Shapes and geometries
as
design, optimization, or control variables

This lecture will provide an overview of recent results on the handling of the geometry as a modeling, optimization, or control variable with illustrative examples and applications. The increased interest in theoretical studies in this general area are motivated by numerous technological developments or phenomenological studies and the fact that such problems are quite different from their analogues involving only vectors of scalars or functions. Special tools and constructions are definitely required. In the past few decades the mathematical and computational communities have made considerable contributions to this general area of activity by nicely intertwining theoretical and numerical methods from optimal design, control theory, optimization, geometry, partial differential equations, free and moving boundary problems, and image processing.

In that context, a good analytical framework and good modeling techniques must be able to handle the mechanics, the physics, or the engineering of the problems at hand. Many basic mathematical ideas and methods have come from very different areas of applications and mathematical activities that have traditionally evolved in parallel directions. This field of research has been steadily broadening because it touches on areas that include classical geometry, modern partial differential equations, geometric measure theory, topological groups, constrained optimization, with applications to classical mechanics of continuous media such as fluid mechanics, elasticity theory, fracture theory, modern theories of optimal design, optimal location and shape of geometric objects, free and moving boundary problems, image processing.

Original issues raised in some applications have forced a new look at the fundamentals of well-established mathematical areas such as boundary value problems, to find suitable relaxation of solutions, or geometry, to relax the basic notions of volume, perimeter, and curvature. In that context Henri Lebesgue was certainly a pioneer when in 1907 he relaxed the intuitive notion of volume to the one of measure on an equivalence class of measurable sets. He was followed in that spirit in the early fifties by the celebrated work of E. De Giorgi who used the relaxed notion of finite perimeter defined on the class of Caccioppoli sets to solve Plateau's problem of minimal surfaces. This school of thoughts known as geometric measure theory considerably expanded under the impulsion and influence of pioneers such as H. Federer.

Reference

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization, SIAM series on Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, USA 2001.