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

TL;DR

This work derives second-order necessary conditions for weak Pareto optimality in multi-objective optimal control on Riemannian manifolds, addressing both fixed and free terminal times. By formulating a Hamiltonian framework on $M$ and introducing singular directions and Lagrangian multipliers, the authors show that the second-order conditions explicitly involve the manifold's curvature through the curvature tensor $R$. The main results, Theorems \ref{'sen'} and \ref{'snt'}, are complemented by Example 2.1 illustrating curvature effects and differentiating these results from Euclidean theory. The analysis relies on convexity, Brouwer fixed-point arguments, and a local coordinate appendix to connect abstract geometric conditions with implementable expressions. Overall, the paper extends classical Euclidean optimal control theory to curved state spaces, highlighting how geometry shapes second-order optimality conditions and solution behavior.

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).