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This study aims to generate a novel set of basis functions called the orthonormal piecewise Chelyshkov functions to solve a certain category of optimal control problems whose dynamical system is go...

In the present paper, we apply the fractional-order Bessel wavelets (FBWs) for solving optimal control problems with variable-order (VO) fractional dynamical system. The VO fractional derivative op...

In this article, a pair of wavelets for Hermite cubic spline bases are presented. These wavelets are in C1 and supported on [−1,1]. These spline wavelets are then adapted to the interval [0,1] and ...

A numerical method for solving time-varying bilinear singular systems is presented. The method is based upon hybrid functions approximations. The properties of hybrid functions consisting of block-pulse functions and Legendre polynomials are presented. The operational matrices of integration and product are then utilized to reduce the solution of time-varying singular bilinear systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

An operational matrix of integration P based on sine and cosine wavelets is presented. A general procedure for forming this matrix is given. The matrix P is then used to reduce a variational problem to the solution of algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.

The optimal control of linear systems with quadratic cost functional with control inequality constraints is obtained by a hybrid function approximations. The properties of hybrid functions that are the combinations of block-pulse functions and Legendre polynomials are presented, and the operational matrix of integration together with the cross-product of two hybrid function vectors are used to reduce the optimal control problem to a set of simultaneous nonlinear equations. An illustrative example is included to demonstrate the validity and applicability of the technique

The optimal control of linear systems with quadratic cost functional is considered. The method is based on shifted Legendre approximation of each state variable that converts the linear quadratic problem to a system of linear algebraic equations. An illustrative example is included to demonstrate the validity and applicability of the technique.

The solution of a linear two-point boundary value problem with time-varying coefficients is obtained by Fourier series approximation. Properties of Fourier series are first briefly discussed and the operational matrix of integration is presented. A transformation matrix relating the back vector to the current time vector together with the operational matrix are utilized to reduce the system to a set of linear algebraic equations and thereby obtain the approximate solution. Illustrative examples are included to demonstrate the validity and applicability of the technique.

A method for solving the linear ordinary differential equations of the two-point boundary-value problem using polynomial series is discussed. The linear ordinary differential equation of the two-point boundary-value problems are reduced to the linear functional differential equation of the initial value problem. It is known that any polynomial series basis vector can be transformed to Taylor polynomials by the use of a suitable transformation and Taylor series have the simplest operational properties. The solution of linear two-point boundary-value problems via Taylor series is first obtained and this solution using other polynomial series is then calculated by transforming the properties of the Taylor series to other polynomial series. The method is simple and convenient for digital computation. Illustrative examples are given.

A method for finding the optimal control of linear distributed-parameter systems using polynomial series is discussed. It is known that any polynomial series basis vector can be transformed into a Taylor polynomial by the use of a suitable transformation. In this paper, the optimal control of a distributed-parameter system is simplified into the solution of a linear two-point boundary value problem, and, as a result, the optimal control is obtained via a Taylor series. It is shown that the implementation of Taylor series for this problem involves the use of an ill-conditioned matrix commonly known as the Hilbert matrix. The optimal control of linear distributed-parameter systems using other polynomial series is then calculated by transforming the properties of the Taylor series into other polynomial series. The formulation is straightforward and convenient for digital computation. An illustrative example is given.

The operational matrix or integration of a Jacobi vector whose elements are Jacobi polynomials is introduced and applied to solve variational problems by a direct method. Illustrative examples are given to demonstrate that only a small number of shifted-Jacobi polynomials are needed to obtain an accurate solution.

A method for finding an approximate solution of a linear two-point boundary value problem with time-varying coefficients is proposed. Properties of Taylor series are first briefly presented and the transformation matrix relating the back vector to the current time vector together with the operation matrix of integration for Taylor series are utilized to solve the linear two-point boundary value problems. The method is computationally very attractive and applications are demonstrated through illustrative examples.