Second-Order Necessary Optimality Conditions for Multi-Objective Optimal Control Problems on Riemannian Manifolds

TL;DR

This paper establishes second-order necessary optimality conditions for multi-objective control problems on Riemannian manifolds, highlighting the influence of manifold curvature on weak Pareto optimal solutions.

Contribution

It extends second-order optimality conditions to multi-objective control on Riemannian manifolds, incorporating curvature effects and addressing fixed and free terminal times.

Findings

Weak Pareto solutions depend on the curvature tensor.

Derived second-order necessary conditions for fixed and free terminal times.

Illustrated results with a specific example contrasting existing theories.

Abstract

In this paper, we investigate the multi-objective optimal control problem of ordinary differential equations on Riemannian manifolds. We first obtain the second-order necessary conditions for weak Pareto optimal solutions for multi-objective optimal control problems with fixed terminal time, and then extend these results to multi-objective optimal control problems with free terminal time, deriving the corresponding second-order necessary conditions for weak Pareto optimal solutions. Our main results (i.e., Theorem 2.2 and Theorem 2.4) show that weak Pareto optimal solutions depend on the curvature tensor of the Riemannian manifold. Finally, we provide an example (i.e., Example 2.1) as an application of our main results, illustrating how our findings differ from existing related results (see Remark 2.2).