In this paper, we examine the computational aspects of a certain class of discrete-time optimal control problems. We propose and analyze two partial conjugate gradient algorithms which operate in cycles ofs+1conjugate gradient steps (s leq n= state space dimension). The algorithms are motivated by the special form of the Hessian matrix of the cost functional. The first algorithm exhibits a linear convergence rate and offers some advantages over steepest descent in certain cases such as when the system is unstable. The second algorithm requires second-order information with respect to the control variables at the beginning of each cycle and exhibitss+1- step superlinear convergence rate. Furthermore, it solves a linear-quadratic problem ins+1steps as compared with them.Nsteps (m= control space dimension,N= number of stages) required by the ordinary conjugate gradient method.