Equivalence and invariants of nonlinear control systems.

*(English)*Zbl 0712.93027
Nonlinear controllability and optimal control, Lect. Workshop, New Brunswick/NJ (USA) 1987, Pure Appl. Math., Marcel Dekker 133, 177-218 (1990).

[For the entire collection see Zbl 0699.00040.]

This paper deals with the problem of identifying complete sets of invariants for nonlinear control systems under various types of equivalence relations. For instance, two systems \(x'=f(x,u)\) and \(y'=g(y,v)\) are said to be state-equivalent if there exists a change of coordinates (a diffeomorphism) which transforms one system to the other, without affecting the control variable (i.e., such that \(u=v)\). A new proof of some well known results about state equivalence of analytic systems is given. Further, a new set of complete invariants is presented. This result can be extended to the \(C^{\infty}\) case.

Weaker equivalence relations can be obtained allowing feedback connections and changes of the time scale, in addition to changes of coordinates. The author presents several results about invariants for each type of equivalence, and applications to the exact linearization problem for affine systems. In particular, a generic classification for one- and two-dimensional families of systems in view of weak and mild feedback equivalences is obtained.

This paper deals with the problem of identifying complete sets of invariants for nonlinear control systems under various types of equivalence relations. For instance, two systems \(x'=f(x,u)\) and \(y'=g(y,v)\) are said to be state-equivalent if there exists a change of coordinates (a diffeomorphism) which transforms one system to the other, without affecting the control variable (i.e., such that \(u=v)\). A new proof of some well known results about state equivalence of analytic systems is given. Further, a new set of complete invariants is presented. This result can be extended to the \(C^{\infty}\) case.

Weaker equivalence relations can be obtained allowing feedback connections and changes of the time scale, in addition to changes of coordinates. The author presents several results about invariants for each type of equivalence, and applications to the exact linearization problem for affine systems. In particular, a generic classification for one- and two-dimensional families of systems in view of weak and mild feedback equivalences is obtained.

Reviewer: A.Bacciotti

##### MSC:

93C10 | Nonlinear systems in control theory |

93B10 | Canonical structure |

93C15 | Control/observation systems governed by ordinary differential equations |

93B18 | Linearizations |