Solving Fractional Variational Problem Via an Orthonormal Function

In the present paper, a direct numerical technique for solving fractional optimal control problems based on an orthonormal wavelet, is introduced. First we approximate the involved functions by Sine-Cosine wavelet basis; then, an operational matrix is used to transform the given problem into a linear system of algebraic equations, which is easier. In fact operational matrix of Riemann-Liouville fractional integration and derivative of Sine-Cosine wavelet are employed to achieve a linear algebraic equation. The mentioned matrices are derived via hat functions. The solution of transformed system, gives us the solution of original problem. Two numerical examples are also given. Finally, the paper is ended with conclusion.


Introduction
In recent years, fractional calculus is one of the interesting issues that attract many scientists.Many realistic models of engineering and physical phenomena can be uttered with fractional calculus.For example they can be applied in nonlinear oscillations of earthquakes [20], fluid-dynamic traffic [21], frequency dependent damping behavior of various viscoelastic materials [4], solid mechanics [35], economics [5], signal processing [32], and control theory [10].Niels Henrik Abel, in 1823, was probably the first to give an application of fractional calculus.Abel applied the fractional calculus in the solution of an integral equation which arises in the formulation of the problem of finding the shape of a frictionless wire lying in a vertical plane such that the time of a bead placed on the wire slides to the lowest point of the wire in the same time regardless of where the bead is placed [6].
A fractional optimal control problem (FOCP) is an optimal control problem in which the performance index or the differential equations in the dynamics of the system or both contain at least one fractional order derivative 448 SOLVING FRACTIONAL VARIATIONAL PROBLEM term [39].Tricaud and Chen have solved a large class of FOCPs (linear, nonlinear, time-invariant, time-variant, SISO, MIMO, state or input constrained, etc.) by converting them into a general and rational form of optimal control problem [40].In addition to the above methods, orthogonal function method is also applicable to solve the fractional order systems and as a result, FOCPs.
Approximation by orthogonal families of basis functions is widely used in science and engineering.The main idea of applying an orthogonal basis is reduction of considered problem into a system of algebraic equations, by truncating series of orthogonal basis functions for the solution of the problem and applying operational matrix of integration and differentiation to eliminate the integral and derivative operations whenever needed, thus greatly simplifying the problem.These matrices can be uniquely determined based on the particular orthogonal functions.
The orthogonal functions are classified into three main cathegories [38], the first one is sets of piecewise constant orthogonal functions such as the Walsh functions and block pulse functions.The second one is orthogonal polynomials such as the Laguerre, Legendre and Chebyshev functions [26], and the last one is sine-cosine functions.On one hand, approximating a continuous function with piecewise constant basis functions results in a piecewise constant approximation, on the other hand, if we approximate a discontinuous function with continuous basis the resulting approximation is continuous which is not proper for modelling the discontinuities.So, neither continuous basis functions nor piecewise constant basis functions, can efficiently model both continuity and discontinuity of phenomena at the same time.In the case that the function under approximation is not analytic, wavelet functions will be more effective.
The operational matrix of fractional integrals has been derived for many types of orthogonal polynomials such as Legendre polynomials [2,15], Jacobi polynomials [8], Laguerre polynomials [7] and so on.In this paper, two new operational matrices are introduced and a direct method based on Sine-Cosine wavelet with their fractional integration and differentiation operational matrix is proposed to solve a FOCP and a variational problem.The main idea is to reduce the problem under consideration into a system of algebraic equations.To this end, we expand the fractional derivative of the state and control variables using the Sine-Cosine wavelet with unknown coefficients.There are many numerical methods to solve the transformed problem.The proposed method can also be applied to systems with time varying coefficients by using operational matrix of production.This matrix could easily obtained by using the sin and cos multiplication properties.
The paper is organized as follows.In next section we will give the preliminaries of fractional calculus, then in section 3 express a brief review of Hat function and the related fractional operational matrix.In section 4, we describe Sine-Cosine wavelets and its application in function approximation.In section 5, operational matrices of fractional integration and differentiation for considered wavelet is given.In section 6, the proposed method is described for solving the underlying FOCP.The proposed method is applied for solving numerical examples, in section 7. Finally, the paper is ended with conclusion.

Preliminaries of fractional calculus
Fractional order calculus deals with the non-integer order differentiation and integration.Now, we give necessary definitions of the fractional calculus theory.The most commonly used definitions for fractional integral and derivative are the Riemann-Liouville and Caputo definitions.The Riemann-Liouville fractional integration operator of a function f of order α ≥ 0 is defined in [34] as as two properties of Riemann-Liouville fractional integration we have Stat., Optim.Inf.Comput.Vol. 7, June 2019 The fractional derivative operator of order α > 0 in the Caputo sense is defined in [34] as: Two useful relation between the Riemann-Liouville and Caputo operators is as follow

Review of Hat functions and the related fractional operational matrix
In this section, we introduce Hat functions (HFs) and its operational matrix of fractional integration.

Definition of HFs
A m− set of HFs basis functions is defined by Tripathi et al. [41] as follows: where h = 1 m − 1 . An arbitrary function like f (t) ∈ L 2 [0, 1] can be expanded by HFs as: where An important property of HFs in approximating the function f (t) is that the coefficients f i in the above equation are stated by:

Operational matrix of fractional integration for HFs
Operational matrix of fractional integration of order α for HFs, which given in [41] is as: where and

Description of Sine-Cosine wavelets and its application in function approximation
Wavelets have been very successful in approximate solution of different types of systems.They constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet.When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets [19].
if we restrict the parameters a and b to discrete values a = a −k 0 , b = nb 0 a −k 0 , where a 0 > 1, b 0 > 0, n and k are positive integers, we have the following family of discrete wavelets: , which are a wavelet basis for L 2 (R).Sine-Cosine wavelets are defined as follows [23] with 1) can be approximated as: where c n,m = ⟨ f (t), ψ n,m ⟩ and ⟨ ., .⟩ denotes the inner product as: Ψ(t) represent the vector of considered wavelet.C and Ψ(t) are 2 k (2l + 1) × 1 matrices which are given by: In this section, we derive the operational matrix of fractional derivative for the considered wavelet using the operational matrix of fractional integration for HFs.

Express Ψ(t) in terms of HFs
Each ψ n,m (t) as a function, can be expanded in terms of HFs function, thus for Ψ(t) we will have In the above equation m = 2 k (2l + 1) and Φ m× m obtain as follow.We choose h = 1 m − 1 , then by considering the property which is given in Eq. ( 6) we have therefore The vector C n,m represent a row of matrix Φ m× m.

Operational matrix of fractional integration and derivative for Sine-Cosine wavelet
Suppose Ψ(t) be the vector defined in (9), then, fractional integration of order α > 0 in the Riemann-Liouville sense of this vector can be expressed as where P α is the operational matrix of fractional integration.By considering Eq. ( 11), P α calculated as follows Using Eqs. ( 13) and ( 14) we obtain Now we calculate operational matrix of derivative using P α , suppose that x(t) ≃ X T Ψ(t) then we have for α ∈ ( 0, 1] we have n = 1 thus Stat., Optim.Inf.Comput.Vol. 7, June 2019 where D is the operational matrix of derivative for Ψ(t) which defined as D = diag(W, W, • • • , W ), which is a 2 k (2l + 1) × 2 k (2l + 1) matrix and W is of size (2l + 1) × (2l + 1) 6. Solution of fractional optimal control problem by Sine-Cosine operational matrix Consider the fractional optimal control problem with quadratic performance index where A and B are constant matrices with the appropriate dimensions, also in cost functional, S and Q are symmetric positive semi-definite matrices and R is a symmetric positive definite matrix.

Conclusion
In this paper, we use a direct numerical method for fractional problems based on two new operational matrices of fractional integration and differentiation.The procedure of constructing these matrices is summarized.This matrices are utilized along with tau method in order to reduce the fractional differential equations into the algebraic equations which can be efficiently solved.The proposed approach is computationally simple.Two examples are given to show the efficiency of method.The result obtained in this paper is more acceptable in comparison with [24], where the fractional operational matrices of the Sine-Cosine wavelet are obtained using block-pulse functions.
The obtained matrices can also be used to solve fractional optimal control with delay or multi-delay.