##
**Singular trajectories and their role in control theory.**
*(English)*
Zbl 1022.93003

MathÃ©matiques & Applications (Berlin). 40. Paris: Springer. xvi, 357 p. (2003).

The authors consider singular optimal control problems when the constraint is a nonlinear control system evolving in a manifold \(M\). This means that the Lagrange multiplier corresponding to this constraint is zero. Closely related is the minimum time problem and the singular trajectories coming from the maximum principle with canonical Hamiltonian which are singularities of the endpoint mapping which associates to a control input the state obtained from the solution to the control system at the given final time.

The authors examine first the simpler linear case and then introduce to optimization (calculus of variations, optimal control) in the general nonlinear case. After that they provide some differential geometric machinery adapted to this context, namely symplectic geometry. They specialize the study to the singular case in each of these situations.

That the singularities of the endpoint mapping are invariant under the feedback group will have strong consequences for the classification problem (normal forms). Obviously, equivalent systems (under the feedback group) will lead to symplectomorphic lifted systems. Here the authors present some characterizations in dimension two or three.

Controllability is analyzed next, still in a differential geometric formulation, and the higher-order maximum principle is presented.

Then conjugate points in singular optimal control problems are introduced. Again the two- and three-dimensional cases are investigated more deeply.

The rest of the book splits into two directions: first an application to a chemical batch reactor with two variables (two chemical concentrations) controlled nonlinearly by the temperature (or its derivative), second more advanced theoretical properties of singular trajectories are developed. One examines the genericity of the normality property for singular extremals of minimal order (single input affine case). Next one discovers the case of optimal control in the context of sub-Riemannian geometry; here mainly specific situations are explored.

The last chapter will be of interest to specialists (and it is the most interesting one): One gets a stratification of the solution to the Hamilton-Jacobi-Bellman equation in the cotangent bundle. Two cases are considered, the single input affine case with drift and the sub-Riemannian case: affine without drift and two inputs which are projections of a point on the circle of unit radius.

Finally some open problems are pointed out: analyticity aspects in sub-Riemannian geometry, constraint issues, computation.

Each chapter has its set of problems. One finds 111 references and an index.

Notice that the authors implicitly assume that the state and controls belong to a direct product of spaces.

The book will be of interest to those interested in singular situations in optimal control.

The authors examine first the simpler linear case and then introduce to optimization (calculus of variations, optimal control) in the general nonlinear case. After that they provide some differential geometric machinery adapted to this context, namely symplectic geometry. They specialize the study to the singular case in each of these situations.

That the singularities of the endpoint mapping are invariant under the feedback group will have strong consequences for the classification problem (normal forms). Obviously, equivalent systems (under the feedback group) will lead to symplectomorphic lifted systems. Here the authors present some characterizations in dimension two or three.

Controllability is analyzed next, still in a differential geometric formulation, and the higher-order maximum principle is presented.

Then conjugate points in singular optimal control problems are introduced. Again the two- and three-dimensional cases are investigated more deeply.

The rest of the book splits into two directions: first an application to a chemical batch reactor with two variables (two chemical concentrations) controlled nonlinearly by the temperature (or its derivative), second more advanced theoretical properties of singular trajectories are developed. One examines the genericity of the normality property for singular extremals of minimal order (single input affine case). Next one discovers the case of optimal control in the context of sub-Riemannian geometry; here mainly specific situations are explored.

The last chapter will be of interest to specialists (and it is the most interesting one): One gets a stratification of the solution to the Hamilton-Jacobi-Bellman equation in the cotangent bundle. Two cases are considered, the single input affine case with drift and the sub-Riemannian case: affine without drift and two inputs which are projections of a point on the circle of unit radius.

Finally some open problems are pointed out: analyticity aspects in sub-Riemannian geometry, constraint issues, computation.

Each chapter has its set of problems. One finds 111 references and an index.

Notice that the authors implicitly assume that the state and controls belong to a direct product of spaces.

The book will be of interest to those interested in singular situations in optimal control.

Reviewer: A.Akutowicz (Berlin)

### MSC:

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

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

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

93C95 | Application models in control theory |

53D25 | Geodesic flows in symplectic geometry and contact geometry |

53C17 | Sub-Riemannian geometry |