Second order necessary conditions for optimal control problems with endpoints-constraints and convex control-constraints

TL;DR

This paper establishes second order necessary conditions for optimal control problems on Riemannian manifolds with endpoint constraints and convex control sets, incorporating curvature effects, and extends applicability beyond Euclidean spaces.

Contribution

It introduces a second order necessary condition involving curvature tensors for control problems on Riemannian manifolds, improving upon existing Euclidean-based results.

Findings

Derived second order necessary condition involving curvature tensor.

Extended applicability to Riemannian manifolds, including Euclidean spaces.

Provided example demonstrating the condition's validity at boundary controls.

Abstract

In this manuscript, we consider a control system governed by a general ordinary differential equation on a Riemannian manifold, with its endpoints satisfying some inequalities and equalities, and its control constrained to a closed convex set. We concern on an optimal control problem of this system, and obtain the second order necessary condition in the sense of convex variation (Theorem 2.2). To this end, we first obtain a second order necessary condition of an optimization problem (Theorem 4.2) via separation theorem of convex sets. Then, we derive our necessary condition by transforming the optimal control problem into an optimization problem. It is worth to point out that, our necessary condtition evolves the curvature tensor, which is trivial in Euclidean case. Moreover, even M is a Euclidean space, our result is still of interest. Actually, we give an example (Example 2.1) which shows that, when an optimal control stays at the boundary of the control set, the existing results are invalid while Theorem 2.2 works.