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Intro Controllability Series expansions Exp-prod Manipulating exponential products Instead of working with complicated concatenations of flows t 9 t 2 t 1 t 8 ( f 0 + f 1 + f 2 ) dt . . . e t 1 ( f 0 + f 1 f 2 ) dt e 0 (

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Intro Controllability Series expansions Exp-prod

Manipulating exponential products

Instead of working with complicated concatenations of flows z(t) = e

t9

t8 (f0+f1+f2) dt ◦ . . . e

t2

t1 (f0+f1−f2) dt ◦ e

t1

0 (f0+f1+f2) dt(p)

it is desirable to rewrite the solution curve using a minimal number of vector fields fπk that 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

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Series expansions: Lift to universal, free system

  • Starting with an affine, real analytic system on Rn

˙ x =

m

  • i=1

ui(t)fi(x(t))

  • or (chronological calculus), work with induced system
  • n the algebra C∞(M) of smooth functions

Basically, from ˙ x = fu(x) to fu : Φ → (fuΦ) = ∇Φ, fu

  • Formally, on free associative algebra ˆ

A(Z) over a set Z = {X1, . . . Xm} of m indeterminates consider system ˙ S = S(t) ·

m

  • i=1

ui(t)Xi

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Intro Controllability Series expansions Exp-prod

Formal solution – Chen-Fliess series

˙ S = S(t) ·

m

  • i=1

ui(t)Xi, S(0) = I

  • n algebra ˆ

A(Z) of formal power series in the noncommuting indeterminates (letters) X1, . . . Xm has the unique solution CF(T, u) =

  • I

T t1 · · · tp−1 uip(tp) . . . ui1(t1) dt1 . . . dtp

  • ΥI(T,u)

Xi1 . . . Xip

  • XI

Use as asymptotic expansion for evolution of output y = ϕ(x) along solution of ˙ x = u1f1(x) + . . . umfm(x). ϕ(x(T, u)) ∼

  • I

T t1 · · · tp−1 uip(tp). . .ui1(t1) dt1 . . . dtp (fi1 . . . fipϕ)(x0)

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Intro Controllability Series expansions Exp-prod

Series expansions, intro

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 → TM smooth vector fields on manifold Mn, u : [0, T] → U ⊂⊂ Rm measurable controls/perturbations, and φ ∈ Cω(M) measured output.

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Series solution by iteration

φ(x(t, u)) = 1 · φ(x0) + t

0ua(s)ds (faφ)(x0)

+ t

0ub(s)ds (fbφ)(x0)

+ 1

2

t s1 s1

0 ua(s1)ua(s2)ds2ds1 (fafaφ)(x0)

+ 1

2

t s1

0 ua(s1)ub(s2)ds2ds1 (fafbφ)(x0)

+ 1

2

t s1

0 ub(s1)ua(s2)ds2ds1 (fbfaφ)(x0)

+ 1

2

t s1

0 ub(s1)ub(s2)ds2ds1 (fbfbφ)(x0)

+ 1

6

t s1 s2

0 ua(s1)ua(s2)ua(s3)ds3ds2ds1 (fafafaφ)(x0)

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

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Intro Controllability Series expansions Exp-prod

Integrate by parts: The wrong way to do it

t s1

0 ua(s1)ub(s2)ds2ds1 fafb+

t s1

0 ub(s1)ua(s2)ds2ds1 fbfa =

= t s1

0 ua(s1)ub(s2)ds2ds1 fafb −

t s1

0 ua(s1)ub(s2)ds2ds1 fbfa

+ t s1

0 ua(s1)ub(s2)ds2ds1 fbfa +

t s1

0 ub(s1)ua(s2)ds2ds1 fbfa

= t s1

0 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+ t

  • ua(s1)

s1

0 ub(s2) + ub(s1)

s1

0 ua(s2)ds2

  • ds1 fbfa

= t s1

0 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+ t

0ua(s) ds

  • ·

t

0ub(s) ds

  • fbfa

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

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Intro Controllability Series expansions Exp-prod

Integrate by parts, smart way

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

  • I∈{a,b}∗ I ⊗ I =

1 ⊗ 1 + a ⊗ a + b ⊗ b + 1

2

aa ⊗ aa + 1

2

ab ⊗ ab + 1

2

ba ⊗ ba + 1

2

bb ⊗ bb + 1

6

aaa ⊗ aaa + . . . = 1 ⊗ 1 + a ⊗ a + b ⊗ b + 1

2

aa ⊗ aa + 1

2

ab ⊗ (ab − ba) + 1

2

(ab + ba) ⊗ ba + 1

2

bb ⊗ bb + 1

6

aaa ⊗ aaa + . . .

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Drop everything except the indices - maps

  • The iterated integral

Υi1i2...in = t t1 · · · tn−1 ui1(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 ◦ . . . fi1

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

  • The Chen series is identified with the identity map on free

associative algebra A(Z) over set of indices Z = {1, . . . m} CF ∼ IdA(Z) =

  • n≥0
  • w∈Z n

w ⊗ w ∈ ˆ A(Z) ⊗ A(Z) with shuffle product on left and concatenation on right

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Recall: definition of the 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) > where ∆: A(Z) → A(Z) ⊗ A(Z) by ∆(a) = 1⊗a+a⊗1 for a ∈ Z

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Shuffles and simplices

On permutations algebras Duchamp and Agrachev consider partially commutative and noncommutative shuffles. Illustration:

✲ ✲ ✲ ✻ ✻ ✻

  • σ1x 2

σ1 σ2 = σ21 ∪ σ12 In the case of three letters {1, 2, 3}

= ∪ ∪

σ(12)x 3 = σ312 ∪ σ132 ∪

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 t t t for better readability):

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Homomorphisms I

  • For fixed smooth vector fields fi

F : A(Z) → partial diff operators on C∞(M) F : (i1i2 . . . in) → fi1 ◦ fi2 ◦ . . . fin associative algebras: concatenation → composition

  • For fixed control u ∈ UZ

Υ(u): A(Z) → AC([0, T], R) Υ(u): (i1i2 . . . in) → T t1 · · · tp−1 uip(tp) . . . ui1(t1) dt1 . . . dtp associative algebras (Ree’s theorem): shuffle of words → pointwise multiplication of functions

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CF-coefficients satisfy shuffle-relations

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

Υ1(t, u) ≡ 1 Υax 1(t, u) = Υa(t, u) = Υa(t, u) · 1 = Υa(t, u) · Υ1(t, u) for any letter a ∈ X

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Induction step

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) = T

0 Υ(wa)x z(t, u) · ub(t)dt +

T

0 Υw x (zb)(t, u) · ua(t)dt

= T

0 (Υwa(t, u) · Υz(t, u) · ub(t) + Υw(t, u) · Υzb(t, u) · ua(t)) dt

= T

  • Υwa(t, u) · d

dt Υzb(t, u) + d dt (Υwa(t, u)) · Υzb(t, u)

  • dt

= Υwa(T, u) · Υzb(T, u)

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Homomorphisms II

  • Restriction is Lie algebra homomorphism

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

  • Do not fix controls: iterated integral functionals

Υ: ∈ A(Z) → IIF(UZ) Υ: (i1i2 . . . in) →

  • u →

T t1 · · · tp−1 uip(tp) . . . ui1(t1) dt1 . . . dtp

  • associative algebras: shuffle of words → pointwise

multiplication of iterated integral functionals

  • Much better: Theorem: If U = L1([0, T], [−1, 1]) then

Υ: (A(Z), ∗) → IIF(UZ) is a Zinbiel algebra isomorphism.

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Zinbiel product and algebra

Abstractly, right Zinbiel identity U ∗ (V ∗ W) = (U ∗ V) ∗ W + (V ∗ U) ∗ W Concrete examples in control:

  • polynomials

X n ∗ X m =

m n+mX n+m

and X n ⋆ X m = 1

nX n+m

  • AC([0, ∞)):

(U ∗ V)(t) = t

0 U(s) V ′(s) ds

and (U ⋆ V)(t) = t

0 U(s)ds V(t)

  • iterated integrals functionals
  • subsets, e.g. exponentials

eimt ∗ eint =

m n+mei(m+n)t

and eimt ⋆ eint = 1

mei(m+n)t

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Intro Controllability Series expansions Exp-prod

Solving DEs by iteration and Zinbiel products

The integrated form of the universal control system ˙ S = S · Φ, S(0) = 1 with Φ =

m

  • i=1

uiXi is compactly rewritten using chronological 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) . . .

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Rewriting the Chen series

  • Recall: Chen series is an exponential Lie series
  • w∈Z ∗

w ⊗ w = exp

  • b∈B

ζb ⊗ b

  • =

←

  • b∈B

exp(ξb ⊗ b) where B is ordered basis of free Lie algebra L(Z) ⊂ A(Z)

  • Coord’s of 2nd kind (classic)

ξHK = ξH ∗ ξK

  • Coord’s of the 1st kind (new)

ζ = π′

1 ◦ ξ

  • Use in control / geometric integration: explicit formula for

iterated integral functionals Υζb and Υξb.

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Intro Controllability Series expansions Exp-prod

Coordinates of the second kind

For a Hall set H over Z and letters a ∈ Z and H, K, HK ∈ H ξa = a, ξHK = µHK ξH ∗ ξK (multifactorial µHK)

  • Schützenberger (1958)
  • Agrachev/Gamkrelidze (?)

(1979) chronological prod

  • Sussmann (1985) exp prod
  • Reutenauer/Melançon (1989)
  • Grayson/Grossmann (1990)
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Intro Controllability Series expansions Exp-prod

Coordinates ξH of 2nd kind, historically

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Intro Controllability Series expansions Exp-prod

Coordinates ξH of 2nd kind, historically

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Zinbiel product in 1958

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Illustration: Normal form for free nilpotent system

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

˙ ξa = ua ˙ ξb = ub ˙ ξab = ξa · ˙ ξb = ξa ub ˙ ξaab = 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 = ξ3

a ub

using ψ−b(aaab) = (a(a(ab))) ˙ ξbaab = ξb · ˙ ξaab = ξbξ2

a ub

using ψ−b(baab) = (b(a(ab))) ˙ ξbbab = ξb · ˙ ξbab = ξ2

bξa ub

using ψ−b(bbab) = (b(b(ab))) ˙ ξaaaab = ξa · ˙ ξaaab = ξ4

a ub

using ψ−b(aaaab) = (a(a(a(ab)))) ˙ ξbaaab = ξb · ˙ ξaaab = ξbξ3

a ub

using ψ−b(baaab) = (b(a(a(ab)))) ˙ ξbbaab = ξb · ˙ ξbaab = ξ2

bξ2 a ub

using ψ−b(bbaab) = (b(b(a(ab)))) ˙ ξabaab = ξab · ˙ ξaab = ξabξ3

a ub

using ψ−b(abaab) = ((ab)(a(ab))) ˙ ξabbab = ξab · ˙ ξbab = ξabξ2

bξa ub

using ψ−b(abbab) = ((ab)(b(ab)))

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

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Dendriform algebras

  • Introduced by Loday[2001], described in control by Rocha

[2004], exploited by Ebrahimi-Fard and Manchon [2006,07]

  • Axioms: A dendriform (di)algebra is a k-vector space with

product x and two bilinear operations ∗, ⋆ satisfying u ∗ (v ∗ w) = (u x v) ∗ w (u ⋆ v) ⋆ w = u ⋆ (v x w) u ⋆ (v ∗ w) = (u ⋆ v) ∗ w where u x v = (u ∗ v) + (u ⋆ v)

  • – we disregarded, did not use, the 3rd axiom!
  • Dendriform structure shows relation between Magnus

and Fer expansions [Ebrahimi-Fard and Manchon, 2008]

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Intro Controllability Series expansions Exp-prod

Selected references

  • L. Loday Dialgebras, Lect. Notes Math. 1763, Springer, Berlin pp. 7-66 (2001).
  • M. Schützenberger Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un

probléme de mathématiques appliquées, Séminaire P . Dubreil, Algèbres et Théorie des Nombres, Faculté des Sciences de Paris (1958/59).

  • K. Ebrahimi-Fard, D. Manchon, and F. Patras

New identities in dendriform algebras [arXiv: 0705.2636v2]

  • K. Ebrahimi-Fard, and D. Manchon

A Magnus- and Fer-type formula in dendriform algebras [arXiv:0707.0607v2]

  • M. Kawski and H. J. Sussmann. Control Theory from the Geometric Viewpoint [Springer, 2004]

Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory, in: Operators, Systems, and Linear Algebra, U. Helmke, D. Prätzel-Wolters and E. Zerz, eds., Teubner (1997),

  • pp. 111–128.
  • M. Kawski, Chronological algebras in nonlinear control, Itogi Nauki i Techniki, vol.68 (2000) pp.144-178.

English translation in J. Math.Sciences, vol. 103 (2001) pp. 725-744.

  • M. Hazewinkel, Endomorphisms of Hopf algebras and a little bit of control LNCIS 321 (2005) pp. 107-122.
  • M. Hazewinkel, The Leibniz-Hopf Algebra and Lyndon Words Centrum voor Wiskunde en Informatica

Report AM-R9612, (1996).

  • A.Agrachëv and R. Gamkrelidze, The shuffle product and symmetric groups, Lect. Notes Pure Appl.

Math., 152, (1994) pp. 365–382.