Block Algorithm for Solving Optimal Control Problems in Ordinary Differential Equation

Abstract

Optimal control problems involve determining a control function that optimizes a specified 
performance index while satisfying a set of dynamical system constraints. These problems 
frequently arise in engineering, economics, and applied sciences, where efficient and stable 
numerical techniques are required for accurate solutions. Traditional single–step or sequential 
integration methods, commonly used for solving the resulting differential equations, often 
suffer from numerical instability, slow convergence rates, and high computational cost, 
particularly when applied to complex or stiff systems. In this study, a block algorithm based 
on Linear Multistep Methods (LMMs) is developed to overcome these limitations. The 
approach partitions the computational interval into sub-blocks, enabling the simultaneous 
computation of solution points within each block. The LMMs are constructed using polynomial 
approximation techniques to derive a continuous approximate solution, which is subsequently 
implemented in block form to enhance computational efficiency and stability. This formulation 
allows multiple solution values to be obtained concurrently, thereby reducing error 
propagation and improving convergence behavior. The theoretical properties of the proposed 
block algorithm are rigorously analyzed. Stability analysis demonstrates that the method 
satisfies the fundamental requirements of zero-stability, consistency, and convergence. To 
solve the optimal control problem, Pontryagin’s Maximum Principle is employed to derive the 
necessary optimality conditions, leading to a coupled system of state and adjoint differential 
equations. The developed block algorithm is implemented using a well-structured MATLAB 
program and applied to several benchmark optimal control problems. Numerical experiments 
indicate that the method provides accurate and stable solutions with improved computational 
efficiency compared to conventional sequential methods. The results demonstrate that the 
proposed block algorithm constitutes an effective and reliable numerical technique for solving 
optimal control problems.