Fractional Optimal Control Problems Research Articles

Regional Fractional Optimal Control Problem of a Bilinear Reaction Diffusion Equation Using Distributed Bounded Controls

Regional Fractional Optimal Control Problem of a Bilinear Reaction Diffusion Equation Using Distributed Bounded Controls

An accurate wavelets‐collocation technique for neutral delay distributed‐order fractional optimal control problems

AbstractIn this study, a new class of optimal control problems called neutral delay distributed‐order fractional optimal control problems is introduced, this problem is solved based on an efficient computational scheme. To solve the problem, we derive an exact formula for the Riemann–Liouville fractional integral operator of Genocchi wavelets based on beta functions for the first time. By taking into account this operator, collocation method, and Gauss–Legendre integration formula, the solution of fractional optimal control problems (FOCPs) under consideration is converted to a nonlinear programming one to which existing well‐developed algorithms may be applied. The mentioned scheme is applied to both FOCPs with or without delay. Error analysis associated with the proposed idea is also investigated under several mild conditions. The effectiveness of the strategy is showed by several illustrative examples, furthermore, a comparison with the previous methods highlights the preference of this scheme.

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

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.

A robust optimisation approach for the 2D-VO fractional optimal control problems

This paper provides a new numerical scheme with the help of discrete Krawtchouk polynomials (DKPs), the Ritz method, and the two-dimensional (2D) Gauss-Legendre quadrature rule for dealing with two-dimensional variable-order (VO) fractional optimal control problems. For this aim, we construct the derivative operational matrices of DKPs. Also, we impose the conditions into the approximation of state and control functions. Then, we expand the functions in the problem given DKPs and operational matrices. Meanwhile, DKPs and the 2D Gauss-Legendre quadrature rule evaluate the performance index function. According to the numerical algorithm process, the proposed problem is reduced to a system of algebraic equations. The chosen polynomial as basis functions and the Ritz method provide a powerful and flexible scheme. Also, we discuss the error of the VO-fractional derivative of the approximate solution and performance index. At last, we include several illustrative examples to indicate the validity and applicability of the present technique.

Numerical simulation for fractional optimal control problems via euler wavelets

Abstract This paper suggests employing Euler wavelets to provide a precise and effective computational approach for certain fractional optimal control problems. The primary objective of the approach is to transform the fractional optimal control problem defined by the dynamical system and performance index into systems of algebraic equations, which then can be readily solved using matrix techniques. Since Euler polynomials are employed to build Euler wavelets and since Euler polynomials have fewer terms than most of commonly used other polynomials to build wavelets, employing them for the numerical approach results in sparser operational matrices. The speed of the recommended numerical method is improved due to the fewer terms in Euler wavelet operational matrices. We obtain the corresponding systems of algebraic equations for state variable, control variable, and Lagrange multipliers (used to determine the essential conditions of optimality) by incorporating operational matrices of Euler wavelets. Subsequently, those systems of algebraic equations are solved to obtain the numerical solution. The suggested method’s efficiency is demonstrated using a few typical examples. The suggested procedure is accurate and efficient, according to the results.

Optimal control for nonlinear time-fractional Schrödinger equation: an application to quantum optics

In this research article, a fractional optimal control problem (FOCP) is applied to a nonlinear time-fractional Schrödinger equation (NTFSE) incorporating a trapping potential. The NTFSE is an innovative mathematical advancement in the field of quantum optics, bridging fractional calculus with nonlinear quantum mechanics and addressing the intricacies of systems involving memory and nonlinearity. This exploration helps with potential technological advancements in quantum optics and related domains. Examining the FOCP within this system allows one to design quantum optical systems with enhanced performance, improved precision stability, and robustness against disturbances. In this work, the performance index for the problem is constructed, and then it is reformulated using the fractional variational principle and the Lagrange multiplier method. Additionally, the Jacobi collocation numerical method is employed to solve the FOCP and numerical simulations are demonstrated across various parameters which offer valuable insights into the implemented methodology.

A gradient-based optimization algorithm to solve optimal control problems with conformable fractional-order derivatives

In this paper, we solve numerically a class of fractional optimal control problems in the conformable sense. We first propose a nonlinear fractional optimal control problem subject to general equality and inequality constraints. Then, based on a novel numerical integration scheme, we present a discretization method for the governed fractional-order system as well as the cost and constraint functionals, which yields a discretized approximate problem. We further derive the gradients of the cost and constraint functionals in regard to the discretized controls. On the basis of this result, a numerical optimization approach to solve the approximate problem is developed. Finally, three non-trivial example problems are solved to illustrate the applicability and effectiveness of the developed approach.

The influence of prevention and isolation measures to control the infections of the fractional Chickenpox disease model

In this paper, we propose a mathematical model using the Caputo fractional derivative (CFD) and two control signals to study the transmission dynamics and control of Chickenpox (Varicella) outbreak. The model consists of six compartments representing susceptible, vaccinated, exposed, infected with complications, infected without complications, and recovered individuals. We analyze the theoretical properties of the model, including existence, uniqueness, and boundedness of solutions, and calculate the basic reproduction number (R0). We identify equilibrium points and establish conditions for their stability. Sensitivity analysis helps identify the most influential parameters on R0. We formulate a fractional optimal control problem (FOCP) by incorporating time-dependent prevention and isolation measures. The necessary optimality conditions are derived using Pontryagin’s maximum principle. Numerical simulations based on the Adams–Bashforth–Moulton (ABM) method illustrate the impact of control measures and fractional order on disease propagation. The results highlight the effectiveness of optimal controls and fractional order in understanding and managing epidemics, enhancing stability conditions. The study contributes to a better understanding of Chickenpox transmission dynamics and provides insights for disease control and management, aiding decision-makers and governments in taking preventive measures.

A new method for solving fractional optimal control problems

In this article, we present a novel approach that utilizes the operational matrix method to solve fractional optimal control problems. We address the challenge of missing initial conditions for the co-state function by incorporating the linear shooting method. Our objective is to provide a method that not only enhances computational efficiency and reduces cost but also improves accuracy and usability. With our new approach, we can calculate the coefficients of the expansion solution without the need to solve an algebraic system. These coefficients can be generated explicitly or through an iterative process. We provide a proof of the uniform convergence of the approximate series of functions to the unique solution of the optimal control problem. To demonstrate the effectiveness of our proposed method, we solve several numerical examples and compare the results with those obtained by other researchers. Additionally, we extend the application of this new method to solve problems with multiple states, thereby expanding its scope to a wider range of problem domains. This allows us to address diverse scenarios using the same efficient and accurate approach. Through our research, we contribute to the development of a powerful and versatile method for solving fractional optimal control problems, offering significant advantages over existing techniques.

Enhancing Water Pollution Management Through a Comprehensive Fractional Modeling Framework and Optimal Control Techniques

Water pollution poses a significant threat to public health, particularly in developing nations. This study aims to enhance our understanding of water pollution transmission dynamics by introducing a novel mathematical model within a general fractional framework. The model offers a comprehensive analysis, including assessing solution feasibility and the stability of equilibrium points. To effectively implement this model, we develop an efficient numerical scheme based on the trapezoidal method. Furthermore, we conduct a thorough error discussion and convergence analysis. Notably, we devise and evaluate a new strategy to effectively tackle the associated fractional optimal control problem concerning water pollutant transmission. Applying this approach to a real case demonstrates its substantial impact, notably in reducing non-soluble contaminants within the water transmission system.

Modified wavelets technique for multitype 2D fractional optimal control problems

Modified wavelets technique for multitype 2D fractional optimal control problems

Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems

This paper introduces a new, highly accurate model for periodic fractional optimal control problems (PFOCPs) based on Riemann–Liouville and Caputo fractional derivatives (FDs) with sliding fixed memory lengths. This new model enables periodic solutions in fractional-order control models by preserving the periodicity of the fractional derivatives. Additionally, we develop a novel numerical solution method based on Fourier and Gegenbauer pseudospectral approaches. Our method simplifies PFOCPs into constrained nonlinear programming problems (NLPs), readily solvable by standard NLP solvers. Some key innovations of this study include: (i) The derivation of a simplified FD calculation formula to transform periodic FD calculations into evaluations of trigonometric Lagrange interpolating polynomial integrals, efficiently computed using Gegenbauer quadratures. (ii) The introduction of the αth-order fractional integration matrix based on Fourier and Gegenbauer pseudospectral approximations, which proves to be highly effective for computing periodic FDs. A priori error analysis is provided to predict the accuracy of the Fourier–Gegenbauer-based FD approximations. Benchmark PFOCP results validate the performance of the proposed pseudospectral method.

Chelyshkov wavelet method for solving multidimensional variable order fractional optimal control problem

Chelyshkov wavelet method for solving multidimensional variable order fractional optimal control problem

The Stability of Solutions of the Variable-Order Fractional Optimal Control Model for the COVID-19 Epidemic in Discrete Time

This article introduces a discrete-time fractional variable order over a SEIQR model, incorporated for COVID-19. Initially, we establish the well-possedness of solution. Further, the disease-free and the endemic equilibrium points are determined. Moreover, the local asymptotic stability of the model is analyzed. We develop a novel discrete fractional optimal control problem tailored for COVID-19, utilizing a discrete mathematical model featuring a variable order fractional derivative. Finally, we validate the reliability of these analytical findings through numerical simulations and offer insights from a biological perspective.

EXPLORING FRACTIONAL DYNAMICAL PROBES IN THE CONTEXT OF GENDER-STRUCTURED HIV–TB COINFECTION: A STUDY OF CONTROL STRATEGIES

The primary objective of this paper is to conduct an in-depth analysis of the intricacies inherent in a gender-structured model depicting the dynamics of HIV–TB coinfection. The model is articulated through a set of nonlinear fractional-order differential equations. Our focus is on elucidating the stability characteristics around equilibrium points, employing advanced fractional stability techniques. We leverage sensitivity analysis as a powerful tool to discern the specific model parameters exerting significant influence on the transmission and control of the coinfection, as gauged by the basic reproduction number. Through this, we aim to identify key factors that drive the spread or containment of the dual affliction. Furthermore, we delve into the realm of optimization by formulating a fractional optimal control problem. The objective is to design and implement two control strategies that effectively mitigate the prevalence of coinfection, demonstrating a state of perfect protection. Notably, our findings reveal that the implementation of control measures at a rate exceeding 80% suffices to eliminate the infection entirely. We anticipate that the insights garnered from this research will prove instrumental for policymakers in crafting strategic interventions to curtail the tragic impact of HIV–TB coinfection. Our scientifically grounded study serves as a foundation upon which comprehensive plans can be devised to stem the devastating consequences of this dual health challenge.

Optimal solution of nonlinear 2D variable-order fractional optimal control problems using generalized Bessel polynomials

This study aims to propose a new optimization method based on the generalized Bessel polynomials (GBPs) as a class of basis functions for a category of nonlinear two-dimensional variable-order fractional optimal control problems (N-2D-VOFOCPs) involved in fractional-order dynamical systems and Caputo derivatives. For the optimal solution of such problems, the optimization method is developed on the basis of operational matrices (OMs) scheme of derivatives, 2D Gauss–Legendre quadrature rule, and Lagrange multiplier technique. The state and control functions are expanded in terms of the GBPs to reduce the complexity of these problems. The proposed method focuses on a system of nonlinear algebraic equations in the process of finding solution to the problems. The convergence of the method based on GBPs is proved, and the accuracy of the method is analyzed by solving several examples.

Optimal control for a variable-order diffusion-wave equation with a reaction term; A numerical study

In this paper, optimal control for a variable-order diffusion-wave equation with a reaction term is numerically studied, where the variable-order operator is defined in the sense of Caputo proportional constant. Necessary optimality conditions for the control problem are derived. Existence and uniqueness for the solutions of fractional optimal control problem are derived. The nonstandard weighted average finite difference method and the nonstandard leap-frog method are developed to study numerically the proposed problem. Moreover, the stability analysis of the methods is proved. Finally, in order to characterise the memory property of the proposed model, three test examples are given. It is found that the nonstandard weighted average finite difference method can be applied to study such variable-order fractional optimal control problems simply and effectively.

Optimal Control Applied to Piecewise-Fractional Ebola Model

A recently proposed fractional-order mathematical model with Caputo derivatives was developed for Ebola disease. Here, we extend and generalize this model, beginning with its correction. A fractional optimal control (FOC) problem is then formulated and numerically solved with the rate of vaccination as the control measure. The research presented in this work addresses the problem of fitting real data from Guinea, Liberia, and Sierra Leone, available at the World Health Organization (WHO). A cost-effectiveness analysis is performed to assess the cost and effectiveness of the control measure during the intervention. We come to the conclusion that the fractional control is more efficient than the classical one only for a part of the time interval. Hence, we suggest a system where the derivative order changes over time, becoming fractional or classical when it makes more sense. This type of variable-order fractional model, known as piecewise derivative with fractional Caputo derivatives, is the most successful in managing the illness.

A direct transcription method for solving distributed-order fractional optimal control problems

This paper considers the numerical solution of distributed-order Fractional Optimal Control Problems (FOCPs). The common approaches for solving these problems involve approximating the problem with a multi-term FOCP and subsequently discretizing the resulting multi-term problem using either a direct or indirect method. However, this paper bypasses the need for approximation with a multi-term FOCP by introducing innovative approximation formulas for distributed-order fractional derivatives. These approximation formulas are directly derived from the conventional Grünwald–Letnikov, L1, and trapezoidal formulas. Building upon these formulas, we develop the relevant fractional-order distributed-order operational matrices. Utilizing the operational matrix, we easily discretize the distributed-order FOCP to a Non-Linear Programming (NLP), which can be solved by a suitable NLP-solver. Additionally, to enhance the efficiency of solving the resulting NLP, we determine the closed-forms of both the Jacobian of constraints and the gradient of the objective function. The presented method is characterized by its speed, simplicity, and ease of implementation. Moreover, it can be used to solve a wide range of distributed-order FOCPs, such as those involving nonlinear dynamics, free final times, free terminal conditions, and path constraints. By means of ample numerical tests, the accuracy and efficiency of the proposed method are assessed.

The Galerkin Bell method to solve the fractional optimal control problems with inequality constraints

In this manuscript, we adopt the Caputo fractional derivative approach and employ the Galerkin-Bell method to tackle fractional optimal control problems (FOCPs) with equality and inequality constraints in multi-dimensional settings. We derive the Riemann–Liouville (RL) operational matrix for Bell polynomials to facilitate our analysis. By leveraging these matrices and utilizing the Galerkin method, we transform the FOCP into a system of algebraic equations that can be readily solved. We delve into the Bell polynomials’ convergence analysis and error estimation and introduce a residual correction procedure for error estimation. To assess the effectiveness and applicability of our proposed method, we conduct experiments on four different examples and compare our results with those previously reported in the literature. Our findings reveal that the obtained results are highly satisfactory, and in some instances, we achieve the exact solution.