TY - JOUR

T1 - k-contraction

T2 - Theory and applications

AU - Wu, Chengshuai

AU - Kanevskiy, Ilya

AU - Margaliot, Michael

N1 - Publisher Copyright:
© 2021 Elsevier Ltd

PY - 2022/2

Y1 - 2022/2

N2 - A dynamical system is called contractive if any two solutions approach one another at an exponential rate. More precisely, the dynamics contracts lines at an exponential rate. This property implies highly ordered asymptotic behavior including entrainment to time-varying periodic vector fields and, in particular, global asymptotic stability for time-invariant vector fields. Contraction theory has found numerous applications in systems and control theory because there exist easy to verify sufficient conditions, based on matrix measures, guaranteeing contraction. We provide a geometric generalization of contraction theory called k-contraction. A dynamical system is called k-contractive if the dynamics contracts k-parallelotopes at an exponential rate. For k=1 this reduces to standard contraction. We describe easy to verify sufficient conditions for k-contraction based on a matrix measure of the kth additive compound of the Jacobian of the vector field. We also describe applications of the seminal work of Muldowney and Li, that can be interpreted in the framework of 2-contraction, to systems and control theory.

AB - A dynamical system is called contractive if any two solutions approach one another at an exponential rate. More precisely, the dynamics contracts lines at an exponential rate. This property implies highly ordered asymptotic behavior including entrainment to time-varying periodic vector fields and, in particular, global asymptotic stability for time-invariant vector fields. Contraction theory has found numerous applications in systems and control theory because there exist easy to verify sufficient conditions, based on matrix measures, guaranteeing contraction. We provide a geometric generalization of contraction theory called k-contraction. A dynamical system is called k-contractive if the dynamics contracts k-parallelotopes at an exponential rate. For k=1 this reduces to standard contraction. We describe easy to verify sufficient conditions for k-contraction based on a matrix measure of the kth additive compound of the Jacobian of the vector field. We also describe applications of the seminal work of Muldowney and Li, that can be interpreted in the framework of 2-contraction, to systems and control theory.

KW - Compound matrices

KW - Contraction analysis

KW - Entrainment

KW - Matrix measures

KW - Stability

KW - Variational equation

UR - http://www.scopus.com/inward/record.url?scp=85119936994&partnerID=8YFLogxK

U2 - 10.1016/j.automatica.2021.110048

DO - 10.1016/j.automatica.2021.110048

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AN - SCOPUS:85119936994

SN - 0005-1098

VL - 136

JO - Automatica

JF - Automatica

M1 - 110048

ER -