Development of a Quality Measure for the Characterization of Guaranteed Solution Sets to ODEs in Engineering

Abstract

Solving initial value problems (IVPs) for ordinary differential equations (ODEs) is a common task in many scientific fields. Over the last decades, several different verified techniques have been developed to compute enclosures of the exact result numerically. The obtained bounds are guaranteed to contain the corresponding solution to the IVP and can be used to verify whether a given control strategy is admissible with respect to state and input constraints. Ideally, we aim at calculating tight enclosures over sufficiently long time intervals for systems with uncertainties given as sets. However, the existing solvers are not always equal in attaining this goal. On the one hand, the quality of the obtained results depends strongly on the types of ODEs that describe a given dynamical system. On the other hand, a great influence of the considered uncertainties can be observed. Our general aim is to provide assistance in choosing an appropriate verified IVP solver with its most suitable ‘tuning parameters’ for the application at hand. In this paper, we describe briefly the basics of verified IVP solvers. After a short introduction to a possible framework for the fair comparison of different approaches, we concentrate on possibilities to quantify the conservativeness introduced by verified solvers for problems with uncertainties.