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
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