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

My first connec:on with JMC: has contributed a paper in the special 60th anniversary issue Rafael Vazquez, U. de Sevilla

(almost) no backstepping today!

IS SS S I � x ( t ) f ( x ( t ), d ( t )) �

IS SS S I � x ( t ) f ( x ( t ), d ( t )) � ISS: Input-to-State Stability (Sontag, 1989) � � � � � � � � x ( t ) x ( 0 ) , t sup d ( s ) � � 0 s t

IS SS S I � x ( t ) f ( x ( t ), d ( t )) � ISS: Input-to-State Stability (Sontag, 1989) � � � � � � � � x ( t ) x ( 0 ) , t sup d ( s ) � � 0 s t � � Effect of Effect of initial condition disturbanc e

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 .

Ch ha al ll le en ng ge e t to o I IS SS S f fo or r s sy ys st te em ms s C s wi it th h u un nb bo ou un nd de ed d i in np pu ut t o op pe er ra at to or rs w � � x Ax Bu ( B -unbounded) �

Ch ha al ll le en ng ge e t to o I IS SS S f fo or r s sy ys st te em ms s C s wi it th h u un nb bo ou un nd de ed d i in np pu ut t o op pe er ra at to or rs w � � x Ax Bu ( B -unbounded) � t � � � � � � x ( t ) exp( A t ) x ( 0 ) exp( A ( t )) Bu ( ) d � 0

Ch ha al ll le en ng ge e t to o I IS SS S f fo or r s sy ys st te em ms s C s wi it th h u un nb bo ou un nd de ed d i in np pu ut t o op pe er ra at to or rs w � � x Ax Bu ( B -unbounded) � t � � � � � � x ( t ) exp( A t ) x ( 0 ) exp( A ( t )) Bu ( ) d � 0 � � � Let exp( At ) M exp( t ) M � � � � x ( t ) M exp( t ) x ( 0 ) B sup u ( s ) � � s � 0 t

Ch ha al ll le en ng ge e t to o I IS SS S f fo or r s sy ys st te em ms s C s wi it th h u un nb bo ou un nd de ed d i in np pu ut t o op pe er ra at to or rs w � � x Ax Bu ( B -unbounded) � t � � � � � � x ( t ) exp( A t ) x ( 0 ) exp( A ( t )) Bu ( ) d � 0 � � � Let exp( At ) M exp( t ) M � � � � x ( t ) M exp( t ) x ( 0 ) B sup u ( s ) � � s � 0 t Unbounded B (in boundary control) potentially makes ISS impossible.

He ea at t e eq qu ua at ti io on n w wi it th h b bo ou 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 on n w wi it th h b bo ou 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 on n w wi it th h b bo ou un nd da ar ry y d di is st tu ur rb ba an nc ce es s H ISS with respect to boundary disturbances? No Lyapunov functional works.

He ea at t e eq qu ua at ti io on n w wi it th h b bo ou un nd da ar ry y d di is st tu ur rb ba an nc ce es s H 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 . � � � � y ( t , z ) x ( t , z ) ( 1 z ) d ( t ) zd ( t ) 0 1

Buggin’ me for about 10 years …

OU UT TL LI IN NE E O Heat Equation

OU UT TL LI IN NE E O Heat Equation Generalization

OU UT TL LI IN NE E O Heat Equation Generalization Quasi-static Thermoelasticity

OU UT TL LI IN NE E O Heat Equation Generalization Quasi-static Thermoelasticity Dynamic Output Feedback

He ea at t e eq qu ua at ti io on n w wi it th h b bo ou un nd da ar ry y d di is st tu ur rb ba an nc ce es s H

1 2 2 � � � � Theorem: For every and ( d , d ) C ( ; ) for which the heat x [ 0 ] C ([ 0 , 1 ]) � 0 1 0 1 � � � � �� � equation has a unique solution x C ( [ 0 , 1 ]) C (( 0 , ) [ 0 , 1 ]) with � 2 � � � x [ t ] C ([ 0 , 1 ]) for all , the following estimates hold for all : t 0 t 0 � � 2 1 1 � � exp t 2 2 � � � x ( t , z ) dz x ( 0 , z ) dz � � 2 � � � 2 exp t L 2 2 n no or rm m ) ( L 0 0 � � 1 � � � max d ( s ) max d ( s ) � 0 1 3 � � � � � � 0 s t 0 s t

max x ( t , z ) � � 0 z 1 � � 1 � � 2 � � � � � max exp ( 2 ) t max x ( 0 , z ) , max d ( s ) , max d ( s ) � � � � 0 1 � � � sin � � � � � � 0 z 1 0 s t 0 s t ∞ L ∞ n no or rm m ) ( L � � � for all ( 0 , / 2 )

max x ( t , z ) � � 0 z 1 � � 1 � � 2 � � � � � max exp ( 2 ) t max x ( 0 , z ) , max d ( s ) , max d ( s ) � � � � 0 1 � � � sin � � � � � � 0 z 1 0 s t 0 s t ∞ L ∞ n no or rm m ) ( L � � � for all ( 0 , / 2 ) � � � For / 4 , we get � � � � 2 � t � � � � � � max x ( t , z ) 2 max exp max x ( 0 , z ) , max d ( s ) , max d ( s ) 0 1 � � � � 4 � � � � � � � � 0 z 1 0 z 1 0 s t 0 s t � � � �

max x ( t , z ) � � 0 z 1 � � 1 � � 2 � � � � � max exp ( 2 ) t max x ( 0 , z ) , max d ( s ) , max d ( s ) � � � � 0 1 � � � sin � � � � � � 0 z 1 0 s t 0 s t ∞ L ∞ n no or rm m ) ( L � � � for all ( 0 , / 2 ) � � � For / 4 , we get � � � � 2 � t � � � � � � max x ( t , z ) 2 max exp max x ( 0 , z ) , max d ( s ) , max d ( s ) 0 1 � � � � 4 � � � � � � � � 0 z 1 0 z 1 0 s t 0 s t � � � � � � � Corollary: For / 2 , we get the maximum principle !! � � � max x ( t , z ) max � max x ( 0 , z ) , max d ( s ) , max d ( s ) � 0 1 � � � � � � � � � � 0 z 1 0 z 1 0 s t 0 s t

Should not be viewed as a restriction!

Analogous finite-dimensional case: Index-1 DAEs Example: � � � x ( t ) x ( t ) x ( t ) � 1 1 2 � � x ( t ) d ( t ) 0 2 2 � � � � � x ( t ) ( x ( t ), x ( t )) , d ( t ) 1 2 � Consistent initialization: x ( 0 ) d ( 0 ) 2

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

Ge en ne er ra al li iz za at ti io on n G in-domain

ISS gains