A decomposition technique for pursuit evasion games with many pursuers

Here we present a decomposition technique for a class of differential games. The technique consists in a decomposition of the target set which produces, for geometrical reasons, a decomposition in the dimensionality of the problem. Using some elements of Hamilton-Jacobi equations theory, we find a relation between the regularity of the solution and the possibility to decompose the problem. We use this technique to solve a pursuit evasion game with multiple agents

Regularity properties of optimal controls for problems with time-varying state and control constraints

Accepted versio

External validation of prognostic models predicting pre-eclampsia : individual participant data meta-analysis

Abstract Background Pre-eclampsia is a leading cause of maternal and perinatal mortality and morbidity. Early identification of women at risk during pregnancy is required to plan management. Although there are many published prediction models for pre-eclampsia, few have been validated in external data. Our objective was to externally validate published prediction models for pre-eclampsia using individual participant data (IPD) from UK studies, to evaluate whether any of the models can accurately predict the condition when used within the UK healthcare setting. Methods IPD from 11 UK cohort studies (217,415 pregnant women) within the International Prediction of Pregnancy Complications (IPPIC) pre-eclampsia network contributed to external validation of published prediction models, identified by systematic review. Cohorts that measured all predictor variables in at least one of the identified models and reported pre-eclampsia as an outcome were included for validation. We reported the model predictive performance as discrimination (C-statistic), calibration (calibration plots, calibration slope, calibration-in-the-large), and net benefit. Performance measures were estimated separately in each available study and then, where possible, combined across studies in a random-effects meta-analysis. Results Of 131 published models, 67 provided the full model equation and 24 could be validated in 11 UK cohorts. Most of the models showed modest discrimination with summary C-statistics between 0.6 and 0.7. The calibration of the predicted compared to observed risk was generally poor for most models with observed calibration slopes less than 1, indicating that predictions were generally too extreme, although confidence intervals were wide. There was large between-study heterogeneity in each model’s calibration-in-the-large, suggesting poor calibration of the predicted overall risk across populations. In a subset of models, the net benefit of using the models to inform clinical decisions appeared small and limited to probability thresholds between 5 and 7%. Conclusions The evaluated models had modest predictive performance, with key limitations such as poor calibration (likely due to overfitting in the original development datasets), substantial heterogeneity, and small net benefit across settings. The evidence to support the use of these prediction models for pre-eclampsia in clinical decision-making is limited. Any models that we could not validate should be examined in terms of their predictive performance, net benefit, and heterogeneity across multiple UK settings before consideration for use in practice. Trial registration PROSPERO ID: CRD42015029349

L∞ estimates on trajectories confined to a closed subset

This paper concerns the validity of estimates on the distance of an arbitrary state trajectory from the set of state trajectories which lie in a given state constraint set. These so called distance estimates have wide-spread application in state constrained optimal control, including justifying the use of the Maximum Principle in normal form and establishing regularity properties of value functions. We focus on linear, L∞ distance estimates which, of all the available estimates have, so far, been the most widely used. Such estimates are known to be valid for general, closed state constraint sets, provided the functions defining the dynamic constraint are Lipschitz continuous, with respect to the time and state variables. We ask whether linear, L∞ distance estimates remain valid when the Lipschitz continuity hypothesis governing t-dependence of the data is relaxed. We show by counter-example that these distance estimates are not valid in general if the hypothesis of Lipschitz continuity is replaced by continuity. We also provide a new hypothesis, ‘absolute continuity from the left’, for the validity of linear, L∞ estimates. The new hypothesis is less restrictive than Lipschitz continuity and even allows discontinuous time dependence in certain cases. It is satisfied, in particular, by differential inclusions exhibiting non-Lipschitz t-dependence at isolated points, governed, for example, by a fractional-power modulus of continuity. The relevance of distance estimates for state constrained differential inclusions permitting fractional-power time dependence is illustrated by an example in engineering design, where we encounter an isolated, square-root type singularity, concerning the t-dependence of the data