A Generalized Shooting Technique

Assume starting initial guesses for the missing initial conditions as 

Now we expand the known final conditions in a linearized Taylor series about the assumed missing initial conditions, 

To calculate the partiais we define shooting system dynamics for each unknown initial condition. These equations come from taking partiais with respect to the missing initial conditions (a and 6), of the describing state differential equations (7.2.1) to (7.2.4). 

To integrate these equations we have the following initial conditions  

Note that the initial conditions for the shooting systems follow directly from the initial conditions on the state system. Solving the state and shooting systems simultaneously gives the information needed to compute new initial conditions a +i and from equation (7.2.7) and 

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