Manipulating exponential products

Instead of working with complicated concatenations of flows like

z(t) = e∫ t9t8

(f0+f1+f2) dt ◦ . . . e∫ t2t1

(f0+f1−f2) dt ◦ e∫ t10 (f0+f1+f2) dt(p)

it is desirable to rewrite the solution curve using a minimal number of

vector fields fπkthat span the tangent space (typically using iterated

Lie brackets of the system fields f0, f1, . . . fm)

Coordinates of the first kind

z(t) = eb1(t,u)fπ1+b2(t,u)fπ1+b3(t,u)fπ3+...+bn(t,u)fπn(p)

Coordinates of the second kind

z(t) = ec1(t,u)fπ1 ◦ ec2(t,u)fπ1 ◦ ec3(t,u)fπ3 ◦ . . . ◦ ecn(t,u)fπn(p)

Using the Campbell-Baker-Hausdorff formula, this is possible, but

a book-keeping nightmare.

Moreover, the CBH formula does not use a basis, but uses linear

combinations of all possible iterated Lie brackets. Yet, by the Jacobi

identity (and anticommutativity), in ever Lie algebra e.g.

[X, [Y, [X, Y ]]] + [Y, [[X, Y ], X ]︸ ︷︷ ︸] + [[X, Y ], [X, Y ]]] = 0

and hence

[X, [Y, [X, Y ]]] = [Y,︷ ︸︸ ︷[X, [X, Y ]]]

Plan:

• Work with bases for (free) Lie algebras.

• Find useful formulae for the coefficients bk(t, u) or ck(t, u).

12

The Chen Fliess series

K. T. Chen, 1957: Geometric invariants of curves in Rn

M. Fliess, 1970s: adaptation to control

The formal control system

S = S

⎛⎝ m∑

i=1uiXi

⎞⎠ , S(0) = I

on the associative algebra A(X1 . . .Xm) of formal power series in the

noncommuting indeterminates (letters) X1, . . .Xm has the unique solution

SCF (T, u) =∑I

∫ T

0

∫ t1

0· · ·

∫ tp−1

0uip(tp) . . . u

i1(t1) dt1 . . . dtp︸ ︷︷ ︸ΥI(T,u)

Xi1 . . . Xip︸ ︷︷ ︸XI

What is the CF-series good for?

For any given control system

x =m∑

i=1ui(x)fi(x), x(0) = p, with “output” y = ϕ(x)

φ(x(T, u)) =∑I

∫ T

0

∫ t1

0· · ·

∫ tp−1

0uip(tp) . . . u

i1(t1) dt1 . . . dtp︸ ︷︷ ︸ΥI(T,u)

(fi1 ◦ . . . ◦ fipϕ)(p)

(uniform convergence for small T and IC’s on compacta [Sussmann, 1983])

The CF-series was basic tool for deriving many high-order conditions for

controllability and optimality. [Hermes, Stefani, Sussmann, Kawski, ...]

13

Inadequacies of the CF-series

The Chen Fliess series is a good starting point, BUT

• It has too many terms (“2∞ when only∞ should do”)

• Lots of duplication: Repetition in the iterated integrals

High-order partial diff operators

where only 1st or low order PDO’s should occur

• It is not geometric:

Its character as exponential Lie series is not obvious

• It is not geometric:

Truncations do not correspond to any systems at all

(not directly useful for obtaining approximating systems)

More desirable alternatives: Expand the series in either of the forms

SCF (T, u) = exp

⎛⎝ ∑B∈B

αB(t, u) B

⎞⎠

or

SCF (T, u) =←∏B∈B

exp (βB(T, u) B)

for suitable bases B of the free Lie algebra L(X1, . . . Xm).Question: Existence?

14

Ree’s theorem and exponential Lie series

Theorem [Ree, 1957]: A formal power-series∑I cIXI is an expo-

nential Lie series iff the coefficients satisfy the shuffle relations

cI x J = cI · cJ for all I, J

Exercise:

The coefficients ΥI(T, u) of the CF-series satisfy the shuffle relations.

(Simple induction. Shuffles correspond to pointwise multiplication of

integrated integral functionals. Recursive definition of shuffle product

corresponds to repeated integration by parts.)

Corollary: Either expansion of the CF-series (exp of sum, or prod-

uct of exp) is possible.

Issues/questions:

• Need explicit basis for the free Lie algebra

• Want explicit formulae for the iterated integral coefficients

αB(T, u) and/or βB(T, u).

15

Shuffle product

Combinatorial definition (for words w, z and letters a, b):

(w a ) X ( z b ) = ((w a ) X z ) b + (w X ( z b )) a

Example: The shuffle product of two words

(ab) X (cd) = a b c d + a c b d + c a b d+

a c d b + c a d b + c d a b

Algebraic definition:

On the free associative algebra A = Ak(X ) (algebra of polynomi-

als, or “words”) over a set X (of noncommuting indeterminates,

or“letters”) define a co-product

Δ:A×A �→ A by Δ(a) = 1⊗ a + a⊗ 1 for a ∈ X

Define the shuffle product X as the transpose of Δ

(on the algebra A = Ak(X ) of formal power series)

< v X w , z >=< v ⊗ w , Δ(z) >

16

Shuffles and simplices

On permutations algebras Duchamp and Agrachev consider

partially commutative and noncommutative shuffles. Illustration:

� � �

� � �

��

��

�

��

��

�

��

��

�

��

��

�

σ1 x 2

σ1

σ2 = σ21∪ σ12

In the case of three letters {1, 2, 3}

= ∪ ∪

σ(12)x 3 = σ312 ∪ σ132 ∪ σ123

E.g. σ(12)x 3 = {t: 0 ≤ t1 ≤ t2 ≤ 1, 0 ≤ t3 ≤ 1}

For multiplicative integrands f(x, y, z) = f1(x) · f2(y) · f3(z)

(using x, y, z, instead of t1, t2, t3 for better readability):(∫ 1

0

∫ y

0( · )dx dy

)·∫ 1

0(·)dz =

∫ 1

0

∫ y

0

∫ x

0( · ) dz dx dy +

∫ 1

0

∫ y

0

∫ z

0( · ) dx dz dy +

∫ 1

0

∫ z

0

∫ y

0( · ) dx dy dz

17

CF-coefficients satisfy shuffle-relations

Sketch of proof of exercise (by induction on the combined lengths ofthe coefficients)

Υ1(t, u) ≡ 1

Υa x 1(t, u) = Υa(t, u)

= Υa(t, u) · 1

= Υa(t, u) ·Υ1(t, u) for any letter a ∈ X

Υ(wa)x (zb)(T, u) =

= Υ((wa)x z)b+(w x (zb))a(T, u)

= Υ((wa)x z)b(T, u) + Υ(w x (zb))a(T, u)

=∫ T0 Υ(wa)x z(t, u) · ub(t)dt+

∫ T0 Υw x (zb)(t, u) · ua(t)dt

=∫ T0 (Υwa(t, u) ·Υz(t, u) · ub(t) + Υw(t, u) ·Υzb(t, u) · ua(t)) dt

=∫ T0

(Υwa(t, u) · d

dtΥzb(t, u) + ddt (Υwa(t, u)) ·Υzb(t, u)

)dt

= Υwa(T, u) ·Υzb(T, u)

(1)

Morale: When working with repeated integrations by parts, omit“integrals and similar notational ballast”. Instead work purely

combinatorially in shuffle algebra.

18

Product expansion of CF-series

Using Ree’s theorem the existence of exponential product expansions

of the CF-series is assured. for suitable bases B of L(X1, . . . Xm).

SCF (T, u) =←∏B∈B

exp (βB(T, u) B)

Recall remaining issues/questions:

• Need explicit basis for the free Lie algebra

• Want explicit formulae for the iterated integral coefficients

αB(T, u) and/or βB(T, u).

Using different bases for the free Lie algebra explicit formulae for the

dual bases (iterated integral functionals βB(T, u) have been redis-

covered several times in different contexts:

• Schutzenberger (1958), Seminaire Dubreil

• Sussmann (1986), Nonlinear control

• Melancon and Reutenauer (1989), Combinatorics

• Grayson and Grossman (1991),

Realizations of free nilpotent Lie algebras

=⇒ See historical slide

19

Lazard elimination

Theorem [Lazard elimination]:

Suppose k a field of scalars, X is a set and c ∈ X .

Then the free Lie algebra Lk(X ) over k gener-

ated by X is the direct sum of the one-dimensional

subspace {λc:λ ∈ k} and of a Lie-subalgebra

of Lk(X ) that is freely generated by the set

{(adjc, b): b ∈ X \ {c}, j ≥ 0}.

This principle is at the heart of constructions involving Hall basesfor free Lie algebras, for Sussmann’s derivation of the exponentialproduct expansion by solving DEs by iteration, and thereby it is

closely connected to Zinbiel structures.

20

Hall and Lyndon bases

Ph. Hall, 1930s, calculus of commutator groups

M. Hall, 1950s, first bases for free Lie algebras

Lyndon, 1950s, different (?) bases

Sirsov, 1950s, different (?) bases

Viennot, 1970s, only one kind of practical basis

A Hall set over a set X is any strictly ordered subset H of the free

magma M(X ) (i.e. the set of all parenthesized words, or labelled

binary trees) that satisfies

• X ⊆ H• Suppose a ∈ X . Then (w, a) ∈ H iff w ∈ H, w < a and

a < (w, a).

• Suppose u, v, w, (u, v) ∈ H.

Then (u, (v, w)) ∈ H iff v ≤ u ≤ (v, w) and u < (u, (v, w)).

Original Hall bases as in Bourbaki require that ordering be compat-

ible with the length. Viennot showed that is not necessary.

The image of a Hall set under the canonical map ϕ:M(X ) �→ Lk(X )

from the free magma into the free Lie algebra is a basis for Lk(X ).

21

Lie brackets and formal brackets

Need to distinguish formal brackets and elements of a Lie algebra.

E.g., consider {x, y, (x, y), (y, (x, (x, y)))} ⊆ H ⊆ M({x, y}). Then

ϕ((x, y)) = [x, y] = −[y, x] , and

ϕ((y, (x, (x, y)))) = [y, [x, [x, y]]] = [x, [y, [x, y]]]

(due to anti-commutativity and Jacobi-identity in Lk(X )).

Consequently,

ϕ−1([x, [y, [x, y]]]) = ϕ−1([y, [x, [x, y]]])

= (y, (x, (x, y)))

�= (x, (y, (x, y))) inM(X )

Similarly,

ϕ−1([−y, x]) = ϕ−1([x, y])

= (x, y)

�= −(y, x) in the algebra overM(X )

But coding of iterated integrals depends critically on the factorization

of the Hall words, requiring well-defined left and right factors.

22

Hall and Lyndon bases, examples

Lyndon basis

b

(((((ab)b)b)b)b)

((((ab)b)b)b)

(a((((ab)b)b)b))

(((ab)b)b)

((ab)(((ab)b)b))

(a(((ab)b)b))

(a(a(((ab)b)b)))

((ab)b)

((ab)((ab)b))

(a((ab)((ab)b)))

(a((ab)b))

((ab)(a((ab)b)))

(a(a((ab)b)))

(a(a(a((ab)b))))

(ab)

((a(ab))(ab))

(a((a(ab))(ab)))

(a(ab))

(a(a(ab)))

(a(a(a(ab))))

(a(a(a(a(ab)))))

b

Lyndon words

L< [LR] <R

Read backwards,

a word is Lyndon if

it is strictly larger

in lexicographical

order than any of its

cyclic rearrangements

Hall words

(in narrow sense)

• a ∈ X ⇒ a ∈ H

• w, z ∈ H, u < v

⇒ |u| < |v|• If a ∈ X , then

(ua) ∈ H ⇔u < a and u < (ua)

• (u, (vw)) ∈ H ⇔u, (vw) ∈ H and

v ≤ u < (vw), u <

(u(vw))

Hall basis

(as in Bourbaki)

((a(ab))(b(ab)))

((ab)(b(b(ab))))

((ab)(b(a(ab))))

((ab)(a(a(ab))))

(b(b(b(b(ab)))))

(b(b(b(a(ab)))))

(b(b(a(a(ab)))))

(b(a(a(a(ab)))))

((ab)(b(ab)))

((ab)(a(ab)))

(b(b(b(ab))))

(b(b(a(ab))))

(b(a(a(ab))))

(a(a(a(ab))))

(b(b(ab)))

(b(a(ab)))

(a(a(ab)))

(b(ab))

(a(ab))

(ab)

b

a

23

Hall-Viennot bases: unique factorization

Bases for free Lie algebras proposed by M. Hall, Lyndon, and Sirsov

were originally considered to be distinct, until Viennot showed that

they all arise from a fundamental unique factorization principle.

By construction: The restriction of the map ϕ

ϕ(a) = a for ; a ∈ X , and

ϕ((w, z)) = [ϕ(w), ϕ(z)] for w, z ∈ M(X ).

to any Hall-set H ⊆ M(X ) is one-to-one by construction [Viennot].

Hence the inverse image ϕ−1(H) ∈ H ⊆ M(X ) of an element

H ∈ ϕ(H) ⊆ Lk(X ) is well-defined

Practically speaking: When working with a fixed Hall set there

is no need to write down any parentheses!

Consequence: Every word w ∈ W (X ) factors uniquely into a

nonincreasing product of Hall words, i.e. there exist unique Hj ∈ H,

such that

w = H1H2 . . . Hs and H1 ≥ H2 ≥ . . . ≥ Hs

24

Zinbiel products

In the most simple basic control system

y1 = u1 |u1| ≤ 1

y2 = u2 |u2| ≤ 1

y3 = y1u2

.

many consider as a virtual third control the function

u3(t) = (u1 � u2)(t)def=

(∫ t0u1(s) ds

)· u2(t)

The ubiquitous occurrence of this product justifies to call it

“THE PRODUCT of control theory”

Exercise: Verify that this product satisfies:

(u1 � u2 + u2 � u1)(t) =d

dt

((∫ t0u1(s) ds

) (∫ t0u2(s) ds

))

and also the three term right Zinbiel identity

(u1 � (u2 � u3)) (t) = (∫ t0 u1(s), ds) · ((∫ t0 u2(s)ds) · u3(t))

= (∫ t0 (∫ s0 u1(σ) dσ) · u2(s)ds) · u3(t)

+ (∫ t0 (∫ s0 u2(σ) dσ) · u1(s)ds) · u3(t)

= ((u1 � u2) � u3) (t) + ((u2 � u1) � u3) (t)

25

Zinbiel algebras

A (right) Zinbiel algebra is a linear space C (over a field k) that is

endowed with a bilinear product ∗:C × C �→ C which satisfies the

(right) Zinbiel identity

v ∗ (w ∗ z) = (v ∗ w) ∗ z + (w ∗ v) ∗ z for all v, w, z ∈ C (2)

Example 1:(repeated)

Lloc([0,∞)) with (u1 � u2)(t)def= (

∫ t0 u1(s) ds) · u2(t)

Example 2:

AC loc([0,∞)) with (U1 ∗ U2)(t)def= (

∫ t0 U1(s)U

′2(s) ds)

Easy exercise:

(F ∗ (G ∗H))(t) =∫ t0 F (s) ·

(dds

∫ s0 G(σ)H ′(σ) dσ

)ds

=∫ t0 F (s)G(s) ·H ′(s) ds

=∫ t0 (∫ s0 F (σ)G′(σ) dσ +

∫ s0 G(σ)F ′(σ) dσ) ·H ′(s) ds

= ((F ∗G) ∗H)(t) + ((F ∗G) ∗H)(t)

Also the symmetrized product is pointwise multiplication

(U1 ∗ U2) + (U2 ∗ U1) = (U1 · U2) for U1, U2 ∈ AC loc,0

26

Zinbiel algebras, examples

There are many familiar Zinbiel subalgebras of the Zinbiel algebras

AC loc,0 and ACloc,0, most notably those of polynomial and of (real,

or complex) exponential functions.

Typical multiplication rules are

xn � xm = 1n+1x

n+m+1 ent � emt = 1ne

(n+m)t

xn ∗ xm = mn+mx

n+m ent ∗ emt = mn+me

(n+m)t

Exercise: In each case verify directly that the right Zinbiel identity

is satisfied. For example:

(xm � xn) � xk = 1m+n+2

· 1m+1

xm+n+k+2

(xn � xm) � xk = 1m+n+2

· 1n+1

xm+n+k+2

xm � (xn � xk) = 1m+1· 1n+1

xm+n+k+2

27

Zinbiel products and solving DEs

Zinbiel products efficiently encode the solution of time-varying linear

differential equations by iteration:

The integrated form of the universal control system

S = S · Φ, §(0) = 1 with Φ =m∑i=1uiXi

is compactly rewritten using Zinbiel products

S = 1 + S ∗ Φ

Iteration yields the explicit series expansion

S = 1 + (1 + S ∗ Φ) ∗ Φ

= 1 + Φ + ((1 + S ∗ Φ) ∗ Φ) ∗ Φ

= 1 + Φ + (Φ ∗ Φ) + (((1 + S ∗ Φ) ∗ Φ) ∗ Φ) ∗ Φ

= 1 + Φ + (Φ ∗ Φ) + ((Φ ∗ Φ) ∗ Φ) + ((((1 + S ∗ Φ) ∗ Φ) ∗ Φ) ∗ Φ) ∗ Φ...

= 1 + Φ + (Φ ∗ Φ) + ((Φ ∗ Φ) ∗ Φ) + (((Φ ∗ Φ) ∗ Φ) ∗ Φ) . . .

Corollaries: Coefficients of the CF-series and in the exp.prod.expansion:

For a word w ∈W (X ) and a letter a ∈ XΥwa(t, u) =

∫ T0

Υw(s, u) ·(ddsΥ

a(s, u))ds = (Υw(·, u) ∗ Υa(·, u)) (t)

CHK(t, u) = CH(t, u) ∗ CK(t, u) if H,K,HK ∈ H

28

Normal forms for nilpotent systems

Zinbiel products provide the most compact way for specifying a normal form

for free nilpotent control systems of rank r. The key is to index the coordinates

by Hall words H ∈ H(r) def= {H ∈ H: |h| ≤ r} (rather than by the integers):

Normal form for a free system (maximally free Lie algebra)

xa = ua if a ∈ XxHK = xH ∗ xK if H,K,HK ∈ H(r)(X )

Example: Normal from for a free nilpotent system (of rank r = 5) using a

typical Hall set on the alphabet X = {0, 1}x0 = u0

x1 = u1

x01 = x0 · x1 = x0 u1

x001 = x0 · x01 = x20 u1 using ψ−1(001) = (0(01))

x101 = x1 · x01 = x1x0 u1 using ψ−1(101) = (1(01))

x0001 = x0 · x001 = x30 u1 using ψ−1(0001) = (0(0(01)))

x1001 = x1 · x001 = x1x20 u1 using ψ−1(1001) = (1(0(01)))

x1101 = x1 · x101 = x21x0 u1 using ψ−1(1101) = (1(1(01)))

x00001 = x0 · x001 = x40 u1 using ψ−1(00001) = (0(0(0(01))))

x10001 = x1 · x0001 = x1x30 u1 using ψ−1(10001) = (1(0(0(01))))

x11001 = x1 · x1001 = x21x

20 u1 using ψ−1(11001) = (1(1(0(01))))

x01001 = x01 · x001 = x01x30 u1 using ψ−1(01001) = ((01)(0(01)))

x01101 = x01 · x101 = x01x21x0 u1 using ψ−1(01101) = ((01)(1(01)))

29

Free Zinbiel algebra

On Ak,0(X ) and Ak,0(X ) noncommutative polynomials and power series with

zero constant term, define a (bilinear, noncommutative, nonassociative) prod-

uct by w ∗ a = wa for any nonempty word w ∈ W0(X ) and a ∈ X , and

w ∗ (z ∗ a) = (w ∗ z) ∗ a + (z ∗ w) ∗ a for a ∈ X , w, z ∈ W0(X )

Exercise:

With this product Ck(X ) def= (Ak,0(X ), ∗) is a Zinbiel algebra that is free in the

usual sense: If C is any Zinbiel algebra then any γ:X �→ C extends uniquely

to a Zinbiel algebra homomorphism γ : Ck(X ) �→ C.

Observe: The shuffle product is (the extension to Ak(X ) of) the symmetriza-

tion of the Zinbiel product (and 1 ∗ 1 cannot be defined meaningfully)

w ∗ z + z ∗ w = w X z for w, z ∈ W0(X )

Lots of useful identities (e.g. to hide unwanted factorials)

For w ∈ Ak,0(X ) define w∗1 = λ1(w) = wx 1 = w, and inductively for n ≥ 1

λn+1(w) = w ∗ λn(w)

w∗(n+1) = w∗n ∗ wwx (n+1) = w X wxn = wxn X w

Theorem: w ∗ w∗(n−1) = (n− 1) · w∗nλn(w) = (n− 1)! · w∗nw x n = n! · w∗n (= nλn(w))

30

Zinbiel products and dual PBW bases

Every Hall-word H ∈ W (X ) \ X , factors uniquely in the form

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

�

��

��

��

��

��

��

��

��

��

��

��

�

��

��

��

��

�

��

��

��

��

��

��

��

��

��

��

H1 ≥ H2 ≥ H3 . . . Hs−2 ≥ Hs−1 ≥ Hs < a

This factorization (due to the connection with Lazard factorization)

perfectly matches Sussmann’s variation of parameters approach to

obtaining the iterated integrals βH(T, u) in the exponential product

expansion of the CF-series

βH =1

(. . .)!(βH1 ∗ (βH2 ∗ . . . (βHs−1 ∗ (βHs∗βa))...))

Compare to the mix of products from different algebras in Reutenauer’s

and Melancon’s formula for the dual-PBW bases.

In the shuffle algebra (A(X ), X ) the transposes of both the left

and right translation by a letter λa:w �→ aw, and �a:w �→ wa are

derivations. However:

On the Zinbiel algebra, only λ†a is a derivation, �†a is not.

λ†a(w ∗ z) = (λ†aw) ∗ z + w ∗ (λ†az)

�†a(w ∗ z) = w ∗ (�†az)

31

Realization of free Zinbiel algebra

Compare the standard realization of polynomial algebras:

k[X1, . . . Xn] polynomials w/ coeff’s in k

↓k[x1, . . . xn] polynomial functions kn �→ k

= the subalgebra of Map(kn,k) = kkn

generated by the projections

xk = πk: (p1, . . . , pn) �→ pk

Similarly realize the free Zinbiel algebra as a Zinbiel algebra of time-

varying scalars. E.g for an index set X of letters

U = ACloc([0,∞),R)

πa:UX �→ U , πa({ub: b ∈ X}) = ua

IIF(X ) ⊆ Map(UX ,U)

subalgebra generated by projections πa, a ∈ XTheorem[Kawski and Sussmann]: The map ΥΥ(:C(X ) �→ IIF(X )

defined by ΥΥ(: a �→ πa is a Zinbiel algebra isomorphism.

Zinbiel algebra surjective homomorphism is rather clear by now. The

one-to-one-ness requires a sufficiently rich coefficient ring and some

analysis (Nagano’s theorem . . . ).

32

CF series as identity map

The Chen Fliess series of iterated integral functionals cor-responds to the identity map under the Zinbiel algebraisomorphism ΥΥ, i.e. it is natural object.

idA ∼= ∑w w ⊗ w

idA⊗ΥΥ←−−→ ∑

w w ⊗ ΥΥ(w)

‖ ‖Schutzenberger, Sussmann

Melancon & R.

Combinatorics Diff Equns proof

‖ ‖←∏H exp ([H ]⊗ SH)

idA⊗ΥΥ←−−→ ←∏

H exp ([H ]⊗ βH)

Hom(A,A) ∼= A⊗A idA⊗ΥΥ←−−→ A⊗AIIF

free Zinbiel Zinbiel.algebra iterated integral

algebra isomorphism functionals

33

Koszul duality and Leibniz operad

The Zinbiel “operad” is “dual” to the Leibniz “operad”.

(“pre-Lie algebra structure”, or “noncommutative Lie algebra”)

(Left) Leibniz identity

v � (w � z) = (v � w) � z + v � (w � z) for all v, w, z (3)

Compare / recall: (right) Zinbiel identity

v ∗ (w ∗ z) = (v ∗ w) ∗ z + (w ∗ v) ∗ z for all v, w, z (4)

and the ( left) chronological identity of Gamkrelidze and Agrachev)

x · (y · z)− (x · y) · z = y · (x · z)− (y · x) · z⇔ x · (y · z)− y · (x · z) = (x · y) · z − (y · x) · z

sometimes suggestively written as L[x,y] = [Lx, Ly]

Puzzle: The anti-commutativity of the Lie-brackets appears so nat-

ural in control – yet algebraically it appears to be only a coincidence.

What in control corresponds geometrically to the Zinbiel algebra – it

must be connections; but they are not much used in controllability

and optimality – should they? What do they add?

For details on Koszul duality of operads see Ginzburg and Kapranov, Duke. J.

Math, 1997. For details on Leibniz algebras, and their role in cyclic homology,

see numerous articles by Loday.

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