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We present some results on the existence and uniqueness of solutions of a two-point nonlinear boundary value problem that arises in the modeling of the flow of the Antarctic Circumpolar Current.
The existence of nontrivial solutions is of considerable interest, since these correspond to azimuthal ows that feature variations in the meridional direction, being thus models that capture the essential geophysical features, conrmed by eld data. In the current research we apply an approach that guarantee existence if the general initial- value problem does not present the nite-time blow-up phenomenon and if a somewhat more general associated boundary-value problem has at most one solution.
We are concerned with optimal control strategies subject to uncertain demands. Taking uncertainty into account becomes more and more important in many areas. In the context of supply chain management, a need for control strategies taking these uncertainties into account naturally arises when it comes to production planning. Deviations from the demand actually realized need to be compensated, which might be very costly and should be avoided. To this end, we consider different approaches to control the produced amount at a given time to meet the stochastic demand in an optimal way (see [1]). Supply chains are represented by transport equations and stochastic differential equations of Ornstein-Uhlenbeck-type are used to model the uncertain demand. Finally, the approaches will be compared in a numerical simulation study.
Acknowledgments: The authors are grateful for the support of the German Research Foundation (DFG) within the project \Novel models and control for networked problems: from discrete event to continuous dynamics" (GO1920/4-1). References [1] S. Göttlich, R. Korn, and K. Lux, Optimal control of electricity input given an uncertain demand. arXiv preprint: 1810.05480, 2018.
Bioreactors are experimental devices where some microorganisms (or species) growth in the presence of some resource, the so-called substrate or nutrient. Among the different types of bioreactors, we will focus on the most common one, the chemostat, a continuously-fed system which has been regarded as an idealization of nature to study microbial ecosystems at steady state, a really important and interesting problem since it can be used to study genetically altered microorganisms, waste water treatment, antibiotic production, fermentation models and play an important role in theoretical ecology.
The chemostat device consists on three interconnected tanks, the feed bottle, the culture vessel and the collection vessel. Typically the nutrient is pumped from the feed bottle to the culture vessel, where the interactions between the substrate and the species take place and another flow is pumped from the second tank to the collection vessel in order to keep the volume in the culture vessel constant.
Typically some restrictions are usually imposed when considering deterministic chemostats, for instance, the availability of the nutrient and its supply rate, the so-called input flow, are fixed. Nevertheless, they are very strong assumptions, specially when taking into account that the real world is often subject to suffer fluctuations.
In this talk, two different ways of modeling disturbances on the input flow in chemostat models will be presented, motivated by biological reasons, in order to get mathematical models which fit better the real ones displayed in laboratories. To this end, we will make use of two well-known stochastic processes: the standard Wiener process and the Ornstein-Uhlenbeck process. In both cases, results concerning the existence and uniqueness of positive global solution will be provided and the asymptotic behavior of the state variables involved in the resulting models will be also studied. Apart from that, different conditions on the parameters of the models to ensure the extinction or the persistence of the microbial biomass will be given, which is the main goal pursued by practitioners. In addition, a comparison between both ways of modeling will be also presented with several numerical simulations to be able to decide the most suitable stochastic process to model real fluctuations on the input flow in chemostat models. Finally, several current and future possible works will be commented.
Elliptic and parabolic boundary value problems are already very well understood in the context of classical Sobolev spaces. However, this theory requires the boundary data to have a certain regularity. This assumption is usually not satisfied if one studies equations with stochastic noise as boundary data. In this talk, we introduce some ideas on how to resolve this issue. The two main approaches will be that we study boundary value problems in function spaces with power weights and in function spaces with dominating mixed smoothness.