An Efficient Algorithm for the Multi-Scale Solution of Nonlinear Fractional Optimal Control Problems

Abstract

An efficient algorithm based on the wavelet collocation method is introduced in order to solve nonlinear fractional optimal control problems (FOCPs) with inequality constraints. By using the interpolation properties of Hermite cubic spline functions, we construct an operational matrix of the Caputo fractional derivative for the first time. Using this matrix, we reduce the nonlinear fractional optimal control problem to a nonlinear programming problem that can be solved with some suitable optimization algorithms. Illustrative examples are examined to demonstrate the important features of the new method.

1. Introduction

A multi-dimensional FOCP with Caputo fractional derivative is defined by

$$\mathrm{Min}\mathbb{J}(\xi ,f\left(\xi \right),u\left(\xi \right))={\int }_0^{1}G(\xi ,f\left(\xi \right),u\left(\xi \right))d\xi ,$$

(1)

subject to

$$\begin{array}{ccc}\hfill & & u\left(\xi \right)=H(\xi ,f\left(\xi \right),{{}_0^{C}D}_{\xi }^{\alpha }f\left(\xi \right)),\alpha \ge 0,n-1<\alpha \le n,0\le \xi \le 1,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & R_j(\xi ,f\left(\xi \right),{{}_0^{C}D}_{\xi }^{\alpha }f\left(\xi \right),u\left(\xi \right))\le 0,j=1,2,\dots ,v,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & f^{\left(i\right)}\left(0\right)=d_i,i=0,1,\dots ,n-1.\hfill \end{array}$$

(4)

where $\mathbb{J}$ is the performance index,

$$u\left(\xi \right)={[u_1\left(\xi \right),u_2\left(\xi \right),\dots ,u_q\left(\xi \right)]}^{\mathsf{T}}$$

and

$$f\left(\xi \right)={[f_1\left(\xi \right),f_2\left(\xi \right),\dots ,f_p\left(\xi \right)]}^{\mathsf{T}}$$

are control and state vector functions, respectively, ${{}_0^{C}D}_{\xi }^{\alpha }f$ denotes the Caputo fractional derivative of a function f, $f^{\left(i\right)}$ denotes the i-th derivative of f, and H, G, and $R_j$ are generally nonlinear and smooth functions.

Finding the optimal solutions of complex systems in engineering, science, operations research, economics, and finance is an important aspect that promotes and encourages research in the optimal control field [1]. It has been an important part of experimental flow science throughout the last century [2]. FOCPs have received special attention from scientists due to their appearance in many branches of science [3,4,5]. Finding exact solutions for these types of problems is difficult or almost impossible. Thus, the search for efficient numerical simulations of these problems has become a major topic for mathematicians. The purpose of numerically solving FOCPs is to find control and state functions so that the performance index can be minimized. In recent years, various algorithms and methods have been employed to obtain approximate solutions to FOCPs.

Alavi et al. [6] used an operational matrix of Mott polynomials to solve fractional optimal control problems. The authors of [7] used the Legendre orthonormal polynomial to solve fractional optimal control problems in the Caputo sense. A spectral method using shifted Chebyshev polynomials was used in [8] to solve fractional optimal control with the Caputo–Katugampola derivative. Ghanbari et al. [9] used generalized fractional-order Chebyshev wavelets to solve fractional optimal control problems. A numerical approach based on Jacobi polynomials and the Ritz method was presented in [10] in order to solve nonlinear fractional optimal control problems. Soradi-Zeid [11] solved a class of fractional optimal control problems via Legendre wavelets. Taherpour et al. [12] used the Bernoulli polynomial method to solve fractional optimal control problems. We refer interested readers to [13,14,15,16,17,18,19,20] in order to study previous research for the numerical solutions of optimal control problems.

Most of above-mentioned references are based on polynomial expansions. Walter Gautschi [21] showed that the condition number of such a conversion grows as ${(1+\sqrt{2})}^N$, where N is the degree of the polynomial. This is satisfactory for the very small $N<10$ employed in the authors’ numerical examples, but is not strongly recommended for larger N. The main advantage of our method is that the order of the polynomials is fixed and never grows; by increasing the level of the space J, we can get more exact results.

Another advantage of our method is the Hermitian interpolation property of cubic Hermite spline functions. This means that one can expand a function using these bases such that not only are the values of the approximated function in the node points equal to the exact function, but the values of the first derivatives of it are also equal to the exact function at the node points. So, one can expect that these functions can solve such a problem effectively.

Recently, due to the fact that wavelet bases have very interesting properties, such as fast algorithms and sparse multi-scale representation, their use in fractional optimization problems has attracted the attention of mathematicians. The authors of [18,22,23,24,25,26,27,28,29,30,31,32,33] extended spectral techniques based on wavelet bases in order to solve FOCPs.

In this article, a different approach to the approximate fractional optimization problem (1)–(4) is introduced. Using biorthogonal Hermite cubic spline multiwavelets (BHCSMs), a collocation method is presented such that it can be implemented efficiently. The operational matrix of the Caputo fractional derivative (OMCFD) for the BHCSMs introduced in this work is our tool for achieving this goal. By using this matrix, the solution of the FOCPs is reduced to the solution of algebraic equations. Some examples are presented, and the results are compared to those of some existing methods in the literature to show and demonstrate the benefits of computing with the presented method. Organization of the paper: In Section 2, some basic definitions of the fractional calculus used for this study are briefly presented. In Section 3, we introduce the BHCSMs and construct the operational matrix of the Caputo fractional derivative for these bases. In Section 4, we give a numerical method based on the BHCSMS for the solution of Equations (1)–(4). An error bound for our expansion and method is given in Section 5. Three examples are examined to show the accuracy of our method. Finally, our conclusions are given.

2. Mathematical Basis of Fractional Calculus

This section of the article is devoted to a brief introduction of the main concepts in the field of fractional calculus that will be used in the rest of the paper.

Definition 1.

The Riemann–Liouville fractional integral of order $\alpha \ge 0$ of a function f is defined as [5]

$${{}_{0}I}_{\xi }^{\alpha }f\left(\xi \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\xi }{(\xi -z)}^{\alpha -1}f\left(z\right)dz,\hfill & \alpha >0,\hfill \\ f\left(\xi \right),\hfill & \alpha =0.\hfill \end{array}\right.$$

Definition 2.

The Caputo fractional derivative of order $\alpha >0$ of a function f is defined as

$${{}_0^{C}D}_{\xi }^{\alpha }f\left(\xi \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_0^{\xi }{(\xi -z)}^{n-\alpha -1}{\displaystyle \frac{d^n}{d{z}^n}}f\left(z\right)dz,\hfill & n-1<\alpha <n,\hfill \\ f^{\left(n\right)}\left(\xi \right),\hfill & \alpha =n.\hfill \end{array}\right.$$

Some properties of the Caputo fractional derivative and Riemann–Liouville fractional integration are described as follows [34]:

$${{}_0^{C}D}_{\xi }^{\alpha }{{}_{0}I}^{\alpha }f\left(\xi \right)=f\left(\xi \right),\alpha >0,f\left(\xi \right)\in C[0,1],$$

(5)

$${{}_{0}I}_{\xi }^{\alpha }f\left(\xi \right)=f\left(\xi \right)-\sum _{i=0}^{\lceil \alpha \rceil -1}\frac{{\xi }^i}{i!}\left(\frac{d^{i}f}{d{\xi }^i}\right)\left(0\right),n-1<\alpha \le n,f\left(\xi \right)\in C^{\lceil \alpha \rceil }[0,1].$$

(6)

$${{}_0^{C}D}_{\xi }^{\alpha }\left({\xi }^i\right)=\left\{\begin{array}{cc}0,\hfill & \alpha \in {\mathbb{N}}_0,i<\alpha ,\hfill \\ \frac{\Gamma (i+1)}{\Gamma (i+1-\alpha )}{\xi }^{i-\alpha },\hfill & otherwise.\hfill \end{array}\right.$$

(7)

3. Multi-Scale Representation

Wavelets have taken an important place in computational mathematics, and especially in the numerical solutions of mathematical equations, such as ODEs and PDEs. As the reader is aware, one of the effective ways of constructing wavelets is the use of multiresolution analysis (MRA). MRA is a family of nested spaces that satisfy certain conditions [35],

$$\left\{0\right\}\subset \dots \subset {\mathcal{V}}_{-1}\subset {\mathcal{V}}_0\subset {\mathcal{V}}_1\subset \dots \subset L^2(\Omega ),$$

where $\Omega$ is a bounded interval or is equal to $\mathbb{R}$. Here, we briefly introduce the BHCSMs.

Given $J\in {\mathbb{Z}}^{+}\cup \left\{0\right\}$, let the subspace ${\mathcal{V}}_J$ be spanned by a set of bases as follows:

$${\mathcal{V}}_J:=\{\sqrt{2}{\varphi }_{J,0}^1{|}_{[0,1]},\sqrt{2}{\varphi }_{J,2^J}^1{|}_{[0,1]}\}\bigcup \left\{{\varphi }_{J,b}^k\right|b\in B,k=1,2\},$$

(8)

where $B:=\{1,\dots ,2^J-1\}$ and ${\varphi }_{J,b}^k:={\varphi }^k(2^J.-b)$, and the scaling functions ${\varphi }^1$ and ${\varphi }^2$ for the multiwavelet system are defined as

$${\varphi }^1\left(\xi \right)=\left\{\begin{array}{cc}1-3{\xi }^2-2{\xi }^3,\hfill & \xi \in [-1,0],\hfill \\ 1-3{\xi }^2+2{\xi }^3,\hfill & \xi \in [0,1],\hfill \\ 0,\hfill & o.w,\hfill \end{array}\right.$$

(9)

and

$${\varphi }^2\left(\xi \right)=\left\{\begin{array}{cc}\xi +2{\xi }^2+{\xi }^3,\hfill & \xi \in [-1,0],\hfill \\ \xi -2{\xi }^2+{\xi }^3,\hfill & \xi \in [0,1],\hfill \\ 0,\hfill & o.w.\hfill \end{array}\right.$$

(10)

We note that ${\varphi }^1,{\varphi }^2\in C^1\left(\mathbb{R}\right)$, and the Hermite interpolation is fulfilled as follows:

$${\varphi }^1\left(\kappa \right)={\delta }_{0,\kappa },{\left({\varphi }^1\right)}^{\prime }\left(\kappa \right)=0,{\varphi }^2\left(\kappa \right)=0,{\left({\varphi }^2\right)}^{\prime }\left(\kappa \right)={\delta }_{0,\kappa },\forall \kappa \in \mathbb{Z},$$

(11)

where ${\delta }_{j,k}$ is the Kronecker delta.

Suppose that $\varphi =({\varphi }^1,{\varphi }^2)$ is a vector of the scaling functions, so it satisfies the matrix form of refinement equations as:

$$\varphi \left(\xi \right)=\sum _{\kappa \in \mathbb{Z}}{H}_{\kappa }\varphi (2\xi -\kappa ),$$

(12)

where the coefficient matrices are given as

$$H_{-1}=\left(\begin{array}{cc}1/2& 3/4\\ -1/8& -1/8\end{array}\right),H_0=\left(\begin{array}{cc}1& 0\\ 0& 1/2\end{array}\right),H_1=\left(\begin{array}{cc}1/2& -3/4\\ 1/8& -1/8\end{array}\right),$$

(13)

and $H_{\kappa }=0,\forall \kappa \notin \{-1,0,1\}$. In addition, it can be shown that the vector of scaling functions $\varphi$ satisfies the symmetry property as follows:

$$\varphi \left(\xi \right)=S\varphi (-\xi ),$$

(14)

where

$$S=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

This means that the scaling function ${\varphi }^1$ is symmetric and ${\varphi }^2$ is antisymmetric. Using (12) and (14), we obtain

$$H_{\kappa }=S{H}_{-\kappa }S,\kappa \in \mathbb{Z}.$$

(15)

Based on the the biorthogonality of our multiwavelets, there exists a vector of new scaling function $\tilde{\varphi }=({\tilde{\varphi }}^1,{\tilde{\varphi }}^2)$, which is called the dual multi-generator and is orthogonal to the vector function $\varphi$ in the following sense:

$$\langle \varphi ,\tilde{\varphi }(.-\kappa )\rangle ={\delta }_{0,\kappa }{I}_2,\kappa \in \mathbb{Z},$$

(16)

where $I_2$ is the identity matrix of size 2. The new vector function $\tilde{\varphi }$ produces another space ${\tilde{\mathcal{V}}}_J\subset L^2\left(\mathbb{R}\right)$ that is orthogonal to ${\mathcal{V}}_J$ in the sense defined in (16). By using the two-scale relations for the primal and dual scaling functions in relation (16), we get the discrete duality relation

$$\sum _{\ell \in \mathbb{Z}}{H}_{\ell }{\tilde{H}}_{\ell +2\kappa }^{\mathsf{T}}=2{\delta }_{0,\kappa }{I}_2,\kappa \in \mathbb{Z}.$$

(17)

The entries of matrices $H_{\ell },\ell \in \mathbb{Z}$ are given as [36]:

$$\begin{gathered} {\tilde{H}}_{-2}=\left(\begin{array}{cc}-\frac{7}{64}& -\frac{5}{64}\\ \frac{87}{128}& \frac{31}{64}\end{array}\right),{\tilde{H}}_{-1}=\left(\begin{array}{cc}\frac{1}{2}& \frac{3}{16}\\ -\frac{99}{32}& -\frac{37}{32}\end{array}\right),{\tilde{H}}_0=\left(\begin{array}{cc}\frac{39}{32}& 0\\ 0& \frac{15}{8}\end{array}\right), \\ {\tilde{H}}_1=\left(\begin{array}{cc}\frac{1}{2}& -\frac{3}{16}\\ \frac{99}{32}& -\frac{37}{32}\end{array}\right),{\tilde{H}}_2=\left(\begin{array}{cc}-\frac{7}{64}& \frac{5}{64}\\ -\frac{87}{128}& \frac{31}{64}\end{array}\right), \end{gathered}$$

and ${\tilde{H}}_{\kappa }=0$ for $\kappa \notin \{-2,\dots ,2\}$.

For simplicity, we rearrange the basis functions of the subspace ${\mathcal{V}}_J$ as

$$\phi =\{{\phi }_1,{\phi }_2,\dots ,{\phi }_{2^{J+1}}\},$$

whose members are defined as

$${\phi }_{2\kappa +(\ell -1)}:={\varphi }_{J,\kappa }^{\ell },for\ell =1,2,\kappa \in B,$$

and ${\phi }_1:=\sqrt{2}{\varphi }_{J,0}^1{|}_{[0,1]}$, ${\phi }_{2^{J+1}}:=\sqrt{2}{\varphi }_{J,2^J}^1{|}_{[0,1]}$. Now, we can project a function $f\in L^2[0,1]$ in the subspace ${\mathcal{V}}_J$ as

$$f\left(\xi \right)\approx P_{J}f\left(\xi \right)=\sum _{\ell =1}^{2^{J+1}}{c}_{\ell }{\phi }_{\ell }\left(\xi \right),$$

(18)

where $P_J$ is the projection operator in the space ${\mathcal{V}}_J$, and the coefficients $c_{\ell },\ell =1,\dots ,2^{J+1}$ are calculated using the Hermitian interpolation property of BHCSMS as

$$\left\{\begin{array}{cc}{c}_{2l}=f\left(\frac{l}{2^J}\right),\hfill & \\ c_{2l+1}=2^{-J}{f}^{\prime }\left(\frac{l}{2^J}\right),\hfill & l=1,\dots ,2^J-1,\hfill \\ c_1:=\frac{1}{\sqrt{2}}f\left(0\right),\hfill & \\ c_{2^{J+1}}:=\frac{1}{\sqrt{2}}f\left(1\right).\hfill \end{array}\right.$$

(19)

Let

$${\Phi }_J\left(\xi \right)={\left[{\phi }_1\left(\xi \right),{\phi }_2\left(\xi \right),\dots ,{\phi }_{2^J+1}\left(\xi \right)\right]}^T,$$

(20)

and

$$C={\left[c_1,c_2,\dots ,c_{2^J+1}\right]}^T,$$

(21)

so the projection relation (18) can be written as

$$f\left(\xi \right)\approx C^T{\Phi }_J\left(\xi \right).$$

(22)

The multiwavelets corresponding to our multi-scaling functions satisfy the following refinement relation [36]:

$$\begin{array}{cc}\hfill & {\psi }^1\left(x\right)={\varphi }^1\left(2x\right)-\frac{1}{2}({\varphi }^1(2x+1)+{\varphi }^1(2x-1))-\frac{23}{12}({\varphi }^2(2x+1)-{\varphi }^2(2x-1)),\hfill \\ \hfill & {\psi }^2\left(x\right)=\frac{37}{22}{\varphi }^2\left(2x\right)+\frac{91}{88}({\varphi }^2(2x+1)+{\varphi }^2(2x-1))+\frac{1}{8}({\varphi }^1(2x+1)-{\varphi }^1(2x-1)).\hfill \end{array}$$

(23)

Note that functions ${\varphi }^1$, ${\varphi }^2$, ${\psi }^1$, and ${\psi }^2$ are supported inside the interval $[-1,1]$. The functions ${\varphi }^1$ and ${\psi }^1$ are symmetric with respect to the origin, and the functions ${\varphi }^2$ and ${\psi }^2$ are antisymmetric with respect to the origin. In addition, the lowest orders of the vanishing moments of both the mother wavelets ${\psi }^1$ and ${\psi }^2$ are 2.

Suppose that the space ${\mathcal{W}}_J$ is generated by the entries of vector ${\Psi }_J$, which is given as

$${\Psi }_J:=\{\sqrt{2}{\psi }_{J,0}^1{|}_{[0,1]},\sqrt{2}{\psi }_{J,2^J}^1{|}_{[0,1]}\}\bigcup \left\{{\psi }_{J,\kappa }^k\right|\kappa \in B,k=1,2\},$$

(24)

where ${\psi }_{J,\kappa }^k:={\psi }^k(2^J.-\kappa )$.

For spaces ${\mathcal{V}}_j$ and ${\mathcal{W}}_j$, we have

$${\mathcal{V}}_{j-1}\subseteq {\mathcal{V}}_{j}and{\mathcal{V}}_j={\mathcal{V}}_{j-1}+{\mathcal{W}}_{j-1}={\mathcal{V}}_{j_0}+{\mathcal{W}}_{j_0}+\cdots +{\mathcal{W}}_{j-1},\forall 0\le j_0<j\in \mathbb{N}.$$

A Riesz wavelet basis for ${\mathcal{V}}_J$ is given by ${\mathcal{E}}_{J_0,J}={\overrightarrow{\Phi }}_{J_0}\cup {\cup }_{j=J_0}^{J-1}{\overrightarrow{\Psi }}_j$, so each function $f\left(\xi \right)$ on $L^2[0,1]$ is approximated with the wavelet basis as

$$f\left(\xi \right)\approx {\mathcal{E}}_{J_0,J}^{\mathsf{T}}\left(\xi \right)\mathcal{D},$$

(25)

Since the calculation of the wavelet coefficients $\mathcal{D}$ in the above expansion is complicated, especially for higher levels of J, to overcome this difficulty, we define the wavelet transformation matrix T as

$$\begin{array}{c}\hfill {\mathcal{E}}_{J_0,J}\left(\xi \right)=T{\Phi }_J\left(\xi \right),\end{array}$$

(26)

where matrix T is given as

$$T=\left[\begin{array}{c}{\mathcal{A}}_{J_0+1}\times {\mathcal{A}}_{J_0+2}\times ...\times {\mathcal{A}}_J\\ ----------\\ {\mathcal{B}}_{J_0+1}\times {\mathcal{A}}_{J_0+2}\times ...\times {\mathcal{A}}_J\\ ----------\\ ⋮\\ {\mathcal{B}}_{J-2}\times {\mathcal{A}}_{J-1}\times {\mathcal{A}}_J\\ ----------\\ {\mathcal{B}}_{J-1}\times {\mathcal{A}}_J\\ ----------\\ {\mathcal{B}}_J\end{array}\right].$$

(27)

and the matrices ${\mathcal{A}}_j,{\mathcal{B}}_j,j=J_0+1,\dots ,J$ can be found by using the two-scale relations (12) and (23), respectively [37].

The Operational Matrix of the Caputo Fractional Derivative

The Caputo fractional derivative of the vector given in relation (20) can be approximated as

$${{}_0^{C}D}_t^{\alpha }{\overrightarrow{\Phi }}_J\left(\xi \right)\approx D_{\alpha }{\overrightarrow{\Phi }}_J\left(\xi \right),$$

(28)

where $D_{\alpha }$ is the Caputo fractional operational matrix of the derivative for Hermite cubic splines.

Let $\alpha >0$. Then, by using the definition of the Caputo fractional derivative of order $\alpha >0$, we have

$$\begin{array}{ccc}& & {{}_0^{C}D}_t^{\alpha }{\varphi }^1(2^{J}t-k)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_0^t{(t-x)}^{n-\alpha -1}\frac{d^n}{d{t}^n}{\varphi }^1(2^{J}x-k)dx,\hfill \\ & & {{}_0^{C}D}_t^{\alpha }{\varphi }^2(2^{J}t-k)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_0^t{(t-x)}^{n-\alpha -1}\frac{d^n}{d{t}^n}{\varphi }^2(2^{J}x-k)dx.\hfill \end{array}$$

Using the definition of ${\varphi }^1\left(x\right)$ and ${\varphi }^2\left(x\right)$, we get

$${\varphi }^1(2^{J}x-k)=\left\{\begin{array}{cc}1-3{(2^{J}x-k)}^2-2{(2^{J}x-k)}^3\hfill & \frac{k-1}{2^J}\le x\le \frac{k}{2^J},\\ \\ 1-3{(2^{J}x-k)}^2+2{(2^{J}x-k)}^3\hfill & \frac{k}{2^J}\le x\le \frac{k+1}{2^J}\\ \\ 0\hfill & \mathrm{o}.\mathrm{w}\end{array}\right.k=0,\dots ,2^J$$

$${\varphi }^2(2^{J}x-k)=\left\{\begin{array}{cc}(2^{J}x-k)+2{(2^{J}x-k)}^2+{(2^{J}x-k)}^3\hfill & \frac{k-1}{2^J}\le x\le \frac{k}{2^J}\\ \\ (2^{J}x-k)-2{(2^{J}x-k)}^2+{(2^{J}x-k)}^3\hfill & \frac{k}{2^J}\le x\le \frac{k+1}{2^J}\\ \\ 0\hfill & \mathrm{o}.\mathrm{w}\end{array}\right.k=1,\dots ,2^J-1$$

If $t\le \frac{k-1}{2^J}$, then the above equations imply that ${{}_0^{C}D}^{\alpha }{\varphi }^1(2^{J}t-k)=0$ and ${{}_0^{C}D}_t^{\alpha }{\varphi }^2(2^{J}t-k)=0$.

If $t\in (\frac{k-1}{2^J},\frac{k}{2^J})$, then we have

$$\begin{array}{cc}{a}_1(t,k)\hfill & :={{}_0^{C}D}_t^{\alpha }{\varphi }^1(2^{J}t-k)\hfill \\ & =\frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k-1}{2^J}}^t{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}\left(1-3{(2^{J}x-k)}^2-2{(2^{J}x-k)}^3\right)dx,\hfill \\ \\ a_2(t,k)\hfill & :={{}_0^{C}D}_t^{\alpha }{\varphi }^2(2^{J}t-k)\hfill \\ & =\frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k-1}{2^J}}^t{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}((2^{J}x-k)+2{(2^{J}x-k)}^2+{(2^{J}x-k)}^3)dx.\hfill \end{array}$$

If $t\in (\frac{k}{2^J},\frac{k+1}{2^J})$, then

$$\begin{array}{cc}{b}_1(t,k):=\hfill & {{}_0^{C}D}_t^{\alpha }{\varphi }^1(2^{J}t-k)\hfill \\ & =\frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k-1}{2^J}}^{\frac{k}{2^J}}{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}(1-3{(2^{J}x-k)}^2-2{(2^{J}x-k)}^3)dx\hfill \\ & +\frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k}{2^J}}^t{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}(1-3{(2^{J}x-k)}^2+2{(2^{J}x-k)}^3)dx,\hfill \\ \\ \\ b_2(t,k)\hfill & :={{}_0^{C}D}_t^{\alpha }{\varphi }^2(2^{J}t-k)=\hfill \\ & \frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k-1}{2^J}}^{\frac{k}{2^J}}{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}((2^{J}x-k)+2{(2^{J}x-k)}^2+{(2^{J}x-k)}^3)dx\hfill \\ \hfill & +\frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k}{2^J}}^t{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}((2^{J}x-k)-2{(2^{J}x-k)}^2+{(2^{J}x-k)}^3)dx.\hfill \end{array}$$

If $t\ge \frac{k+1}{2^J}$, then

$$\begin{array}{cc}{c}_1(t,k)\hfill & :={{}_0^{C}D}_t^{\alpha }{\varphi }^1(2^{J}t-k)\hfill \\ \hfill & =\frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k-1}{2^J}}^{\frac{k}{2^J}}{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}(1-3{(2^{J}x-k)}^2-2{(2^{J}x-k)}^3)dx\hfill \\ \hfill & +\frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k}{2^J}}^{\frac{k+1}{2^J}}{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}(1-3{(2^{J}x-k)}^2+2{(2^{J}x-k)}^3)dx,\hfill \\ \\ c_2(t,k)\hfill & :={{}_0^{C}D}_t^{\alpha }{\varphi }^2(2^{J}t-k)\hfill \\ & =\frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k-1}{2^J}}^{\frac{k}{2^J}}{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}((2^{J}x-k)+2{(2^{J}x-k)}^2+{(2^{J}x-k)}^3)dx\hfill \\ \hfill & +\frac{1}{\Gamma (n-\alpha )}{\int }_{\frac{k}{2^J}}^{\frac{k+1}{2^J}}{(t-x)}^{n-\alpha -1}\frac{d^n}{d{x}^n}((2^{J}x-k)-2{(2^{J}x-k)}^2+{(2^{J}x-k)}^3)dx\hfill \end{array}$$

The above integrals can be evaluated explicitly in terms of $\alpha$, J, and k for all values of $k=0,\dots ,2^J$ for a given J. We used the library function $\mathit{int}$ that is available in Maple to evaluate the above integrals analytically. So, the Caputo fractional derivatives ${{}_0^{C}D}_t^{\alpha }{\varphi }^1(2^{J}t-k)$ and ${{}_0^{C}D}_t^{\alpha }{\varphi }^2(2^{J}t-k)$ are obtained as follows:

$$\begin{array}{c}\hfill {\varpi }_1(t,k):={{}_0^{C}D}_t^{\alpha }{\varphi }^1(2^{J}t-k)=\left\{\begin{array}{cc}0,\hfill & t\le \frac{k-1}{2^J},\hfill \\ a_1(t,k),\hfill & \frac{k-1}{2^J}\le t<\frac{k}{2^J},\hfill \\ b_1(t,k),\hfill & \frac{k}{2^J}\le t<\frac{k+1}{2^J},\hfill \\ c_1(t,k),\hfill & t\ge \frac{k+1}{2^J},\hfill \end{array}\right.\end{array}$$

(29)

and

$$\begin{array}{c}\hfill \varpi 2(t,k):={{}_0^{C}D}_t^{\alpha }{\varphi }^2(2^{J}t-k)=\left\{\begin{array}{cc}0,\hfill & t\le \frac{k-1}{2^J},\hfill \\ a_2(t,k),\hfill & \frac{k-1}{2^J}\le t<\frac{k}{2^J},\hfill \\ b_2(t,k),\hfill & \frac{k}{2^J}\le t<\frac{k+1}{2^J},\hfill \\ c_2(t,k),\hfill & t\ge \frac{k+1}{2^J}.\hfill \end{array}\right.\end{array}$$

(30)

Employing Equations (29) and (30), we have

$${{}_0^{C}D}_t^{\alpha }{\Phi }_J\left(t\right)=\left[\begin{array}{c}{\varpi }_1(t,0)\\ {\varpi }_1(t,1)\\ {\varpi }_2(t,1)\\ ⋮\\ \\ {\varpi }_1(t,2^J-1)\\ {\varpi }_2(t,2^J-1)\\ {\varpi }_1(t,2^J)\end{array}\right]\approx D_{\alpha }{\Phi }_J\left(t\right)$$

(31)

Now, by expanding the entries of the vector function ${{}_0^{C}D}_t^{\alpha }{\Phi }_J\left(t\right)$ with the multiscaling functions, we can get the operational matrix of the Caputo fractional derivative.

4. Multiwavelet Collocation Method for FOCPs with Inequality Constraints

Let us now develop our method to discretize the FOCPs with inequality constraints. To this end, let

$$\begin{array}{ccc}& & f\left(\xi \right)={[f_1\left(\xi \right),f_2\left(\xi \right),\dots ,f_l\left(\xi \right)]}^{\mathsf{T}},u\left(\xi \right)={[u_1\left(\xi \right),u_2\left(\xi \right),\dots ,u_q\left(\xi \right)]}^{\mathsf{T}},\hfill \end{array}$$

(32)

$$\begin{array}{ccc}& & {\widehat{\Phi }}_J\left(\xi \right)=I_l\otimes {\Phi }_J\left(\xi \right),{\widehat{\Phi }}_{J,q}\left(\xi \right)=I_q\otimes {\Phi }_J\left(\xi \right),\hfill \end{array}$$

(33)

where we used Kronecker product [38] ⊗, identity matrices $I_l\in {\mathbb{R}}^{l\times l}$ and $I_q\in {\mathbb{R}}^{q\times q}$, and matrices ${\widehat{\Phi }}_J\left(\xi \right)$ and ${\widehat{\Phi }}_{J,q}\left(\xi \right)$ with dimension $Nl\times l$ and $Nq\times q$, respectively. Applying Equation (22),each of $f_i\left(\xi \right),i=1,\cdots ,l$ and each of $u_j\left(\xi \right),j=1,\dots ,q$ can be approximated in terms of Hermite cubic splines scaling functions as

$$f_i\left(\xi \right)\approx {\Phi }_J^{\mathsf{T}}\left(\xi \right)F_i,i=1,\dots ,l$$

(34)

$$u_j\left(\xi \right)\approx {\Phi }_J^{\mathsf{T}}\left(\xi \right)U_j,j=1,\dots ,q,$$

(35)

Applying the operational matrix of the fractional Caputo derivative $D_{\alpha }$ and Equation (33), each of the fractional state rates ${{}_0^{C}D}_{\xi }^{\alpha }{f}_i\left(\xi \right),i=1,\dots ,l$ can be approximated as

$${{}_0^{C}D}_{\xi }^{\alpha }{f}_i\left(\xi \right)\approx {\widehat{\Phi }}_J^{\mathsf{T}}\left(\xi \right)D_{\alpha }^{\mathsf{T}}F.$$

(36)

Now, by considering Equations (32) and (33) and the approximations given in (34)–(36), we get

$$\begin{array}{ccc}& & f\left(\xi \right)\approx {\widehat{\Phi }}_J^{\mathsf{T}}\left(\xi \right)F,\hfill \end{array}$$

(37)

$$\begin{array}{ccc}& & {{}_0^{C}D}_{\xi }^{\alpha }f\left(\xi \right)\approx {\widehat{\Phi }}_J^{\mathsf{T}}\left(\xi \right){\widehat{D}}_{\alpha }^{\mathsf{T}}F,\hfill \end{array}$$

(38)

$$\begin{array}{ccc}& & u\left(\xi \right)\approx {\widehat{\Phi }}_{J,q}^{\mathsf{T}}\left(\xi \right)U,\hfill \end{array}$$

(39)

where F and U are vectors of order $Ml\times 1$ and $Mq\times 1$, respectively, $M=2(2^J+1)$, ${\widehat{D}}_{\alpha }=D_l\otimes D_{\alpha }$, and ${\overline{F}}_i$, $i=0,\dots ,\lceil \alpha \rceil -1$ are known matrices.

In order to discretize the performance index $\mathbb{J}$ in Equation (1), we approximate $G(\xi ,f(\xi ),u(\xi \left)\right)$ by usingbiorthogonal wavelets as

$$G(\xi ,{\widehat{\Phi }}_J^{\mathsf{T}}\left(\xi \right)F,{\widehat{\Phi }}_{J,q}^{\mathsf{T}}\left(t\right)U)={\Phi }_J^{\mathsf{T}}\left(\xi \right)V,$$

(40)

where

$$V={[{\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_N]}^{\mathsf{T}},$$

and the elements ${\vartheta }_i$ are obtained as in Equation (19). We substitute the approximation given in (40) into (1) and get

$$\begin{array}{c}\hfill \mathbb{J}={\int }_0^1{\overrightarrow{\Phi }}_J^{\mathsf{T}}\left(\xi \right)Vd\xi =PV,\end{array}$$

(41)

where $P={\int }_0^1{\overrightarrow{\Phi }}_J^{\mathsf{T}}\left(\xi \right)d\xi$. Using Equation (39), the dynamic constraint (2) may be written in the form

$$\begin{array}{cc}\hfill {\widehat{\Phi }}_{J,q}^{\mathsf{T}}\left(\xi \right)U& =H(\xi ,{\widehat{\Phi }}_J^{\mathsf{T}}\left(\xi \right)F,{\widehat{\Phi }}_J^{\mathsf{T}}\left(\xi \right){\widehat{D}}_{\alpha }^{\mathsf{T}}F).\hfill \end{array}$$

(42)

We substitute the approximations given in Equations (39) and (38) into (3) and obtain

$$R_j(\xi ,{\widehat{\Phi }}_J^{\mathsf{T}}\left(\xi \right){\widehat{D}}_{\alpha }^{\mathsf{T}}F,{\widehat{\Phi }}_J^{\mathsf{T}}\left(\xi \right)F,{\widehat{\Phi }}_{J,q}^{\mathsf{T}}\left(\xi \right)U)\le 0,j=1,\dots ,v.$$

(43)

It should be noted that we do not need to change the inequality constraints by adding slack variables, which increases the number of decision variables. Actually, in nonlinear programming problems, handling inequalities is generally easier than handling equalities [15]. So, we collocate Equation (43) at the shifted Chebyshev–Radau points ${\xi }_i$ as

$${\xi }_i=\frac{1}{2}cos\left(\frac{(i-1)\pi }{M}\right)+\frac{1}{2},i=1,2,\dots ,M.$$

Therefore, using (41)–(43), the minimization problem (1), together with the conditions (2)–(4), is reduced to a nonlinear programming problem (NLP):

$$\begin{array}{ccc}& & Min\mathbb{J}=PV,\hfill \\ & & s.t{\widehat{\Phi }}_{J,q}^{\mathsf{T}}\left(\xi \right)U=H({\xi }_i,{\widehat{\Phi }}_J^{\mathsf{T}}\left({\xi }_i\right)F,{\widehat{\Phi }}_J^{\mathsf{T}}\left({\xi }_i\right){\widehat{D}}_{\alpha }^{\mathsf{T}}F)\hfill \\ & & R_j({\xi }_i,{\widehat{\Phi }}_J^{\mathsf{T}}\left({\xi }_i\right){\widehat{D}}_{\alpha }^{\mathsf{T}}F,{\widehat{\Phi }}_J^{\mathsf{T}}\left({\xi }_i\right)F,{\widehat{\Phi }}_{J,q}^{\mathsf{T}}\left({\xi }_i\right)U)\le 0,j=1,\dots ,v.\hfill \end{array}$$

(44)

Many well-developed NLP techniques can be used to solve this minimization problem (see, for example, [39,40,41]). After finding the unknown vectors U and F, an approximate expansion of the state and control functions based on Hermite cubic spline multiwavelets is gained by using (37) and (39), respectively. Moreover, we get an approximate value for the optimal value of the performance index by substituting the results in (41).

Remark 1.

To solve the problem using Hermite cubic spline multiwavelets, it is necessary to replace vector ${\Phi }_J\left(\xi \right)$ with vector $T^{-1}{\mathcal{E}}_{J_0,J}\left(\xi \right)$ in all obtained approximations.

5. Estimation of Error

It follows from Theorem 2 in [42], the expansion error for relation (18) is bounded as follows.

Theorem 1.

Suppose that the function $f:[0,1]\to \mathbb{R}$ is continuously differentiable four times; then, we have

$$\begin{array}{c}\hfill e_J\left(\xi \right):=|f\left(\xi \right)-P_{J}f\left(\xi \right)|=O\left(2^{-J}\right).\end{array}$$

Proof. 

See [42]. □

As in Theorem (1), for the control function $u\left(\xi \right)$, we can conclude that

$$\begin{array}{c}\hfill |u\left(\xi \right)-P_{J}u\left(\xi \right)|=O\left(2^{-J}\right).\end{array}$$

To obtain an estimation for the error of the performance index, we provide the following theorem.

Theorem 2.

Let $G:{\mathbb{R}}^3\to \mathbb{R}$ be sufficiently continuously differentiable; further let there be a constant Ω such that

$$|G(\xi ,f_1,u)-G(\xi ,f_2,u)|\le \Omega |f_1-f_2|$$

(45)

Suppose that $u^{*}\left(\xi \right)$ and $f^{*}\left(\xi \right)$ are the optimal control and state functions, respectively, which minimize the performance index $\mathbb{J}$ with the optimal value ${\mathbb{J}}^{*}$. If $\widehat{u}\left(\xi \right),\widehat{f}\left(\xi \right)$, and $\widehat{\mathbb{J}}$ are the approximations of the optimal control function, state function, and the performance index obtained with the proposed technique, one thus obtains

$$\begin{array}{c}\hfill |{\mathbb{J}}^{*}-\widehat{\mathbb{J}}|=O\left(2^{-J}\right).\end{array}$$

Proof. 

Suppose that one has

$${\mathbb{J}}^{*}={\int }_0^{1}G(\xi ,f^{*}\left(\xi \right),u^{*}\left(\xi \right))d\xi ,$$

(46)

and

$$\widehat{\mathbb{J}}=PV.$$

(47)

By subtracting (47) from (46) and employing Theorem 1 and Equation (45), we have

$$\begin{array}{cc}\hfill |{\mathbb{J}}^{*}-\widehat{\mathbb{J}}|& =|{\int }_0^{1}G(\xi ,f^{*}\left(\xi \right),u^{*}\left(\xi \right))d\xi -PV|=\hfill \\ \hfill & =|{\int }_0^{1}G(\xi ,f^{*}\left(\xi \right),u^{*}\left(\xi \right))d\xi -{\int }_0^{1}G(\xi ,\widehat{f}\left(\xi \right),u^{*}\left(\xi \right))d\xi \hfill \\ \hfill & +{\int }_0^{1}G(\xi ,\widehat{f}\left(\xi \right),u^{*}\left(\xi \right))d\xi -PV|\hfill \\ \hfill & \le |{\int }_0^{1}G(\xi ,f^{*}\left(\xi \right),u^{*}\left(\xi \right))d\xi -{\int }_0^{1}G(\xi ,\widehat{f}\left(\xi \right),u^{*}\left(\xi \right))d\xi |\hfill \\ \hfill & +|{\int }_0^{1}G(\xi ,\widehat{f}\left(\xi \right),u^{*}\left(\xi \right))d\xi -PV|\hfill \\ \hfill & \le {\int }_0^1\Omega \left|\right(f^{*}\left(\xi \right)-\widehat{f}\left(\xi \right)|d\xi +O\left(2^J\right)\hfill \\ \hfill & \le O\left(2^J\right)+O\left(2^J\right)\hfill \\ \hfill & \le O\left(2^J\right).\hfill \end{array}$$

(48)

This completes the proof. □

6. Numerical Experiments

Here, we discuss the numerical solution of some examples to show that our method supports our theoretical results.

Example 1.

First, we examine the following time-invariant problem:

$$Min\mathbb{J}(f,u)=\frac{1}{2}{\int }_0^1(f^2\left(\xi \right)+u^2\left(\xi \right))d\xi ,$$

subject to

$${}_0^C{D}^{\alpha }f\left(\xi \right)=-f\left(\xi \right)+u\left(\xi \right),f\left(0\right)=1.$$

The purpose of solving this example is to find the control variable $u\left(\xi \right)$ that can find the minimum value of the quadratic performance index J. For $\alpha =1$, the exact solution of this problem is given in [42] as:

$$\begin{array}{ccc}& & f\left(\xi \right)=Ch\left(\sqrt{2}\xi \right)+\beta Sh\left(\sqrt{2}\xi \right),\hfill \\ & & u\left(\xi \right)=(1+\sqrt{2}\beta )Ch\left(\sqrt{2}\xi \right)+(\sqrt{2}+\beta )Sh\left(\sqrt{2}\xi \right),\hfill \\ & & \beta =-\frac{Ch\left(\sqrt{2}\right)+\sqrt{2}Sh\left(\sqrt{2}\right)}{\sqrt{2}Ch\left(\sqrt{2}\right)+Sh\left(\sqrt{2}\right)}\simeq -0.98.\hfill \end{array}$$

The solution of this problem is obtained by employing our method at different levels of J. In Figure 1, the approximate solutions of the problem for various values of $\alpha$ with $J=6$ are demonstrated. In Figure 1, it can be seen that as $\alpha$ tends to 1, the approximate solutions for $f\left(\xi \right)$ and $u\left(\xi \right)$ approach the exact solutions of the problem in the case of $\alpha =1$.

In

Table 1. Absolute errors of $x\left(t\right)$ in comparison to the results from [25,32,42].

t	Legendre Basis [25]	Bernoulli Basis [32]	Multiwavelet Method [42]	Present	Method
	$\mathrm{M}=5$	$\mathrm{M}=5$	$J=8$	J = 4	J = 5
0	$6.25\times {10}^{-6}$	$6.25\times {10}^{-6}$	$0.0$	0.0	0.0
0.1	$1.34\times {10}^{-5}$	$2.39\times {10}^{-6}$	$1.26\times {10}^{-6}$	$6.81\times {10}^{-7}$	$3.98\times {10}^{-7}$
0.2	$2.12\times {10}^{-5}$	$1.21\times {10}^{-6}$	$1.03\times {10}^{-6}$	$1.01\times {10}^{-6}$	$1.82\times {10}^{-7}$
0.3	$3.24\times {10}^{-5}$	$1.72\times {10}^{-6}$	$8.35\times {10}^{-7}$	$9.50\times {10}^{-7}$	$1.52\times {10}^{-7}$
0.4	$4.73\times {10}^{-5}$	$6.82\times {10}^{-7}$	$6.68\times {10}^{-7}$	$4.08\times {10}^{-7}$	$2.70\times {10}^{-7}$
0.5	$6.20\times {10}^{-5}$	$1.93\times {10}^{-6}$	$5.23\times {10}^{-7}$	$1.90\times {10}^{-8}$	$7.12\times {10}^{-9}$
0.6	$7.49\times {10}^{-5}$	$3.11\times {10}^{-7}$	$3.99\times {10}^{-7}$	$2.41\times {10}^{-7}$	$1.78\times {10}^{-7}$
0.7	$8.88\times {10}^{-5}$	$1.90\times {10}^{-6}$	$2.90\times {10}^{-7}$	$3.72\times {10}^{-7}$	$7.27\times {10}^{-8}$
0.8	$1.07\times {10}^{-5}$	$9.17\times {10}^{-7}$	$1.94\times {10}^{-7}$	$2.53\times {10}^{-7}$	$3.90\times {10}^{-8}$
0.9	$1.31\times {10}^{-5}$	$2.49\times {10}^{-6}$	$1.09\times {10}^{-7}$	$7.90\times {10}^{-8}$	$4.63\times {10}^{-8}$

, we show the distances between the results obtained by our method for the state variable $x\left(t\right)$ and its exact value at some points when $\alpha =1$, and we compare them with the results obtained in [25,32,42] for J = 4 and 5.

Table 2. Estimated values of $\mathbb{J}$ for Example 1.

Method	$\mathbb{J}$
Method [42]
J = 7	0.192909
Method [27]
M = 8	0.192909
Present method
J = 2	0.19290931206
J = 3	0.19290929658
J = 4	0.19290929819
J = 5	0.19290929807
Exact solution	0.19290929809

reports the minimum value of $\mathbb{J}$ for different values J when $\alpha =1$.

Example 2.

Consider the two-dimensional minimization problem described by [27] as

$$Min\mathbb{J}(\xi ,f\left(\xi \right),u\left(\xi \right))={\int }_0^1{(f_1\left(\xi \right)-1-{\xi }^{\frac{3}{2}})}^2+{(f_2\left(\xi \right)-{\xi }^{\frac{5}{2}})}^2+{(u\left(\xi \right)-\frac{3\sqrt{\pi }}{4}\xi +{\xi }^{\frac{5}{2}})}^{2}d\xi$$

subject to

$$\begin{array}{ccc}& & {}_0^C{D}^{0.5}{f}_1\left(\xi \right)=f_2\left(\xi \right)+u\left(\xi \right),\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & {}_0^C{D}^{0.5}{f}_2\left(\xi \right)=f_1\left(\xi \right)+\frac{15\sqrt{\pi }}{16}{\xi }^2-{\xi }^{\frac{3}{2}}-1,\hfill \end{array}$$

(50)

with the initial condition

$$f_1\left(0\right)=1,f_2\left(0\right)=0.$$

The exact optimal solutions for this problem are given as

$$f_1\left(\xi \right)=1+{\xi }^{\frac{3}{2}},f_2\left(\xi \right)={\xi }^{\frac{5}{2}},u\left(\xi \right)=\frac{3\sqrt{\pi }}{4}\xi -{\xi }^{\frac{5}{2}}.$$

and the minimum value of $\mathbb{J}$ is 0.

Here, we approximate the conditions (49) and (50) as

$$\begin{array}{c}\hfill {\Phi }_J^{\mathsf{T}}\left(\xi \right)D_{0.5}^{\mathsf{T}}{F}_1={\Phi }_J^{\mathsf{T}}\left(\xi \right)F_2+{\Phi }_J^{\mathsf{T}}\left(\xi \right)U,\\ \hfill {\Phi }_J^{\mathsf{T}}\left(\xi \right)D_{0.5}^{\mathsf{T}}{F}_1={\Phi }_J^{\mathsf{T}}\left(\xi \right)F_2+{\Phi }_J^{\mathsf{T}}\left(\xi \right)U.\end{array}$$

So, we get

$$\begin{array}{c}\hfill D_{0.5}^{\mathsf{T}}{F}_1=F_2+U,\end{array}$$

(51)

$$\begin{array}{c}\hfill D_{0.5}^{\mathsf{T}}{F}_1=F_2+U.\end{array}$$

(52)

For this problem, because of the linearity of the constraints, we replace relation (42) with (51) and (52). We apply our method in various levels of J, and we get the approximation results reported in

Table 3. The computational results for Example 2.

Methods	$\mathbb{J}$
Epsilon–Ritz Method [27]
$k=5,ϵ=0.001$,	$0.000053424$
$k=8,ϵ=0.0001$,	$8.0027\times {10}^{-6}$
Biorthogonal multiwavelets [37]
$J=4$	$9.49\times {10}^{-7}$
$J=5$	$6.30\times {10}^{-8}$
$J=6$	$4.15\times {10}^{-9}$
Present method
$J=1$	$2.60\times {10}^{-5}$
$J=2$	$3.35\times {10}^{-6}$
$J=3$	$3.37\times {10}^{-7}$
$J=4$	$2.99\times {10}^{-8}$
$J=5$	$2.43\times {10}^{-9}$
$J=6$	$1.86\times {10}^{-10}$

. In this table, we provide a comparison between the optimal value of $\mathbb{J}$ obtained with this new method for $J=1,2,\dots ,6$ and the values of $\mathbb{J}$ reported in [27,37]. In addition, the plots of the approximate and exact state and control functions are shown in Figure 2.

Example 3.

Now, let us find the optimal solutions of a nonlinear FOCP in which

$$Min\mathbb{J}={\int }_0^1-ln2f\left(\xi \right)d\xi ,$$

subject to

$${}_0^C{D}^{\alpha }f\left(\xi \right)=ln\left(2\right)(f\left(\xi \right)+u\left(\xi \right)),$$

and the inequality constraints

$$\left|u\right(\xi \left)\right|\le 1,u\left(\xi \right)+f\left(\xi \right)\le 2,$$

with the initial condition

$$f\left(0\right)=0.$$

Here, we apply the method proposed in Section 4 in order to solve this nonlinear FOCP. We evaluated the approximate solutions for the state variable with $\alpha =0.8,0.9,1$ and $J=1$, and these are presented in Figure 3. In this figure, one can find that, as $\alpha$ tends to 1, the numerical state variable solutions tend to the analytical solution. In the particular case, the exact performance index for $\alpha =1$ is $\mathbb{J}=-0.30682$.

Table 4. Computational results for Example 3.

Methods	$\mathbb{J}$
Bernstein polynomials [43]
$M_2=3$	$-0.30685$
Hybrid of block-pulse and Bernoulli polynomials [28]
$M=3,N=1$	$-0.30683$
Fractional order of Chebyshev polynomials [29]	$-0.30685380$
$M=3,N=1$
Biorthogonal multiwavelets [37]
$J=5$	$-0.30683$
$J=6$	$-0.30682$
Present Method
$J=3$	$-0.30683$
$J=4$	$-0.30682$
Exact	$-0.30682$

shows the numerical results obtained with the proposed method, the results obtained with the methods presented in [28,29,37,43], and the exact solution.

For future work, we will use our idea to solve FDEs, try to find operational matrix for other definitions of fractional derivatives, and solve similar problems.

7. Concluding Remarks

In this research, we proposed a multiwavelet collocation method for solving nonlinear FOCPs with inequality constraints. To this end, the operational matrix of the Caputo fractional derivative is first constructed for Hermite cubic biorthogonal multiwavelets; such a matrix is introduced for the first time in this work. This matrix allows us to reduce a nonlinear FOCP to an algebraic system of nonlinear equations, which greatly simplifies the problem. We considered three numerical examples in order to test the proposed method. The efficiency and accuracy of the introduced method in comparison with those of analytical solutions and some other numerical methods were determined.

In future work, we will apply our method in order to solve fractional differential equations and fractional optimal control problems with other definitions of fractional derivatives.