Exponential Stability of Semigroups Related to Operator Models in ...

Mathematical Notes, vol. 73, no. 5, 2003, pp. 618–624. Translated from Matematicheskie Zametki, vol. 73, no. 5, 2003, pp. 657–664. c Original Russian Text Copyright 2003 by R. O. Griniv, A. A. Shkalikov.

Exponential Stability of Semigroups Related to Operator Models in Mechanics R. O. Griniv and A. A. Shkalikov Received October 28, 2002

Abstract—In this paper, we consider equations of the form x ¨ + B x˙ + Ax = 0 , where x = x(t) is a function with values in the Hilbert space H , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in H . The linear operator T generating the C0 -semigroup in the energy space H1 × H is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms. Key words: self-adjoint operator, C0 -semigroup, exponential stability, energy space, dissipative operator, Hilbert space, generalized spectrum.

1. INTRODUCTION Linearized equations arising in various problems of elasticity theory and hydromechanics often take the following form: x ¨ + B x˙ + Ax = 0, (1) where x(t) is a function with values in the Hilbert space H and A and B are (unbounded) operators in H . In the general case (see [1, 2]), the operator A is self-adjoint and uniformly positive (it corresponds to the potential energy of the system) and the operator B is symmetric and nonnegative (it corresponds to a damping). The energy of the solution x(t) to Eq. (1) is defined by the functional E(x(t)) :=

1 (x ˙ 2 + A1/2 x2 ). 2

Equation (1) implies that for B ≥ 0 , the energy of the system decreases in time: d E(x(t)) = −(B x(t), ˙ x(t)) ˙ ≤ 0. dt For a number of specific problems in mechanics, one can prove that energy decreases exponentially. In the literature, particular attention is given to abstract models with this property (see, for example, [1–6] and the references therein). The present paper is devoted to the same problem. We will give a relatively simple proof for the exponential decrease of energy under assumptions on the operators A and B that are much less restrictive than those considered up to now. We denote by Hθ , θ ∈ R , the scale of Hilbert spaces generated by a uniformly positive and self-adjoint operator A . Hence for all θ > 0 , the space Hθ coincides with the domain D(Aθ/2 ) of the operator Aθ/2 equipped by the norm xθ := Aθ/2 x , and for θ < 0 this space coincides with the completion of the initial space H in this norm. We can rewrite Eq. (1) in the following form ˙ (2) x(t) = T0 x(t), 618

0001-4346/2003/7356-0618$25.00

c 2003 Plenum Publishing Corporation

EXPONENTIAL STABILITY OF SEMIGROUPS RELATED TO OPERATOR MODELS

where x(t) =

x1 (t) x2 (t)

T0 :=

619

is a function with value in the energy space E := H1 × H , and

0 I −A −B

D(T0 ) =

,

x1 x2

,

x1 ∈ D(A),

x2 ∈ D(B) .

(3)

It can be easily seen that x(t) is a solution of Eq. (2) if and only if x1 (t) is a solution of Eq. (1) and x2 (t) = x˙ 1 (t) ; then we have x2E = 2E(x1 ) . It is well known (see [6]) that if B ≥ 0 and the operator A dominates B in the usual sense, i.e., if D(B) ⊃ D(A),

(4)

the closure T of the operator T0 is a maximal dissipative operator in the space E . Hence the operator T generates the contractive C0 -semigroup exp(tT ) in H , and any solution of Eq. (2) has the form x(t) = exp(tT )x0 , where x0 = x(0) is the initial state. Hence E(x) decreases exponentially for any solution of the Eq. (1) if and only if the semigroup exp(tT ) is exponentially stable , i.e., if the exponential type ω(T ) of the C0 -semigroup exp(tT ) (see [7, 8]) defined by the equality 1 ω(T ) := lim sup exp(tT )E , t→∞ t is negative. In [1–3, 5], it was proved that ω(T ) < 0 if, besides Eq. (4), one of the following conditions hold: (a) there exists an ε > 0 such that Bx ≥ εA1/2 x for all x ∈ D(A1/2 ) ; (b) the operators B and Aα are comparable for α ∈ [1/2, 1] , i.e., c1 (Aα x, x) ≤ (Bx, x) ≤ c2 (Aα x, x)

for all

x ∈ D(Aα/2 ),

where c1 , c2 > 0.

Actually, under such assumptions a stronger statement was proved: the semigroup exp(tT ) is analytic, and in this case the exponential type ω(T ) coincides with the spectral abscissa s(T ) := sup{Re λ | λ ∈ σ(T )}, where σ(T ) is the spectrum of the operator T . (Note that the inequality s(T ) ≤ ω(T ) holds in the general case.) In [4], it was proved that the coincidence of s(T ) and ω(T ) also occurs if B and Aα for any α ∈ (0, 1/2) ; however, in this case, the semigroup is not analytic and belongs to the Gevrey class of order 1/2α . The analyticity of the semigroup exp(tT ) for sectorial operators B was studied in [6]. In this paper, we relax the condition (4) (because it does not hold for some relevant problems) and replace it by a weaker assumption: the operator A dominates B in the sense of quadratic forms. That is, the lineal D(B) ∩ H1 is dense in the space H1 = D(A1/2 ) , and the following estimate holds: for all x ∈ D(B) ∩ H1 , (5) |(Bx, x)| ≤ c(A1/2 x, A1/2 x) where c > 0 is a constant. This condition is equivalent to the assumption that the operator B has an extension by continuity as an operator from H1 to H−1 . Indeed, the estimate (5) implies that the operator K = A−1/2 BA−1/2 is densely defined in H and its numerical range is bounded. Hence K can be extended by continuity to a bounded operator from H1 to H−1 . The converse assertion is obvious. Note that in the case B ≥ 0 , the domination property in the sense of quadratic 1/2 forms is equivalent to the inclusion D(BF ) ⊃ D(A1/2 ) , where BF is the Friedrichs extension of the operator B (see [6]). MATHEMATICAL NOTES

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R. O. GRINIV, A. A. SHKALIKOV

Let us explain why condition (5) is less restrictive than (4). It is well known (see, for example, [9]), that condition (4) implies the boundedness of the operator B : H2 → H ; hence B ∗ : H−2 → H . Taking into account the fact that the operator B ⊂ B ∗ is symmetric, the interpolation theorem (see, for example, [10, Chap. I]) implies that the operator B : H1 → H−1 is also bounded. Another purpose of this paper is to prove that the semigroup exp(tT ) is well defined if the operator A dominates the operator B ≥ 0 in the sense of quadratic forms, and the condition B 0 implies that the exponential type ω(T ) of the semigroup is negative. Moreover, as was proved in Remark 1, in some sense these conditions are necessary for the exponential stability of the semigroup. It is obvious that the condition B 0 does not ensure the analyticity of the semigroup, and we cannot say that ω(T ) coincides with the spectral abscissa s(T ) . However, we will obtain an efficient estimate for the constant ω(T ) . 2. THE STATEMENT OF THE THEOREM Further, we denote by (x, y) the inner product of the elements x, y ∈ H , and this notation is used for the value of the functional x ∈ H−1 on the element y ∈ H1 . Let us associate the symbol L(λ) = λ2 + λB + A to Eq. (1). Condition (5) does not ensure that the lineal L = D(B) ∩ D(A) is dense in H (it may happen even that L = {0}); hence the operator-valued function L(λ) is not densely defined in the space H . However, A admits an isometric extension to an operator from H1 to H−1 , and condition (5) implies that B has a bounded extension from H1 to H−1 . We denote these , respectively. Then the operator extensions by A˜ and B + A˜ ˜ L(λ) = λ2 + λB is well defined for λ ∈ C as a bounded operator from H1 to H−1 or as an unbounded operator ˜ = H1 . Now we can introduce the operator L(λ) acting in the space H in H−1 with domain D(L) ˜ by the rule L(λ)x = L(λ)x on the domain ˜ ∈ H}. D(L(λ)) = {x ∈ H1 | L(λ)x It is obvious that D(L(λ)) can depend on λ . According to [11], the range of λ ∈ C for which ˜ the operator L(λ) (operator L(λ)) has no bounded inverse operator in H (in H−1 ) is called a spectrum of the pencil L(λ) (the generalized spectrum of L(λ)); it will be denoted by σ(L) (respectively σ ˜ (L)). In order to formulate the main theorem, we introduce the numbers β := inf{(Bx, x) | x ∈ D(B), x = 1}, ˜ r(L) := inf{r ∈ R | L(λ) A for all λ > r}, ˜ where L(λ) A means that

˜ (L(λ)x, x) ≥ ε(Ax, x)

for some ε > 0 (depending on λ). It is obvious that (5) implies that r(L) < 0 . Theorem. Suppose that A = A∗ 0 , B ≥ 0 and the operator A dominates B in the sense quadratic forms. Then the operator T0 , defined by Eq. (3), is essentially m-dissipative, i.e., its closure T is an m-dissipative operator in the energy space E = H1 ×H . The operator T is defined by the equality x2 x1 = (6) T ˜ 1 − Bx 2 x2 −Ax MATHEMATICAL NOTES

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on the domain D(T ) =

x1 x2

x1 , x2 ∈ H1 ,

˜ Ax1 + Bx2 ∈ H .

621

(7)

The spectrum of the operator T is contained in the generalized spectrum σ ˜ (L) of the pencil L(λ) . −1 has the following representation For λ ∈ /σ ˜ (L) and λ = 0 , the resolvent (T − λ) −1 −1 ˜ (λ)A˜ − I) −L ˜ −1 (λ) λ (L −1 (8) (T − λ) = ˜ −1 (λ) . ˜ −1 (λ)A˜ −λL L The exponential type ω(T ) of the semigroup exp(tT ) admits the following estimate: β . ω(T ) ≤ ω0 := max r(L), − 2

(9)

In particular, if B 0 , then ω(T ) < 0 , i.e., the semigroup exp(tT ) is exponentially stable. 3. PRELIMINARY RESULTS An extensive literature exists on conditions necessary and sufficient for operators in Hilbert or Banach spaces to be generators of uniformly stable C0 -semigroups (see, for example, [8, 12–19] and references there). For a Hilbert space, we will use the Gearhart result [13]. We recall that the Hardy space H ∞ (α) in the half-plane Π(α) := {z ∈ C | Re z > α} consists of holomorphic uniformly bounded functions in Π(α) . This notation will be used also for the space of holomorphic uniformly bounded operator-valued functions in Π(α) . Proposition 1 (see [13]). Let T be the generator of a C0 -semigroup S(t) in the Hilbert space H . Then the exponential type of the semigroup is defined by the equality ω(T ) = inf{α ∈ R | (T − λ)−1 ∈ H ∞ (α)}. Now we will use the notion of a closed quadratic form and an operator associated to this form. We refer the reader to the book [20, Chap. VI] for the main facts concerning these objects. ˜ x) defined on Proposition 2. Let λ ∈ (r(L), ∞) . Then the quadratic form lλ [x] := (L(λ)x, elements x ∈ H1 is closed and the operator associated to this form by the first representation theorem coincides with L(λ) . Moreover, the operator L(λ) is self-adjoint, uniformly positive, ˜ is a bijection between H1 and H−1 . D([L(λ)]1/2 ) = H1 , and the operator L(λ) Proof. Suppose for a while that the closedness of the quadratic form lλ is established. Since the quadratic form lλ is uniformly positive for λ > r(L) , the first representation theorem associates to 1/2 this semigroup a uniformly positive self-adjoint operator Lλ acting in H so that D(Lλ ) = H1 . For x ∈ D(Lλ ) and arbitrary y ∈ H1 , we have ˜ y) ; (Lλ x, y) = lλ [x, y] = (L(λ)x, ˜ = L(λ)x . Thus, L(λ) is a symmetric extension of Lλ , and hence we have hence Lλ x = L(λ)x 1/2 Lλ = L(λ) . Note that the operator Lλ is a bijection between H1 and H , and the adjoint 1/2 1/2 operator (Lλ )∗ (the extension of Lλ ) defines a bijection between H and H−1 = (H1 )∗ . Hence 1/2 1/2 ˜ L(λ) = (Lλ )∗ Lλ is a bijection between H1 and H−1 . Thus, what is left to prove is that the quadratic form lλ on H1 is closed. This is obvious if λ = 0 . The theorem on the closedness of the sum of operators [20, Chap. VI, Theorem 1.33] and condition (5) imply that the form lλ is closed on H1 if λ is in a neighborhood of the origin. Let MATHEMATICAL NOTES

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(a, b) be a maximal interval such that the quadratic forms lλ are closed on H1 for all λ ∈ (a, ∞) . Suppose that a > r(L) . According to the definition, there exists an ε > 0 such that ˜ x) ≥ ε(Ax, x), la [x] = (L(a)x,

x ∈ H1 .

Let c be a constant from Eq. (5). By using the theorem on the closability of the sum of forms again, we see that the quadratic forms la and la+δ with arbitrary δ ∈ (−ε/2c, ε/2c) are closed or unclosed on H1 simultaneously. Since la+δ is closed on H1 for all δ > 0 , we obtain a contradiction with the choice of a . Hence the assumption a > r(L) is wrong, i.e., a ≤ r(L) . One can similarly prove that b = ∞ . 4. PROOF OF THE THEOREM Let us split the proof into several steps. Step 1. The proof of the dissipativity condition Re(T0 x, x) = −(Bx2 , x2 ),

x = (x1 , x2 )T ,

is straightforward. Thus, T0 admits a closure T . According to the definition, the operator T0 is essentially m-dissipative if the image of T0 − λ is dense in E for a certain (and hence for all) λ in the complex right half-plane. Let us prove that for any y1 ∈ D(B) , y2 ∈ H , λ > 0 the equation (T0 − λ)(x1 , x2 )T = (y1 , y2 )T has a solution. Proposition 2 implies that, for λ > 0 , the operator L(λ) is invertible, and so the solution of this equation is given by x1 = −L−1 (λ)(y2 + (B + λ)y1 ),

x2 = λx1 + y1 .

Since D(B) is dense in H1 , the image of T0 − λ is dense in E = H1 × H . Now we see that the operator T defined by equality (6) on the domain (7) is a dissipative extension of T0 . Since T0 does not admit nontrivial dissipative extensions in E , the equality T = T0 follows from the closedness of T . The closedness of T can be derived from the definition, as was done in the proof of Theorem 1 [6]. Step 2. Note that the operator T has a bounded inverse in E and −A˜−1 −A˜−1 B −1 . = T I 0 Next, for λ = 0 , λ ∈ / σ ˜ (L) , we can calculate explicitly the resolvent (T − λ)−1 expressed by Eq. (8). This representation and Proposition 1 imply the following result. Proposition 3. The C0 -semigroup exp(tT ) has exponential type ω(T ) ≤ ω0 if and only if for any α > ω0 the four operator-valued functions ˜ −1 (λ)A˜1/2 , ˜ −1 (λ), f2 (λ) := A1/2 L f1 (λ) := (λ − ω0 )−1 A1/2 L ˜ −1 (λ)A˜1/2 , ˜ −1 (λ) f4 (λ) := λL f3 (λ) := L

(10)

˜ −1/2 belong to the space H ∞ (α) in the uniform operator topology of the space H . Here A˜1/2 = AA 1/2 is the extension of the operator A acting from H to H−1 . Step 3. Let us fix an arbitrary number δ satisfying the condition δ > ω0 := max{−β/2, r(L)} . 1 and A˜1 be the corresponding extensions of B1 and A1 Set B1 = B + 2δ and A1 = L(δ) . Let B as operators mapping H1 to H−1 . Consider the operator-valued function 1 + A˜1 = L(λ ˜ + δ) ˜ 1 (λ) := λ2 + λB L MATHEMATICAL NOTES

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623

mapping H1 to H−1 and the corresponding operator-valued function L1 (λ) = L(λ + δ) in H . According to Proposition 2, we have A1 0 , D(A1 ) = H1 . The operator A1 clearly dominates B1 in the sense of quadratic forms, and the definition of β implies that B1 > 0 . Consider the operator 0 I . T1 = −A1 B1 Step 1 of the proof justifies that the operator T1 is essentially m-dissipative in E and its closure T1 generates a contractive C0 -semigroup in E . Proposition 3 implies that all four of the functions 1/2 ˜ −1 ˜1/2 g1 (λ) := (λ − ω0 )−1 A1 L 1 (λ)A1 ,

1/2 ˜ −1 g2 (λ) := A1 L 1 (λ),

˜ −1 (λ)A˜1/2 , g3 (λ) := L 1 1

˜ −1 (λ) g4 (λ) := λL 1 1/2

belong to the Hardy spaces H ∞ (α) for any α > 0 . Since both operators A1/2 and A1 are 1/2 bijections between H1 and H , the operator K = A1 A−1/2 is bounded and invertible in H . Taking into account the equality 1/2

A1

1/2 A˜1 = A˜1/2 K ∗ ,

= KA1/2 ,

˜ 1 (λ) = L(λ ˜ + δ), L

we see that the four functions in Eq. (10) belong to the Hardy spaces H ∞ (α) for any α > δ . By applying Proposition 3 again, we can conclude that the semigroup exp(tT ) has exponential type ω(T ) ≤ δ . This completes the proof of the theorem. Remark 1. To conclude the paper, we note that if B is a symmetric operator, conditions (5) and B 0 are in some sense necessary for the exponential stability of the semigroup exp(tT ) . Indeed, if condition (5) is violated, the operator T has no bounded inverse in E (cf. Step 2 of the proof), i.e., 0 ∈ σ(T ) and ω(T ) ≥ s(T ) = 0 . If the assumption on the uniform positivity of the operator B is omitted, for example, if B = A−1 , then ω(T ) ≥ s(T ) = 0 . ACKNOWLEDGMENTS This research was supported by the Russian Foundation for Basic Research under grants no. 0015-96100 and 01-01-00691. REFERENCES 1. G. Chen and D. L. Russel, “A mathematical model for linear elastic systems with structural damping,” Quart. Appl. Math. (1982), no. 1, 433–454. 2. S. Chen and R. Triggiani, “Proof of extensions of two conjectures on structural damping for elastic systems,” Pacific J. Math., 136 (1989), no. 1, 15–55. 3. F. Huang, “On the mathematical model for linear elastic systems with analytic damping,” SIAM J. Control Optim., 26 (1988), no. 3, 714–724. 4. S. Chen and R. Triggiani, “Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0 < α < 1/2 ,” Proc. Amer. Math. Soc., 110 (1990), no. 2, 401–415. 5. F. Huang, “Some problems for linear elastic systems with damping,” Acta Math. Sci., 10 (1990), no. 3, 316–326. 6. R. O. Griniv and A. A. Shkalikov, “Operator models in elasticity theory and hydrodynamics, and related analytical semigroups,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], (1999), no. 5, 5–14. 7. Yu. I. Lyubich, “Classical and local Laplace transformation for the abstract Cauchy problem,” Uspekhi Mat. Nauk [Russian Math. Surveys], 21(129) (1966), no. 3, 3–51. 8. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, in: Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York–Berlin, 1983. MATHEMATICAL NOTES

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9. T. Kato, “A generalization of the Heinz inequality,” Proc. Japan. Acad., 37 (1961), 305–308. 10. J. L. Lions and E.Magenes, Non-Homogeneous Boundary-Value Problems and Applications, SpringerVerlag, 1972. 11. A. A. Shkalikov, “Operator pencils arising in elasticity and hydrodynamics: the instability index formula,” in: Operator Theory: Adv. and Appl., vol. 87, Birkh¨ auser-Verlag, 1996, pp. 358–385. 12. R. Datko, “Extending a theorem of A. M. Liapunov to Hilbert space,” J. Math. Anal. Appl., 32 (1970), 610–616. 13. L. Gearhart, “Spectral theory for contraction semigroups on Hilbert spaces,” Trans. Amer. Math. Soc., 236 (1978), 385–394. 14. Casteren J. A. Van, “Operators similar to unitary or self-adjoint ones,” Pacific J. Math., 104 (1983), no. 1, 241–255. 15. S. N. Naboko, “On similarity criteria for unitary and self-adjoint operators,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 18 (1984), no. 1, 16–27. 16. M. M. Malamud, “The criterion for similarity between closed and self-adjoint operators,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 37 (1985), no. 1, 49–56. 17. S. Rolewicz, “On uniform N -equistability,” J. Math. Anal. Appl., 115 (1986), 431–441. 18. W. Littman, “A Generalization of a Theorem of Datko and Pazy,” in: Lecture Notes in Control and Inform. Sci., vol. 130, Springer-Verlag, Berlin–New York, 1989, pp. 318–323. 19. A. M. Gomilko, “On characterization of generators of uniformly bounded C0 -semigroups,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 33 (2000), no. 4, 66–69. 20. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1976. (R. O. Griniv) Institute of Applied Problems in Mathematics and Mechanics, Lviv, Ukraina E-mail: [email protected] (A. A. Shkalikov) M. V. Lomonosov Moscow State University E-mail: [email protected]

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