Partial conjugate gradient methods for a class of optimal control problems

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 of
s+1
conjugate 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 exhibits
s+1
- step superlinear convergence rate. Furthermore, it solves a linear-quadratic problem in
s+1
steps as compared with the
m.N
steps (
m
= control space dimension,
N
= number of stages) required by the ordinary conjugate gradient method.