Monge Parametrizations and Integration ofRectangular Linear Differential Systems

Thomas Cluzeau

University of Limoges ; CNRS ; XLIM (France)

Joint work with M. Barkatou, C. El Bacha (University ofLimoges ; CNRS ; XLIM) and A. Quadrat (INRIA Saclay)

MEGA 2011, Stockholm, May 31

Introduction / Motivation

� Starting points:

Most algorithms for computing local or global solutions oflinear differential systems handle only square systems;

Systems appearing in many applications such as controltheory are in general non-square.

� Contribution: reduce the integration of a rectangular system tothat of a square one.

� Main tool: constructive algebraic analysis techniques, Mongeparametrizations

Outline of the talk

1 Constructive algebraic analysis approach to systems theory

2 Monge parametrizations of linear systems

3 Main result

4 Extensions and perspectives

I

Constructive algebraic analysisapproach to systems theory

Methodology

1 A linear system is defined by a matrix R with coefficients in aring D of functional operators:

R y = 0. (?)

2 To (?) we associate a left D-module M (finitely presented).

3 There exists a dictionary between the properties of (?) and M.

4 Homological algebra allows to check the properties of M.

5 Effective algebra (non-commutative Grobner/Janet bases)gives algorithms.

6 Implementation : Maple (OreModules, OreMorphisms),Singular:Plural, GAP4 (homalg), . . . .

Example: Wind tunnel model

� The wind tunnel model (Manitius, IEEE TAC 84):x1(t) + a x1(t)− k a x2(t − h) = 0,

x2(t)− x3(t) = 0,

x3(t) + ω2 x2(t) + 2 ζ ω x3(t)− ω2 u(t) = 0.

� Let us consider D = Q(a, k , ω, ζ)[∂, δ]

� The system can then be written as R y = 0 with

R =

∂ + a −k a δ 0 0

0 ∂ −1 0

0 ω2 ∂ + 2 ζ ω −ω2

∈ D3×4.

The left D-module M

� D ring of functional operators, R ∈ Dq×p and F a left D-module(the functional space):

∀P1, P2 ∈ D, ∀ η1, η2 ∈ F : P1 η1 + P2 η2 ∈ F .

� Consider the system kerF (R.) = {η ∈ Fp | R η = 0}.

� As in number theory or algebraic geometry, to kerF (R.) weassociate the left D-module:

M = D1×p/(D1×q R).

given by the finite presentation:

D1×q .R−→ D1×p π−→ M −→ 0,λ = (λ1, . . . , λq) 7−→ λR

Theorem [Malgrange]:

kerF (R.) ∼= homD(M,F) = {f : M → F , f is left D-linear}.

Definitions of module theory

� M ′f−→ M

g−→ M ′′ is called a complex if im f ⊆ ker g .

� M ′f−→ M

g−→ M ′′ is said exact if im f = ker g .

Definition

1. M is free if ∃ r ∈ Z+ such that M ∼= Dr .

2. M is projective if ∃ r ∈ Z+ and a D-module P such that:

M ⊕ P ∼= Dr .

3. M is torsion-free if:

t(M) = {m ∈ M | ∃ 0 6= d ∈ D : d m = 0} = 0.

(Ex.: x1 = x2 + u, x2 = x1 + u defines a torsion module sincez = x1 − x2 satisfies z + z = 0 so that t(M) 6= 0)

Syzygy module computations

� Let D be a ring of functional operators which is a noetheriandomain and R ∈ Dq×p.

� There exist

P ∈ Dr×q such that kerD(.R) = D1×r P:

D1×r .P−→ D1×q .R−→ D1×p is exact.

Q ∈ Dp×m such that kerD(R.) = Q Dm:

Dq R.←− Dp Q.←− Dm is exact, R Q = 0.

� Such matrices P and Q can be computed by means ofnon-commutative Grobner/Janet bases computations.

� Implementation in the Maple library OreModules available at

http://wwwb.math.rwth-aachen.de/OreModules/

Injective modules

Let F be a left D-module, applying homD(.,F) to the exactsequence

D1×r .P−→ D1×q .R−→ D1×p,

we get the following complex with R Fp ⊆ kerF (P.)

F r P.←− Fq R.←− Fp. (1)

Definition: A left D-module F is injective if for every injective leftD-homomorphism f : A→ B from a left D-module A to a leftD-module B and for every ψ ∈ homD(A,F), there exists a leftD-homomorphism φ ∈ homD(B,F) such that ψ = φ ◦ f .

Theorem: F is injective iff the functor homD(.,F) is exact.

� If P ∈ Dr×q is such that kerD(.R) = D1×r P and F is aninjective left D-module, then R Fp = kerF (P.), i.e., (1) is exact.

II

Monge parametrizations

Monge parametrization of linear functional systems

� Let D be a ring of functional operators which is a noetheriandomain.

� Let F be a left D-module.

� Let us consider R ∈ Dq×p and the linear functional system:

kerF (R.) , {η ∈ Fp |R η = 0}.

� Quadrat-Robertz’ algorithm: compute, if they exist, 4 matricesQ ∈ Dp×m,R ′′ ∈ Dq×q′

,T ∈ Dr ′×q′and S ∈ Dp×q′

such that:

kerF (R.) = {Q µ+ S ζ | µ ∈ Fm,

(R ′′

T

)ζ = 0}.

� Key point: integrating R η = 0⇐⇒ integrating

(R ′′

T

)ζ = 0.

Monge parametrization of LFS: QR’s algorithm (1)

� Compute Q ∈ Dp×m and R ′ ∈ Dq′×p such that

kerD(R.) = Q Dm, kerD(.Q) = D1×q′R ′.

� R Q = 0 implies D1×qR ⊆ kerD(.Q) = D1×q′R ′

⇒ there exists R ′′ ∈ Dq×q′such that R = R ′′ R ′.

� Compute T ∈ Dr ′×q′such that kerD(.R ′) = D1×r ′

T .⇒ we thus have R ′Fp ⊆ kerF (T .).

� We have:

R η = 0⇐⇒ R ′′ R ′ η = 0⇐⇒

R ′′ ζ = 0,T ζ = 0,R ′ η = ζ.

� Links with t(M):

M/t(M) = D1×p/(D1×q′R ′),

t(M) = (D1×q′R ′)/(D1×q R) ∼= D1×q′

/(D1×(q+r ′) (R ′′

TTT )T

).

Monge parametrization of LFS: QR’s algorithm (2)

� We now consider the system: R ′ η = ζ, ζ ∈ kerF

((R ′′

T

).

).

� Since kerD(.Q) = D1×q′R ′, if F is an injective left D-module,

then kerF (R ′.) = Q Fm.

⇒ General solution of the homogeneous system R ′ η = 0 given byQ µ, for all µ ∈ Fm

� Quadrat-Robertz: particular solution given by S ζ if there existsS ∈ Dp×q′

satisfying that there exists V ∈ Dq′×q such thatR ′ − R ′ S R ′ = V R.

S is called a generalized inverse of R ′ modulo D1×q R;

Existence ⇔ M ∼= t(M)⊕M/t(M) (ok if M/t(M) projective);

Algorithm implemented in OreModules.

Monge parametrization: QR’s result

Theorem: If F is an injective left D-module andM ∼= t(M)⊕M/t(M), then we can compute 4 matricesQ ∈ Dp×m, R ′′ ∈ Dq×q′

, T ∈ Dr ′×q′and S ∈ Dp×q′

such that:

kerF (R.) = {Q µ+ S ζ | µ ∈ Fm,

(R ′′

T

)ζ = 0}.

Remark 1: t(M) ∼= D1×q′/(D1×(q+r ′) (R ′′T TT )T

). The

system to be integrated is (over)determined: algorithms exist.

Remark 2: M is torsion-free kerF (R.) = {Q µ |µ ∈ Fm}.

� Implemented on the Maple library OreModules available at

http://wwwb.math.rwth-aachen.de/OreModules/

III

Main result

The case of a linear differential system: D = C[x ][ ddx ]

� Let R ∈ Dq×p: the matrix defining our linear differential system.

� Let F be a given space of functions (e.g., regular formal powerseries, polynomial, rational or exponential functions, . . . ).

The solution spaces of interest are not injective left D-modules!

However

� The ring D = C[x ][ ddx ] has strong properties (hereditary ring,

Stafford’s theorem)

generalization of the previous Quadrat-Robertz’ algorithm

Hereditary rings

Definition: A ring D is left hereditary if every left ideal isprojective.

Properties:

1 D is left hereditary iff every submodule of a projective moduleis projective;

2 If D is left hereditary, then every torsion free left D-module isprojective; we always have M ∼= t(M)⊕M/t(M).

Main theorem

Theorem: Let D = C[x ][ ddx ] and R ∈ Dq×p full row-rank. Assume

that q ≥ 2 and let F be a given space of functions (e.g., regularformal power series, polynomial, rational or exponential functions,. . . ) having a left D-module structure.Then, there exist Q ∈ Dp×m, a full row-rank matrix R ′ ∈ Dq×p, asquare matrix R ′′ ∈ Dq×q and S ∈ Dp×q satisfying

R = R ′′ R ′, kerF (R ′.) = Q Fm, R ′ S = Iq.

Moreover, we have

kerF (R.) = {Q µ+ S ζ | µ ∈ Fm, R ′′ ζ = 0}.

⇒ Integrating the rectangular linear differential system R η = 0 isreduced to integrating the square linear differential system

R ′′ ζ = 0.

Sketch of the proof

� Follow Quadrat-Robertz’ method in our case:

1 Compute Q ∈ Dp×m such that kerD(R.) = Q Dm;

2 D hereditary ring & R full row rank ⇒ kerD(.Q) projective ofrank q;

3 q ≥ 2 & Stafford’s theorem ⇒ kerD(.Q) free of rank q;

4 Compute a basis: R ′ ∈ Dq×p full row-rank such thatkerD(.Q) = D1×q R ′;⇒ T = 0 and R ′′ square of size q such that R = R ′′ R ′;

5 D hereditary ⇒ kerF (R ′.) = Q Fm;

6 D hereditary & R ′ full row rank ⇒ R ′ admits a right inverseS ∈ Dp×q.

� Algorithm has been implemented in Maple using OreModules.

Example (1)

� Let D = C[t][d ] with d = ddt and consider the rectangular linear

differential system given by

R =

1 t3d2 − 2 t3d + t3 0 1

t2 2 td + t2d − 2 t − t2 1 + t −t

0 0 −1 t

∈ D3×4.

� Using OreModules, we compute

Q =(−1 0 t 1

)Tthat satisfies kerD(R.) = Q D and

R ′ =

1 0 0 1

0 1 0 0

0 0 −1 t

, R ′′ =

1 t3d2 − 2 t3d + t3 0

t2 2 td + t2d − 2 t − t2 −t − 1

0 0 1

,

so that kerD(.Q) = D1×3 R ′ and R = R ′′ R ′.

Example (2)

� Computing a right inverse S of R ′, i.e., R ′ S = I3 we find:

S =

1 0 0

0 1 0

0 0 −1

0 0 0

.

� Theorem ⇒ Solutions of R η = 0 in F given by Q µ+ S ζ whereµ ∈ F and ζ ∈ F3 satisfies the square linear differential systemR ′′ ζ = 0.

Example (3)

� Regular formal solutions (use algorithms developed in C. ElBacha’s PhD for square systems) of R η = 0:

η(t) = S ζ(t)+Q µ(t) =

−µ (t)

1 + t + 12 t2 + 1

6 t3 + 124 t4 + O(t5)

tµ (t)

µ (t)

,

for all µ(t) = tλ0 z(t) where λ0 ∈ C, z(t) ∈ C[log(t)][[t]].

IV

Extensions and perspectives

Extensions / Perspectives

� In our theorem, we can replace D = C[x ][ ddx ] by D = kJxK〈∂〉,

where k is a field of characteristic zero, or by D = k{x}〈∂〉, wherek = R or C.

Indeed, Quadrat-Robertz proved recently that those rings have theneeded algebraic properties.

� In the future:

Gather our implementation with those of algorithmscomputing global or local solutions;(e.g., Isolde - http://isolde.sourceforge.net/)

Comparisons to Jacobson normal form, B.-El B.-Pfluegel andGrigoriev: singularities may be introduced!

Can this be extended to other rings D?