Path constraints like "do not go below the surface of the Earth" are also not feasible to implement. In more plain English this means that atmospheric modelling is highly problematic to nearly impossible, particularly for any generalized solver that isn't constrained to specific problems. The limitations of PVG are that the indirect method that it uses is sensitive to initial conditions, and is not usable for dissipative systems. The closest real world algorithms to PVG are OPGUID and SWITCH which also used the Calculus of Variations approach, with Runge-Kutta-Fehlberg and a quasi-Newtonian method to solve the root finding problem but with numerical integration of the Jacobian, or analytical computation for coast arcs. There are no linear approximations made, so it is considerably more accurate than PEG and other guidance models which use linear gravity models and other approximations, at the cost of substantially more computational overhead (as a result PVG runs in its own dedicated thread, and may take multiple KSP time ticks in order to solve the optimal control problem). It uses the Dormand-Prince Runge Kutta method as a forward integrator (equivalent to Matlab's "ode45") to do multiple shooting and then Levenburg-Marquardt as a quasi-Newtonian root finding algorithm (equivalent to Matlab's "lsqnonlin") to find the optimal solution along with a numerically computed estimate of the Jacobian of the boundary value problem. Primer Vector Guidance is an implementation of a trajectory optimizer that is based around the "indirect" method of optimal control theory using Pontryagin's Minimum Principle and the calculus of variations as synthesized in Lawden's Primer Vectory Theory.
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