TY - JOUR
T1 - Curvature of optimal control
T2 - Deformation of scalar-input planar systems
AU - Kawski, Matthias
AU - Maxwell, Parnell Ted
N1 - Funding Information:
We wish to thank T. F. Lynch, from our laboratory, and D. O. Boyd and P. Hudson, from the Ft. Wayne Astronomical Society, who helped get the pictures of M13. Also, our thanks to Dr. R. K. Honeycutt for this preliminary testing and evaluation of the #F4548 instrument at the Goethe Link Observatory.
PY - 2008
Y1 - 2008
N2 - The Pontryagin Maximum Principle and high-order open-mapping theorems generalize elementary first-derivative tests to nonlinear optimal control. They provide necessary conditions for a trajectory-control-pair to be optimal, or sufficient conditions for local controllability. Sufficient conditions for optimality (and necessary conditions for nonlinear controllability) are harder to obtain. Like the Legendre-Clebsch condition, they generally take the form of tests for definiteness of second order derivatives. Recently, Agrachev introduced an attractive alternative by developing a notion of curvature of optimal control that generalizes classical Gauss (and Ricci) curvatures. This theory naturally applies to systems whose controls take values on a circle or sphere. In this article we present initial studies of how this notion of curvature provides insight into the limiting case when the circles become degenerate ellipses in the form of closed intervals. Of particular interest are well studied accessible, but uncontrollable, nonlinear systems, and systems that exhibit conjugate points, in which the control takes values in a closed interval u = (u1,u2) ∈ [-1, 1 × {0} ⊆ R2. We focus on systems that are well-known models for the analysis of small-time local controllability and time-optimal control.
AB - The Pontryagin Maximum Principle and high-order open-mapping theorems generalize elementary first-derivative tests to nonlinear optimal control. They provide necessary conditions for a trajectory-control-pair to be optimal, or sufficient conditions for local controllability. Sufficient conditions for optimality (and necessary conditions for nonlinear controllability) are harder to obtain. Like the Legendre-Clebsch condition, they generally take the form of tests for definiteness of second order derivatives. Recently, Agrachev introduced an attractive alternative by developing a notion of curvature of optimal control that generalizes classical Gauss (and Ricci) curvatures. This theory naturally applies to systems whose controls take values on a circle or sphere. In this article we present initial studies of how this notion of curvature provides insight into the limiting case when the circles become degenerate ellipses in the form of closed intervals. Of particular interest are well studied accessible, but uncontrollable, nonlinear systems, and systems that exhibit conjugate points, in which the control takes values in a closed interval u = (u1,u2) ∈ [-1, 1 × {0} ⊆ R2. We focus on systems that are well-known models for the analysis of small-time local controllability and time-optimal control.
KW - Curvature
KW - Optimal control
UR - http://www.scopus.com/inward/record.url?scp=62449139428&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=62449139428&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:62449139428
SN - 0324-8569
VL - 37
SP - 353
EP - 367
JO - Control and Cybernetics
JF - Control and Cybernetics
IS - 2
ER -