A remark on nonlinear accessibility conditions and infinite prolongations

Abstract

The Lie rank condition for strong nonlinear accessibility is interpreted via the differential geometry of jets and prolongations of infinite order. It yields to a new Lie algebraic criterion and to the consideration of first integrals, which apply to nonlinear systems in quite general form; the latter characterization in particular is valid without distinguishing between the system variables and is independent of time-scalings. Weak accessibility is also considered.

Diffieties and Liouvillian Systems

“…Proof. Since σ S is flat, σ S ≃ σ F , hence ker ∂ M = ker ∂ S = R, i.e., any local first integral of σ M is trivial, and it follows that σ M is locally controllable (cf [8]).…”

On a Characteristic Vector Field for Systems Reducible to Order Two

“…Controllability (strong accessibility) is known to be directly related to the existence of invariants (first integrals) (Chow, 1940;Rashevsky, 1938;Pavlovsky and Yakovenko, 1982;Schipanov, 1939;Fliess et al, 1997). Here, the attention is put on the connection of these concepts with equation 7, or otherwise stated with the characteristic prolongation.…”

Trajectory Optimization for Flat Dynamic Systems

“…9 This property refers to the linearizability of systems subject to an endogenous state feedback transformation and a state-dependent time-scaling transformation. It was shown by Fliess et al 11 and Guay 15 that this concept can be viewed as a generalization of the differential flatness property to weakly locally accessible nonlinear systems. The orbital flatness of a simple chemical reactor due to Kravaris and Chung 20 was demonstrated in Guay 15 .…”