NUMERICAL SOLUTION TO ONE-DIMENSIONAL MULTI-DELAY SYSTEM

USING MODIFIED ALTERNATING DIRECTION METHOD OF MULTIPLIERS

Olusegun Olotu and Kazeem A. Dawodu

Dept. of Mathematical Sciences, Federal University of Technology Akure, Nigeria

Abstract

This research paper presents an Algorithm for the numerical solution of the Optimal Control model constrained by Ordinary Differential Equation with multiple constant delays on the state and control variables using the Modified Alternating Direction Method of Multipliers (MADMM). The constraint and objective functionals of the derived optimal control model were discretized using the Composite Simpson’s Methods and then re-reformulated into a discrete form amenable to the M-ADMM structured for convex optimization problems. The Gauss Seidel acceleration variant was introduced to speed up the rate of convergence of the M-ADMM algorithm. The spectral analysis was carried out on the matrix operators to ascertain their well-posessness for the M-ADMM while the convergence analysis of the algorithm was carried out to ascertain its convergence to a solution at a faster rate. Hypothetical example was carried out to test the efficiency and degree of accuracy of the algorithm. The result was found favorable and comparable with those of existing numerical algorithms.

Keywords: Optimal Control model, Modified Alternating Direction Method of Multipliers, Multiple Constant Delays, Convex Optimization, Composite Simpson’s, Matrix Operators, Consistency and Stability analyses.