Numerical solution of a class of two‐dimensional quadratic optimal control problems by using Ritz method

Mamehrashi, Kamal; Yousefi, S. A.

doi:10.1002/oca.2191

Cited by 17 publications

(13 citation statements)

References 32 publications

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“…Example Consider the following 2D FOCP with the dynamical system constraint of Darboux equation. The dynamical processes in gas absorption, air drying, and water steam heating can be described by using Darboux equation

min J false[u false] = \frac{1}{2} \int_{0}^{3} \int_{0}^{3} ([], f^{T} false(x, t false) Q f false(x, t false) + u^{T} false(x, t false) R u false(x, t false)) normald x normald t,

subject to

\frac{\partial^{2} f}{\partial x \partial t} false(x, t false) = - \frac{\partial^{α} f}{\partial x^{α}} false(x, t false) - 3 \frac{\partial^{β} f}{\partial t^{β}} false(x, t false) + 0.2 f false(x, t false) + 0.3 u false(x, t false),

f false(x, 0 false) = e^{- 3 x} \cos false(2 π x false), f false(0, t false) = e^{- 2 t},

where

Q = ([], \begin{matrix} center \\ array 2 & array 1 \\ array 1 & array 1 \end{matrix})

and R = 1.…”

Section: Illustrative Test Problemsmentioning

confidence: 99%

“…By substituting the approximate state function and control function into the functional J and calculating the integral by using 2D LG quadrature rule by taking l = M , l ′ = M ′ , a nonlinear optimization problem is obtained. In Table , we compare values of

\overset{J}{true}

obtained using the present method and the other methods for α = β = 1. Figure illustrates the values of state and control function for

α = β = 1 and γ = \frac{1}{2}

with k = k ′ = 2; M = M ′ = 2.…”

Section: Illustrative Test Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Rahimkhani

Ordokhani

2018

Optim Control Appl Methods

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In this paper, a method for finding an approximate solution of a class of 2D fractional optimal control problems with fractional-order dynamical system is discussed. In the proposed method, the fractional derivative is expressed in the Caputo sense. The method consists of expanding unknown functions as the elements of two-dimensional (2D) Müntz-Legendre wavelets. The 2DMüntz-Legendre wavelets are constructed and their properties are presented.The operational matrix of fractional-order integration for these wavelets is utilized to reduce the solution of 2D fractional optimal control problem to an optimization problem, which can then be solved easily. Some results concerning the error analysis are obtained. Finally, two illustrative test problems are included to demonstrate the validity and applicability of the technique. Moreover, our achievements are compared with the previous results to show the superiority of the proposed method. KEYWORDSCaputo derivative, fractional optimal control problem, numerical solution, Riemann-Liouville fractional integral operator, wavelet method INTRODUCTIONNowadays, fractional calculus have a wide range of applications in fields of fluid-dynamic traffic, 1 frequency dependent damping behavior of various viscoelastic materials, 2 colored noise, 3 solid mechanics, 4 economics, 5 signal processing, 6 and control theory. 7 Therefore, many researchers have been interested in finding the approximate solutions of these problems, and some of these methods include spectral Tau method, 8 finite difference method, 9 wavelet method, 10 least square approximation method, 11 finite element method, 12 spline collocation method, 13 fractional wavelet method, 14 shifted fractional-order Jacobi orthogonal functions, 15 Jacobi Tau method, 16,17 and Legendre Tau method. 18 An optimal control problem (OCP) is defined to minimize the functional over an admissible set of control functions subject to dynamic constraints on the state and control variables. A fractional OCP (FOCP) is an optimal control problem in which the performance index or the differential equations governing the dynamics of the system or both contains at least one fractional-order derivative term. In recent the years, several numerical methods have been devoted to solve of one-dimensional FOCPs. We list here some of these numerical methods, as follows: • Eigen functions method (Agrawal 19 ); • Quadratic numerical scheme (Agrawal 20 ); • Rational approximation method (Tricaud and Chen 21 ); • Legendre multiwavelet collocation method (Yousefi et al 22 ); 1916

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min J false[u false] = \frac{1}{2} \int_{0}^{3} \int_{0}^{3} ([], f^{T} false(x, t false) Q f false(x, t false) + u^{T} false(x, t false) R u false(x, t false)) normald x normald t,

subject to

\frac{\partial^{2} f}{\partial x \partial t} false(x, t false) = - \frac{\partial^{α} f}{\partial x^{α}} false(x, t false) - 3 \frac{\partial^{β} f}{\partial t^{β}} false(x, t false) + 0.2 f false(x, t false) + 0.3 u false(x, t false),

f false(x, 0 false) = e^{- 3 x} \cos false(2 π x false), f false(0, t false) = e^{- 2 t},

where

Q = ([], \begin{matrix} center \\ array 2 & array 1 \\ array 1 & array 1 \end{matrix})

and R = 1.…”

Section: Illustrative Test Problemsmentioning

confidence: 99%

\overset{J}{true}

obtained using the present method and the other methods for α = β = 1. Figure illustrates the values of state and control function for

α = β = 1 and γ = \frac{1}{2}

with k = k ′ = 2; M = M ′ = 2.…”

Section: Illustrative Test Problemsmentioning

confidence: 99%

Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Rahimkhani

Ordokhani

2018

Optim Control Appl Methods

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“…The minimal cost function reported by Li et al by taking 30 discrete points in both variables with integer order derivative ( = = 1) was J 3,3 = 64.2611. Recently, for the same problem, Mamehrashi et al [29] achieved the approximated value J = 1.289178 by applying the Ritz method for m = n = 3.…”

Section: Examplementioning

confidence: 99%

“…In [25,26], multidimensional chaotic autonomous systems are also described. The two-dimensional optimal control with discrete-time and continuous-time for both linear and nonlinear systems can be found in the monographs [27][28][29]. The two-dimensional discrete-time systems of fractional order were first introduced by Kaczorek [30] and developed in the monographs [31,32].…”

Section: Introductionmentioning

confidence: 99%

The Use of the Ritz Method and Laplace Transform for Solving 2D Fractional‐Order Optimal Control Problems Described by the Roesser Model

Nemati

Mamehrashi

2018

Asian Journal of Control

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In this article a numerical solution is presented for a class of two-dimensional fractional-order optimal control problems (2D-FOOCPs) with one input and two outputs. To implement the numerical method, the Legendre polynomial basis is used with the aid of the Ritz method and the Laplace transform. By taking the Ritz method as a basic scheme into account and applying a new constructed fractional operational matrix to estimate the fractional and integer order derivatives of the basis, the given 2D-FOOCP is reduced to a system of algebraic equations. One of the advantages of the proposed method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, satisfactory results are obtained in just a small number of polynomials order. The convergence of the method is extensively investigated and finally two illustrative examples are included to show the validity and applicability of the novel proposed technique in the current work.

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“…In Tsai et al (2002) and Li et al (2002), the authors have considered some continuous-time performance index problems suited to continuous time 2D system of the Roesser's model and they have used the discretization approach for solving the problems. However, recently the Ritz method has been employed in Mamehrashi and Yousefi (2016) for a class of 2DOCPs based on the Legendre polynomials. The aim of the current paper is to study the systems similar to the problems in Mamehrashi and Yousefi (2016) but with distributed and boundary controls.…”

Section: Introductionmentioning

confidence: 99%

On the numerical scheme of a 2D optimal control problem with the hyperbolic system via Bernstein polynomial basis

Mamehrashi

Yousefi

Soltanian

2018

Transactions of the Institute of Measurement and Control

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A numerical method for solving a 2D optimal control problem (2DOCP) governed by a linear time-varying constraint is presented in this paper. The method is based upon the Bernstein polynomial basis. The properties of Bernstein polynomial functions are presented. These properties, together with the Ritz method, are then utilized to reduce the given 2DOCP to the solution of an algebraic system of equations. By solving this system, the solution of the proposed problem is achieved. The main advantage of this scheme is that the approximate solutions satisfy all initial and boundary conditions of the problem. We extensively discuss the convergence of the method. Finally, an illustrative example is included to demonstrate the validity and applicability of the new technique.

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Numerical solution of a class of two‐dimensional quadratic optimal control problems by using Ritz method

Cited by 17 publications

References 32 publications

Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

Numerical solution a class of 2D fractional optimal control problems by using 2D Müntz‐Legendre wavelets

The Use of the Ritz Method and Laplace Transform for Solving 2D Fractional‐Order Optimal Control Problems Described by the Roesser Model

On the numerical scheme of a 2D optimal control problem with the hyperbolic system via Bernstein polynomial basis

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