The link you gave appears to be dead so I'm not 100% familiar with some of these terms, however I believe I know what's going on.
These extra multiple shooting parameters your dealing with are a result of the method used, not the problem formulation. Since you haven't transcribed the problem definition to anything new, the additional parameters used in multiple shooting should not appear in the cost function. If this is the case, then the sensitivities are just 0. You can probably stop here.
HOWEVER, why are these new multiple shooting parameters used as explicit NLP parameters in the first place? If something is not a free parameter, then the NLP solver shouldn't have direct access to it. NLP solvers can usually handle equality constraints without any problem. For a trajectory $\gamma$ that has been sliced up into $\{\gamma_1, \gamma_2, \cdots, \gamma_n\}$ each with time segments $\{[\tau_0, \tau_1], [\tau_1, \tau_2], \cdots\, [\tau_{n-1}, \tau_n]\}$ in multiple shooting, extra boundary conditions become
$$
\gamma_i(\tau_{i}) - \gamma_{i+1}(\tau_i) = 0 \; \forall \; i \in [1,n-1]
$$
If you append these extra boundary conditions directly as nonlinear equality constraints, you shouldn't have a problem, but again the link you gave was dead.
TL;DR (short answer): If you want to control an inverted pendulum to balance itself you can use linear controllers for stabilizing the system. If you want to land on the moon then you will have to use nonlinear control. Depending on the design requirements it might be sufficient to use a linear controller, but for high-end applications, you will most likely have to use nonlinear control design.
Long answer:
Linear controllers simply work for many nonlinear plants. This is possible because many nonlinear systems can be described pretty accurate by a linear approximation at a specific operating point. There are control architectures like gain scheduling which are based on this principle. The downside of linearization is that the control effort is often not very efficient and that the performance of the system is not as good as it could be. An example is given in robotics in which the end effector (for example a cutting tool) has to follow a circular trajectory. A simple PID controller is not able to provide a sufficient quality for the reference tracking task. Another example is given in time optimal control. If you want to move a point mass form an initial position to the origin in minimal time, while the input is $u$ is constrained. The solution to this problem is given by a switching bang-bang control, which is nonlinear.
At the same time, linear control theory is almost a completed field of research (there are only a few open questions), that means control designers have a lot of knowledge about what and how they can achieve something. Hence, linear control theory is something like a quite universal toolbox to many control problems and it can be easily applied.
Nonlinear control theory is still a very active field of research. Hence, there are only a few methods like exact feedback linearization, backstepping and sliding mode control which can be applied to more general nonlinear systems. But there are still nonlinear systems that cannot be controlled by these nonlinear control architectures. These methods also have drawbacks. For example, exact feedback linearization requires the designer to have very precise knowledge about the system parameters and it also eliminates useful nonlinearities that would reduce the control effort. Backstepping and sliding mode control can be made very robust against plant uncertainties. Backstepping has the drawback that only works for systems in strict-feedback form and you have to come up with a Control Lyapunov function for the first subsystem and that the resulting closed-loop dynamics are nonlinear and difficult to predict. Whereas sliding mode control shows chattering close to the switching surface.
Best Answer
for the indirect approach, you solve as you already mentioned, the neccessary condition for optimality of the optimal control problem (minimum principle ,...). This forms a boundary value problem (BVP) which is often nonlinear and has several local optimal solutions (optimal state and optimal costate). The problem here is, you can solve the BVP but you obtain only ONE local optimal solution (you don't know if its global optimal). In order to find the global optimal one, you would have to determine all local optimal solutions and then use the HJB equation to check, which one of the local opt. solutions are also global optimal.
In generall, the solution depends on your initial guess (for solving numerically). For two different initial guesses you might obtain two different solutions. This holds for both methods (direct, indirect)
i don't understand your question! Please reformulate!
i am not sure why they call it like that. The main difference of direct and indirect methods are:
general optimal control: minimize a functional (performance measure) subject to constraints/bounds and of course the dynamics
direct method: solve DIRECTLY this optimal control problem as it is written there.
indirect method: solve the neccessary condition (BVP) which INDIRECTLY represents the original optimal control problem. you just rewrite the optimal control problem (which is an optimization problem) in another optimization problem. Both problems
then, you can use the same methods (e.g. collocation, shooting) for both direct and indirect methods. you only solve different sytems of ODE with different constraints.