Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main...

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Algebra and the geometryof nonlinear control systems

Matthias Kawski †

School of Mathematical and Statistical SciencesArizona State UniversityTempe, AZ 85287, USA

http://math.asu.edu/~kawski

Workshop in Celebration of the Life, Mathematics,and Memories of Christopher I. Byrnes

Lubbock, TX, September 2010

1This work was partially supported by the National Science Foundationthrough the grants DMS 09-08204 and DMS 09-60589,.

Trimester: Combinatorics & Control 2010, Madrid

Outline

• Background: From control to algebra and combinatorics

• Recall: Hall sets and coord’s of 2nd kind

• Hopf algebras and convolutions

• Coord’s of 1st kind w/ examples

• Conclusion and outlook

In a nut-shell

• Recall: Chen Fliess series is an exponential Lie series

∑w∈Z∗

w ⊗ w = exp

(∑b∈B

ζb ⊗ b

)=←∏

b∈Bexp(ξb ⊗ b)

where B is ordered basis of free Lie algebra L(Z ) ⊂ A(Z )

• Known (coord’s of 2nd kind): ξHK = ξH ∗ ξK

• New/recent (coord’s of the 1st kind): ζ = π′1 ◦ ξ

• Use in control / geometric integration: explicit formula foriterated integral functionals Υζb and Υξb .

In a nut-shell

∑w∈Z∗

w ⊗ w = exp

(∑b∈B

ζb ⊗ b

)=←∏

b∈Bexp(ξb ⊗ b)

In a nut-shell

∑w∈Z∗

w ⊗ w = exp

(∑b∈B

ζb ⊗ b

)=←∏

b∈Bexp(ξb ⊗ b)

In a nut-shell

∑w∈Z∗

w ⊗ w = exp

(∑b∈B

ζb ⊗ b

)=←∏

b∈Bexp(ξb ⊗ b)

Basic set-up: splitting

Splitting into geometric state-dependent and time-varying parts

x = u1(t)f1(x) + . . .+ um(t)fm(x) = F (t , x)y = ϕ(x)

fi : M 7→ TM smooth vector fields on manifold Mn,

u : [0,T ] 7→ U ⊂⊂ Rm measurable controls/perturbations

φ ∈ Cω(M) measured output

not only control, but also geometric numerical integration,stochastic DE (rough paths [Lyons]) ldots.

u : [0,T ] 7→ U ⊂⊂ Rm measurable controls/perturbations

u : [0,T ] 7→ U ⊂⊂ Rm measurable controls/perturbations,

u : [0,T ] 7→ U ⊂⊂ Rm measurable controls/perturbations, and

φ ∈ Cω(M) measured output.

Long history

Combinatorics and algebra behind integration of ODEs

Just a few names

Campbell-Baker-Hausdorff, Magnus, Chen,

Schützenberger, Reutenauer, Fer, Ferfera,

Fliess, Lamnabhi-Lagarrigue, Crouch, Gray,

Grossmann, Larson, Rota and Baxter, Ebrahimi-Fard, . . .

A closer look at manipulating series expansions

φ(x(t ,u)) = 1 · φ(x0)

+∫ t

0ua(s)ds (faφ)(x0)

+∫ t

0ub(s)ds (fbφ)(x0)

∫ t0

∫ s10

∫ s10 ua(s1)ua(s2)ds2ds1 (fafaφ)(x0)

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafbφ)(x0)

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 (fbfaφ)(x0)

∫ t0

∫ s10 ub(s1)ub(s2)ds2ds1 (fbfbφ)(x0)

∫ t0

∫ s10

∫ s20 ua(s1)ua(s2)ua(s3)ds3ds2ds1 (fafafaφ)(x0)

+ . . .

Objective: Collect first order differential operators, and minimizenumber of higher order differential operators involved

A closer look at manipulating series expansions

φ(x(t ,u)) = 1 · φ(x0)

+∫ t

0ua(s)ds (faφ)(x0)

+∫ t

0ub(s)ds (fbφ)(x0)

∫ t0

∫ s10

∫ s10 ua(s1)ua(s2)ds2ds1 (fafaφ)(x0)

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafbφ)(x0)

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 (fbfaφ)(x0)

∫ t0

∫ s10 ub(s1)ub(s2)ds2ds1 (fbfbφ)(x0)

∫ t0

∫ s10

∫ s20 ua(s1)ua(s2)ua(s3)ds3ds2ds1 (fafafaφ)(x0)

+ . . .Objective: Collect first order differential operators, and minimizenumber of higher order differential operators involved

Integrate by parts: The wrong way to do it∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb+

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa =

=∫ t

∫ s10 ua(s1)ub(s2)ds2ds1 fafb −

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa

+∫ t

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa +

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa

=∫ t

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+∫ t

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

Lie brackets together w/ iterated integrals in right orderhigher order deriv’s (wrong order) w/ pointwise prod’s of int’s

∫ t0

=∫ t

∫ t0

+∫ t

∫ t0

=∫ t

+∫ t

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

∫ t0

=∫ t

∫ t0

+∫ t

∫ t0

=∫ t

+∫ t

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

∫ t0

=∫ t

∫ t0

+∫ t

∫ t0

=∫ t

+∫ t

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

∫ t0

=∫ t

∫ t0

+∫ t

∫ t0

=∫ t

+∫ t

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

∫ t0

=∫ t

∫ t0

+∫ t

∫ t0

=∫ t

+∫ t

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

Integrate by parts, smart way. Hopf algebra preview

Do not manipulate iterated integrals and iterated Lie brackets ofvector fields by hand – work on level of “words” (their indices)∑

I∈{a,b}∗I ⊗ I = 1⊗ 1

+ a ⊗ a

+ b ⊗ b

+ 12 aa ⊗ aa

+ 12 ab ⊗ ab

+ 12 ba ⊗ ba

+ 12 bb ⊗ bb

+ 16 aaa ⊗ aaa

+ . . .

= 1⊗ 1

+ a ⊗ a

+ b ⊗ b

+ 12 aa ⊗ aa

+ 12 ab ⊗ (ab − ba)

+ 12 (ab + ba) ⊗ ba

+ 12 bb ⊗ bb

+ 16 aaa ⊗ aaa

+ . . .

Drop everything except the indices

• The iterated integral

Υi1i2...in =

∫ t1

0· · ·∫ tn−1

0ui1(t1)ui2(t2) · · · uin (tn) dtn dtn−1 · · · dt1

is uniquely identified by the multi-index (“word”) i1i2 . . . in

• The n-th order partial differential operator fin ◦ fin−1 ◦ . . . fi1is uniquely identified by the multi-index (“word”) i1i2 . . . in

• The Chen series is identified with the identity map on freeassociative algebra A(Z ) over set of indices Z = {1, . . .m}

CF ∼ IdA(Z ) =∑n≥0

∑w∈Z n

w ⊗ w ∈ A(Z )⊗ A(Z )

with shuffle product on left and concatenation on right

Υi1i2...in =

∫ t1

0· · ·∫ tn−1

CF ∼ IdA(Z ) =∑n≥0

∑w∈Z n

w ⊗ w ∈ A(Z )⊗ A(Z )

Υi1i2...in =

∫ t1

0· · ·∫ tn−1

CF ∼ IdA(Z ) =∑n≥0

∑w∈Z n

w ⊗ w ∈ A(Z )⊗ A(Z )

Homomorphisms I

• For fixed smooth vector fields fi

F : A(Z ) 7→ partial diff operators on C∞(M)

F : (i1i2 . . . in) 7→ fi1 ◦ fi2 ◦ . . . finassociative algebras: concatenation 7→ composition

• For fixed control u ∈ UZ

Υ(u) : A(Z ) 7→ AC([0,T ],R)

Υ(u) : (i1i2 . . . in) 7→∫ T

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

associative algebras (Ree’s theorem):shuffle of words 7→ pointwise multiplication of functions

Homomorphisms I

• For fixed smooth vector fields fi

F : A(Z ) 7→ partial diff operators on C∞(M)

F : (i1i2 . . . in) 7→ fi1 ◦ fi2 ◦ . . . finassociative algebras: concatenation 7→ composition

• For fixed control u ∈ UZ

Υ(u) : A(Z ) 7→ AC([0,T ],R)

Υ(u) : (i1i2 . . . in) 7→∫ T

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

associative algebras (Ree’s theorem):shuffle of words 7→ pointwise multiplication of functions

Homomorphisms II

• Restriction is Lie algebra homomorphism

F : L(Z ) ⊆ A(Z ) 7→ Γ∞(M) (vector fields)

• Do not fix controls: iterated integral functionals

Υ: ∈ A(Z ) 7→ IIF(UZ )

Υ: (i1i2 . . . in) 7→

(u 7→

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

)associative algebras: shuffle of words 7→ pointwisemultiplication of iterated integral functionals

• Much better:

Theorem: If U = L1([0,T ], [−1,1]) thenΥ: (A(Z ), ∗) 7→ IIF(UZ ) is a Zinbiel algebra isomorphism.

Homomorphisms II

Υ: ∈ A(Z ) 7→ IIF(UZ )

Υ: (i1i2 . . . in) 7→

(u 7→

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

• Much better:

Homomorphisms II

Υ: ∈ A(Z ) 7→ IIF(UZ )

Υ: (i1i2 . . . in) 7→

(u 7→

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

• Much better:

Homomorphisms II

Υ: ∈ A(Z ) 7→ IIF(UZ )

Υ: (i1i2 . . . in) 7→

(u 7→

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

• Much better: Theorem: If U = L1([0,T ], [−1,1]) thenΥ: (A(Z ), ∗) 7→ IIF(UZ ) is a Zinbiel algebra isomorphism.

Zinbiel product, Hall sets, and formula for ξ

The product of control theory (U ∗ V )(t) =∫ t

0 U(s) V ′(s) dssatisfies the right Zinbiel identity

U ∗ (V ∗W ) = (U ∗ V ) ∗W + (V ∗ U) ∗W

Theorem (Schützenberger, Sussmann, Grossman,Reutenauer-Melançon)For a Hall set H over Z and letters a ∈ Z and H,K ,HK ∈ H

ξa = aξHK = µHK ξH ∗ ξK

Zinbiel product, Hall sets, and formula for ξ

The product of control theory (U ∗ V )(t) =∫ t

0 U(s) V ′(s) dssatisfies the right Zinbiel identity

U ∗ (V ∗W ) = (U ∗ V ) ∗W + (V ∗ U) ∗W

Theorem (Schützenberger, Sussmann, Grossman,Reutenauer-Melançon)For a Hall set H over Z and letters a ∈ Z and H,K ,HK ∈ H

ξa = aξHK = µHK ξH ∗ ξK

Illustration: Normal form for nilpotent systemNormal from for a free nilpotent system (of rank r = 5)using a typical Hall set on the alphabet Z = {a,b}

ξa = ua

ξb = ubξab = ξa · ξb = ξa ubξaab = ξa · ξab = ξ2

a ub using ψ−b(aab) = (a(ab))

ξbab = ξb · ξab = ξbξa ub using ψ−b(bab) = (b(ab))

ξaaab = ξa · ξaab = ξ3a ub using ψ−b(aaab) = (a(a(ab)))

ξbaab = ξb · ξaab = ξbξ2a ub using ψ−b(baab) = (b(a(ab)))

ξbbab = ξb · ξbab = ξ2bξa ub using ψ−b(bbab) = (b(b(ab)))

ξaaaab = ξa · ξaaab = ξ4a ub using ψ−b(aaaab) = (a(a(a(ab))))

ξbaaab = ξb · ξaaab = ξbξ3a ub using ψ−b(baaab) = (b(a(a(ab))))

ξbbaab = ξb · ξbaab = ξ2bξ

2a ub using ψ−b(bbaab) = (b(b(a(ab))))

ξabaab = ξab · ξaab = ξabξ3a ub using ψ−b(abaab) = ((ab)(a(ab)))

ξabbab = ξab · ξbab = ξabξ2bξa ub using ψ−b(abbab) = ((ab)(b(ab)))

Cost: accept label coord’s not by integers, but by Hall wordsBenefit: One-line formula for all iterated integrals! ξHK = ξH ∗ ξK

ξa = ua

Cost: accept label coord’s not by integers, but by Hall words

Benefit: One-line formula for all iterated integrals! ξHK = ξH ∗ ξK

ξa = ua

Cost: accept label coord’s not by integers, but by Hall wordsBenefit: One-line formula for all iterated integrals! ξHK = ξH ∗ ξK

Hopf algebra and dynamical systems

Long history of using combinatorial Hopf algebras for analysisof dynamical systems.

E.g. Grossmann and Larson, Hopf algebra structure of labeledtrees, and of differential operators

Main reference for this work: Reutenauer, Free Lie Algebras

Recent related work, numerical “geometric” integration:Murua, Ebrahimi-Fard, Munthe-Kaas, . . .

Easy intro to Hopf algebra: shuffle

Combinatorially: for words w , z ∈ Z ∗ and letters a,b ∈ Z

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

Example: (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

Algebraically: transpose of the coproduct ∆

< v X w , z >=< v ⊗ w , ∆(z) >

∆: A(Z ) 7→ A(Z )⊗ A(Z ) by ∆(a) = 1⊗a+a⊗1 for a ∈ Z

Easy intro to Hopf algebra: shuffle

Combinatorially: for words w , z ∈ Z ∗ and letters a,b ∈ Z

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

Example: (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

Algebraically: transpose of the coproduct ∆

< v X w , z >=< v ⊗ w , ∆(z) >

∆: A(Z ) 7→ A(Z )⊗ A(Z ) by ∆(a) = 1⊗a+a⊗1 for a ∈ Z

Two Hopf algebra structures on A(Z )

• Products: Concatenation and shuffle

• Coproducts:∆: A(Z ) 7→ A(Z )⊗ A(Z ) concatenation homo∆′ : A(Z ) 7→ A(Z )⊗ A(Z ) shuffle homomorphismFor a ∈ Z , ∆(a) = ∆′(a) = 1⊗ a + a⊗ 1, and for w ∈ Z ∗

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

• Antipode (same for both): linear map α : A(Z ) 7→ A(Z )if w = a1a2 . . . ak ∈ Z k then α(w) = (−)k ak . . . a2a1.

• Example: If a,b ∈ Z then

∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and

∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and

∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and

∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

• Example: If a,b ∈ Z then ∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and ∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

• Example: If a,b ∈ Z then ∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and ∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

Two convolution productsEach Hopf algebra structure yields an associative convolutionproduct of linear endomorphisms f ,g : A(Z ) 7→ A(Z )

f ? g = conc ◦ (f ⊗ g) ◦∆, and

f ?′ g = shu ◦ (f ⊗ g) ◦∆′,

For w ∈ A(Z )

(f ? g)(w) =∑

u,v∈Z+

〈w ,u x v〉 f (u)g(v) and

(f ?′ g)(w) =∑

u,v∈Z+

〈w ,uv〉 f (u) x g(v)

Note: Apparent confusion of ? and ?′ led to incorrect formulasfor ζh in the literature.

Fundamental subspaces and projections• Direct sum decomposition A(Z ) =

⊕∞k=1 Uk where for

k ∈ Z+, Uk ⊆ A(Z ) is linear subspace spanned by `k

(concatenation) for ` ∈ L(Z ). Alternatively, Uk is spannedby symmetric products of Lie polynomials `1, . . . `k ∈ L(Z )

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

• associated projections (NOT orthogonal) πk : A(Z ) 7→ Uk

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

(−)k−1 1k conck ◦ I⊗k ◦∆k , and

πk = 1k!π

where I : A(Z ) 7→ A(Z ), I(1) = 0, I(w) = w for w ∈ Z ∗.

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

πk = 1k!π

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

πk = 1k!π

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

πk = 1k!π

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

(−)k−1 1k conck ◦ I⊗k ◦∆k

πk = 1k!π

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

πk = 1k!π

The adjoint π′1

Define the adjoint map π′1 : A(Z ) 7→ A(Z ) by

〈w , π1(z)〉 = 〈π′1(w), z〉 for all w , z ∈ Z.

Since I is selfadjoint, conc and ∆′ are adjoints of each other, asare x and ∆. Hence, using the second convolution product ?′

π′1 =∑k≥1

(−)k−1 1k I?

′k . =∑k≥1

(−)k−1 1k shuk ◦ I⊗k ◦∆′k

On words w ∈ Z , calculate (sum over uj 6= 1)

π′1(w) =∑k≥1

1k (−)k−1

∑u1...uk=w

u1 x . . . x uk

Example: π′1(ξab) = π′1(ab) = ab − 12 a x b = 1

2 (ab − ba).

Formulas for the ζh

Theorem[E. Gehrig].On every Hall set ζ = π′1 ◦ ξ.

Moreover: (recall PBW basis and Viennot) every w ∈ Z ∗ factorsuniquely w = h1h2 · · · hk with Hall words h1 ≥ h2 ≥ . . . ≥ hk

Naturally extending ξ and ζ to A(Z ),

ξw = 1r1!···rk !ξ

x r1h1

x ξx r2h2

x . . . x ξx rkhk

(known)

ζw = π′|r |(ξw ) = 1r1!···rk !ζ

x r1h1

x . . . x ζx rkhk

In general the maps π′k do not commute with shuffles.But surprisingly they do on products of this special form.

Formulas for the ζh

Theorem[E. Gehrig].On every Hall set ζ = π′1 ◦ ξ.

Moreover: (recall PBW basis and Viennot) every w ∈ Z ∗ factorsuniquely w = h1h2 · · · hk with Hall words h1 ≥ h2 ≥ . . . ≥ hk

Naturally extending ξ and ζ to A(Z ),

ξw = 1r1!···rk !ξ

x r1h1

x ξx r2h2

x . . . x ξx rkhk

(known)

ζw = π′|r |(ξw ) = 1r1!···rk !ζ

x r1h1

x . . . x ζx rkhk

In general the maps π′k do not commute with shuffles.But surprisingly they do on products of this special form.

Sample calculation π′1(aaabb) and ζabaab

π′1(aaabb) = aaabb − 12 (aaab x b + aaa x bb + aa x aab

+a x aabb) + 13(aaa x b x b + aa x ab x b

+2aa x a x bb + a x aab x b + a x a x abb)

−14(aa x a x b x b + a x aa x b x b + a x a x ab x b

+a x a x a x bb) + 15a x a x a x b x b.

Similarly, starting fromξabaab = 1

2 (ab ∗ (a ∗ (a ∗ b))) = 3aaabb + 2aabab + abaab

ζabaab = π′1(ξabaab) = π′1(3aaabb + 2aabab + abaab)

= 110aaabb + 1

15aabab + 115aabba + 1

15abaab − 110ababa

+ 115abbaa− 1

10baaab + 115baaba + 1

15babaa− 110bbaaa.

The first 14 terms as nilpotent cascade system I

Iterated integrals zH(t ,U ′) = µH Υ(ζH)(U)(t)

za = ua

zb = ub

zab = −16zbua + 1

zaab =(−1

2zab − 112zazb

)ua + 1

12z2aub

zbab = − 112z2

bua +(−1

2zab + 112zazb

zaaab =(−1

2zaab − 112zabza

zbaab =(−1

2zbab − 112zabzb

)ua −

(12zaab + 1

12zabza)

zbbab =(−1

2zbab − 112zabzb

The first 14 terms as nilpotent cascade system II

zaaaab =(−1

2zaaab − 112zazaab − 1

720z3azb)

ua − 1720z4

zbbbab = 1720z4

bua −(1

2zbbab + 112zbzbab + 1

720zaz3b

zbaaab =( 1

240z2ay2

b −12zbaab − 1

12zbzaab − 112zazbab

2zaaab + 112zazaab + 1

240z3azb)

zabaab =(−1

2zbaab + 112zbzaab − 1

12zazbab − 112y2

+ 1360z2

ua +(−1

6zazaab − 1360z3

zabbab =(−1

2zbbab1

12zbzbab + 1720zaz3

6zazbab + 112y2

ab + 1720z2

zbbaab =(−1

2zbbab − 112zbzbab + 1

240zaz3b

2zbaab + 112zbzaab + 1

12zazbab + 1240z2

Summary and outlook

• Formulas to obtain coord’s of the first kind ζhfrom known formulas for ξh in terms of Zinbiel product

• The main contribution is not the existence of the map,but expressing it via elementary objects in Hopf algebra.

• Formulas for ζh are explicit and easily computable,and amenable for implementation for path planning,or for geometric integration procedures.

• Combinatorial Hopf algebra: Connections betweennonlinear control, geometric numerical integration,stochastic DEs (rough paths), more to come.Next: geometric, no letters, but use full tensor algebra

For more information, visit http://math.asu.edu/~kawski

Thank you.

This presentation was prepared with LaTeX using the beamer package.

Summary and outlook

Thank you.

Summary and outlook

Thank you.

Summary and outlook

Thank you.

Summary and outlook

Thank you. This presentation was prepared with LaTeX using the beamer package.

Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main...

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