Neighboring extremal optimal control for mechanical systems on Riemannian manifolds (Q2358456)

The authors present an extension of the Neighboring Extremal Optimal Control (NEOC) to Optimal Control Problems (OPC) for mechanical systems evolving on Riemannian manifolds. After reviewing some background material concerning the NEOC in the setting of \(\mathbb{R}^n\), the authors begin by recalling some notations used in the paper. Then they state the OPC consisting in minimization of a cost functional, subject to some conditions fulfilled by some control inputs. First, they study a particular OPC which is used to illustrate NEOC. Then, the OPC is solved using Lagrange multipliers and the corresponding variational equations are derived. The same OCP is solved as a variational problem, obtaining the corresponding variational equation. The results are specialized to the case when the base Riemannian manifold \(\mathcal{Q}\) is a compact semi-simple Lie group. The authors conclude their work by presenting a numerical example. They present simulation results for the case when \(n=3\) and \(T=10\). Some figures are obtained, representing the trajectories of \(u\) obtained by re-solving the OCP and using NEOC.