Entry Trajectory Optimization Using Generalized Lagrange Multiplier

We begin with the standard point-mass dimensionless equations of motion in the vertical plane over a spherical nonrotating Earth. The negative specific energy were used in stead of time in the motion equations, both the equations and the optimization problem were simplified. The control variables, namely drag and lift, were treated as continuous piecewise-linear functions of the negative specific energy. By the Runge-Kutta methods, the trajectory optimization problem was transferred to nonlinear programming, which were solved by the Generalized Lagrange Multiplier. The optimal entry trajectories with different downrange distance of minimal accumulated heat load were gained, which satisfy the constraints of heating-rate, dynamic pressure and load factor. The characters of the optimal entry trajectories were discussed.