References
The indirect approach to the continuous-time optimal control problem (OCP) formulates the necessary conditions of optimality as a two-point boundary value problem (TP-BVP), which generally requires numerical methods. The direct approach to the continuous-time OCP relies heavily on numerical methods too, namely the methods for solving nonlinear programs (NLP) and methods for solving ordinary differential equations (ODE). Numerical methods for both approaches share a lot of common principles and tools, and these are collectively presented in the literature as called numerical optimal control. A recommendable (and freely online available) introduction to these methods is [1]. Shorter version of this is in chapter 8 of [2], which is also available online. A more comprehensive treatment is in [3].
Some survey papers such as [4] and [5] can also be useful, although now primarily as historical accounts. Similarly with the classics [6] and [7], which cover the indirect approach only.
Another name under which the numerical methods for the direct approach are presented is trajectory optimization. There are quite a few tutorials and surveys such as [8] and [9].
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