Second-order necessary optimality conditions for multi-objective optimal control problems on Riemannian manifolds

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 show that weak Pareto optimal solutions depend on the curvature tensor of the Riemannian manifold. Finally, we provide an example as an application of our main results, illustrating how our findings differ from existing related results.