Talk:Direct multiple shooting method

My 21 June edit
I removed some parts which perhaps needs more explanation than fits in an edit summary: Additionally, I'm wondering whether we should move the article to multiple shooting method, because that's the name that I'm familiar with. I'm not sure what the direct refers to. My guess is that in the context of optimal control or differential-equation-constrained optimization, there is a direct multiple shooting method that uses the direct equation and an indirect multiple shooting method that uses the adjoint equation. But multiple shooting can be used for any boundary value problem, not just those arising in optimal control, and if the optimal control setting is not discussed in the article then I don't think the article should be called  direct multiple shooting method. But I'm not an expert on optimal control, so I may well be mistaken. -- Jitse Niesen (talk) 14:27, 21 June 2008 (UTC)
 * I don't think shooting methods of any kind are used in initial value problems. That just does not make sense.
 * I also removed that multiple shooting has better "distribution of nonlinearity" over single shooting because I have no idea what that is supposed to mean.

Re: My 21 June edit
The phrase "distribution of nonlinearity" refers to the solution of nonlinear boundary value problems (BVP), which is a natural application of the multiple shooting method. Here, the nonlinearity exhibited by the ODE system over time is significantly reduced by having to solve it on shorter time intervals only.

I'm also going to rewrite the article to use first-order ODE notation only. Shooting methods are in no way special to second-order ODEs, which can always be written as a system of first order ODEs.

The phrase "direct" in "direct multiple shooting" refers to the solution of optimal control problems using a direct approach (discretizing the controls) as opposed to an indirect approach (relying on a maximum principle). I think we should not separate direct multiple shooting from (simply) multiple shooting, as the first really is a special (but nonetheless the most prevailing) application of the latter.

-- Agb0ck (talk) 18:52, 1 July 2008 (UTC)

Re Re: My 21 June edit
Afaik, if the DE is only first order, you only need one initial condition, so there is no need to use the shooting method at all. Forcing the solution through 2 points will leave the system overdetermined. —Preceding unsigned comment added by 41.245.180.214 (talk) 10:42, 7 October 2009 (UTC)

Re
You didn't understand. The order of the DE is unimportant, any DE can be transformed to a first order DE. In shooting methods you split the time horizon into multiple segments. You solve an IVP on each segment, and splice the trajectories together by continuity conditions. The system is well determined as you have N initial values on N segments and N-1 continuity condition, leaving you with 1 degree of freedom, your original initial value. —Preceding unsigned comment added by 78.42.73.106 (talk) 21:44, 18 March 2010 (UTC)