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International Journal of Stochastic Analysis
Volume 2013 (2013), Article ID 868301, 16 pages
Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion
Department of Mathematics, West Chester University of Pennsylvania, 25 University Avenue, West Chester, PA 19343, USA
Received 1 October 2013; Accepted 9 November 2013
Academic Editor: Ciprian A. Tudor
Copyright © 2013 Mark A. McKibben. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model.
The mathematical description of the dynamic buckling has been the subject of investigation for decades and is of interest to the engineering world. Dynamic buckling arises in various ways including vibrations, single load pulses of large amplitude, occurrence of a suddenly applied load, and flutter enhanced bending (see ). For the purpose of our study we restrict our attention to the dynamic buckling of a hinged extensible beam which is stretched or compressed by an axial force.
Dickey  initiated an investigation of the hyperbolic partial differential equation where is Young’s modulus, is the cross-sectional moment of inertia, is the density, is an axial force, is the natural length of the beam, and is the cross-sectional area; these are all positive parameters. Here, describes the transverse deflection of an extensible beam at point at time . The nonlinear term in (1) accounts for the change in tension of the beam due to extensibility. The ends of the beam are held at a fixed distance apart and the ends are hinged; this translates to the following boundary conditions: The initial conditions given by describe the initial deflection and initial velocity at each pointof the beam.
Fitzgibbon  and Ball  established the general existence theory for IBVP (1)–(3). Patcheu  subsequently incorporated a nonlinear friction force term into the mathematical model to account for dissipation; this was done by replacing the right side of (1) by the term , where is a bounded linear operator. More recently, Balachandran and Park  further introduced the term into (1) to account for the fact that during vibration, the elements of the beam not only undergo translator motion but also rotate.
The above results were established for the deterministic case and did not account for environmental noise. As Kannan and Bharucha-Reid  points out, the variability in measurement obtained experimentally suggests that studying a stochastic version of the model is advantageous. Indeed, doing so enables us to understand the effects of noise on the behavior of the phenomenon. To this end, Mahmudov and McKibben  studied a stochastic version of (1) (with damping), assuming that noise was incorporated into the model via a standard one-dimensional Brownian motion process.
The purpose of the present work is to generalize those results in two ways. One of the directions for this generalization is that we broaden the class of nonlinear forcing terms incorporated into the model. We do so in two ways, one of which is to consider more general functional-type forcing terms that capture general integral terms, semilinear terms, and others under the same abstract form. For the second way, we point out that accounting for certain types of nonlinearities in the mathematical modeling of some phenomena—for instance, nonlinear waves and the dynamic buckling of a hinged extensible beam—requires that the nonlinearities depend on the probability distribution of the solution process. This notion was first studied in the finite-dimensional setting [9, 10], was initiated in the infinite-dimensional setting by Ahmed and Ding , and subsequently was studied by various authors [12, 13].
The second direction of the generalization is to incorporate noise into the model via a more general fractional Brownian motion process. Regarding the mathematical description of the noise term, it is often the case that the standard Brownian motion is insufficient. Indeed, it has been shown that certain processes arising in a broad array of applications—such as communication networks, self-similar protein dynamics, certain financial models, and nonlinear waves—exhibit a self-similarity property in the sense that the processes and have the same law (see [14–16]). Indeed, while the case when generates a standard Brownian motion, concrete data from a variety of applications have exhibited other values of , and it seems that this difference enters in a nonnegligible way in the modeling of this phenomena. In fact, since is not a semimartingale unless , the standard stochastic calculus involving the Itó integral cannot be used in the analysis of related stochastic evolution equations. Several authors have studied the formulation of stochastic calculus for fBm and differential/evolution equations driven by fBm in the past decade [14–18].
Specifically, we consider the following stochastic IBVP: where , , (a complete probability space), is a fractional Brownian motion, and the nonlinear forcing term will take on various forms including where is a family of probability measures, and , , and are appropriate mappings.
The plan of the paper is as follows. We will collect some preliminary information on cosine function theory, special function spaces, probability measures, and fractional Brownian motion in Section 2. Then we present abstract formulations of the IBVPs under consideration in Section 3. The main theoretical results together with corollaries illustrating the applicability to the IBVPs introduced above are gathered in Section 4. The proofs of the main results are provided in Section 5, followed by an interpretation of these results for the specific nonlinear beam model in Section 6. An extensive reference list then follows.
For details of this section, we refer the reader to [4, 16, 17, 19, 20] and the references therein. Throughout this paper, and are real separable Hilbert spaces with norms and , and inner products and equipped with a complete orthonormal basis . Also, is a complete probability space. Henceforth, for brevity, we will suppress the dependence of random variables on .
We make use of several different function spaces throughout this paper. is the space of all bounded linear operators on, while stands for the space of all -valued random variables for which . Also, stands for the space of -continuous -valued random variables such that The space of Hilbert Schmidt operators from into is denoted by and is equipped with the norm .
The remaining function spaces coincide with those used in ; we recall them here for convenience. First, stands for the Borel class on and represents the space of all probability measures defined on equipped with the weak convergence topology. Define by , , and consider the space For , we let where is the Jordan decomposition of and . Then define the space equipped with the metric given by It is known that is a complete metric space. The space of all continuous -valued functions defined on is denoted by is complete when equipped with the metric We recall some facts about cosine families of operators defined as follows.
Definition 1. (i) The one-parameter family , satisfying(a),(b) is continuous in on , for all ,(c), for all , is called a strongly continuous cosine family.
(ii) The corresponding strongly continuous sine family is defined by , for all , for all .
(iii) The (infinitesimal) generator of is given by , for all .
It is known that the infinitesimal generator is a closed, densely-defined operator on (see ). Such a cosine and (corresponding sine) family and its generator satisfy the following properties.
Proposition 2. Suppose thatis the infinitesimal generator of a cosine family of operators (cf. Definition 1). Then the following hold.(i)There exist and such that and hence, ,(ii), for all ,(iii)There exists such that , for all .
The Uniform Boundedness Principle, together with (i) above, implies that both and are uniformly bounded by .
Next, we make precise the definition of a -valued fBm and related stochastic integral used in this paper. The approach we use coincides with the one formulated and analyzed in [15, 17]. Let be a sequence of independent, one-dimensional fBms with Hurst parameter such that for all (i),(ii),(iii),(iv).
In such case, , so that the following definition is meaningful.
Definition 3. For every , is a -valued fBm, where the convergence is understood to be in the mean-square sense.
It has been shown in  that the covariance operator of is a positive nuclear operator such that We now outline the discussion leading to the definition of the stochastic integral associated with for bounded, measurable functions. To begin, assume that is a simple function; that is, there exists such that where and .
Definition 4. The -valued stochastic integral is defined by
As argued in Lemma 2.2 of , this integral is well-defined since
Since the set of simple functions is dense in the space of bounded, measurable -valued functions, a standard density argument can be used to extend Definition 4 to the case of a general bounded measurable integrand.
Finally, in addition to the familiar Young, Hölder, and Minkowski inequalities, the inequality of the form , where is a nonnegative constant and , will be used to establish various estimates.
3. Abstract Formulation
The goal of this section is to reformulate the stochastic IBVPs introduced in Section 1 as abstract stochastic evolution equations. There will be two related abstract formulations depending on the nature of the nonlinearity. We begin with the reformulation (4)–(6) assuming that the forcing term is given by (9).
We impose the following conditions:(A1) satisfies the Caratheodory conditions (i.e., measurable in and continuous in the third variable) such that(i)there exists for which , for all , , and ;(ii)there exists for which , for all , , and ;(A2) satisfies the Caratheodory conditions such that(i)there exists for which , for all , , and ;(ii) belongs to , for every , ;(A3) is a bounded measurable function;(A4) is the probability law of , for each ;(A5), .
We reformulate the associated IBVP by making the following identifications as in ; we recall the highlights of that formulation here for completeness of the discussion.
Let (note they are real separable Hilbert spaces) and identify the solution process by Define the operator by It is known that is a positive, self-adjoint operator on (cf. [21, 22]). As such, generates a strongly continuous cosine family of operators on . We will express the other terms on the left side of (4) using fractional powers of , as in . To this end, the eigenvalues of are with corresponding eigenvalues Then has spectral representation The following operators are well defined because the fractional powers of are positive and self-adjoint: Also, observe that It can be shown using standard computations involving the inner product and properties of fractional powers of that and are uniformly bounded on .
Define the mappings , , , and , and the operator as follows: for all and .
These identifications can be used to formulate (4)–(6), with nonlinear forcing term given by (9) as the following abstract second-order stochastic evolution equation in a real separable Hilbert space : in a real separable Hilbert space . (By the probability distribution of , we mean , for each .) Here, is a linear (possibly unbounded) operator which generates a strongly continuous cosine family on with associated sine family ; ; is a bounded, measurable mapping; is a bounded linear operator; is a -valued fBm with Hurst parameter ; and.
We next consider the same IBVP, with the exception that the forcing term is given by a mapping of the form (7) or (8). While all other identifications remain identical, we will describe such nonlinear forcing terms more generally as a so-called functional. Precisely, we consider the following abstract second-order functional stochastic evolution equation in a real separable Hilbert space : where is a given functional and all other identifications are the same as for (26). We impose the following conditions for (7) and (8):(A6);(A7) are mappings that satisfy the Caratheodory conditions and are such that there exists such that for all , , and ;(A8) satisfies the Caratheodory conditions and is such that there exists such that for all , , and ;(A9);(A10), where satisfies for all and . Identify the solution process by It can be shown that under these assumptions, the following are well-defined mappings from into : The strategy is to formulate a general theory for these two abstract second-order problems and then to apply the results to the specific IBVPs formulated in Section 1.
4. Statement of Results
We consider mild solutions of (26) in the following sense.
Definition 5. A stochastic process is a mild solution of (26) if(i)(ii) is the probability distribution of , for all .
The following conditions on (26) are imposed on the data and mappings in (26):(A11) is the infinitesimal generator of a strongly continuous cosine family on with associated sine family on ;(A12) satisfies the following.(i)There exists a positive constant such that globally on ,(ii)there exists a positive constant such that globally on ;(A13) is a bounded, measurable mapping;(A14) is a bounded linear operator;(A15) is a -valued fBm;(A16) are -measurable, where is the family of -algebras generated by .
Henceforth, we write which is finite by (A11).
The following lemma, introduced in (me and dave) and stated here without proof, is critical in establishing several estimates.
Lemma 6. Assume that satisfies (A13). Then, for all, where is a positive constant depending on , , and the growth bound on g, and is defined as in the discussion leading to Definition 4.
Let be fixed and define the solution map by The first two integrals on the right side of (39) are taken in the Bochner sense, while the third is defined in Section 2. The operator satisfies the following properties.
Lemma 7. If (A11)–(A16) hold, then is a well-defined -continuous mapping.
Our first main result is as follows.
Theorem 8. If (A11)–(A16) hold, then (26) has a unique mild solution on with corresponding probability law . Further, for every , there exists a positive constant (depending on , , and ) such that
Mild solutions of (26) depend continuously on the initial data and probability distribution of the state process in the following sense.
Proposition 9. Assume that (A11)–(A16) hold, and let and be the mild solutions of (26) (as guaranteed to exist by Theorem 8) corresponding to initial data , and , with respective probability distributions and . Then there exists a positive constant such that
Next, we formulate a result in which a deterministic initial-value problem is approximated by a sequence of stochastic equations of a particular form of (26) arising frequently in applications. Specifically, consider the deterministic initial-value problem where , , is a bounded linear operator, satisfies (A11), and satisfies the following conditions:
(A17)(i)there exists a positive constant such that globally on ;(ii)there exists a positive constant such that globally on .
It is known that under these conditions, (42) has a unique mild solution given by the representation formula For each , consider the stochastic initial-value problem in . Here, we assume that and , for all and that are given mappings. We impose the following conditions on the data and mappings in (44), for each .(A18) generates a strongly continuous cosine family satisfying strongly as , uniformly in . Also, the corresponding sine family satisfying strongly as , uniformly in . Moreover, for all and is the same bound used for the cosine and sine families generated by .(A19) is a bounded linear operator such that in and , for all .(A20) is Lipschitz in the second variable (with the same Lipschitz constant used for in (A17)), and as , for all , uniformly in .(A21) is a continuous mapping such that uniformly in as .(A22) is a bounded, measurable function such that as , uniformly in .
Under these assumptions, the following result holds.
Definition 11. A stochastic process is a mild solution of (27) if for all .
We assume that conditions (A11), (A13)–(A16) hold, in addition to the following condition on the functional forcing term:(A23) is such that there exists for which Define the solution map by (cf. (39)). The operator satisfies the following properties.
Lemma 12. If (A11), (A13)–(A16), and (A23) hold, then is a well-defined -continuous mapping.
The proof of the following theorem follows almost immediately from this result.
Theorem 13. If (A11), (A13)–(A16), and (A23) hold, then (27) has a unique mild solution on provided that . Further, for every , there exists a positive constant (depending on , , and ) such that
Corollary 14. Assume that (A11) and (A13)–(A16) hold.(i)If (A6) and (A7) are satisfied, then (27) where the forcing term is given by (33) has a unique mild solution on [0,T] provided that Moreover, for every , estimate (49a) holds.(ii)If (A8)–(A10) are satisfied, then (27) where the forcing term is given by (32) has a unique mild solution on provided that Moreover, for every , estimate (49a) holds.
5. Proofs of Main Results
Proof of Lemma 7. Let be fixed and consider the solution map defined in (39).
One can see from the discussion in Section 2 and the properties of that for any , is a well-defined stochastic process, for each . In order to verify the -continuity of on , let and consider and sufficiently small so that all terms are well defined. Observe that The strong continuity of and implies that Using (A11) and (A14), together with Hölder’s inequality and properties of cosine operators, yields The strong continuity of , along with the Lebesgue dominated convergence theorem, implies that the right side of (52) goes to 0 as . Regarding the fourth term of (50), similar computations involving (A11) and (A12) yield the estimate Since and the right side is independent of , it follows immediately from (A11) that the right side of (53) goes to 0 as .
It remains to show that as . Observe that For the moment, assume that is a simple function as defined in (16). Observe that for , arguing as in Lemma 6 of  yields where . Hence, and the right side of (57) goes to 0 as . Next, observe that Using the property with and , we can argue as above to conclude that the right side of (58) goes to 0 as . Consequently, as when is a simple function. Since the set of all such simple functions is dense in , a standard density argument can be used to extend this conclusion to a general bounded, measurable function . This establishes the -continuity of .
Finally, we assert that . Indeed, the necessary estimates can be established as above, and when used in conjunction with Lemma 6, one can readily verify that , for any . Thus, we conclude thatis well defined, and the proof of Lemma 7 is complete.
Proof of Theorem 8. Let be fixed and consider the operator as defined in (39). We know that is well defined and -continuous from Lemma 7. We now prove that has a unique fixed point in . Indeed, for any , (39) implies that
where . For any natural number , it follows from successive iteration of (59) that
Since for large enough values of , we can conclude from (60) that is a strict contraction on and so for a given , has a unique fixed point (by the Banach contraction mapping principle), and this coincides with a mild solution of (26) as desired.
To complete the proof, we must show that is, in fact, the probability law of . To this end, let represent the probability law of and define the map by . It is not difficult to see that , for all since . In order to verify the continuity of the map , we first comment that an argument similar to the one used to establish Lemma 7 can be used to show that for sufficiently small Consequently, since for all and , it is the case that and hence for any . Thus, is a continuous map, so that , thereby showing that is well defined. In order to show that has a unique fixed point in , let and let be the corresponding mild solutions of (26). A standard computation yields Observe that . Thus, continuing the inequality in (64) yields Applying Gronwall’s lemma now yields We can choose to ensure , so that taking the supremum above yields Since (which follows directly from (62) and (63)), we have so that is a strict contraction on . Thus, (26) has a unique mild solution on with probability distribution . The solution can then be extended, by continuity, to the entire interval in finitely many steps, thereby completing the proof of the existence-uniqueness portion of the theorem.
Next, we establish the boundedness of th moments of the mild solutions of (26). Let and observe that the standard computations yield, for all ,
Applying the Jensen and Hölder inequalities enables us to continue the string of inequalities above as follows: