A discretized scheme, Discretized Continuous Algorithm (DCA), for solving constrained quadratic optimal control problems was developed to ease the computational cumbersomeness inherent in some existing algorithms, particularly, the Function Space A lgorithm (FSA) by replacing the integral by a series of summation. In order to accomplish this numerical scheme, we resort to a finite approximation of it by discretizing its time interval and using finite difference method for its differential constraint. Using the penalty function method, an unconstrained formulation of the problem was obtained. With the bilinear form expression of the problem, an associated operator was constructed which aided the scheme for the solution of such class of problems. A sample problem was examined to test the effectiveness of the scheme as to convergence with relation to other existing schemes such as Extended Conjugate Gradient Method (ECGM), Multiplier Imbedding Extended Conjugate Gradient Method (MECGM) and Function Space Algorithm (FSA) for solving penalized functional of optimal control problem characterized by non-linear integral quadratic nature.

This study examines the effect of variable Winkler foundation on the natural frequencies of a prestressed nonuniform Rayleigh beam. In this work, the elastic coefficients of the foundations are assumed to vary along the length direction of the beam. A semi-analytical approach known as Differential Transform Method (DTM) is applied to the non-dimensional form of the governing equations of motion of the prestressed non-uniform Rayleigh beam and a set of recursive algebraic equations are obtained. Evaluating these derived equations and using some computer codes written and implemented in MAPLE 18, the non-dimensional frequencies and the associated mode shapes of the beam are obtained. The effects of variable Winkler foundation variations and axial force for various values of the slenderness ratio on the non-dimensional frequencies are investigated. The clamped-clamped and simply supported boundary conditions are considered to illustrate the accuracy and efficiency of this method. Finally, the results obtained are validated and are found to compare favorably well with those in the open literature.

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