Calculation of Elements of a Guiding Program Package for Singular Clusters of the Set of Initial States in the Package Guidance Problem

A fixed-time package guidance problem is considered for a linear controlled dynamic system with a finite set of initial states. The control set is convex and compact and the target set is convex and closed. The paper focuses on the case where the set of initial states has singular clusters for which the existing algorithm for estimating the elements of a guiding program package is not applicable. It is suggested to consider a perturbed problem of extended program guidance with a smoothed control set. It is proved that the motions of the original and perturbed problems are close to each other at the terminal time; the corresponding estimates are provided. In the case of an extended target set with nonempty interior, it is also shown that a solution of the extended program guidance problem that is precisely guiding to the target set can be obtained by applying the existing algorithm for the perturbed problem with compressed target set. The suggested theoretical constructions are illustrated with a model example.