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Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms Stephan Trenn ∗ and Fabian Wirth ∗∗ ∗ Technomathematics group, University of Kaiserslautern, Germany ∗∗ Department for Mathematics, University of W¨ urzburg, Germany 51st IEEE Conference on Decision and Control Tuesday, December 11, 2012, 11:40–12:00, Maui, USA

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Content Introduction 1 Evolution operator and its semigroup 2 Lyapunov and Barabanov norm 3 Conclusions 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Switched DAEs Linear switched DAE (differential algebraic equation) (swDAE) E σ ( t ) ˙ x ( t ) = A σ ( t ) x ( t ) or short E σ ˙ x = A σ x with switching signal σ : R → { 1 , 2 , . . . , p } piecewise constant, right-continuous locally finitely many jumps (no Zeno behavior) matrix pairs ( E 1 , A 1 ) , . . . , ( E p , A p ) E p , A p ∈ R n × n , p = 1 , . . . , p ( E p , A p ) regular, i.e. det( E p s − A p ) �≡ 0 impulse-free solutions (but jumps are allowed!) Question ? x = A σ x asymp. stable ∀ σ ⇒ E σ ˙ common Lyapunov function Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Lyapunov norms More general approach: Definition (Lyapunov norm) ~ · ~ is a λ -Lyapunov norm , λ ∈ R , ~ x ( t ) ~ ≤ e λ t ~ x (0 − ) ~ ∀ solutions x of E σ ˙ : ⇔ ∀ σ : x = A σ x In particular: λ < 0 ⇒ V = ~ · ~ defines Lyapunov function New question Find Lyapunov norm for E σ ˙ x = A σ x (stable or unstable) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Solution formula Theorem ( A diff and Π ( E , A ) , Tanwani & T. 2010) Let ( E , A ) be regular and consider E ˙ x = Ax on [0 , ∞ ) ⇒ ∃ unique consistency projector Π ( E , A ) and unique flow matrix A diff : x = A diff x x (0) = Π ( E , A ) x (0 − ) ˙ on (0 , ∞ ) Furthermore, A diff Π ( E , A ) = Π ( E , A ) A diff . Corollary (Solution formula for switched DAE) Any solution of the switched DAE E σ ˙ x = A σ x has the form x ( t ) = e A diff k ( t − t k ) Π k e A diff k − 1 ( t k − t k − 1 ) Π k − 1 · · · e A diff 1 ( t 2 − t 1 ) Π 1 e A diff 0 ( t 1 − t 0 ) Π 0 x ( t 0 − ) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Evolution operator x ( t ) = e A diff k ( t − t k ) Π k e A diff k − 1 ( t k − t k − 1 ) Π k − 1 · · · e A diff 1 ( t 2 − t 1 ) Π 1 e A diff 0 ( t 1 − t 0 ) Π 0 x ( t 0 − ) � �� � =: Φ σ ( t , t 0 ) � � ( A diff Let M := p , Π p ) | corresponding to ( E p , A p ) , p = 1 , . . . , p . Definition (Set of all evolutions with fixed time span ∆ t > 0 ) � { Φ σ ( t 0 + ∆ t , t 0 ) | t 0 ∈ R } S ∆ t := σ � � � k k � � � e A diff τ i Π i � � ( A diff = , Π i ) ∈ M , τ i = ∆ t , τ i > 0 i � i i =0 i =0 Note that ∀ t 0 ∈ R ∀ ∆ t > 0: x solves E σ ˙ x = A σ x ⇔ ∃ Φ ∆ t ∈ S ∆ t : x ( t 0 + ∆ t ) = Φ ∆ t x ( t 0 − ) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Semi group property Lemma (Semi group) The set � S := S ∆ t ∆ t > 0 is a semi group with S s + t = S s S t := { Φ s Φ t | Φ s ∈ S s , Φ t ∈ S t } Need commutativity to show “ ⊆ ”: e A diff τ Π = e A diff ( τ − τ ′ ) e A diff τ ′ ΠΠ = e A diff ( τ − τ ′ ) Π e A diff τ ′ Π for any ( A diff , Π) ∈ M and 0 < τ ′ < τ Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Exponential growth bound Definition (Exponential growth bound) For t > 0 the exponential growth bound of E σ ˙ x = A σ x is ln � Φ t � λ t ( S t ) := sup ∈ R ∪ {−∞ , ∞} t Φ t ∈S t Definition implies for all solutions x of E σ ˙ x = A σ x : � x ( t ) � = � Φ t x (0 − ) � ≤ � Φ t � � x (0 − ) � ≤ e λ t ( S t ) t � x (0 − ) � Difference to switched ODEs without jumps λ t ( S t ) = ±∞ is possible! All jumps are trivial, i.e. Π p = 0 ⇒ λ t ( S t ) = −∞ Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Infinite exponential growth bound Example: �� 0 � � 1 �� �� 0 � � − 1 �� 0 − 1 0 0 ( E 1 , A 1 ) = ( E 2 , A 2 ) = , , 0 1 0 − 1 1 1 0 − 1 � x � � x � x 2 x 1 t t � � k � � 1 1 1 1 For small dwell times: Φ t ≈ (Π 1 Π 2 ) k = = 2 k − 1 1 1 1 1 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Existence of exponential growth rate Theorem (Boundedness of S t ) S t is bounded ⇔ the set of consistency projectors is product bounded Reminder: � � � k k � � � e A diff τ i Π i � � ( A diff S t := , Π i ) ∈ M , τ i = ∆ t , τ i > 0 i � i i =0 i =0 Theorem (Exponential growth rate well defined) Let the consistency projectors be product bounded and not all be trivial, then the (upper) Lyapunov exponent � Φ t � λ ( S ) := lim t →∞ λ t ( S t ) = lim t →∞ sup t Φ t ∈S t of E σ ˙ x = A σ x is well defined and finite. Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions A converse Lyapunov Theorem Theorem (Lyapunov norm) Assume λ ( S ) is finite. Then for each ε > 0 e − ( λ ( S )+ ε ) t � Φ t x � ~ x ~ ε := sup sup t > 0 Φ t ∈S t defines a ( λ ( S ) + ε ) -Lyapunov norm for E σ ˙ x = A σ x. Corollary (Converse Lyapunov Theorem) E σ ˙ x = A σ x is uniformly exp. stable ⇒ V = ~ · ~ ε is Lyapunov function In particular: V (Π x ) ≤ V ( x ) for all consistency projectors Π Non-smooth Lyapunov function ~ · ~ ε in general non-smooth. “Smoothification” as in Yin, Sontag & Wang 1996 might violate jump condition! Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Barabanov norm Definition (Barabanov norm) ~ · ~ is called Barabanov norm for E σ ˙ x = A σ x , iff ~ x ( t ) ~ = ~ Φ t x (0 − ) ~ ≤ e λ t ~ x (0 − ) ~ , Φ t ∈ S t 1 ∀ x 0 ∈ R n ∃ Φ t ∈ S t : ~ Φ t x 0 ~ = e λ t ~ x 0 ~ 2 In particular, every Barabanov norm is also a λ -Lyapunov norm, hence if λ < 0 we have an optimal Lyapunov function Theorem (Existence of Barabanov norm) Assume S is irreducible, i.e. SM ⊆ M implies M = ∅ or M = R n . Then the following are equivalent: The consistency projectors are product bounded 1 The Lyapunov exponent λ ( S ) is bounded 2 There exists a Barabanov norm with λ = λ ( S ) 3 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms

Introduction Evolution operator and its semigroup Lyapunov and Barabanov norm Conclusions Construction of Barabanov norm Construction of Barabanov norm similar as in (Wirth 2002, LAA): � � e − λ ( S ) t S t S ∞ := T ≥ 0 t ≥ T is a compact nontrivial semigroup, the limit semigroup. ~ x ~ := max { � Sx � | S ∈ S ∞ } is the sought Barabanov norm. Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Linear switched DAEs: Lyapunov exponents, a converse Lyapunov theorem, and Barabanov norms