סמינר מחלקה של עומרי דלין - "תנאי מספיק ל- k-contraction שאיננו דורש חישוב של k-compound"

21 בנובמבר 2022, 14:00 - 15:00 
פקולטה להנדסה 
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סמינר מחלקה של עומרי דלין - "תנאי מספיק ל- k-contraction שאיננו דורש חישוב של k-compound"

 

 

 

School of Mechanical Engineering Seminar

Monday, November 21, 2022 at 14:00
Wolfson Building of Mechanical Engineering, Room 206

"Verifying k-contraction without computing k-compounds"

 

Omri Dalin

MSc of Lea Belkin

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 of the equilib-
rium for time-invariant vector fields.

A dynamical system is called k-contractive if the dynamics contracts k-parallelotopes
at an exponential rate. A sufficient condition for k-contraction is that a matrix measure
(also called logarithmic norm) of the k-additive compound of the Jacobian of the vector
field is uniformly negative. However, this may be difficult to check in practice because
the k-additive compound of an
n × n matrix has dimensions nk×nk.

For an n×n matrix A, we prove a duality relation between the k and (n−k) compounds
of A. We use this duality relation to derive a sufficient condition for k-contraction that does
not require the computation of any k-compounds. We demonstrate our results by deriving
a sufficient condition for k-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 this implies 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.

 

Join Zoom Meetin https://us02web.zoom.us/j/82108132163?pwd=Z2h4UzNzUS9mbXplT0lMU1pZenFEQT09

 

   

 

 
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