  This study focuses on the numerical discretization methods for the
continuous-time discounted linear-quadratic optimal control problem (LQ-OCP)
with time delays. By assuming piecewise constant inputs, we formulate the
discrete system matrices of the discounted LQ-OCPs into systems of differential
equations. Subsequently, we derive the discrete-time equivalent of the
discounted LQ-OCP by solving these systems. This paper presents three numerical
methods for solving the proposed differential equations systems: the
fixed-time-step ordinary differential equation (ODE) method, the step-doubling
method, and the matrix exponential method. Our numerical experiment
demonstrates that all three methods accurately solve the differential equation
systems. Interestingly, the step-doubling method emerges as the fastest among
them while maintaining the same level of accuracy as the fixed-time-step ODE
method.
