Fractional Order Optimal Control Research Articles

Fractional-order optimal control model for the equipment management optimization problem with preventive maintenance

Fractional-order optimal control model for the equipment management optimization problem with preventive maintenance

SARS-CoV-2 infection with lytic and non-lytic immune responses: A fractional order optimal control theoretical study

In this research article, we establish a fractional-order mathematical model to explore the infections of the coronavirus disease (COVID-19) caused by the novel SARS-CoV-2 virus. We introduce a set of fractional differential equations taking uninfected epithelial cells, infected epithelial cells, SARS-CoV-2 virus, and CTL response cell accounting for the lytic and non-lytic effects of immune responses. We also include the effect of a commonly used antiviral drug in COVID-19 treatment in an optimal control-theoretic approach. The stability of the equilibria of the fractional ordered system using qualitative theory. Numerical simulations are presented using an iterative scheme in Matlab in support of the analytical results.

Control of COVID-19 dynamics through a fractional-order model

We investigate, through a fractional mathematical model, the effects of physical distance on the SARS-CoV-2 virus transmission. Two controls are considered in our model for eradication of the spread of COVID-19: media education, through campaigns explaining the importance of social distancing, use of face masks, etc., towards all population, while the second one is quarantine social isolation of the exposed individuals. A general fractional order optimal control problem, and associated optimality conditions of Pontryagin type, are discussed, with the goal to minimize the number of susceptible and infected while maximizing the number of recovered. The extremals are then numerically obtained.

Optimal surveillance mitigation of COVID'19 disease outbreak: Fractional order optimal control of compartment model

In present time, the whole world is in the phase of war against the deadly pandemic COVID'19 and working on different interventions in this regard. Variety of strategies are taken into account from ground level to the state to reduce the transmission rate. For this purpose, the epidemiologists are also augmenting their contribution in structuring such models that could depict a scheme to diminish the basic reproduction number. These tactics also include the awareness campaigns initiated by the stakeholders through digital, print media and etc. Analyzing the cost and profit effectiveness of these tactics, we design an optimal control dynamical model to study the proficiency of each strategy in reducing the virulence of COVID'19. The aim is to illustrate the memory effect on the dynamics of COVID'19 with and without prevention measures through fractional calculus. Therefore, the structure of the model is in line with generalized proportional fractional derivative to assess the effects at each chronological change. Awareness about using medical mask, social distancing, frequent use of sanitizer or cleaning hand and supportive care during treatment are the strategies followed worldwide in this fight. Taking these into consideration, the optimal objective function proposed for the surveillance mitigation of COVID'19, is contemplated as the cost function. The effect analysis is supported through graphs and tabulated values. In addition, sensitivity inspection of basic reproduction number is also carried out with respect to different values of fractional index and cost function. Ultimately, social distancing and supportive care of infected are found to be significant in decreasing the basic reproduction number more rapidly.

Implementation of Fractional Optimal Control Problems in Real-World Applications

Abstract In this article, we aim to analyze a mathematical model of tumor growth as a problem of fractional optimal control. The considered fractional-order model describes the interaction of effector-immune cells and tumor cells, including combined chemo-immunotherapy. We deduce the necessary optimality conditions together with implementing the Adomian decomposition method on the suggested fractional-order optimal control problem. The key motive is to perform numerical simulations that shall facilitate us in understanding the behavior of state and control variables. Further, the graphical interpretation of solutions effectively validates the applicability of the present analysis to investigate the growth of cancer cells in the presence of medical treatment.

A hybrid fractional optimal control for a novel Coronavirus (2019-nCov) mathematical model

IntroductionCoronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann–Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo’s fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions. ObjectivesA novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived. MethodsThe numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grünwald-Letnikov nonstandard finite difference method. ResultsThe proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin’s maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grünwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator. ConclusionsThe combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists.

Numerical simulation of fractional order optimal control problem

A simple and straightforward numerical method based on reflection operator is proposed in this paper for solving an optimal control problem (OCP) of continuous- time fractional order singular system (FOSS) with fixed final state end point condition by keeping final time fixed. Singular system (SS) dynamics is described by the fractional order differential equations (FDEs) in the sense of either Riemann-Liouville fractional derivative (RLFD) or Caputo fractional derivative (CFD). By using reflection operator we convert right RLFD to its equivalent left RLFD and then approximate the necessary conditions using Grünwald-Letnikov approximation (GLA). An example is illustrated to check the efficacy of proposed algorithm.

Analysis and Simulation of a Fractional Order Optimal Control Model for HBV

In this paper, we propose a fractional optimal control model of anti-HBV infection based on saturation incidence and logistic proliferation of uninfected cells for the first time. We derive the basic reproduction number R0 and the cytotoxic T lymphocyte immune response reproductive number R1 and give the stable analysis based on R0 and R1. We analyse the optimal control condition and give two optimal control strategies about entecavir monotherapy and combination therapy of traditional Chinese medicine and entecavir with different fractional orders by simulation. The simulation shows that combination therapy may be a good choice in anti-HBV infection therapy. We also compare the objective function values of the optimal control strategies with other constant control strategies; the comparison shows that the optimal therapy can get similar or better treatment effect with less drug dose and side effects.

A fractional power series neural network for solving a class of fractional optimal control problems with equality and inequality constraints

ABSTRACTThis paper solved fractional order optimal control problems, in which the dynamic control system involves integer and fractional order derivatives with equality and inequality constraints. According to the Pontryagin minimum principle (PMP) for fractional optimal control problem (FOCP) with fractional derivative in the Caputo sense and by constructing a suitable error function, an unconstrained minimization problem is defined. In the optimization problem, trial solutions are used for the states, Lagrange multipliers and control functions where these trial solutions are constructed by fractional power series neural network models. An error function is then minimized using a numerical optimization scheme where weight parameters (or coefficients of the series) and biases associated with all neurons are unknown. Some computational simulations are discussed in details.

Optimal synchronization of non-smooth fractional order chaotic systems with uncertainty based on extension of a numerical approach in fractional optimal control problems

In this paper, a control mechanism is presented for optimal synchronization of two non-smooth fractional order chaotic systems with parametric uncertainty based on nonlinear fractional order proportional derivative (NLFPD) controller combined with optimal periodic control signals. Unlike synchronization methods based on FPID controllers, in this approach, optimal tuning of the NLFPD controller and determination of optimal periodic control signals for synchronization process are presented in the form of non-smooth fractional order optimal control (FOC) problems. Optimal periodic control signals are demonstrated as a generalized expansion in the sense of Fourier expansion that is used to accelerate the synchronization process. Using the generalization of a numerical method in nonlinear optimal control problems and the Grunwald-Letnikov(GL) fractional derivative definition, the synchronization problem based on the suggested mechanism is transformed into the form of a smooth FOC problem. By defining a base-time soft switch, a supervisory approach is added to the proposed control mechanism for desired and robust performance against parametric uncertainty in the slave system. Finally, to illustrate the proposed control mechanism, the synchronization of two identical fractional order Chua systems with simulation results is presented. The Results show that using the suggested control mechanism, the synchronization is fast and robust against parametric uncertainty in Chua slave system.

Solution of Some Types for Composition Fractional Order Differential Equations Corresponding to Optimal Control Problems

The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems. The approximate solutions are defined on interval [0,1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.

Fractional-Order Optimal Control of Fractional-Order Linear Vibration Systems with Time Delay

Vibration control of fractional-order linear systems in the presence of time delays has been dealt in this article. Considering a delayed n-degree-of freedom linear structure that is modeled by fractional order equations, a fractional-order optimal control is provided to minimize both control input and output of delayed system via quadratic objective function. To do this, first the fractional order model of system that is subject to time delay is rewritten into a non-delay form through a particular transformation. Then, a fractional order optimal controller is provided using the classical optimal control theory to find an optimal input control. A delayed viscose system is then presented as a practical worked-out example. Numerical simulation results are given to confirm the efficiency of the proposed control method.

Formulation of Euler–Lagrange Equations for Multidelay Fractional Optimal Control Problems

In this paper, a new numerical scheme is proposed for multidelay fractional order optimal control problems where its derivative is considered in the Grunwald–Letnikov sense. We develop generalized Euler–Lagrange equations that results from multidelay fractional optimal control problems (FOCP) with final terminal. These equations are created by using the calculus of variations and the formula for fractional integration by parts. The derived equations are then reduced into system of algebraic equations by using a Grunwald–Letnikov approximation for the fractional derivatives. Finally, for confirming the accuracy of the proposed approach, some illustrative numerical examples are solved.

Solution a Class of Riemann-Liouville Derivative-Composition Fractional Order Optimal Control Problems

Solution a Class of Riemann-Liouville Derivative-Composition Fractional Order Optimal Control Problems

Numerical Solution of 1D and 2D Fractional Optimal Control of System via Bernoulli Polynomials

In this current paper, we present a numerical method to solve a class of fractional optimal control problems of system described by integer-fractional derivative. The fractional derivative is explained in the Caputo sense and a new formulation for the Bernoulli Caputo fractional derivative operational matrix has been achieved for the first time. Then this matrix and operational matrix of multiplication of these polynomials are applied to solve the fractional-order optimal control problems by the direct method. As a matter of fact, the functions of the problem are approximated by Bernoulli polynomials with unknown coefficients in the functional and conditions. Therefore, a fractional optimal control problem is converted to an unconstrained optimization problem and then it could be solved via Newton’s iterative method. Also we demonstrate that if the number of Bernoulli basis is increased, this method is convergent. The effectiveness and validity of the new methodology are illustrated by four examples, in addition our findings in comparison with the existing results show the preference of presented method.

Comparative studies for the fractional optimal control in transmission dynamics of West Nile virus

In this paper, optimal control for a novel West Nile virus (WNV) model of fractional order derivative is presented. The proposed model is governed by a system of fractional differential equations (FDEs), where the fractional derivative is defined in the Caputo sense. An optimal control problem is formulated and studied theoretically using the Pontryagin maximum principle. Two numerical methods are used to study the fractional-order optimal control problem. The methods are, the iterative optimal control method (OCM) and the generalized Euler method (GEM). Positivity, boundedness and convergence of the IOCM are studied. Comparative studies between the proposed methods are implemented, it is found that the IOCM is better than the GEM.

Suboptimal control of fractional-order dynamic systems with delay argument

In this paper, an efficient linear programming formulation is proposed for a class of fractional-order optimal control problems with delay argument. By means of the Lagrange multiplier in the calculus of variations and using the formula for fractional integration by parts, the Euler–Lagrange equations are derived in terms of a two-point fractional boundary value problem including an advance term as well as the delay argument. The derived equations are then reduced into a linear programming problem by using a Grünwald–Letnikov approximation for the fractional derivatives and introducing a new transformation in the calculus of variations. The new scheme is also effective for the delay fractional optimal control problems influenced by the external persistent disturbances. Numerical simulations and comparative results verify that the proposed approach is efficient and easy to implement.

Fractional order optimal control problems with free terminal time

We consider fractional order optimal control problems in whichthe dynamic control system involves integerand fractional order derivatives and the terminal time is free.Necessary conditions for a state/control/terminal-time tripletto be optimal are obtained. Situations with constraints presentat the end time are also considered.Under appropriate assumptions, it is shown that theobtained necessary optimality conditions become sufficient.Numerical methods to solve the problems are presented,and some computational simulations are discussed in detail.

A Numerical Scheme to Solve Fractional Optimal Control Problems

We review recent results obtained to solve fractional order optimal control problems with free terminal time and a dynamic constraint involving integer and fractional order derivatives. Some particular cases are studied in detail. A numerical scheme is given, based on expansion formulas for the fractional derivative. The efficiency of the method is illustrated through examples.

Optimal Control of a Fractional-Order HIV-Immune System With Memory

The fact that fractional-order models possess memory leads to modeling a fractional-order HIV-immune system. We discuss the necessary conditions for the optimality of a general fractional optimal control problem whose fractional derivative is described in the Caputo sense. Using an objective function that minimizes the infectious viral load and count of infected T cells, the optimal control problem is solved for the fractional-order optimality system with minimal dosage of anti-HIV drugs and the effects of mathematically optimal therapy are demonstrated. Simulation results show that the fractional-order optimal control scheme can achieve improved quality of the treatment.