Topological optimization via cost penalization

We consider general shape optimization problems governed by Dirichlet boundary value problems. The proposed approach may be extended to other boundary conditions as well. It is based on a recent representation result for implicitly defined manifolds, due to the authors, and it is formulated as an optimal control problem. The discretized approximating problem is introduced and we give an explicit construction of the associated discrete gradient. Some numerical examples are also indicated

Maximal monotonicity and convex programming

We introduce an explicit constraint qualification condition which is necessary and sufficient for the nondegenerate Lagrange multipliers rule to hold. We compare it with metric regularity conditions and we show that it is strictly weaker than the Slater assumption. Under certain weak smoothness hypotheses, our condition, the Slater condition and the existence of nondegenerate Lagrange multipliers are equivalent. The basic ingredient in the proof of the main result is the theory of maximal monotone operators (Minty's theorem). Another approach using a direct exact penalization argument yields a modified nondegenerate Lagrange multipliers rule involving the positive part of the constraint mapping. Examples and applications to abstract optimal control problems are also indicated

Optimal design of mechanical structures

We prove new properties for the linear isotropic elasticity system and for thickness minimization problems. We also present very recent results concerning shape optimization problems for three-dimensional curved rods and for shells. The questions discussed in this paper are related to the control variational method and to control into coefficients problems

Generalized bang-bang properties in the optimization of plates

For a simly supported plate, we consider two optimization problems: the volume minimization and the identification of a coefficient. Via a transformation recently introduced by the authors, we obtain the optimality conditions in a qualified form, and their analysis yields the bang-bang properties for the optimal thickness. Non consid{\'e}rons pour une plaque pos{\'e}e deux probl{\`e}mes d'optimisation: la minimisation du volume et l'identification d'un coeffcient. Via une transformation r{\'e}cemment introduite par les auteurs, on obtient les conditions d'optimalit{\'e} dans une forme qualifi{\'e}e et leur analyse entraine les propri{\'e}t{\'e}s de boum-boum pour l'{\'e}paisseur optimale

On the necessity of some constraint qualification conditions in convex programming

In this paper, we realize a study of various constraint qualification conditions for the existence of Lagrange multipliers for convex minimization problems in general normed vector spaces, it is based on a new formula for the normal cone to the constraint set, on local metric regularity and a metric regularity property on bounded subsets. As a by-product, we obtain a characterization of the metric regularity of a finite family of closed convex sets

Optimization of ordinary differential systems with hysteresis

We investigate general control problems governed by ordinary differential systems involving hysteresis operators. Our main hypotheses are of continuity type, and we discuss existence results, discretization methods, and approximation approaches

Sur les arches lipschitziennes

We study the Kirchhoff-Love model in the case when the middle curve of the arch has corners. Our approach does not use the Dirichlet principle or the Korn inequality. We propose a variational formulation based on optimal control theory and we obtain explicit formulas for the deformation

A general asymptotic model for Lipschitzian curved rods

In this paper we show that the asymptotic methods provide an advantageous approach to obtain models of thin elastic bodies under minimal regularity assumptions on the geometry. Our investigation is devoted to clamped curved rods with a nonsmooth line of centroids and the obtained model is a generalization of results already available in the literature

Shape optimization in free boundary systems

We analyze existence results in constrained optimal design problems governed by variational inequalities of obstacle type. The main applications that we discuss concern the optimal packaging problem and the electrochemical machining process. Our assumptions, in order to obtain the existence of at least one optimal domain, are just boundedness and uniform continuity (the uniform segment property) for the boundaries of the unknown regions where the free boundary problems are defined. No restrictions on the dimension are imposed