We consider optimal two-impulse space interception problems with multiple constraints. The multiple constraints are imposed on the terminal position of a space interceptor, impulse and impact instants, and the component-wise magnitudes of velocity impulses. These optimization problems are formulated as multi-point boundary value problems and solved by the calculus of variations. Slackness variable methods are used to convert all inequality constraints into equality constraints so that the Lagrange multiplier method can be used. A new dynamic slackness variable method is presented. As a result, an indirect optimization method is developed. Subsequently, our method is used to solve the two-impulse space interception problems of free-flight ballistic missiles. A number of conclusions for local optimal solutions have been drawn based on highly accurate numerical solutions. Specifically, by numerical examples, we show that when time and velocity impulse constraints are imposed, optimal two-impulse solutions may occur; if two-impulse instants are free, then a two-impulse space interception problem with velocity impulse constraints may degenerate to a one-impulse case.

我们考虑了具有多个约束的最优两脉冲空间拦截问题。多重约束施加在空间拦截器的终端位置，脉冲和冲击瞬间以及速度脉冲的各个分量上。这些优化问题被公式化为多点边值问题，并通过变化演算解决。松弛度变量方法用于将所有不等式约束转换为等式约束，以便可以使用拉格朗日乘数法。提出了一种新的动态松弛度可变方法。结果，开发了一种间接优化方法。随后，我们的方法被用来解决自由飞行弹道导弹的两脉冲空间拦截问题。基于高度精确的数值解，已经得出了许多局部最优解的结论。具体来说，通过数值示例，我们表明，当施加时间和速度脉冲约束时，可能会出现最优的两脉冲解；这可能会导致最优解。如果两个脉冲的时刻是自由的，则具有速度脉冲约束的两个脉冲的空间拦截问题可能会退化为一个脉冲的情况。