References
The indirect approach to the continuous-time optimal control problem (OCP), which formulates the necessary conditions of optimality as a two-point boundary value problem (TP-BVP), generally calls for numerical methods that iterate over trajectories. Besides some simple textbook problems allowing closed-form solutions, an regular exception is the LQR problem, for which a proportional feedback control is determined by solving (albeit also numerically) the continuous-time algebraic Riccati equation (CARE).
The direct approach to the continuous-time OCP proceeds by transcribing (discretizing) the continuous-time problem into a nonlinear programming problem (NLP), which is then solved numerically for the optimal (discretized) trajectories.
Numerical methods for both direct and indirect approaches share a lot of common principles and tools. 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 [2, Ch. 8], which is also freely available online. A more comprehensive treatment by another authors 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]. Recommendable is surely the dedicated chapter of the popular course [10, Ch. 10].
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