Optimization in elliptic problems with applications to mechanics of deformable bodies and fluid mechanics.

*(English)*Zbl 0947.49001
Operator Theory: Advances and Applications. 119. Basel: Birkhäuser. xxii, 522 p. (2000).

The monograph is devoted to mathematical models of optimal control problems from mechanics of deformable bodies (plates, shells,…) and fluid mechanics. Considerable attention is given to the mathematical setting of problems, to the relations between continuous models and discretized ones, to the dependence of solutions of the corresponding equations on controls, to the convergence of approximate solutions.

A distinctive feature of the monograph is that mostly compact sets of control, for instance, bounded sets from Sobolev spaces embedded in Lebesgue spaces, are considered if controls appear in coefficients of elliptic operators or if the control is the shape of a body. The cause for that is the fact that in many cases smooth controls comply with the physical and engineering aspects of the problem.

The monograph consists of six chapters. The Chapter 1 gives the functional-analytical framework of the book. The Chapters 2 and 3 are devoted to optimization problems for elliptic systems with controls in coefficients or in the right-hand side, among them various problems of eigenvalue or shape optimization are studied. In Chapter 4, direct problems for various mathematical models of plates and shells are investigated. The Chapter 5 is devoted to the optimization of structures (including some structures of composite materials), mostly the thickness of a plate or a shell being the control. In Chapter 6, various optimization problems for steady flows of viscous and nonlinear viscous fluids are studied, in particular, the optimization by body and surface forces, by the distribution of velocities on the inlet, by perforated walls, by the shape of domains or canals.

The monograph is written in an accessible and self-contained manner. It would be of a great interest to researchers working in optimization theory for partial differential equations and its applications as well as for graduate students with interests in applications of the theory of optimal control to real problems from mechanics.

A distinctive feature of the monograph is that mostly compact sets of control, for instance, bounded sets from Sobolev spaces embedded in Lebesgue spaces, are considered if controls appear in coefficients of elliptic operators or if the control is the shape of a body. The cause for that is the fact that in many cases smooth controls comply with the physical and engineering aspects of the problem.

The monograph consists of six chapters. The Chapter 1 gives the functional-analytical framework of the book. The Chapters 2 and 3 are devoted to optimization problems for elliptic systems with controls in coefficients or in the right-hand side, among them various problems of eigenvalue or shape optimization are studied. In Chapter 4, direct problems for various mathematical models of plates and shells are investigated. The Chapter 5 is devoted to the optimization of structures (including some structures of composite materials), mostly the thickness of a plate or a shell being the control. In Chapter 6, various optimization problems for steady flows of viscous and nonlinear viscous fluids are studied, in particular, the optimization by body and surface forces, by the distribution of velocities on the inlet, by perforated walls, by the shape of domains or canals.

The monograph is written in an accessible and self-contained manner. It would be of a great interest to researchers working in optimization theory for partial differential equations and its applications as well as for graduate students with interests in applications of the theory of optimal control to real problems from mechanics.

Reviewer: U.Raitums (Riga)

##### MSC:

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

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74P10 | Optimization of other properties in solid mechanics |

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

35J45 | Systems of elliptic equations, general (MSC2000) |