arXiv subjects

Francesco Bartaloni

Publications and source records attributed to Francesco Bartaloni.

4 recordsLinked to original sources

arXiv2026

Regularization and Asymptotic Behaviour of Ornstein-Uhlenbeck Evolution Operators in Infinite Dimension

We are concerned with the properties of Ornstein-Uhlenbeck evolution operators acting on functions defined in a Hilbert space and p-integrable with respect to a suitable Gaussian measure. These operators provide solutions to the infinite dimensional, non-autonomous backward Kolmogorov equation. The first part of the paper focuses on some general regularization properties, while the second part carries on a deep analysis of the asymptotic behaviour in the periodic case. In particular, we identify the optimal convergence rate of the Ornstein-Uhlenbeck operator and we give an optimality criterion depending only on the drift term of the Kolmogorov equation.

math.PR↗

arXiv2024

An optimal uniqueness result for Riccati equations arising in abstract parabolic control problems

An abstract nonautonomous parabolic linear-quadratic regulator problem with very general final cost operator P_T is considered, subject to the same assumptions under which a classical solution of the associated differential Riccati equation was shown to exist, in two papers appeared in 1999 and 2000, by Terreni and the first named author. We prove an optimal uniqueness result for the integral Riccati equation in a wide and natural class, filling a gap existing in the autonomous case, too. In addition, we give a regularity result for the optimal state.

math.OC↗

arXiv2017

Existence of the optimum for Shallow Lake type models

We consider the optimal control problem associated with a general version of the well known shallow lake model, and we prove the existence of an optimum in the class $L_{loc}^{1}\left(0,+\infty\right)$. Any direct proof seems to be missing in the literature. Dealing with admissible controls that can be unbounded (even locally) is necessary in order to represent properly the concrete optimization problem; on the other hand, the non-compactness of the control space together with the infinite horizon setting prevents from having good \emph{a priori} estimates - and this makes the existence problem considerably harder. We present an original method which is in a way opposite to the classical control theoretic approach used to solve finite horizon Mayer or Bolza problems. Synthetically, our method is based on the following scheme: i) two uniform localization lemmas providing, given $T\geq1$ and a maximizing sequence of controls, another sequence of controls which is bounded in $L^{\infty}\left(\left[0,T\right]\right)$ and still maximizing. ii) A special diagonal procedure dealing with sequences which are not extracted one from the other. iii) A 'standard' diagonal procedure. The optimum results to be locally bounded by construction. Keywords: Optimal control, infinite horizon, non compact control space, uniform localization, convex-concave dynamics, logarithmic utility.

math.OC↗

arXiv2014

A utility maximization problem with state constraint and non-concave technology

We consider an optimal control problem arising in the context of economic theory of growth, on the lines of the works by Skiba (1978) and Askenazy - Le Van (1999). The economic framework of the model is intertemporal infinite horizon utility maximization. The dynamics involves a state variable representing total endowment of the social planner or average capital of the representative dynasty. From the mathematical viewpoint, the main features of the model are the following: (i) the dynamics is an increasing, unbounded and not globally concave function of the state; (ii) the state variable is subject to a static constraint; (iii) the admissible controls are merely locally integrable in the right half-line. Such assumptions seem to be weaker than those appearing in most of the existing literature. We give a direct proof of the existence of an optimal control for any initial value of the state variable and we carry on a qualitative study of the value function; moreover, using dynamic programming methods, we show that the value function is a continuous viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

math.OC↗