[Paper Review] Second Order Necessary Conditions for Optimal Control Problems on Riemannian Manifolds

Qing Cui, Li Deng+1 more|arXiv (Cornell University)|Aug 25, 2015

Geometric Analysis and Curvature Flows19 references3 citations

TL;DR

This paper establishes second-order necessary and sufficient optimality conditions for optimal control problems on Riemannian manifolds, incorporating the curvature tensor via dual equations and variational analysis. It derives integral and pointwise second-order conditions for both free and fixed endpoint cases, recovering classical geodesic energy minimization as a special case where standard control methods fail.

ABSTRACT

This work is concerned with an optimal control problem on a Riemannian manifold, for which two typical cases are considered. The first case is when the endpoint is free. For this case, the control set is assumed to be a separable metric space. By introducing suitable dual equations, which depend on the curvature tensor of the manifold, we establish the second order necessary and sufficient optimality conditions of integral form. In particular, when the control set is a Polish space, the second order necessary condition is reduced to a pointwise form. As a key preliminary result and also an interesting byproduct, we derive a geometric lemma, which may have some independent interest. The second case is when the endpoint is fixed. For this more difficult case, the control set is assumed to be open in an Euclidian space. We obtain the second order necessary and sufficient optimality conditions, in which the curvature tensor also appears explicitly. Our optimality conditions can be used to recover the following famous geometry result: Any geodesic connecting two fixed points on a Riemannian manifold satisfies the second variation of energy; while the existing optimality conditions in control literatures fail to give the same result.

Motivation & Objective

• To develop second-order necessary and sufficient optimality conditions for optimal control problems on Riemannian manifolds, particularly when the state evolves on a curved manifold.
• To address the limitation of existing control-theoretic optimality conditions in capturing geodesic energy minimization, a fundamental result in Riemannian geometry.
• To extend second-order analysis to cases with free and fixed endpoints, with distinct assumptions on the control set (separable metric space vs. open subset of Euclidean space).
• To derive a geometric lemma involving curvature that may have independent interest beyond the main optimality results.
• To demonstrate that the proposed conditions can recover the second variation of energy for geodesics, a result not captured by classical control-theoretic methods.

Proposed method

• Introduce dual equations dependent on the Riemannian curvature tensor to characterize the second-order variation of the cost functional.
• Use variational analysis on the state manifold, including first and second-order variational equations for perturbed trajectories.
• Apply Taylor expansion to the cost functional and endpoint constraints, expanding up to second order in the control variation.
• Distinguish between normal and abnormal cases via the Lagrange multiplier structure, with the abnormal case relying on sign-definiteness of the Hessian on the kernel of the endpoint constraint derivative.
• Employ compactness and convergence arguments (e.g., weak convergence in L²) to pass to the limit in sequences of perturbations and derive necessary conditions.
• Utilize the exponential map and its inverse to linearize the state trajectory variation, enabling the derivation of second-order terms involving the curvature tensor.

Experimental results

Research questions

• RQ1 Can second-order optimality conditions be derived for optimal control problems on Riemannian manifolds when the control set is a general separable metric space?
• RQ2 How does the Riemannian curvature tensor influence the second-order necessary and sufficient conditions in optimal control on manifolds?
• RQ3 Can the proposed second-order conditions recover the classical result that geodesics minimize the energy functional?
• RQ4 What is the role of the endpoint constraint (fixed vs. free) in shaping the structure of second-order conditions?
• RQ5 Under what conditions does the second-order condition reduce from integral to pointwise form?

Key findings

• Second-order necessary and sufficient optimality conditions are derived in integral form for the free endpoint case, with the curvature tensor explicitly appearing in the dual equations.
• When the control set is a Polish space, the second-order necessary condition reduces to a pointwise form, improving the regularity of the optimality condition.
• For the fixed endpoint case, the second-order conditions are established with the control set open in Euclidean space, and the curvature tensor again appears explicitly in the Hessian terms.
• The proposed optimality conditions successfully recover the second variation of energy for geodesics, a result that existing control-theoretic methods fail to capture.
• The analysis reveals that the Hessian of the endpoint map must be negative definite on the kernel of the derivative of the endpoint constraint, ensuring local optimality.
• A geometric lemma involving the curvature tensor is derived as a byproduct, which may have independent interest in differential geometry.