A gradient flow equation for optimal control problems with end-point cost

In this paper we consider a control system of the form $\dot x = F(x)u$, linear in the control variable $u$. Given a fixed starting point, we study a finite-horizon optimal control problem that consists in minimizing a weighted sum of an end-point cost and the squared $2$-norm of the control. We study the gradient flow induced by this functional on the Hilbert space of the admissible controls, and we prove a convergence result by means of the Simon-Lojasiewicz inequality. Finally, we prove that, if we let the weight of the end-point cost tend to infinity, a $\Gamma$-convergence result holds, and it turns out that the limiting problem consists in joining the starting point and a minimizer of the end-point cost with a horizontal length-minimizer path.Comment: 24 pages. Corrections in Section

Ensembles of affine-control systems with applications to Deep Learning

This thesis is devoted to the study of optimal control problems of ensembles of dynamical systems, where the dynamics has an affine dependence in the controls. By means of $\Gamma$-convergence arguments, we manage to approximate infinite ensembles with a sequence of growing-in-size finite ensembles. The advantage of this approach is that, under a suitable change of the states space, finite ensembles of control systems can be treated as a \textit{single} control system. Motivated by this fact, in the first part of the thesis we formulate a gradient flow equation on the space of admissible controls related to \textit{single} optimal control problems with end-point cost. Then, this is applied to the case of finite ensembles, where it is used to derive an implementable algorithm for the numerical resolution of ensemble optimal control problems. We also consider an iterative method based on the Pontryagin Maximum Principle. Finally, in the last part of the thesis, we formulate the task of the interpolation of a diffeomorphism with a Deep Neural Network as an ensemble optimal control problem. Therefore, we can take advantage the algorithms developed before to \textit{train} the network

Overview of optical technology for aerosols characterization

The study of the optical properties of particles in micro and macro environments is a field that collects valuable information on the chemistry and microphysics of aerosols (referring to particles and gases). Constant improvements in instrument and detector designs and mechanisms enable measurements with ever-increasing precision and resolution. Moreover, in recent years low-cost open-source technology has emerged as a massive offer, with very good features, which has allowed low-income laboratories and institutions to carry out their own optical measurements on samples and systems of different natures. This review aims to present the main current advances in optical measurements of air samples using optical properties. Emphasis is placed on low-cost technology and its characterization and applications due to its current boom and mass marketing, with the added value of new scientific articles showing advances in its uses and applications. On the one hand, the theory behind spectral photometric measurements is described, and on the other, the bases for detection from the scattering of a coherent light beam. Finally, the main advances in the literature making use of these phenomena are described, complemented with general results of own measurements. In order to focus the analysis, these aspects are presented based on atmospheric measurements.Fil: Scagliotti, Ariel Fabricio. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mendoza; Argentina. Universidad Tecnológica Nacional; Argentin

Optimizing chemotherapy for patients with advanced non-small cell lung cancer

AbstractPlatinum-based therapy remains the standard of care for the first-line treatment of patients with advanced non-small cell lung cancer (NSCLC). When combined with a third-generation agent, platinum-based doublets improve survival compared with the third-generation agent given alone. Controversy remains, however, regarding the relative risks and benefits of the third-generation agents. Four large phase III trials have addressed this question, with only one trial finding a survival benefit in one of the treatment arms. TAX 326 compared docetaxel-based therapy with vinorelbine/cisplatin, and found that survival, response, and quality of life outcomes all favoured the docetaxel/cisplatin regimen. Consistent benefits have been reported with this regimen in other studies. The non-platinum-based docetaxel/gemcitabine combination is an alternative for patients who are not suitable candidates for platinum-based therapy. Other results have shown that single-agent docetaxel is an appropriate option for elderly patients and those with poor performance status. Overall, the wealth of data with docetaxel in advanced NSCLC suggests that it plays an important role in first-line treatment and, as a single agent, can be considered a reasonable approach in elderly and frail patients

Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs

The term "Normalizing Flows" is related to the task of constructing invertible transport maps between probability measures by means of deep neural networks. In this paper, we consider the problem of recovering the $W_2$-optimal transport map $T$ between absolutely continuous measures $\mu,\nu\in\mathcal{P}(\mathbb{R}^n)$ as the flow of a linear-control neural ODE. We first show that, under suitable assumptions on $\mu,\nu$ and on the controlled vector fields, the optimal transport map is contained in the $C^0_c$-closure of the flows generated by the system. Assuming that discrete approximations $\mu_N,\nu_N$ of the original measures $\mu,\nu$ are available, we use a discrete optimal coupling $\gamma_N$ to define an optimal control problem. With a $\Gamma$-convergence argument, we prove that its solutions correspond to flows that approximate the optimal transport map $T$. Finally, taking advantage of the Pontryagin Maximum Principle, we propose an iterative numerical scheme for the resolution of the optimal control problem, resulting in an algorithm for the practical computation of the approximated optimal transport map.Comment: Correction of typos and new bibliographical references. 32 pages, 1 figur