##
**Controllability of evolution equations.**
*(English)*
Zbl 0862.49004

Lecture Notes Series, Seoul. 34. Seoul: Seoul National Univ. 163 p. (1996).

This monograph is mainly based on the authors’ research activity in the period 1990-1996 in the field of optimal control theory, more exactly, in controllability of evolution equations which is one of the most developing areas of the domain.

The book is structured in four chapters as follows:

Chapter 1 is devoted to controllability of parabolic equations. Firstly is proved a Carleman estimate for adjoint parabolic equations which is then applied to exact controllability of linear parabolic equation. There are also studied exact and locally exact controllability of semilinear parabolic equations, of Burgers equation and there are proved some results on uncontrollability of semilinear parabolic equations.

The second Chapter deals with the local controllability problem for the Boussinesq system that describes the incompressible fluid flow coupled to thermal dynamics. Using a variant of the implicit function theorem the authors reduce the study of the controllability property for the nonlinear problem to the solvability of the analogous problem for its linearization.

Chapter III is concerned with the local exact controllability of the 2-D Navier-Stokes system, defined in a bounded domain \(\Omega \subset R^2\) for the control distributed on the whole boundary \(\partial \Omega\), or on its part. The case of local distributed control is also studied.

In the last Chapter, problems of exact boundary controllability of second order hyperbolic equations are studied.

Each chapter begins with an introductory section in which the authors provide an extensive guide to the literature, pointing out their own contributions. The printing is excellent and the appearance of the book is pleasing. Making up a modern account of some of the latest research in this area it is expected that the book to be of major interest to applied mathematicians and research engineers.

The book is structured in four chapters as follows:

Chapter 1 is devoted to controllability of parabolic equations. Firstly is proved a Carleman estimate for adjoint parabolic equations which is then applied to exact controllability of linear parabolic equation. There are also studied exact and locally exact controllability of semilinear parabolic equations, of Burgers equation and there are proved some results on uncontrollability of semilinear parabolic equations.

The second Chapter deals with the local controllability problem for the Boussinesq system that describes the incompressible fluid flow coupled to thermal dynamics. Using a variant of the implicit function theorem the authors reduce the study of the controllability property for the nonlinear problem to the solvability of the analogous problem for its linearization.

Chapter III is concerned with the local exact controllability of the 2-D Navier-Stokes system, defined in a bounded domain \(\Omega \subset R^2\) for the control distributed on the whole boundary \(\partial \Omega\), or on its part. The case of local distributed control is also studied.

In the last Chapter, problems of exact boundary controllability of second order hyperbolic equations are studied.

Each chapter begins with an introductory section in which the authors provide an extensive guide to the literature, pointing out their own contributions. The printing is excellent and the appearance of the book is pleasing. Making up a modern account of some of the latest research in this area it is expected that the book to be of major interest to applied mathematicians and research engineers.

Reviewer: G.Aniculăesei (Iaşi)

### MSC:

49J20 | Existence theories for optimal control problems involving partial differential equations |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

35Q30 | Navier-Stokes equations |

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