Involutive subdistributions and canonical forms for distributions and control systems.

*(English)*Zbl 0621.93023
Theory and applications of nonlinear control systems, Sel. Pap. 7th Int. Symp. Math. Theory Networks Syst. Stockholm 1985, 123-135 (1986).

[For the entire collection see Zbl 0592.00038.]

This interesting paper deals with the problem to construct canonical forms for regular local distributions \(D^ k\) described by k linearly independent smooth vector fields on an n-manifold. The problem is to find a basis of vector fields for \(D^ k\) which have an affine representation relative to some appropriate local coordinate system. The existence of such a basis is shown for regular k-distributions \(D^ k\) on a \((k+2)\)- dimensional manifold which have a (k-1)-dimensional involutive subdistribution and a first derived distribution of dimension \(k+2\). The approach of the paper is inspired by control theory, in particular the problem of finding a canonical form for the control system \(\dot x=\sum u_ iX^ i(x)\) under the feedback group. The relation to results of M. E. Cartan on the classification of Pfaffian systems is also explained. The paper ends with some results and comments concerning the classification of the feedback equivalence classes of nilpotent affine control systems described by vector fields which are homogeneous relative to a given dilation on \({\mathbb{R}}^ n\).

This interesting paper deals with the problem to construct canonical forms for regular local distributions \(D^ k\) described by k linearly independent smooth vector fields on an n-manifold. The problem is to find a basis of vector fields for \(D^ k\) which have an affine representation relative to some appropriate local coordinate system. The existence of such a basis is shown for regular k-distributions \(D^ k\) on a \((k+2)\)- dimensional manifold which have a (k-1)-dimensional involutive subdistribution and a first derived distribution of dimension \(k+2\). The approach of the paper is inspired by control theory, in particular the problem of finding a canonical form for the control system \(\dot x=\sum u_ iX^ i(x)\) under the feedback group. The relation to results of M. E. Cartan on the classification of Pfaffian systems is also explained. The paper ends with some results and comments concerning the classification of the feedback equivalence classes of nilpotent affine control systems described by vector fields which are homogeneous relative to a given dilation on \({\mathbb{R}}^ n\).

Reviewer: D.Hinrichsen

##### MSC:

93C10 | Nonlinear systems in control theory |

58A30 | Vector distributions (subbundles of the tangent bundles) |

93B10 | Canonical structure |

58A17 | Pfaffian systems |