International Conference on Nonlinear Differential Equations and Applications

Dear Participants, Colleagues and Friends It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA). This conference takes place at the Colégio Espírito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics. The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area

Nonlinear Stochastic Systems And Controls: Lotka-Volterra Type Models, Permanence And Extinction, Optimal Harvesting Strategies, And Numerical Methods For Systems Under Partial Observations

This dissertation focuses on a class of stochastic models formulated using stochastic differential equations with regime switching represented by a continuous-time Markov chain, which also known as hybrid switching diffusion processes. Our motivations for studying such processes in this dissertation stem from emerging and existing applications in biological systems, ecosystems, financial engineering, modeling, analysis, and control and optimization of stochastic systems under the influence of random environments, with complete observations or partial observations. The first part is concerned with Lotka-Volterra models with white noise and regime switching represented by a continuous-time Markov chain. Different from the existing literature, the Markov chain is hidden and canonly be observed in a Gaussian white noise in our work. We use a Wonham filter to estimate the Markov chain from the observable evolution of the given process, and convert the original system to a completely observable one. We then establish the regularity, positivity, stochastic boundedness, and sample path continuity of the solution. Moreover, stochastic permanence and extinction using feedback controls are investigated. The second part develops optimal harvest strategies for Lotka-Volterra systems so as to establish economically, ecologically, and environmentally reasonable strategies for populations subject to the risk of extinction. The underlying systems are controlled regime-switching diffusions that belong to the class of singular control problems. We construct upper bounds for the value functions, prove the finiteness of the harvesting value, and derive properties of the value functions. Then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time. We further show that this is a reasonable candidate for the best lower bound that one can expect. In the last part, we study optimal harvesting problems for a general systems in the case that the Markov chain is hidden and can only be observed in a Gaussian white noise. The Wonham filter is employed to convert the original problem to a completely observable one. Then we treat the resulting optimal control problem. Because the problem is virtually impossible to solve in closed form, our main effort is devoted to developing numerical approximation algorithms. To approximate the value function and optimal strategies, Markov chain approximation methods are used to construct a discrete-time controlled Markov chain. Convergence of the algorithm is proved by weak convergence method and suitable scaling

Optimal Control and Coordination of Small UAVs for Vision-based Target Tracking

Small unmanned aerial vehicles (UAVs) are relatively inexpensive mobile sensing platforms capable of reliably and autonomously performing numerous tasks, including mapping, search and rescue, surveillance and tracking, and real-time monitoring. The general problem of interest that we address is that of using small, fixed-wing UAVs to perform vision-based target tracking, which entails that one or more camera-equipped UAVs is responsible for autonomously tracking a moving ground target. In the single-UAV setting, the underactuated UAV must maintain proximity and visibility of an unpredictable ground target while having a limited sensing region. We provide solutions from two different vantage points. The first regards the problem as a two-player zero-sum game and the second as a stochastic optimal control problem. The resulting control policies have been successfully field-tested, thereby verifying the efficacy of both approaches while highlighting the advantages of one approach over the other. When employing two UAVs, one can fuse vision-based measurements to improve the estimate of the target's position. Accordingly, the second part of this dissertation involves determining the optimal control policy for two UAVs to gather the best joint vision-based measurements of a moving ground target, which is first done in a simplified deterministic setting. The results in this setting show that the key optimal control strategy is the coordination of the UAVs' distances to the target and not of the viewing angles as is traditionally assumed, thereby showing the advantage of solving the optimal control problem over using heuristics. To generate a control policy robust to real-world conditions, we formulate the same control objective using higher order stochastic kinematic models. Since grid-based solutions are infeasible for a stochastic optimal control problem of this dimension, we employ a simulation-based dynamic programming technique that relies on regression to form the optimal policy maps, thereby demonstrating an effective solution to a multi-vehicle coordination problem that until recently seemed intractable on account of its dimension. The results show that distance coordination is again the key optimal control strategy and that the policy offers considerable advantages over uncoordinated optimal policies, namely reduced variability in the cost and a reduction in the severity and frequency of high-cost events

SDEs, Jumps and Estimates

Long Title: Stochastic Ordinary Differential Equations with Jumps: Theory and Estimates. Chapters: Stochastic Integrals - Initial Approach to SDEs - Estimates of SDEs - Other Formulations of SDEs - SDEs with Reflection - PDE Connections

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described