A contact covariant approach to optimal control with applications to sub-Riemannian geometry (1509.01628v3)

Abstract: We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of [Ohsawa, 2015], providing simple and elegant characterizations of normal and abnormal sub-Riemannian extremals.

• The paper presents a novel contact geometry framework for analyzing optimal control problems, including sub-Riemannian geometry.
• It provides a unified geometric interpretation of the Pontryagin Maximum Principle, covering both normal and abnormal solutions via contact dynamics.
• The approach yields theoretical insights for sub-Riemannian geometry and suggests potential for new numerical algorithms based on geometric invariants.

A Contact Covariant Approach to Optimal Control with Applications to Sub-Riemannian Geometry

The paper by Michał J{\'o}zwikowski and Witold Respondek explores a novel contact geometry framework for handling optimal control problems, particularly focusing on sub-Riemannian (SR) geometry. The work builds upon previous discoveries by Ohsawa and attempts to establish a more intrinsic geometric understanding and characterization of optimal control solutions using contact geometry.

Overview of the Approach

The authors revisit the Pontryagin Maximum Principle (PMP), a cornerstone in optimal control theory traditionally expressed using Hamiltonian formalism. They propose a dual view, leveraging the intrinsic properties of contact geometry—represented by curves of hyperplanes rather than covectors. This contact formulation incorporates both normal and abnormal solutions in a unified framework, potentially offering a deeper geometric insight into the PMP and expanding its applicability.

Key Contributions

1. Contact Structure and PMP: The authors articulate a contact interpretation of the Pontryagin Maximum Principle. They theorize that the PMP's dynamics in its Hamiltonian formulation mirror the contact evolution within the projectivized cotangent bundle, thereby facilitating the identification of optimal trajectories through these geometric equivalents.
2. Contact Dynamic Equivalence: By proving that the PMP's Hamiltonian evolution corresponds to a contact evolution, the work reduces the complexity of analyzing abnormal and normal solutions using a contact perspective. It shows that this equivalence holds uniformly across both solution types, thus avoiding the traditional bifurcation in analysis between normal and abnormal cases.
3. Applications to Sub-Riemannian Geometry: The paper extends the contact covariant approach to SR geometry, a significant context where traditional metrics are replaced by distributions and control-driven mechanics. The authors provide a comprehensive character characterization of normal and abnormal SR extremals, grounded in geometric distribution relations rather than just algebraic Hamiltonians.

Numerical and Theoretical Implications

The theoretical insights offered by the contact covariant approach yield promising numerical algorithms for solving SR geodesic problems. This paper emphasizes:

• The construction of hyperlayer sequences along extremals to discern optimal trajectories.
• Simplified criteria within the SR context that rely heavily on invariant geometric properties rather than non-unique covector strategies.

Noteworthy is the potential for this framework to inform precise computational algorithms inherently aligned with the geometric properties of control systems, potentially advancing both theoretical understanding and practical implementations.

Future Prospects

The research opens considerable pathways in control theory, especially in domains like robotics and navigation systems where SR geometries are prevalent. Future work could expand beyond sub-Riemannian scenarios to more generalized Hamiltonian systems, particularly within mechanical systems with non-holonomic constraints.

In conclusion, this paper challenges traditional paradigms by equipping optimal control investigations with the nuanced tools of contact geometry, fostering a comprehensive understanding of control and optimization problems in non-linear domains. This not only introduces novel theoretical perspectives but aligns computational practices with the intrinsic geometry of control models.