Mohsen Razzaghi

The pseudospectral Legendre method for discretizing optimal control problems

This paper presents a computational technique for optimal control problems including state and control inequality constraints. The technique is based on spectral collocation methods used in the solution of differential equations. The system dynamics are collocated at Legendre-Gauss-Lobatto points. The derivative x/spl dot/(t) of the state x(t) is approximated by the analytic derivative of the corresponding interpolating polynomial. State and control inequality constraints are collocated at Legendre-Gauss-Lobatto nodes. The integral involved in the definition of the performance index is discretized based on the Gauss-Lobatto quadrature rule. The optimal control problem is thereby converted into a mathematical programming program. Thus existing, well-developed optimization algorithms may be used to solve the transformed problem. The method is easy to implement, capable of handling various types of constraints, and yields very accurate results. Illustrative examples are included to demonstrate the capability of the proposed method, and a comparison is made with existing methods in the literature. >

The Legendre wavelets operational matrix of integration

An operational matrix of integration P based on Legendre wavelets is presented. A general procedure for forming this matrix is given. Illustrative examples are included to demonstrate the validity and applicability of the matrix P.

Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations

A numerical method for solving the nonlinear Volterra-Fredholm integral equations is presented. The method is based upon Legendre wavelet approximations. The properties of Legendre wavelet are first presented. These properties together with the Gaussian integration method are then utilized to reduce the Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Legendre wavelets direct method for variational problems

A direct method for solving variational problems using Legendre wavelets is presented. An operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Optimal control of linear delay systems via hybrid of block-pulse and Legendre polynomials

A method for finding the optimal control of a linear time varying delay system with quadratic performance index is discussed. The properties of the hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. The operational matrices of integration, delay and product are utilized to reduce the solution of optimal control to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

RATIONAL CHEBYSHEV TAU METHOD FOR SOLVING HIGHERORDER ORDINARY DIFFERENTIAL EQUATIONS

An approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev (RC) tau method. The operational matrices of the derivative and product of RC functions are presented. These matrices together with the tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Application of the Adomian decomposition method for the Fokker-Planck equation

In this work we will discuss the solution of an initial value problem of parabolic type. The main objective is to propose an alternative method of solution, one not based on finite difference or finite element or spectral methods. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the Fokker-Planck equation and some similar equations. This method can successfully be applied to a large class of problems. The Adomian decomposition method needs less work in comparison with the traditional methods. This method decreases considerable volume of calculations. The decomposition procedure of Adomian will be obtained easily without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent series with easily computed components. In this work we are concerned with the application of the decomposition method for the linear and nonlinear Fokker-Planck equation. To give overview of methodology, we have presented several examples in one and two dimensional cases.

Rational Chebyshev tau method for solving Volterra's population model

An approximate method for solving Volterras population model for population growth of a species in a closed system is proposed. Volterras model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. The approach is based on a rational Chebyshev tau method. The Volterras population model is first converted to a nonlinear ordinary differential equation. The operational matrices of derivative and product of rational Chebyshev functions are presented. These matrices together with the tau method are then utilized to reduce the solution of the Volterras model to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results.

Short communication: A collocation-type method for linear quadratic optimal control problems

This communication presents a spectral method for solving time-varying linear quadratic optimal control problems. Legendre-Gauss-Lobatto nodes are used to construct the mth-degree polynomial approximation of the state and control variables. The derivative x(t) of the state vector x(t) is approximated by the analytic derivative of the corresponding interpolating polynomial. The performance index approximation is based on Gauss-Lobatto integration. The optimal control problem is then transformed into a linear programming problem. The proposed technique is easy to implement, efficient and yields accurate results. Numerical examples are included and a comparison is made with an existing result.

Legendre wavelets method for the solution of nonlinear problems in the calculus of variations

A numerical technique for solving the nonlinear problems of the calculus of variations is presented. Two nonlinear examples are considered. In the first example, the brachistochrone problem is formulated as a nonlinear optimal control problem, and in the second example, a higher-order nonlinear problem is given. An operational matrix of integration is introduced and is utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement, and yields very accurate results.