On State Constrained Optimal Shape Design Problems

This paper is concerned with the following optimal design problem with constraints both on the state and on the control: $$MinimizeJ(y,u)$$ (P) subject to $$A\left( u \right)y + \partial \varphi \left( y \right) \mathrel\backepsilon Bu + f,$$ (1.1) $$y \in K,$$ (1.2) $$u \in {U_{ad}}.$$ (1.3)

Sur les problèmes d'optimisation structurelle

We discuss existence theorems for shape optimization and material distribution problems. The conditions that we impose on the unknown sets are continuity of the boundary, respectively a certain measurability hypothesis. peerReviewed

Fixed domain approaches in shape optimization problems

This work is a review of results in the approximation of optimal design problems, defined in variable/unknown domains, based on associated optimization problems defined in a fixed ?hold-all? domain, including the family of all admissible open sets. The literature in this respect is very rich and we concentrate on three main approaches: penalization?regularization, finite element discretization on a fixed grid, controllability and control properties of elliptic systems. Comparison with other fixed domain approaches or, in general, with other methods in shape optimization is performed as well and several numerical examples are included.

Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions

Fixed domain methods have well-known advantages in the solution of variable domain problems including inverse interface problems. This paper examines two new control approaches to optimal design problems governed by general elliptic boundary value problems with Dirichlet boundary conditions. Numerical experiments are also included peerReviewed

A Fixed Domain Approach in Shape Optimization Problems with Neumann Boundary Conditions

Fixed domain methods have well-known advantages in the solution of variable domain problems, but are mainly applied in the case of Dirichlet boundary conditions. This paper examines a way to extend this class of methods to the more difficult case of Neumann boundary conditions.

A boundary controllability approach in optimal shape design problems

We indicate a formulation of optimal shape design problems as boundary control problems, based on some approximate controllability-type results. Numerical examples and a comparison with the standard method are included.

Controllability-type properties for elliptic systems and applications

We consider approximate and exact controllability results for elliptic problems. These results enable one to formulate optimal shape design problems in a fixed domain with certain boundary conditions.

A variational inequality approach to the problem of the design of the optimal covering of an obstacle

A Boundary Control Approach to an Optimal Shape Design Problem

Abstract We consider the problem of controlling the coincidence set in connection with an obstacle problem. We shall transform the obtained optimal shape design problem into a boundary control problem with Dirichlet boundary conditions.

A variational inequality approach to constrained control problems for parabolic equations

A distributed optimal control problem for parabolic systems with constraints in state is considered. The problem is transformed to control problem without constraints but for systems governed by parabolic variational inequalities. The new formulation presented enables the efficient use of a standard gradient method for numerically solving the problem in question. Comparison with a standard penalty method as well as numerical examples are given.

Existence for shape optimization problems in arbitrary dimension

We discuss some existence results for optimal design problems governed by second order elliptic equations with the homogeneous Neumann boundary conditions or with the interior transmission conditions. We show that our continuity hypotheses for the unknown boundaries yield the compactness of the associated characteristic functions, which, in turn, guarantees convergence of any minimizing sequences for the first problem. In the second case, weaker assumptions of measurability type are shown to be sufficient for the existence of the optimal material distribution. We impose no restriction on the dimension of the underlying Euclidean space.

Optimal control for state constrained two-phase Stefan problems

We give a new approach to state constrained control problems associated to non-degenerate nonlinear parabolic equations of Stefan type. We obtain uniform estimates for the violation of the constraints.