Equivariant moving frame method and the local equivalence of u xx = r(x, u, v, u x , v x ) under fiber-preserving transformations

Abstract

We use the equivariant moving frame method to study the local equivalence problem of scalar control equations of the form uxx = r(x, u, v, ux, vx) under the pseudo-group of fiber-preserving transformations X = ϕ(x), U = β(x, u), V = α(x, u, v). Three typical branches of the equivalence problem are considered: the degenerate case, which contains the control systems with largest fiber-preserving symmetry group, the branch containing the Hilbert–Cartan equation, and the generic case.