Classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems

Necessary and sufficient conditions for convexity and strong convexity, respectively, of sublevel sets that are defined by finitely many real-valued $C^{1,1}$-maps are presented. A novel characterization of strongly convex sets in terms of the so-called local quadratic support is proved. The results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous-time nonlinear systems to be strongly convex. An application of these conditions is a novel method to over-approximate attainable sets when strong convexity is present.Comment: 20 pages, 3 figure

Knowledge Does Not Fall Far from the Tree - A Case Study on the Diffusion of Solar Cells in Germany

The purpose of this paper is to illuminate the geographical diffusion of photovoltaic installations in Germany quantitatively and to test if preexisting photovoltaic systems stimulate further installations nearby; thus we investigate to which extent knowledge flows depend on geographic proximity. We develop an econometric model, which is discrete in time and space, but the level of geographical agglomeration is adjustable in arbitrarily small steps. We find that the probability to install a photovoltaic system dependents on the geographic proximity to agents, who have previously installed a photovoltaic system. In conclusion, our results confirm that knowledge exchange attenuates with distance.

Sample path large deviations for Laplacian models in (1+1)(1+1)-dimensions

For Laplacian models in dimension $(1+1)$ we derive sample path large deviations for the profile height function, that is, we study scaling limits of Gaussian integrated random walks and Gaussian integrated random walk bridges perturbed by an attractive force towards the zero-level, called pinning. We study in particular the regime when the rate functions of the corresponding large deviation principles admit more than one minimiser, in our models either two, three, or five minimiser depending on the pinning strength and the boundary conditions. This study complements corresponding large deviation results for gradient systems with pinning for Gaussian random walk bridges in $(1+1)$-dimension (\cite{FS04}) and in $(1+d)$-dimension (\cite{BFO}), and recently in higher dimensions in \cite{BCF}. In particular it turns out that the Laplacian cases, i.e., integrated random walks, show richer and more complex structures of the minimiser of the rate functions which are linked to different phases.Comment: 37, 5 figure

Approximately Optimal Controllers for Quantitative Two-Phase Reach-Avoid Problems on Nonlinear Systems

The present work deals with quantitative two-phase reach-avoid problems on nonlinear control systems. This class of optimal control problem requires the plant's state to visit two (rather than one) target sets in succession while minimizing a prescribed cost functional. As we illustrate, the naive approach, which subdivides the problem into the two evident classical reach-avoid tasks, usually does not result in an optimal solution. In contrast, we prove that an optimal controller is obtained by consecutively solving two special quantitative reach-avoid problems. In addition, we present a fully-automated method based on Symbolic Optimal Control to practically synthesize for the considered problem class approximately optimal controllers for sampled-data nonlinear plants. Experimental results on parcel delivery and on an aircraft routing mission confirm the practicality of our method.Comment: 14 pages, 7 figure

Creature Care

Caring for a pet is a formidable responsibility that requires intense care and monitoring in order to protect and prolong the life of the animal. Traditionally, caring for a reptile presents a unique challenge for pet owners, who not only need to care for the animal but for the environment as well. Individual species distinctly require precise monitoring and control of environmental parameters such as temperature, humidity, and air quality; and deviation from the nominal values often leads to serious health problems for the reptile. To simplify reptile ownership, a system was designed that measures the critical parameters of a reptile enclosure and automatically regulates the local environment, notifying the user of any issues that require manual intervention. The system provides live and historical data, accessible on the Web via any computer or mobile device, and is capable of alerting the user by email or text message when any problems occur within the system. Additional reminders for feeding and cleaning are also configurable based on the requirements of the species. This design gives peace of mind for the user, ensuring that the animal within the enclosure is being provided the most ideal local environment unique to the species