##
**Lie brackets, real analyticity and geometric control.**
*(English)*
Zbl 0545.93002

Differential geometric control theory, Proc. Conf., Mich. Technol. Univ. 1982, Prog. Math. 27, 1-116 (1983).

[For the entire collection see Zbl 0503.00014.]

On over one hundred pages the author gives something between a survey and an introduction to a large part of the subject called now ”differential geometric methods in control theory”. The subject has been developed in the last fifteen years and already deserves a monograph. The paper is a first step in this direction.

The main problems discussed in the paper are: controllability, maximum principle, system equivalence, realization theory, local controllability, bang-bang theorem, singular controls, regular synthesis and others. One should mention here that the author contributed much to these subjects, many of the theorems are his own theorems. The reader gets then an expert exposition of many important problems in the theory of nonlinear control. Some of the results discussed in the paper are new results in other areas of mathematics as differential geometry or differential topology (e.g. orbit theorem, theorem on quotients of manifolds).

The paper is divided into ten chapters. The two introductory chapters introduce the necessary definitions and state some of the problems. The maximum principle is formulated in a geometric setting (on a manifold). In chapter 3 the author introduces the orbits of polysystems and discusses the question of the existence of integral manifolds of a family of vector fields. As a result he gets a Lie algebra criterion for accessibility.

Chapter 4 is concerned with the equivalence of polysystems under a change of coordinates in the state. Cartan’s techniques of the graph is used to prove that, roughly, the Lie brackets determine an analytic polysystem up to a local diffeomorphism. It is also proved that small-time local controllability in \({\mathbb R}^ 2\) is a finitely determined property. Chapter 5 deals with linear systems from the point of view of Lie brackets. In chapter 6 the author formulates his two results concerning global realization theory and mentions a result by M. Fliess.

Chapter 7 deals with small-time local controllability. Except of an introductory result a sufficient condition for local controllability due to H. Hermes and the author is given. Chapter 8 gives bang-bang theorems for linear and nonlinear systems and a theorem describing the structure of time-optimal trajectories in \({\mathbb R}^ 2\). The last chapter discusses the question of existence of optimal control in a form of feedback (regular synthesis of Boltyanskii). A technique of applying subanalytic sets to the problem of synthesis (developed by P. Brunovský and the author) is presented. Deep theorems on the existence of regular synthesis and the sufficiency of the maximum principle are obtained in the way.

Obviously, the choice of the material is influenced by the author’s interest. Among the subjects of geometric control which are not included in the paper one could mention: use of algebraic-geometric methods for studying families of linear systems, disturbance decoupling, global controllability, feedback equivalence and feedback linearization, higher order necessary conditions of optimality and others.

The author only sketches the proofs of many results. However, all the technical details concerning the formulations and hypotheses are given accurately. Thus, the paper is easy to read and, apart from specialists, it should be particularly useful for beginners with some preliminary experience in the subject.

On over one hundred pages the author gives something between a survey and an introduction to a large part of the subject called now ”differential geometric methods in control theory”. The subject has been developed in the last fifteen years and already deserves a monograph. The paper is a first step in this direction.

The main problems discussed in the paper are: controllability, maximum principle, system equivalence, realization theory, local controllability, bang-bang theorem, singular controls, regular synthesis and others. One should mention here that the author contributed much to these subjects, many of the theorems are his own theorems. The reader gets then an expert exposition of many important problems in the theory of nonlinear control. Some of the results discussed in the paper are new results in other areas of mathematics as differential geometry or differential topology (e.g. orbit theorem, theorem on quotients of manifolds).

The paper is divided into ten chapters. The two introductory chapters introduce the necessary definitions and state some of the problems. The maximum principle is formulated in a geometric setting (on a manifold). In chapter 3 the author introduces the orbits of polysystems and discusses the question of the existence of integral manifolds of a family of vector fields. As a result he gets a Lie algebra criterion for accessibility.

Chapter 4 is concerned with the equivalence of polysystems under a change of coordinates in the state. Cartan’s techniques of the graph is used to prove that, roughly, the Lie brackets determine an analytic polysystem up to a local diffeomorphism. It is also proved that small-time local controllability in \({\mathbb R}^ 2\) is a finitely determined property. Chapter 5 deals with linear systems from the point of view of Lie brackets. In chapter 6 the author formulates his two results concerning global realization theory and mentions a result by M. Fliess.

Chapter 7 deals with small-time local controllability. Except of an introductory result a sufficient condition for local controllability due to H. Hermes and the author is given. Chapter 8 gives bang-bang theorems for linear and nonlinear systems and a theorem describing the structure of time-optimal trajectories in \({\mathbb R}^ 2\). The last chapter discusses the question of existence of optimal control in a form of feedback (regular synthesis of Boltyanskii). A technique of applying subanalytic sets to the problem of synthesis (developed by P. Brunovský and the author) is presented. Deep theorems on the existence of regular synthesis and the sufficiency of the maximum principle are obtained in the way.

Obviously, the choice of the material is influenced by the author’s interest. Among the subjects of geometric control which are not included in the paper one could mention: use of algebraic-geometric methods for studying families of linear systems, disturbance decoupling, global controllability, feedback equivalence and feedback linearization, higher order necessary conditions of optimality and others.

The author only sketches the proofs of many results. However, all the technical details concerning the formulations and hypotheses are given accurately. Thus, the paper is easy to read and, apart from specialists, it should be particularly useful for beginners with some preliminary experience in the subject.

Reviewer: B. Jakubczyk

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

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

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

17B99 | Lie algebras and Lie superalgebras |

32B20 | Semi-analytic sets, subanalytic sets, and generalizations |

37N35 | Dynamical systems in control |

49L20 | Dynamic programming in optimal control and differential games |

93B03 | Attainable sets, reachability |

57R27 | Controllability of vector fields on \(C^\infty\) and real-analytic manifolds |

58C35 | Integration on manifolds; measures on manifolds |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

37C80 | Symmetries, equivariant dynamical systems (MSC2010) |

93B05 | Controllability |

93B15 | Realizations from input-output data |

93B50 | Synthesis problems |

93C05 | Linear systems in control theory |

93C10 | Nonlinear systems in control theory |

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