Geometric Properties of Time-Optimal Controls With State Constraints Using Strong Observability

This article considers minimum time optimal control problems with linear dynamics subject to state equality constraints and control inequality constraints. In the absence of state constraints, there are well-known sufficient conditions to guarantee that optimal controls are at the boundary or extreme points of the control set. With strong observability as the key tool, analogous conditions are derived for problems subject to both extrinsic and intrinsic state constraints. Understanding these geometric properties enables exact convex relaxations. The relaxation technique is used to convert a nonconvex quadratic program to a second-order cone program and a mixed integer linear program to a linear program. The relaxations accelerate numerical solution times by factors of 18 000 and 150, respectively. As such, the theorems and relaxations are seen as important tools for real-time optimization-based control.