  This paper aims to address the nonlinear optimal guidance problem with
impact-time and impact-angle constraints, which is fundamentally important for
multiple pursuers to collaboratively achieve a target. Addressing such a
guidance problem is equivalent to solving a nonlinear minimum-effort control
problem in real time. To this end, the Pontryagain's maximum principle is
employed to convert extremal trajectories as the solutions of a parameterized
differential system. The geometric property for the solution of the
parameterized system is analyzed, leading to an additional optimality
condition. By incorporating this optimality condition and the usual
disconjugacy condition into the parameterized system, the dataset for optimal
trajectories can be generated by propagating the parameterized system without
using any optimization methods. In addition, a scaling invariance property is
found for the solutions of the parameterized system. As a consequence of this
scaling invariance property, a simple feedforward neural network trained by the
solution of the parameterized system, selected at any fixed time, can be used
to generate the nonlinear optimal guidance within milliseconds. Finally,
numerical examples are presented, showing that the nonlinear optimal guidance
command generated by the trained network can not only ensure the expected
impact angle and impact time are precisely met but also requires less control
effort compared with existing guidance methods.
