Fractional Optimal Control Problems Research Articles

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.

A cardinal-based numerical method for fractional optimal control problems with Caputo–Katugampola fractional derivative in a large domain

In this paper, the piecewise Chebyshev cardinal functions (PCCFs) are handled to design a highly accurate direct method to solve a category of fractional optimal control problems (FOCPs) involving the Caputo–Katugampola fractional derivative (as a generalisation of the Caputo derivative) in a large domain. These cardinal functions have many useful features, such as cardinality, exponential accuracy and orthogonality. Notice that the exponential accuracy makes it possible to obtain an approximate solution with very high accuracy for the problem under consideration by applying only a small number of basis functions. In this regard, first, the ordinary and fractional integral matrices of the PCCFs are obtained to construct a direct method using them. Then, the constructed approach turns to solve the FOCP under consideration into deriving the solution of an algebraic system (by the aid of approximating the state and control variables based on the PCCFs). To show the validity of the established technique some illustrative examples are provided.

The effect of curative and preventive optimal control measures on a fractional order plant disease model

In this paper, we present a novel mathematical model of fractional order for epidemic dynamics in plants, this fractional order model (FOM) in the sense of Caputo derivatives governing fractional differential equations (FDEs), introducing modified parameters to coincide both sides dimensions for FOM that means enhancing its accuracy in representing the real-life scenarios to control the spread of the epidemic. Our contributions include a comprehensive qualitative analysis of the proposed FOM, such as proving the existence and uniqueness of the projected solution by transforming our problem into a fixed point problem and applying a Banach contraction principle and Schauder’s fixed-point theorem. The positivity and boundedness of this solution are also demonstrated. We precisely evaluate all equilibrium points (EPs), examine their local and global stability by using Routh–Hurwitz conditions and fractional LaSalle’s invariance principle (LIP), respectively and shed light on the dynamic behavior of the FOM. Furthermore, we discuss the sensitivity analysis for parameters of the control reproduction number (CRN), with insightful plots to illustrate the model’s response to parameter changes (replanting and roguing parameters). From this foundation, we formulate a fractional optimal control problem (FOCP) by incorporating preventive u1 and curative u2 controls to effectively eliminate the propagation of the epidemic in plants. The fractional necessary optimality conditions (FNOCs) are derived attribution to Pontryagin’s maximum principle (PMP). We employ the forward–backward sweep method (FBSM) based on fractional Euler method (FEM) in order to facilitate control implementation and show the different suggested strategies. By enhancing this method, optimal control efforts are projected onto the FOM and yield three various strategies to impact the epidemic’s trajectory. For each strategy, we explain how the presence (i.e. with control cases) and absence (without control cases) of the proposed controls affected the susceptible, protected and infected plants. Some attractive figures with various values of the weight factor and fractional order α for the CRN are also presented, allowing a visual assessment of their impact on the dynamics of the epidemic in plants. In addition, we calculate the objective function for controlled and uncontrolled cases to provide a quantitative measure of its effectiveness in containing the outbreak in the plants. The results gained from our analyses and simulations provide valuable guidance for the management and elimination of epidemics in plants by offering various scenarios to implement the proposed controls.

Mittag-Leffler wavelets and their applications for solving fractional optimal control problems

Herein, we design a new scheme for finding approximate solutions to fractional optimal control problems (OCPs) with and without delay. In this strategy, we introduce Mittag-Leffler wavelet functions and develop a new Riemann–Liouville fractional integral operator for these functions utilizing the hypergeometric function. The properties of the operational matrix have reflected well in the process of the numerical method and affect the accuracy of the proposed method directly. Employing the Riemann–Liouville fractional integral operator, delay operational matrix, and Galerkin method, the considered problems lead to systems of algebraic equations. An error analysis is proposed. Finally, some illustrative numerical tests are given to show the precision and validity of the suggested technique. The proposed method is very efficient for solving the OCPs with delay and without delay, and gives very accurate results.

Integral constraint regularization method for fractional optimal control problem with pointwise state constraint

This paper investigates integral constraint regularization approximation of fractional optimal control problem with pointwise state constraint. Firstly the continuous first-order optimality condition is derived. In order to deal with the irregular multiplier, a regularized optimal control problem with finitely many integral state constraints is proposed based on integral constraint regularization method. The error between the original solution and the regularized solution is deduced. Continuous piecewise linear finite element combined with variational discretization approach is used to discretize the regularized optimal control problem. Finite element discretization error combined with the regularization error leads to the final error estimate. Finally, numerical examples are presented to verify the theoretical findings.

Fractional Optimal Control Model and Bifurcation Analysis of Human Syncytial Respiratory Virus Transmission Dynamics

In this paper, the Caputo-based fractional derivative optimal control model is looked at to learn more about how the human respiratory syncytial virus (RSV) spreads. Model solution properties such as boundedness and non-negativity are checked and found to be true. The fundamental reproduction number is calculated by using the next-generation matrix’s spectral radius. The fractional optimal control model includes the control functions of vaccination and treatment to illustrate the impact of these interventions on the dynamics of virus transmission. In addition, the order of the derivative in the fractional optimal control problem indicates that encouraging vaccination and treatment early on can slow the spread of RSV. The overall analysis and the simulated behavior of the fractional optimum control model are in good agreement, and this is due in large part to the use of the MATLAB platform.

Fractional optimal control analysis of Covid-19 and dengue fever co-infection model with Atangana-Baleanu derivative

<abstract><p>A co-infection with Covid-19 and dengue fever has had worse outcomes due to high mortality rates and longer stays either in isolation or at hospitals. This poses a great threat to a country's economy. To effectively deal with these threats, comprehensive approaches to prevent and control Covid-19/dengue fever co-infections are desperately needed. Thus, our focus is to formulate a new co-infection fractional model with the Atangana-Baleanu derivative to suggest effective and feasible approaches to restrict the spread of co-infection. In the first part of this paper, we present Covid-19 and dengue fever sub-models, as well as the co-infection model that is locally asymptotically stable when the respective reproduction numbers are less than unity. We establish the existence and uniqueness results for the solutions of the co-infection model. We extend the model to include a vaccination compartment for the Covid-19 vaccine to susceptible individuals and a treatment compartment to treat dengue-infected individuals as optimal control strategies for disease control. We outline the fundamental requirements for the fractional optimal control problem and illustrate the optimality system for the co-infection model using Pontraygin's principle. We implement the Toufik-Atangana approximating scheme to simulate the optimality system. The simulations show the effectiveness of the implemented strategy in determining optimal vaccination and treatment rates that decrease the cost functional to a minimum, thus significantly decreasing the number of infected humans and vectors. Additionally, we visualize a meaningful decrease in infection cases with an increase in the memory index. The findings of this study will provide reasonable disease control suggestions to regions facing Covid-19 and dengue fever co-infection.</p></abstract>

Novel closed-loop controllers for fractional linear quadratic time-varying systems

A novel method for finding the closed-loop optimal controllers of the finite-time fractional linear quadratic regulator systems is introduced. The necessary optimality conditions for the fractional optimal control problems are obtained. The matrix fractional differential Riccati equation is presented. A new method for solving the fractional differential Riccati equation is given for the first time which can also be applied to the integer order Riccati equation. Several illustrative examples (including a high order system) are presented to show the applicability and capabilities of the method.

A necessary optimality condition for extended weighted generalized fractional optimal control problems

Using the recent weighted generalized fractional order operators of Hattaf, a general fractional optimal control problem without constraints on the values of the control functions is formulated and a corresponding (weak) version of Pontryagin’s maximum principle is proved. As corollaries, necessary optimality conditions for Caputo–Fabrizio, Atangana–Baleanu and weighted Atangana–Baleanu fractional dynamic optimization problems are trivially obtained. As an application, the weighted generalized fractional problem of the calculus of variations is investigated and a new more general fractional Euler–Lagrange equation is given.

An efficient optimization algorithm for nonlinear 2D fractional optimal control problems

An efficient optimization algorithm for nonlinear 2D fractional optimal control problems

A numerical method for solving quadratic fractional optimal control problems

The objective of this article is to present a novel algorithm that can efficiently address fractional quadratic optimal control problems (FQOCPs) through the application of the generalized differential transform method, in conjunction with a Vandermonde matrix. The algorithm’s performance, in terms of solution accuracy, reliability, and efficiency, is exemplified by a range of illustrative examples. This paper introduces an innovative methodology for the numerical resolution of FQOCPs, demonstrating its inherent capabilities and efficacy.

Fixed terminal time fractional optimal control problem for discrete time singular system

Fixed terminal time fractional optimal control problem for discrete time singular system

Existence of optimal control for multi-order fractional optimal control problems

Existence of optimal control for multi-order fractional optimal control problems

A comprehensive review on fractional-order optimal control problem and its solution

Abstract This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems

We introduce a new optimal control problem where the controlled dynamical system depends on multi-order (incommensurate) fractional differential equations. The cost functional to be maximized is of Bolza type and depends on incommensurate Caputo fractional-orders derivatives. We establish continuity and differentiability of the state solutions with respect to perturbed trajectories. Then, we state and prove a Pontryagin maximum principle for incommensurate Caputo fractional optimal control problems. Finally, we give an example, illustrating the applicability of our Pontryagin maximum principle.

Atangana-Baleanu Fractional Dynamics of Predictive Whooping Cough Model with Optimal Control Analysis

Whooping cough, or pertussis, is an infectious disease that causes serious threats to people of all ages, specifically to infant and young children, all over the world. Due to the severe impact on health, it is necessary to construct a mathematical model that can be used to predict future dynamics of the disease, as well as to suggest strategies to eliminate the disease in an optimal way. For this, we constructed a new Atangana–Baleanu fractional model for whooping cough disease to predict the future dynamics of the disease, as well as to suggest strategies to eliminate the disease in an optimal way. We prove that the proposed model has a unique solution that is positive and bounded. To measure the contagiousness of the disease, we determined the reproduction number R0 and used it to examine the local and global stability at equilibrium points that have symmetry. Through sensitivity analysis, we determined parameters of the model that are most sensitive to R0. The ultimate aim of this research was to analyze different disease prevention approaches in order to find the most suitable one. For this, we included the vaccination and quarantine compartments in the proposed model and formulated an optimal control problem to assess the effect of vaccination and quarantine rates on disease control in three distinct scenarios. Firstly, we study the impact of vaccination strategy and conclude the findings with a graphical presentation of the results. Secondly, we examine the impact of quarantine strategy on whooping cough infection and its possible elimination from society. Lastly, we implement vaccination and quarantine strategies together to visualize their combined effect on infection control. In addition to the study of the optimal control problem, we examine the effect of the fractional order on disease dynamics, as well as the impact of constant vaccination and quarantine rates on disease transmission and control. The numerical results reveal that the optimal control strategy with vaccination and quarantine together would be more effective in reducing the spread of whooping cough infection. The implementation of the Toufik–Atangana-type numerical scheme for the solution of the fractional optimal control problem is another contribution of this article.

Piecewise fractional Legendre functions for nonlinear fractional optimal control problems with ABC fractional derivative and non‐smooth solutions

AbstractIn this study, to solve fractional problems with non‐smooth solutions (which include some terms in the form of piecewise or fractional powers), a new category of basis functions called the orthonormal piecewise fractional Legendre functions is introduced. The upper bound of the error of the series expansion of these functions is obtained. Two explicit formulas for computing the Riemann–Liouville and Atangana–Baleanu fractional integrals of these functions are derived. A direct method based on these functions and their fractional integral is proposed to solve a family of optimal control problems involving the ABC fractional differentiation whose solutions are non‐smooth in the above expressed forms. By the proposed technique, solving the original fractional problem turns into solving an equivalent system of algebraic equations. The established method accuracy is studied by solving some examples.

Kinetic Behavior and Optimal Control of a Fractional-Order Hepatitis B Model

The fractional-order calculus model is suitable for describing real-world problems that contain non-local effects and have memory genetic effects. Based on the definition of the Caputo derivative, the article proposes a class of fractional hepatitis B epidemic model with a general incidence rate. Firstly, the existence, uniqueness, positivity and boundedness of model solutions, basic reproduction number, equilibrium points, and local stability of equilibrium points are studied employing fractional differential equation theory, stability theory, and infectious disease dynamics theory. Secondly, the fractional necessary optimality conditions for fractional optimal control problems are derived by applying the Pontryagin maximum principle. Finally, the optimization simulation results of fractional optimal control problem are discussed. To control the spread of the hepatitis B virus, three control variables (isolation, treatment, and vaccination) are applied, and the optimal control theory is used to formulate the optimal control strategy. Specifically, by isolating infected and non-infected people, treating patients, and vaccinating susceptible people at the same time, the number of hepatitis B patients can be minimized, the number of recovered people can be increased, and the purpose of ultimately eliminating the transmission of hepatitis B virus can be achieved.

A study on fractional tumor-immune interaction model related to lung cancer via generalized Laguerre polynomials

BackgroundCancer, a complex and deadly health concern today, is characterized by forming potentially malignant tumors or cancer cells. The dynamic interaction between these cells and their environment is crucial to the disease. Mathematical models can enhance our understanding of these interactions, helping us predict disease progression and treatment strategies.MethodsIn this study, we develop a fractional tumor-immune interaction model specifically for lung cancer (FTIIM-LC). We present some definitions and significant results related to the Caputo operator. We employ the generalized Laguerre polynomials (GLPs) method to find the optimal solution for the FTIIM-LC model. We then conduct a numerical simulation and compare the results of our method with other techniques and real-world data.ResultsWe propose a FTIIM-LC model in this paper. The approximate solution for the proposed model is derived using a series of expansions in a new set of polynomials, the GLPs. To streamline the process, we integrate Lagrange multipliers, GLPs, and operational matrices of fractional and ordinary derivatives. We conduct a numerical simulation to study the effects of varying fractional orders and achieve the expected theoretical results.ConclusionThe findings of this study demonstrate that the optimization methods used can effectively predict and analyze complex phenomena. This innovative approach can also be applied to other nonlinear differential equations, such as the fractional Klein–Gordon equation, fractional diffusion-wave equation, breast cancer model, and fractional optimal control problems.

Characterization Results of Solution Sets Associated with Multiple-Objective Fractional Optimal Control Problems

Characterization Results of Solution Sets Associated with Multiple-Objective Fractional Optimal Control Problems