Fractional logarithmic Chebyshev cardinal functions: Application for Caputo–Hadamard fractional optimal control problems

Abstract

This study investigates a specific type of fractional optimization problems, subject to a dynamical system including the Caputo–Hadamard fractional differentiation. In this way, a novel class of basis functions known as the fractional logarithmic Chebyshev cardinal functions is introduced. An operational matrix regarding the Hadamard fractional integral of these fractional functions is derived and employed to develop an efficient numerical method for the provided problem. The established algorithm converts the solution of the primary fractional optimization problem into the solution of an algebraic system of equations by representing the state and control variables via the introduced fractional functions. Two test problems are considered to confirm the effectiveness of the suggested method. The derived outcomes of solving these examples highlight the satisfactory accuracy and efficiency of the designed method.

Keywords

Caputo–Hadamard fractional derivative fractional optimal control problem fractional logarithmic Chebyshev cardinal functions operational matrix of Hadamard fractional integral

1. Introduction

Control theory is concerned with how a dynamic system can be influenced to achieve a desired goal. Control theory has made significant progress since the pioneering works of Pontryagin and colleagues in the late 1950s, and is now recognized as an important area of applied mathematics. In optimal control, the aim is to minimize or maximize a numerical value of a specified quantity that depends on the behavior of the system. Optimal control has already found its way into diverse fields of science and engineering (Bai et al., 2021; Guo et al., 2023a, 2023b; Luo et al., 2023; Mohammadzadeh et al., 2024).

In recent decades, fractional calculus has become a valuable topic of research because of its diverse applications in different fields of science and engineering, like, computer science, biology, microchemistry, mechanics and physics (Deng et al., 2020; Hilfer, 2000; Ionescu et al., 2017; Li, 2020; Tarasov, 2011). The most celebrated definitions of fractional operators are the Riemann–Liouville derivative and integral, the Grunwald–Letnikov differentiation, the Caputo differentiation, and the Hadamard integral and derivative (Kilbas et al., 2006; Li and Cai, 2019; Podlubny, 1998). Another interesting fractional operator, originally introduced by Hadamard in 1892 (Hadamard, 1892), has been used in a wide range of practical problems in mechanics and engineering (Cai et al., 2022; Garra et al., 2017, 2018). In recent years, several texts have been devoted to describe the properties and applications of Hadamard integral and differential operators. One of the remarkable texts is the book written by Kilbas et al. (2006). In 2012, Jarad et al. (2012) proposed a new derivative by modifying the Hadamard derivative with the Caputo one, known as the Caputo–Hadamard derivative, and studied the properties of such an operator. By changing the order of its differential and integral parts, the Caputo–Hadamard derivative can be obtained from Hadamard derivative. Like other fractional differential equations, expressing solutions of Caputo–Hadamard fractional differential equations in explicit form is challenging. Recently, some numerical techniques have been developed to solve fractional differential equations including Caputo–Hadamard fractional derivative. Some of the techniques outlined are as follows: The predictor-corrector method (Gohar et al., 2020), incomplete Gamma function via Whittaker M function (Toh et al., 2021), logarithmic Jacobi collocation method (Zaky et al., 2022), mapped Jacobi logarithmic orthogonal functions (Zhao et al., 2023), and nonuniform L1-type formula method (Wang and Sun, 2024).

It is difficult to think of an area of science or engineering where fractional calculus has not been applied. One of these areas is optimal control problems. Note that the analytical solution of nonlinear fractional optimal control problems (FOCPs) is quite challenging and almost impossible in all cases. As a result, it is very important to establish numerical methods to deal with this kind of problems. Recently, different numerical strategies have been established to solve FOCPs. Here is a brief overview of some of these methods: Agrawal who was the first to introduce a general formulation and a solution approach for FOCPs (Agrawal, 2004). The author derived the Euler–Lagrange equations for these problems by employing the calculus of variations, the Lagrange multiplier method and the fractional integration by parts formula. Additionally, a numerical method was devised to solve the resulting equations effectively. As earlier research regarding the formulation of problems, derivation of optimality conditions and development of solution methods for FOCPs, we can refer to Baleanu et al. (2009), Pooseh et al. (2014) and Ezz-Eldien et al. (2017). Following these early studies, extensive research has been conducted to establish and implement computational techniques that effectively address these problems. Some of these techniques are the ones based on the Genocchi polynomials (Phang et al., 2018), orthonormal wavelets (Sahu and Saha Ray, 2018), generalized Chebyshev polynomials (Hassani et al., 2019), radial basis functions (Soradi-Zeid, 2020), shifted Chebyshev polynomials (Abdelhakem et al., 2019), orthonormal piecewise Chelyshkov functions (Heydari and Razzaghi, 2022), piecewise Chebyshev cardinal functions (Heydari and Razzaghi, 2021), fractional Boubaker wavelets (Rabiei and Razzaghi, 2023), piecewise fractional Legendre functions (Zhagharian et al., 2023), Mittag–Leffler wavelet functions (Ghasempour et al., 2024), and modified fractional homotopy method (Qing and Pan, 2024).

There are significantly fewer numerical methods for problems with the Hadamard fractional derivative (Fan et al., 2022), Gohar et al. (2020) compared to those developed for Riemann–Liouville and Caputo derivatives (Diethelm et al., 2005; Li and Cai, 2019), particularly in the field of FOCPs. Zguaid et al. (2023) investigated the Hadamard FOCPs and proposed an algorithm to solve the resulting equations from the necessary optimality conditions. The lack of different numerical approaches for the Hadamard FOCPs prompted us to establish a numerical strategy for a category of such problems. A set of studies on the Hadamard-type fractional derivatives have revealed that logarithmic functions can play a crucial role in solving fractional problems involving such derivatives. For instance, see Istafa and Rehman (2023), Zaky et al. (2022) and Zhao et al. (2023). The application of this idea for FOCPs involving the Caputo–Hadamard derivative has the potential to produce favorable outcomes.

In this work, we aim to develop a highly accurate method that uses the fractional logarithmic Chebyshev cardinal functions, as the basis functions, for solving Caputo–Hadamard FOCPs. The major contributions of this study are briefly provided as follows:

• The fractional logarithmic Chebyshev cardinal functions, as a suitable basis for the Hadamard-type fractional problems, are introduced for the first time.

• A new Hadamard fractional integral matrix for the fractional logarithmic functions is derived.

• A numerical method based on the introduced fractional functions is developed for a class of the Caputo–Hadamard FOCPs.

• The accuracy of the proposed method is investigated on some numerical examples.

The structure of the ongoing research is as follows: Some definitions and features regarding the Hadamard-type fractional calculus are reviewed in Section 2. Section 3 is devoted to the problem formulation. The fractional logarithmic Chebyshev cardinal functions are defined in Section 4. The Hadamard fractional integral matrix of these functions is derived in Section 5. Section 6 presents a numerical procedure for the introduced FOCPs. Section 7 provides some numerical examples to display the performance of the suggested scheme. At the end, a conclusion is provided in Section 8.

2. Preliminaries

Beginning with Hadamard’s original work (Hadamard, 1892), numerous studies have concentrated on analyzing fractional operators with logarithmic kernels. To see more details in this regard, see Kilbas (2001) and Kilbas et al. (2006). This section provides a concise overview of the concepts and fundamental features of the Hadamard fractional calculus.

Definition 2.1

(Hadamard, 1892) The Hadamard fractional integral with order ρ > 0 for a given function $f (ξ) \in L^{1} (a, b)$ can be defined in the form of

{}_{a}^{H}I_{ξ}^{ρ} f (ξ) = \frac{1}{Γ (ρ)} \int_{a}^{ξ} {(\ln \frac{ξ}{τ})}^{ρ - 1} f (τ) \frac{d τ}{τ}, ξ \in (a, b) .

(2.1)

Property 1

(Hadamard, 1892) If ρ > 0, β > 0 and a > 0, then we have

{}_{a}^{H}I_{ξ}^{ρ} {(\ln \frac{ξ}{a})}^{β - 1} = \frac{Γ (β)}{Γ (β + ρ)} {(\ln \frac{ξ}{a})}^{β + ρ - 1} .

(2.2)

Definition 2.2

(Kilbas, 2001) Let $[a, b]$ be a finite interval where −∞ < a < b < ∞ and $A C [a, b]$ denotes the absolutely continuous functions space defined on $[a, b]$ . We indicate δ = ξd/dξ and define for any $\hat{n} \in Z^{+}$ the space

{A C}_{δ}^{\hat{n}} [a, b] = \{f : [a, b] \to R| (δ^{\hat{n} - 1} f) \in A C [a, b]\} .

(2.3)

Definition 2.3

(Jarad et al., 2012) The Caputo–Hadamard fractional differentiation with order $\hat{n} - 1 < ρ < \hat{n} \in N$ for a given function $f (ξ) \in {A C}_{δ}^{\hat{n}} [a, b]$ can be defined as

{}_{a}^{C H}D_{ξ}^{ρ} f (ξ) = {}_{a}^{H}I_{ξ}^{\hat{n} - ρ} [δ^{\hat{n}} f (ξ)] = \frac{1}{Γ (\hat{n} - ρ)} \int_{a}^{ξ} {(\ln \frac{ξ}{τ})}^{\hat{n} - ρ - 1} δ^{\hat{n}} f (τ) \frac{d τ}{τ}, ξ \in (a, b) .

(2.4)

Property 2

(Jarad et al., 2012) If $\hat{n} - 1 < ρ < \hat{n} \in N$ , β > 0 and a > 0, then we have

{}_{a}^{C H}D_{ξ}^{ρ} {(\ln \frac{ξ}{a})}^{β - 1} = \frac{Γ (β)}{Γ (β - ρ)} {(\ln \frac{ξ}{a})}^{β - ρ - 1}, β > \hat{n},

(2.5)

and as a direct consequence, we get

{}_{a}^{C H}D_{ξ}^{ρ} {(\ln \frac{ξ}{a})}^{k} = 0, k = 0,1, \dots, \hat{n} - 1 .

(2.6)

Lemma 2.4

(Jarad et al., 2012) Let $f (ξ) \in {A C}_{δ}^{\hat{n}} [a, b]$ and ρ > 0, then

{}_{a}^{H}I_{ξ}^{ρ} [{}_{a}^{C H}D_{ξ}^{ρ} f (ξ)] = f (ξ) - \sum_{k = 0}^{\hat{n} - 1} \frac{δ^{k} f (a)}{k!} {(\ln \frac{ξ}{a})}^{k} .

(2.7)

Definition 2.5

(Kilbas et al., 2006) The generalized hypergeometric series can be defined by the following expression

{}_{p}F_{q} (c_{1}, \dots, c_{p}; d_{1}, \dots, d_{q}; ξ) = \sum_{k = 0}^{\infty} \frac{{(c_{1})}_{k} \dots {(c_{p})}_{k}}{{(d_{1})}_{k} \dots {(d_{q})}_{k}} \frac{ξ^{k}}{k!}, | ξ | < 1,

(2.8)

where

c_{i}, d_{j} \in C

with d_j ≠ 0, − 1, − 2, … for i = 1, 2, …, p and j = 1, 2, …, q, and (c)_k is the Pochhammer symbol given as

{(c)}_{k} = \{\begin{cases} 1, & k = 0, \\ c (c + 1) \dots (c + k - 1), & k \in N . \end{cases}

In the special case of p = 2 and q = 1, this function is referred to the Gaussian hypergeometric function

{}_{2}F_{1} (c, d; e; ξ) = \sum_{k = 0}^{\infty} \frac{{(c)}_{k} {(d)}_{k}}{{(e)}_{k}} \frac{ξ^{k}}{k!}, | ξ | < 1 .

3. Problem statement

In this study, we investigate a class of nonlinear FOCPs defined as

\min \hat{J} (u) = \int_{a}^{b} F (ξ, x (ξ), u (ξ)) d ξ,

(3.1)

subject to the dynamical system

\begin{align} {}_{a}^{C H}D_{ξ}^{ρ} x (ξ) & = P (ξ, x (ξ), u (ξ)), \hat{n} - 1 < ρ \leq \hat{n}, \end{align}

(3.2)

\begin{align} δ^{k} x (a) & = x_{k a}, k = 0,1, \dots, \hat{n} - 1, \end{align}

(3.3)

where

{}_{a}^{C H}D_{ξ}^{ρ}

denotes the Caputo–Hadamard derivative of order ρ. Moreover,

x \in R^{\bar{n}}

and

u \in R^{\bar{s}}

are the state and control variables, respectively. In addition, the scalar function

F : R \times R^{\bar{n}} \times R^{\bar{s}} \to R

and the vector function

P : R \times R^{\bar{n}} \times R^{\bar{s}} \to R^{\bar{n}}

are smooth functions.

4. Fractional logarithmic cardinal functions

In the sequel, we introduce a general strategy to construct the fractional logarithmic Chebyshev cardinal functions. Let a and b are positive real numbers, and $θ_{r} = - \cos ((2 r - 1) π / 2 (n + 1))$ , r = 1, 2, …, n + 1 represent the roots of the Chebyshev polynomial of degree $(n + 1)$ . A set of $(n + 1)$ fractional logarithmic Chebyshev cardinal functions can be defined on $[a, b]$ as

ϕ_{m} (ξ) = \prod_{\begin{array}{c} r = 1 \\ r \neq m \end{array}}^{n + 1} \frac{{(\ln ξ)}^{μ} - {(\ln ξ_{r})}^{μ}}{{(\ln ξ_{m})}^{μ} - {(\ln ξ_{r})}^{μ}}, m = 1,2, \dots, n + 1,

(4.1)

where

ξ_{r} = ((b - a) θ_{r} + a + b) / 2

Remark 4.1

The fractional logarithmic Chebyshev cardinal functions expressed in (4.1) can also be represented in a more appropriate form as

ϕ_{m} (ξ) = \frac{1}{η_{m}} \sum_{r = 0}^{n} ψ_{m r} {(\ln ξ)}^{μ (n - r)}, m = 1,2, \dots, n + 1,

(4.2)

where

η_{m} = \prod_{\begin{array}{c} l = 1 \\ l \neq m \end{array}}^{n + 1} [{(\ln ξ_{m})}^{μ} - {(\ln ξ_{l})}^{μ}], ψ_{m r} = \{\begin{cases} 1, & r = 0, \\ - \frac{1}{r} \sum_{l = 0}^{r} ν_{m l} ψ_{m r - l}, & r \neq 0, \end{cases} ν_{m l} = \sum_{\begin{array}{c} k = 1 \\ k \neq m \end{array}}^{n + 1} {(\ln ξ_{k})}^{μ l} .

For instance, in the case of n = 3 and μ = 3/4, the fractional logarithmic Chebyshev cardinal functions defined on $[1,2]$ are as follows

\begin{array}{l} ϕ_{1} (ξ) & = - 9.8824 {(\ln ξ)}^{\frac{9}{4}} + 17.1417 {(\ln ξ)}^{\frac{3}{2}} - 9.5610 {(\ln ξ)}^{\frac{3}{4}} + 1.6946, \\ ϕ_{2} (ξ) & = 38.4035 {(\ln ξ)}^{\frac{9}{4}} - 55.5316 {(\ln ξ)}^{\frac{3}{2}} + 22.0721 {(\ln ξ)}^{\frac{3}{4}} - 1.4980, \\ ϕ_{3} (ξ) & = - 60.9019 {(\ln ξ)}^{\frac{9}{4}} + 73.2224 {(\ln ξ)}^{\frac{3}{2}} - 22.7022 {(\ln ξ)}^{\frac{3}{4}} + 1.4376, \\ ϕ_{4} (ξ) & = 32.3808 {(\ln ξ)}^{\frac{9}{4}} - 34.8325 {(\ln ξ)}^{\frac{3}{2}} + 10.1911 {(\ln ξ)}^{\frac{3}{4}} - 0.6343 . \end{array}

Using the above basis functions, any function $f (ξ) \in C ([a, b])$ can be expanded as

f (ξ) ≃ \sum_{m = 1}^{n + 1} f (ξ_{m}) ϕ_{m} (ξ) ≜ F^{T} Φ (ξ),

(4.3)

where

F = {[f (ξ_{1}), f (ξ_{2}), \dots, f (ξ_{n + 1})]}^{T}

, and

Φ (ξ) = {[ϕ_{1} (ξ), ϕ_{2} (ξ), \dots, ϕ_{n + 1} (ξ)]}^{T} .

(4.4)

5. Hadamard fractional integral matrix

This section presents a formula for calculating the Hadamard fractional integral matrix of the fractional logarithmic Chebyshev cardinal functions.

Theorem 5.1

For a specific value ρ > 0, the fractional integral in the Hadamard sense of order ρ of the vector $Φ (ξ)$ given in (4.4) can be expressed as

{}_{a}^{H}I_{ξ}^{ρ} Φ (ξ) ≃ Q^{(ρ)} Φ (ξ),

(5.1)

where

Q^{(ρ)} = [q_{m, j}^{(ρ)}]

is an

(n + 1) \times (n + 1)

matrix with entries

\begin{array}{l} q_{m, j}^{(ρ)} & = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \frac{{(\ln ξ_{j})}^{μ (n - r) + ρ}}{(μ (n - r) + 1)} {}_{2}F_{1} (1 - ρ, μ (n - r) + 1; μ (n - r) + 2; 1) \\ - \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \frac{{(\ln ξ_{j})}^{ρ - 1} {(\ln a)}^{μ (n - r) + 1}}{(μ (n - r) + 1)} {}_{2}F_{1} (1 - ρ, μ (n - r) + 1; μ (n - r) + 2; \frac{\ln a}{\ln ξ_{j}}) . \end{array}

Proof

For any 1 ≤ m ≤ n + 1, we get

\begin{matrix} {}_{a}^{H}I_{ξ}^{ρ} ϕ_{m} (ξ) & = \frac{1}{Γ (ρ)} \int_{a}^{ξ} {(\ln \frac{ξ}{s})}^{ρ - 1} ϕ_{m} (s) \frac{d s}{s} = \frac{1}{Γ (ρ)} \int_{a}^{ξ} {(\ln \frac{ξ}{s})}^{ρ - 1} (\frac{1}{η_{m}} \sum_{r = 0}^{n} ψ_{m r} {(\ln s)}^{μ (n - r)}) \frac{d s}{s} \\ = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \int_{a}^{ξ} {(\ln \frac{ξ}{s})}^{ρ - 1} {(\ln s)}^{μ (n - r)} \frac{d s}{s} . \end{matrix}

(5.2)

Considering the relation

{(\ln \frac{ξ}{s})}^{ρ - 1} = \sum_{i = 0}^{\infty} {(- 1)}^{i} (\binom{ρ - 1}{i}) {(\ln s)}^{i} {(\ln ξ)}^{ρ - i - 1} = \sum_{i = 0}^{\infty} \frac{Γ (i - ρ + 1)}{Γ (i + 1) Γ (1 - ρ)} {(\ln s)}^{i} {(\ln ξ)}^{ρ - i - 1},

(5.3)

and substituting (5.3) into (5.2), gives

\begin{array}{l} {}_{a}^{H}I_{ξ}^{ρ} ϕ_{m} (ξ) & = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \int_{a}^{ξ} (\sum_{i = 0}^{\infty} \frac{Γ (i - ρ + 1)}{Γ (i + 1) Γ (1 - ρ)} {(\ln ξ)}^{ρ - i - 1} {(\ln s)}^{i}) {(\ln s)}^{μ (n - r)} \frac{d s}{s} \\ = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \sum_{i = 0}^{\infty} \frac{Γ (i - ρ + 1)}{Γ (i + 1) Γ (1 - ρ)} {(\ln ξ)}^{ρ - i - 1} (\int_{a}^{ξ} {(\ln s)}^{μ (n - r) + i} \frac{d s}{s}) \\ = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \sum_{i = 0}^{\infty} \frac{Γ (i - ρ + 1)}{Γ (i + 1) Γ (1 - ρ)} {(\ln ξ)}^{ρ - i - 1} \frac{{(\ln s)}^{μ (n - r) + i + 1}}{μ (n - r) + i + 1} |_{a}^{ξ} \\ = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \sum_{i = 0}^{\infty} \frac{Γ (i - ρ + 1)}{Γ (i + 1) Γ (1 - ρ)} \frac{Γ (μ (n - r) + i + 1)}{Γ (μ (n - r) + i + 2)} {(\ln ξ)}^{ρ - i - 1} {(\ln s)}^{μ (n - r) + i + 1} |_{a}^{ξ} \end{array}

\begin{array}{l} = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \sum_{i = 0}^{\infty} \frac{Γ (i - ρ + 1)}{Γ (i + 1) Γ (1 - ρ)} \frac{Γ (μ (n - r) + i + 1)}{Γ (μ (n - r) + i + 2)} {(\ln ξ)}^{ρ - 1} {(\ln s)}^{μ (n - r) + 1} \frac{{(\ln s)}^{i}}{{(\ln ξ)}^{i}} |_{a}^{ξ} \\ = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \frac{Γ (μ (n - r) + 2) {(\ln ξ)}^{ρ - 1} {(\ln s)}^{μ (n - r) + 1}}{(μ (n - r) + 1) Γ (μ (n - r) + 1) Γ (1 - ρ)} \sum_{i = 0}^{\infty} \frac{Γ (1 - ρ + i) Γ (μ (n - r) + 1 + i)}{Γ (μ (n - r) + 2 + i)} \frac{{(\frac{\ln s}{\ln ξ})}^{i}}{Γ (i + 1)} |_{a}^{ξ} \\ = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \frac{{(\ln ξ)}^{ρ - 1} {(\ln s)}^{μ (n - r) + 1}}{(μ (n - r) + 1)} {}_{2}F_{1} (1 - ρ, μ (n - r) + 1; μ (n - r) + 2; \frac{\ln s}{\ln ξ}) |_{a}^{ξ} \\ = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \frac{{(\ln ξ)}^{μ (n - r) + ρ}}{(μ (n - r) + 1)} {}_{2}F_{1} (1 - ρ, μ (n - r) + 1; μ (n - r) + 2; 1) \\ - \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \frac{{(\ln ξ)}^{ρ - 1} {(\ln a)}^{μ (n - r) + 1}}{(μ (n - r) + 1)} {}_{2}F_{1} (1 - ρ, μ (n - r) + 1; μ (n - r) + 2; \frac{\ln a}{\ln ξ}) . \end{array}

One can approximate the above result as

{}_{a}^{H}I_{ξ}^{ρ} ϕ_{m} (ξ) ≃ \sum_{j = 1}^{n + 1} q_{m, j}^{(ρ)} ϕ_{j} (ξ),

where

\begin{array}{l} q_{m, j}^{(ρ)} & = {}_{a}^{H}I_{ξ}^{ρ} ϕ_{m} (ξ) |_{ξ = ξ_{j}} = \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \frac{{(\ln ξ_{j})}^{μ (n - r) + ρ}}{(μ (n - r) + 1)} {}_{2}F_{1} (1 - ρ, μ (n - r) + 1; μ (n - r) + 2; 1) \\ - \frac{1}{η_{m} Γ (ρ)} \sum_{r = 0}^{n} ψ_{m r} \frac{{(\ln ξ_{j})}^{ρ - 1} {(\ln a)}^{μ (n - r) + 1}}{(μ (n - r) + 1)} {}_{2}F_{1} (1 - ρ, μ (n - r) + 1; μ (n - r) + 2; \frac{\ln a}{\ln ξ_{j}}) . \end{array}

Thus, we obtain

{}_{a}^{H}I_{ξ}^{ρ} Φ (ξ) = {[{}_{a}^{H}I_{ξ}^{ρ} ϕ_{1} (ξ), {}_{a}^{H}I_{ξ}^{ρ} ϕ_{2} (ξ), \dots, {}_{a}^{H}I_{ξ}^{ρ} ϕ_{n + 1} (ξ)]}^{T} ≃ [\begin{matrix} q_{1,1}^{(ρ)} & q_{1,2}^{(ρ)} & \dots & q_{1, n + 1}^{(ρ)} \\ q_{2,1}^{(ρ)} & q_{2,2}^{(ρ)} & \dots & q_{2, n + 1}^{(ρ)} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ q_{n + 1,1}^{(ρ)} & q_{n + 1,2}^{(ρ)} & \dots & q_{n + 1, n + 1}^{(ρ)} \end{matrix}] Φ (ξ),

that completes the proof.

6. The suggested approach

To solve the FOCP presented in (3.1)–(3.3) using fractional logarithmic Chebyshev cardinal functions, we assume

{}_{a}^{C H}D_{ξ}^{ρ} x_{i} (ξ) ≃ {\bar{X}}_{i}^{T} Φ (ξ), 1 \leq i \leq \bar{n},

(6.1)

where

x_{i} (ξ)

represents the ith component of the state variable X. Thus, we get

x_{i} (ξ) ≃ {\bar{X}}_{i}^{T} Q^{(ρ)} Φ (ξ) + \sum_{k = 0}^{\hat{n} - 1} \frac{δ^{k} x_{i} (a)}{k!} {(\ln \frac{ξ}{a})}^{k}, 1 \leq i \leq \bar{n},

and consequently

x_{i} (ξ) ≃ {\bar{X}}_{i}^{T} Q^{(ρ)} Φ (ξ) + {\bar{G}}_{i}^{T} Φ (ξ) ≜ {\bar{\bar{X}}}_{i} Φ (ξ), 1 \leq i \leq \bar{n},

(6.2)

where

Φ (ξ)

and

Q^{(ρ)}

are given in (4.4) and (5.1), respectively. Additionally,

{\bar{X}}_{i}

is an unknown vector and

{\bar{G}}_{i}

is a known vector defined as follows

{\bar{G}}_{i} = {[{\bar{G}}_{i, 1}, {\bar{G}}_{i, 2}, \dots, {\bar{G}}_{i, n + 1}]}^{T}, {\bar{X}}_{i} = {[{\bar{x}}_{i, 1}, {\bar{x}}_{i, 2}, \dots, {\bar{x}}_{i, n + 1}]}^{T}, 1 \leq i \leq \bar{n},

with

{\bar{G}}_{i, r} = \sum_{k = 0}^{\hat{n} - 1} δ^{k} x_{i} (a) / k! {(\ln ξ_{r} / a)}^{k}

for r = 1, 2, …, n + 1. In addition, we assume

u_{j} (ξ) ≃ {\bar{U}}_{j}^{T} Φ (ξ), 1 \leq j \leq \bar{s},

(6.3)

such that

u_{j} (ξ)

denotes the jth component of the control variable U. Moreover,

{\bar{U}}_{j}

is an unknown vector that is defined as follows

{\bar{U}}_{j} = [{\bar{u}}_{j, 1}, {\bar{u}}_{j, 2}, \dots, {\bar{u}}_{j, n + 1}], 1 \leq j \leq \bar{s} .

Utilizing the interpolation attribute of the fractional logarithmic Chebyshev cardinal functions and substituting relations (6.2) and (6.3) into relation (3.1), yield

\bar{J} ≃ \hat{J} (U) = \int_{a}^{b} F (ξ, {\bar{\bar{X}}}_{1}^{T} Φ (ξ), {\bar{\bar{X}}}_{2}^{T} Φ (ξ), \dots, {\bar{\bar{X}}}_{\bar{n}}^{T} Φ (ξ), {\bar{U}}_{1}^{T} Φ (ξ), {\bar{U}}_{2}^{T} Φ (ξ), \dots, {\bar{U}}_{\bar{s}}^{T} Φ (ξ)) d ξ = {\bar{S}}^{T} \bar{Φ},

(6.4)

such that

\bar{S} = {[{\bar{S}}_{1}, {\bar{S}}_{2}, \dots, {\bar{S}}_{n + 1}]}^{T}

with

{\bar{S}}_{j} = F (ξ_{j}, {\bar{x}}_{1, j}, {\bar{x}}_{2, j}, \dots, {\bar{x}}_{\bar{n}, j}, {\bar{u}}_{1, j}, {\bar{u}}_{2, j}, \dots, {\bar{u}}_{\bar{s}, j})

and

\bar{Φ} = {[{\bar{Φ}}_{1}, {\bar{Φ}}_{2}, \dots, {\bar{Φ}}_{n + 1}]}^{T}

in which

{\bar{Φ}}_{j} = \int_{a}^{b} ϕ_{j} (ξ) d ξ

. Inserting equations (6.1), (6.2) and (6.3) into equation (3.2), yields

{\bar{X}}_{i}^{T} Φ (ξ) ≃ P_{i} (ξ, {\bar{\bar{X}}}_{1}^{T} Φ (ξ), {\bar{\bar{X}}}_{2}^{T} Φ (ξ), \dots, {\bar{\bar{X}}}_{\bar{n}}^{T} Φ (ξ), {\bar{U}}_{1}^{T} Φ (ξ), {\bar{U}}_{2}^{T} Φ (ξ), \dots, {\bar{U}}_{\bar{s}}^{T} Φ (ξ)), 1 \leq i \leq \bar{n} .

(6.5)

We can simplify relation (6.5) by utilizing the cardinality of the fractional logarithmic Chebyshev cardinal functions as follows

({\bar{X}}_{i}^{T} - {\bar{R}}_{i}^{T}) Φ (ξ) ≜ {\bar{ϒ}}_{i}^{T} Φ (ξ) ≃ 0, 1 \leq i \leq \bar{n},

(6.6)

where

{\bar{R}}_{i} = {[{\bar{R}}_{i, 1}, {\bar{R}}_{i, 2}, \dots, {\bar{R}}_{i, n + 1}]}^{T}, 1 \leq i \leq \bar{n},

in which

{\bar{R}}_{i, j} = P_{i} (ξ_{j}, {\bar{x}}_{1, j}, {\bar{x}}_{2, j}, \dots, {\bar{x}}_{\bar{n}, j}, {\bar{u}}_{1, j}, {\bar{u}}_{2, j}, \dots, {\bar{u}}_{\bar{s}, j}), 1 \leq i \leq \bar{n}, 1 \leq j \leq n + 1 .

One can derive an algebraic system from equation (6.6) as

{\bar{H}}_{i, j} ≜ {[{\bar{ϒ}}_{i}]}_{j}, 1 \leq i \leq \bar{n}, 1 \leq j \leq n + 1 .

By applying the technique of Lagrange multipliers, we get

J ≃ J (\bar{X}, \bar{U}, \bar{Λ}) = \bar{J} + \bar{H} \bar{Λ},

where

\bar{H} = [{\bar{H}}_{1,1}, {\bar{H}}_{1,2}, \dots, {\bar{H}}_{1, n + 1} | {\bar{H}}_{2,1}, {\bar{H}}_{2,2}, \dots, {\bar{H}}_{2, n + 1} | \dots | {\bar{H}}_{\bar{n}, 1}, {\bar{H}}_{\bar{n}, 2}, \dots, {\bar{H}}_{\bar{n}, n + 1}], \bar{Λ} = [{\bar{λ}}_{1}, {\bar{λ}}_{2}, \dots, {\bar{λ}}_{\bar{n} (n + 1)}],

so that the Lagrange multipliers vector is denoted by

\bar{Λ}

. Therefore, we use the following system to get the required conditions for the optimality

\begin{align} \frac{\partial J}{\partial {\bar{x}}_{i, k}} & = 0, 1 \leq i \leq \bar{n}, 1 \leq k \leq n + 1, \\ \frac{\partial J}{\partial {\bar{u}}_{i, k}} & = 0, 1 \leq i \leq \bar{s}, 1 \leq k \leq n + 1, \\ \frac{\partial J}{\partial {\bar{λ}}_{j}} & = 0, 1 \leq j \leq \bar{n} (n + 1) . \end{align}

(6.7)

By solving the nonlinear algebraic system generated in (6.7), the elements of ${\bar{X}}_{i}, i = 1,2, \dots, \bar{n}$ and ${\bar{U}}_{j}, j = 1,2, \dots, \bar{s}$ (as the unknown vectors) are determined. Eventually, a numerical solution for the main FOCP defined in (3.1)–(3.3) is obtained with the help of approximations provided in (6.2) and (6.3).

7. Test problems

To evaluate the accuracy and efficiency of the suggested approach, we investigate two examples. MAPLE 18 (with 50 decimal digits) is used to do all required numerical simulations. Additionally, the following formulas are employed to verify the accuracy of the results.

\begin{align} e_{x} & = {(\int_{a}^{b} {(x (ξ) - x^{⋆} (ξ))}^{2} d ξ)}^{\frac{1}{2}}, \end{align}

(7.1)

\begin{align} e_{u} & = {(\int_{a}^{b} {(u (ξ) - u^{⋆} (ξ))}^{2} d ξ)}^{\frac{1}{2}}, \end{align}

(7.2)

where x^⋆ and u^⋆ are the approximate solutions, and x and u are the exact solutions. Moreover, the absolute error of the performance index is obtained by

e_{\hat{J}} = | \hat{J} - J^{⋆} |

where

\hat{J}

and J^⋆ represent the exact and approximate values of the performance index, respectively.

Example 1

Consider the following FOCP

\min \hat{J} (u) = \int_{1.5}^{2.5} [ξ^{\frac{3}{2}} {(x (ξ) - ({(\ln \frac{ξ}{1.5})}^{\frac{3}{2}} + {(\ln \frac{ξ}{1.5})}^{\frac{7}{3}}))}^{2} + e^{- 2 ξ} {(u (ξ) - {(\ln \frac{ξ}{1.5})}^{\frac{3}{2}})}^{2}] d ξ,

subject to dynamical system

\begin{array}{l} {}_{1.5}^{C H}D_{ξ}^{ρ} x (ξ) & = \sin (ξ) x (ξ) + u (ξ) - \sin (ξ) ({(\ln \frac{ξ}{1.5})}^{\frac{3}{2}} + {(\ln \frac{ξ}{1.5})}^{\frac{7}{3}}) - {(\ln \frac{ξ}{1.5})}^{\frac{3}{2}} \\ + \frac{Γ (\frac{5}{2})}{Γ (\frac{5}{2} - ρ)} {(\ln \frac{ξ}{1.5})}^{\frac{3}{2} - ρ} + \frac{Γ (\frac{10}{3})}{Γ (\frac{10}{3} - ρ)} {(\ln \frac{ξ}{1.5})}^{\frac{7}{3} - ρ}, 0 < ρ < 1, \\ x (1.5) = 0 . \end{array}

The problem’s optimal solution is ${x (ξ) = {(\ln ξ / 1.5)}^{3 / 2} + {(\ln ξ / 1.5)}^{7 / 3}, u (ξ) = {(\ln ξ / 1.5)}^{3 / 2}, \hat{J} = 0}$ . The numerical approach described in the preceding section is employed to this example. Figure 1 presents the approximate solutions obtained using the developed method for $x (ξ)$ and $u (ξ)$ whenever ρ = 1/2, μ = 1/2 and n = 10. Furthermore, in Figure 2, the absolute errors of $x (ξ)$ and $u (ξ)$ are plotted. Table 1 reports the errors of $x (ξ)$ , $u (ξ)$ and $\hat{J}$ for different values of ρ, μ and n. This also provides evidence of the convergence rate of the suggested approach. The results show the proposed method accurately approximates the optimal state and control variables. Moreover, when the number of n increases, the errors decrease.

Table 1.

The errors of $x (ξ)$ , $u (ξ)$ , and $\hat{J}$ for several values of ρ, μ, and n in Example 1.

		ρ = 1/4			ρ = 1/2			ρ = 3/4
μ	n	e _x	e _u	$e_{\hat{J}}$	e _x	e _u	$e_{\hat{J}}$	e _x	e _u	$e_{\hat{J}}$
1/2	4	1.1397 × 10⁻⁰⁴	1.2161 × 10⁻⁰⁴	4.1904 × 10⁻¹⁰	1.1416 × 10⁻⁰⁴	1.1651 × 10⁻⁰⁴	2.7465 × 10⁻¹¹	1.3213 × 10⁻⁰⁴	6.1174 × 10⁻⁰⁴	1.3560 × 10⁻⁰⁸
	6	2.6194 × 10⁻⁰⁵	3.0505 × 10⁻⁰⁵	5.1046 × 10⁻¹¹	2.6564 × 10⁻⁰⁵	2.4942 × 10⁻⁰⁵	1.2235 × 10⁻¹²	3.8318 × 10⁻⁰⁵	2.3740 × 10⁻⁰⁴	1.7521 × 10⁻⁰⁹
	8	8.1542 × 10⁻⁰⁶	1.6543 × 10⁻⁰⁵	1.0172 × 10⁻¹¹	8.7780 × 10⁻⁰⁶	8.6540 × 10⁻⁰⁶	1.1063 × 10⁻¹³	1.8833 × 10⁻⁰⁵	1.0533 × 10⁻⁰⁴	3.4679 × 10⁻¹⁰
	10	7.1196 × 10⁻⁰⁶	8.7755 × 10⁻⁰⁶	2.6875 × 10⁻¹²	7.8975 × 10⁻⁰⁶	7.7179 × 10⁻⁰⁶	1.4349 × 10⁻¹⁴	1.6673 × 10⁻⁰⁵	9.7989 × 10⁻⁰⁵	8.5406 × 10⁻¹¹
3/4	4	1.3861 × 10⁻⁰⁴	1.5217 × 10⁻⁰⁴	5.6492 × 10⁻¹⁰	1.3885 × 10⁻⁰⁴	1.4229 × 10⁻⁰⁴	4.5145 × 10⁻¹¹	1.5739 × 10⁻⁰⁴	7.0654 × 10⁻⁰⁴	1.7805 × 10⁻⁰⁸
	6	3.2000 × 10⁻⁰⁵	3.7421 × 10⁻⁰⁵	6.6136 × 10⁻¹¹	3.2335 × 10⁻⁰⁵	3.0926 × 10⁻⁰⁵	1.9384 × 10⁻¹²	4.2249 × 10⁻⁰⁵	2.5256 × 10⁻⁰⁴	2.3986 × 10⁻⁰⁹
	8	9.2384 × 10⁻⁰⁶	1.8477 × 10⁻⁰⁵	1.2889 × 10⁻¹¹	9.8292 × 10⁻⁰⁶	9.6076 × 10⁻⁰⁶	1.7303 × 10⁻¹³	1.9343 × 10⁻⁰⁵	1.0649 × 10⁻⁰⁴	4.8702 × 10⁻¹⁰
	10	8.4372 × 10⁻⁰⁶	1.0605 × 10⁻⁰⁵	3.4155 × 10⁻¹²	8.9059 × 10⁻⁰⁶	8.7842 × 10⁻⁰⁶	2.3779 × 10⁻¹⁴	1.5038 × 10⁻⁰⁵	7.8367 × 10⁻⁰⁵	1.2427 × 10⁻¹⁰
1	4	1.6788 × 10⁻⁰⁴	1.9853 × 10⁻⁰⁴	7.9262 × 10⁻¹⁰	1.6816 × 10⁻⁰⁴	1.8014 × 10⁻⁰⁴	9.0497 × 10⁻¹¹	1.8859 × 10⁻⁰⁴	8.2299 × 10⁻⁰⁴	2.3628 × 10⁻⁰⁸
	6	3.9335 × 10⁻⁰⁵	4.7147 × 10⁻⁰⁵	8.7566 × 10⁻¹¹	3.9681 × 10⁻⁰⁵	3.8857 × 10⁻⁰⁵	3.2376 × 10⁻¹²	4.8971 × 10⁻⁰⁵	2.7460 × 10⁻⁰⁴	3.2885 × 10⁻⁰⁹
	8	1.0642 × 10⁻⁰⁵	2.0779 × 10⁻⁰⁵	1.6553 × 10⁻¹¹	1.1235 × 10⁻⁰⁵	1.0890 × 10⁻⁰⁵	2.7350 × 10⁻¹³	2.0893 × 10⁻⁰⁵	1.0892 × 10⁻⁰⁴	6.6559 × 10⁻¹⁰
	10	9.6908 × 10⁻⁰⁶	1.2572 × 10⁻⁰⁵	4.3111 × 10⁻¹²	1.0101 × 10⁻⁰⁵	1.0048 × 10⁻⁰⁵	3.7219 × 10⁻¹⁴	1.4968 × 10⁻⁰⁵	6.2548 × 10⁻⁰⁵	1.7131 × 10⁻¹⁰

Example 2

Consider the following FOCP

\min \hat{J} (u) = \int_{2}^{3} [ξ^{2} {(x (ξ) - ({(\ln \frac{ξ}{2})}^{2.4} + \ln \frac{ξ}{2}))}^{2} + e^{- ξ} {(u (ξ) - ({(\ln \frac{ξ}{2})}^{2.4} + {(\ln \frac{ξ}{2})}^{1.3}))}^{2}] d ξ,

subject to dynamical system

\begin{array}{l} {}_{2}^{C H}D_{ξ}^{ρ} x (ξ) & = - x^{2} (ξ) + (ξ + 3) u (ξ) + {({(\ln \frac{ξ}{2})}^{2.4} + \ln \frac{ξ}{2})}^{2} - (ξ + 3) ({(\ln \frac{ξ}{2})}^{2.4} + {(\ln \frac{ξ}{2})}^{1.3}) \\ + \frac{Γ (3.4)}{Γ (3.4 - ρ)} {(\ln ξ)}^{2.4 - ρ}, 1 < ρ < 2, \\ x (2) & = 0, δ x (2) = 1 . \end{array}

The problem’s optimal solution is ${x (ξ) = {(\ln ξ / 2)}^{2.4} + \ln ξ / 2, u (ξ) = {(\ln ξ / 2)}^{2.4} + {(\ln ξ / 2)}^{1.3}, \hat{J} = 0}$ . The numerical method outlined in the preceding section is utilized for this example. Figure 3 shows the approximate solutions obtained using the proposed method for $x (ξ)$ and $u (ξ)$ when ρ = 5/4, μ = 3/4 and n = 9. The absolute errors of $x (ξ)$ and $u (ξ)$ are depicted in Figure 4. The errors of $x (ξ)$ , $u (ξ)$ and $\hat{J}$ for three values of ρ, μ and n are rendered in Table 2. It also demonstrates the convergence of the suggested approach. The reported outcomes confirm the obtained optimal state and control variables are enough accurate.

Table 2.

The errors of $x (ξ)$ , $u (ξ)$ , and $\hat{J}$ for several values of ρ, μ, and n in Example 2.

		ρ = 5/4			ρ = 3/2			ρ = 7/4
μ	n	e _x	e _u	$e_{\hat{J}}$	e _x	e _u	$e_{\hat{J}}$	e _x	e _u	$e_{\hat{J}}$
1/4	3	8.7130 × 10⁻⁰⁵	3.9324 × 10⁻⁰⁴	6.9962 × 10⁻¹⁰	8.7128 × 10⁻⁰⁵	4.5262 × 10⁻⁰⁴	5.6763 × 10⁻¹⁰	9.2633 × 10⁻⁰⁵	5.8764 × 10⁻⁰⁴	1.0187 × 10⁻⁰⁸
	5	3.1584 × 10⁻⁰⁶	6.3818 × 10⁻⁰⁵	2.6086 × 10⁻¹¹	3.9063 × 10⁻⁰⁶	7.8166 × 10⁻⁰⁵	3.6303 × 10⁻¹¹	8.2005 × 10⁻⁰⁶	1.3528 × 10⁻⁰⁴	1.3302 × 10⁻⁰⁹
	7	6.0926 × 10⁻⁰⁷	1.5465 × 10⁻⁰⁵	2.1358 × 10⁻¹²	7.7477 × 10⁻⁰⁷	2.1322 × 10⁻⁰⁵	4.3665 × 10⁻¹²	3.4342 × 10⁻⁰⁶	5.0046 × 10⁻⁰⁵	2.3889 × 10⁻¹⁰
	9	2.7710 × 10⁻⁰⁷	1.1866 × 10⁻⁰⁵	2.8422 × 10⁻¹³	2.7594 × 10⁻⁰⁷	1.3757 × 10⁻⁰⁵	8.2432 × 10⁻¹³	1.7212 × 10⁻⁰⁶	2.6853 × 10⁻⁰⁵	5.9229 × 10⁻¹¹
1/2	3	1.2272 × 10⁻⁰⁵	3.4965 × 10⁻⁰⁴	8.9720 × 10⁻¹⁰	1.6412 × 10⁻⁰⁵	4.0912 × 10⁻⁰⁴	7.2589 × 10⁻¹⁰	2.6775 × 10⁻⁰⁵	5.6724 × 10⁻⁰⁴	1.2803 × 10⁻⁰⁸
	5	3.8528 × 10⁻⁰⁶	7.0116 × 10⁻⁰⁵	3.2576 × 10⁻¹¹	4.6074 × 10⁻⁰⁶	8.3852 × 10⁻⁰⁵	4.1707 × 10⁻¹¹	9.0112 × 10⁻⁰⁶	1.4202 × 10⁻⁰⁴	1.4959 × 10⁻⁰⁹
	7	6.8818 × 10⁻⁰⁷	1.7608 × 10⁻⁰⁵	2.5097 × 10⁻¹²	8.6483 × 10⁻⁰⁷	2.2834 × 10⁻⁰⁵	4.7623 × 10⁻¹²	3.6000 × 10⁻⁰⁶	5.1241 × 10⁻⁰⁵	2.5746 × 10⁻¹⁰
	9	2.9875 × 10⁻⁰⁷	1.2596 × 10⁻⁰⁵	3.2408 × 10⁻¹³	2.9103 × 10⁻⁰⁷	1.4147 × 10⁻⁰⁵	8.8105 × 10⁻¹³	1.7814 × 10⁻⁰⁶	26749 × 10⁻⁰⁵	6.2773 × 10⁻¹¹
3/4	3	4.9778 × 10⁻⁰⁵	3.4249 × 10⁻⁰⁴	1.2060 × 10⁻⁰⁹	5.1427 × 10⁻⁰⁵	4.0377 × 10⁻⁰⁴	9.2136 × 10⁻¹⁰	5.2622 × 10⁻⁰⁵	5.8311 × 10⁻⁰⁴	1.5963 × 10⁻⁰⁸
	5	4.6341 × 10⁻⁰⁶	7.7246 × 10⁻⁰⁵	4.0667 × 10⁻¹¹	5.3621 × 10⁻⁰⁶	9.0474 × 10⁻⁰⁵	4.7688 × 10⁻¹¹	9.8616 × 10⁻⁰⁶	1.4969 × 10⁻⁰⁴	1.6686 × 10⁻⁰⁹
	7	7.9416 × 10⁻⁰⁷	1.9995 × 10⁻⁰⁵	2.9443 × 10⁻¹²	9.7021 × 10⁻⁰⁷	2.4718 × 10⁻⁰⁵	5.1704 × 10⁻¹²	3.7597 × 10⁻⁰⁶	5.2968 × 10⁻⁰⁵	2.7586 × 10⁻¹⁰
	9	3.2591 × 10⁻⁰⁷	1.3298 × 10⁻⁰⁵	3.6762 × 10⁻¹³	3.0588 × 10⁻⁰⁷	1.4672 × 10⁻⁰⁵	9.3626 × 10⁻¹³	1.8340 × 10⁻⁰⁶	2.7167 × 10⁻⁰⁵	6.6142 × 10⁻¹¹
1	3	8.5778 × 10⁻⁰⁵	3.6241 × 10⁻⁰⁴	1.6158 × 10⁻⁰⁹	8.6659 × 10⁻⁰⁵	4.2804 × 10⁻⁰⁴	1.1901 × 10⁻⁰⁹	8.6344 × 10⁻⁰⁵	6.2519 × 10⁻⁰⁴	1.9794 × 10⁻⁰⁸
	5	5.7571 × 10⁻⁰⁶	8.5234 × 10⁻⁰⁵	5.0903 × 10⁻¹¹	6.3965 × 10⁻⁰⁶	9.8032 × 10⁻⁰⁵	5.4390 × 10⁻¹¹	1.0863 × 10⁻⁰⁵	1.5812 × 10⁻⁰⁴	1.8481 × 10⁻⁰⁹
	7	9.3728 × 10⁻⁰⁷	2.2696 × 10⁻⁰⁵	3.4698 × 10⁻¹²	1.0992 × 10⁻⁰⁶	2.6997 × 10⁻⁰⁵	5.6036 × 10⁻¹²	3.9198 × 10⁻⁰⁶	5.5067 × 10⁻⁰⁵	2.9427 × 10⁻¹⁰
	9	3.5881 × 10⁻⁰⁷	1.3999 × 10⁻⁰⁵	4.1878 × 10⁻¹³	3.2155 × 10⁻⁰⁷	1.5282 × 10⁻⁰⁵	9.9317 × 10⁻¹³	1.8831 × 10⁻⁰⁶	2.7800 × 10⁻⁰⁵	6.9456 × 10⁻¹¹

Example 3

In this example, we consider the vibration of a mass-spring-damper system subjected to an external force. In particular, the response to harmonic excitations is examined. In many environments, rotating machinery, motors, and so on, cause periodic motions of structures to induce vibrations into other mechanical devices and structures nearby. On summing the forces, the equation for the forced vibration of the system illustrated in Figure 5 is given as follows.

m \ddot{x} (ξ) + C \dot{x} (ξ) + K x (ξ) = F (ξ),

(7.3)

where m, C and K are given constants (Inman, 2017). The force

B (ξ)

results from the impact of the road bump or potholes on the car. Because the weight of the vehicle is constant during the motion, the resultant force applied to the car from the road,

F (ξ)

, is obtained from the combination of the force of the car’s weight (mg) and force

B (ξ)

. We remind that the mass-spring-damper system is a fundamental model for understanding the dynamics of various mechanical systems. It is instrumental in analyzing the behavior of systems subject to external forces, allowing us to explore the properties of materials that exhibit both elastic and viscous characteristics. By adjusting the quantity and configuration of its constituent elements (mass, spring, and damper) either in parallel or series, one can tailor the mass-spring-damper system to mimic a wide array of applications including structural vibrations (modeling the response of buildings and structures to seismic activity or other dynamic loads), vehicle suspension systems (designing a system to absorb shocks and maintain vehicle stability over uneven terrain), biomechanics (modeling the complex interactions within the human body especially tendon movements and muscle deformations leverages). Furthermore, the mass-spring-damper model is pivotal in developing control systems that require precise manipulation of dynamic variables. It aids engineers in creating systems that can compensate for disturbances and maintain desired performance levels. The versatility of this model makes it an indispensable tool in mechanical engineering, biomedical research, and computer animation. Using the change of variables

\begin{matrix} x_{1} (ξ) = x (ξ), x_{2} (ξ) = \dot{x} (ξ), \end{matrix}

the problem introduced in (7.3) is converted into the following system

\begin{array}{l} {\dot{x}}_{1} (ξ) & = x_{2} (ξ), \\ {\dot{x}}_{2} (ξ) & = - \frac{K}{m} x_{1} (ξ) - \frac{C}{m} x_{2} (ξ) + \frac{1}{m} F (ξ) . \end{array}

It is common to approximate the driving forces, $F (ξ)$ , as periodic of the form $F (ξ) = F_{0} \sin (ω ξ)$ where F₀ and ω represent the amplitude and the frequency (rad/s) of the applied force, respectively. With the specific application of the linear regulator problem in vibration suppression (Yari, 2021), the following optimal control problem can be designed

\min \hat{J} (u) = \frac{1}{2} \int_{a}^{b} [a_{1} x_{1}^{2} (ξ) + a_{2} u^{2} (ξ)] d ξ,

subject to dynamical system

\begin{array}{l} {\dot{x}}_{1} (ξ) & = x_{2} (ξ), \\ {\dot{x}}_{2} (ξ) & = - \frac{K}{m} x_{1} (ξ) - \frac{C}{m} x_{2} (ξ) + \frac{1}{m} F (ξ) + u (ξ), \\ x_{1} (a) & = x_{1 a}, x_{2} (a) = x_{2 a} . \end{array}

By substituting the classical derivative with the Caputo–Hadamard fractional derivative and employing the parameters provided in Table 3, we obtain the following FOCP

\min \hat{J} (u) = \frac{1}{2} \int_{1}^{2} [0.001 x_{1}^{2} (ξ) + u^{2} (ξ)] d ξ,

subject to dynamical system

\begin{matrix} {}_{1}^{C H}D_{ξ}^{ρ} x_{1} (ξ) & = ξ x_{2} (ξ), 0 < ρ \leq 1, \\ {}_{1}^{C H}D_{ξ}^{ρ} x_{2} (ξ) & = ξ (- 4 x_{1} (ξ) - 0.4 x_{2} (ξ) + u (ξ) + \frac{1}{\sqrt{2}} \sin (3 ξ)), \\ x_{1} (1) & = 0.7029, x_{2} (1) = 3.3876 . \end{matrix}

The problem’s optimal solution in the case of ρ = 1 is as follows

\begin{array}{l} x_{1} (ξ) & = - e^{- 0.2 ξ} (0.008 \sin (1.99 ξ) + 2.089 \cos (1.99 ξ)) - 0.134 \sin (3 ξ) - 0.032 \cos (3 ξ), \\ x_{2} (ξ) & = 0.2 e^{- 0.2 ξ} (0.008 \sin (1.99 ξ) + 2.089 \cos (1.99 ξ)) - e^{- 0.2 ξ} (0.01592 \cos (1.99 ξ) - 4.15711 \sin (1.99 ξ)) \\ - 0.402 \cos (3 ξ) + 0.096 \sin (3 ξ), \\ u (ξ) & = 0, \hat{J} = 0.0009 . \end{array}

The numerical method described in the previous section is applied to this example. Table 4 lists the approximate solutions of $x_{1} (ξ)$ and $x_{2} (ξ)$ for several values of ρ with μ = 3/4, 1 and n = 8. Figure 6 compares the exact solutions and the approximate solutions of $x_{1} (ξ)$ and $x_{2} (ξ)$ for some values of ρ. From this figure, it is clear that as ρ approaches 1, the approximate solutions of $x_{1} (ξ)$ and $x_{2} (ξ)$ converge to the corresponding exact solutions. Moreover, the approximate solutions at ρ = 1 are in a very good agreement with the corresponding exact solutions. The absolute errors of $x_{1} (ξ)$ and $x_{2} (ξ)$ (for the case ρ = 1) are shown in Figure 7. The results indicate that the established method accurately approximates the optimal state variables.

Table 3.

The values of parameters in Example 3.

Parameters	a	b	a ₁	a ₂	m	K	C	F ₀	ω	x _1a	x _2a
Values	1	2	0.001	1	1	4	0.4	$1 / \sqrt{2}$	3	0.7029	3.3876

Table 4.

The approximate solutions of $x_{1} (ξ)$ and $x_{2} (ξ)$ for several values of ρ when μ = 3/4, 1 and n = 8 in Example 3.

		$x_{1} (ξ)$				$x_{2} (ξ)$
μ	ξ	ρ = 1/4	ρ = 1/2	ρ = 3/4	ρ = 1	ρ = 1/4	ρ = 1/2	ρ = 3/4	ρ = 1
3/4	1.0	1.2332	0.7893	0.6935	0.6950	2.0999	3.2126	3.3334	3.3614
	1.2	0.8560	1.2321	1.4426	1.2824	0.0298	0.3108	1.3711	2.3269
	1.4	0.5990	0.8366	1.3840	1.6110	−0.2229	−0.4800	−0.2514	0.9234
	1.6	0.4407	0.5141	0.9889	1.6472	−0.2860	−0.6356	−1.2422	−0.5497
	1.8	0.3697	0.3307	0.5099	1.4047	−0.2611	−0.5424	−1.5384	−1.8204
	2.0	0.3696	0.2698	0.1295	0.9475	−0.1965	−0.3681	−1.2904	−2.6673
1	1.0	1.2457	0.9045	0.7353	0.7028	1.8401	3.1196	3.3512	3.3876
	1.2	0.8542	1.2302	1.4418	1.2819	0.0359	0.3105	1.3712	2.3257
	1.4	0.5998	0.8381	1.3838	1.6103	−0.2265	−0.4799	−0.2506	0.9228
	1.6	0.4404	0.5134	0.9886	1.6464	−0.2848	−0.6361	−1.2421	−0.5501
	1.8	0.3700	0.3313	0.5100	1.4039	−0.2611	−0.5410	−1.5366	−1.8193
	2.0	0.3695	0.2701	0.1301	0.9471	−0.1960	−0.3683	−1.2891	−2.6650

8. Conclusion

This study introduced a novel set of cardinal functions known as fractional logarithmic Chebyshev cardinal functions to devise a numerical strategy for a family of Hadamard FOCPs. To achieve this, the Hadamard integral operational matrix of this new family of fractional functions was derived. The suggested approach transformed solving the original problem into finding the solution of an algebraic system of equations which was extracted by utilizing the specified integration operational matrix and representing the unknown control and state variables via the introduced fractional logarithmic functions. The validity of the expressed scheme was verified by solving two illustrative examples. The obtained results confirmed the high accuracy of the established algorithm. In addition, it was demonstrated that increasing the number of fractional logarithmic Chebyshev cardinal functions used in the developed scheme improves the accuracy of the outcomes.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) did not receive any financial support about the research, authorship, and/or publication of this article.

ORCID iD

Mohammad Hossein Heydari

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