Characterizing positively invariant sets : Inductive and topological methods

Ghorbal, K. and Sogokon, A. (2022) Characterizing positively invariant sets : Inductive and topological methods. Journal of Symbolic Computation, 113. pp. 1-28. ISSN 0747-7171

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We present two characterizations of positive invariance of sets under the flow of systems of ordinary differential equations. The first characterization uses inward sets which intuitively collect those points from which the flow evolves within the set for a short period of time, whereas the second characterization uses the notion of exit sets, which intuitively collect those points from which the flow immediately leaves the set. Our proofs emphasize the use of the real induction principle as a generic and unifying proof technique that captures the essence of the formal reasoning justifying our results and provides cleaner alternative proofs of known results. The two characterizations presented in this article, while essentially equivalent, lead to two rather different decision procedures (termed respectively LZZ and ESE) for checking whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations. The procedure LZZ improves upon the original work by Liu, Zhan and Zhao (Liu et al., 2011). The procedure ESE, introduced in this article, works by splitting the problem, in a principled way, into simpler sub-problems that are easier to check, and is shown to exhibit substantially better performance compared to LZZ on problems featuring semi-algebraic sets described by formulas with non-trivial Boolean structure.

Item Type:
Journal Article
Journal or Publication Title:
Journal of Symbolic Computation
Uncontrolled Keywords:
?? decision proceduresdynamical systemsordinary differential equationspolynomial vector fieldspositively invariant setsalgebra and number theorycomputational mathematics ??
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Deposited On:
24 Mar 2023 16:40
Last Modified:
15 Jul 2024 22:21