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Erschienen in:
2014 | OriginalPaper | Buchkapitel
Characterizing Algebraic Invariants by Differential Radical Invariants
verfasst von : Khalil Ghorbal, André Platzer
Erschienen in: Tools and Algorithms for the Construction and Analysis of Systems
Verlag: Springer Berlin Heidelberg
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We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This so-called
differential radical characterization
relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it. The characterization leads to a differential radical invariant proof rule that is sound and complete, which implies that invariance of algebraic equations over real-closed fields is decidable. Furthermore, the problem of generating invariant varieties is shown to be as hard as minimizing the rank of a symbolic matrix, and is therefore NP-hard. We investigate symbolic linear algebra tools based on Gaussian elimination to efficiently automate the generation. The approach can, e.g., generate nontrivial algebraic invariant equations capturing the airplane behavior during take-off or landing in longitudinal motion.