Global topological aspects of continuous-time linear dynamically varying (LDV) control

Linear dynamically varying (LDV) systems are a subset of linear parameter varying (LPV) systems characterized by parameters that are dynamically modeled. An LDV system is, in most cases of practical interest, a family of linearized approximations of a nonlinear dynamical system indexed by the point around which the system is linearized. Special attention is devoted to nonlinear dynamical systems running over a Riemannian manifold. Such (local) differential geometric concepts as curvature play a crucial role in defining the LDV approximation. Furthermore, such (global) topological properties as parallelizability, Euler characteristics-and a global “flatness” concept-are crucially involved in defining the problem in a computationally attractive coordinate-dependent fashion. Finally, an LQ trajectory tracking problem is formulated, revealing a partial differential Riccati equation, itself related to a linear PDO, for which an index theorem can be formulated