Exponential Stability for Linear Evolutionary Equations
aa r X i v : . [ m a t h . A P ] J un Exponential Stability for Linear EvolutionaryEquations.
Sascha TrostorffInstitut für Analysis, Fachrichtung MathematikTechnische Universität DresdenGermanysascha.trostorff@tu-dresden.deJune 22, 2018
Abstract.
We give an approach to exponential stability within the frameworkof evolutionary equations due to [R. Picard. A structural observation for linearmaterial laws in classical mathematical physics. Math. Methods Appl. Sci.,32(14):1768–1803, 2009]. We derive sufficient conditions for exponential stabilityin terms of the material law operator which is defined via an analytic and boundedoperator-valued function and give an estimate for the expected decay rate. Theresults are illustrated by three examples: differential-algebraic equations, partialdifferential equations with finite delay and parabolic integro-differential equations.
Keywords and phrases: exponential stability, evolutionary equations, causality, differential-algebraic equa-tions, delay-equations, integro-differential equations
Mathematics subject classification 2010: ontents Contents
References 20 Introduction
The article gives an approach to the exponential stability of equations of the form ∂ V + AU = F, (1)where we denote the derivative with respect to the temporal variable by ∂ . The (unbounded)linear operator A , acting on some Hilbert space, is assumed to be maximal monotone andwe refer to the monographs [4, 15] for the topic of monotone operators on Hilbert spaces.Equation (1) is completed by a constitutive relation of the form V = M U, where M is a bounded linear operator acting in time and space. Thus, we end up with anequation of the form ( ∂ M + A ) U = F, which we refer to as an evolutionary equation. This class of problems was introduced byPicard in [16], where A was assumed to be skew-selfadjoint, and it was illustrated that manyequations of classical mathematical physics are covered by this abstract class. The well-posedness for such problems is proved, by showing that the operator ∂ M + A is boundedlyinvertible in a suitable Hilbert space. More precisely, the derivative ∂ is established as anormal, continuously invertible operator in an exponentially weighted L -space and M isdefined as a function of ∂ − (see Section 2) and the well-posedness is shown under a positivedefiniteness constraint on the operator M (see [16, Solution Theory] and Theorem 2.4 of thisarticle). Moreover, the question of causality, which can be seen as a characterizing property forevolutionary processes, was addressed which leads to additional constraints on the operator M , namely that M is defined via the Fourier-Laplace transformation of an analytic and boundedfunction M : B C ( r, r ) → L ( H ) (for more details see Section 2). Especially the analyticityof M is crucial for the causality, due to the correlation of supports of L -functions and theanalyticity of their Laplace transforms by the Paley-Wiener Theorem (see [21, Theorem 19.2]).Later on these results were generalized to the case of A being a maximal monotone relationin [22, 24].In this work we give sufficient criteria for the exponential stability of the evolutionary problemin terms of the function M . The study of stability issues for differential equations, which goesback to Lyapunov (see [14] for a survey), has become a very active field of research for manydecades and there exist numerous works dealing with this topic. We just like to mention somestandard approaches to exponential stability. The first strategy goes back to Lyapunov. Theaim is to find a suitable function (a so-called Lyapunov function) yielding a certain differentialinequality which allows to derive statements about the asymptotic behavior of solutions of thedifferential equation. A second approach, which applies to linear differential equations, is basedon the theory of semigroups. In this framework different criteria for exponential stability werederived in terms of the semigroup or its generator, e.g. Datko’s Lemma ([8] or [9, p. 300]),Gearharts Theorem ([11] or [9, p. 302]) or the Spectral Mapping Theorem (see [9, p. 302,Theorem 1.10]). A third approach uses the Fourier or the Laplace transform of a solution toderive statements of their asymptotics. These methods are sometimes referred to as FrequencyDomain Methods. In our framework it seems to be appropriate to employ the last approach,3 The framework of evolutionary equations since, by the definition of M , methods of vector-valued complex analysis are at hand throughthe Fourier-Laplace transformation. Note that, due to the general structure of evolutionaryequations, the results apply to a broad class of differential equations, such as differential-algebraic equations, equations with memory effects or integro-differential equations, wheresemigroup methods may be difficult to apply.The article is structured as follows. In Section 2 we recall the framework of evolutionaryequations and its solution theory (Theorem 2.4). Section 3 provides an abstract conditionfor exponential stability for evolutionary equations in terms of the function M and the proofof the main stability result (Theorem 3.7). To illustrate the versatility of the previous re-sults, we discuss several examples in Section 4. We begin with studying differential-algebraicequations of mixed type and derive a condition for their exponential stability. Moreover,we give a concrete example for such an equation, which seems hard to be tackled by otherapproaches, since the type of the differential equation switches on different parts of the un-derlying domain. Furthermore, we give a possible approach of how to deal with initial valueproblems. In Subsection 4.2 we consider an example of a partial differential equation withmemory effect. A similar problem was also treated by Batkai and Piazerra in [2], using asemigroup approach. In contrast to their result our approach directly extends to the case ofdifferential-algebraic equations with delay. We conclude the article by discussing a parabolicintegro-differential equation with an operator-valued kernel, where we adopt the ideas of [23]to derive the exponential stability.Throughout let H be a complex Hilbert space. We denote its inner product by h·|·i which isassumed to be linear in the second and conjugate linear in the first argument. We denote itsinduced norm by | · | . We recall some basic notions and results on linear evolutionary equations, i.e. equations ofthe form (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) u = f, (2)where A : D ( A ) ⊆ H → H is a linear, maximal monotone operator, ∂ ,̺ denotes the time-derivative, established in a suitable Hilbert space and M ( ∂ − ,̺ ) is a bounded linear operatorin time and space, a so-called linear material law. We refer to [16, 17, 13, 22, 24] for moredetails and proofs of the following statements. First we begin by introducing the Hilbert spacesetting, where we want to consider equation (2).For ̺ ∈ R we define the space H ̺, ( R ; H ) as the space of (equivalence classes of) measurablefunction f : R → H which are square-integrable with respect to the exponentially weightedLebesgue measure e − ̺t d t, equipped with the inner product h f | g i H ̺, ( R ; H ) := Z R h f ( t ) | g ( t ) i e − ̺t d t. Note that H , ( R ; H ) is just the space L ( R ; H ) . We define the derivative ∂ ,̺ as the closure4f the operator ∂ ,̺ | C ∞ c ( R ; H ) : C ∞ c ( R ; H ) ⊆ H ̺, ( R ; H ) → H ̺, ( R ; H ) φ φ ′ , where we denote by C ∞ c ( R ; H ) the space of function φ : R → H which are arbitrarily oftendifferentiable and have compact support. Remark . (a) For each ̺ ∈ R the operator ∂ ,̺ is normal with Re ∂ ,̺ = (cid:16) ∂ ,̺ + ∂ ∗ ,̺ (cid:17) = ̺. Moreover weobtain that ∂ ∗ ,̺ = − ∂ ,̺ + 2 ̺. In particular, for ̺ = 0 the operator ∂ , is skew-selfadjointand coincides with the usual weak derivative on L ( R ; H ) with domain H , ( R ; H ) = H ( R ; H ) = W ( R ; H ) .(b) For ̺ = 0 the operators ∂ ,̺ on H ̺, ( R ; H ) and ∂ , + ̺ on L ( R ; H ) are unitarily equivalentand ∂ ,̺ is boundedly invertible with k ∂ − ,̺ k = | ̺ | . For ̺ = 0 the operator ∂ , is notboundedly invertible, since lies in its continuous spectrum.(c) For ̺ = 0 the inverse operator ∂ − ,̺ is given by (cid:16) ∂ − ,̺ u (cid:17) ( t ) = (R t −∞ u ( s ) d s if ̺ > , − R ∞ t u ( s ) d s if ̺ < for u ∈ H ̺, ( R ; H ) and almost every t ∈ R . This representation yields that for ̺ > theoperator ∂ − ,̺ is forward causal whereas it is backward causal for ̺ < . (d) Let ̺ ∈ R and denote by L ̺ : H ̺, ( R ; H ) → L ( R ; H ) the so-called Fourier-Laplacetransformation, defined as the unitary extension of the operator given by ( L ̺ φ ) ( x ) = 1 √ π Z R e − i xt e − ̺t φ ( t ) d t for φ ∈ C ∞ c ( R ; H ) and x ∈ R . Then ∂ ,̺ = L ∗ ̺ (i m + ̺ ) L ̺ , (3)where by m : D ( m ) ⊆ L ( R ; H ) → L ( R ; H ) we denote the multiplication-by-the-argument operator with maximal domain, i.e., ( mf ) ( t ) = tf ( t ) for almost every t ∈ R and every f ∈ D ( m ) := { f ∈ L ( R ; H ) | ( t tf ( t )) ∈ L ( R ; H ) } . Note that in the case ̺ = 0 , (3) is just the usual spectral representation for the weak derivative on L ( R ; H ) via the Fourier transformation (see [1, p. 112]).We consider (2) as an equation in the Hilbert space H ̺, ( R ; H ) . As a matter of physicalrelevance, we require that the corresponding solution operator (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − , if it A mapping F : D ( F ) ⊆ H ̺, ( R ; H ) → H ̺, ( R ; H ) is called forward causal if for each f, g ∈ D ( F ) with f = g on some interval ] − ∞ , a [ for a ∈ R the functions F ( f ) and F ( g ) coincide on the same interval ] − ∞ , a [ . Analogously F is called backward causal if for each f, g ∈ D ( F ) with f = g on some interval ] a, ∞ [ for a ∈ R the functions F ( f ) and F ( g ) coincide on the same interval ] a, ∞ [ . The framework of evolutionary equations exists, is forward causal, that means, roughly speaking, that the present state of the solution u does not depend on the future behavior of the source term f. With Remark 2.1 (c) in mind,we therefore assume that ̺ > . We now define the operator M ( ∂ − ,̺ ) for ̺ > with the help of formula (3). Definition 2.2.
Let r > and M : B C ( r, r ) → L ( H ) . For ̺ > r we define M ( ∂ − ,̺ ) := L ∗ ̺ M (cid:18) m + ̺ (cid:19) L ̺ , where M (cid:16) m + ̺ (cid:17) is defined as the multiplication operator (cid:16) M (cid:16) m + ̺ (cid:17) f (cid:17) ( t ) := M (cid:16) t + ̺ (cid:17) f ( t ) on L ( R ; H ) with domain n f ∈ L ( R ; H ) (cid:12)(cid:12)(cid:12) (cid:16) t M (cid:16) t + ̺ (cid:17) f ( t ) (cid:17) ∈ L ( R ; H ) o . We call M ( ∂ − ,̺ ) a linear material law if the function M belongs to H ∞ ( B C ( r, r ); L ( H )) , i.e., M is boundedand analytic. Remark . The notion material law is motivated by several examples of mathematicalphysics, since it turns out that all material parameters, such as mass density, conductivity,permeability etc. can be incorporated into the operator M ( ∂ − ,̺ ) (see [16] for several exam-ples). Thus, it is natural that M ( ∂ − ,̺ ) is forward causal. Since the operator M ( ∂ − ,̺ ) is linearand it commutes with the translation operators τ h , mapping u ∈ H ̺, ( R ; H ) to t u ( t + h ) for h ∈ R , causality can be characterized via the requirement that spt M ( ∂ − ,̺ ) u ⊆ ]0 , ∞ [ if spt u ⊆ ]0 , ∞ [ , where by spt g we denote the support of a function g ∈ L , loc ( R ; H ) . This, how-ever, can be characterized by the analyticity and boundedness of the mapping M , employinga Paley-Wiener-type result (see e.g. [21, Theorem 19.2]). Moreover, note that due to theboundedness of M , the operator M (cid:16) ∂ − ,̺ (cid:17) becomes a bounded operator on H ̺, ( R ; H ) . We are now able to state the solution theory for evolutionary equations. For the proof werefer to [16, 24].
Theorem 2.4 (Solution Theory) . Let A : D ( A ) ⊆ H → H be a maximal monotone linearoperator. Moreover, let r > and M ∈ H ∞ ( B C ( r, r ); L ( H )) and assume that the solvabilitycondition is satisfied: ∃ c > ∀ z ∈ B C ( r, r ) , x ∈ H : Re h z − M ( z ) x | x i ≥ c | x | . (4) Then for each ̺ > r the problem (2) is well-posed in H ̺, ( R ; H ) , i.e. for each ̺ > r theoperator ∂ ,̺ M ( ∂ − ,̺ )+ A is boundedly invertible and has a dense range. Moreover, the solutionoperator (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − is forward causal.Remark . Note that the solution operator (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − commutes with the time-derivative ∂ ,̺ . This yields, that for right hand sides f ∈ D ( ∂ k ,̺ ) for k ∈ N in (2) the cor-responding solution u also lies in D ( ∂ k ,̺ ) , i.e. the solution operator preserves “temporal”regularity . We denote by B C ( x, s ) the open ball in C with center x ∈ C and radius s > Indeed, one can show that the solution operator (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − extends to the whole Sobolev-chain ( H k ( ∂ ,̺ )) k ∈ Z associated with the derivative ∂ ,̺ , see [17] which yields a solution theory for distributionalright-hand sides. An abstract condition for exponential stability
In this section we show that under certain constraints on the function M , the correspondingevolutionary problem is exponentially stable. Although in the literature, exponential stabilityusually means that the solution of an initial value problem decays with an exponential rateas time tends to infinity, we like to introduce a slightly weaker notion within our frameworkwhich, however, yields the desired decay if the source term f is regular enough (see Remark3.2 (a)). Moreover, since the framework of evolutionary equations covers different types ofequations, where initial values do not make sense, such as differential-algebraic or even purealgebraic equations or equations with memory effect, where a given pre-history would bemore appropriate then an initial value, we cannot treat initial value problems within thisgeneral approach. However, in concrete examples we can reformulate initial value problemsas evolutionary equations with a modified right hand side (see Subsection 4.1) such that thefollowing results still apply. Definition 3.1.
Let A : D ( A ) ⊆ H → H be a maximal monotone linear operator and M ∈ H ∞ ( B C ( r, r ); L ( H )) for some r > which satisfies the solvability condition (4). Let ̺ > r . We call the solution operator (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − of (2) exponentially stable withstability rate ν > if for each ≤ ν < ν and f ∈ H − ν, ( R ; H ) ∩ H ̺, ( R ; H ) the solution u of (2) satisfies u = (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − f ∈ \ − ν<µ ≤ ̺ H µ, ( R ; H ) , which especially implies R R e µt | u ( t ) | d t < ∞ for all ≤ µ < ν. Remark . (a) We show that our notion of exponential stability indeed yields an exponential decay ofthe solution if the given right hand side is regular enough. For doing so, let ̺ > r and (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − exponentially stable with stability rate ν > . Moreover, assumethat f ∈ H − ν, ( R ; H ) ∩ H ̺, ( R ; H ) and f ∈ D ( ∂ ,̺ ) such that ∂ ,̺ f ∈ H − ν, ( R ; H ) ∩ H ̺, ( R ; H ) for some < ν < ν . Then u = (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − f also lies in D ( ∂ ,̺ ) and ∂ ,̺ u = (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − ∂ ,̺ f (compare Remark 2.5). By the assumed exponentialstability, we get that e µm u ∈ L ( R ; H ) and e µm ∂ ,̺ u ∈ L ( R ; H ) for each ≤ µ < ν. Thelatter yields e µm u ∈ W ( R ; H ) . Indeed, for φ ∈ C ∞ c ( R ; H ) we compute h µe µm u + e µm ∂ ,̺ u | φ i L ( R ; H ) = Z R h u ( t ) | µe µt φ ( t ) i d t + Z R D ( ∂ ,̺ u ) ( t ) | e (2 ̺ + µ ) t φ ( t ) e − ̺t E d t = Z R h u ( t ) | µe µt φ ( t ) i d t + h ∂ ,̺ u | e (2 ̺ + µ ) m φ i H ̺, ( R ; H ) = Z R h u ( t ) | µe µt φ ( t ) i d t + D u (cid:12)(cid:12)(cid:12) − ∂ ,̺ (cid:16) e (2 ̺ + µ ) m φ (cid:17) E H ̺, ( R ; H ) An abstract condition for exponential stability + D u (cid:12)(cid:12)(cid:12) ̺ (cid:16) e (2 ̺ + µ ) m φ (cid:17) E H ̺, ( R ; H ) = Z R h u ( t ) | µe µt φ ( t ) i d t − Z R (2 ̺ + µ ) h u ( t ) | e (2 ̺ + µ ) t φ ( t ) i e − ̺t d t − Z R h u ( t ) | e (2 ̺ + µ ) t φ ′ ( t ) i e − ̺t d t + Z R ̺ h u ( t ) | e (2 ̺ + µ ) t φ ( t ) i e − ̺t d t = − Z R h u ( t ) | e (2 ̺ + µ ) t φ ′ ( t ) i e − ̺t d t = −h e µm u | φ ′ i L ( R ; H ) . Thus, we obtain e µt | u ( t ) | → as t tends to infinity for each ≤ µ < ν due to Sobolev’sembedding theorem (see e.g. [9, p. 408]), i.e. u decays exponentially with a decay rate lessthan ν. (b) If f ∈ H µ, ( R ; H ) ∩ H ̺, ( R ; H ) for some µ, ̺ ∈ R with µ > ̺ , then f ∈ H ν, ( R ; H ) for all ν ∈ [ ̺, µ ] . Indeed, we estimate Z R | f ( t ) | e − νt d t = Z −∞ | f ( t ) | e − νt d t + ∞ Z | f ( t ) | e − νt d t ≤ Z −∞ | f ( t ) | e − µt d t + ∞ Z | f ( t ) | e − ̺t d t ≤ | f | H µ, ( R ; H ) + | f | H ̺, ( R ; H ) . We now give conditions for the function M and show that they yield the well-posedness andexponential stability for the corresponding evolutionary problem. Hypotheses.
Let ν > . We assume that(a) M : C \ B C h − ν , ν i → L ( H ) is analytic ,(b) for each r > and ≤ ν < ν the function B C ( r, r ) \ { ν − } ∋ z (1 − νz ) M (cid:0) z (1 − νz ) − (cid:1) has a bounded and analytic extension to B C ( r, r ) ,(c) for every < ν < ν there exists c > such that for all z ∈ C \ B C (cid:2) − ν , ν (cid:3) Re z − M ( z ) ≥ c. Remark . Let M satisfy the assumptions above and let r > . Then the restriction of M to B C ( r, r ) is an element of H ∞ ( B C ( r, r ); L ( H )) . Indeed, the analyticity is clear from (a) and We denote by B C [ x, s ] the closed ball in C with center x ∈ C and radius s ≥ . Here and further on we set B C ( r, r ) \ { − } := B C ( r, r ) . ̺ > according to Theorem 2.4.We now state some auxiliary results which in particular imply that the solution operator (cid:18) ∂ ,̺ M (cid:16) ∂ − ,̺ (cid:17) + A (cid:19) − for an evolutionary problem does not depend on the particular choiceof ̺. These results can also be found in [17, p. 429 f.]. However, for sake of completeness wepresent them again with a slightly modified proof.
Lemma 3.4.
Let ̺, µ ∈ R with µ > ̺ and set U := { z ∈ C | Re z ∈ [ ̺, µ ] } . Moreover, let f : U → H be continuous on U and analytic in the interior of U , such that R µ̺ f (i R + s ) d s → as R → ±∞ . Then lim sup R →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R Z − R f (i t + µ ) d t − R Z − R f (i t + ̺ ) d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . Proof.
According to Cauchy’s integral theorem we have i R Z − R f (i t + ̺ ) d t + µ Z ̺ f (i R + s ) d s − i R Z − R f (i t + µ ) d t + µ Z ̺ f ( − i R + s ) d s = 0 for each R > . The assertion now follows by taking the limes superior as R tend to ∞ . The next lemma shows that we can approximate a function which belongs to two differentexponentially weighted L -spaces by the same sequence of test functions with respect to bothtopologies. Lemma 3.5.
Let ̺, µ ∈ R and f ∈ H ̺, ( R ; H ) ∩ H µ, ( R ; H ) . Then, for each ε > there exists φ ∈ C ∞ c ( R ; H ) such that max (cid:8) | f − φ | H ̺, ( R ; H ) , | f − φ | H µ, ( R ; H ) (cid:9) ≤ ε. Proof.
Let ε > . Then we choose N ∈ N such that f N := f χ [ − N,N ] satisfies max (cid:8) | f − f N | H ̺, ( R ; H ) , | f − f N | H µ, ( R ; H ) (cid:9) ≤ ε . We denote by ( ψ k ) k ∈ N ∈ C ∞ c ( R ) N the Friedrichs mollifier (see e.g. [10, Chapter C.4]). Then,for each k ∈ N we have that spt ψ k ∗ f N ⊆ [ − N − , N + 1] . Now, we choose k large enoughsuch that | ψ k ∗ f N − f N | L ([ − N − ,N +1]; H ) ≤ e − {| ̺ | , | µ |} ( N +1) ε . Then the function ψ k ∗ f N ∈ C ∞ c ( R ; H ) has the desired property. Lemma 3.6.
Let ̺, µ ∈ R with µ > ̺ and set U := { z ∈ C | Re z ∈ [ ̺, µ ] } . Moreover, let f ∈ H ̺, ( R ; H ) ∩ H µ, ( R ; H ) and T ∈ H ∞ (˚ U ; L ( H )) ∩ C b ( U ; L ( H )) (i.e. T is bounded andcontinuous on U and analytic in the interior of U ). Then (cid:0) L ∗ ̺ T (i m + ̺ ) L ̺ f (cid:1) ( t ) = (cid:0) L ∗ µ T (i m + µ ) L µ f (cid:1) ( t ) An abstract condition for exponential stability for almost every t ∈ R . Proof.
According to Lemma 3.5 it suffices to prove the assertion for test functions. So let φ ∈ C ∞ c ( R ; H ) . We show that the function U ∋ z e zt T ( z ) (( L Re z φ ) ( Im z )) ∈ H satisfies the assumptions of Lemma 3.4. Indeed it is continuous in U and analytic in the interiorof U as a composition of analytic functions. Furthermore, we estimate for s ∈ [ ̺, µ ] , ξ ∈ R : |L s φ ( ξ ) | ≤ √ π Z R e − rs | φ ( r ) | d r ≤ C √ π Z R | φ ( r ) | d r, where C := sup { e − rs | s ∈ [ ̺, µ ] , r ∈ spt φ } , which shows that the function U ∋ z ( L Re z φ ) ( Im z ) is bounded. Moreover, due to the Riemann-Lebesgue lemma, we get that ( L s φ ) ( R ) → as R → ±∞ for every s ∈ R . Therefore, according to Lebesgue’s dominated convergence theorem,we deduce that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ Z ̺ e (i R + s ) t T (i R + s ) ( L s φ ) ( R ) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ max { e ̺t , e µt }| T | ∞ µ Z ̺ | ( L s φ )( R ) | d s → R → ±∞ ) . Thus, by Lemma 3.4, we get that Z R e (i s + ̺ ) t T (i s + ̺ ) ( L ̺ φ ) ( s ) d s = Z R e (i s + µ ) t T (i s + µ ) L µ φ ( s ) d s, which yields the assertion.We are now able to prove our main theorem. Theorem 3.7.
Let A : D ( A ) ⊆ H → H be a maximal monotone linear operator and M amapping satisfying the hypotheses above for some ν > . Then for each ̺ > the solutionoperator (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − is exponentially stable with stability rate ν .Proof. Let ̺ > , ≤ ν < ν and take f ∈ H − ν, ( R ; H ) ∩ H ̺, ( R ; H ) . We set u := (cid:16) ∂ ,̺ M ( ∂ − ,̺ ) + A (cid:17) − f and we have to show that u ∈ H µ, ( R ; H ) for each µ ∈ ] − ν, ̺ ] . Let < η < ̺ + ν. We define e N ( z ) = (1 − νz ) M (cid:16) z (1 − νz ) − (cid:17) z ∈ B C (cid:16) η , η (cid:17) \ { ν − } . Note that according to hypotheses (b), e N has an extension N ∈ H ∞ (cid:16) B C (cid:16) η , η (cid:17) ; L ( H ) (cid:17) . Moreover, Re z − N ( z ) = Re z − e N ( z ) = Re (1 − νz ) z M (cid:18) z − νz (cid:19) ≥ c (5)for every z ∈ B C (cid:16) η , η (cid:17) \ { ν − } and some suitable c > , since z (1 − νz ) − / ∈ B C (cid:2) − ν , ν (cid:3) .Due to the continuity of N , estimate (5) also holds for z = ν . Thus, according to Theorem2.4, we obtain a solution v := (cid:16) ∂ ,η N ( ∂ − ,η ) + A (cid:17) − e νm f ∈ H η, ( R ; H ) , where we have used that e νm f ∈ H , ( R ; H ) ∩ H ̺ + ν, ( R ; H ) ⊆ H η, ( R ; H ) (see Remark 3.2(b)). We apply Lemma 3.6 to T ( z ) = ( zN ( z − ) + A ) − for z ∈ C with Re z ≥ η and get that v = L ∗ η (i m + η ) N (cid:18) m + η (cid:19) + A ! − L η e νm f = L ∗ η + ν (i m + η + ν ) N (cid:18) m + η + ν (cid:19) + A ! − L η + ν e νm f. Using (i t + η + ν ) N (cid:16) t + η + ν (cid:17) = (i t + η ) M (cid:16) t + η (cid:17) for t ∈ R , we derive e − νm v = L ∗ η (i m + η ) M (cid:18) m + η (cid:19) + A ! − L η f. Again, applying Lemma 3.6 to T ( z ) = (cid:0) zM ( z − ) + A (cid:1) − for z ∈ C with Re z ≥ min { ̺, η } weget that e − νm v = L ∗ η (i m + η ) M (cid:18) m + η (cid:19) + A ! − L η f = L ∗ ̺ (i m + ̺ ) M (cid:18) m + ̺ (cid:19) + A ! − L ̺ f = u, which gives u ∈ H η − ν ( R ; H ) . Since η ∈ ]0 , ̺ + ν [ was chosen arbitrarily, we get the assertion. In this section we illustrate our results of the previous section by three different types ofdifferential equations which, however, are all covered by the abstract notion of evolutionaryequations. We emphasize that we do not claim that in the forthcoming examples the stability11
Examples rates are optimal under the given constraints nor that an exponential decay could not beobtained under lesser constraints. But we emphasize that our approach provides a unifiedway to study exponential stability of a broad class of differential equations.We begin to study a class of differential-algebraic equations, where the material law is of thesimplest form. Moreover, we provide a strategy of how to deal with initial values problems forthis class. As a second example we treat a partial differential-algebraic equation with finitedelay. We conclude this section with an example of a parabolic integro-differential equationwith an operator-valued kernel.
It turns out that in applications, the material law is often of the form M ( ∂ − ,̺ ) = M + ∂ − ,̺ M (see for instance [16, 17, 22]), where M , M ∈ L ( H ) . The corresponding evolutionary equationis then of the form ( ∂ ,̺ M + M + A ) u = f, (6)where A : D ( A ) ⊆ H → H is a maximal monotone linear operator. In order to obtain thewell-posedness of this evolutionary equation we require that M is selfadjoint and strictlypositive definite on its range, while Re M := ( M + M ∗ ) is strictly positive definite on thekernel of M (see [16, 22, 24] for the proof of well-posedness). In order to obtain exponentialstability for this problem, we require that Re M is strictly positive definite on the whole space H. Theorem 4.1.
Let A : D ( A ) ⊆ H → H be a maximal monotone linear operator and M , M ∈ L ( H ) such that M is selfadjoint and strictly positive definite on its range, and Re M ≥ c > .Then for each ̺ > the solution operator (cid:0) ∂ ,̺ M + M + A (cid:1) − is exponentially stable withstability rate c k M k .Proof. We have to verify the hypotheses (a)-(c) for the function M ( z ) = M + zM ( z ∈ C ) . Obviously the assumption (a) holds. Let now r > and ν ≥ . Then we compute (1 − νz ) M (cid:0) z (1 − νz ) − (cid:1) = (1 − νz ) M + zM for z ∈ B C ( r, r ) \ { ν − } , which shows (b). Let now ν ∈ ]0 , c k M k [ . In order to show (c) we note,that for z ∈ C \ B C (cid:2) − ν , ν (cid:3) there exists t ∈ R and ̺ > − ν such that z − = i t + ̺. Thus, for ̺ ≥ we can estimate Re z − M ( z ) = ̺M + Re M ≥ c, while for ̺ ∈ ] − ν, we estimate Re z − M ( z ) = ̺M + Re M ≥ − ν k M k + c > . Thus, the assertion follows by Theorem 3.712 .1 Differential-algebraic equations of mixed type
To illustrate the versatility of our approach, we discuss the following simple example of anevolutionary equation, whose type (elliptic, parabolic or hyperbolic) changes on different partsof the underlying domain. A similar example was also discussed in [18, 19].
Example 4.2.
Let Ω ⊆ R n and Ω , Ω ⊆ Ω be measurable disjoint subsets with positiveLebesgue measure. We consider the evolutionary equation (cid:18) ∂ ,̺ (cid:18) χ Ω + χ Ω χ Ω (cid:19) + (cid:18) c c (cid:19) + (cid:18) c grad 0 (cid:19)(cid:19) (cid:18) vq (cid:19) = (cid:18) fg (cid:19) , (7)where c > . The differential operator div c is defined as the closure of the operator div | C ∞ c (Ω) n : C ∞ c (Ω) n ⊆ L (Ω) n → L (Ω)( φ i ) i ∈{ ,...,n } n X i =1 ∂ i φ i , where we denote by ∂ i the partial derivative with respect to the i -th coordinate. The operator grad is defined as the negative adjoint of div c , i.e. grad := − (div c ) ∗ . This operator is just the usual weak gradient on L (Ω) with domain H (Ω) . Note that then theoperator matrix (cid:18) c grad 0 (cid:19) is a skew-selfadjoint operator (and hence maximal monotone)on H := L (Ω) ⊕ L (Ω) n . Moreover, the operators M = (cid:18) χ Ω + χ Ω χ Ω (cid:19) , M = (cid:18) c c (cid:19) satisfy the assumptions of Theorem 4.1 and hence, the solution operator is exponentially stablewith stability rate c. Although this example seems to be quite easy, it seems hard to attack the problem of solving(6) by using semigroup techniques. The reason for that is that (6) changes its type on differentparts of the domain Ω . Indeed, on Ω we obtain a hyperbolic problem of the form (cid:18) ∂ ,̺ (cid:18) (cid:19) + (cid:18) c c (cid:19) + (cid:18) c grad 0 (cid:19)(cid:19) (cid:18) vq (cid:19) = (cid:18) fg (cid:19) , while on Ω the problem becomes parabolic, namely (cid:18) ∂ ,̺ (cid:18) (cid:19) + (cid:18) c c (cid:19) + (cid:18) c grad 0 (cid:19)(cid:19) (cid:18) vq (cid:19) = (cid:18) fg (cid:19) , which yields, in case of g = 0 the parabolic differential equation ∂ ,̺ v + cv − c − div c grad v = f. Examples
On the remaining part Ω \ (Ω ∪ Ω ) the problem is elliptic (cid:18)(cid:18) c c (cid:19) + (cid:18) c grad 0 (cid:19)(cid:19) (cid:18) vq (cid:19) = (cid:18) fg (cid:19) . Note that we can treat this problem, without requiring any explicit transmission conditions onthe interfaces ∂ Ω and ∂ Ω and without imposing regularity assumptions on these boundaries. Initial value problems
Now, we present a possible way to tackle initial value problems for equations of the form (6).Consider the following initial value problem ( ∂ ,̺ M + M + A ) u = f on ]0 , ∞ [ (8) M u (0+) = M u for M , M , A as before, f ∈ H − ν, ( R ; H ) ∩ H ̺, ( R ; H ) for some ν, ̺ > with spt f ⊆ [0 , ∞ [ and u ∈ D ( A ) . One way to deal with such problems is to consider the evolutionary equation ( ∂ ,̺ M + M + A ) e v = f − χ [0 , ∞ [ ( m ) M u − χ [0 , ∞ [ ( m ) Au on R , for the new unknown e v := u − χ [0 , ∞ [ ( m ) u given by e v ( t ) = u ( t ) − χ [0 , ∞ [ ( t ) u for almost every t ∈ R . Then the right-hand side belongs to H ̺, ( R ; H ) for positive ̺, but does not decayif f decays. Hence, this approach can be used for the issue of well-posedness but it is notappropriate for exponential stability.Instead, we consider an alternative problem for the unknown v := u − φ ( m ) u , where φ is given by φ ( t ) := if t ∈ [0 , , − t if t ∈ ]1 , , otherwise.It is clear that if u satisfies (8), then v satisfies the equation ( ∂ ,̺ M + M + A ) v = f + χ ]1 , ( m ) M u − φ ( m ) M u − φ ( m ) Au =: g (9)and vice versa. Since the function χ ]1 , ( m ) M u − φ ( m ) M u − φ ( m ) Au belongs to H µ, ( R ; H ) for every µ ∈ R , we obtain g ∈ H ̺, ( R ; H ) ∩ H − ν, ( R ; H ) . Thus, Theorem 4.1 applies to (9)and we get that (9) is well-posed and v ∈ H µ, ( R ; H ) for each µ ∈ ] − ν, ̺ ] . This gives that u ∈ H µ, ( R ; H ) for each µ ∈ ] − ν, ̺ ] , since φ ( m ) u ∈ T µ ∈ R H µ, ( R ; H ) . It is left to show inwhich sense M u attains the initial value M u . By (9) we get that ∂ ,̺ M v = g − M v − Av Note that it only makes sense to prescribe an initial value for M u yielding an initial value for the part of u lying in the range of M . .2 Linear partial differential equations with finite delay and the right hand side belongs to H ̺, ( R ; H − ( A + 1)) , where H − ( A + 1) is the extrapolationspace associated with A + 1 . Thus, a version of Sobolev’s embedding theorem ([17, Lemma3.1.59], [13, Lemma 5.2]) yields that M v is continuous as a function which attains values in H − ( A + 1) . Furthermore, due to the causality of the solution operator (cid:0) ∂ ,̺ M + M + A (cid:1) − , v is supported on [0 , ∞ [ , since spt g ⊆ [0 , ∞ [ . This yields M v (0+) = M v (0 − ) = 0 and hence,since φ (0+) = 1 we get that M u (0+) = M u , where the equality holds in H − ( A + 1) . As a second example we study a differential equation with finite delay of the form ( ∂ ,̺ M + τ h + M + A ) u = f, (10)where M , M ∈ L ( H ) such that M is selfadjoint and non-negative, Re M ≥ c > , A : D ( A ) ⊆ H → H is linear and maximal monotone and τ h is the translation operator withrespect to time, i.e. ( τ h u ) ( t ) = u ( t + h ) for t ∈ R and some h ≤ . We will prove that underthese assumptions, the corresponding solution operator is exponentially stable and we givean estimate for the stability rate. A similar problem is treated in [2, Example 4.14] for aparticular operator A , where the well-posedness is shown via semigroups and a criterion forthe exponential stability is given, using the Spectral Mapping Theorem for eventually normcontinuous semigroups (cf. [9, p. 280]).Before we state our stability result for (10), we need to inspect the operator τ h a bit closer. Lemma 4.3.
Let ̺, h ∈ R . We define the operator τ h : H ̺, ( R ; H ) → H ̺, ( R ; H ) u ( t u ( t + h )) . Then τ h ∈ L ( H ̺, ( R ; H )) with k τ h k = e ̺h and τ h = L ∗ ̺ e (i m + ̺ ) h L ̺ .Proof. Obviously, τ h defines a bounded linear operator on H ̺, ( R ; H ) . For φ ∈ C ∞ c ( R ; H ) wecompute L ̺ ( τ h φ ) ( t ) = 1 √ π Z R e − i st e − ̺s φ ( s + h ) d s = e (i t + ̺ ) h √ π Z R e − i st e − ̺s φ ( s ) d s = e (i t + ̺ ) h ( L ̺ φ ) ( t ) for each t ∈ R , which gives τ h = L ∗ ̺ e (i m + ̺ ) h L ̺ . Moreover, by this unitary equivalence we getthat k τ h k = k e (i m + ̺ ) h k = e ̺h . For a boundedly invertible linear operator C : D ( C ) ⊆ H → H the extrapolation space H − ( C ) is given asthe completion of H with respect to the norm | · | H − ( C ) defined as | x | H − ( C ) := | C − x | H for x ∈ H (see[17, Section 2.1]). Examples
Using this lemma, we are able to write (10) as an evolutionary equation with M ( z ) = M + ze z − h + zM , (11)which is clearly analytic and bounded on balls of the form B C ( r, r ) for r > if we require that h ≤ . This restriction is natural, since for h ≤ the operator τ h is forward causal, while for h > it is backward causal. Theorem 4.4.
Let A : D ( A ) ⊆ H → H be a maximal monotone linear operator, M , M ∈ L ( H ) such that M is selfadjoint and non-negative and Re M ≥ c > . Moreover let h < . Then for each ̺ > the solution operator (cid:0) ∂ ,̺ M + τ h + M + A (cid:1) − is exponentially stablewith stability rate ν > satisfying ν k M k + e − ν h = c. Proof.
We have to show that M given by (11) satisfies the hypotheses (a)-(c) of Section 3.Obviously M is analytic on C \ { } , which shows (a). Let now r > and ν ≥ . As (1 − νz ) M (cid:0) z (1 − νz ) − (cid:1) = (1 − νz ) M + ze ( z − − ν ) h + zM for z ∈ B C ( r, r ) \ { ν − } , we see that z (1 − νz ) M (cid:0) z (1 − νz ) − (cid:1) has an analytic extensionto B C ( r, r ) . Moreover we estimate sup z ∈ B C ( r,r ) | e ( z − − ν ) h | = sup ̺> r e ( ̺ − ν ) h , which is finite, since h < . Thus, hypothesis (b) is also satisfied. For showing (c), let ν ∈ ]0 , ν [ and z ∈ C \ B C (cid:2) − ν , ν (cid:3) . Then there exists ̺ > − ν and t ∈ R such that z − = i t + ̺. If ̺ ≥ we estimate Re z − M ( z ) = ̺M + e ̺h cos( th ) + Re M ≥ c − > , and for the case ̺ ∈ ] − ν, we get Re z − M ( z ) = ̺M + e ̺h cos( th ) + Re M ≥ − ν k M k − e − νh + c > , since ν < ν . This proves (c) and thus, the assertion follows by Theorem 3.7.
Remark . Note that M in (10) is allowed to have a non-trivial kernel which could alsodepend on the spatial variable (compare Subsection 4.1). Thus, Theorem 4.4 also covers acertain class of differential-algebraic equations with delay. We consider the following parabolic integro-differential equation ∂ ,̺ u + Bu − C ∗ Bu = f, (12)where B : D ( B ) ⊆ H → H is linear such that A := B − c is maximal monotone for some c > , C : [0 , ∞ [ → L ( H ) is a weakly measurable function such that t
7→ k C ( t ) k is measurable16 .3 Parabolic integro-differential equations and there exists ν > with R ∞ k C ( t ) k e ν t d t < . We set U := { z ∈ C | Im z ≤ ν } anddefine the complex Fourier transform of C by b C ( z ) := 1 √ π ∞ Z e − i tz C ( t ) d t ( z ∈ U ) , where the integral is meant in the weak sense. Note that b C : U → L ( H ) is continuous andbounded on U and analytic in the interior of U . The well-posedness and asymptotic behaviorfor equations of the form (12), including non-linear perturbations, were discussed in severalworks (e.g. [5, 25, 23] and [6, 20, 3, 7, 23] for a hyperbolic version of the problem), imposingadditional constraints on the kernel C .Following [23] we are led to assume that C satisfies the following conditions:1. C ( t ) is selfadjoint for almost every t ∈ R , C ( t ) and C ( s ) commute for almost every t, s ∈ R ,
3. for all t ∈ R we have t Im b C ( t + i ν ) ≤ . (13) Remark . (a) Note that (13) is equivalent to Im b C ( t + i ν ) ≤ t ∈ ]0 , ∞ [) . (b) A typical example for a kernel satisfying the conditions above is a real-valued, differentiablefunction k : [0 , ∞ [ → [0 , ∞ [ with R ∞ k ( t ) e ν t d t < and k ′ ( t ) ≤ − k ( t ) ν for every t ≥ .Similar kernels were considered by Prï¿ ss [20] under a weaker constraint on k ′ . Indeed,the conditions 1. and 2. are trivially satisfied, since k is real-valued. For showing condition3. we note that e ν t k ( t ) − e ν s k ( s ) ≤ sup ξ ∈ [ s,t ] e ν ξ ( ν k ( ξ ) + k ′ ( ξ )) ≤ , for every t ≥ s ≥ . Thus, the function t e ν t k ( t ) is non-increasing and we estimate Im b k ( t + i ν ) = 1 √ π ∞ Z e ν s sin( − ts ) k ( s ) d s = 1 √ π ∞ X k =0 (2 k +1) πt Z k πt e ν s sin( − ts ) k ( s ) d s + k +1) πt Z (2 k +1) πt e ν s sin( − ts ) k ( s ) d s = 1 √ π ∞ X k =0 (2 k +1) πt Z k πt sin( − ts ) (cid:16) e ν s k ( s ) − e ν ( s + πt ) k (cid:16) s + πt (cid:17)(cid:17) d s Here we use the fact that scalar analyticity and local boundedness on a norming set yields analyticity (see[12, Theorem 3.10.1]). Examples ≤ for every t ∈ ]0 , ∞ [ (compare [23, Remark 3.6 (b)]).(c) In [6] the authors considered real-valued kernels k : [0 , ∞ [ → R such that R ∞ k ( s ) e ν s d s < and the integrated kernel [0 , ∞ [ ∋ t R ∞ t k ( s ) e ν s d s gives rise to a positive definiteconvolution operator on L ([0 , ∞ [) . Again, the conditions 1. and 2. are satisfied, since k is real-valued and condition 3. holds according to [6, Proposition 2.2 (a)].Before we can state a stability result for problems of the form (12), we recall some propertiesof the convolution operator C ∗ . Lemma 4.7.
We denote by S ( R ; H ) the space of simple H -valued functions. Then for ̺ ≥ − ν the operator C ∗ : S ( R ; H ) ⊆ H ̺, ( R ; H ) → H ̺, ( R ; H ) u t Z R C ( t − s ) u ( s ) d s is bounded and linear with k C ∗ k L ( H ̺, ( R ; H )) ≤ R ∞ k C ( t ) k e ν t d t and can therefore be extendedto H ̺, ( R ; H ) . Moreover, for u ∈ H ̺, ( R ; H ) , ̺ ≥ − ν we have ( L ̺ ( C ∗ u )) ( t ) = √ π b C ( t − i ̺ ) ( L ̺ u ( t )) (14) for almost every t ∈ R .Proof. A proof for the first assertion can be found in [23, Lemma 3.1]. The proof of formula(14) is straight forward and we omit it.According to Lemma 4.7, the operator (1 − C ∗ ) is boundedly invertible in H ̺, ( R ; H ) for each ̺ ≥ − ν . Therefore, instead of considering (12) we can study (cid:0) ∂ ,̺ (1 − C ∗ ) − + B (cid:1) u = (1 − C ∗ ) − f, (15)or equivalently (cid:0) ∂ ,̺ (1 − C ∗ ) − + c + A (cid:1) u = (1 − C ∗ ) − f. Note, that this as an evolutionary equation of the form (2) where M is defined by M ( z ) = (1 − √ π b C ( − i z − )) − + cz, (cid:18) z ∈ C \ B C (cid:18) − ν , ν (cid:19)(cid:19) (16)where we have used Lemma 4.7. The next lemma shows that (13) already implies that thesame condition holds if one replaces − ν by ̺ for arbitrary ̺ ≥ − ν . Lemma 4.8.
Assume that C satisfies the conditions 1., 2. and 3. above. Then for every ̺ ≥ − ν we have t Im b C ( t − i ̺ ) ≤ . Proof.
The proof can be done analogously to the one of [23, Lemma 3.7].18 .3 Parabolic integro-differential equations
We now state our stability result for (15).
Theorem 4.9.
Let A : D ( A ) ⊆ H → H be maximal monotone and linear and let c > .Moreover, let C : [0 , ∞ [ → L ( H ) be weakly measurable, such that t
7→ k C ( t ) k is measurableand there exists ν > such that R ∞ k C ( t ) k e ν t < and C satisfies the conditions 1., 2. and3. from above. Then for each ̺ > the solution operator (cid:16) ∂ ,̺ (1 − C ∗ ) − + c + A (cid:17) − existsand is exponentially stable with a stability rate ν ∈ ]0 , ν ] satisfying ν − ∞ Z k C ( s ) k e ν s d s − ≤ c. Proof.
Let ν ∈ ]0 , ν ] such that ν (cid:0) − R ∞ k C ( s ) k e ν s d s (cid:1) − ≤ c. We prove that M given by(16) satisfies the hypotheses of Section 3. The assumption (a) is clear, since k√ π b C ( − i z − ) k < for each z ∈ C \ B C h − ν , ν i . Let now r > and ≤ ν < ν . Then for z ∈ B C ( r, r ) \{ ν − } we compute (1 − νz ) M ( z (1 − νz ) − ) = (1 − νz ) (cid:18)(cid:16) − √ π b C (cid:0) − i z − (1 − νz ) (cid:1)(cid:17) − + cz (1 − νz ) − (cid:19) = (1 − νz ) (cid:16) − √ π b C (cid:0) − i( z − − ν ) (cid:1)(cid:17) − + cz, which has a holomorphic extension in ν − . Noting that for each z ∈ B C ( r, r ) we have that z − = i t + ̺ for some ̺ > r , t ∈ R , we estimate k√ π b C ( − i( z − − ν )) k ≤ ∞ Z e − ( ̺ − ν ) s k C ( s ) k d s ≤ ∞ Z e ν s k C ( s ) k d s < . Hence, the extension of (1 − νz ) M ( z (1 − νz ) − ) to B C ( r, r ) is indeed bounded for each r > , ν ∈ [0 , ν [ . We now show that M satisfies the assumption (c) on C \ B C h − ν , ν i . Wefollow the strategy of the proof of [23, Lemma 3.8]. Let z ∈ C \ B C h − ν , ν i . Note thatthen there exists ̺ > − ν and t ∈ R such that z − = i t + ̺. We set D := | − √ π b C ( t − i ̺ ) | − ,which is well-defined according to Lemma 4.8. Note that (1 − √ π b C ( t − i ̺ )) − = D (1 − √ π b C ( − t − i ̺ )) , where we have used assumption 1. Moreover, due to assumption 2., we get that D (1 − √ π b C ( − t − i ̺ )) = D (1 − √ π b C ( − t − i ̺ )) D. eferences Thus, we obtain for x ∈ H Re h z − M ( z ) x | x i = Re D z − (cid:16) (1 − √ π b C ( − i z − )) − + cz (cid:17) x (cid:12)(cid:12)(cid:12) x E = Re D (cid:16) ̺ Re (cid:16) − √ π b C ( − t − i ̺ ) (cid:17) + √ πt Im b C ( − t − i ̺ ) (cid:17) Dx (cid:12)(cid:12)(cid:12) Dx E + c | x | ≥ ̺ Re D D (cid:16) − √ π b C ( − t − i ̺ ) (cid:17) Dx (cid:12)(cid:12)(cid:12) x E + c | x | . If ̺ is non-negative, the latter term can be estimated by c . For negative ̺ we observe that (cid:13)(cid:13)(cid:13) D Re (cid:16) − √ π b C ( − t − i ̺ ) (cid:17) D (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) − √ π b C ( − t − i ̺ ) (cid:17) − (cid:13)(cid:13)(cid:13)(cid:13) ≤ − k√ π b C ( − t − i ̺ ) k≤ − ∞ Z e ν s k C ( s ) k d s − and hence, ̺ Re D D (cid:16) − √ π b C ( − t − i ̺ ) (cid:17) Dx (cid:12)(cid:12)(cid:12) x E + c | x | ≥ ̺ − ∞ Z e ν s k C ( s ) k d s − + c | x | > − ν − ∞ Z e ν s k C ( s ) k d s − + c | x | , which shows that M satisfies hypothesis (c), according to the choice of ν . Thus, the assertionfollows by Theorem 3.7. Remark . Theorem 4.9 gives the exponential stability for equation (15). This also yieldsthe exponential stability of the original problem (12), since the operator (1 − C ∗ ) − leaves thespace H − ν, ( R ; H ) for all ν ≤ ν invariant. Indeed, observing that e νm (1 − C ∗ ) − = (1 − ( e νm C ) ∗ ) − e νm we obtain for f ∈ H − ν, ( R ; H ) ∩ H ̺, ( R ; H ) e νm (1 − C ∗ ) − f = (1 − ( e νm C ) ∗ ) − e νm f ∈ L ( R ; H ) . References [1] N. Akhiezer and I. Glazman.
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