Journal
JOURNAL OF GLOBAL OPTIMIZATION
Volume 62, Issue 3, Pages 575-613Publisher
SPRINGER
DOI: 10.1007/s10898-014-0235-6
Keywords
Interval analysis; Ellipsoidal calculus; Taylor models; Ordinary differential equations; Differential inequalities; Convergence analysis; Dynamic optimization; Global optimization
Funding
- Engineering and Physical Sciences Research Council (EPSRC) [EP/J006572/1]
- Marie Curie Career Integration Grant [PCIG09-GA-2011-293953]
- Centre of Process Systems Engineering (CPSE) of Imperial College
- CONACYT
- Engineering and Physical Sciences Research Council [EP/J006572/1] Funding Source: researchfish
- EPSRC [EP/J006572/1] Funding Source: UKRI
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This paper presents a framework for constructing and analyzing enclosures of the reachable set of nonlinear ordinary differential equations using continuous-time set-propagation methods. The focus is on convex enclosures that can be characterized in terms of their support functions. A generalized differential inequality is introduced, whose solutions describe such support functions for a convex enclosure of the reachable set under mild conditions. It is shown that existing continuous-time bounding methods that are based on standard differential inequalities or ellipsoidal set propagation techniques can be recovered as special cases of this generalized differential inequality. A way of extending this approach for the construction of nonconvex enclosures is also described, which relies on Taylor models with convex remainder bounds. This unifying framework provides a means for analyzing the convergence properties of continuous-time enclosure methods. The enclosure techniques and convergence results are illustrated with numerical case studies throughout the paper, including a six-state dynamic model of anaerobic digestion.
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