IJPAM: Volume 42, No. 2 (2008)

LEFT INVARIANT OPTIMAL CONTROL SYSTEMS
AND SUB-RIEMANNIAN GEOMETRY

Felipe Monroy-Pérez$^1$, Alfonso Anzaldo-Meneses$^2$
$^1$Mathematical Analysis Research Area
Universidad Autonoma Metropolitana - Azcapotzalco
Av. San Pablo 180, Azcapotzalco, México D.F., 022000, MÉXICO
e-mail: fmp@correo.azc.uam.mx
$^2$Theoretical Physics Research Area
Universidad Autonoma Metropolitana - Azcapotzalco
Av. San Pablo 180, Azcapotzalco, México D.F., 022000, MÉXICO
e-mail: alfons_rex@hotmail.com


Abstract.In this paper we present the optimal control problem consisting in the minimization of the functional $\int u_1^2+\cdots+ u_n^2$ among the solutions $t\mapsto(g,u_1,\ldots,u_n)$ of the control system $\dot
g=u_1 X_1+\cdots u_n X_n, g\in G$ and $u_i\in L^2(\R)$. Here $G$ is a Lie group, and $\Delta=\{X_1,\ldots,X_n\}$ with $n<\dim G$, is a family of left invariant vector fields on $G$. This is a generalization to the Lie-group theoretical framework of the classical affine-in-controls-quadratic optimal control problem. We establish the problem as the sub-Riemannian geodesic problem defined on $G$ by $\Delta$ and a smooth varying metric defined on the planes $\Delta(g)$.

Received: August 17, 2007

AMS Subject Classification: 49K30, 49Q99, 93B29

Key Words and Phrases: optimal control, Lie groups and algebras, sub-Riemannian geodesics, Hamiltonian system, extremal curves

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 42
Issue: 2