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

2018 ◽  
Vol 13 (6) ◽  
Author(s):  
Sohrab Effati ◽  
Seyed Ali Rakhshan ◽  
Samane Saqi
Keyword(s):  
Optimal Control ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Control Problems ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
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.

A Quadratic Numerical Scheme for Fractional Optimal Control Problems

2007 ◽  
Vol 130 (1) ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Numerical Scheme ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Classical Solutions ◽  
Control Problems ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

This paper presents a quadratic numerical scheme for a class of fractional optimal control problems (FOCPs). The fractional derivative is described in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of fractional differential equations. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic systems. The formulation is used to derive the control equations for a quadratic linear fractional control problem. For a linear system, this method results into a set of linear simultaneous equations, which can be solved using a direct or an iterative scheme. Numerical results for a FOCP are presented to demonstrate the feasibility of the method. It is shown that the solutions converge as the number of grid points increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs.

A numerical approach for solving fractional optimal control problems with mittag-leffler kernel

2021 ◽  
pp. 107754632110169
Author(s):  
Hossein Jafari ◽  
Roghayeh M Ganji ◽  
Khosro Sayevand ◽  
Dumitru Baleanu
Keyword(s):  
Optimal Control ◽  
Approximate Solution ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Numerical Approach ◽  
Control Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this work, we present a numerical approach based on the shifted Legendre polynomials for solving a class of fractional optimal control problems. The derivative is described in the Atangana–Baleanu derivative sense. To solve the problem, operational matrices of AB-fractional integration and multiplication, together with the Lagrange multiplier method for the constrained extremum, are considered. The method reduces the main problem to a system of nonlinear algebraic equations. In this framework by solving the obtained system, the approximate solution is calculated. An error estimate of the numerical solution is also proved for the approximate solution obtained by the proposed method. Finally, some illustrative examples are presented to demonstrate the accuracy and validity of the proposed scheme.

Numerical solution for fractional optimal control problems by Hermite polynomials

2020 ◽  
pp. 107754632093312
Author(s):  
Ayatollah Yari
Keyword(s):  
Optimal Control ◽  
Numerical Method ◽  
Optimal Control Problems ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Control Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this study, a numerical method based on Hermite polynomial approximation for solving a class of fractional optimal control problems is presented. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Operational matrices of integration by using such known formulas as Caputo and Riemann–Liouville operators for computing fractional derivatives and integration of polynomials is introduced and used to reduce the problem of a system of algebraic equations. The convergence of the proposed method is analyzed, and the error upper bound for the operational matrix of the fractional integration is obtained. To confirm the validity and accuracy of the proposed numerical method, three numerical examples are presented along with a comparison between our numerical results and those obtained using Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

An iterative approach for solving fractional optimal control problems

2016 ◽  
Vol 24 (1) ◽  
pp. 18-36 ◽  
Author(s):  
Ali Alizadeh ◽  
Sohrab Effati
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Necessary Optimality Conditions ◽  
Variational Iteration Method ◽  
Successive Approximations ◽  
Classical Solutions ◽  
Control Problems ◽  
Lagrange Equations ◽  
Fractional Optimal Control ◽  
Fractional Optimal Control Problems

In this work, the variational iteration method (VIM) is used to solve a class of fractional optimal control problems (FOCPs). New Lagrange multipliers are determined and some new iterative formulas are presented. The fractional derivative (FD) in these problems is in the Caputo sense. The necessary optimality conditions are achieved for FOCPs in terms of associated Euler–Lagrange equations and then the VIM is used to solve the resulting fractional differential equations. This technique rapidly provides the convergent successive approximations of the exact solution and the solutions approach the classical solutions of the problem as the order of the FDs approaches 1. To achieve the solution of the FOCPs using VIM, four illustrative examples are included to demonstrate the validity and applicability of the proposed method.

Fractional Optimal Control of Continuum Systems

2009 ◽  
Vol 131 (2) ◽  
Author(s):  
X. W. Tangpong ◽  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Numerical Scheme ◽  
Numerical Technique ◽  
Fractional Integration ◽  
Scalar Case ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Time Step ◽  
Fractional Optimal Control ◽  
Continuum Systems

This paper presents a formulation and a numerical scheme for fractional optimal control (FOC) of a class of continuum systems. The fractional derivative is defined in the Caputo sense. The performance index of a fractional optimal control problem is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a partial fractional differential equation. The scheme presented relies on reducing the equations of a continuum system into a set of equations that have no space parameter. Several strategies are pointed out for this task, and one of them is discussed in detail. The numerical scheme involves discretizing the space domain into several segments, and expressing the spatial derivatives in terms of variables at spatial node points. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain the Euler–Lagrange equations for the problem. The numerical technique presented in the work of Agrawal (2006, “A Formulation and a Numerical Scheme for Fractional Optimal Control Problems,” Proceedings of the Second IFAC Conference on Fractional Differentiations and Its Applications, FDA ‘06, Porto, Portugal) for the scalar case is extended for the vector case. In this method, the FOC equations are reduced to the Volterra type integral equations. The time domain is also discretized into a number of subintervals. For the linear case, the numerical technique results in a set of algebraic equations that can be solved using a direct or an iterative scheme. An example problem is solved for various orders of fractional derivatives and different spatial and temporal discretizations. For the problem considered, only a few space grid points are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other continuum systems.

Comparison on wavelets techniques for solving fractional optimal control problems

2016 ◽  
Vol 24 (6) ◽  
pp. 1185-1201 ◽  
Author(s):  
PK Sahu ◽  
S Saha Ray
Keyword(s):  
Optimal Control ◽  
Performance Index ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Algebraic Equations ◽  
Fractional Optimal Control ◽  
Dynamic Constraints ◽  
Orthonormal Wavelets ◽  
Fractional Optimal Control Problems ◽  
Wavelet Methods

This paper presents efficient numerical techniques for solving fractional optimal control problems (FOCP) based on orthonormal wavelets. These wavelets are like Legendre wavelets, Chebyshev wavelets, Laguerre wavelets and Cosine And Sine (CAS) wavelets. The formulation of FOCP and properties of these wavelets are presented. The fractional derivative considered in this problem is in the Caputo sense. The performance index of FOCP has been considered as function of both state and control variables and the dynamic constraints are expressed by fractional differential equation. These wavelet methods are applied to reduce the FOCP as system of algebraic equations by applying the method of constrained extremum which consists of adjoining the constraint equations to the performance index by a set of undetermined Lagrange multipliers. These algebraic systems are solved numerically by Newton's method. Illustrative examples are discussed to demonstrate the applicability and validity of the wavelet methods.

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