non-euclidean contraction and its extensions with

Non-Euclidean Contraction and its Extensionswith Application to Network Systems

Saber Jafarpour

Center for Control, Dynamical Systems & ComputationUniversity of California at Santa Barbara

Georgia Institute of Technology

May 3, 2021

Acknowledgment

Francesco BulloUCSB

Alexander DavydovUCSB

Pedro Cisneros-VelardeUCSB

SJ and P. Cisneros-Velarde and F. Bullo. Weak and Semi-Contraction for Network Systems andDiffusively-Coupled Oscillators. IEEE Transactions on Automatic Control, accepted, Mar. 2021.

SJ and A. Davydov and F. Bullo. Non-Euclidean Contraction Theory for Monotone and PositiveSystems. Submitted, Apr. 2021.

P. Cisneros-Velarde and SJ and F. Bullo. Distributed and time-varying primal-dual dynamics viacontraction analysis. Submitted, May 2020.

SJ (UCSB) Non-Euclidean Contraction Theory May 3, 2021 2 / 39

Motivation: Large-scale Nonlinear Networks

Power grids Brain neural network Transportation network

Nonlinearity:

Multiple equilibria

Transient stability

Cluster synchronization

Large-scale:

Stochastic

Distributed

“... As [power] systems become more heavily loaded,nonlinearities play an increasingly important role in powersystem behavior ... ” [I. Hiskens,1995]

“... in Oahu, Hawaii, at least 800,000 micro-invertersinterconnect photovoltaic panels to the grid... ”[IEEESpectrum, 2015]


Presentation outline

non-Euclidean contraction theory

one-sided Lipschitz constant

characterization of contraction wrt non-Euclidean norms

contraction-based small gain theorem

weakly-contracting systems

definition and examples

dichotomy in asymptotic behavior

example: distributed primal-dual

semi-contracting systems


convergence to invariant subspaces

example: diffusively-coupled oscillators


Contraction theory: a brief reviewDefinition

Definition: Contracting systems

x = f (t, x) is contracting wrt to ‖ · ‖ with rate c > 0:

‖x(t)− y(t)‖ ≤ e−ct‖x(0)− y(0)‖.

Contracting system: flow is a contracting map.

x0

y0

y(t)

x(t)

circle with radius e�ct


Contraction theory: a brief reviewHistorical notes

B. P. Demidovich. Dissipativity of a nonlinear system of differential equations.Uspekhi Matematicheskikh Nauk, 16(3(99)):216, 1961

Application in control theory: W. Lohmiller and J.-J. E. Slotine. On contraction analysisfor non-linear systems.Automatica, 34(6):683–696, 1998

Differential framework: F. Forni and R. Sepulchre. A differential Lyapunov framework forcontraction analysis.IEEE Trans. Autom. Control, 59(3):614–628, 2014

Non-Euclidean contraction: S. Coogan. A contractive approach to separable Lyapunovfunctions for monotone systems.Automatica, 106:349–357, 2019

Review: M. Di Bernardo, D. Fiore, G. Russo, and F. Scafuti. Convergence, consensus andsynchronization of complex networks via contraction theory.In Complex Systems and Networks: Dynamics, Controls and Applications, pages 313–339.Springer, 2016

Review: Z. Aminzare and E. D. Sontag. Contraction methods for nonlinear systems: Abrief introduction and some open problems.In Proc CDC, pages 3835–3847, Dec. 2014


Contraction theory: a brief reviewProperties of contracting systems

Highly ordered asymptotic and transient behaviors:

1 initial conditions are forgotten

2 no overshoot in distance between trajectories

3 time-invariant f : unique globally stable equilibrium

4 periodic f : unique globally stable periodic solution

5 robustness properties: input-to-state stability even in the presence ofunmodeled dynamics.


Contraction theory: a brief reviewRole of non-Euclidean norms

Why non-Euclidean norms?

1 systematic and efficient stability analysis:

conservation law: 1>n x = c

geometric symmetry: f (x + 1n) = f (x)

2 `2-norm gives conservative estimates:

contraction: lack of symmetry in dynamics: ∂fi∂xj6= ∂fj

∂xi

nonlinearity: frequency instability in power grids, cluster synchronization inBrain neural networks

3 error analysis for large-scale networks:

x ∈ N (0, σIn) =⇒ E(‖x‖22) = nσ2,

E(‖x‖2∞) ∼ 2 ln(n)σ2


Characterization of contracting systemsContraction tests

Fundamental Question

How to check if a system is contracting?

wrt `2-norm

1 Differential condition: Linear matrix inequality2 Integral condition: one-sided Lipschitz constant

wrt non-Euclidean norms

1 Differential condition: matrix measure

Differential conditions computationally intensive

Differential conditions challenging for switching systems


One-sided Lipschitz constant: scalar vector fields

Let x = f (x), where f : R→ R:

Lipschitz constant ` ∈ R

|f (x)− f (y)| ≤ `|x − y | =⇒ −` ≤ f ′(x) ≤ `

One-sided Lipschitz constant b ∈ R

(x − y)(f (x)− f (y)) ≤ b(x − y)2 =⇒ f ′(x) ≤ b

〈f (x)− f (y), x − y〉 ≤ b|x − y |2

By the Growall-Bellman Lemma: |x(t)− y(t)| ≤ ebt |y(0)− x(0)|.Numerical analysis: E. Hairer, S. P. Nørsett, and G. Wanner. Solving OrdinaryDifferential Equations I. Nonstiff Problems.

Springer, 1993


Contraction for `2 norm

For x ∈ Rn and differentiable f :

x = f (t, x)

For P = P> � 0, define ‖x‖2P = x>Px

equivalent properties:

1 osL: (f (t, x)− f (t, y))>P(x − y) ≤ −c‖x − y‖2P , for all x , y , t

2 LMI: PDf (t, x) + Df (t, x)>P � −2cP for all x , t,

3 dIS: D+‖x(t)− y(t)‖P ≤ −c‖x(t)− y(t)‖P , for all soltns x(·), y(·)4 IS: ‖x(t)− y(t)‖P ≤ e−c(t−t0)‖x(t0)− y(t0)‖P , for all soltns x(·), y(·)

If f not differentiable, then osL ⇐⇒ dIS ⇐⇒ IS

Question: How to extend osL and LMI to non-Euclidean norms?


Matrix measuresDefinition and properties

The matrix measure of A ∈ Rn×n wrt to ‖ · ‖:

µ‖·‖(A) := limh→0+

‖In + hA‖ − 1

h.

Directional derivative of norm ‖ · ‖ in direction of A,

Logarithmic norm:T. Strom. On logarithmic norms.

SIAM Journal on Numerical Analysis, 12(5):741–753, 1975

spectral property: −‖A‖ = <(λ) ≤ µ‖·‖(A) ≤ ‖A‖, for every λ ∈ spec(A)

<latexit sha1_base64="vOrRKPBiVk8xg8hJIHP4AKYwBzE=">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</latexit>

� 2 spec(A)

<latexit sha1_base64="PVarAe+K0M9BqBqm3WkEpzwluV8=">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</latexit>

µ(A)<latexit sha1_base64="+2YI3P4S02yI693kM1YWgCI1pDs=">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</latexit>kAk<latexit sha1_base64="KsOMA6ML84GanpF2NvPqQL08GII=">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</latexit>�kAk <latexit sha1_base64="YxGalPK4r7wyX3o0p7Jq9dJoizQ=">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</latexit>�µ(�A)

<latexit sha1_base64="HSNx8IVlHW3cdYTPbQXGy0ebtu0=">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</latexit>

↵(A) = max<(�)

<latexit sha1_base64="V+M2/Tg9DKokEeJ5QAur254UeBk=">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</latexit>

min<(�)

<latexit sha1_base64="TfFGiSI/Owjpnlfprkm3TijV/IU=">AAACvXicdVHbahRBEO0dL4njbaOPvjQuig+yzASCeTOiD74IEbNJYHtZuntqZpv0ZeiuiS7D4A/4Nb7qj/g39k5WNMla0HC6TtU5VJWotQqYZb8GyY2bt25vbd9J7967/+DhcOfRcXCNlzCRTjt/KngArSxMUKGG09oDN0LDiTh7u+JPzsEH5ewRLmuYGV5ZVSrJMabmw+dMgkXwylZUu4p7hQujJGUsjYplLFRg5XI+HGXjrA96HeRrMCLrOJzvDL6ywsnGRHWpeQjTPKtx1nKPSmroUtYEqLk84xVMI7TcQHgZzqsezNov/WgdfRa5gpbOx2eR9tl/e1tuQlgaESsNx0W4yq2Sm7hpg+X+rFW2bjAOeGFUNpqio6s90UJ5kKiXEXAZtxJ3IhfccxmXddlFmC5N2TuIY3r4EI3e6HrBBWDLettSYdce5V3LQllAyRsd/8wUf3HQf3DKPgH+T0M4XWxQEuVGpXiy/OqBroPj3XG+N84+7o4OsvXxtskT8pS8IDl5RQ7Ie3JIJkSSb+Q7+UF+Jq8TSHRiL0qTwbrnMbkUyeffGCHgfQ==</latexit>

logarithmicine�ciency


Weak semi-inner productsDefinition and properties

A weak semi-inner product (WSIP) is J·, ·K : Rn × Rn → R satisfying

1 Jx1 + x2, yK ≤ Jx1, yK + Jx2, yK and x 7→ Jx , yK is continuous,

2 Jαx , yK = Jx , αyK = α Jx , yK for α ≥ 0 and J−x ,−yK = Jx , yK,

3 Jx , xK > 0, for all x 6= 0,

4 | Jx , yK | ≤ Jx , xK1/2 Jy , yK1/2,

5 (compatibility) Jx , xK = ‖x‖2 for all x

Extension of semi-inner product: G. Lumer. Semi-inner-product spaces.

Transactions of the American Mathematical Society, 100:29–43, 1961

Properties:

Matrix measure: µ(A) = sup‖x‖=1

JAx , xK ,

Norm derivative formula: ‖x(t)‖D+‖x(t)‖ = Jx(t), x(t)K .


Norm WSIP Matrix measure

‖x‖P =√x>Px Jx , yK2,P1/2 = x>Py

µ2,P1/2 (A) = min{b ∈ R | A>P + PA � 2bP}

= max‖x‖P=1

x>PAx

‖x‖p =(∑

i

|xi |p)1/p

p ∈ ]1,∞[

Jx , yKp = ‖y‖2−pp (y ◦ |y |p−2)>x µp(A) = max

‖x‖p=1(x ◦ |x |p−2)>Ax

‖x‖1 =∑i

|xi | Jx , yK1 = ‖y‖1sign(y)>x

µ1(A) = maxj∈{1,...,n}

(ajj +

∑i 6=j

|aij |)

= sup‖x‖1=1

sign(x)>Ax

‖x‖∞ = maxi|xi | Jx , yK∞ = max

i∈I∞(y)yixi

µ∞(A) = maxi∈{1,...,n}

(aii +

∑j 6=i

|aij |)

= max‖x‖∞=1

maxi∈I∞(x)

xi (Ax)i

Table of norms, WSIPs, and matrix measures for weighted `2, `p for p ∈ (1,∞), `1, and`∞ norms. Note: I∞(x) = {i ∈ {1, . . . , n} | |xi | = ‖x‖∞}.


Contraction for arbitrary norm

For x ∈ Rn and differentiable f :

x = f (t, x)

For norm ‖ · ‖ with matrix measure µ(·) and compatible WSIP J·, ·K,equivalent properties:

1 osL: Jf (t, x)− f (t, y), x − yK ≤ −c‖x − y‖2 for all x , y , t ≥ 0,

2 MM: µ(Df (t, x)) ≤ −c , for all x , t ≥ 0,

3 dIS: D+‖x(t)− y(t)‖ ≤ −c‖x(t)− y(t)‖, for soltns x(·), y(·),

4 IS: ‖x(t)− y(t)‖ ≤ e−c(t−t0)‖x(t0)− y(t0)‖, for all soltns x(·), y(·)


Measure Demidovich One-sided Lipschitzbound condition condition

µ2,P(Df (x)) ≤ −c PDf (x) + Df (x)>P � −2cP (x − y)>P(f (x)− f (y)

)≤ −c ‖x − y‖2

P

µp(Df (x)) ≤ −c (v ◦ |v |p−2)>Df (x)v ≤ −c‖v‖pp ((x − y) ◦ |x − y |p−2)>(f (x)− f (y)) ≤ −c‖x − y‖pp

µ1(Df (x)) ≤ −c sign(v)>Df (x)v ≤ −c ‖v‖1 sign(x − y)>(f (x)− f (y)) ≤ −c ‖x − y‖1

µ∞(Df (x)) ≤ −c maxi∈I∞(v)

vi (Df (x)v)i ≤ −c ‖v‖2∞ max

i∈I∞(x−y)(xi − yi )(fi (x)− fi (y)) ≤ −c ‖x − y‖2

∞

Table of equivalences between measure bounded Jacobians, differential Demidovich andone-sided Lipschitz conditions. Note: I∞(v) = {i ∈ {1, . . . , n} | |vi | = ‖v‖∞}.


Contraction-based small-gain theoremSetting

n interconnected systems xi = fi (xi , x−i ) where xi ∈ RNi

Lyapunov functions Vi : RNi → R≥0:

LfiVi (xi ) =∂Vi

∂xifi (xi , x−i ) ≤ gi (Vi (xi ),V−i (x−i ))

Classical small-gain:

gi (v) = −αi (vi ) +∑j 6=i

γij(vj)

for class K∞ functions αi , γij .

Comparison system

Study properties of

v = g(v), g = (g1, . . . , gn)>

for v ∈ Rn≥0.


Contraction-based small-gain theoremMain result

V (x) = (V1(x1), . . . ,Vn(xn))>

J·, ·Kp,R is associated with ‖R(·)‖p, for p ∈ [1,∞], R ∈ Rn×n non-negative.

Theorem: Contraction-based small-gain

If there exists c > 0 such that for all v ≥ w ≥ 0n,

Jg(v)− g(w), v −wKp,R ≤ −c‖v −w‖2p,R ,

Then

‖V (x(t))‖p,R ≤ e−ct‖V (x(0))‖p,R

For R = diag(η) where η ∈ Rn>0

if p ∈ [1,∞), sum-separable Lyapunov function:∑n

i=1 ηpi V

pi (xi )

if p =∞, max-separable Lyapunov function: maxi{ηiVi (xi )}


Contraction-based small-gain theoremProof of the main result

Key Lemma

Jx , yKp,R ≤ Jz , yKp,R , for every x ≤ z and y ≥ 0n.

‖V (x(t))‖p,RD+‖V (x(t))‖p,R =rV (x(t)),V (x(t))

z

p,RNorm derivative formula

≤ Jg(V (x(t))),V (x(t))Kp,R Key Lemma

≤ −c‖V (x(t))‖2p,R osL

Unlike classical small-gain theorems we do not need g to be monotone.


Presentation outline

non-Euclidean contraction theory

one-sided Lipschitz constant


contraction-based small gain theorem

weakly-contracting systems


dichotomy in asymptotic behavior

example: distributed primal-dual

semi-contracting systems


convergence to invariant subspaces

example: diffusively-coupled oscillators


Contraction theory for networks

Challenge: many real-world networks are not contracting.

Network flow system x = f (x) preserving commodity x :

constant = 1>n x(t)

=⇒ 0 = 1>n x(t) = 1>n f (x(t))

=⇒ 0n = 1>n Df (x(t))

If additionally f has Metzler Jacobian, then µ1(Df (x)) = 0.


Weakly-contracting systemsDefinition and examples

Definition: Weakly-contracting systems

x = f (t, x) with f continuously differentiable in x is weakly-contracting wrt ‖ · ‖:

µ‖·‖(Df (t, x)) ≤ 0

1 Lotka-Volterra population dynamics (Lotka, 1920; Volterra, 1928) (`1-norm)

2 Kuramoto oscillators (Kuramoto, 1975) and coupled swing equations (Bergen and Hill,1981) (`1-norm and `∞-norm)

3 Daganzo’s cell transmission model for traffic networks (Daganzo, 1994), (`1-norm)

4 compartmental systems in biology, medicine, and ecology (Sandberg, 1978; Maeda et al.,1978). (`1-norm)

5 saddle-point dynamics for optimization of weakly-convex functions (Arrow et al., 1958).(`2-norm)


Weakly-contracting systemsPart I: Dichotomy in asymptotic behavior

Theorem: Dichotomy for weakly-contracting systems

For a weakly-contracting system x = f (x), either

1 f has no equilibrium and every trajectory is unbounded, or

2 f has at least one equilibrium x∗ and every trajectory is bounded.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6

x1 = �x2

x2 = x1

x1 = �2x1 + x2

x2 = �x1

x1 = �x1 + sin(x2) + 3

x2 = x1


Weakly-contracting systemsPart II: bounded trajectory case

Theorem

If x = f (x) is weakly-contracting and f has at least one equilibrium x∗ then:

(i) each equilibrium x∗∗ is stable with weak Lyapunov function x 7→ ‖x − x∗∗‖,

(ii) if the norm ‖ · ‖ is a (p,R)-norm, p ∈ {1,∞} and f is piecewise real analytic,then every trajectory converges to the set of equilibria,

(iii) x∗ is locally asy stable =⇒ x∗ is globally asy stable.

Idea of the proof

µ(Df(x)) 0

x⇤ x0xtmp

B(x⇤, ✏)x⇤

is locally asym stable

x(t)


Example: Primal-dual algorithmDistributed implementation over networks

Optimization problem

minx∈Rk

f (x) = minx∈Rk

n∑i=1

fi (x)

Distributed implementation

n agents communicate over a undirected weighted graph G ,

agent i have access to function fi and can exchange xi with its neighbors.

minx∈Rk

n∑i=1

fi (xi )

x1 = x2 = . . . = xn

In matrix form by assuming x = (x>1 , . . . , x>n )> ∈ Rnk :

minx∈Rk

n∑i=1

fi (xi )

(L⊗ Ik)x = 0nN


Example: Primal-dual algorithmDistributed implementation over networks

If each fi is continuously differentiable in xi :

Lagrangian

L(x , ν) =∑n

i=1 fi (xi ) + ν>(L⊗ Ik)x

Distributed primal-dual algorithm (component form):

xi = −∂L∂xi

= −∇fi (xi )−n∑

j=1

aij(νi − νj),

νi =∂L∂νi

=n∑

j=1

aij(xi − xj)

Distributed primal-dual algorithm (vector form):

x = −∇f (x)− (L⊗ Ik)ν,

ν = (L⊗ Ik)x


Example: Primal-dual algorithmStability and rate of convergence

Assume

1 f has a minimum x∗ ∈ Rk ,

2 for each i ∈ {1, . . . , n}, fi is twice differentiable, ∇2fi (x) � 0 for all x , and∇2fi (x

∗) � 0, and

3 the undirected weighted graph G is connected with Laplacian L.

Theorem: Distributed primal-dual dynamics

The distributed primal-dual algorithm

1 is weakly-contracting wrt `2-norm,

2 (x(t), ν(t))→ (1n ⊗ x∗,1n ⊗ ν∗), with ν∗ =∑n

i=1 νi (0),

3 exponential convergence rate is −αess

([−∇2f (x∗) −L⊗ IkL⊗ Ik 0

])where

αess(A) := max{<(λ) | λ ∈ spec(A) \ {0}}.


Semi-contracting systemsSemi-norms

Idea: flows converges to each other only in certain directions.

Definition: Semi-norms

|||·||| is a semi-norm if

1 |||cv ||| = |c ||||v |||, for every v ∈ Rn and c ∈ R;

2 |||v + w ||| ≤ |||v |||+ |||w |||, for every v ,w ∈ Rn.

Define the subspace Ker |||·||| = {v ∈ Rn | |||v ||| = 0}.Example: for k < n, R ∈ Rk×n, and norm ‖ · ‖, we get |||x |||R = ‖Rx‖.Example: for a network G with edge set E and incidence matrix B:

|||x |||E := max(i,j)∈E

|xi − xj | = ‖B>x‖∞

For strongly connected graphs Ker |||·|||E = span{1n}


Semi-contracting systemsMatrix semi-measures

Definition: Matrix semi-measures

The matrix semi-measure of A ∈ Rn×n wrt |||·|||:

µ|||·|||(A) = limh→0+

|||In + hA||| − 1

h.

Directional derivative of |||·||| in direction of A.

if Ker |||·||| is invariant under A then <(λ) ≤ µ|||·|||(A), for every

λ ∈ specKer |||·|||⊥(A>).


Semi-contracting systemsDefinition and examples

Definition: Semi-contracting systems

x = f (t, x) with f continuously differentiable in x is semi-contracting wrt thesemi-norm |||·||| with rate c > 0:

µ|||·|||(Df (t, x)) ≤ −c

1 Kuramoto oscillators (Kuramoto, 1975) and coupled swing equations (Bergen and Hill,1981), (`1-norm)

2 Chua’s diffusively-coupled circuits (Wu and Chua, 1995), (`2-norm)

3 morphogenesis in developmental biology (Turing, 1952), (`1-norm)

4 Goodwin model for oscillating auto-regulated gene (Goodwin, 1965). (`1-norm)


Semi-contracting systemsSemi-contraction and asymptotic behavior

Theorem: Semi-contracting systems

Consider x = f (t, x) with f continuously differentiable in x and assume

f is semi-contracting wrt the semi-norm |||·||| with rate c > 0, and

(Affine invariance): f (t, x∗ + Ker |||·|||) ⊆ Ker |||·||| for every t

Then,

1 for every trajectory x(t),

|||x(t)− x∗||| ≤ e−ct |||x(0)− x∗|||, for every t ≥ 0.

2 every trajectory converges to x∗ + Ker |||·|||.

partial contraction and horizontal contraction


Example: Diffusively-coupled oscillators

n agents connected by a weighted undirected graph G ,

identical internal dynamics f : R≥0 × Rk → Rk

xi = f (t, xi )−∑n

j=1 aij(xi − xj), i ∈ {1, . . . , n}

synchronization:

limt→∞ ‖xi − xj‖ = 0 for every i , j

synchronization of diffusively-coupled oscillators:

1 contractivity of the internal dynamics2 strength of the diffusive coupling



Introduce a local-global mixed norm: (2, p)-tensor norm on Rnk = Rn ⊗ Rk

‖u‖(2,p) = inf{( r∑

i=1

‖v i‖22‖w i‖2

p

) 12 ∣∣ u =

r∑i=1

v i ⊗ w i}.

closely related to, but different from, the projective tensor product norm

R. A. Ryan. Introduction to Tensor Products of Banach Spaces.

Springer, 2002

different from the mixed global norm

G. Russo, M. Di Bernardo, and E. D. Sontag. A contraction approach to the hierarchicalanalysis and design of networked systems.

IEEE Trans. Autom. Control, 58(5):1328–1331, 2013

Global norm: `2-norm for the interactions between agents

Local norm: `p-norm for internal dynamics of each agent



The orthogonal projection P ∈ Rn×n

P = In − 1n1n1>n =

n−1n − 1

n . . . − 1n

− 1n

n−1n . . . − 1

n...

.... . .

...− 1

n − 1n . . . n−1

n

(P ⊗ Ik)x measures dissimilarity of the states xi

x =1n ⊗ x∗ =⇒

(P ⊗ Ik)x = (P ⊗ Ik)(1n ⊗ x∗) = P1n ⊗ x∗ = 0(n−1)×k .



G is an undirected weighted graph with Laplacian L,

p ∈ [1,∞], Q ∈ Rk×k

xi = f (t, xi )−∑n

j=1 aij(xi − xj), i ∈ {1, . . . , n}

Theorem: diffusively-coupled oscillators are semi-contracting

Suppose that

µp,Q(Df (t, x)) ≤ λ2(L)− c , for every t, x

then

1 the dynamics is semi-contracting wrt ‖ · ‖(2,p),(P⊗Q);

2 for every trajectory x(t),

‖x(t)− 1n ⊗ xave(t)‖(2,p),(P⊗Q) ≤ e−ct‖x(0)− 1n ⊗ xave(0)‖(2,p),(P⊗Q).

3 the system achieves synchronization: limt→∞ x(t) = 1n ⊗ xave(t)

where xave(t) = 1n

∑ni=1 xi (t)



µp,Q(Df (t, x)) ≤ λ2(L)− c , for every t, x

trade off between internal dynamics and coupling strength

f time-invariant: every trajectory converges to the unique equilibrium point.

f periodic: every trajectory converges to the unique periodic orbit.

Unstable dynamics f , sufficiently strong coupling =⇒ λ2(L) large =⇒ thenetwork synchronizes.


Summary

notion of one-sided Lipschitz constant and weak semi-inner product


contraction-based small-gain theorem

two extensions of classical contraction:

weak contractionsemi-contraction

properties of weakly-contracting and semi-contracting systems


Future research

contraction-based compositional analysis of interconnected systems

scalable stability certificates using non-Euclidean contraction.

computing equilibra of contracting and weakly-contracting systems

explicit and implicit integration algorithms

accelerated convergence.

optimization algorithms using contraction theory

extension to gradient descent algorithms and time-varying algorithms.

connection with discrete-time algorithms for optimization.

implicit deep learning using contraction theory

use contraction condition for well-posedness in the optimization problem.


Contraction-based small gain theoremA simple example

Consider the following system on R2:

x1 = −x1 + 3x42 x1 − x3

1

x2 = −x2 − 3x41 x2 − x5

2

and pick Vi (xi ) = x2i for i ∈ {1, 2}. Then

V1 = −2V1 + 6V1V22 − 2V 2

1 ≤ −2V1 + 6V1V22 := g1(V1,V2)

V2 = −2V2 − 6V 21 V2 − 2V 3

2 ≤ −2V2 − 6V 21 V2 := g2(V1,V2)

but g = (g1, g2)> is not monotone. However, for every v ≥ w ≥ 0n,

Jg(v)− g(w), v −wK2

=[v1 − w1 v2 − w2

] [−2v1 + 2w1 + 6v1v22 − 6w1w

22

−2v2 + 2w2 − 6v21 v2 + 6w2

1w2

]≤ −2|v1 − w1|2 − 2|v2 − w2|2 = −2‖v −w‖2

2


non-euclidean contraction and its extensions with

Documents