I. Capuzzo Dolcetta

Approximate solutions of the bellman equation of deterministic control theory

We consider an infinite horizon discounted optimal control problem and its time discretized approximation, and study the rate of convergence of the approximate solutions to the value function of the original problem. In particular we prove the rate is of order 1 as the discretization step tends to zero, provided a semiconcavity assumption is satisfied. We also characterize the limit of the optimal controls for the approximate problems within the framework of the theory of relaxed controls.

Liouville theorems for semilinear equations on the Heisenberg group

Abstract In this paper we consider problems of the type Δ H u + h(ϰ)u p ⩾ 0, in D ⊂ R 2n + 1 u≥0 in D , where ΔH is the Heisenberg Laplacian, D is an unbounded domain and h is a non negative function. We prove that, under suitable conditions on h, p and D, the only solution of (1) is u ≡ 0.

Discrete Dynamic Programming and Viscosity Solutions of the Bellman Equation

Abstract This paper presents a technique for approximating the viscosity solution of the Bellman equation in deterministic control problems. This technique, based on discrete dynamic programming, leads to monotonically converging schemes and allows to prove a priori error estimates. Several computational algorithms leading to monotone convergence are reviewed and compared.

Hadamard and Liouville Type Results for Fully Nonlinear Partial Differential Inequalities

In this paper we prove some Hadamard and Liouville type properties for nonnegative viscosity supersolutions of fully nonlinear uniformly elliptic partial differential inequalities in the whole space.

ABP and global Hölder estimates for fully nonlinear elliptic equations in unbounded domains

We prove global Holder estimates for solutions of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions a la Cabre.

Finite element approximation of some indefinite elliptic problems

We consider the finite element approximation of some indefinite Neumann problems in a domain of IRN. From the Fredholm Alternative this kind of problem admits a solution if and only if the right hand term has zero mean value with respect to a measure whose density m is the solution of a homogeneous adjoint problem. The first step consists in the construction of piecewise linear finite element approximations mh of m, showing their optimal rate of convergence both in energy and Lp norms. The functions mh are then shown to be crucial in testing admissible data for the Neumann problem and also in its numerical resolution (actually, the standard Galerkin approximation may not be solvable without suitable perturbations of the data).

The impact of new technologies on teaching calculus: a report of an experiment

We present the methodology and results of a three‐year experiment using personal computers in calculus and advanced calculus courses. The main features of this experiment are: the use of computers during lectures to present and visualize different topics and the addition of working sessions on computers to the standard timetable of the courses. Different aspects related to the choice of the programming language, the organization of working sessions and to consequent changes in course programs are also discussed in view of the results of a test.