The reach control problem (RCP) characterizes a control design approach, based on computer science notions of object triangulation, that has been extensively developed as a means of guiding the complete transient response of a system, entirely within a desired polytopic region of state-space operation characterized by linear constraints on its states. This thesis expands upon results achieved in the area of RCP problem solvability under continuous feedback, identifying new necessary conditions. It accomplishes this using algebraic topology constructs, mapping the reach control problem to an equivalent topological one to successfully demonstrate conditions under which topological obstructions are generated. These obstructions, which render the RCP unsolvable by continuous feedback are then used to characterize equivalent conditions necessary for solvability of the problem. This thesis also serves to formally demonstrate the substantial advantages of the RCP design approach over more conventional industry techniques, by solving real-world problems with complex specifications.

The reach control problem (RCP) characterizes a control design approach, based on computer science notions of object triangulation, that has been extensively developed as a means of guiding the complete transient response of a system, entirely within a desired polytopic region of state-space operation characterized by linear constraints on its states. This thesis expands upon results achieved in the area of RCP problem solvability under continuous feedback, identifying new necessary conditions. It accomplishes this using algebraic topology constructs, mapping the reach control problem to an equivalent topological one to successfully demonstrate conditions under which topological obstructions are generated. These obstructions, which render the RCP unsolvable by continuous feedback are then used to characterize equivalent conditions necessary for solvability of the problem. This thesis also serves to formally demonstrate the substantial advantages of the RCP design approach over more conventional industry techniques, by solving real-world problems with complex specifications.

As control systems become more integrated with high-end engineering systems as well as consumer products, they are expected to achieve specifications that may include logic rules, safety constraints, startup procedures, and so forth. Control design for such complex specifications is a relatively unexplored research area. One possible design approach is based on partitioning the state space into polytopic regions, and then formulating a certain control problem on each polytope, with the intention that the set of all controllers so obtained would collectively achieve the specification. The control problem which must be solved for each polytope is called the reach control problem, and it has been identified as turnkey to the further development of this approach. The reach control problem (RCP) is to find a state feedback to make the closed-loop trajectories of an affine (or linear) control system defined on a polytope reach and exit a prescribed facet of the polytope in finite time. This dissertation studies a number of aspects of the reach control problem, and it uses tools from convex analysis, nonsmooth analysis, and computational geometry for this study. The dissertation has three main themes. First, we formulate and solve a variant of RCP in which trajectories exit the polytope in a monotonic sense; this provides a triangulation-independent solution of RCP. Second, we develop a Lyapunov-like theory for verifying if RCP is solved using a given candidate controller. This involves the introduction of the notion of generalized flow functions, a LaSalle Principle for RCP, and several converse theorems on existence of generalized flow functions. Third, we study the relationship between affine feedbacks and continuous state feedbacks for RCP on simplices. Although the two feedback classes have been shown to be equivalent under an assumption on the triangulation of the state space, we show by a counterexample that the equivalence is no longer true under arbitrary triangulations. Then we provide for single-input systems a constructive method for the synthesis of multi-affine feedbacks for RCP on simplices.

As control systems become more integrated with high-end engineering systems as well as consumer products, they are expected to achieve specifications that may include logic rules, safety constraints, startup procedures, and so forth. Control design for such complex specifications is a relatively unexplored research area. One possible design approach is based on partitioning the state space into polytopic regions, and then formulating a certain control problem on each polytope, with the intention that the set of all controllers so obtained would collectively achieve the specification. The control problem which must be solved for each polytope is called the reach control problem, and it has been identified as turnkey to the further development of this approach. The reach control problem (RCP) is to find a state feedback to make the closed-loop trajectories of an affine (or linear) control system defined on a polytope reach and exit a prescribed facet of the polytope in finite time. This dissertation studies a number of aspects of the reach control problem, and it uses tools from convex analysis, nonsmooth analysis, and computational geometry for this study. The dissertation has three main themes. First, we formulate and solve a variant of RCP in which trajectories exit the polytope in a monotonic sense; this provides a triangulation-independent solution of RCP. Second, we develop a Lyapunov-like theory for verifying if RCP is solved using a given candidate controller. This involves the introduction of the notion of generalized flow functions, a LaSalle Principle for RCP, and several converse theorems on existence of generalized flow functions. Third, we study the relationship between affine feedbacks and continuous state feedbacks for RCP on simplices. Although the two feedback classes have been shown to be equivalent under an assumption on the triangulation of the state space, we show by a counterexample that the equivalence is no longer true under arbitrary triangulations. Then we provide for single-input systems a constructive method for the synthesis of multi-affine feedbacks for RCP on simplices.