C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations On Stability theory for C 0 -Semigroups and applications Francis Félix Córdova Puma Departamento de Ciências Exatas e Educação / UFSC 23 − 06 − 17
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Outline C 0 -Semigroup 1 2 Stability Theory Strong Stability Polynomial stability Exponential Stability Applications 3
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations � u ′ in = A u X u = e t A x 0 = ⇒ u ( 0 ) = x 0
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations � u ′ in = A u X u = e t A x 0 = ⇒ u ( 0 ) = x 0 - S ( 0 ) = I d , - S ( t ) S ( r ) = S ( t + r ) , ∀ t , r ≥ 0. (The semigroup property)
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations � u ′ in = A u X u = e t A x 0 = ⇒ u ( 0 ) = x 0 - S ( 0 ) = I d , - S ( t ) S ( r ) = S ( t + r ) , ∀ t , r ≥ 0. (The semigroup property) - lim t → 0 + || S ( t ) x − x || = 0 , ∀ x ∈ X . (Strong continuity)
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations � u ′ in = A u X u = e t A x 0 = ⇒ u ( 0 ) = x 0 - S ( 0 ) = I d , - S ( t ) S ( r ) = S ( t + r ) , ∀ t , r ≥ 0. (The semigroup property) - lim t → 0 + || S ( t ) x − x || = 0 , ∀ x ∈ X . (Strong continuity) - || S ( t ) || L ( X ) ≤ 1 , ∀ t ≥ 0 . (Contractions )
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Geration of Semigroups S ( t ) x − x S ( t ) x − x lim , D ( A ) = { x ∈ H : ∃ lim Ax = } t t t → 0 + t → 0 +
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Geration of Semigroups S ( t ) x − x S ( t ) x − x lim , D ( A ) = { x ∈ H : ∃ lim Ax = } t t t → 0 + t → 0 + - are necessarily closed operators, - have dense domain, and - have their spectrum contained in some proper left half-plane.
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations The Hille - Yosida theorem A linear (unbounded) operator A is the infinitesimal generator of a C 0 -Semigroup of contractions S ( t ) , t ≥ 0, if and only if (i) A is closed and D ( A ) = H (ii) the resolvent set ̺ ( A ) of A contains R + and for every λ > 0, � ( λ I − A ) − 1 � L ( H ) ≤ 1 λ
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations The Hille - Yosida theorem A linear (unbounded) operator A is the infinitesimal generator of a C 0 -Semigroup of contractions S ( t ) , t ≥ 0, if and only if (i) A is closed and D ( A ) = H (ii) the resolvent set ̺ ( A ) of A contains R + and for every λ > 0, � ( λ I − A ) − 1 � L ( H ) ≤ 1 λ In the general case the Hille - Yosida theorem is mainly of theoretical importance since the estimates on the powers of the resolvent operator that appear in the statement of the theorem can usually not be checked in concrete examples.
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations The Lumer - Phillips theorem A generates a contraction semigroup in H - Hilbert if and only if - D ( A ) is dense in H - A is closed. - A is dissipative: Re ( A U , U ) H ≤ 0 , ∀ U ∈ D ( A ) . - ( A − λ 0 I ) is surjective for some λ 0 > 0.
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations The Lumer - Phillips theorem A generates a contraction semigroup in H - Hilbert if and only if - D ( A ) is dense in H - A is closed. - A is dissipative: Re ( A U , U ) H ≤ 0 , ∀ U ∈ D ( A ) . - ( A − λ 0 I ) is surjective for some λ 0 > 0. An operator satisfying the last two conditions is called maximally dissipative. Energy of system E ( t ) = 1 d 2 � U � 2 dt E ( t ) = Re ( A U , U ) H H ,
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Collorary Let A be a linear operator with dense domain D ( A ) in H . If A is dissipative and 0 ∈ ̺ ( A ) , then A is the infinitesimal generator of a C 0 -Semigroup of contractions S ( t ) ∈ L ( H ) . (i) ∀ F ∈ H , ∃ ! U ∈ D ( A ) : ( λ I − A ) U = F (ii) � U �≤ C � F � 0 ∈ ̺ ( A ) = ⇒
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Type of the Semigroup: ln || S ( t ) || ω 0 ( A ) := lim . t t →∞ The spectral bound: � sup { Re ( λ ) : λ ∈ σ ( A ) } If σ ( A ) � = ∅ , ω σ ( A ) := If σ ( A ) = ∅ −∞ , For every ω > ω 0 ( A ) there exists M ω ≥ 1 such that � S ( t ) � ≤ M ω e ω t , ∀ t ≥ 0 . r σ ( S ( t )) = e ω 0 t .
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations dim H < ∞
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations dim H < ∞
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations dim H = ∞
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations dim H = ∞
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Semigroups and their generators dim H < ∞ - For every ω > ω σ ( A ) there exists M ω ≥ 1 such that � e tA �≤ M ω e ω t , ∀ t ≥ 0 . - ω σ ( A ) = ω 0 ( A ) . - For every λ ∈ C with Re λ > ω σ ( A ) we have � ∞ e − λ t e tA dt = ( λ I − A ) − 1 . 0
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Semigroups and their generators dim H = ∞ - For every ω > ω 0 ( A ) there exists M ω ≥ 1 such that � S ( t ) � L ( H ) ≤ M ω e ω t , ∀ t ≥ 0 . - ω σ ( A ) ≤ ω 0 ( A ) . - For every λ ∈ C with Re λ > ω 0 ( A ) we have � ∞ e − λ t S ( t ) dt = ( λ I − A ) − 1 . 0
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Principle of linear stability: ω σ ( A ) = ω 0 ( A ) . This condition is important because it gives a practical criterion for exponential stability. Exponential stability : ω 0 ( A ) < 0 Superstability : ω 0 ( A ) = −∞ It is well know that the Principle of linear stability holds for wide classes of semigroups: - Uniformly Continuous Semigroups - Analytic Semigroups - Eventually Compact Semigroups
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Principle of linear stability: ω σ ( A ) = ω 0 ( A ) . Let A = A 0 + B be the infinitesimal generator of a C 0 -Semigroup of operators in H . Assume that A 0 is a normal and B is a bounded. Assume that there exists a number M > 0 and an integer n 0 such that the following holds: If λ ∈ σ ( A 0 ) and | λ | > M − 1, then λ is an isolated eigenvalue of finite multiplicity. If | z | > M , then the number of eigenvalues of A 0 in the unit disk centred at z (counted by multiplicity) does not exceed n 0 . Then the principle of linear stability holds.
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Principle of linear stability: ω σ ( A ) = ω 0 ( A ) . Let A = A 0 + B be the infinitesimal generator of a C 0 -Semigroup of operators in H . Assume that A 0 is a normal and B is a bounded. Assume that there exists a number M > 0 and an integer n 0 such that the following holds: If λ ∈ σ ( A 0 ) and | λ | > M − 1, then λ is an isolated eigenvalue of finite multiplicity. If | z | > M , then the number of eigenvalues of A 0 in the unit disk centred at z (counted by multiplicity) does not exceed n 0 . Then the principle of linear stability holds. RENARDY, M. On the Type of Certain C 0 -Semigroups. Commun. in Partial Diferential Equations, 18(7-8), pp 1299-1307, 1993.
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Strong Stability Let A be the infinitesimal generator of a C 0 -Semigroup e t A of contractions in H with compact resolvent. Then e t A is strongly stable ⇐ ⇒ i R ∩ σ ( A ) = ∅
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Strong Stability Let A be the infinitesimal generator of a C 0 -Semigroup e t A of contractions in H with compact resolvent. Then e t A is strongly stable ⇐ ⇒ i R ∩ σ ( A ) = ∅ Huang Falun. Strong Asymptotic Stability of Linear Dynamical System in Banach Spaces (1993)
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Polynomial stability ∃ α > 0 such that: || e tA U 0 || H ≤ c t α || U 0 || D ( A ) , ∀ t > 0 .
C 0 -Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Exponential Stability ∃ ω > 0 such that: || e tA || L ( H ) ≤ Me − ω t , ∀ t ≥ 0 . Let S A ( t ) be a C 0 -semigroup of contractions of linear operators on Hilbert space H with infinitesimal generator A . Then S A ( t ) is exponentially stable if and only if || ( i λ I − A ) − 1 || L ( H ) < ∞ and lim sup i R ⊂ ρ ( A ) | λ |→ + ∞
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