Reachability Analysis and Deterministic Global Optimization of DAE Models

This article provides a tutorial overview of recent progress in numerical methods for the reachability analysis and deterministic global optimization of DAE models. These problems are highly interrelated, and are global problems in the sense that they concern the parametric solutions of a DAE model over a potentially large range of model parameters, rather than locally about a single value. Many techniques are available for computing such global information for functions known in closed-form (i.e., factorable functions). Two of the simplest and most flexible techniques are interval arithmetic and McCormick’s relaxation technique. The methods reviewed herein extend these techniques to functions defined as the solutions of DAE models, which are notably non-factorable. In doing so, we repeatedly exploit the idea that the factorable representations of the DAE governing equations, combined with insights from dynamical systems theory, can be used to infer global information about the DAE solutions. This concept is first used to derive methods for computing interval bounds and more general convex enclosures of the solutions of DAE models over a range of model parameters. Subsequently, these enclosures are employed in a branch-and-bound algorithm for deterministic global dynamic optimization with DAE’s embedded. The article closes with an illustrative case study in parameter estimation and some prospects for future work.