Hi,
Direct method used for optimal control problem discretize the control problem into a nonlinear constrained optimization problem. bvp4c also divide the range of integration in case of multipoint Boundary value problem. The collocation technique uses a mesh for dividing the interval of integral to subintervals. Thus, solver solves the algebraic equations arising from the boundary condition and this process is repeated until error in numerical solution doesn’t exceed the tolerance criteria.
Thus, the boundary condition and the algorithm used in bvp4c makes it a direct method as final equation to solve on the subinterval are algebraic (can be linear or non-linear depending upon the integration problem).
You may refer to the following examples and reference for exploring more:
