Formal solution Chen-Fliess series

Transcription

1 Formal solution Chen-Fliess series Ṡ = S(t) u a (t) a, S() = 1 Z = m a Z on algebra Â(Z ) of formal power series in aset Z of noncommuting indeterminates (letters) has the unique solution CF(T, u) = w Z T t1 tp 1 u ap (t p )... u a 1 (t 1 ) dt 1... dt p } {{ } Υ w (T,u) Use as asymptotic expansion for evolution of output y = ϕ(x) along solution of ẋ = a Z ua f a (x). ϕ(x(t, u)) w Z T t1 tp 1 a 1... a p }{{} =w u ap (t p )...u a 1 (t 1 ) dt 1... dt p (f a1... a ip ϕ)

2 Series solution by iteration φ(x(t, u)) = 1 φ(x ) + t ua (s)ds (f a φ)(x ) + t ub (s)ds (f b φ)(x ) t t t t t ua (s 1 )u a (s 2 )ds 2 ds 1 (f a f a φ)(x ) ua (s 1 )u b (s 2 )ds 2 ds 1 (f a f b φ)(x ) ub (s 1 )u a (s 2 )ds 2 ds 1 (f b f a φ)(x ) ub (s 1 )u b (s 2 )ds 2 ds 1 (f b f b φ)(x ) s2 ua (s 1 )u a (s 2 )u a (s 3 )ds 3 ds 2 ds 1 (f a f a f a φ)(x ) +... Objective: Collect first order differential operators, and minimize number of higher order differential operators involved

3 Integrate by parts: The wrong way to do it t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a = = t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a

4 Integrate by parts: The wrong way to do it t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a = = t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a

5 Integrate by parts: The wrong way to do it t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a = = t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a = t + t ua (s 1 )u b (s 2 )ds 2 ds 1 (f a f b f b f a ) ( u a (s 1 ) s 1 ub (s 2 ) + u b (s 1 ) ) s 1 ua (s 2 )ds 2 ds 1 f b f a

6 Integrate by parts: The wrong way to do it t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a = = t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a = t + t ua (s 1 )u b (s 2 )ds 2 ds 1 (f a f b f b f a ) ( u a (s 1 ) s 1 ub (s 2 ) + u b (s 1 ) ) s 1 ua (s 2 )ds 2 ds 1 f b f a = t ua (s 1 )u b (s 2 )ds 2 ds 1 (f a f b f b f a ) ( ) ( ) t t + ua (s) ds ub (s) ds f b f a

7 Integrate by parts: The wrong way to do it t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a = = t ua (s 1 )u b (s 2 )ds 2 ds 1 f a f b t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ua (s 1 )u b (s 2 )ds 2 ds 1 f b f a + t ub (s 1 )u a (s 2 )ds 2 ds 1 f b f a = t + t ua (s 1 )u b (s 2 )ds 2 ds 1 (f a f b f b f a ) ( u a (s 1 ) s 1 ub (s 2 ) + u b (s 1 ) ) s 1 ua (s 2 )ds 2 ds 1 f b f a = t ua (s 1 )u b (s 2 )ds 2 ds 1 (f a f b f b f a ) ( ) ( ) t t + ua (s) ds ub (s) ds f b f a Lie brackets together w/ iterated integrals in right order higher order deriv s (wrong order) w/ pointwise prod s of int s

8 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 = a a + b b aa aa ab ab ba ba bb bb aaa aaa +... = a a + b b aa aa ab (ab ba) (ab + ba) ba bb bb aaa aaa +...

9 Drop everything except the indices - maps The iterated integral Υ a 1a 2...a n = t t1 tn 1 u a 1 (t 1 )u a 2 (t 2 ) u an (t n ) dt n dt n 1 dt 1 is uniquely identified by the multi-index ( word ) a 1 a 2... a n The n-th order partial differential operator f an f an 1... f a1 is uniquely identified by the multi-index ( word ) a 1 a 2... a n The Chen series is identified with the identity map on free associative algebra A(Z ) over set Z of with Z = m CF Id A(Z ) = w w Â(Z ) A(Z ) n w Z n with shuffle product on left and concatenation on right

10 Homomorphisms I For fixed smooth vector fields f i F : A(Z ) partial diff operators on C (M) F : (a 1 a 2... a n ) f a1 f a2... f an associative algebras: concatenation composition For fixed control u U Z Υ(u): A(Z ) AC([, T ], R) Υ(u): (a 1 a 2... a n ) T t1 tp 1 u ap (t p )... u a1 (t 1 ) dt 1... dt p associative algebras (Ree s theorem): shuffle of words pointwise multiplication of functions

11 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 where < v X w, z > = < v w, (z) > SKIP : A(Z ) A(Z ) A(Z ) by (a) = 1 a+a 1 for a Z

12 Shuffles and simplices SKIP 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 = σ (12) x 3 = σ 312 σ 132 E.g. σ (12) x 3 = {t : t 1 t 2 1, t 3 1} For multiplicative integrands f (x, y, z) = f 1 (x) f 2 (y) f 3 (z) ( 1 y ( )dx dy ) 1 ( )dz = 1 y x ( ) dz dx dy+ 1 y z ( ) dx dz dy+ 1 z y ( ) dx dy dz

13 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(U Z ) ( Υ: (a 1 a 2... a n ) u T t1 tp 1 associative algebras: shuffle of words pointwise multiplication of iterated integral functionals u ap (t p )... u a 1 (t 1 ) dt 1... dt Much better: Theorem: If U = L 1 ([, T ], [ 1, 1]) then Υ: (A(Z ), ) IIF(U Z ) is a Zinbiel algebra isomorphism.

14 Zinbiel: The product & algebra of iterated integrals Abstractly, right Zinbiel identity U (V W ) = (U V ) W + (V U) W Concrete examples in control: polynomials X n X m = m n+m X n+m and X n X m = 1 n X n+m AC([, )): and iterated integrals functionals (U V )(t) = t U(s) V (s) ds (U V )(t) = t U(s)ds V (t) subsets, e.g. exponentials e imt e int = m n+m ei(m+n)t and e imt e int = 1 m ei(m+n)t

15 Abstract solution of DE by iteration Rewrite universal control system Ṡ = S Φ, S() = 1 with Φ = m as integral equation using right Zinbiel product S = 1 + S Φ and iterate to obtain Chen Fliess series i=1 u i X i or Φ = a Z a a SKIP 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)...

16 Rewriting the Chen series Recall: Chen series is an exponential Lie series ( ) w w = exp ζ b b = exp(ξ b b) w Z b B b B where B is ordered basis of free Lie algebra L(Z ) A(Z ) Coord s of 2 nd kind (classic) Coord s of the 1 st kind (new) ξ HK = ξ H ξ K ζ = π 1 ξ Use in control / geometric integration: explicit formula for iterated integral functionals Υ ζ b and Υ ξ b.

17 Hall words Hall words (in narrow sense as in Bourbaki) a Z a H w, z H, u < v u < v If a Z, then (ua) H u < a and u < (ua) (u, (vw)) H u, (vw) H and v u < (vw), u < (u(vw)) ((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

18 Iterated integral functionals ξ H (coord. s of 2 nd kind Presented as 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} ξ a = u a ξ b = u b ξ ab = ξ a ξ b = ξ a u b ξ aab = ξ a ξ ab = ξa 2 u b using ψ b (aab) = (a(ab)) ξ bab = ξ b ξ ab = ξ b ξ a u b using ψ b (bab) = (b(ab)) ξ aaab = ξ a ξ aab = ξa 3 u b using ψ b (aaab) = (a(a(ab))) ξ baab = ξ b ξ aab = ξ b ξa 2 u b using ψ b (baab) = (b(a(ab))) ξ bbab = ξ b ξ bab = ξb 2ξ a u b using ψ b (bbab) = (b(b(ab))) ξ aaaab = ξ a ξ aaab = ξa 4 u b using ψ b (aaaab) = (a(a(a(ab)))) ξ baaab = ξ b ξ aaab = ξ b ξa 3 u b using ψ b (baaab) = (b(a(a(ab)))) ξ bbaab = ξ b ξ baab = ξb 2ξ2 a u b using ψ b (bbaab) = (b(b(a(ab)))) ξ abaab = ξ ab ξ aab = ξ ab ξa 3 u b using ψ b (abaab) = ((ab)(a(ab))) ξ abbab = ξ ab ξ bab = ξ ab ξb 2ξ a u b using ψ b (abbab) = ((ab)(b(ab))) Analyze orbits of all nilpotent systems under feedback group