Optimal control of continuous and discontinuous processes in a Riemannian tangent bundle

Abstract

In this paper some formulations of stochastic systems with values in the tangent bundle of a Riemannian manifold will be given in terms of stochastic differential equations that contain jump processes and these formulations will serve to model stochastic control problems for which necessary and sufficient conditions for optimality will be given. In stochastic systems both discontinuous and continuous processes often appear and it frequently occurs that the system evolves in a smooth manifold that is not a linear space. With suitable conditions on the manifold there are continuous and discontinuous processes that respect the geometry and these properties can be lost by an abstract formulation. Traditionally, many problems in control have been modelled by differential equations in Euclidean spaces to show the dynamical property of the physical systems. The differential geolrletric formulation preserves the geometric interpretations of the differential equations while also providing a more mathematically reasonable formulation of the physical system.