Implicit differential equations and Lie-Backlund mappings

In engineering, as well as in physics, one often encounters implicit systems of ordinary differential equations of the form F_i (t,y˙₁,…,y˙_m,y₁ ,…,y_m)=0, i=1,…,m, in the unknowns y₁,…,y_m, where the Jacobian matrix (∂F _i/∂y˙_j)_i,j is identically singular. We state a condition of well-posedness and provide a formula for the gauge degree of freedom, which is important in physics. A decomposition is established, which gives as a byproduct a clear-cut definition of the index. Implicit control systems, on the other hand, are often differentially flat. We employ tools stemming from the differential geometry of infinite jets and prolongations and Lie-Backlund applications, since the F_i´s must be differentiated an arbitrary number of times