High-order small-time local controllability.

*(English)*Zbl 0703.93013
Nonlinear controllability and optimal control, Lect. Workshop, New Brunswick/NJ (USA) 1987, Pure Appl. Math., Marcel Dekker 133, 431-467 (1990).

[For the entire collection see Zbl 0699.00040.]

A smooth control system \(\dot x=f(x)+\sum u_ jg^ j(x)\) on \({\mathbb{R}}^ n\) is called small-time locally controllable (STLC) if for all positive times t the rest point \(x=0\) (assuming \(f(0)=0)\) is contained in the interior of the attainable set A(t), that is the set of all points reachable from \(x=0\) in time t by means of solution curves of the system resulting from measurable controls u (with values in a specified compact subset of \({\mathbb{R}}^ m).\)

This article surveys the recent progress in characterizations of the STLC property, of finding both necessary and sufficient conditions for a given system to be STLC, and it provides a wealth of simple examples that illustrate each of these conditions.

It follows the mainly differential-geometric analytical approach taken earlier by Sussmann and many others. The man tool is a Lie series expansion (Chen-Fliess series) of solutions of the system. Conditions for STLC are expressed in terms of the readily computable iterated Lie brackets of the vector fields f and \(g_ j.\)

Starting with the accessibility rank condition, over families of control variations, high order approximating cones of tangent-vectors to the attainable set, and families of dilations on \({\mathbb{R}}^ n\), this article develops the main ideas behind the most recent conditions for STLC (in several series of papers by Hermes, Sussmann, Stefani, a.o.). Emphasis is placed on simple examples illustrating these conditions. Finally, new results on new good brackets and on the neutralization and balancing of bad brackets are given. The article concludes with introducing families of control variations that are parametrized by the number of switchings, and that give rise to very unexpected controllability, and thus also optimality results.

A smooth control system \(\dot x=f(x)+\sum u_ jg^ j(x)\) on \({\mathbb{R}}^ n\) is called small-time locally controllable (STLC) if for all positive times t the rest point \(x=0\) (assuming \(f(0)=0)\) is contained in the interior of the attainable set A(t), that is the set of all points reachable from \(x=0\) in time t by means of solution curves of the system resulting from measurable controls u (with values in a specified compact subset of \({\mathbb{R}}^ m).\)

This article surveys the recent progress in characterizations of the STLC property, of finding both necessary and sufficient conditions for a given system to be STLC, and it provides a wealth of simple examples that illustrate each of these conditions.

It follows the mainly differential-geometric analytical approach taken earlier by Sussmann and many others. The man tool is a Lie series expansion (Chen-Fliess series) of solutions of the system. Conditions for STLC are expressed in terms of the readily computable iterated Lie brackets of the vector fields f and \(g_ j.\)

Starting with the accessibility rank condition, over families of control variations, high order approximating cones of tangent-vectors to the attainable set, and families of dilations on \({\mathbb{R}}^ n\), this article develops the main ideas behind the most recent conditions for STLC (in several series of papers by Hermes, Sussmann, Stefani, a.o.). Emphasis is placed on simple examples illustrating these conditions. Finally, new results on new good brackets and on the neutralization and balancing of bad brackets are given. The article concludes with introducing families of control variations that are parametrized by the number of switchings, and that give rise to very unexpected controllability, and thus also optimality results.

Reviewer: M.Kawski

##### MSC:

93B05 | Controllability |

93B29 | Differential-geometric methods in systems theory (MSC2000) |

49K15 | Optimality conditions for problems involving ordinary differential equations |

93C10 | Nonlinear systems in control theory |

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