Giuseppe Buttazzo

An optimal design problem with perimeter penalization

We study the optimal design problem of finding the minimal energy configuration for a mixture of two conducting materials when a perimeter penalization of the unknown domain is added. We show that in this situation an optimal domain exists and that, under suitable assumptions on the data, it is an open set.

Rate monotonic analysis: the hyperbolic bound

We propose a novel schedulability analysis for verifying the feasibility of large periodic task sets under the rate monotonic algorithm when the exact test cannot be applied on line due to prohibitively long execution times. The proposed test has the same complexity as the original Liu and Layland (1973) bound, but it is less pessimistic, thus allowing it to accept task sets that would be rejected using the original approach. The performance of the proposed approach is evaluated with respect to the classical Liu and Layland method and theoretical bounds are derived as a function of n (the number of tasks) and for the limit case of n tending to infinity. The analysis is also extended to include aperiodic servers and blocking times due to concurrency control protocols. Extensive simulations on synthetic tasks sets are presented to compare the effectiveness of the proposed test with respect to the Liu and Layland method and the exact response time analysis.

An existence result for a class of shape optimization problems

Given a bounded open subset Ω of Rn, we prove the existence of a minimum point for a functional F defined on the family A(Ω) of all “quasiopen” subsets of Ω, under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A(Ω) with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal kth eigenvalue (or with minimal capacity) with respect to a given elliptic operator.

A variational definition of the strain energy for an elastic string

Using the variational point of view, the constitutive equations of an elastic one-dimensional string are deduced from the stress-strain relations of nonlinear three-dimensional elasticity, by passing to the limit when the other dimensions go to zero. The assumptions made on the three-dimensional model are not very restrictive.

General existence theorems for unilateral problems in continuum mechanics

The problem of minimizing a possibly non-convex and non-coercive functional is studied. Either necessary or sufficient conditions for the existence of solutions are given, involving a generalized recession functional, whose properties are discussed thoroughly. The abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region.

Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions

We study an optimal design problem for the domain of an elliptic equation with Dirichlet boundary conditions. We introduce a relaxed formulation of the problem which always admits a solution, and we prove some necessary conditions for optimality both for the relaxed and for the original problem.

Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands

The study of regularity properties for minimizers of problems of the form min , u, u’) dt: u E IV’-‘(Q; R”), u(a) = a, u(b) = B (1.1) has been undertaken for a long time in Calculus of Variations, and a number of results are by now available on this subject. When f(t, s, z) is a smooth coercive function, problem ( 1.1) was considered by Tonelli (see [ 18, 19]), who proved that every minimizer u(t) of the problem (1.1) is locally Lipschitz on a relatively open set of full measure in [a, b]. Under some additional assumptions on f, it is also possible to prove global regularity theorems (see for instance Cesari [3, Chap 2, Sect. 61). In recent years, much attention has been devoted to problems of the form (l.l), on the one hand in weakening the hypotheses of Tonelli’s regularity results, and on the other in finding counterexamples of nonregular minimizers for coercive functionals. With respect to this last subject, we recall that it has been shown (see Ball and Mizel [l] and Clarke and Vinter [6]) that there exist polynomial functions f(t, s, z) such that, for suitable C > 0 and p,

On Newton’s problem of minimal resistance

In 1685, Sir Isaac Newton studied the motion of bodies through an inviscid and incompressible medium. In his words (from his Principia Mathematica): If in a rare medium, consisting of equal particles freely disposed at equal distances from each other, a globe and a cylinder described on equal diameter move with equal velocities in the direction of the axis of the cylinder, (then) the resistance of the globe will be half as great as that of the cylinder.... I reckon that this proposition will be not without application in the building of ships.

Γ-convergence and optimal control problems

In this paper, we give some applications ofG-convergence and Γ-convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (Ph) of optimal control problems and a control problem (P∞), we determine some general conditions, involvingG-convergence and Γ-convergence, under which the sequence of the optimal pairs of the problems (Ph) converges to the optimal pair of problem (P∞).

Reinforcement by a Thin Layer with Oscillating Thickness

We study a singular perturbation problem involving variational functionals that nearly degenerate on a thin layer with rapidly varying thickness. One interpretation is the reinforcement of a rod in torsion by a thin, oscillatory coating of a very strong material.