On the Optimal Control of Impulsive Hybrid Systems On Riemannian Manifolds

TL;DR

This paper derives a geometric version of the Hybrid Minimum Principle for impulsive hybrid systems on Riemannian manifolds, addressing systems with state jumps and switching manifolds in a differential geometric framework.

Contribution

It introduces a geometrical derivation of the Hybrid Minimum Principle for impulsive hybrid systems on Riemannian manifolds, extending optimal control theory to manifold-valued states with jumps.

Findings

Derived the HMP for autonomous impulsive hybrid systems on Riemannian manifolds.

Expressed extremal trajectories on the cotangent bundle of the manifold.

Established conditions for time-invariant switching manifolds and smooth jump functions.

Abstract

This paper provides a geometrical derivation of the Hybrid Minimum Principle (HMP) for autonomous impulsive hybrid systems on Riemannian manifolds, i.e. systems where the manifold valued component of the hybrid state trajectory may have a jump discontinuity when the discrete component changes value. The analysis is expressed in terms of extremal trajectories on the cotangent bundle of the manifold state space. In the case of autonomous hybrid systems, switching manifolds are defined as smooth embedded submanifolds of the state manifold and the jump function is defined as a smooth map on the switching manifold. The HMP results are obtained in the case of time invariant switching manifolds and state jumps on Riemannian manifolds.