Manipulating exponential products Instead of working with - PDF document

Manipulating exponential products Instead of working with complicated concatenations of flows like � 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 ( f 0 + f 1 + f 2 ) dt ( p ) z ( t ) = e it is desirable to rewrite the solution curve using a minimal number of vector fields f π k that span the tangent space (typically using iterated Lie brackets of the system fields f 0 , f 1 , . . . f m ) Coordinates of the first kind z ( t ) = e b 1 ( t,u ) f π 1 + b 2 ( t,u ) f π 1 + b 3 ( t,u ) f π 3 + ... + b n ( t,u ) f πn ( p ) Coordinates of the second kind z ( t ) = e c 1 ( t,u ) f π 1 ◦ e c 2 ( t,u ) f π 1 ◦ e c 3 ( t,u ) f π 3 ◦ . . . ◦ e c n ( t,u ) f πn ( 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 b k ( t, u ) or c k ( t, u ). 12

The Chen Fliess series K. T. Chen, 1957: Geometric invariants of curves in R n M. Fliess, 1970s: adaptation to control The formal control system ⎛ ⎞ m ˙ � ⎠ , u i X i S = S S (0) = I ⎝ i =1 on the associative algebra ˆ A ( X 1 . . . X m ) of formal power series in the noncommuting indeterminates (letters) X 1 , . . . X m has the unique solution � T � t 1 � t p − 1 � u i p ( t p ) . . . u i 1 ( t 1 ) dt 1 . . . dt p S CF ( T, u ) = 0 · · · X i 1 . . . X i p 0 0 I � �� � � �� � X I Υ I ( T,u ) What is the CF-series good for? For any given control system m � x = ˙ u i ( x ) f i ( x ) , x (0) = p, with “output” y = ϕ ( x ) i =1 � T � t 1 � t p − 1 � u i p ( t p ) . . . u i 1 ( t 1 ) dt 1 . . . dt p φ ( x ( T, u )) = 0 · · · ( f i 1 ◦ . . . ◦ f i p ϕ )( p ) 0 0 I � �� � Υ I ( T,u ) (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 1 st 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 ⎛ ⎞ ⎝ � S CF ( T, u ) = exp B ∈B α B ( t, u ) B ⎠ or ← � S CF ( T, u ) = B ∈B exp ( β B ( T, u ) B ) for suitable bases B of the free Lie algebra L ( X 1 , . . . X m ). Question: Existence? 14

Ree’s theorem and exponential Lie series Theorem [Ree, 1957]: A formal power-series � I c I X I is an expo- nential Lie series iff the coefficients satisfy the shuffle relations c I x J = c I · c J for all I, J Exercise : The coefficients Υ I ( 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 α B ( T, u ) and/or β B ( 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 = A k ( X ) (algebra of polynomi- als, or “words” ) over a set X (of noncommuting indeterminates, or “letters” ) define a co-product ∆: A × A �→ A by ∆( a ) = 1 ⊗ a + a ⊗ 1 for a ∈ X Define the shuffle product X as the transpose of ∆ (on the algebra ˆ A = ˆ A k ( X ) of formal power series) < v X w , z > = < v ⊗ w , ∆( z ) > 16

Shuffles and simplices On permutations algebras Duchamp and Agrachev consider partially commutative and noncommutative shuffles. Illustration: ✻ ✻ ✻ � � � � � � � σ 12 � σ 2 σ 1 x 2 � = � � ∪ � σ 21 � � � � � � � � ✲ ✲ ✲ σ 1 In the case of three letters { 1 , 2 , 3 } = ∪ ∪ σ (12) x 3 = σ 312 ∪ σ 132 ∪ σ 123 E.g. σ (12) x 3 = { t : 0 ≤ t 1 ≤ t 2 ≤ 1 , 0 ≤ t 3 ≤ 1 } For multiplicative integrands f ( x, y, z ) = f 1 ( x ) · f 2 ( y ) · f 3 ( z ) (using x, y, z , instead of t 1 , t 2 , t 3 for better readability): �� 1 � 1 � y � 0 ( · ) dx dy · 0 ( · ) dz = 0 � 1 � 1 � 1 � y � x � y � z � z � y 0 ( · ) dz dx dy + 0 ( · ) dx dz dy + 0 ( · ) dx dy dz 0 0 0 0 0 0 17

CF-coefficients satisfy shuffle-relations Sketch of proof of exercise (by induction on the combined lengths of the coefficients) Υ 1 ( t, u ) ≡ 1 Υ a x 1 ( t, u ) = Υ a ( t, u ) = Υ a ( t, u ) · 1 = Υ a ( t, u ) · Υ 1 ( 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 ) � T � T (1) = 0 Υ ( wa ) x z ( t, u ) · u b ( t ) dt + 0 Υ w x ( zb ) ( t, u ) · u a ( t ) dt � T = 0 (Υ wa ( t, u ) · Υ z ( t, u ) · u b ( t ) + Υ w ( t, u ) · Υ zb ( t, u ) · u a ( t )) dt � T � � Υ wa ( t, u ) · d dt Υ zb ( t, u ) + d = dt (Υ wa ( t, u )) · Υ zb ( t, u ) dt 0 = Υ wa ( T, u ) · Υ zb ( T, u ) 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 ( X 1 , . . . X m ). ← � S CF ( T, u ) = B ∈B exp ( β B ( T, u ) B ) Recall remaining issues/questions : • Need explicit basis for the free Lie algebra • Want explicit formulae for the iterated integral coefficients α B ( T, u ) and/or β B ( T, u ). Using different bases for the free Lie algebra explicit formulae for the dual bases (iterated integral functionals β B ( T, u ) have been redis- covered several times in different contexts: • Sch¨ utzenberger (1958), S´ eminaire Dubreil • Sussmann (1986), Nonlinear control • Melan¸ con 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 L k ( 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 L k ( X ) that is freely generated by the set { (ad j c, b ): b ∈ X \ { c } , j ≥ 0 } . This principle is at the heart of constructions involving Hall bases for free Lie algebras, for Sussmann’s derivation of the exponential product 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 Sˇ irsov, 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 ) �→ L k ( X ) from the free magma into the free Lie algebra is a basis for L k ( 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 L k ( X )). Consequently, ϕ − 1 ([ x, [ y, [ x, y ]]]) = ϕ − 1 ([ y, [ x, [ x, y ]]]) = ( y, ( x, ( x, y ))) � = ( x, ( y, ( x, y ))) in M ( X ) Similarly, ϕ − 1 ([ − y, x ]) = ϕ − 1 ([ x, y ]) = ( x, y ) � = − ( y, x ) in the algebra over M ( X ) But coding of iterated integrals depends critically on the factorization of the Hall words, requiring well-defined left and right factors. 22