On Stability theory for C 0 -Semigroups and applications Francis - - PowerPoint PPT Presentation

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

On Stability theory for C0-Semigroups and applications

Francis Félix Córdova Puma

Departamento de Ciências Exatas e Educação / UFSC

23 − 06 − 17

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Outline

1

C0-Semigroup

2

Stability Theory Strong Stability Polynomial stability Exponential Stability

3

Applications

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

u = etAx0 = ⇒

• u′

= Au in X u(0) = x0

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

u = etAx0 = ⇒

• u′

= Au in X u(0) = x0

• S(0) = Id,
• S(t)S(r) = S(t + r), ∀ t, r ≥ 0. (The semigroup property)

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

u = etAx0 = ⇒

• u′

= Au in X u(0) = x0

• S(0) = Id,
• S(t)S(r) = S(t + r), ∀ t, r ≥ 0. (The semigroup property)
• limt→0+ ||S(t)x − x|| = 0, ∀ x ∈ X. (Strong continuity)

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

u = etAx0 = ⇒

• u′

= Au in X u(0) = x0

• S(0) = Id,
• S(t)S(r) = S(t + r), ∀ t, r ≥ 0. (The semigroup property)
• limt→0+ ||S(t)x − x|| = 0, ∀ x ∈ X. (Strong continuity)
• ||S(t)||L(X) ≤ 1, ∀ t ≥ 0. (Contractions )

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Geration of Semigroups Ax = lim

t→0+

S(t)x − x t , D(A) = { x ∈ H : ∃ lim

t→0+

S(t)x − x t }

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Geration of Semigroups Ax = lim

t→0+

S(t)x − x t , D(A) = { x ∈ H : ∃ lim

t→0+

S(t)x − x t }

• are necessarily closed operators,
• have dense domain, and
• have their spectrum contained in some proper left

half-plane.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

The Hille - Yosida theorem A linear (unbounded) operator A is the infinitesimal generator

• f a C0-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, (λI − A)−1L(H) ≤ 1 λ

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

The Hille - Yosida theorem A linear (unbounded) operator A is the infinitesimal generator

• f a C0-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, (λI − A)−1L(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.

C0-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(AU, U)H ≤ 0, ∀ U ∈ D(A).

• (A − λ0I) is surjective for some λ0 > 0.

C0-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(AU, U)H ≤ 0, ∀ U ∈ D(A).

• (A − λ0I) is surjective for some λ0 > 0.

An operator satisfying the last two conditions is called maximally dissipative. Energy of system E(t) = 1 2 U 2

H ,

d dt E(t) = Re (AU, U)H

C0-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 C0-Semigroup of contractions S(t) ∈ L(H). (i) ∀F ∈ H, ∃!U ∈ D(A) : (λI − A) U = F (ii) U ≤ C F = ⇒ 0 ∈ ̺(A)

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Type of the Semigroup: ω0(A) := lim

t→∞

ln ||S(t)|| t . The spectral bound: ωσ(A) := sup { Re (λ) : λ ∈ σ(A)} , If σ(A) = ∅ −∞ , If σ(A) = ∅ For every ω > ω0(A) there exists Mω ≥ 1 such that S(t) ≤ Mωeωt, ∀ t ≥ 0. rσ (S(t)) = eω0t.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

dim H < ∞

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

dim H < ∞

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

dim H = ∞

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

dim H = ∞

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Semigroups and their generators dim H < ∞

• For every ω > ωσ(A) there exists Mω ≥ 1 such that

etA ≤ Mωeωt, ∀ t ≥ 0.

• ωσ(A) = ω0(A).
• For every λ ∈ C with Re λ > ωσ(A) we have

∞ e−λtetAdt = (λI − A)−1.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Semigroups and their generators dim H = ∞

• For every ω > ω0(A) there exists Mω ≥ 1 such that

S(t) L(H)≤ Mωeωt, ∀ t ≥ 0.

• ωσ(A) ≤ ω0(A).
• For every λ ∈ C with Re λ > ω0(A) we have

∞ e−λtS(t)dt = (λI − A)−1.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Principle of linear stability: ωσ(A) = ω0(A). This condition is important because it gives a practical criterion for exponential stability. Exponential stability : ω0(A) < 0 Superstability : ω0(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

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Principle of linear stability: ωσ(A) = ω0(A). Let A = A0 + B be the infinitesimal generator of a C0-Semigroup of operators in H. Assume that A0 is a normal and B is a bounded. Assume that there exists a number M > 0 and an integer n0 such that the following holds: If λ ∈ σ(A0) and |λ| > M − 1, then λ is an isolated eigenvalue of finite multiplicity. If |z| > M, then the number of eigenvalues of A0 in the unit disk centred at z (counted by multiplicity) does not exceed n0. Then the principle of linear stability holds.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Principle of linear stability: ωσ(A) = ω0(A). Let A = A0 + B be the infinitesimal generator of a C0-Semigroup of operators in H. Assume that A0 is a normal and B is a bounded. Assume that there exists a number M > 0 and an integer n0 such that the following holds: If λ ∈ σ(A0) and |λ| > M − 1, then λ is an isolated eigenvalue of finite multiplicity. If |z| > M, then the number of eigenvalues of A0 in the unit disk centred at z (counted by multiplicity) does not exceed n0. Then the principle of linear stability holds. RENARDY, M. On the Type of Certain C0-Semigroups.

• Commun. in Partial Diferential Equations, 18(7-8), pp

1299-1307, 1993.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Strong Stability

Let A be the infinitesimal generator of a C0-Semigroup etA of contractions in H with compact resolvent. Then etA is strongly stable ⇐ ⇒ iR ∩ σ(A) = ∅

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Strong Stability

Let A be the infinitesimal generator of a C0-Semigroup etA of contractions in H with compact resolvent. Then etA is strongly stable ⇐ ⇒ iR ∩ σ(A) = ∅ Huang Falun. Strong Asymptotic Stability of Linear Dynamical System in Banach Spaces (1993)

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Polynomial stability

∃ α > 0 such that: ||etAU0||H ≤ c tα ||U0||D(A), ∀t > 0.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Exponential Stability

∃ ω > 0 such that: ||etA||L(H) ≤ Me−ωt, ∀t ≥ 0. Let SA(t) be a C0-semigroup of contractions of linear operators

• n Hilbert space H with infinitesimal generator A. Then SA(t)

is exponentially stable if and only if iR ⊂ ρ(A) and lim sup

|λ|→+∞

||(iλI − A)−1||L(H) < ∞

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Exponential Stability

∃ ω > 0 such that: ||etA||L(H) ≤ Me−ωt, ∀t ≥ 0. Let SA(t) be a C0-semigroup of contractions of linear operators

• n Hilbert space H with infinitesimal generator A. Then SA(t)

is exponentially stable if and only if iR ⊂ ρ(A) and lim sup

|λ|→+∞

||(iλI − A)−1||L(H) < ∞ PRÜSS, J. On the Spectrum of C0-Semigroups. Transaction of the American Mathematical Society, 284, Nro. 2, pp 847-857, 1984. Exponential Stability

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Exponential Stability

∃ ω > 0 such that: ||etA||L(H) ≤ Me−ωt, ∀t ≥ 0. Let SA(t) be a C0-semigroup of contractions of linear operators

• n Hilbert space H with infinitesimal generator A. Then SA(t)

is exponentially stable if and only if iR ⊂ ρ(A) and lim sup

|λ|→+∞

||(iλI − A)−1||L(H) < ∞ PRÜSS, J. On the Spectrum of C0-Semigroups. Transaction of the American Mathematical Society, 284, Nro. 2, pp 847-857, 1984. Exponential Stability → Polynomial stability

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations Exponential Stability

∃ ω > 0 such that: ||etA||L(H) ≤ Me−ωt, ∀t ≥ 0. Let SA(t) be a C0-semigroup of contractions of linear operators

• n Hilbert space H with infinitesimal generator A. Then SA(t)

is exponentially stable if and only if iR ⊂ ρ(A) and lim sup

|λ|→+∞

||(iλI − A)−1||L(H) < ∞ PRÜSS, J. On the Spectrum of C0-Semigroups. Transaction of the American Mathematical Society, 284, Nro. 2, pp 847-857, 1984. Exponential Stability → Polynomial stability → Strong Stability

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Mixture of two linear isotropic one-dimensional elastic materials

• R. Quintanilla. Exponential decay in mixtures with localized

dissipative term, Applied mathematics Letters 18(2005) 1381-1388. U = (u1(x, t), u2(x, t))⊤. ρj is the mass density of each component. By ξ we denote the coefficient of the relative velocity. ρ1u1

tt

= a11u1

xx + a12u2 xx−ξu1 t +ξu2 t

ρ2u2

tt

= a12u1

xx + a22u2 xx+ξu1 t −ξu2 t

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Mixture of two linear isotropic one-dimensional elastic materials

• R. Quintanilla. Exponential decay in mixtures with localized

dissipative term, Applied mathematics Letters 18(2005) 1381-1388. U = (u1(x, t), u2(x, t))⊤. ρj is the mass density of each component. By ξ we denote the coefficient of the relative velocity. ρ1u1

tt

= a11u1

xx + a12u2 xx−ξu1 t +ξu2 t

ρ2u2

tt

= a12u1

xx + a22u2 xx+ξu1 t −ξu2 t

RUtt = AUxx − BUt

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Ternary Mixtures with Frictional Dissipation Filippo Dell’Oro and Jaime E. Muñoz Rivera. Stabilization

• f ternary mixtures with frictional dissipation, Asymptotic

Analysis, 3(2014), No 2, 1 - 28. U = (u1(x, t), u2(x, t), u3(x, t))⊤. ρj is the mass density of each component. By ξj we denote the coefficient of the relative velocity. ρ1u1

tt

= a11u1

xx + a12u2 xx + a13u3 xx−αu1+αu2−αu3 − ξ1u1 t

ρ2u2

tt

= a12u1

xx + a22u2 xx + a23u3 xx+αu1−αu2+αu3 − ξ2u2 t

ρ3u1

tt

= a13u1

xx + a23u2 xx + a33u3 xx−αu1+αu2−αu3 − ξ3u3 t

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Ternary Mixtures with Frictional Dissipation Filippo Dell’Oro and Jaime E. Muñoz Rivera. Stabilization

• f ternary mixtures with frictional dissipation, Asymptotic

Analysis, 3(2014), No 2, 1 - 28. U = (u1(x, t), u2(x, t), u3(x, t))⊤. ρj is the mass density of each component. By ξj we denote the coefficient of the relative velocity. ρ1u1

tt

= a11u1

xx + a12u2 xx + a13u3 xx−αu1+αu2−αu3 − ξ1u1 t

ρ2u2

tt

= a12u1

xx + a22u2 xx + a23u3 xx+αu1−αu2+αu3 − ξ2u2 t

ρ3u1

tt

= a13u1

xx + a23u2 xx + a33u3 xx−αu1+αu2−αu3 − ξ3u3 t

RUtt = AUxx − αNU − BUt

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Ternary Mixtures with Frictional Dissipation Dell’Oro-Rivera model with α = 0 U = (u1(x, t), u2(x, t), u3(x, t))⊤. ρj is the mass density of each component. By ξj ≥ 0 we denote the coefficient of the relative velocity, η1 = ξ1 + ξ2 and η2 = ξ2 + ξ3 . ρ1u1

tt

= a11u1

xx + a12u2 xx + a13u3 xx − ξ1u1 t − ξ2u2 t + η1u3 t

ρ2u2

tt

= a12u1

xx + a22u2 xx + a23u3 xx − ξ2u1 t − ξ3u2 t + η2u3 t

ρ3u1

tt

= a13u1

xx + a23u2 xx + a33u3 xx + η1u1 t + η2u2 t − (η1 + η2)u3 t

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Ternary Mixtures with Frictional Dissipation Dell’Oro-Rivera model with α = 0 U = (u1(x, t), u2(x, t), u3(x, t))⊤. ρj is the mass density of each component. By ξj ≥ 0 we denote the coefficient of the relative velocity, η1 = ξ1 + ξ2 and η2 = ξ2 + ξ3 . ρ1u1

tt

= a11u1

xx + a12u2 xx + a13u3 xx − ξ1u1 t − ξ2u2 t + η1u3 t

ρ2u2

tt

= a12u1

xx + a22u2 xx + a23u3 xx − ξ2u1 t − ξ3u2 t + η2u3 t

ρ3u1

tt

= a13u1

xx + a23u2 xx + a33u3 xx + η1u1 t + η2u2 t − (η1 + η2)u3 t

RUtt = AUxx − BUt ;

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Ternary Mixtures with Frictional Dissipation Dell’Oro-Rivera model with α = 0 U = (u1(x, t), u2(x, t), u3(x, t))⊤. ρj is the mass density of each component. By ξj ≥ 0 we denote the coefficient of the relative velocity, η1 = ξ1 + ξ2 and η2 = ξ2 + ξ3 . ρ1u1

tt

= a11u1

xx + a12u2 xx + a13u3 xx − ξ1u1 t − ξ2u2 t + η1u3 t

ρ2u2

tt

= a12u1

xx + a22u2 xx + a23u3 xx − ξ2u1 t − ξ3u2 t + η2u3 t

ρ3u1

tt

= a13u1

xx + a23u2 xx + a33u3 xx + η1u1 t + η2u2 t − (η1 + η2)u3 t

RUtt = AUxx − BUt ; Hip : ξ1 ξ2 ξ2 ξ3

• ≻ 0

= ⇒ Rank B = 2

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Stability to mixtures with boundary frictional damping    RUtt − AUxx = 0, U(0, t) = 0 and AUx(1, t) + BUt(1, t) = 0, U(x, 0) = U0(x), Ut(x, 0) = U1(x), (1) where U0(x) = 0 and U1(x) = (u1

1(x), u2 1(x), u3 1(x))T is given

by: ui

1(x) =

       if 0 ≤ x ≤ 0.4 10i(x − 0.4) if 0.4 ≤ x ≤ 0.5 10i(0.6 − x) if 0.5 ≤ x ≤ 0.6 if 0.6 ≤ x ≤ 1 . (2)

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Stability to mixtures with boundary frictional damping    RUtt − AUxx = 0, U(0, t) = 0 and AUx(1, t) + BUt(1, t) = 0, U(x, 0) = U0(x), Ut(x, 0) = U1(x), (1) where U0(x) = 0 and U1(x) = (u1

1(x), u2 1(x), u3 1(x))T is given

by: ui

1(x) =

       if 0 ≤ x ≤ 0.4 10i(x − 0.4) if 0.4 ≤ x ≤ 0.5 10i(0.6 − x) if 0.5 ≤ x ≤ 0.6 if 0.6 ≤ x ≤ 1 . (2) Numerical Experiments: Rafael Aleixo, Louise Reips

Departamento de Ciências Exatas e Educação, UFSC, Blumenau.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Si(t) Semigroups Families

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Si(t) Semigroups Families

• S1(t) R = A = I, B1

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Si(t) Semigroups Families

• S1(t) R = A = I, B1
• S2(t) R = A = I, B2

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Si(t) Semigroups Families

• S1(t) R = A = I, B1
• S2(t) R = A = I, B2
• S3(t) R = I, A3, B3

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Si(t) Semigroups Families

• S1(t) R = A = I, B1
• S2(t) R = A = I, B2
• S3(t) R = I, A3, B3

B1 =   2 1 1 1 1   ≻ 0 and B2 =   6 −2 3 −2 −2 −2 2   0. A3 =   2 1 1 1 1 1 4   ≻ 0 and B3 =   6 −2 3 −2 −2 −2 2   0.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Numerical Experiments t ∈ [0, 100]

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Numerical Experiments t ∈ [0, 100]

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Numerical Experiments t ∈ [0, 100]

• S1(t)x0 Exponential stability

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Numerical Experiments t ∈ [0, 100]

• S1(t)x0 Exponential stability
• S2(t)x0 Non-strong stability

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Numerical Experiments t ∈ [0, 100]

• S1(t)x0 Exponential stability
• S2(t)x0 Non-strong stability
• S3(t)x0 Polynomial stability

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

t ∈ [0, 1000]

• S1(t)x0 Exponential stability
• S2(t)x0 Non-strong stability
• S3(t)x0 Polynomial stability

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

t ∈ [0, 1000]

• S1(t)x0 Exponential stability
• S2(t)x0 Non-strong stability
• S3(t)x0 Polynomial stability

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Table: Summary of the experiments.

Experiment rank B (B, R−1A) OBSV Stability 1 √ √ Exponential stability 2 × × Oscillatory solutions 3 × √ Strong stability

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

c2utt = uxx , 0 < t , 0 < x < 1. u(t, 0) = 0, cux(t, 1) + ut(t, 1) = 0

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

c2utt = uxx , 0 < t , 0 < x < 1. u(t, 0) = 0, cux(t, 1) + ut(t, 1) = 0 u(0, x) = 0, ut(0, x) = ui(x), i = 1, 2, 3.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

c2utt = uxx , 0 < t , 0 < x < 1. u(t, 0) = 0, cux(t, 1) + ut(t, 1) = 0 u(0, x) = 0, ut(0, x) = ui(x), i = 1, 2, 3.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.25 −0.2 −0.15 −0.1 −0.05 displacement U1(0,x) Initial Conditions 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 displacement U1(0,x) Initial Conditions 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 displacement U1(0,x) Initial Conditions

Initial Conditions

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

c2utt = uxx , 0 < t , 0 < x < 1. u(t, 0) = 0, cux(t, 1) + ut(t, 1) = 0 u(0, x) = 0, ut(0, x) = ui(x), i = 1, 2, 3.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

c2utt = uxx , 0 < t , 0 < x < 1. u(t, 0) = 0, cux(t, 1) + ut(t, 1) = 0 u(0, x) = 0, ut(0, x) = ui(x), i = 1, 2, 3. Solutions

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

c2utt = uxx , 0 < t , 0 < x < 1.

0.5 1 1.5 2 2.5 3 0.005 0.01 0.015 0.02 0.025 0.03 0.035 time Energy System Energy 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 time Energy System Energy 0.5 1 1.5 2 2.5 3 0.01 0.02 0.03 0.04 0.05 0.06 0.07 time Energy System Energy

Energy of System The Semigroup is exponentially stable!

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Mixture of n elastic materials with frictional damping ρ1u1

tt

= a11u1

xx + ... + a1JuJ xx + a1(J+1)uJ+1 xx

+ ... + a1nun

xx − b1u1 t

ρ2u2

tt

= a21u1

xx + ... + a2JuJ xx + a2(J+1)uJ+1 xx

+ ... + a2nun

xx − b2u2 t

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ρJuJ

tt

= aJ1u1

xx + ... + aJJuJ xx + aJ(J+1)uJ+1 xx

+ ... + aJnun

xx − bJuJ t

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ρnun

tt

= an1u1

xx + ... + anJuJ xx + an(J+1)uJ+1 xx

+ ... + annun

xx

RUtt = AUxx − BUt , U(0) = U(l) = 0 .

Stabilization Results

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Mixture of n Thermoelastic Materials ρ1u1

tt

= a11u1

xx + a12u2 xx + ... + a1nun xx − b1θx

ρ2u2

tt

= a12u1

xx + a22u1 xx + ... + a2nun xx − b2θx

· · · · · · · · · · · · · · · · · · · · · · · · · · · ρnun

tt

= a1nu1

xx + a2nu2 xx + ... + annun xx − bnθx

ςθt = κθxx − b1u1

xt − b2u2 xt − ... − bnun xt

β = (b1, b2, ..., bn)⊤ = 0 RUtt = AUxx − βθx ςθt = κθxx − β⊤Uxt U(0) = U(l) = 0 , θ(0) = θ(l) = 0

Stabilization Results

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

The Hilbert space Hf = [H1

0(0, l)]n × [L2(0, l)]n ,

||(U, V)||2

Hf =

l U∗

xAUxdx +

l V ∗RVdx The Frictional Operator Af Af U V

• =
• V

R−1AUxx

• :=A0U

+

• −R−1BV
• :=BU

(3) with domain D(Af) = [H1

0(0, l) ∩ H2(0, l)]n × [H1 0(0, l)]n.

RESULT:

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

The Hilbert space Hf = [H1

0(0, l)]n × [L2(0, l)]n ,

||(U, V)||2

Hf =

l U∗

xAUxdx +

l V ∗RVdx The Frictional Operator Af Af U V

• =
• V

R−1AUxx

• :=A0U

+

• −R−1BV
• :=BU

(3) with domain D(Af) = [H1

0(0, l) ∩ H2(0, l)]n × [H1 0(0, l)]n.

RESULT: Af is the infinitesimal generator of a C0-semigroup of contractions, and satisfies the principle linear stability.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Stabilization Result Theorem A Let etAf be a C0-semigroup generated by Af. etAf is exponentially stable if and only if iR ⊂ ̺(Af). Otherwise, the operator Af has infinite imaginary eigenvalues.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

The Hilbert space Hι = [H1

0(0, l)]n × [L2(0, l)]n × [L2(0, l)] ,

||(U, V, θ)||2

Hι =

l U∗

xAUxdx +

l V ∗RVdx + ς l |θ|2dx, The Thermoelastic operator Aι Aι   U V θ   =   V R−1AUxx − R−1βθx ς−1κθxx − ς−1β⊤Vx   (4) D(Aι) = [H1

0(0, l) ∩ H2(0, l)]n × [H1 0(0, l)]n × [H1 0(0, l) ∩ H2(0, l)].

RESULT:

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

The Hilbert space Hι = [H1

0(0, l)]n × [L2(0, l)]n × [L2(0, l)] ,

||(U, V, θ)||2

Hι =

l U∗

xAUxdx +

l V ∗RVdx + ς l |θ|2dx, The Thermoelastic operator Aι Aι   U V θ   =   V R−1AUxx − R−1βθx ς−1κθxx − ς−1β⊤Vx   (4) D(Aι) = [H1

0(0, l) ∩ H2(0, l)]n × [H1 0(0, l)]n × [H1 0(0, l) ∩ H2(0, l)].

RESULT: Aι is the infinitesimal generator of a C0-semigroup of contractions.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Stabilization Result Theorem B The Semigroup associated to the thermoelastic problem, etAι , is exponentially stable if and only if iR ⊂ ̺(A). Otherwise, the

• perator Aι has infinite imaginary eigenvalues.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Stability Exponential for Semigroup associated with Mixture Materials I : iR ⊂ ̺(A) R : ||R(iλ, A)|| ≤ C Hf :

Mixture with frictional damping

Hι :

Mixture Thermoelastic

k = n − J E : etA Exponential Stability R Hi E I Hf : dim spam

• Nj, NjM, ..., NjMk−1; N = [Nj]J×k
• = k

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Stability Exponential for Semigroup associated with Mixture Materials I : iR ⊂ ̺(A) R : ||R(iλ, A)|| ≤ C Hf :

Mixture with frictional damping

Hι :

Mixture Thermoelastic

k = n − J E : etA Exponential Stability R Hi E I Hf : dim spam

• Nj, NjM, ..., NjMk−1; N = [Nj]J×k
• = k

Hι : dim spam

• β⊤, β⊤M, ..., β⊤Mn−1

= n

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

(iλ − Af)U = F U ∈ D(Af), F ∈ Hf and λ ∈ R.

• Lemma1. For all ǫ > 0 there exists cǫ > 0 such that

l |V J|2 + |UJ

x |2dx ≤ ǫ||U||2 H + cǫ||F||2 H.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

(iλ − Af)U = F U ∈ D(Af), F ∈ Hf and λ ∈ R.

• Lemma1. For all ǫ > 0 there exists cǫ > 0 such that

l |V J|2 + |UJ

x |2dx ≤ ǫ||U||2 H + cǫ||F||2 H.

• Lemma2. For all ǫ > 0 there exists cǫ > 0 such that

l |NMmV k|2 + |NMmUk

x |2dx ≤ ǫ||U||2 Hf + cǫ||F||2 Hf , m < k.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

If dim spam

• Nj, NjM, ..., NjMk−1; N = [Nj]J×k
• = k

= ⇒ UHf ≤ CFHf ⇒ etAf is exponentially stable.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Mixture of three elastic materials with partial frictional damping ρ1u1

tt

= au1

xx + bu2 xx + bu3 xx − gu1 t

ρ2u2

tt

= bu1

xx + du2 xx + eu3 xx

ρ2u3

tt

= bu1

xx + eu2 xx + du3 xx

A ≻ 0 ↔ a > 0, ad > b2 and a(d + e)(d − e) > 2db2

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Mixture of three elastic materials with partial frictional damping ρ1u1

tt

= au1

xx + bu2 xx + bu3 xx − gu1 t

ρ2u2

tt

= bu1

xx + du2 xx + eu3 xx

ρ2u3

tt

= bu1

xx + eu2 xx + du3 xx

A ≻ 0 ↔ a > 0, ad > b2 and a(d + e)(d − e) > 2db2 J = 1 and k = 2 N = R−1

J AS =

• b

ρ1 b ρ1

• 1×2

M = R−1

k Ak =

• d

ρ2 e ρ2 e ρ2 d ρ2

• 2×2

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

= ⇒ N NM

• 2×2

= b ρ1

• 1

1

d+e ρ2 d+e ρ2

• 2×2

and dim spam {N, NM} = 1. σ(M) = d + e ρ2 , d − e ρ2

• , λν =
• d − e

ρ2

• ν

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

= ⇒ N NM

• 2×2

= b ρ1

• 1

1

d+e ρ2 d+e ρ2

• 2×2

and dim spam {N, NM} = 1. σ(M) = d + e ρ2 , d − e ρ2

• , λν =
• d − e

ρ2

• ν

Uν = (Uν, Vν) Uν =   sin νx − sin νx   , Vν =   iλν sin νx −iλν sin νx   iλνUν − AfUν = 0.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

2u1

tt

= 8u1

xx + 3u2 xx + 1u4 xx − 1u1 t

u2

tt

= 3u1

xx + 10u2 xx + 7u3 xx + 1u4 xx

u3

tt

= 7u2

xx + 8u3 xx

u4

tt

= 1u1

xx + 1u2 xx + 1u4 xx

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

2u1

tt

= 8u1

xx + 3u2 xx + 1u4 xx − 1u1 t

u2

tt

= 3u1

xx + 10u2 xx + 7u3 xx + 1u4 xx

u3

tt

= 7u2

xx + 8u3 xx

u4

tt

= 1u1

xx + 1u2 xx + 1u4 xx

A =     8 3 1 3 10 7 1 7 8 1 1 1     , B =     1     , R =     2 1 1 1     , then J = 1, k = 3 , RUtt = AUxx − BUt

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

M = R−1

k Ak =

  10 7 1 7 8 1 1   , N = R−1

J AS =

• 1.5

0.5

• .

= ⇒ det   N NM NM2   = det   1.5 0.5 15.5 10.5 2 230.5 192.5 17.5   = 0

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

then dim spam {N, NM, NM2} = 3. Applying the Theorem A, we have that the semigroup associated is exponentially stable.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

Theorem Let us denote by W = R−1A and let B be a positive semidefinite matrix, then the following statements are equivalents. SA(t) is exponentially stable. SA(t) is strongly stable. Denoting by Bj the j-row vector of B then dim span

• Bj, BjW, BjW2, ..., BjWn−1, j = 1, 2..., n
• = n.

F . F . Córdova Puma and J. E. Muñoz Rivera, The lack of Polynomial stability to mixtures with frictional dissipation,

• J. Math. Anal. Appl. (2017)

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

References

1.

• M. Alves, J.E. Muñoz Rivera, M. Sepúlveda and O. Vera

Villagrán The lack of exponential stability in certain transmission problems with localized Kelvin-Voigt Dissipation, SIAM J. APPL. MATH. Vol. 74, No. 2, pp. 345-365. 2014

2.

• R. Quintanilla, M. Alves, J.E. Muñoz Rivera Exponential

decay in a thermoelastic mixture of solids, INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES 46, pp. 1659-1666. 2009. LIU Z. and ZHENG S. Semigroups associated with dissipative systems. CHAPMAN and HALL/CRC, 1999.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

• A. Borichev and Y. Tomilov. Optimal polynomial decay of

functions and operator semigroups. mathematische

• Annalen. Vol. 347. 2 (2009) 455-478.

C0-Semigroup Stability Theory Applications Stabilization Results - Ideas of demonstrations

OBRIGADO!