Resolving Control to Facet Problems for Affine Hypersurface Systems on Simplices

We study control to facet problems for affine hypersurface systems on simplices. An easily verifiable necessary and sufficient condition is derived that allows one to determine whether there is a linear affine feedback driving all trajectories of the closed-loop system out of the simplex through a desired facet. In cases where the condition is violated, methods are developed to establish the largest subset of the simplex such that there is still a feedback control (not necessarily linear) which can steer all trajectories starting from this set to the desired facet without crossing other facets. The results are demonstrated on a circuit with a piecewise linear resistor