The development of a Conjugate Gradient Method (CGM) algorithm has immensely contributed to the solution of optimization problems due to its quadratic convergence property. Hinging on the CGM algorithm convergence property for optimization problems, this paper stresses the construction of a control operator that is introduced to a CGM algorithm that makes it amenable to solve optimal control problems. The introduction of the operator in CGM algorithm gave rise to an Extended Conjugate Gradient Method (ECGM) algorithm adopted for solving Continuous Time Regulator Problems (CLRP) that is constrained by delay differential equation. Unlike the similar control operators constructed in time past designated to solve either the Mayer or the Lagrange cost forms of the CLRP, the adoption of this control operator in ECGM algorithm will effectively and robustly takes care of the Mayer form, the Lagrange cost forms, and the Bolza cost form of the CLRP. The resulting algorithm on the introduction of the control operator to ECGM was tried on a number CLRP exhibiting an improved convergence profile over the classical methods hence widening the range of problems to which the ECGM algorithm can be employed to solve.

Index Terms: Delay Differential State Equation, Extended Conjugate Gradient Method, Linear Operator, Optimal Control, Regulator Problem.