Survey of Numerical Methods for Trajectory Optimization

The techniques for numerically solving trajectory optimization problems are classified as either indirect or direct. Indirect methods are characterized by explicitly solving the optimality conditions stated in terms of the adjoint differential equations, the maximum principle, and associated boundary conditions. The indirect approach usually requires the solution of nonlinear multipoint boundary value problem. In contrast, a direct method does not require an analytic expression for the necessary conditions and typically does not require initial guesses for the adjoint variables. Instead, the dynamic variables are adjusted to directly optimize the objective function. All direct methods introduce some parametric representation for the control variables.