Necessary and sufficient conditions for reachability on a simplex

In this paper we solve the general problem of designing a feedback controller to reach a set of facets of an n-dimensional simplex in finite time, for a system evolving with linear affine dynamics. Necessary and sufficient conditions are presented in the form of bilinear inequalities on the vertices of the simplex. By exploiting the structure of the problem, the bilinear inequalities are converted to a series of linear programming problems.