Passive DAEs and maximal monotone operators Stephan Trenn AG - PowerPoint PPT Presentation

Passive DAEs and maximal monotone operators Stephan Trenn AG Technomathematik, TU Kaiserslautern joint work with K. Camlibel (U Groningen, NL), L. Iannelli (U Sannio in Benevento, IT), A. Tanwani (LAAS-CNRS, Toulouse, FR) 7th European Congress of Mathematics Berlin, 21.07.2016, 10:00–10:30

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Motivation: Electrical circuits with ideal diodes v D i Linear complementarity systems Theorem ( Camlibel et al. 1999 ) ( A , B , C , D ) passive x = Ax + Bz ˙ v L L ⇓ w = Cx + Dz Existence & uniqueness of solutions 0 ≤ z ⊥ w ≥ 0   ∅ , i < 0 ,  d d t i = Lv D Reformulation: 0 ≤ i ⊥ v D ≥ 0 ⇔ v D ∈ [0 , ∞ ) , i = 0 ,  0 ≤ i ⊥ v D ≥ 0  { 0 } , i > 0 . v D Set-valued constraints Theorem ( Camlibel et al. 2015 ) x = Ax + Bz ˙ (A,B,C,D) passive and F maximal-monotone i w = Cx + Dz ⇓ w ∈ F ( − z ) Existence & Uniqueness of solutions Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Question Generalization to DAEs E ˙ x = Ax + Bz w = Cx + Dz w ∈ F ( − z ) ? ( E , A , B , C , D ) passive & F maximal-monotone ⇒ existence & uniqueness of solutions Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Maximal-monotone operators Definition (Monotonicity) M : R n ⇒ R n is monotone : ⇔ ∀ y 1 ∈ M ( x 1 ) , y 2 ∈ M ( x 2 ) : � y 2 − y 1 , x 2 − x 1 � ≥ 0 A monotone M : R n ⇒ R n is maximal : ⇔ ∀ � � M ⊃ M : M is not monotone Examples for scalar maximal-monotone operators: y y y y x x x x Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Maximal-monotone operators Definition (Monotonicity) M : R n ⇒ R n is monotone : ⇔ ∀ y 1 ∈ M ( x 1 ) , y 2 ∈ M ( x 2 ) : � y 2 − y 1 , x 2 − x 1 � ≥ 0 A monotone M : R n ⇒ R n is maximal : ⇔ ∀ � � M ⊃ M : M is not monotone Non-monotone example: Non-maximal example: y y x x Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Maximal-monotone operators and differential inclusions Theorem ( Brezis 1973 ) M : R n ⇒ R n max.-monotone ⇒ x ∈ −M ( x ) , x (0) = x 0 ∈ dom( M ) , is uniquely solvable ˙ Global solutions (Philipov-solutions) No global solution:   − 1 , x > 0 , �  − 1 , x ≥ 0 , x ∈ − sign ∗ ( x ) := ˙ [ − 1 , 1] , x = 0 , x = − sign( x ) := ˙   1 , x < 0 1 , x < 0 sign( x ) sign ∗ ( x ) x x Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Linear systems with set-valued constraints We have: x = Ax + Bz ˙ w = Cx + Dz ⇐ ⇒ x ∈ −M ( x ) ˙ w ∈ F ( − z ) where M ( x ) := − Ax + B ( F + D ) − 1 ( Cx ) . Passivity and maximal-monotonicity ( A , B , C , D ) passive & F maximal-monotone ⇒ M is maximal-monotone Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Linear systems with set-valued constraints We have: E ˙ x = Ax + Bz x ∈ − E − 1 M ( x ) w = Cx + Dz ⇐ ⇒ ˙ w ∈ F ( − z ) where M ( x ) := − Ax + B ( F + D ) − 1 ( Cx ) . Maximal-monotonicity is lost E − 1 M ( x ) is maximal-monotone ( E , A , B , C , D ) passive & F maximal-monotone �⇒   x 1 = z ˙ x 3 x 3 = x 2 + z ˙   x ∈ − , for x 3 = max { 0 , x 1 } , ∅ otherwise ˙ R 0 = x 3 + z ⇐ ⇒ − x 2 + x 3 � 0 � �� 0 �� w = x 1 ∈ E − 1 M not monotone, consider e.g. − 1 1 − z ∈ F − 1 ( w ) := max { 0 , w } − 1 0 Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Passivity: Definitions and important consequences Definition (Passivity) � t 1 E ˙ x = Ax + Bz ∃ V : R n → R + : z ⊤ w passive : ⇔ V ( x ( t 1 )) ≤ V ( x ( t 0 )) + w = Cx + Dz t 0 Lemma (Passivity & special quasi-Weierstrass-form, Freund & Jarre 2004 ) ⇒ ∃ S , T invertible: ( E , A , B , C , D ) passive (and minimal)       I 0 0 A 1 0 0    ,    ( SET , SAT ) = 0 0 I 0 I 0 0 0 0 0 0 I In particular, a (minimal) passive DAE is either an ODE or an index-2-DAE. Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Passivity: Charakterisation Theorem (Passivity & LMIs, Camlibel & Frasca 2009 ) �� I 0 0 � � � � � A 1 0 0 � B 1 is passive with V ( x ) = x ⊤ Kx ⇔ ( E , A , B , C , D ) = , [ C 1 C 2 C 3 ] , D , , B 2 0 0 I 0 I 0 0 0 0 0 0 I B 3   K 11 0 0 K = K ⊤ =   ≥ 0 0 0 0 1 0 0 K 33 ( A 1 , B 1 , C 1 , D − C 2 B 2 − C 3 B 3 ) is passive, i.e. the following LMI holds: 2 � A ⊤ � K 11 B 1 − C ⊤ 1 K 11 + K 11 A 1 where � 1 ≤ 0 , D = D − C 2 B 2 − C 3 B 3 − ( � D + � B ⊤ D ⊤ ) 1 K 11 − C 1 B ⊤ 3 K 33 = − C 2 3 Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Elimination of variables Theorem (Camlibel, Iannelli, Tanwani, T. 2016) ( x , z , w ) solves passive and minimal ( E , A , B , C , D ) with constraint w ∈ F ( − z ) ⇐ ⇒ � � x 1 P ˙ x := solves x ∈ −M ( x ) , − z where � K 11 � 0 P := symmetric and positive-semidefinit B ⊤ 0 3 K 33 B 3 and � K 11 A 1 � � � − K 11 B 1 0 M ( x ) := − x + maximal-monotone − � F ( − z ) C 1 D Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) New class of differential-inclusions Differential-algebraic-inclusions (DAIs) P ˙ x ∈ −M ( x ) ( DAI ) Theorem ( Camlibel, Iannelli, Tanwani, T. 2016 ) Consider ( DAI ) with P ≥ 0 and max.-mon. M .Then: For every initial condition x (0) = x 0 with x 0 ∈ M − 1 (im P ) a global solution 1 x : [0 , ∞ ) → R n with absolute-continuos Px exists Stability in the following sense holds: 2 � � � Px 1 ( t ) − Px 2 ( t ) � ≤ c � x 1 (0) − x 2 (0) � , in particular, Px uniquely determined by initial value. Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators

Introduction Maximal-monotone operators Passivity Differential-algebraic inclusions (DAIs) Summary x = Ax + Bz ˙ E ˙ x = Ax + Bz w = Cx + Dz w = Cx + Dz w ∈ F ( − z ) w ∈ F ( − z ) P ˙ x ∈ −M ( x ) ˙ x ∈ −M ( x ) Passivity preserves maximal-monotonicity DAEs lead to maximal-monotene differential-algebraic-inclusion Uniqueness of solutions is lost, but global existence is guaranteed Further questions: External inputs (works for ODE case) How important is positive-semi-definitness and symmetry of P ? Physical interpretation of non-uniqueness? Stephan Trenn AG Technomathematik, TU Kaiserslautern Passive DAEs and maximal monotone operators