| Reach control theory is an approach to satisfying complex control objectives on a constrained state space. It relies on triangulating the state space into simplices, and devising a separate controller on each simplex to satisfy the control specifications. A fundamental element of reach control theory is the Reach Control Problem (RCP). The goal of the RCP is to drive system trajectories of an affine control system on a simplex to leave this simplex through a predetermined facet. This thesis discusses a number of issues pertaining to reach control theory and the RCP. Its central part is a discussion of the solvability of the RCP. We identify strong necessary conditions for the solvability of the RCP by affine feedback and continuous state feedback, and provide elegant characterizations of these conditions using methods from linear algebra and algebraic topology. Additionally, using the theory of positive systems, Z-matrices, and graph theory, we obtain several new interpretations of the currently known set of necessary and sufficient conditions for the solvability of the RCP by affine feedback. The thesis also provides a rigorous foundation for the notion of exiting a simplex through a facet, and discusses uniqueness and existence of trajectories in reach control with discontinuous feedback. Finally, building on previous results, the thesis includes novel applications of reach control theory to parallel parking and adaptive cruise control. These applications serve to motivate new directions of theoretical research in reach control. |
