formula for the function in (22) is given byUsing (12) and (13), in (18) we havethen Therefore, for and by (12) and (13), in for formula (18), we getNow, can be expressed approximately in terms of shifted pseudo-spectral Chebyshev polynomial series, so we havewhere is obtained from (21) with . If only the first -terms from pseudo-spectral shifted Chebyshev polynomials in (23) are considered, the approximate formula for the fractional derivative of the shifted pseudo-spectral Chebyshev polynomials is as follows:From (29) and (32), we haveFrom (33), the fractional derivative of order for the function at the point leads to the desired result.
Theorem 4 (see [2]). Let be the approximation of the fractional derivative of the function as given by (24). Then it holdswhere
3.2. Approximation of the Right Riemann-Liouville Fractional Derivative
Using the right Caputo fractional derivative (RCFD) of a function is defined in (4) when and ,From relation between the RLFD and the CFD (11), when , we haveUse (37) in (38) to obtainLet be a sufficiently smooth function in and let be defined as follows: Substituting (40) in (39), we deduce Now, we approximate , by a sum of shifted Chebyshev polynomials according to where
Lemma 5 (see [7]). Let be the polynomial of degree as given by (42); then there exists a polynomial of degree such that
Proof. Let be expanded in a Taylor series at Then, The assertion follows, if we choose , with an arbitrary constant From (44), we have Moreover, can be approximated by means of We express in (47) by a sum of Chebyshev polynomials and provide the recurrence relation satisfied by the Chebyshev coefficients. Differentiating both sides of (44) with respect to yields To evaluate in (47), we expand in terms of the shifted Chebyshev polynomials: where (′) on the summation means that the first term is to be taken with a factor where On the other hand, we have By using the relation and from (51), it follows that Such that
LetInserting and as given (52) and (54) into (50) and taking (55) into account, we get The Chebyshev coefficients of as given by (55) can be evaluated by integrating and comparing it with (42):with starting values , where are the Chebyshev coefficients of
4. The Necessary Optimality Conditions of Composition Order Fractional Optimal Control Problems
In this problem, the constraint has multiorder and composition for the three fractional derivatives , and , where and The multi-order fractional optimal control problems refers to the minimization of an objective functional subject to dynamical constraints on the state and the control which have three fractional derivatives order models.
The necessary optimality conditions of this type are introduced as follows.
Let and be real numbers and and , and let and be two differentiable functions with domain . We consider a general form of multicomposition fractional optimal control problem: which subject to the multi derivative-composite fractional dynamic control system,and the boundary conditions,where , are fixed real numbers.
Theorem 6. If is a minimizer of (58), (59), and (60), then there exists a function for which satisfies the multi-composite fractional optimality conditions:
(i) The amiltonian system is defined as
(ii) The stationary condition is
Proof. We consider the following multi-composite fractional optimal control problem: Thus, Suppose that Using (8) in (65), we getEvaluating the Euler equation for of (66),From (67), we get , as follows:Now, evaluating Euler's equation for of (65):Finally, the Euler's equation for of (65) isFrom (65) it is clear that , then we get From (69), (71), and (73) we have that
5. Illustrative Example
In this section, we consider the following linear–quadratic of multi-composite fractional optimal control problem:which is subject to the multi-composite fractional dynamical system, and the boundary conditions,The exact solution for is given by the following:Now, we develop algorithm for solution (75), (76), and (77). It is based on the necessary optimality conditions from Theorem 6 and implements the following steps.
Step 1. Compute the amiltonian function:Step 2. Derive the necessary optimality conditions from Theorem 6:Step 3. Find the coupled system.
(i) From substitution (82) in (80), we get
(ii) Find control variable from (82) and substitution in (80) to obtain Using property (6) for (83) and (84), we get the coupled system:
Step 4a. Solve (85a) by Chebyshev expansion method, and use (48) to approximate., where and is defined in (47).
Sum both (86) and (87) to get, where and is defined in (47).
Step 4b. Solve (85b) by Chebyshev expansion method, and use (24) to approximate ., where is defined in (25).
Using (89) and (90) in (85a) to get where and are defined in (25).
Step 5. Find the results at the shifted Gauss-Lobatto nodes , from (22) when , and and , in two cases.
Case 1. Let , and .
, and since , then .
Now, substituting the value of and , in (88), we get where , where and are defined in (47).
After calculating (92) at the shifted Gauss-Lobatto nodes and substitution in (88) we get
Now, we calculate (91) and, using the boundary conditions (78), we have Use (25) to evaluate the computation of and , which is shown in Table 1.
when and
The results of Table 1 and boundary conditions from (78) can be substituted in (93); we get
Step 6. Compute approximation solution of and at .To find and from (23), we get Now, we substitute , and , in (98), to getTo compute the approximation solution of the control from (76), we get Using (102), (103), and (100) in (101), we have:
Case 2. Let , and .
, and since , then
Now, we calculate (91) and using the boundary conditions (78), we have To find and from Table 1 and we get , in the same steps of Case 1.Now, compute approximation solution of and at , in Case 2:To find and , from (23), we getWe substitute , and , in (108), to getTo compute the approximation solution of the control , we get Now, using (110) and (112) in (111), we get In Table 2, we give the computation of and when values t = 0…1 with different values of orders , for fractional derivative (); see Figures 1(b) and 2(b) and (); see Figures 1(c) and 2(c) for , which converge to exact values with (); see Figures 1(a) and 2(a).
(a) Exact solution of for ( and
(b) Approximate solutions of with ( and
(c) Approximate solutions of with (
(a) Exact solution of with ( and
(b) Approximate solutions of for ( and
(c) Approximate solutions of with (
6. Conclusions
In this paper, we have presented algorithm for solution a class of multi-composite order fractional optimal control problems, in two cases one when and , and the other one when at ( In both cases, the solution is approximated by Chebyshev series. Numerical results for illustrative example show that the algorithm converges from the exact solution when , and we note that the convergent to the exact solution is dependent on increasing of the fractional order of derivative.
Conflicts of Interest
The authors declare that they have no conflicts of interest.