Minicourse on behavioral systems theory Lecture 5: Multidimensional systems

The Behavioral Approach

to

Systems Theory

Paolo Rapisarda, Un. of Southampton, U.K. &

Jan C. Willems, K.U.Leuven, Belgium

MTNS 2006

Lecture 5: Multidimensional systems

Outline

Motivation and aim

Basic definitions

Examples

Elimination of latent variables

Motivation

In many situations, system variables depend not only on time but also onspace:

• Heat diffusion processes

• Electromagnetism

• Vibration of structures

• ...

Aim

Develop a behavioral framework for systems described byPartialDifferentialEquations (PDEs).

Issues:

• Definitions consistent with 1-D case and basic tenets of behavioral approach

• Calculus of representations

Outline

Motivation and aim

Basic definitions

Examples

Elimination of latent variables

Linear differential distributed systems

Σ = (_T,_W,B)

T: independent variables,T = Rn withn> 1

W: external variables,W = Rw

B⊆ (_W)T_{: behavior, solution set of system} oflinear, constant-coefficient PDEs

The behavior

Bis a n-D linear differential behavior if it is

linear: w1,w2 ∈ B⇒α1w1+α2w2 ∈ B∀

α1, α2 ∈R;

shift-invariant: w ∈B ⇒σxw ∈B, where

x = (x1, . . . ,xn)and

(σxw)(x₁0, . . . ,x_n0) := w(x1+x₁0, . . . ,xn+xn0)

differential: Bis solution set of a system of PDEs.

Outline

Motivation and aim

Basic definitions

Examples

Elimination of latent variables

Vibrating membrane (2

−

D

wave) equation

T(independent variables): R×R2

W(dependent variables): R

B:={w satisfyingρ0∂

2_w

∂t2 −τ

2_∇2_{w = 0}}

Maxwell’s equations

Set_Tof independent variables: _R×_R3

Set_Wof dependent variables: _R10

B:= {(E~, ~B,~j, ρ)satisfying Maxwell’s equations}

∇ ·~E− 1

ε0

ρ = 0

∇ ×~E+ ∂

∂t

~

B = 0

∇ ·~B = 0

c2∇ ×~B− 1

ε0

~_{j −} ∂

∂t

~

n

-variable polynomial matrices

R ∈ _R•×w

[ξ1, . . . , ξn]induces

R(_∂x∂

1,· · · ,

∂

∂xn)w = 0

akernel representation of

B:= {w ∈C∞(_Rn

,_Rw

) |R(_∂x∂

1,· · · ,

∂

∂xn)w = 0}

Example:

2

−

D

wave equation

B= {w ∈C∞ _Rn,_R1 satisfyingρ0

∂2w

∂t2 −τ 2_∇2_w

= 0}

Heren = 3(time, space),w= 1. Consequently,

R[ξt, ξx, ξy]

R(ξt, ξx, ξy) =ρ0ξ_t2−τ2ξ_x2−τ2ξ_y2

(ρ0

∂2 ∂t2 −τ

2 ∂ 2

∂x2 −τ 2 ∂

2

∂y2

| {z }

R(_∂∂_t,_∂∂_x,_∂∂_y)

Example: Maxwell’s equations

w = (~E, ~B,~j, ρ)∈ C∞(_R4_,

R10), 8 equations

R(ξt, ξx, ξy, ξz) =



   

ξx ξy ξz 0 0 0 0 0 0 _ε1

0

0 −ξz ξy ξt 0 0 0 0 0 0

ξz 0 −ξx 0 ξt 0 0 0 0 0

−ξy ξx 0 0 0 ξt 0 0 0 0

0 0 0 ξx ξy ξz 0 0 0 0

ξt 0 0 0 ξz −ξy ε1₀ 0 0 0

0 ξt 0 −ξz 0 ξx 0 _ε1

0 0 0

0 0 ξt ξy −ξx 0 0 0 _ε1

0 0     

R(∂

∂t, ∂

∂x, ∂

∂y, ∂

Linear differential latent variable distributed systems

L: latent variables,L= Rl

Bf ⊆ Lwn×l: full behavior

{(w, `)|R( ∂ ∂x1

,· · · , ∂ ∂xn

)w = M( ∂ ∂x1

,· · · , ∂ ∂xn

)`}

Σinduces Σe = (T,W,B), themanifest system

B: manifest behavior

Linear differential latent variable distributed systems

L: latent variables,L= Rl

Bf ⊆ Lwn×l: full behavior

{(w, `)|R( ∂ ∂x1

,· · · , ∂ ∂xn

)w = M( ∂ ∂x1

,· · · , ∂ ∂xn

)`}

Σinduces Σe = (T,W,B), themanifest system

B: manifest behavior

Maxwell’s equations and latent variables

Dependent variables: (~E,~j, ρ)i.e. _W= _R7

Latent variable: ~Bi.e. _L = _R3

Bf := {(~E,~j, ρ, ~B)satisfying Maxwell’s equations}

B := {(~E,~j, ρ) | ∃~Bs.t. (~E,~j, ρ, ~B) ∈Bf}

¿IsBalso described

Algebraic characterization of behaviors

Differentn-variable polynomial matrices may represent the same behavior

NB := {r ∈ R1×w[ξ1,· · · , ξn] |r(_∂∂_x

1,· · · ,

∂

∂xn)B = 0}

Module of annihilators ofB

<R >:= module generated by the rows ofR.

Of course[B = ker(R)] =⇒[< R >⊆ NB];

forC∞trajectories, also converse holds:

Calculus of representations

Lw

n is one-one with

modules ofR1×w[ξ1,· · · , ξn] :

• ker(R1) =ker(R2)iff< R1 >=< R2 >

• ker(R1)⊆ ker(R2)iff<R1 >⊇< R2 >

• ker(R1)∩ker(R2);<R1 >∪ < R2 >

R[ξ1,· · · , ξn]is not a Euclidean domain!

Outline

Motivation and aim

Basic definitions

Examples

Elimination of latent variables

Elimination of latent variables

B_f =n(w, `)|R(_∂∂

x1,· · ·, ∂

∂xn)w =M(

∂ ∂x1,· · · ,

∂ ∂xn)`

o

|

πw

↓

B=nw | ∃`s.t.R(_∂∂

x1,· · · , ∂

∂xn)w =M(

∂ ∂x1,· · · ,

∂ ∂xn)`

o

¿∃R0 ∈_R•×w_[ξ

1,· · · , ξn]s.t. B = ker(R0(_∂∂_x1,· · · ,_∂∂_xn))?

Yes! B∈ Lw

The Fundamental Principle for static equations

¿ GivenM ∈ _R•×•_{andy ∈}

R•, is therex s.t. Mx = y ?

There existsx

s.t. Mx = y ⇐⇒

v ∈NM ⇒ vTy = 0

for allv ∈NM

The Fundamental Principle for static equations

¿ GivenM ∈ _R•×•_{andy ∈}

R•, is therex s.t. Mx = y ?

Assumex exists, and consider

NM :={v ∈ R• | v>M = 0}

Then

v ∈ NM ⇒v>y = 0

There existsx

s.t. Mx = y ⇐⇒

v ∈NM ⇒ vTy = 0

for allv ∈NM

The Fundamental Principle for static equations

¿ GivenM ∈ _R•×•_{andy ∈}

R•, is therex s.t. Mx = y ?

Conversely, assume

v ∈ NM ⇒v>y = 0

for allv ∈ NM. Then

(Im(M))⊥ ⊆ y⊥

which implies

y ∈Im(M)

i.e. the existence ofx such thatMx = y.

There existsx

s.t. Mx = y ⇐⇒

v ∈NM ⇒ vTy = 0

for allv ∈NM

The Fundamental Principle for static equations

¿ GivenM ∈ _R•×•_{andy ∈}

R•, is therex s.t. Mx = y ?

There existsx

s.t. Mx = y ⇐⇒

v ∈NM ⇒ vTy = 0

for allv ∈NM

The fundamental principle (Ehrenpreis-Palamodov)

,_Rf_{). There exists}

`inC∞(_Rn ,_Rl

)s.t.

M( ∂

∂x1

,· · · , ∂ ∂xn

)`= f

if and only if

n( ∂

∂x1

,· · · , ∂ ∂xn

)f = 0

for alln ∈

v ∈_R1ו_[ξ

1,· · · , ξn] | v ·M = 0 .

v ∈_R1ו_[ξ

1,· · · , ξn] | v ·M = 0 :

The fundamental principle and elimination

Exists`s.t. M(_∂x∂

1,· · · ,

∂

∂xn)`= R(

∂

∂x1,· · · ,

∂

∂xn)w IFF

n( ∂ ∂x1

,· · · , ∂ ∂xn

)R( ∂ ∂x1

,· · · , ∂ ∂xn

)w = 0

for allnin the left syzygy of M.

How: compute, e.g. with Gröbner bases, a generator matrix F for the left syzygy of M. Then w ∈ B if and only if

(FR)( ∂ ∂x1

,· · · , ∂ ∂xn

Example: Maxwell’s equations

Eliminating~B andρ: compute left syzygy of

2 6 6 6 6 6 6 6 6 6 6 4

0 0 0 1

ε₀

ξt 0 0 0

0 ξt 0 0

0 0 ξt 0

ξx ξy ξz 0 0 ξz −ξy 0

−ξz 0 ξx 0 ξy −ξx 0 0

3 7 7 7 7 7 7 7 7 7 7 5 Leads to ε0 ∂ ∂t∇

~

E +∇~j = 0

ε0

∂2 ∂t2∇

~

E +ε0c2∇ × ∇ ×~E +

∂ ∂t

Outline

Motivation and aim

Basic definitions

Examples

Elimination of latent variables

Image representations

w = M(_∂x∂

1,· · · ,

∂ ∂xn)`

From Fundamental Principle∃ R•×•[ξ1,· · · , ξn] s.t.

R( ∂ ∂x1

,· · · , ∂ ∂xn

)w = 0

¿What kernels of polynomial

Controllable systems

B∈Lw

niscontrollableif for every w1,w2 ∈ Band any openO1,O2 ⊆ Rn such thatO1∩O2 = ∅, there

existsw ∈Bsuch that

w|O1 = w1andw|O2 = w2

“Patching"of trajectories is key:

O 1 W R R 1 w2 O2 w W R R 1 O2 O

w1 _w

2

Controllable systems

B∈Lw

niscontrollableif for every w1,w2 ∈ Band any openO1,O2 ⊆ Rn such thatO1∩O2 = ∅, there

existsw ∈Bsuch that

w|O1 = w1andw|O2 = w2

“Patching"of trajectories is key:

O 1 W R R 1 w2 O2 w W R R 1 O2 O

w1 _w

2

Characterizations of controllable

n

−

D

systems

Theorem: Let B ∈ Lw

n. The following statements are equivalent:

1. Bis controllable;

2. Badmits an image representation; 3. _R1×w

[ξ1,· · · , ξn]/NB is torsion-free.

¡Controllability ≡image representation!

Characterizations of controllable

n

−

D

systems

Theorem: Let B ∈ Lw

n. The following statements are equivalent:

1. Bis controllable;

2. Badmits an image representation; 3. _R1×w

[ξ1,· · · , ξn]/NB is torsion-free.

¡Controllability ≡image representation!

Characterizations of controllable

n

−

D

systems

Theorem: Let B ∈ Lw

n. The following statements are equivalent:

1. Bis controllable;

2. Badmits an image representation; 3. _R1×w

[ξ1,· · · , ξn]/NB is torsion-free.

¡Controllability ≡image representation!

Outline

B- and QDFs forn−D systems

The calculus ofn−D B/QDFs

Losslessness

Example: vibrating string

∂2

∂t2w −c 2 ∂

2

∂x2w = 0

d dt      1 2 _∂

∂tw

2

| {z } kinetic energy

+ c

2

2

_∂

∂xw

2

| {z } potential energy      = 0

Bilinear differential forms

LΦ : C∞(Rn,Rw1)×C∞(Rn,Rw2)→ C∞(Rn,R)

LΦ(v,w) :=

P

k,`

dk

dxkw1

> Φk,`

d`

dx`w2

k := (k1, . . . ,kn) ∈Nn `:= (`1, . . . , `n) ∈Nn

dk dxk :=

∂k1+...+kn

∂xk1

1 ···∂x

kn

n

d`

dx` :=

∂`1+...+`n

∂x`1

1 ···∂x `n

n

2

n

-variable polynomial representation

LΦ(v,w) :=

P

k,`

dk

dxkv

>

Φk,`

d` dx`w

k:= (k1, . . . ,kn)∈Nn `:= (`1, . . . , `n) ∈Nn

ζ := (ζ1, . . . , ζn) η := (η1, . . . , ηn)

P

Quadratic differential forms

QΦ : C∞(Rn,Rw)→ C∞(Rn,R)

QΦ(w) :=

P

k,`

dk

dxkw

>

Φk,`

d` dx`w

W.l.o.g. symmetry: Φk,` = Φ><sub>k,`</sub>

P

k,`Φk,`ζkη`

Outline

B- and QDFs forn−D systems

The calculus ofn−D B/QDFs

Losslessness

Divergence

Vector of QDFs(VQDF)QΦ = col(QΦi)i=1,...,n

(div(QΦ)) (w) := ∂ ∂x1

QΦ1(w) +. . .+

∂ ∂xn

QΦn(w)

¿What 2n-variable polynomial corresponds to divQΦ?

(ζ1+η1)Φ1(ζ, η) +. . .+ (ζn+ηn)Φn(ζ, η)

Divergence

Vector of QDFs(VQDF)QΦ = col(QΦi)i=1,...,n

(div(QΦ)) (w) := ∂ ∂x1

QΦ1(w) +. . .+

∂ ∂xn

QΦn(w)

¿What 2n-variable polynomial corresponds to divQΦ?

(ζ1+η1)Φ1(ζ, η) +. . .+ (ζn+ηn)Φn(ζ, η)

Divergence

Vector of QDFs(VQDF)QΦ = col(QΦi)i=1,...,n

(div(QΦ)) (w) := ∂ ∂x1

QΦ1(w) +. . .+

∂ ∂xn

QΦn(w)

¿What 2n-variable polynomial corresponds to divQΦ?

(ζ1+η1)Φ1(ζ, η) +. . .+ (ζn+ηn)Φn(ζ, η)

Path independence

bounded.

R

ΩQΦ(w)dx is independent of path if it depends only on the values ofw and its derivatives on∂Ω.

in a force field

B A

possible paths between A and B

Theorem:The following statements are equivalent.

1. R

ΩQΦ path independent ∀closed boundedΩ ⊆ R n

2. R

QΦ = 0 (on compact support trajectories)

3. Φ(−ξ, ξ) =0

4. ∃ VQDFΨ ∈_Rw×w

[ζ, η]n_{, s.t. div}

Outline

B- and QDFs forn−D systems

The calculus ofn−D B/QDFs

Losslessness

Lossless systems

Supply rateQΦ: “energy" delivered to the system, positive when absorbed.

A controllableB∈ Lw

nislossless with respect toQΦ if

R

QΦ(w)dx = 0

for allw ∈Bof compact support.

R

Algebraic characterization

Theorem.LetB = im(M(_dxd )). LetΦ ∈_Rw×w_{[ζ, η], and} defineΦ0(ζ, η) := M(ζ)>Φ(ζ, η)M(η). The following statements are equivalent:

1. Bis lossless w.r.t. QΦ; 2. R

ΩQΦ(w)dx is independent of path

for all bounded and closedΩ ⊆ _Rn_{and allw ∈}

B;

3. R

QΦ0 is a path integral;

4. ∃ VQDFΨ s.t. for all(w, `)s.t. w = M(<sub>dx</sub>d )`, holds

div(QΨ)(w) = QΦ0(`) =Q_Φ(w)

Algebraic characterization

Theorem.LetB = im(M(_dxd )). LetΦ ∈_Rw×w_{[ζ, η], and} defineΦ0(ζ, η) := M(ζ)>Φ(ζ, η)M(η). The following statements are equivalent:

1. Bislossless w.r.t. QΦ;

2. R

ΩQΦ(w)dx is independent of path

for all bounded and closedΩ ⊆ _Rn_{and allw ∈}

B;

3. R

QΦ0 is a path integral;

4. ∃ VQDFΨ s.t. for all(w, `)s.t. w = M(<sub>dx</sub>d )`, holds

div(QΨ)(w) = QΦ0(`) =Q_Φ(w)

Example: vibrating string

ρ0

∂2

∂t2w1−T0

∂2

∂x2w1 = w2

ρ0 density,T0tension w1position,w2(vertical) force

2 x ρ₀ mass density T0 Tension vertical displacement w(t,x) 1

vertical component of body force per unit length w

R(ξt, ξx) =

ρ0ξ2_t −T0ξ_x2 −1

Image representationw = M(_dxd )`induced by

M(ξt, ξx) :=

1 ρ0ξ_t2−T0ξ2_x

Example: vibrating string

ρ0

∂2

∂t2w1−T0

∂2

∂x2w1 = w2

ρ0 density,T0tension w1position,w2(vertical) force

2 x ρ₀ mass density T0 Tension vertical displacement w(t,x) 1

vertical component of body force per unit length w

R(ξt, ξx) =

ρ0ξ2_t −T0ξ_x2 −1

Image representationw = M(_dxd )`induced by

M(ξt, ξx) :=

1 ρ0ξ_t2−T0ξ2_x

Example: vibrating string

1 2

1 ρ0ζ_t2−T0ζ_x2

0 ζt

ηt 0

1 ρ0ζ_t2−T0ζ_x2

Φ(ζ, η) = 1₂ ρ0ζt2ηt −T0ζx2ηt +ρ0ζtηt2−T0ζtηx2

= (ζt +ηt)1₂(ρ0ζtηt +T0ζxηx)

+(ζx +ηx)1₂(−T0ζtηx −T0ηtζx)

QΦ(w1) =

∂ ∂t      1 2ρ0

_∂

∂tw1

2

| {z }

kinetic energy

+ 1 2T0

_∂

∂xw1

2

| {z }

potential energy      + ∂ ∂x     −1

2T0

_∂

∂xw1

∂

∂tw1

| {z }

flux



  

Example: vibrating string

Supply rateis _∂∂_tw1·w2, represented by

1 2

1 ρ0ζ_t2−T0ζ_x2

0 ζt

ηt 0

1 ρ0ζ_t2−T0ζ_x2

Φ(ζ, η) = 1₂ ρ0ζt2ηt −T0ζx2ηt +ρ0ζtηt2−T0ζtηx2

= (ζt +ηt)1₂(ρ0ζtηt +T0ζxηx)

+(ζx +ηx)1₂(−T0ζtηx −T0ηtζx)

QΦ(w1) =

∂ ∂t      1 2ρ0

_∂

∂tw1

2

| {z }

kinetic energy

+ 1 2T0

_∂

∂xw1

2

| {z }

potential energy      + ∂ ∂x     −1

2T0

_∂

∂xw1

∂

∂tw1

| {z }

flux



  

Outline

B- and QDFs forn−D systems

The calculus ofn−D B/QDFs

Losslessness

Dissipative systems

n be controllable, and let Φ ∈ R

w×w_{[ζ, η].}

Bisdissipative w.r.t. QΦ if

Z

QΦ(w)dx ≥ 0

for allw ∈B∩C∞(_Rn_,

Rw)of compact support.

power energy

Energy isdissipated, but local flow can be negative.

¡Energy must be locally stored!

SUPPLY

DISSIPATION

FLUX

Dissipative systems

n be controllable, and let Φ ∈ R

w×w_{[ζ, η].}

Bisdissipative w.r.t. QΦ if

Z

QΦ(w)dx ≥ 0

for allw ∈B∩C∞(_Rn_,

Rw)of compact support.

power

energy Energy isdissipated,

but local flow can be negative.

¡Energy must be locally stored!

SUPPLY

DISSIPATION

FLUX

Dissipative systems

n be controllable, and let Φ ∈ R

w×w_{[ζ, η].}

Bisdissipative w.r.t. QΦ if

Z

QΦ(w)dx ≥ 0

for allw ∈B∩C∞(_Rn_,

Rw)of compact support.

power

energy

Energy isdissipated, but local flow can be negative.

¡Energy must be locally stored!

SUPPLY

DISSIPATION

FLUX

Dissipative systems

n be controllable, and let Φ ∈ R

w×w_{[ζ, η].}

Bisdissipative w.r.t. QΦ if

Z

QΦ(w)dx ≥ 0

for allw ∈B∩C∞(_Rn_,

Rw)of compact support.

power energy

Energy goesintothe system

Energy isdissipated, but local flow can be negative.

¡Energy must be locally stored!

SUPPLY

DISSIPATION

FLUX

Dissipative systems

n be controllable, and let Φ ∈ R

w×w_{[ζ, η].}

Bisdissipative w.r.t. QΦ if

Z

QΦ(w)dx ≥ 0

for allw ∈B∩C∞(_Rn_,

Rw)of compact support.

power energy

Energy isdissipated, but local flow can be negative.

¡Energy must be locally stored!

SUPPLY

DISSIPATION

FLUX

Storage and dissipation functions

Brepresented asw = M _dxd

`, letΦ ∈ _Rw×w_{[ζ, η].}

VQDFΨ = (Ψ1, . . . ,Ψn)isstorage function(flux) forB

w.r.t. QΦ if

divQΨ(`)≤ QΦ(w)

∀` ∈C∞(_Rn_,

Rl)of compact support and(w, `) ∈Bf.

∆ ∈ _Rl×l

[ζ, η]isdissipation functionforBw.r.t. QΦ if

Q∆ ≥ 0 and

Z

Q∆(`) =

Z

QΦ(w)

∀` ∈C∞(_Rn ,_Rl

Storage and dissipation functions

Brepresented asw = M _dxd

`, letΦ ∈ _Rw×w_{[ζ, η].}

VQDFΨ = (Ψ1, . . . ,Ψn)isstorage function(flux) forB

w.r.t. QΦ if

divQΨ(`)≤ QΦ(w)

∀` ∈C∞(_Rn_,

Rl)of compact support and(w, `) ∈Bf.

∆ ∈ _Rl×l

[ζ, η]isdissipation functionforBw.r.t. QΦ if

Q∆ ≥ 0 and

Z

Q∆(`) =

Z

QΦ(w)

∀` ∈C∞(_Rn ,_Rl

Characterizations of dissipativity

Theorem:LetBbe controllable, andΦ ∈_Rw×w_{[ζ, η].} ThenBadmits an image representationw = M(_dxd )` s.t. the following conditions are equivalent:

• Bis dissipative w.r.t. QΦ (acting onw); • ∃ a storage functionQΨ (acting on`); • ∃ a dissipation functionQ∆ (acting on`). Also, the followingdissipation equality holds:

divQΨ(`) +Q∆(`) = QΦ(w)

Example: damped vibrating string

ρ0

∂2

∂t2w1−T0

∂2

∂x2w1+β

∂

∂tw1 = w2

β > 0 friction coefficient,

w1position,w2(vertical) force

R(ξt, ξx) =

ρ0ξ2t −T0ξx2+βξt −1

Image representationw = M(_dxd )`induced by

M(ξt, ξx) :=

1

ρ0ξ_t2−T0ξ2_x +βξt

Example: damped vibrating string

ρ0

∂2

∂t2w1−T0

∂2

∂x2w1+β

∂

∂tw1 = w2

β > 0 friction coefficient,

w1position,w2(vertical) force

R(ξt, ξx) =

ρ0ξ2t −T0ξx2+βξt −1

Image representationw = M(_dxd )`induced by

M(ξt, ξx) :=

1

ρ0ξ_t2−T0ξ2_x +βξt

Example: damped vibrating string

ρ0∂

2

∂t2 −T0

∂2

∂x2 +β

∂ ∂t

`

Supply rateis _∂∂_tw1·w2, represented by

1 2 ρ0ζ

2

tηt −T0ζx2ηt +2βζtηt +ρ0ζtηt2−T0ζtη2x

=: Φ(ζt, ζx, ηt, ηx)

Φ(−ξt,−ξx, ξt, ξx) = −2βξ_t2=⇒dissipation rateis √

2βζt √

Example: damped vibrating string

Simple algebra leads to thestorage function

(ζt+ηt)1₂(ρ0ζtηt+T0ζxηx)+(ζx+ηx)1₂(−T0ζtηx−T0ηtζx)

corresponding to ∂ ∂t      1 2ρ0

_∂

∂tw1

2

| {z }

kinetic energy

+ 1 2T0

_∂

∂xw1

2

| {z }

potential energy      + ∂ ∂x     −1

2T0

_∂

∂xw1

∂

∂tw1

| {z }

flux



Factorization of multivariable polynomial matrices

Q∆(`)≥ 0 for all `∈ C∞(_Rn

,_Rl )

of compact support

⇐⇒ ∆(−ξ, ξ) =D(−ξ)>D(ξ)

Ifn > 1, it is not possible in general to factorize with a polynomialD.

Factorization of multivariable polynomial matrices

Q∆(`)≥ 0 for all `∈ C∞(_Rn

,_Rl )

of compact support

⇐⇒ ∆(−ξ, ξ) =D(−ξ)>D(ξ)

Forn= 1, this is aspectral factorizationproblem, with known solvability conditions.

Ifn > 1, it is not possible in general to factorize with a polynomialD.

Factorization of multivariable polynomial matrices

Q∆(`)≥ 0 for all `∈ C∞(_Rn

,_Rl )

of compact support

⇐⇒ ∆(−ξ, ξ) =D(−ξ)>D(ξ)

Forn= 1, this is aspectral factorizationproblem, with known solvability conditions.

Hilbert’s 17th problem:

givenp ∈ _R[ξ1, . . . , ξn],

write it as the sum-of-squares

p = p₁2+. . .+p_k2

Ifn > 1, it is not possible in general to factorize with a polynomialD.

Factorization of multivariable polynomial matrices

Q∆(`)≥ 0 for all `∈ C∞(_Rn

,_Rl )

of compact support

⇐⇒ ∆(−ξ, ξ) =D(−ξ)>D(ξ)

Ifn > 1, it is not possible in general to factorize with a polynomialD.

On the storage function

Storage function is not unique; in the damped vibrat-ing strvibrat-ing example, another choice is

(ζt +ηt)

1

2(ρ0ζtηt −T0ζ

2

x −T0η2x −T0ζxηx)

+(ζx +ηx)

1

2(T0ζtζx +T0ηtηx)

On the storage function

Storage function is not unique; in the damped vibrat-ing strvibrat-ing example, another choice is

(ζt +ηt)

1

2(ρ0ζtηt −T0ζ

2

x −T0η2x −T0ζxηx)

+(ζx +ηx)

1

2(T0ζtζx +T0ηtηx)

Non-uniqueness of storage function arises from

• The non-uniqueness ofD(ξ)in the factorization of ∆(−ξ, ξ) =D(−ξ)>D(ξ);

• Ifn >1, there is no one-one correspondence between storage- and dissipation function

On the storage function

Storage function is not unique; in the damped vibrat-ing strvibrat-ing example, another choice is

(ζt +ηt)

1

2(ρ0ζtηt −T0ζ

2

x −T0η2x −T0ζxηx)

+(ζx +ηx)

1

2(T0ζtζx +T0ηtηx)

Summary

• Basic definitions for systems described by PDEs;

• Representation via polynomial matrices;

• The fundamental principle and the elimination of latent variables ;

• Bilinear and quadratic differential forms;

Summary

• Basic definitions for systems described by PDEs;

• Representation via polynomial matrices;

• The fundamental principle and the elimination of latent variables ;

• Bilinear and quadratic differential forms;

Summary

• Basic definitions for systems described by PDEs;

• Representation via polynomial matrices;

• The fundamental principle and the elimination of latent variables ;

• Bilinear and quadratic differential forms;

Summary

• Basic definitions for systems described by PDEs;

• Representation via polynomial matrices;

• The fundamental principle and the elimination of latent variables ;

• Bilinear and quadratic differential forms;

Summary

• Basic definitions for systems described by PDEs;

• Representation via polynomial matrices;

• The fundamental principle and the elimination of latent variables ;

• Bilinear and quadratic differential forms;