Optimal reduction of controlled Hamiltonian system with Poisson structure and symmetry

In this paper, our goal is to study the optimal reduction theory of controlled Hamiltonian (CH) systems with Poisson structure and symmetry, and this reduction is an extension of optimal reduction theory of Hamiltonian systems under controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly CH systems defined on a cotangent bundle and on the optimal reduced spaces, we first define a kind of CH systems on a Poisson fiber bundle. Then we introduce the optimal point, optimal orbit, and regular Poisson reducible CH systems with symmetry by using the optimal momentum map and reduced Poisson tensors (or reduced symplectic forms). Moreover, we give some optimal reduction theorems for CH systems to explain the relationships between OpCH-equivalence, OoCH-equivalence, RPR-CH-equivalence for optimal reducible CH systems with symmetry and CH-equivalence for associated optimal reduced CH systems. Finally, we describe the CH system and CH-equivalence from the viewpoint of port Hamiltonian system with a Poisson structure, and give two examples to state theoretical results of optimal point reduction of CH systems.