Solving optimal control problems with integral equations or integral equations - differential with the help of cubic B-spline scaling functions and wavelets

In this paper, a numerical method based on cubic B-spline scaling functions and wavelets for solving optimal control problems with the dynamical system of the integral equation or the differential-integral equation is discussed. The Operational matrices of derivative and integration of the product of two cubic B-spline wavelet vectors, collocation method and Gauss-Legendre integration rule for the discretization of the continuous optimal control problem and its transformation into a problem of non-linear programming is used. The convergence of control and state functions and the performance index of the optimal approximation of the proposed method and also the upper bound of the error are obtained. Illustrative examples show the effectiveness, accuracy, and usefulness of the proposed idea.