duality and fundamental principle for analytic linear ...september 15-19, 2008 henri bourlès,...

Duality and fundamental principle for analyticlinear systems of partial differential-difference

equationsTalk at the workshop Linear Systems Theory

Organizer: Professor Paul FuhrmannSeptember 15-19, 2008

Henri Bourlès, SATIE, ENS de Cachan, ParisUlrich Oberst, Institut für Mathematik, Universität Innsbruck

September 13, 2008

Linear ordinary delay-differential equations withconstant coefficients

Standard example:y ′(t) + y(t − 1) = u(t), t ∈ R , y ∈ C1(R , C )

Ingredients:1. delays τ ∈ H ⊂ R , H :=finitely generated subgroup of R ,

number of delays:= m := dimZ (H),

for instance H := Z ⊕ Z√

2⊕ Z π, m = 3.

2. Delayed derivatives y (k)(t + τ), τ ∈ H, in particular forτ := 0.

3. C -linear combinations of these delayed derivatives.

Aim of this talk

1. Extension to partial differential equations with delays=partial differential-difference equations

2. Replacement of time domain R by C n and C∞-signals bymultivariable analytic signals

3. Extensions to the MIMO case (Multiple input/multipleoutput), linear systems

4. Outlook: Extension to R n and C∞-signals or evendistributions, convolution equations

History I: SISO case

Reference only to work in context with our duality,extensive literature on ordinary delay-differential equations,also systems theoretic.

1. Ehrenpreis, Malgrange ca. 1953-1957:

SISO convolution equations T ∗ y = u,T =distribution of compact supportnot always solvable

2. Outstanding results:2.1 Solvability of T ∗ y = δ, T =differential operator,

fundamental solution y2.2 Ehrenpreis: Solvability of T ∗ y = u,

T =partial differential-difference operator, y , u C∞

3. Definitive exposition: Hörmander [3, Ch. XVI]: TheAnalysis of Linear Partial Differential Operators II

History II: Behavioral theory, MIMO systems

1. Gluesing-Luerssen 1997- : Behavioral theory of ordinarydelay-differential systems with one delay

2. Habets, Gluesing-Luerssen, Rocha, Vettori, Willems,Zampieri et al.1994-: Behavioral theory of ordinarydelay-differential systems with several delays

Basic data for this talk1. Complex field C , n > 0

2. Indeterminates or complex variabless = (s1, · · · , sn), x = (x1, · · · , xn)

3. Polynomial algebras A := C [s], C [x ], polynomials

f =∑

µ∈N n

fµsµ, µ = (µ1, · · · , µn) ∈ N n, sµ = sµ11 ∗ · · · ∗ sµn

where fµ = 0 for almost all µ

4. formal power series algebras C [[s]], C [[x ]],

f =∑

µ∈N n

fµsµ ∈ C [[s]], w =∑

µ∈N n

wµxµ ∈ C [[x ]].

Standard duality I1. Non-degenerate bilinear form

< −,− >: C [s]× C [[x ]] → C

< f , w >=<∑

µ∈N n

fµsµ,∑

µ∈N n

wµxµ >:=∑

µ∈N n

fµwµµ!

µ!, xν >=< sµ,

ν!>= δµ,ν , dual bases .

HomC (C [s], C ) ∼= C [[x ]]

φ =< −, w >↔ w =∑

µ∈N n

wµxµ =∑

µ∈N n

(∂|µ|w/∂xµ

µ!xµ

wµ = φ

(∂|µ|w/∂xµ

Standard duality I1. Non-degenerate bilinear form

< −,− >: C [s]× C [[x ]] → C

< f , w >=<∑

µ∈N n

fµsµ,∑

µ∈N n

wµxµ >:=∑

µ∈N n

fµwµµ!

µ!, xν >=< sµ,

ν!>= δµ,ν , dual bases .

HomC (C [s], C ) ∼= C [[x ]]

φ =< −, w >↔ w =∑

µ∈N n

wµxµ =∑

µ∈N n

(∂|µ|w/∂xµ

µ!xµ

wµ = φ

(∂|µ|w/∂xµ

Standard duality II1. module structure, action by partial differentiation

◦ : C [s]× C [[x ]] → C [[x ]], < fg, w >=< f , g ◦ w >,

si ◦ w = ∂w/∂xi

2. C [s]C [[x ]] injective cogenerator, i.e., categorical duality

Finitely generated C [s]−modules ⇐⇒ subbehaviors of C [[x ]]`.

In colloquial terms:

2.1 Elimination is possible (injectivity)2.2 Equations determine solutions and solutions determine

equations (cogenerator).

3. Aim as far as possible: Analogous results for partialdifferential-difference equations

◦ : C [s]× C [[x ]] → C [[x ]], < fg, w >=< f , g ◦ w >,

Basic data for analytic partial differential-differenceequations I

1. C -algebra D of entire analytic functions or everywhereconvergent power series

D = O(C n) :=f =∑

µ∈N n

fµsµ ∈ C [[s]]; sum convergent for all s ∈ C n

O(C n)= Frechet space with topology of compactconvergence,topological algebra with dense subalgebra A = C [s].

2. subalgebra of polynomial-exponential functions

B := PE(s) := ⊕τ∈C nC [s]eτ•s, τ • s := τ1s1 + · · ·+ τnsn

A = C [s] ⊂ B = PE(s) ⊂ D = O(C n)

Basic data for analytic partial differential-differenceequations I

1. C -algebra D of entire analytic functions or everywhereconvergent power series

D = O(C n) :=f =∑

µ∈N n

fµsµ ∈ C [[s]]; sum convergent for all s ∈ C n

O(C n)= Frechet space with topology of compactconvergence,topological algebra with dense subalgebra A = C [s].

2. subalgebra of polynomial-exponential functions

B := PE(s) := ⊕τ∈C nC [s]eτ•s, τ • s := τ1s1 + · · ·+ τnsn

A = C [s] ⊂ B = PE(s) ⊂ D = O(C n)

Basic data for analytic partial differential-differenceequations II

1. D = O(C n) and B = PE(s) have good properties, but arenot noetherian

2. space of entire analytic signals of at most exponentialgrowth in the variables x :

W := O(C nx ; exp) :=

w =∑

µ∈N n

wµxµ ∈ O(C n); w with (∗)

(∗) : ∃C > 0, Mi > 0 with |w(x)| ≤ CeM1|x1|+···+Mn|xn|.

3. subspace

PE(x) := ⊕τ∈C nC [x ]eτ•x ⊂ W = O(C nx ; exp).

W := O(C nx ; exp) :=

w =∑

µ∈N n

3. subspace

W := O(C nx ; exp) :=

w =∑

µ∈N n

3. subspace

Non-degenerate bilinear form I

1. Non-degenerate C -bilinear form like C [s]× C [[x ]] → C

< −,− >: O(C n)×O(C nx ; exp) → C

µ∈N n

fµsµ,∑

µ∈N n

wµxµ >:=∑

µ∈N n

fµwµµ!

D′ = O(C n)′ := {φ ∈ HomC (O(C n), C ); φ continuous}D′ ∼= W = O(C n

x ; exp)

φ =< −, w >↔ w =∑

µ∈N n

wµxµ = φ(es•x), φ

)= wµ

φ ∈ D′: analytic functional,w = φ(es•x) :=Laplace transform of φ

Non-degenerate bilinear form I

1. Non-degenerate C -bilinear form like C [s]× C [[x ]] → C

< −,− >: O(C n)×O(C nx ; exp) → C

µ∈N n

fµsµ,∑

µ∈N n

wµxµ >:=∑

µ∈N n

fµwµµ!

D′ = O(C n)′ := {φ ∈ HomC (O(C n), C ); φ continuous}D′ ∼= W = O(C n

x ; exp)

φ =< −, w >↔ w =∑

µ∈N n

wµxµ = φ(es•x), φ

)= wµ

φ ∈ D′: analytic functional,w = φ(es•x) :=Laplace transform of φ

Non-degenerate bilinear form II

1. module structure

◦ : O(C n)×O(C nx ; exp) → O(C n

x ; exp), < fg, w >=< f , g ◦ w >,

(si ◦ w)(x) = ∂w/∂xi , (eτ•s ◦ w) (x) = w(x + τ).

Hence BW =PE(s) O(C nx ; exp): action by partial

differential-difference operators.

2. Standard extensions to vectors, D1×` =rows, W `=columns:

< −,− >: D1×` ×W ` → C< (f1, · · · , f`), (w1, · · · , w`)

> >:=< f1, w1 > + · · ·+ < f`, w` >,

◦ : D1×` ×W ` → W

(f1, · · · , f`) ◦ (w1, · · · , w`)> := f1 ◦ w1 > + · · ·+ f` ◦ w`

< gf , w >=< g, f ◦ w >, g ∈ D.

Non-degenerate bilinear form II

1. module structure

◦ : O(C n)×O(C nx ; exp) → O(C n

x ; exp), < fg, w >=< f , g ◦ w >,

(si ◦ w)(x) = ∂w/∂xi , (eτ•s ◦ w) (x) = w(x + τ).

Hence BW =PE(s) O(C nx ; exp): action by partial

differential-difference operators.2. Standard extensions to vectors, D1×` =rows, W `=columns:

< −,− >: D1×` ×W ` → C< (f1, · · · , f`), (w1, · · · , w`)

> >:=< f1, w1 > + · · ·+ < f`, w` >,

◦ : D1×` ×W ` → W

(f1, · · · , f`) ◦ (w1, · · · , w`)> := f1 ◦ w1 > + · · ·+ f` ◦ w`

< gf , w >=< g, f ◦ w >, g ∈ D.

Behaviors I

1. The behavior of U ≤D D1×`:

B := Uo := {w ∈ W `;< U, w >= 0} (polar space) =

U⊥ := {w ∈ W `; U ◦ w = 0} (solution module).

2. The module of equations of B ≤D W `:

Bo :={

f ∈ D1×`;< f ,B >= 0}

(polar space) =

B⊥ :={

f ∈ D1×`; f ◦ B = 0}

(module of all equations)

3. The correspondence U 7→ U⊥, B 7→ B⊥ is a Galoiscorrespondence, especially U ⊆ U⊥⊥.

Behaviors II

If U = D1×kR, R ∈ Dk×`, is finitely generated then

U is closed and B := U⊥ = {w ∈ W `; R ◦ w = 0}.

Cogenerator properties I

Theorem

1. If U ≤D D1×` and B := U⊥ then

closure of U = B⊥ = U⊥⊥ =(B

⋂PE(x)`

2. {w ∈ B; w polynomial-exponential} is dense in B.3. Order reversing bijection{

U ⊆ D1×` closed}∼=

{B ⊆ W `

}, U = B⊥ ↔ B = U⊥

4. If Ui = D1×ki Ri , Ri ∈ Dki×`, i = 1, 2, and Bi = U⊥i then

B2 ⊆ B1 ⇐⇒ ∃X ∈ Dk1×k2 with R1 = XR2.

Theorem

⋂PE(x)`

2. {w ∈ B; w polynomial-exponential} is dense in B.

3. Order reversing bijection{U ⊆ D1×` closed

{B ⊆ W `

}, U = B⊥ ↔ B = U⊥

B2 ⊆ B1 ⇐⇒ ∃X ∈ Dk1×k2 with R1 = XR2.

Theorem

⋂PE(x)`

{B ⊆ W `

}, U = B⊥ ↔ B = U⊥

B2 ⊆ B1 ⇐⇒ ∃X ∈ Dk1×k2 with R1 = XR2.

Theorem

⋂PE(x)`

{B ⊆ W `

}, U = B⊥ ↔ B = U⊥

B2 ⊆ B1 ⇐⇒ ∃X ∈ Dk1×k2 with R1 = XR2.

Cogenerator properties II

1. The module O(C n)O(C nx ; exp) is not a cogenerator like

C [s]C [[x ]].

2. The implication(D1×k2R2

)⊥ ⋂PE(x)` ⊆

(D1×k1R1

)⊥ ⋂PE(x)` =⇒

∃X with R1 = XR2

was shown in Malgrange’s thesis (1955).

Cogenerator properties II

1. The module O(C n)O(C nx ; exp) is not a cogenerator like

C [s]C [[x ]].2. The implication(

D1×k2R2

)⊥ ⋂PE(x)` ⊆

(D1×k1R1

)⊥ ⋂PE(x)` =⇒

∃X with R1 = XR2

was shown in Malgrange’s thesis (1955).

Fundamental principle and elimination

Theorem

1. Elimination: Let U ≤D D1×` be closed, B := U⊥ ⊆ W ` andP ∈ Dm×`. Then the image P ◦ B ⊆ W m is also a behavior,and indeed

P ◦ U⊥ = V⊥ with V := {g ∈ D1×m; gP ∈ U}

where V is also closed.

2. If U = D1×kR is finitely generated (f.g) and henceB = {w ∈ W `; R ◦ w = 0} then V is not necessarily f.g..

3. Fundamental principle: If U = 0, hence B = W ` then

V = {g ∈ D1×m; gP = 0} and P ◦W ` = V⊥, i.e.,

P ◦ y = u ∈ W m is solvable for y ∈ W ` ⇐⇒ V ◦ u = 0.

Theorem

P ◦ U⊥ = V⊥ with V := {g ∈ D1×m; gP ∈ U}

V = {g ∈ D1×m; gP = 0} and P ◦W ` = V⊥, i.e.,

Theorem

P ◦ U⊥ = V⊥ with V := {g ∈ D1×m; gP ∈ U}

V = {g ∈ D1×m; gP = 0} and P ◦W ` = V⊥, i.e.,

Distributions and C∞-functions I

1. W = O(C nx ; exp) ⊂ Ex := C∞(R n

x , C ), w = w |R n

2. E ′ = E ′(R n) := {distributions with compact support} ⊂D′ = D′(R n),

3. module structure

◦ : E ′ × Ex → Ex , (T1 ∗ T2)(w) = T1(T2 ◦ w),

T2 ◦ w = T2 ∗ w , T2(x) = T2(−x).

4. Laplace transform of T ∈ E ′:

T (s) := Tx(es•x) ∈ O(C n),

T ◦ w = T ◦ w for w ∈ O(C nx ; exp) ⊂ Ex

1. W = O(C nx ; exp) ⊂ Ex := C∞(R n

x , C ), w = w |R n

3. module structure

◦ : E ′ × Ex → Ex , (T1 ∗ T2)(w) = T1(T2 ◦ w),

T2 ◦ w = T2 ∗ w , T2(x) = T2(−x).

T (s) := Tx(es•x) ∈ O(C n),

1. W = O(C nx ; exp) ⊂ Ex := C∞(R n

x , C ), w = w |R n

3. module structure

◦ : E ′ × Ex → Ex , (T1 ∗ T2)(w) = T1(T2 ◦ w),

T2 ◦ w = T2 ∗ w , T2(x) = T2(−x).

T (s) := Tx(es•x) ∈ O(C n),

1. W = O(C nx ; exp) ⊂ Ex := C∞(R n

x , C ), w = w |R n

3. module structure

◦ : E ′ × Ex → Ex , (T1 ∗ T2)(w) = T1(T2 ◦ w),

T2 ◦ w = T2 ∗ w , T2(x) = T2(−x).

T (s) := Tx(es•x) ∈ O(C n),

Distributions and C∞-functions II1. E ′E (convolution equations) =⇒O(C n) O(C n

x ; exp) (this talk):more operators, but less signals

2. Gurevic 1973: Example with R ∈ (E ′)6,

O(C n) =6∑

O(C n)Ri or 1 = f1R1 + · · ·+ f6R6, hence

{w ∈ O(C nx ; exp); R ◦ w = R ◦ w = 0} = 0, but

{w ∈ Ex ; R ◦ w = 0} 6= 0 =⇒6∑

E ′ ∗ Ri ( E ′

3. Consequences and Problems3.1 Open: Cogenerator properties for E′Ex .3.2 Open: Fundamental principle for E′Ex .3.3 But Ehrenpreis 1956 (see Hörmander 1983): T ∗ y = u is

solvable if T◦ is a partial differential-difference operator.

O(C n) =6∑

{w ∈ Ex ; R ◦ w = 0} 6= 0 =⇒6∑

E ′ ∗ Ri ( E ′

O(C n) =6∑

{w ∈ Ex ; R ◦ w = 0} 6= 0 =⇒6∑

E ′ ∗ Ri ( E ′

Finitely many delays

1. H = ⊕mj=1Z τ j ⊂ C n finitely generated subgroup with m

delays τ j .Gluesing-Luerssen 1997-: n = m = 1.

A ⊂ C = ⊕τ∈HAeτ•s = C [s1, · · · , sn, σ1, σ−11 · · · , σm, σ−1

m ] ⊂ B,

A = C [s] ⊂ B = PE(s), σj := eτ j•s,

C noetherian, dim(C) = n + m > n = dim(C n)

3. DW = O(C nx ; exp), C ⊂ D =⇒C W : Action by partial

differential-difference operators with m delays.

m ] ⊂ B,

A = C [s] ⊂ B = PE(s), σj := eτ j•s,

m ] ⊂ B,

A = C [s] ⊂ B = PE(s), σj := eτ j•s,

C-behaviors I

1. If V = C1×kR ⊆ C1×`, R ∈ Ck×`, then C-behavior

B := V⊥ := (DV )⊥ = {w ∈ W `; R ◦ w = 0}

i.e., Equations in C =differential-difference operators with finitely many delaysC1×` with induced topology

2. Theorem1. closure of V = C1×`

⋂DV = B⊥C := {ξ ∈ C1×`; ξ ◦ B = 0}.

2. Order reversing bijection{V ⊆ C1×`closed

{C-behavior B ⊆ W `

V = B⊥C ↔ V⊥ = B

3. C-algebraic characterization of closed submodules V .

C-behaviors I

1. If V = C1×kR ⊆ C1×`, R ∈ Ck×`, then C-behavior

B := V⊥ := (DV )⊥ = {w ∈ W `; R ◦ w = 0}

i.e., Equations in C =differential-difference operators with finitely many delaysC1×` with induced topology

2. Theorem1. closure of V = C1×`

⋂DV = B⊥C := {ξ ∈ C1×`; ξ ◦ B = 0}.

2. Order reversing bijection{V ⊆ C1×`closed

{C-behavior B ⊆ W `

V = B⊥C ↔ V⊥ = B

3. C-algebraic characterization of closed submodules V .

C-behaviors II

1. Problems1.1 Computation of C1×`

⋂DV , V = D1×k R.

1.2 If P ∈ Cm×` and B ⊆ W ` is a C-behavior then P ◦ B is abehavior (elimination), but not a C-behavior in general.

2. Gluesing-Luerssen, Habets et al. m = 1: C = C [s, σ, σ−1]replaced by

H(C) := D⋂

quot(C) =

{fg∈ O(C n); f ∈ C, g ∈ C [s]

}Advantage for m = n = 1: H(C) is Bezout domain, i.e., allfinitely generated ideals are principal whereas dim(C) = 2.

Proof technique

1. Locally convex vector spaces: bipolar theorem,Hahn-Banach theorem, Banach’s open mapping theorem

2. Coherent analytic sheaves on Stein manifold C n

H. Grauert, R. Remmert, Theorie der Steinschen Räume,Springer, 1977

H. Grauert, R. Remmert, Coherent Analytic Sheaves,Springer, 1984

L. Hörmander, The Analysis of Linear Partial DifferentialOperators II, Springer, Berlin, 1983

L. Hörmander, Complex Analysis in Several Variables, 3.Ed., North Holland, Amsterdam, 1990

duality and fundamental principle for analytic linear ...september 15-19, 2008 henri bourlès,...

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