Controllability and positivity constraints in population dynamics with age structuring and diffusion

This Accepted Manuscript will be available for reuse under a CC BY-NC-ND licence after 24 months of embargo periodIn this article, we study the null controllability of a linear system coming from a population dynamics model with age structuring and spatial diffusion (of Lotka–McKendrick type). The control is localized in the space variable as well as with respect to the age. The first novelty we bring in is that the age interval in which the control needs to be active can be arbitrarily small and does not need to contain a neighbourhood of 0. The second one is that we prove that the whole population can be steered into zero in a uniform time, without, as in the existing literature, excluding some interval of low ages. Moreover, we improve the existing estimates of the controllability time and we show that our estimates are sharp, at least when the control is active for very low ages. Finally, we show that the system can be steered between two positive steady states by controls preserving the positivity of the state trajectory. The method of proof, combining final-state observability estimates with the use of characteristics and with L∞ estimates of the associated semigroup, avoids the explicit use of parabolic Carleman estimatesThe research of Enrique Zuazua was supported by the Advanced Grant DyCon (Dynamical Control) of the European Research Council Executive Agency (ERC), the MTM2014-52347 and MTM2017-92996 Grants of the MINECO (Spain) and the ICON project of the French ANR-16-ACHN-001

Controllability of the one-dimensional fractional heat equation under positivity constraints

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Communications on Pure and Applied Analysis following peer review. The definitive publisher-authenticated version Biccari, U., Warma, M., & Zuazua, E. (2020). Controllability of the one-dimensional fractional heat equation under positivity constraints. Communications on Pure & Applied Analysis, 19(4), 1949-1978. is available online at https://www.aimsciences.org/article/doi/10.3934/cpaa.2020086In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian (−dx2)s (0 < s < 1) on the interval (−1, 1). We prove the existence of a minimal (strictly positive) time Tmin such that the fractional heat dynamics can be controlled from any initial datum in L2(−1, 1) to a positive trajectory through the action of a positive control, when s > 1/2. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical resultsThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement NO. 694126-DyCon). The work of the three authors is partially supported by the Air Force Office of Scientific Research under Award NO: FA9550-18-1-0242. The work of the first and of the third author was partially supported by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain) and by the ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government. The work of the third author was partially supported by the Alexander von Humboldt-Professorship program, the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement NO. 765579-ConFlex, and by the Grant ICON-ANR-16-ACHN-0014 of the French AN

Controllability for a population equation with interior degeneracy

We deal with a degenerate model in divergence form describing the dynamics of a population depending on time, on age and on space. We assume that the degeneracy occurs in the interior of the spatial domain and we focus on null controllability. To this aim, first we prove Carleman estimates for the associated adjoint problem, then, via cut off functions, we prove the existence of a null control function localized in the interior of the space domain. We consider two cases: either the control region contains the degeneracy point $x_0$, or the control region is the union of two intervals each of them lying on one side of $x_0$. This paper complement some previous results, concluding the study of the subject.Comment: arXiv admin note: text overlap with arXiv:1812.11416, arXiv:1611.0602

Equations of structured population dynamics.

Guo Bao Zhu.Thesis (Ph.D.)--Chinese University of Hong Kong.Includes bibliographical references.Abstract --- p.1Introduction --- p.3Chapter Chapter 1. --- Semigroup for Age-Dependent Population Equations with Time DelayChapter 1.1 --- Introduction --- p.13Chapter 1.2 --- Problem Statement and Linear Theory --- p.14Chapter 1.3 --- Spectral Properties of the Infinitesimal Generator --- p.20Chapter 1.4 --- A Nonlinear Semigroup of the Logistic Age-Dependent Model with Delay --- p.26References --- p.34Chapter Chapter 2. --- Global Behaviour of Logistic Model of Age-Dependent Population GrowthChapter 2.1 --- Introduction --- p.35Chapter 2.2 --- Global Behaviour of the Solutions --- p.37Chapter 2.3 --- Oscillatory Properties --- p.47References --- p.51Chapter Chapter 3. --- Semigroups for Age-Size Dependent Population Equations with Spatial DiffusionChapter 3. 1 --- Introduction --- p.52Chapter 3.2 --- Properties of the Infinitesimal Generator --- p.54Chapter 3.3 --- Properties of the Semigroup --- p.59Chapter 3.4 --- Dynamics with Age-Size Structures --- p.61Chapter 3.5 --- Logistic Model with Diffusion --- p.66References --- p.70Chapter Chapter 4. --- Semi-Discrete Population Equations with Time DelayChapter 4. 1 --- Introduction --- p.72Chapter 4.2 --- Linear Semi-Discrete Model with Time Delay --- p.74Chapter 4.3 --- Nonlinear Semi-Discrete Model with Time Delay --- p.88References --- p.98Chapter Chapter 5. --- A Finite Difference Scheme for the Equations of Population DynamicsChapter 5.1 --- Introduction --- p.99Chapter 5.2 --- The Discrete System --- p.102Chapter 5.3 --- The Main Results --- p.107Chapter 5.4 --- A Finite Difference Scheme for Logistic Population Model --- p.113Chapter 5.5 --- Numerical Simulation --- p.116References --- p.119Chapter Chapter 6. --- Optimal Birth Control Policies IChapter 6. 1 --- Introduction --- p.120Chapter 6.2 --- Fixed Horizon and Free Point Problem --- p.120Chapter 6.3 --- Time Optimal Control Problem --- p.129Chapter 6.4 --- Infinite Horizon Problem --- p.130Chapter 6.5 --- Results of the Nonlinear System with Logistic Term --- p.143Reference --- p.148Chapter Chapter 7. --- Optimal Birth Control Policies IIChapter 7. 1 --- Free Final Time Problems --- p.149Chapter 7.2 --- Systems with Phase Constraints --- p.160Chapter 7.3 --- Mini-Max Problems --- p.166References --- p.168Chapter Chapter 8. --- Perato Optimal Birth Control PoliciesChapter 8.1 --- Introduction --- p.169Chapter 8.2 --- The Duboviskii-Mi1yutin Theorem --- p.171Chapter 8.3 --- Week Pareto Minimum Principle --- p.172Chapter 8.4 --- Problem with Nonsmooth Criteria --- p.175References --- p.181Chapter Chapter 9. --- Overtaking Optimal Control Problems with Infinite HorizonChapter 9. 1 --- Introduction --- p.182Chapter 9.2 --- Problem Statement --- p.183Chapter 9.3 --- The Turnpike Property --- p.190Chapter 9.4 --- Existence of Overtaking Optimal Solutions --- p.196References --- p.198Chapter Chapter 10. --- Viable Control in Logistic Populatiuon ModelChapter 10. 1 --- Introduction --- p.199Chapter 10. 2 --- Viable Control --- p.200Chapter 10.3 --- Minimum Time Problem --- p.205References --- p.208Author's Publications During the Candidature --- p.20

Risk Minimization for Spreading Processes over Networks via Surveillance Scheduling and Sparse Control

Spreading processes, such as epidemics and wildfires, have an initial localized outbreak that spreads rapidly throughout a network. The real-world risks associated with such events have stressed the importance and current limitations of methods to quickly map and monitor outbreaks and to reduce their impact by planning appropriate intervention strategies. This thesis is, therefore, concerned with risk minimization of spreading processes over networks via surveillance scheduling and sparse control. This is achieved by providing a flexible optimization framework that combines surveillance and intervention to minimize the risk. Here, risk is defined as the product of the probability of an outbreak occurring and the future impact of that outbreak. The aim is now to bound or minimize the risk by allocation of resources and use of persistent monitoring schedules. When setting up an optimization framework, four other aspects have been found to be of importance. First of all, being able to provide targeted risk estimation and minimization for more vulnerable or high cost areas. Second and third, scalability of algorithms and sparsity of resource allocation are essential due to the large network structures. Finally, for wildfires specifically, there is a gap between the information embedded in fire propagation models and utilizing it for path planning algorithms for efficient remote sensing. The presented framework utilizes the properties of positive systems and convex optimization, in particular exponential cone programming, to provide flexible and scalable algorithms for both surveillance and intervention purposes. We demonstrate with different spreading process examples and scenarios, focusing on epidemics and wildfires, that the presented framework gives convincing and scalable results. In particular, we demonstrate how our method can include persistent monitoring scenarios and provide more targeted and sparse resource allocation compared to previous approaches

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

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal