TY - JOUR

T1 - Verifying κ-Contraction Without Computing κ-Compounds

AU - Dalin, Omri

AU - Ofir, Ron

AU - Bar-Shalom, Eyal

AU - Ovseevich, Alexander

AU - Bullo, Francesco

AU - Margaliot, Michael

N1 - Publisher Copyright:
© 1963-2012 IEEE.

PY - 2024/3/1

Y1 - 2024/3/1

N2 - Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition for κ-contraction is that a logarithmic norm (also called matrix measure) of the κ-additive compound of the Jacobian is uniformly negative. However, this computation may be difficult to perform analytically and expensive numerically because the κ-additive compound of an n\× n matrix has dimensions binom n k\× \binom n k. This article establishes a duality relation between the κ and (n-k) compounds of an n\× n matrix A. This duality relation is used to derive a sufficient condition for κ-contraction that does not require the computation of any κ-compounds. These theoretical results are demonstrated by deriving a sufficient condition for κ-contraction of an n-dimensional Hopfield network that does not require to compute any compounds. In particular, for k=2 this sufficient condition implies that the network is 2-contracting and thus admits a strong asymptotic property: every bounded solution of the network converges to an equilibrium point, that may not be unique. This is relevant, for example, when using the Hopfield network as an associative memory that stores patterns as equilibrium points of the dynamics.

AB - Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition for κ-contraction is that a logarithmic norm (also called matrix measure) of the κ-additive compound of the Jacobian is uniformly negative. However, this computation may be difficult to perform analytically and expensive numerically because the κ-additive compound of an n\× n matrix has dimensions binom n k\× \binom n k. This article establishes a duality relation between the κ and (n-k) compounds of an n\× n matrix A. This duality relation is used to derive a sufficient condition for κ-contraction that does not require the computation of any κ-compounds. These theoretical results are demonstrated by deriving a sufficient condition for κ-contraction of an n-dimensional Hopfield network that does not require to compute any compounds. In particular, for k=2 this sufficient condition implies that the network is 2-contracting and thus admits a strong asymptotic property: every bounded solution of the network converges to an equilibrium point, that may not be unique. This is relevant, for example, when using the Hopfield network as an associative memory that stores patterns as equilibrium points of the dynamics.

KW - Contracting systems

KW - Hopfield networks

KW - matrix measure

KW - stability

KW - κ-shifted logarithmic norm

UR - http://www.scopus.com/inward/record.url?scp=85174823313&partnerID=8YFLogxK

U2 - 10.1109/TAC.2023.3326058

DO - 10.1109/TAC.2023.3326058

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AN - SCOPUS:85174823313

SN - 0018-9286

VL - 69

SP - 1492

EP - 1506

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

IS - 3

ER -