Second Order Necessary Conditions for Endpoints-Constrained Optimal Control Problems on Riemannian manifolds

TL;DR

This paper develops second order necessary conditions for endpoint-constrained optimal control problems on Riemannian manifolds, incorporating curvature effects and applicable to various control constraints.

Contribution

It introduces new second order necessary conditions involving curvature tensors for control problems on Riemannian manifolds with endpoint constraints.

Findings

Derived integral and quasi-pointwise second order conditions

Conditions apply to Pontryagin critical controls

Applied to Bolza problem with endpoint constraints

Abstract

In this paper, we are concerned with optimal control problems evolved on Riemannian manifolds, where the initial and final states satisfy some inequality and equality type constraints, and the control set is a separable metric space. We obtain the second order necessary conditions of integral and quasi-pointwise forms, both of which work for Pontryagin type critical controls and involve the curvature tensor. Also, we apply the condition of integral form to the Bolza problem, where the initial and final states are subject to equality's type constraint.