Stabilization and asymptotic behavior of a generalized telegraph - PowerPoint PPT Presentation

Stabilization and asymptotic behavior of a generalized telegraph equation Serge Nicaise Université de Valenciennes et du Hainaut-Cambresis Laboratoire de Mathematiques et leurs Applications de Valenciennes, LAMAV Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 1 / 32

Outline Introduction 1 Well Posedness 2 Strong stability 3 Energy decay 4 A spectral analysis 5 Optimality 6 Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 2 / 32

Introduction Motivation Consider the evolution system:  V t + gV + aI x + kW = 0 in Ω = ( 0 , 1 ) × ( 0 , + ∞ ) ,   I t + rI + bV x = 0 in Ω × ( 0 , + ∞ ) ,    W t + cW = V in Ω × ( 0 , + ∞ ) , (1) V ( 0 , · ) = V ( 1 , · ) = 0 , on ( 0 , + ∞ ) ,     V ( x , 0 ) = V 0 ( x ) , I ( x , 0 ) = I 0 ( x ) , W ( x , 0 ) = W 0 ( x ) , in Ω .  V electric potential, I electric current, W auxiliary variable (representing the non local effects) Coupling between the usual telegraph equation and a first order differential equation of parabolic type. Such problem was first introduced in S. Imperiale and P . Joly. Error estimates for 1D asymptotic models in coaxial cables with non-homogeneous cross-section. Adv. Appl. Math. Mech. , 4(6):647–664, 2012. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 3 / 32

Introduction Particular cases If r = k = 0, eliminating I , we find the wave equation with internal damping: V tt − abV xx + gV t = 0 , Exponential decay if ab > 0 and g > 0. If r = g = 0 by eliminating I and setting p = W t , we arrive at  V tt − abV xx + kp = 0 , in Ω × ( 0 , + ∞ ) ,   p t + cp = V t in Ω × ( 0 , + ∞ ) ,  V ( 0 , · ) = V ( L , · ) = 0 , on ( 0 , + ∞ ) ,   V ( x , 0 ) = V 0 ( x ) , p ( x , 0 ) = V 0 ( x ) − cW 0 ( x ) , in Ω .  Coupling between the wave equation in V with a first order differential equation in p for which a polynomial stability was established in S. Nicaise. Stabilization and asymptotic behavior of dispersive medium models. Systems Control Lett. , 61(5):638–648, 2012. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 4 / 32

Introduction Main questions Strong stability of the solution. Uniform Stability. Polynomial Stability. Optimality. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 5 / 32

Well Posedness The energy space We assume that all the coefficients involved in (1) are in L ∞ (Ω) , real valued and non-negative. Moreover, we suppose that there exists a positive constant δ such that a ≥ δ, b ≥ δ, c ≥ δ, k + g ≥ δ, a. e. in Ω . (2) These assumptions are in agreement with physical settings. Consider the energy space H = L 2 (Ω) 3 , equipped with the inner product � � ( V , I , W ) ⊤ , ( V ∗ , I ∗ , W ∗ ) ⊤ � V ∗ + β I ¯ I ∗ + γ W ¯ ( α V ¯ W ∗ ) dx , H = (3) Ω with α, β, γ ∈ L ∞ (Ω) fixed appropriately such that α ( x ) ≥ δ 0 , β ( x ) ≥ δ 0 , γ ( x ) ≥ δ 0 , a. e. in Ω , (4) for some δ 0 > 0. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 6 / 32

Well Posedness Cauchy problem/Maximal dissipativity Problem (1) can be written as a Cauchy problem: ˙ u = A u , u ( 0 ) = u 0 , (5) with D ( A ) = H 1 0 (Ω) × H 1 (Ω) × L 2 (Ω) , and       V aI x + gV + kW V  := −  , ∀  ∈ D ( A ) . A I bV x + rI I    W cW − V W There exist α, β, γ ∈ L ∞ (Ω) satisfying (4) such that A is m-dissipative, i.e., λ I − A is surjective for some λ > 0 and ℜ ( A U , U ) ≤ − 1 � ( α g | V | 2 + 2 β r | I | 2 + γ c | W | 2 ) ≤ 0 . 2 Ω Using Lumer-Phillips’ thm, A generates a C 0 -semigroup of contraction. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 7 / 32

Strong stability Arendt-Batty’s thm One simple way to prove the strong stability is to use the following theorem. Theorem (Arendt-Batty) Let X be a reflexive Banach space and ( T ( t )) t ≥ 0 be a semigroup generated by A on X. Assume that ( T ( t )) t ≥ 0 is bounded and that no eigenvalues of A lies on the imaginary axis. If σ ( A ) ∩ i R is countable, then ( T ( t )) t ≥ 0 is stable. Since the resolvent of our operator is not compact, we have to analyze the full spectrum on the imaginary axis. W. Arendt and C. J. K. Batty. Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. , 305(2):837–852, 1988. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 8 / 32

Strong stability Eigenvalue problem Lemma (Le 1) For all ξ ∈ R ∗ := R \ { 0 } , we have ker ( i ξ − A ) = { 0 } . If r is not identically equal to 0 , we have ker A = { 0 } . otherwise 0 is an eigenvalue of A whose associated eigenvector is ( 0 , 1 , 0 ) ⊤ . Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 9 / 32

Strong stability Proof Let ξ ∈ R and U = ( V , I , W ) ⊤ ∈ D ( A ) be such that ( i ξ − A ) U = 0 , or equivalently satisfying  i ξ V + aI x + gV + kW = 0 ,  i ξ I + bV x + rI = 0 , (6) i ξ W + cW − V = 0 .  Hence by the dissipativeness of A , we get gV = rI = W = 0 . Coming back to the third identity of (6), we obtain that V = 0 and by the first identity of (6) I x = 0 (hence I is constant). By the second identity of (6) we obtain ( i ξ + r ) I = 0 . Hence ξ � = 0 ⇒ I = 0; otherwise if ξ = 0, this identity reduces to rI = 0 . If r � = 0 ⇒ I = 0, otherwise if r ≡ 0: I = constant allowed. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 10 / 32

Strong stability Surjectivity Lemma (Le 2) For all ξ ∈ R ∗ , i ξ − A is surjective. If r is not identically equal to 0 , then A is surjective, otherwise the range R ( A ) of A is equal to � H 0 := { ( V , I , W ) ⊤ ∈ H : β I = 0 } . Ω The proof is based on a compact perturbation argument and the use of the previous Lemma. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 11 / 32

Strong stability Strong stability of A In view of the previous result, if r = 0, we need to introduce the operator A 0 from H 0 into itself defined by D ( A 0 ) = D ( A ) ∩ H 0 and A 0 U = A U , ∀ U ∈ D ( A 0 ) . Corollary (Coro 3) If r is not identically equal to 0 , A is strongly stable, otherwise A 0 is strongly stable. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 12 / 32

Energy decay Frequency domain approach: exponential decay Lemma (Pruss, Huang) A C 0 semigroup ( e tA ) t ≥ 0 of contractions on a Hilbert space H is exponentially stable, i.e., satisfies || e tA U 0 || ≤ C e − ω t || U 0 || H , ∀ U 0 ∈ H , ∀ t ≥ 0 , for some positive constants C and ω if � β ∈ R � � � ρ ( A ) ⊃ ≡ i R , i β (7) � ( i β − A ) − 1 � L ( H ) < ∞ . sup (8) β ∈ R Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 13 / 32

Energy decay Frequency domain approach: polynomial decay Lemma (Borichev-Tomilov) A C 0 semigroup ( e tA ) t ≥ 0 of contractions on a Hilbert space H satisfies || e tA U 0 || ≤ C t − 1 l || U 0 || D ( A ) , ∀ U 0 ∈ D ( A ) , ∀ t > 1 , for some constant C > 0 and for some positive integer l if (7) holds and if 1 β l � ( i β − A ) − 1 � L ( H ) < ∞ . lim sup (9) | β |→∞ A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Math. Ann. , 347(2):455–478, 2010. Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 14 / 32

Energy decay Resolvent estimate Since condition (7) was already treated, it remains to analyze the behaviour of the resolvent on the imaginary axis. Let us start with the exponential decay. Lemma (Le 4) In addition to the previous assumption (2), suppose that r ∈ W 1 , ∞ (Ω) and that there exists δ 1 > 0 such that r + g ≥ δ 1 a. e. in Ω . (10) Then the resolvent of the operator of A satisfies condition (8), i.e., � ( i β − A ) − 1 � L ( H ) < ∞ . sup β ∈ R Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 15 / 32

Energy decay Proof We use a contradiction argument, i.e., we suppose that (8) is false. Then there exist a sequence of real numbers ξ n → + ∞ and a sequence of vectors U n = ( V n , I n , W n ) ⊤ in D ( A ) such that � U n � H = 1 , (11) � ( i ξ n + g ) V n + aI n , x + kW n � → 0 , (12) � ( i ξ n + r ) I n + bV n , x � → 0 , (13) � ( i ξ n + c ) W n − V n � → 0 . (14) By the dissipativeness of A , we directly have √ gV n → 0 in L 2 (Ω) , (15) √ rI n → 0 in L 2 (Ω) , (16) W n → 0 in L 2 (Ω) . (17) Serge Nicaise (LAMAV) Stabilization telegraph equation 18 June 2015, MIS 2015 16 / 32