Page 24 - IJOCTA-15-3

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.3, pp.396-406 (2025)
https://doi.org/10.36922/IJOCTA025090036

RESEARCH ARTICLE

Using infinitesimal symmetries for determining the first Maxwell
time of geometric control problem on SH(2)

*
Soukaina Ezzeroual and Brahim Sadik

Department of Mathematics, Cadi Ayyad University, Faculty of Sciences Semlalia, Marrakesh, Morocco
s.ezzeroual.ced@uca.ac.ma, sadik@uca.ac.ma

ARTICLE INFO ABSTRACT

Article History:
Received: February 25, 2025 In this work, we utilize infinitesimal symmetries to compute Maxwell points
Revised: April 1, 2025 which play a crucial role in studying sub-Riemannian control problems. By ex-
Accepted: April 14, 2025 amining the infinitesimal symmetries of the geometric control problem on the
Published Online: April 29, 2025 SH(2) group, particularly through its Lie algebraic structure, we identify invari-
Keywords: ant quantities and constraints that streamline the Maxwell point computation.
Geometric control theory
Lie algebra
Maxwell time
Special hyperbolic group
Sub-Riemannian geometry
AMS Classification 2010:
93C85; 17B66; 53C17;
22E60; 53C22

1. Introduction using Pontryagin’s maximum principle (PMP).
Additionally, the author analyzes the discrete
Describing locally optimal trajectories is crucial
symmetries of this problem to understand the
in dynamical systems, control theory, robotics,
optimality of geodesics.
and other areas where precise motion control is
1
necessary. In Sachkov, Moiseev and Sachkov, 2 Infinitesimal symmetries are transformations
3
Sachkov, several optimal control problems are in- that leave the fundamental equations of a system
vestigated to determine optimal solutions and an- invariant. They are also powerful tools for study-
alyze the optimality of geodesics, specifically fo- ing problems in theoretical physics and dynamic
cusing on the first Maxwell time, which indicates systems, providing insight into conserved quanti-
how long a geodesic trajectory remains optimal. ties and invariant structures within a system. By
For more details on sub-riemannian geometry and studying the Lie algebra of these symmetries, we
geometric control theory, we refer the reader to can often derive conserved quantities that serve
1
Sachkov and Agrachev et al. 4 in the computation of some properties of some
An optimal control problem concerns finding control systems.
controls that steer the studied system from one In this work, we extend the analysis presented
5
state to another while minimizing certain quan- in Butt by considering a control problem primar-
tities, often related to energy or time. A simple ily addressed through geometric methods. We
and interesting example is the geometric control build upon both algebraic and geometric tech-
6
problem on the Lie group SH(2) that involves niques Gallier and Quaintance to further inves-
controlling a system whose state space is the spe- tigate this problem. First, we compute the infini-
cial Euclidean group in two dimensions. In Butt, 5 tesimal symmetries of the sub-Riemannian prob-
the author makes significant contributions to this lem and then apply these results to the geometric
problem by presenting optimal solutions derived control problem on the Lie group SH(2), whose
*Corresponding Author
396