I n h ono r og Jean-Miche l Co r on o n h is 60 t h bj rthda y My - - PowerPoint PPT Presentation

Input-to-State Stability and Small-Gain Theorems for Parabolic PDEs

Iasson Karafyllis and Miroslav Krstic

In honor og Jean-Michel Coron on his 60th bjrthday

Rafael Vazquez, U. de Sevilla

My first connec:on with JMC: has contributed a paper in the special 60th anniversary issue

(almost) no backstepping today!

I IS SS S

)) ( ), ( ( ) ( t d t x f t x

I IS SS S

)) ( ), ( ( ) ( t d t x f t x

• ISS: Input-to-State Stability (Sontag, 1989)
• )

( sup , ) ( ) ( s d t x t x

t s

I IS SS S

)) ( ), ( ( ) ( t d t x f t x

• ISS: Input-to-State Stability (Sontag, 1989)
• e

disturbanc condition initial

• f

Effect

• f

Effect s d t x t x

t s

• )

( sup , ) ( ) (

Papers on ISS for PDEs

Mazenc and Prieur (2011), “Strict Lyapunov functions for semilinear parabolic partial differential equations”, Mathematical Control and Related Fields. Prieur and Mazenc (2012), “ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws”, Math. Control, Signals, & Syst. Bribiesca Argomedo, Witrant, and Prieur (2012), “D1-input-to-state stability of a time- varying nonhomogeneous diffusive equation subject to boundary disturbances”,

• Am. Control Conf.

Bribiesca Argomedo, Prieur, Witrant, and Bremond (2013), “A strict control Lyapunov function for a diffusion equation with time-varying distributed coefficients”, IEEE Trans. Automatic Control, 2013. Dashkovskiy and Mironchenko (2013), “Input-to-state stability of infinite-dimensional control systems”,

• Math. Control, Signals, & Syst.

Mironchenko and Ito (2015), “Construction of Lyapunov Functions for Interconnected Parabolic Systems: An iISS Approach”, SICON. Mironchenko (2016), “Local input-to-state stability: Characterizations and counterexamples”, Syst. &

• Contr. Lett.

C Ch ha al ll le en ng ge e t to

• I

IS SS S f fo

• r

r s sy ys st te em ms s w wi it th h u un nb bo

• u

un nd de ed d i in np pu ut t

• p

pe er ra at to

• r

rs s

Bu Ax x

• (B-unbounded)

C Ch ha al ll le en ng ge e t to

• I

IS SS S f fo

• r

r s sy ys st te em ms s w wi it th h u un nb bo

• u

un nd de ed d i in np pu ut t

• p

pe er ra at to

• r

rs s

Bu Ax x

• (B-unbounded)
• t

d Bu t A x t A t x ) ( )) ( exp( ) ( ) exp( ) (

C Ch ha al ll le en ng ge e t to

• I

IS SS S f fo

• r

r s sy ys st te em ms s w wi it th h u un nb bo

• u

un nd de ed d i in np pu ut t

• p

pe er ra at to

• r

rs s

Bu Ax x

• (B-unbounded)
• t

d Bu t A x t A t x ) ( )) ( exp( ) ( ) exp( ) (

• Let

) exp( ) exp( t M At

• )

( sup ) ( ) exp( ) ( s u B M x t M t x

t s

C Ch ha al ll le en ng ge e t to

• I

IS SS S f fo

• r

r s sy ys st te em ms s w wi it th h u un nb bo

• u

un nd de ed d i in np pu ut t

• p

pe er ra at to

• r

rs s

Bu Ax x

• (B-unbounded)
• t

d Bu t A x t A t x ) ( )) ( exp( ) ( ) exp( ) (

• Let

) exp( ) exp( t M At

• )

( sup ) ( ) exp( ) ( s u B M x t M t x

t s

• Unbounded B (in boundary control) potentially makes ISS impossible.

H He ea at t e eq qu ua at ti io

• n

n w wi it th h b bo

• u

un nd da ar ry y d di is st tu ur rb ba an nc ce es s

H He ea at t e eq qu ua at ti io

• n

n w wi it th h b bo

• u

un nd da ar ry y d di is st tu ur rb ba an nc ce es s

H He ea at t e eq qu ua at ti io

• n

n w wi it th h b bo

• u

un nd da ar ry y d di is st tu ur rb ba an nc ce es s

ISS with respect to boundary disturbances? No Lyapunov functional works.

H He ea at t e eq qu ua at ti io

• n

n w wi it th h b bo

• u

un nd da ar ry y d di is st tu ur rb ba an nc ce es s

ISS with respect to boundary disturbances? No Lyapunov functional works. The transformation of boundary disturbances to distributed disturbances gives ISS with respect to boundary disturbances and their time derivatives. ) ( ) ( ) 1 ( ) , ( ) , (

1

t zd t d z z t x z t y

Buggin’ me for about 10 years…

O OU UT TL LI IN NE E

Heat Equation

O OU UT TL LI IN NE E

Heat Equation Generalization

O OU UT TL LI IN NE E

Heat Equation Generalization Quasi-static Thermoelasticity

O OU UT TL LI IN NE E

Heat Equation Generalization Quasi-static Thermoelasticity Dynamic Output Feedback

H He ea at t e eq qu ua at ti io

• n

n w wi it th h b bo

• u

un nd da ar ry y d di is st tu ur rb ba an nc ce es s

Theorem: For every ]) 1 , ([ ] [

2

C x

• and

) ; ( ) , (

2 1 1

• C

d d for which the heat equation has a unique solution

]) 1 , [ ) , (( ]) 1 , [ (

1

• C

C x with ]) 1 , ([ ] [

2

C t x

• for all
• t

, the following estimates hold for all

• t

:

• )

( max ) ( max 3 1 ) , ( exp 2 exp ) , (

1 1 2 2 2 1 2

s d s d dz z x t t dz z t x

t s t s

• (L

L2

2

n no

• r

rm m)

• )

( max , ) ( max , ) , ( max ) 2 ( exp max sin 1 ) , ( max

1 1 2 1

s d s d z x t z t x

t s t s z z

• for all

) 2 / , (

• (L

L∞

∞

n no

• r

rm m)

• )

( max , ) ( max , ) , ( max ) 2 ( exp max sin 1 ) , ( max

1 1 2 1

s d s d z x t z t x

t s t s z z

• for all

) 2 / , (

• (L

L∞

∞

n no

• r

rm m)

For 4 /

• , we get
• )

( max , ) ( max , ) , ( max 4 exp max 2 ) , ( max

1 1 2 1

s d s d z x t z t x

t s t s z z

• )

( max , ) ( max , ) , ( max ) 2 ( exp max sin 1 ) , ( max

1 1 2 1

s d s d z x t z t x

t s t s z z

• for all

) 2 / , (

• (L

L∞

∞

n no

• r

rm m)

For 4 /

• , we get
• )

( max , ) ( max , ) , ( max 4 exp max 2 ) , ( max

1 1 2 1

s d s d z x t z t x

t s t s z z

• Corollary: For

2 /

• , we get the maximum principle!!
• )

( max , ) ( max , ) , ( max max ) , ( max

1 1 1

s d s d z x z t x

t s t s z z

Should not be viewed as a restriction!

Analogous finite-dimensional case: Index-1 DAEs Example:

• )

( , )) ( ), ( ( ) ( ) ( ) ( ) ( ) ( ) (

2 2 1 2 2 1 1

t d t x t x t x t d t x t x t x t x

• Consistent initialization:

) ( ) (

2

d x

G Ge en ne er ra al li iz za at ti io

• n

n

G Ge en ne er ra al li iz za at ti io

• n

n

in-domain

ISS gains

I IS SS S i in n L L2

2

n no

• r

rm m

Theorem: Let (H1), (H2), (H3) hold. Then for every ]) 1 , ([ ] [

2

C x

• ,

]) 1 , [ (

1

• C

u and ) ], [ ( ) , (

1

u x d d

• , the following estimate holds for all

,

• and
• t

:

• )

( max ) 1 )( 1 ( ) ( max ) 1 )( 1 ( ) , ( max max 1 ~ ] [ exp 2 exp ] [

1 1 1 1 1 , 2 1 1 , 2

s d C s d C z s u C x t t t x

t s t s z t s r r

• where

2 / 1 1 2 , 2

) , ( ) ( : ] [

• dz

z t x z r t x

r

( ) (

• z

r : weight function on diffusion)

I IS SS S i in n L L2

2

n no

• r

rm m

r n n n n

x v g v z d d g v g p C

, 2 2 2 1 2 2 2 2

~ 1 ) ( ) ( 1 ) ( :

• where

]) 1 , ([ ~

2

C x is the solution of the BVP ) ( ~ ) ( ) ( ~ ) (

• z

x z q z dz x d z p dz d for ] 1 , [

• z

with

2 2

) ( ~ ) ( ~ v g dz x d v x g

• ,

) 1 ( ~ ) 1 ( ~

1 1

• dz

x d v x g ,

I IS SS S i in n L L2

2

n no

• r

rm m

r n n n n

x v g v z d d g v g p C

, 2 2 2 1 2 2 2 2

~ 1 ) ( ) ( 1 ) ( :

• where

]) 1 , ([ ~

2

C x is the solution of the BVP ) ( ~ ) ( ) ( ~ ) (

• z

x z q z dz x d z p dz d for ] 1 , [

• z

with

2 2

) ( ~ ) ( ~ v g dz x d v x g

• ,

) 1 ( ~ ) 1 ( ~

1 1

• dz

x d v x g , Analogous for 1 C

I IS SS S i in n L L2

2

n no

• r

rm m

r n n n n

x v g v z d d g v g p C

, 2 2 2 1 2 2 2 2

~ 1 ) ( ) ( 1 ) ( :

• where

]) 1 , ([ ~

2

C x is the solution of the BVP ) ( ~ ) ( ) ( ~ ) (

• z

x z q z dz x d z p dz d for ] 1 , [

• z

with

2 2

) ( ~ ) ( ~ v g dz x d v x g

• ,

) 1 ( ~ ) 1 ( ~

1 1

• dz

x d v x g , Analogous for 1 C

1 0,C

C

c ca an n b be e c co

• m

mp pu ut te ed d w wi it th ho

• u

ut t k kn no

• w

wl le ed dg ge e

• f

f e ei ig ge en nv va al lu ue es s a an nd d e ei ig ge en nf fu un nc ct ti io

• n

ns s

I IS SS S i in n L L2

2

n no

• r

rm m

• 1

2 1 2

) ( ) ( 1 : ~

n n n

dz z z r C

S Sk ke et tc ch h

• f

f p pr ro

• f

f

Eigenfunction expansion of solution x

S Sk ke et tc ch h

• f

f p pr ro

• f

f

Eigenfunction expansion of solution x Infinitely many ODEs for (gen.) Fourier coefficients

S Sk ke et tc ch h

• f

f p pr ro

• f

f

Eigenfunction expansion of solution x Infinitely many ODEs for (gen.) Fourier coefficients Obtain estimates for each (gen.) Fourier coefficient

S Sk ke et tc ch h

• f

f p pr ro

• f

f

Eigenfunction expansion of solution x Infinitely many ODEs for (gen.) Fourier coefficients Obtain estimates for each (gen.) Fourier coefficient Parseval’s identity

(existence of a posi:ve eigenfunc:on for the Sturm-Liouville operator)

I IS SS S i in n L L∞

∞

n no

• r

rm m

Theorem: Let (H1)-(H4) hold. Then for every ]) 1 , ([ ] [

2

C x

• ,

]) 1 , [ (

1

• C

u and ) ], [ ( ) , (

1

u x d d

• , the following estimate holds for all
• t

:

I IS SS S i in n L L∞

∞

n no

• r

rm m

Theorem: Let (H1)-(H4) hold. Then for every ]) 1 , ([ ] [

2

C x

• ,

]) 1 , [ (

1

• C

u and ) ], [ ( ) , (

1

u x d d

• , the following estimate holds for all
• t

:

• )

( ) , ( max max ) 1 ( ) 1 ( ) ( max , ) ( ) ( ) ( max , ] [ exp max ] [

1 1 1 1 1 , ,

z z s u v g s d v g s d x t t x

z t s t s t s

I IS SS S i in n L L∞

∞

n no

• r

rm m

Theorem: Let (H1)-(H4) hold. Then for every ]) 1 , ([ ] [

2

C x

• ,

]) 1 , [ (

1

• C

u and ) ], [ ( ) , (

1

u x d d

• , the following estimate holds for all
• t

:

• )

( ) , ( max max ) 1 ( ) 1 ( ) ( max , ) ( ) ( ) ( max , ] [ exp max ] [

1 1 1 1 1 , ,

z z s u v g s d v g s d x t t x

z t s t s t s

• where

) ( ) , ( max : ] [

1 ,

z z t x t x

z

• .

S Sk ke et tc ch h

• f

f p pr ro

• f

f

Finite-difference discretization on uniform grid: ) , ( ) ( z i t x t xi

• ,

) , ( ) ( z i t u t ui

• ,

N i ,...,

• with

1

• N

z

S Sk ke et tc ch h

• f

f p pr ro

• f

f

Finite-difference discretization on uniform grid: ) , ( ) ( z i t x t xi

• ,

) , ( ) ( z i t u t ui

• ,

N i ,...,

• with

1

• N

z

• )

( ) ( ) ( ) (

, ,

error t d B t x A t t x

N N

S Sk ke et tc ch h

• f

f p pr ro

• f

f

Finite-difference discretization on uniform grid: ) , ( ) ( z i t x t xi

• ,

) , ( ) ( z i t u t ui

• ,

N i ,...,

• with

1

• N

z

• )

( ) ( ) ( ) (

, ,

error t d B t x A t t x

N N

• Parameterized family of uncertain linear discrete-time systems

(with parameters N ,

• ,

2

) /( z t

• ).

S Sk ke et tc ch h

• f

f p pr ro

• f

f

Finite-difference discretization on uniform grid: ) , ( ) ( z i t x t xi

• ,

) , ( ) ( z i t u t ui

• ,

N i ,...,

• with

1

• N

z

• )

( ) ( ) ( ) (

, ,

error t d B t x A t t x

N N

• Parameterized family of uncertain linear discrete-time systems

(with parameters N ,

• ,

2

) /( z t

• ).

ISS Lyapunov function:

• )

( ) , ( max : ) (

1 ,..., 1

z i z i t x t V

N i N

• Lyapunov analysis with V(t+dt)-V(t) but for con:nuous t.

Component-wise es:mates, exploi:ng proper:es of heat eqn, lead to max-norm es:mate.

S Sk ke et tc ch h

• f

f p pr ro

• f

f

Finite-difference discretization on uniform grid: ) , ( ) ( z i t x t xi

• ,

) , ( ) ( z i t u t ui

• ,

N i ,...,

• with

1

• N

z

• )

( ) ( ) ( ) (

, ,

error t d B t x A t t x

N N

• Parameterized family of uncertain linear discrete-time systems

(with parameters N ,

• ,

2

) /( z t

• ).

ISS Lyapunov function:

• )

( ) , ( max : ) (

1 ,..., 1

z i z i t x t V

N i N

• Finally, we use estimates of (error) and obtain the required estimate.

1 1-

• D

D q qu ua as si i-

• s

st ta at ti ic c t th he er rm mo

• e

el la as st ti ic ci it ty y

) , ( ) , (

2 2

z t z x a z t t x

1 1-

• D

D q qu ua as si i-

• s

st ta at ti ic c t th he er rm mo

• e

el la as st ti ic ci it ty y

) , ( ) , (

2 2

z t z x a z t t x

• BCs
• 1

) , ( ) ( ) , ( dz z t x z g t x ,

• 1

1

) , ( ) ( ) 1 , ( dz z t x z g t x

1 1-

• D

D q qu ua as si i-

• s

st ta at ti ic c t th he er rm mo

• e

el la as st ti ic ci it ty y

) , ( ) , (

2 2

z t z x a z t t x

• BCs
• 1

) , ( ) ( ) , ( dz z t x z g t x ,

• 1

1

) , ( ) ( ) 1 , ( dz z t x z g t x

• x

entropy

1 1-

• D

D q qu ua as si i-

• s

st ta at ti ic c t th he er rm mo

• e

el la as st ti ic ci it ty y

) , ( ) , (

2 2

z t z x a z t t x

• BCs
• 1

) , ( ) ( ) , ( dz z t x z g t x ,

• 1

1

) , ( ) ( ) 1 , ( dz z t x z g t x

• x

entropy Model derived by singular perturbation of (modified) Euler-Bernoulli beam eqn. (with thermal expansion), solution in space for displacement in terms of temperature, and linear combination of temperature and stress to form entropy.

F Fe ee ed db ba ac ck k C Co

• n

nf fi ig gu ur ra at ti io

• n

n

familiar picture:

F Fe ee ed db ba ac ck k C Co

• n

nf fi ig gu ur ra at ti io

• n

n

familiar picture:

F Fe ee ed db ba ac ck k C Co

• n

nf fi ig gu ur ra at ti io

• n

n

F Fe ee ed db ba ac ck k C Co

• n

nf fi ig gu ur ra at ti io

• n

n

two feedback loops!

Gains of static maps ) ( ] [ t d t x

i

Gains of static maps ) ( ] [ t d t x

i

• 1

2 1 2 1 1 2 2

) ( ] [ ) ( ) ( ] [ ) ( dz z g t x t d dz z g t x t d

Gains of static maps ) ( ] [ t d t x

i

• 1

2 1 2 1 1 2 2

) ( ] [ ) ( ) ( ] [ ) ( dz z g t x t d dz z g t x t d

• r
• 1

1 1 1

) ( ] [ ) ( ) ( ] [ ) ( dz z g t x t d dz z g t x t d

Recall ISS properties:

• )

( max ) ( max 3 1 ] [ exp 2 exp ] [

1 2 2 2 2

s d s d x t a t a t x

t s t s

• ,

Recall ISS properties:

• )

( max ) ( max 3 1 ] [ exp 2 exp ] [

1 2 2 2 2

s d s d x t a t a t x

t s t s

• ,
• r
• )

( max , ) ( max , ] [ 4 exp max 2 ] [

1 2

s d s d x t a t x

t s t s

L L2

2

l lo

• p

p

(block diagram in terms of es:mates, not signals)

L L∞

∞

l lo

• p

p

S St ta ab bi il li it ty y i in n L L2

2

b by y S Sm ma al ll l-

• G

Ga ai in n A Ar rg gu um me en nt t

Theorem: Suppose that

3 ) ( ) (

2 / 1 1 2 1 2 / 1 1 2

• dz

z g dz z g

S St ta ab bi il li it ty y i in n L L2

2

b by y S Sm ma al ll l-

• G

Ga ai in n A Ar rg gu um me en nt t

Theorem: Suppose that

3 ) ( ) (

2 / 1 1 2 1 2 / 1 1 2

• dz

z g dz z g

Then there exist constants ,

• M

such that for every allowable ] [ x the following estimate holds for all

• t

:

• 2

2

] [ exp ] [ x t M t x

S St ta ab bi il li it ty y i in n L L∞

∞

b by y S Sm ma al ll l-

• G

Ga ai in n A Ar rg gu um me en nt t

Theorem: Suppose that 2 2 ) (

1

• dz

z g and 2 2 ) (

1 1

• dz

z g

S St ta ab bi il li it ty y i in n L L∞

∞

b by y S Sm ma al ll l-

• G

Ga ai in n A Ar rg gu um me en nt t

Theorem: Suppose that 2 2 ) (

1

• dz

z g and 2 2 ) (

1 1

• dz

z g Then there exist constants ,

• M

such that for every allowable ] [ x the following estimate holds for all

• t

:

• ]

[ exp ] [ x t M t x

C Ca as sc ca ad de e S Sy ys st te em m A Ap pp pl li ic ca at ti io

• n

n

) , ( ) , ( ) , (

2 2

z t x z t z x a z t t x

• )

, ( ) , ( t x q t z x

• ,

) ( ) 1 , ( t u t x

C Ca as sc ca ad de e S Sy ys st te em m A Ap pp pl li ic ca at ti io

• n

n

) , ( ) , ( ) , (

2 2

z t x z t z x a z t t x

• )

, ( ) , ( t x q t z x

• ,

) ( ) 1 , ( t u t x

• Measurement:

) , ( ) ( t x t y

• (first result on output-Ak stabiliza:on in L-inf norm)

C Ca as sc ca ad de e S Sy ys st te em m A Ap pp pl li ic ca at ti io

• n

n

C Ca as sc ca ad de e S Sy ys st te em m A Ap pp pl li ic ca at ti io

• n

n

• ]

[ ] [

• t

Me t

C Ca as sc ca ad de e S Sy ys st te em m A Ap pp pl li ic ca at ti io

• n

n

• ]

[ ] [

• t

Me t

• ]

[ max ] [ ˆ ] [ ˆ s X e R t X

t s t

C Ca as sc ca ad de e S Sy ys st te em m A Ap pp pl li ic ca at ti io

• n

n

• ]

[ ] [

• t

Me t

• ]

[ max ] [ ˆ ] [ ˆ s X e R t X

t s t

• Then

,

• s.t.

C Ca as sc ca ad de e S Sy ys st te em m A Ap pp pl li ic ca at ti io

• n

n

• ]

[ ] [

• t

Me t

• ]

[ max ] [ ˆ ] [ ˆ s X e R t X

t s t

• Then

,

• s.t.
• ]

[ ] [ ˆ ] [ ] [ ˆ

• X

e t t X

t

(exponential stability in L∞ norm for the overall system)

C Ca as sc ca ad de e S Sy ys st te em m A Ap pp pl li ic ca at ti io

• n

n

• ]

[ ] [

• t

Me t

• ]

[ max ] [ ˆ ] [ ˆ s X e R t X

t s t

• Then

,

• s.t.
• ]

[ ] [ ˆ ] [ ] [ ˆ

• X

e t t X

t

(exponential stability in L∞ norm for the overall system) Then ˆ

• s.t.
• ]

[ ] [ ˆ ] [ ] [ e x e t e t x

t

• (output-Ak stabiliza:on in L-inf norm)

More to do: Lp ISS estimates for

• p

1 ( 1

• p

results available),

More to do: Lp ISS estimates for

• p

1 ( 1

• p

results available), ISS estimates from generalized ISS Lyapunov functionals,

More to do: Lp ISS estimates for

• p

1 ( 1

• p

results available), ISS estimates from generalized ISS Lyapunov functionals, Stability study for interconnected parabolic systems.

BON ANNIVERSAIRE, JEAN-MICHEL!