Janet's Algorithm and Systems Theory

Janet’s Algorithm and Systems Theory

Daniel Robertz

Lehrstuhl B f¨ur Mathematik

http://wwwb.math.rwth-aachen.de/~daniel

28.11.2012

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Outline

Janet’s algorithm

Module-theoretic approach to linear systems

Parametrizing linear systems

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet’s algorithm

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet’s algorithm for linear PDEs

∂2u ∂u − = 0 ∂x∂y ∂y  find: u = u(x,y) analytic  ∂2u ∂u  − = 0 ∂x2 ∂y  

2 x2 xy y u(x,y)= a0,0 + a1,0 x + a0,1 y + a2,0 2! + a1,1 1!1! + a0,2 2! + ...

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet’s algorithm for linear PDEs

∂2u ∂u − = 0 ∂x∂y ∂y  find: u = u(x,y) analytic  ∂2u ∂u  − = 0 ∂x2 ∂y  

2 x2 xy y u(x,y)= a0,0 + a1,0 x + a0,1 y + a2,0 2! + a1,1 1!1! + a0,2 2! + ...

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet’s algorithm for linear PDEs

∂2u ∂u − = 0 ∂x∂y ∂y  find: u = u(x,y) analytic  ∂2u ∂u  − = 0 ∂x2 ∂y   ∂ ∂2u ∂u ∂ ∂2u ∂u ∂2u ∂u ∂x ∂x∂y − ∂y − ∂y ∂x2 − ∂y = ∂y2 − ∂y = 0 2 x2 xy y u(x,y)= a0,0 + a1,0 x + a0,1 y + a2,0 2! + a1,1 1!1! + a0,2 2! + ...

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet’s algorithm for linear PDEs

Janet’s algorithm computes a vector space basis for power series solutions

(Maurice Janet, ∼ 1920)

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet’s algorithm for linear PDEs

uy,y = 0 A   ux,x − yuz,z = 0 B  is equivalent to

uy,y = 0 A   ux,x − yuz,z = 0 B    1 2 − 2 − 1 2  uy,z,z = 0 2 (∂x y∂z )A 2 ∂y B     ux,y,y = 0 ∂xA  1 4 − 2 2 2 4 − 1 2 2 − 2 2 2 uz,z,z,z = 0 2 (∂x 2y∂x∂z + y ∂z )A 2 (∂x∂y y∂y ∂z +2∂y ∂z )B    1 3 − 2 − 1 2  ux,y,z,z = 0 2 (∂x y∂x∂z )A 2 ∂x∂y B    1 5 − 3 2 2 4 − 1 3 2 2 2 − 2  ux,z,z,z,z = 0 2 (∂x 2y∂x∂z + y ∂x∂z )A 2 (∂x∂y + y∂x∂y ∂z 2∂x∂y ∂z )B   Taylor coeﬀ’s for 1, z, y, x, z2, yz, xz, xy, z3, xz2, xyz, xz3 arbitrary, all other coeﬀ’s determined by linear equations

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Outline

Decomposition of multiple closed sets of monomials into disjoint cones

Janet’s algorithm

(Generalized) Hilbert series, Hilbert polynomial

Construction of a free resolution

Janet bases over Z

Janet bases for Ore algebras

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Strategy

R = K[x1,...,xn], K field

I ideal of R (or, more generally: M submodule of Rm)

goal: compute “good” generating set for I (Janet basis)

sort terms in a polynomial in a way compatible with multiplication

i n total ordering < on M := Mon(x1,...,xn) := {x | i ∈ (Z≥0) }

use highest terms lm(p), lm(q) to decide “divisibility”

∂2u ∂u represent − as ∂ ∂ − ∂ in Q[∂ ,∂ ] ∂x∂y ∂y x y y x y

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Multiple closed sets of monomials

i n M := Mon(x1,...,xn) := {x | i ∈ (Z≥0) }

S ⊆ M is M-multiple closed if

ms ∈ S ∀ m ∈ M, s ∈ S

M-multiple closed set 2 3 4 generated by x1x2, x1x2, x1

2 3 4 =: h x1x2, x1x2, x1 iM

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Multiple closed sets of monomials

Lemma Every M-multiple closed set S ⊆ M has a finite generating set. Proof.

Every seq. F : m1,m2,m3 ... ∈ M s.t. mi 6| mj ∀ i

Induction: n = 1: clear.

a1 an n → n + 1: Let m1 = x1 ··· xn .

(j,d) b1 d bn Define subsequence F : mi = x1 ··· xj ··· xn

We have: {F (j,d)} = {F } j n d a 1≤[≤ 0≤[≤ j By induction, the {F (j,d)} are finite.

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Multiple closed sets of monomials

Lemma Every M-multiple closed set S ⊆ M has a finite generating set.

Cor. Every ascending sequence of M-multiple closed sets becomes stationary.

Given a finite generating set {p1,...,pr} for I E K[x1,...,xn], Janet’s algorithm computes

S0 ⊆ S1 ⊆ ... ⊆ Sk = lm(I) (all M-multiple closed)

where S0 is generated by lm(p1), ..., lm(pr)

⇒ termination

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Decomposition into disjoint cones

Def.

Let C ⊆ Mon(x1,...,xn), µ ⊆ {x1,...,xn}.

(C,µ) is a cone if ∃ v ∈ C s.t. C = Mon(µ) v

Variables in µ: multiplicative variables (for v)

Variables in {x1,...,xn}− µ: non-multiplicative variables (for v)

Dimension of (C,µ): |µ|

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Decomposition into disjoint cones

Def.

Let S ⊆ M = Mon(x1,...,xn).

{ (C1,µ1), ..., (Cr,µr) }⊂P(M) ×P({x1,...,xn})

is a decomposition of S into disjoint cones if

˙ r each (Ci,µi) is a cone and S = i=1Ci. S

{ (v1,µ1), ..., (vr,µr) } is a decomp. of S into disj. cones

if { (Mon(µ1) v1,µ1), ..., (Mon(µr) vr,µr) } is one.

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Decomposition into disjoint cones

Strategy of Janet’s algorithm:

Decompose M-multiple closed sets S into disjoint cones.

2 3 4 S = h x1x2, x1x2, x1 iM

decomposition: 2 2 2 { (x1x2, {x2}), (x1x2, {x2}), 3 4 (x1x2, {x2}), (x1, {x1,x2}) }

This can also be done for Mon(x1,...,xn) − S.

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet division

The possible ways of decomposing M-multiple closed sets into disjoint cones are studied as involutive divisions (Gerdt, Blinkov et. al.)

Janet division:

Let G ⊂ M = Mon(x1,...,xn) be finite.

a1 an For a cone with vertex v = x1 ··· xn ∈ G

xi is a multiplicative variable iﬀ

b ai = max{ bi | x ∈ G; bj = aj ∀ j

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet division

a1 an For v = x1 ··· xn :

b xi ∈ µ ⇐⇒ ai = max{ bi | x ∈ G; bj = aj ∀ j

Example: G = { yz, xyz, x2 yz, x2 y2 }

x yz

x2 yz

x2 y2

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet division

a1 an For v = x1 ··· xn :

b xi ∈ µ ⇐⇒ ai = max{ bi | x ∈ G; bj = aj ∀ j

Example: G = { yz, xyz, x2 yz, x2 y2 }

yz ∗ y z

x yz ∗ y z

x2 yz x ∗ z

x2 y2 x y z

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Decomposition into disjoint cones

Decompose(G, η) G ⊂ Mon(x1,...,xn), ∅ 6= η ⊆ {x1,...,xn} G ←{g ∈ G |6∃ h ∈ G : h | g} if |G| ≤ 1 or |µ| =1 then return {(m, η) | m ∈ G} else

y ← ya with a = min{i | 1 ≤ i ≤ n, yi ∈ η}

d ← max{degy(g) | g ∈ G}

Gi ←{g ∈ G | degy(g) = i}, i =0,...,d i−1 i−j Gi ← Gi ∪ Sj=0 {y g | g ∈ Gj }, i =1,...,d

Td ←{ (m,ζ ∪{y}) | (m,ζ) ∈ Decompose(Gd, η −{y}) }

Ti ← Decompose(Gi, η −{y}), i =0,...,d − 1 d return Si=0 Ti ﬁ

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet reduction

NF(p, T , ≺) p ∈ K[x1,...,xn], T = { (d1,µ1),..., (ds,µs) } r ← 0 while p 6=0 do if ∃ (d,µ) ∈ T : lm(p) ∈ Mon(µ) lm(d) then lc(p) lm(p) p ← p − lc(d) lm(d) d else r ← r + lc(p) lm(p) p ← p − lc(p) lm(p) fi od return r Disj. cones ⇒ course of Alg. is uniquely determined

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet’s algorithm

JanetBasis(F , ≺) F ⊆ K[x1,...,xn] finite G ← F do G ← auto-reduce G

J ←{ (p1,µ1),..., (pr,µr) } s.t. { (lm(p1),µ1),..., (lm(pr),µr) }

decomp. into disj. cones of h lm(G) iM P ←{ NF(x · p,J) | (p,µ) ∈ J, x 6∈ µ } G ←{ p | (p,µ) ∈ J }∪ P while P 6= {0}

return J

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet basis

Janet basis J = { (p1,µ1), ..., (pr,µr) } for I = hF i

Invariant of the loop:

G (or {p1,...,pr}) always forms a gen. set for I

r Mon(µi)pi is a K-basis of I. i[=1 Linear independence: clear.

r p ∈ I: p = ci pi Xi=1

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Example

2 Let I := h g1, g2 i E K[x,y], g1 := x − y, g2 := xy − y.

Let > be degrevlex, x>y.

Decomposition into disjoint cones of h lm(g1), lm(g2) i:

{ (x2, {x,y}), (xy, {y}) }

2 2 f := x · g2 = x y − xy ∈ I, f = ci gi? i=1 X

2 Reduction of f modulo g1, g2 yields: g3 := y − y ∈ I

{ (g1, {x,y}), (g2, {y}), (g3, {y}) } (minimal) Janet basis for I

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 (Generalized) Hilbert series

Janet basis J = { (p1,µ1), ..., (pr,µr) } for I

We have lm(I)= h lm(p1), ..., lm(pr) iM.

Generalized Hilbert series r 1 H (x ,...,x )= lm(p ) I 1 n i 1 − x x ∈µ j Xi=1 Yj i enumerates a K-basis of lm(I).

HI (t,...,t) is the usual Hilbert series.

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Example

Janet basis for I

2 2 2 3 4 J = { (x1x2, {x2}), (x1x2, {x2}), (x1x2, {x2}), (x1, {x1,x2}) }

generalized Hilbert series: x x2 x2x2 x3x x4 1 2 + 1 2 + 1 2 + 1 1 − x2 1 − x2 1 − x2 (1 − x1)(1 − x2)

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Hilbert polynomial

Janet basis of M: { (p1,µ1), ..., (pr,µr) }

k HM (t,...,t) = dimK Mk t k X≥0 r 1 = tdeg(pi) (1 − t)|µi| i=1 Xr |µi| + j − 1 = tdeg(pi) tj j i=1 j X X≥0 k Coeﬀ. of t in HM (t,...,t) ? For k ≥ max{deg(pi) | i = 1,...,r}:

r |µi| + k − deg(pi) − 1 dimK Mk = k − deg(pi) i=1 X 

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Example

S = lm(I)

Decomp. of Mon(x1,...,xn) − S into disjoint cones

generalized Hilbert series enum. a K-basis of K[x1,...,xn]/I

generalized Hilbert series:

1 2 2 3 + x1 + x1x2 + x1 + x1x2 + x1 1 − x2

Hilbert polynomial: 6 |µ | + k − deg(p ) − 1 i i = 1 k − deg(pi) i=1 X 

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Free resolution

Janet basis J = { (p1,µ1), ..., (pr,µr) } for I

We have xj pi = k αi,j,k pk, xj 6∈ µi, αi,j,k ∈ K[µk]

|J| P Define π : R → R :p ˆi 7→ pi. (pˆi std. gen.)

Prop.

xj pˆi − αi,j,k pˆk, xj 6∈ µi, i = 1,...,r, k X form a Janet basis of ker π for a suitable monomial ordering.

construction of a free resolution of R/I.

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Example

R = K[x1,x2,x3], x1 >x2 >x3, I =(x1,x2,x3)

Janet basis: x1 x1 x2 x3 x2 ∗ x2 x3 x3 ∗ ∗ x3

normal form computation: x1 · x2 − x2 · x1 x1 · x3 − x3 · x1 x2 · x3 − x3 · x2

−x x 0 x  2 1   1  −x3 0 x1 x2      0 −x3 x2   x3  R1×3 −−−−−−−−−−−−−−−→ R1×3 −−−−−−→ R −→ R/I −→ 0

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Example

R = K[x1,x2,x3], x1 >x2 >x3, I =(x1,x2,x3)

Janet basis: [−x2 x1 0] x1 x2 x3 [−x3 0 x1 ] x1 x2 x3 [ 0 −x3 x2 ] ∗ x2 x3

normal form computation:

x1 · [ 0 −x3 x2] − x2 · [−x3 0 x1]+ x3 · [−x2 x1 0]

−x x 0 x  2 1   1  −x3 0 x1 x2     ( x3 −x2 x1 )  0 −x3 x2   x3  0 → R −−−−−−−−−−−−→ R1×3 −−−−−−−−−−−−−−−→ R1×3 −−−−−−→ R → R/I → 0

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Free resolution

Prop.

xj pˆi − αi,j,k pˆk, xj 6∈ µi, i = 1,...,r, k X form a Janet basis of ker π w.r.t. ≺.

x2 x1

Choose a total order ≪ on J s.t. x1 x2 pk ≪ pl if ∃ path from pl to pk in the Janet graph. x1 x3

i j x lm(pk)

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Consequences

{ (p1,µ1), ..., (pr,µr) } Janet basis for I E R can decide ideal membership normal form for residue classes modulo I enumeration of a K-basis of I and a K-basis of R/I (generalized Hilbert series) can easily determine Hilbert polynomial can read oﬀ a free resolution of R/I every Janet basis is a Gr¨obner basis

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet bases over Z

NF(p, T , ≺) p ∈ Z[x1,...,xn], T = { (d1,µ1),..., (dl,µl) } r ← 0 while p 6=0 do if ∃ (d,µ) ∈ T : lm(p) ∈ Mon(µ) d then write lc(p) = a · lc(d) + b, |b| < | lc(d)| if a 6=0 then lm(p) p ← p − a lm(d) d else move leading term from p to r fi else move leading term from p to r fi od return r

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet bases for Ore algebras Skew polynomial ring A[∂; σ, δ]:

A domain and K-algebra

σ : A → A K-algebra endomorphism

δ : A → A σ-derivation, i.e.

δ(a b)= σ(a) δ(b)+ δ(a) b, a, b ∈ A

i A[∂; σ, δ]= ai ∂ | ai ∈ A, i ∈ Z≥0 ( ) Xﬁn. with commutation rule

∂a = σ(a) ∂ + δ(a), a ∈ A

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet bases for Ore algebras

Ore algebra D = A[∂1; σ1,δ1] ... [∂m; σm,δm]:

A = K or A = K[x1,...,xn]

σi : D → D K-algebra endomorphisms

δi : D → D σi-derivations

i m A[∂1; σ1,δ1] ... [∂m; σm,δm]= ai ∂ | ai ∈ A, i ∈ (Z≥0) ( ) Xﬁn. with commutation rules

∂i a = σi(a) ∂i + δi(a), a ∈ A,

∂i ∂j = ∂j ∂i

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet bases for Ore algebras

Weyl algebra: ordinary diﬀerential equations

d d d da A1 = K[t][ dt ] dt a = a dt + dt Weyl algebra: partial diﬀerential equations

An = K[x1,...,xn][∂1,...,∂n] ∂i xj = xj ∂i + δij

Bn = K(x1,...,xn)[∂1,...,∂n] Shift operators: diﬀerence equations

Sh = K[t][δh] δh t =(t − h) δh

combinations ...

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet bases for Ore algebras

D = K[x1,...,xn][∂1,...,∂m]

I left ideal of D generated by p1, ..., pr

normal form for elements of D:

use ∂i xj = σi(xj) ∂i + ...

to move all ∂i to the right of every xj

i j n m M := { x ∂ | i ∈ (Z≥0) , j ∈ (Z≥0) } consider M-multiple closed set generated by

the normal forms of lm(pi), i = 1,...,r decomp. into disj. cones as before

reduction: all multiplications from the left

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Janet bases for Ore algebras

D = K[x1,...,xn][∂1,...,∂m]

I left ideal of D generated by p1, ..., pr

For termination of the algorithm, assume that

∂i xj =(ci,j xj + di,j) ∂i + ei,j

where ci,j ∈ K − {0}, di,j ∈ K,

ei,j ∈ K[x1,...,xn] with deg(ei,j) ≤ 1

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Maple packages...

...at Lehrstuhl B f¨ur Mathematik, RWTH Aachen University

implementing the involutive basis technique:

Involutive / Janet

JanetOre

LDA (Linear Diﬀerence Algebra)

in cooperation with V. P. Gerdt & Y. A. Blinkov

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Involutive

Janet (-like Gr¨obner) bases for submodules of free modules over a commutative polynomial ring coeﬀicients: rationals or finite fields and field extensions, and rational integers Janet division, Janet-like division term orderings: degrevlex, plex TOP / POT block / elimination orderings

web: http://wwwb.math.rwth-aachen.de/Janet

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Involutive

Analogues of Buchberger’s criteria can be selected Interface to C++: call fast routines when needed or switch to fast routines for the whole Maple session Syzygies, Hilbert series, etc. Applications: commutative algebra solving systems of algebraic equations

web: http://wwwb.math.rwth-aachen.de/Janet

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Main procedures of Involutive

InvolutiveBasis compute Janet(-like Gr¨obner) basis PolInvReduce involutive reduction modulo Janet basis FactorModuleBasis vector space basis of residue class module Syzygies syzygy module PolResolution free resolution PolHilbertSeries, PolHilbertPolynomial, etc. combinatorial devices PolMinPoly, PolRepres, etc. computing in residue class rings

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 ginv

C++ module for Python comp. of Gröbner bases using involutive algorithms polynomials, differential / difference equations open source software originated by V. P. Gerdt, Y. A. Blinkov contributions by LBfM coefficients: rationals or finite fields and some algebraic and transcendental field extensions term orderings: degrevlex (TOP / POT), lex, product orderings see web page for timings

web: http://invo.jinr.ru

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 ginv

import ginv st = ginv.SystemType("Polynomial") im = ginv.MonomInterface("DegRevLex", st, [’x’, ’y’]) ic = ginv.CoeffInterface("GmpZ", st) ip = ginv.PolyInterface("PolyList", st, im, ic) iw = ginv.WrapInterface("CritPartially", ip) iD = ginv.DivisionInterface("Janet", iw) eqs = ["x^2+y^2", ...] basis = ginv.basisBuild("TQ", iD, eqs)

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Module-theoretic approach to linear systems

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Linear Systems

D ring (field, integral domain, Ore algebra, ...)

R ∈ Dq×p, F left D-module

Ry = 0, y ∈ F p.

“optimal” answer for us: P ∈ Dp×r s.t. ker(R.) = im(P.)

not possible in general

Example. D = k a (skew) field. Gaussian elimination singles out parameters injective parametrization

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Example

de Rham complex:

∞ 3 D = R[∂x,∂y,∂z], e.g. F = C (Ω), Ω ⊆ R convex

∂x 0 ∂z −∂y  ∂y  .  −∂z 0 ∂x  . ∂z ∂y −∂x 0 ∂x ∂y ∂z .   3×1   3×1 0 → R →F −−−−−−→F −−−−−−−−−−−−−−−→F −−−−−−−−−−→F → 0

∂x 0 ∂z −∂y .  ∂y  .  −∂z 0 ∂x  ∂z ∂y −∂x 0 . ∂x ∂y ∂z   1×3   1×3 0 ← M ← D ←−−−−−− D ←−−−−−−−−−−−−−−− D ←−−−−−−−−−− D ← 0

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Module-theoretic approach to linear systems Σ: Ry = 0, R ∈ Dq×p

D = ring of functional operators

M = D1×p/D1×q R independent of eq.’s chosen for Σ

F = signal space

If F is an injective cogenerator for DM then

homD( · ,F) M / SolF (M)

is a categorical duality.

Malgrange, Sato, Kashiwara, Oberst, Fliess, Mounier, Pommaret, Quadrat, Willems, Zerz, . . .

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Module-theoretic approach to linear systems Σ: Ry = 0, R ∈ Dq×p, M = D1×p/D1×q R

F injective cogenerator for DM:

.R .S 0 o M o D1×p o D1×q o D1×r exact

ifandonlyif (apply homD( · , F))

p R. q S. r 0 / SolF (M) / F / F / F exact.

Fundamental principle (Ehrenpreis, Malgrange, Palamodov)

for D = C[∂1,...,∂n] acting by diﬀerentiation on F:

e.g. F ∈ { complex-valued C∞-functions on Rn, complex-valued distributions on Rn, formal / convergent power series }

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Injective Cogenerator

v1 ∂x 2×1 R1 u = , R1 := ∈ D , D = K[∂x,∂y] v ∂y 2 compatibility condition:

v R 1 = 0, R := (∂ − ∂ ) ∈ D1×2, 2 v 2 y x 2 

.R1 .R2 0 o M o D o D1×2 o D o 0 exact

(R1). (R2). 2×1 0 / SolF (M) / F / F / F / 0 exact

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Parametrizing linear systems

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Parametrization and torsion-freeness Let M be given by the finite presentation

ρ .R 0 o M o D1×p o D1×q.

Assume P is a parametrization, i.e.

.P .R D1×r o D1×p o D1×q

is an exact sequence of left D-modules. Then M is torsion-free:

.P .R F r D1×r o D1×p o D1×q ✈ O ✈✈ ∗ ✈✈ ι ι ✈✈ ✈✈ ρ z✈ SolF (M) M ③③ O ③③ ③③ ③③ |③ 0 0 0

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Example

F. Dubois, N. Petit, and P. Rouchon, Motion Planning and Nonlinear Simulations for a Tank Containing a Fluid, Proc. ECC, Karlsruhe (Germany), 1999.

φ1(t − 2) + φ2(t) − 2 φ˙3(t − 1) = 0,  ˙  φ1(t)+ φ2(t − 2) − 2 φ3(t − 1) = 0.  D := Q [∂,δ] (diﬀerential time-delay operators)

2 δ 1 −2 δ∂ 2×3 R := 2 ∈ D , 1 δ −2 δ∂ !

M := D1×3/D1×2 R

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Parametrizability test M = D1×p/D1×q R M ⊤ = Dq×1/R Dp×1

2 δ∂  2 δ∂  . δ2 1 −2 δ∂ . δ2 + 1 1 δ2 −2 δ∂ D1×1   / D3×1 / D2×1 / M ⊤ / 0

2 δ∂ .  2 δ∂  δ2 1 −2 δ∂ . δ2 + 1 1 δ2 −2 δ∂ D1×1 o   D1×3 o D1×2 not exact

2 δ∂ .  2 δ∂  δ2 1 −2 δ∂ . δ2 + 1 1 δ2 −2 δ∂ D1×1 o   D1×3 o❚ D1×2 j ❚❚❚❚ ❚❚❚❚ ❚❚❚❚ 1 −1 0 ❚❚❚ R′ := ❚❚ 0 −δ2 − 1 2 δ∂ D1×2

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Parametrizability test

We have t(M)= D1×2 R′/D1×2 R 6= 0.

2 In particular, (δ − 1) (φ1(t) − φ2(t))=0, φ1 − φ2 is 2-periodic.

2 δ∂ 2 δ∂ is a parametrization of the subsystem  δ2 + 1    φ1(t) − φ2(t) = 0,  ˙  −φ2(t − 2) − φ2(t) + 2 φ3(t − 1) = 0.

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Parametrizability test

1 ⊤ ∼ In fact, we compute extD(M ,D) = t(M).

0 ↓ t(M) 0 0 0 ↓ ↓ ↓ ↓ ⊤ 0 ←− M ←− F0 ←− F1 ←− homD(M ,D) ←− 0 ↓ ↓ ↓ ↓ ⊤ 0 ←− K ⊗D M ←− K ⊗D F0 ←− K ⊗D F1 ←− homD(M , K) ←− 0 ↓ ↓ ↓ ↓ ⊤ 0 ←− (K/D) ⊗D M ←− (K/D) ⊗D F0 ←− (K/D) ⊗D F1 ←− homD(M ,K/D) ←− 0 ↓ ↓ ↓ ↓ 1 ⊤ 0 0 0 extD(M ,D) ↓ 0

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 Parametrizability test

We can compute

2 ′ 1 −1 0 ′′ δ −1 R := 2 , R := 0 −δ − 1 2 ∂δ ! 1 −1 !

which satisfy R = R′′ R′. Herewehave ker(R′) = 0.

⇒ t(M) =∼ D1×2/(D1×2 R′′), M/t(M) =∼ D1×3/(D1×2 R′).

M/t(M) corresponds to the parametrizable subsystem

φ1(t) − φ2(t) = 0,  ˙  −φ2(t − 2) − φ2(t) + 2 φ3(t − 1) = 0. 

Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012 System Module Homological Algebra

autonomous t(M) 6= 0 ext1 (M T ,D) 6= 0 elements D

controllable, t(M) = 0 ext1 (M T ,D) = 0 parametrizable D

i T parametrization reflexive extD(M ,D) = 0, is parametrizable i = 1, 2

......

i T ... projective extD(M ,D) = 0, 1 ≤ i ≤ gld(D)

ﬂatness free ...

Contributions to this classification: Pommaret-Quadrat, Oberst, Fliess, Mounier, . . .

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Inria Saclay – ˆIle-de-France / L2S, Sup´elec, 28.11.2012