Given an affine system on a full-dimensional polytope, the problem of reaching a particular facet of the polytope, using continuous piecewise-affine state feedback is studied. Necessary conditions and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the polytope. Special attention is paid to affine systems on full-dimensional simplices. In this case, the necessary and sufficient conditions are equivalent and a constructive procedure yields an affine feedback control law, that solves the reachability problem under consideration.
Habets, L. C. G. J. M., & Schuppen, van, J. H. (2004). A control problem for affine dynamical systems on a full-dimensional polytope. Automatica, 40(1), 21-35. https://doi.org/10.1016/j.automatica.2003.08.001