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Stability verification for monotone systems using homotopy algorithms. (English) Zbl 1229.93115

Summary: A monotone self-mapping of the nonnegative orthant induces a monotone discrete-time dynamical system which evolves on the same orthant. If with respect to this system the origin is attractive then there must exist points whose image under the monotone map is strictly smaller than the original point, in the component-wise partial ordering. Here it is shown how such points can be found numerically, leading to a recipe to compute order intervals that are contained in the region of attraction and where the monotone map acts essentially as a contraction. An important application is the numerical verification of so-called generalized small-gain conditions that appear in the stability theory of large-scale systems.

MSC:

93C55 Discrete-time control/observation systems
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
93B28 Operator-theoretic methods

Software:

Bertini; HOM4PS; PHCpack
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Full Text: DOI arXiv

References:

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