Numerical methods for initial value problems for differential inclusions usually require a discretization of time as well as of the set valued right hand side. In this paper, two numerical fixed grid methods for the approximation of the full solution set are proposed and analyzed. Convergence results are proved which show the combined influence of time and (phase) space discretization.
The quartz crystal microbalance (QCM) was used to study the variability of acoustic properties of living cells on the sub-second time scale. A confluent cell layer of rat cardiac myocytes was grown onto the electrode of quartz crystal resonator. The cell layer performed periodic, synchronous contractions at a rate of about 1.5 Hz. In order to monitor these rather fast changes in the state of the cells, the QCM was operated in a "fast mode", which allows sampling of the shift of the resonance frequency and energy dissipation with a rate of up to 100 Hz. The contractions were clearly reflected in periodic variations of the resonance frequency and the bandwidth. The rate of the contractions, in particular, could be easily detected in this way. Building on the rate of contraction, the setup can be used to monitor the response of the cell layer to heart stimulating drugs like isoproterenol. Depending on the concentration of isoproterenol, the beat rate was found to increase by up to a factor of two.
We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of a function chosen in a reproducing kernel Hilbert space. In some sense, the method can be considered as an extension to the optimal control algorithm approach recently developed by Baier, Gerdts and Xausa. The convergence of the method is illustrated numerically for selected examples
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.