ABSTRACT
Shape Optimization was introduced around 1970 by Jean Ce´a [31], who understood, after several engineering studies [127, 12, 35, 110, 102, 83, 84, 7], the future issues in the context of optimization problems. At that time, he proposed a list of open problems at the French National Colloquium in Numerical Analysis. These new problems were formulated in terms of minimization of functionals (referred as open loop control or passive control) governed by partial differential boundary value problems where the control variable was the geometry of a given boundary part [103, 76]. From the beginning, the terminology shape optimization was not connected to the structural mechanical sciences in which elasticity and optimization of the compliance played a central role. Furthermore, these research studies were mainly addressed in the context of the numerical analysis of the finite element methods. At the same time, there was some independent close results concerning fluid mechanics by young researchers such as O. Pironneau [123, 124, 78], Ph. Morice [107] and also several approaches related to perturbation theory by P.R. Garabedian [74, 75] and D.D Joseph [91, 92]. Very soon, it appeared that the shape controlof Boundary Value Problems (BVP) was at the crossroads of several disciplines such as PDE analysis, non-autonomous semi-group theory, numerical approximation (including finite element methods), control and optimization theory, geometry and even physics. Indeed several classical modeling in both structural and fluid mechanics (among other fields) needed to be extended. An illustrative example concerns a very popular problem in the 80’s concerning the thickness optimization of a plate modeled by the classical Kirchoff biharmonic equation. This kind of solid model is based on the assumption that the thickness undergoes only small variations. Therefore, many pioneering works were violating the validity of this assumption, leading to strange results, e.g., the work presented in the Iowa NATO Study [85] stating the existence of optimal beams having zero cross section values. In the branch which followed the passive control approach, we shall mention the work of G. Chavent [32, 34] based on the theory of distributed system control introduced by J-L. Lions [98]. Those results did not address optimization problems related to the domain but instead related to the coefficients inside the PDE. At that time, it was hoped that the solution of elliptic problems would be continuous with respect to the weak convergence of the coefficients.
