Riccati transformations for control optimization using the second variation

Two new numerical methods which may be used to calculate solutions to optimal control problems are developed. These methods involve guessing initial values for unknown Lagrange multipliers and a control sequence. They are similar to the successive sweep method in that Riccati equations are used to calculate Corrections to these guessed variables. They are however substantially different in other aspects. Two example problems, the Brachistachrone and an earth to Mars low thrust transfer are solved to illustrate the application of the methods.