Naive implementations of Newton's method for unconstrained N-stage discrete-time optimal control problems with Bolza objective functions tend to increase in cost like N3 as N increases. However, if the inherent recursive structure of the Bolza problem is properly exploited, the cost of computing a Newton step will increase only linearly with N. The efficient Newton implementation scheme proposed here is similar to Mayne's DDP (differential dynamic programming) method but produces the Newton step exactly, even when the dynamical equations are nonlinear. The proposed scheme is also related to a Riccati treatment of the linear, two-point boundary-value problems that characterize optimal solutions. For discrete-time problems, the dynamic programming approach and the Riccati substitution differ in an interesting way; however, these differences essentially vanish in the continuous-time limit.
- dynamic programming
- Newton's method
- Unconstrained optimal control
ASJC Scopus subject areas
- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics