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Curvatures of single-input control systems

Jakubczyk B.

Control and Cybernetics

2009

Vol. 38, no 4B

1375-1391

We introduce invariants of control-affine systems which we call curvatures. They are defined by the drift and the control distribution, given by the system. The curvatures allow us to analyse the variational equation along a given trajectory, as well as existence of conjugate points.

Phase portraits of planar control-affine systems

Jakubczyk B., Respondek W.

Control and Cybernetics

2005

Vol. 34, no 3

819-847

We study nonlinear control systems in the plane, affine with respect to control. We introduce two sets of feedback equivariants forming a phase portrait PP and a parameterized phase portrait PPP of the system. The phase portrait PP consists of an equilibrium set E, a critical set C (parameterized, for PPP), an optimality index, a canonical foliation and a drift direction. We show that under weak generic assumptions the phase portraits determine, locally, the feedback and orbital feedback equivalence class of a system. The basic role is played by the critical set C and the critical vector field on C. We also study local classification problems for systems and their families.