##
**Infinite dimensional optimization and control theory.**
*(English)*
Zbl 0931.49001

Encyclopedia of Mathematics and Its Applications. 62. Cambridge: Cambridge University Press. xv, 798 p. (1999).

This book is an impressive monograph on infinite dimensional optimal control theory. It is divided into three big parts. The first one deals with control problems of ordinary differential equations. The second one is the most important one, both by the size and the quantity of results and techniques. It is devoted to partial differential equations with control and state constraints. The last part is concerned with relaxed controls both for ordinary and partial differential equations.

In Chapter 1 many examples are presented to give an overview of problems and questions dealt in the book. In Chapter 2, questions related to existence of optimal controls, the Hamiltonian formalism for optimality conditions, and some results for linear quadratic problems are presented in a pleasant way. In order to give a unified presentation of necessary optimality conditions for the different problems, control problems are written as abstract nonlinear programming problems. The first applications of this abstract theory are introduced in Chapter 3 for time optimal problems. Optimality conditions are obtained for solutions of the abstract problems thanks to approximate solutions obtained by the Ekeland variational principle. This abstract nonlinear programming theory, due to the author and Frankowska, is first presented in Chapter 4, in the context of finite dimensional problems. A minimum principle is next derived for control problems governed by ordinary differential equations. Many examples are presented in Chapter 4. Next these results are extended in Part 2, firstly in Chapter 6 for constraints defined in Hilbert spaces, and next in Chapter 7 for constraints defined in Banach spaces. In particular the case when the norm of the space is not GĂ˘teaux-differentiable off the origin is studied in detail. Chapter 5 is devoted to the semigroup theory and its applications to semilinear equations in Banach spaces. Existence results for semilinear wave equations are obtained. A minimum principle for time optimal problems, and for general control problems, with applications to semilinear wave equations are presented in Chapter 6. In order to deal with problems in the presence of pointwise state constraints, additional tools concerning the Phillips adjoint semigroup are developed in Chapter 7. Fractional powers of generators of analytic semigroups, and interpolation theory are next applied to prove local existence results for the Navier-Stokes equations in Chapter 8. Some controllability results for linear systems are reviewed in Chapter 9. Pontryagin minimum principle for problems with state constraints, including the case of pointwise state constraints, are derived in Chapter 10 and 11. The case of a point target is also considered under the condition that zero is an interior point of the control set. Applications to control problems for the Navier-Stokes equations are given. The last part begins with an introduction to spaces of relaxed controls (Chapter 12). The existence of relaxed controls and the minimum principle for problems governed by ordinary differential equations are studied in Chapter 13. Existence of relaxed controls for semilinear equation in Banach spaces is explored in the case where the corresponding semigroup \(S(t)\) is compact for \(t>0\) in the first part of Chapter 14. The case where \(S(t)\) is not compact is considered in the last part of Chapter 14.

The book contains many original results from recent publications, detailed comments and extended bibliographical remarks are provided at the end of each part. It is a very valuable guide into infinite-dimensional control theory and many related topics in functional analysis, functional equations, semigroup theory. This is an original and extensive contribution which is not covered by other recent books in the control theory.

In Chapter 1 many examples are presented to give an overview of problems and questions dealt in the book. In Chapter 2, questions related to existence of optimal controls, the Hamiltonian formalism for optimality conditions, and some results for linear quadratic problems are presented in a pleasant way. In order to give a unified presentation of necessary optimality conditions for the different problems, control problems are written as abstract nonlinear programming problems. The first applications of this abstract theory are introduced in Chapter 3 for time optimal problems. Optimality conditions are obtained for solutions of the abstract problems thanks to approximate solutions obtained by the Ekeland variational principle. This abstract nonlinear programming theory, due to the author and Frankowska, is first presented in Chapter 4, in the context of finite dimensional problems. A minimum principle is next derived for control problems governed by ordinary differential equations. Many examples are presented in Chapter 4. Next these results are extended in Part 2, firstly in Chapter 6 for constraints defined in Hilbert spaces, and next in Chapter 7 for constraints defined in Banach spaces. In particular the case when the norm of the space is not GĂ˘teaux-differentiable off the origin is studied in detail. Chapter 5 is devoted to the semigroup theory and its applications to semilinear equations in Banach spaces. Existence results for semilinear wave equations are obtained. A minimum principle for time optimal problems, and for general control problems, with applications to semilinear wave equations are presented in Chapter 6. In order to deal with problems in the presence of pointwise state constraints, additional tools concerning the Phillips adjoint semigroup are developed in Chapter 7. Fractional powers of generators of analytic semigroups, and interpolation theory are next applied to prove local existence results for the Navier-Stokes equations in Chapter 8. Some controllability results for linear systems are reviewed in Chapter 9. Pontryagin minimum principle for problems with state constraints, including the case of pointwise state constraints, are derived in Chapter 10 and 11. The case of a point target is also considered under the condition that zero is an interior point of the control set. Applications to control problems for the Navier-Stokes equations are given. The last part begins with an introduction to spaces of relaxed controls (Chapter 12). The existence of relaxed controls and the minimum principle for problems governed by ordinary differential equations are studied in Chapter 13. Existence of relaxed controls for semilinear equation in Banach spaces is explored in the case where the corresponding semigroup \(S(t)\) is compact for \(t>0\) in the first part of Chapter 14. The case where \(S(t)\) is not compact is considered in the last part of Chapter 14.

The book contains many original results from recent publications, detailed comments and extended bibliographical remarks are provided at the end of each part. It is a very valuable guide into infinite-dimensional control theory and many related topics in functional analysis, functional equations, semigroup theory. This is an original and extensive contribution which is not covered by other recent books in the control theory.

Reviewer: J.-P.Raymond (Toulouse)

### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49K20 | Optimality conditions for problems involving partial differential equations |

93C20 | Control/observation systems governed by partial differential equations |

49K27 | Optimality conditions for problems in abstract spaces |

93B05 | Controllability |