Control of Some Linear Equations in a Hilbert ... - Semantic Scholar

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Control of Some Linear Equations in a Hilbert Space with Fractional Brownian Motions T.E. Duncan ∗ B. Maslowski ∗∗ B. Pasik-Duncan ∗∗∗ ∗

Department of Mathematics, University of Kansas, Lawrence, KS 66045 USA, (e-mail: [email protected]) ∗∗ Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic, (e-mail: [email protected]) ∗∗∗ Department of Mathematics, University of Kansas, Lawrence, KS 66045 USA, (e-mail: [email protected]) Abstract: A linear-quadratic control problem for some infinite-dimensional controlled stochastic differential equations driven by a fractional Gaussian noise is solved. The feedback form of the optimal control and the optimal cost are given. The optimal control is the sum of the well known linear feedback control for the associated deterministic linear-quadratic control problem and a suitable prediction of an optimal system response to the future noise. The covariance of the noise as well as the control operator may in general be unbounded, so the results can also be applied where the noise or the control are on the boundary of the domain or at discrete points in the domain. Some examples of controlled stochastic partial differential equations are given. Keywords: linear-quadratic control, stochastic control, stochastic distributed systems, fractional Brownian motion, stochastic partial differential equations. 1. INTRODUCTION The linear-quadratic Gaussian (LQG) control problem for the control of a finite dimensional linear stochastic system with a Brownian motion (white Gaussian noise) and a quadratic cost functional of the state and the control (e.g. Fleming et al. [1975]) is the most well known and basic solvable stochastic control problem for stochastic systems with continuous sample paths. Similarly the linearquadratic Gaussian control of an infinite dimensional linear stochastic system with a Brownian motion and a quadratic cost functional of the state and the control is the most well known and basic solvable stochastic control problem for infinite dimensional stochastic systems with continuous sample paths. This latter problem is not an immediate generalization of the finite dimensional problem because there are unbounded operators in the system equation and the interpretation of the solutions of the linear system and a Riccati equation need more refined definitions than for a finite dimensional linear system. The noise or perturbations of a system are typically modeled by a Brownian motion because such a process is Gauss-Markov and has independent increments. However empirical data from many physical phenomena suggest that Brownian motion is often inappropriate to use in the mathematical models of these phenomena. A family of processes that has empirical evidence of wide physical applicability is the collection of fractional Brownian motions. ? Research supported in part by NSF grant DMS 0808138, AFOSR grant FA9550-09-1-0554, ARO grant W911NF-10-1-0248, GACR grant no 201/07/0237 and MSM 0021620839.

Copyright by the International Federation of Automatic Control (IFAC)

Fractional Brownian motions are a family of Gaussian processes that were defined by Kolmogorov (Kolmogorov [1940]) in his study of turbulence. While this family of processes includes Brownian motion, it also includes other processes that describe behavior that is bursty or has a long range dependence. These other processes are neither Markov nor semimartingales. The first empirical evidence of the usefulness of these latter processes was provided by Hurst (Hurst, [1951]) in his statistical analysis of rainfall along the Nile River. Subsequently empirical justifications for modeling with fractional Brownian motions have been noted for a wide variety of physical phenomena, such as economic data, flicker noise in electronic devices, turbulence, internet traffic, biology, and medicine. The study of the solutions of stochastic equations in an infinite-dimensional space with a (cylindrical) fractional Brownian motion (for example, stochastic partial differential equations) has been relatively limited. Linear and semilinear equations with an additive fractional Gaussian noise (the formal derivative of a fractional Brownian motion), are considered in Duncan [2000], Duncan et al. [2002], Grecksch et al. [1999], Pasik-Duncan et al. [2006]. Strong solutions of bilinear evolution equations with a fractional Brownian motion are obtained in Duncan et al. [2006, 2005], and the same type of equation is studied in Tindel et al. [2003], where a fractional Feynman-Kac formula is obtained. Since fractional Brownian motions (FBMs) have a wide variety of potential applications, it is natural to consider the control of a linear stochastic system with an FBM and a

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

quadratic cost functional. It is natural to call such problems, (A1) One of the following two conditions is satisfied for B linear-quadratic fractional Gaussian (LQFG) control. Some and C in (1): initial work has been done on these problems. The control (a) B ∈ L(U, H), C ∈ L(V, H), where U = (U, < ·, · >U problems for stochastic equations driven by fractional , | · |U ) (the state space of controls) is a Hilbert space. noise have been studied only recently (cf. Kleptsyna et al. (b) (S(t), t ≥ 0) is an analytic semigroup and there [2003] where a one-dimensional problem is investigated are constants α ∈ (0, 1) and β ∈ (0, 1] such that and Duncan et al. [2009], Duncan et al. [2010] where a β−1 α−1 B ∈ L(U, DA ) and C ∈ L(V, DA ). multidimensional problem is investigated) and no results seem to be available for infinite-dimensional systems (e.g. stochastic partial differential equations) that are considered Note that in the case (A1)(b), B and C may be unbounded in this paper. as operators U → H and V → H, respectively (but the formulation requires the semigroup (S(t), t ≥ 0) to be 2. PRELIMINARIES analytic). Let U, V and H denote real, separable Hilbert spaces and For the family of controls the following assumption is introduced. consider the state equation H dX(t) = (AX(t) + Bu(t))dt + CdB (t) (1) (A2) u ∈ U := LpF = LpF ((0, T )×Ω; U ), where p > α1 , p ≥ 2 X0 = x ∈ H, is fixed and LpF denotes the linear subspace of all (Ft )in the space H, where t ≥ 0, A : H → H is a linear, progressively measurable processes in Lp ((0, T )×Ω; U ). (in general) unbounded operator that is the infinitesimal If B ∈ L(U, H) then p can be chosen to satisfy only generator of a strongly continuous semigroup (S(t), t ≥ 0) p ≥ 2. H and (B (t), t ≥ 0) is a cylindrical fractional Brownian motion on the space V , defined on a filtered probability space (Ω, F, (Ft )t≥0 , P), that is, Obviously, if B ∈ L(U, H) or if α > 1/2, then it is appropriate to choose p = 2 because it is not reasonable ∞ to reduce the space of controls. If, however, α ≤ 1/2 the X p B H (t) = ei λi βi (t), t > 0 (2) operator B is too singular and it is necessary to restrict the space of controls so that the solution to the controlled i=1 where {ei , i ∈ N} is a complete orthonormal basis in equation would be well defined. V , (βi (t), i ∈ N, t ≥ 0) is a family of (stochastically) The following condition is used for the stochastic convoindependent, real-valued, standard fractional Brownian lution integral (appearing in the variation of constants motions with the Hurst parameter H ∈ ( 21 , 1) fixed and formula to obtain the solution for the controlled system) λi ≥ 0, (λi , i ∈ N) is a bounded sequence in R+ . It can to be well defined. be assumed that the filtration (Ft , t ≥ 0) satisfies the so-called usual conditions (e.g. Karatzas et al. [1988]). exist some constants T0 > 0 and η > 0 such For the control problem here it is natural to assume that (A3) There RT RT e 1/2 |L (V,H) (Ft , t ≥ 0) is the P-completion of σ(B(s), s ≤ t). The that 0 0 0 0 r−η s−η |Sr C Q 2 H e of (B (t), t ≥ 0) is defined by e 1/2 |L (V,H) φH (r − s)drds < ∞ incremental covariance Q |Ss C Q 2 e n = λn en , n ∈ N where φH (r) := H(2H − 1)|r|2H−2 . Qe (3) e is a trace class operator on V , (it is not required that Q so the series in (2) may diverge in the space V , cf.Duncan Consider the equation (1) with feedback controls of the et al. [2006], Duncan et al. [2002] for the basic theory form u(t) = K(t)X(t) + h(t), where (X(t), t ≥ 0) satisfies of fractional Brownian motions and stochastic integrals the equation for t ∈ [0, T ] driven by fractional Brownian motions that is relevant in dX(t) = (AX(t) + B(K(t)X(t) + h(t)))dt + CdB H (t) the present case. (4) In this paper the following notation is used: If Y, Z are X0 = x ∈ H Hilbert spaces, let < ·, · >Y and | · |Y denote the inner h ∈ U and K ∈ C ([0, T ], L(H, U )) (where the space s product and norm on the space Y , respectively, let L(Y, Z) C ([0, T ], L(Y, X)) is the space of strongly continuous s and L2 (Y, Z) denote the spaces of bounded linear and L(Y, X)-valued operators and Y and Z are Banach spaces). Hilbert-Schmidt operators from Y to Z, respectively and The (mild) solution to the equation (4) is defined in the let L(Y ) = L(Y, Y ) and L2 (Y ) = L2 (Y, Y ). usual way by the mild formula for t ∈ [0, T ] Z t In a part of this paper it is assumed that the semigroup X(t) = S(t)x + S(t − r)B(K(r)X(r) + h(r))dr + Z(t) (S(t), t ≥ 0) is analytic (Pazy, [1984]). In that case there 0 ˆ is a strictly negative exists βˆ ≥ 0 such that Aˆ := A − βI (5) α operator. Let DA , α ≥ 0 be the domain of the fractional (which coincides with the mild solution of (1) for u(t) = ˆ α equipped with the norm |x|Dα := |(−A) ˆ α x|H K(t)X(t) + h(t)) where Z is the stochastic convolution. power (−A) A α ∗ Since the mapping (t, x) 7→ f (t, x) := K(t)x is continuous (and similarly DA∗ , | · |DAα ∗ for the adjoint A ). as a map [0, T ] × H → U and f (t, ·) is Lipschitz for each Some assumptions are given now. t ∈ [0, T ], it is easy to see that (5) has a pathwise unique H-continuous solution. 3241

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The cost functional is defined as Z T 1 (< QXs , Xs >H + < Rus , us >U )ds J(x, u) := E 2 0 (6) 1 + E < GXT , XT >H 2 for x ∈ H and u ∈ U , where Q, R and G are linear operators satisfying (A4) Q, G ∈ L(H), Q ≥ 0, G ≥ 0, R ∈ L(U ), R ≥ 0, Q, G and R are self-adjoint. The problem is to minimize the cost functional J(x, u), that is, e J(x) := inf J(x, u) (7)

for all t ∈ (0, T ), x, y ∈ DA , and V (0) = V0 . The mapping V satisfying (9) (or equivalently, (10)) is called the mild (or weak) solution to the equation (8) with V0 = G. Subsequently let P (t) = V (T − t), where V is the mild solution to (8) with V0 = G, and let e := P (t)BR−1 B ∗ . B(t) (11) 1−α ∗ By the fact that P ∈ Cs ([0, T ], L(H, DA ∗ )) and P = P on H there is a unique extension of P (denoted again α−1 e ∈ by P ) such that P ∈ Cs ([0, T ], L(DA , H)), hence B 1−α Cs ([0, T ], L(DA∗ , H)).

For t ≥ s consider the equation e y(t) ˙ = A∗ y(t) − B(t)y(t) y(s) = x ∈ H.

u∈U

and (for given x ∈ H) to find an optimal control u ˆ∈U e that achieves the infimum in (7), that is, J(x, u ˆ) = J(x). 3. OPTIMAL CONTROLS

Consider the stochastic integral Z T ϕ(t) := UP (s, t)P (s)CdB H (s).

(12)

(13)

t

The hypotheses (A1)–(A4) are restricted slightly. In addition, the following assumption is introduced.

for t ∈ [0, T ], which will later play an important role in the formulation of our main result and UP is the fundamental solution of the homogeneous adjoint of (12).

(A5) The following three conditions are satisfied. e 1−H, where α, β are defined in (A1)(b) (c) R has a bounded inverse, that is R−1 ∈ L(U ), 1−α and G ∈ L(H, DA ∗ ).

Lemma 1. If (A1), (A3)–(A5) are satisfied, then the process ϕ given by (13) is a well-defined centered Gaussian 1−α process in Lp (Ω × (0, T ), DA ∗ ).

The condition (A5)(a) implies that (B H (t), t ≥ 0) is a ”genuine” V -valued fractional Brownian motion, not merely a cylindrical fractional Brownian motion. The inequality β ≥ α means, roughly speaking, that the diffusion operator C is not more ”unbounded” than the control operator B. The formal Riccati equation in this case is V˙ = A∗ V + V A − V BR−1 B ∗ V + Q

(8)

V (0) = G which does not include the noise linear operator C, so the well-known deterministic result for this case ( Flandoli, [1986]) can be used. Let Σ+ = {V ∈ L(H), V = V ∗ ; V ≥ 0}. This deterministic result is given in the following theorem. The proof of this result and the following results are given in Duncan et al. [2011].

Note that the solution of the controlled equation is well defined by the mild formula even if the control u ∈ V := Lp ([0, T ] × Ω, U ) is not adapted. Using the operator-valued mapping P and the process ϕ the following result for a nonadapted control is obtained. It may be of independent interest but it also will be used later to prove a corresponding result for adapted controls. Theorem 2. Let (A1)–(A5) be satisfied, let x ∈ H and u ∈ U be arbitrary, let ϕ be given by (13) and let (Xt , t ≥ 0) be the solution of the controlled equation (1). Let V = Lp ([0, T ]×Ω, U ) be the linear space of nonadapted controls. The optimal control v¯ ∈ V for the control problem (1)–(6) with U replaced by V is v¯(t) = −R−1 B ∗ (P (t)X(t) + ϕ(t))

(14)

where (X(t), t ∈ [0, T ]) is the solution of the controlled equation (1) with u = v¯ and ϕ is given by (13). The optimal cost J˜ is

Theorem 1. If (A1), (A4) and (A5) are satisfied, then for 1−α an arbitrary V0 ∈ Σ+ ∩ L(H, DA ∗ ) there exists a unique 1−α operator-valued function V ∈ Cs ([0, T ], L(H, DA ∗ )) ∩ + Cs ([0, T ], Σ ) such that V (t) = S ∗ (t)V0 S(t) (9) Z t + S ∗ (t − s)(Q − (B ∗ V (s))∗ R−1 B ∗ V (s))S(t − s)ds 0

for all t ∈ [0, T ] or (equivalently) d < V (t)x, y >H =< V (t)x, Ay >H + < Ax, V (t)y >H dt + < Qx, y >H − < R−1 B ∗ V (t)x, B ∗ V (t)y >U (10) 3242

Z T ˜ = 1 < P (0)x, x >H − 1 E J(x) |R−1 B ∗ ϕ(s)|2U ds 2 2 0 Z TZ s e H (r − s)drds + T r[C ∗ P (s)U ∗ (s, r)C Q]φ 0

0

1 = < P (0)x, x >H (15) 2 Z TZ TZ T 1 e ∗ P (q) − T r(R−1 B ∗ U (r, s)P (r)C QC 2 0 s s U ∗ (q, s)R−1 B)]φH (r − q)dqdrds Z TZ s e H (r − s)drds + T r[C ∗ P (s)U ∗ (s, r)C Q]φ 0

0

Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

Theorem 3. If (A1)–(A5) are satisfied, then there exists a unique optimal control (¯ u(t), t ∈ [0, T ]) in U for the control problem (1)–(6) that is given by u ¯(t) = −R−1 B ∗ P (t)X(t) − R−1 B ∗ ψ(t) (16) and ψt := E[ϕ(t)|Ft ] (17) Z t −(H−1/2) = s−(H−1/2) (It− 0 (H−1/2)

(IT−

uH−1/2 (UP (·, t)P (·)C)))(s)dB H (s)

where t ∈ [0, T ], ua (s) = sa I for s > 0, a ∈ R, and Ita− denotes the left-sided fractional Riemann-Liouville integral (?), Z t 1 a (It− h)(s) := (λ − s)a−1 h(λ)dλ Γ(a) s s ∈ (0, t), h ∈ L1 (0, t, L(U, H)), a > 0 and Γ is the (Euler) Gamma function. The optimal cost is 1 e J(x) = J(x, u ¯) = < P (0)x, x >H (18) 2 Z T 1 − E |R−1 B ∗ ψ(s)|2U ds 2 0 Z TZ s ˜ H (r − s)drds. + T r[C ∗ P (s)U ∗ (s, r)C Q]φ 0

0

It can be verified (Duncan et al. [2006]) that the RHS of (17) is a zero mean Gaussian random variable. While it is assumed that H > 21 , the optimal control given in Theorem 3 is also valid for the well known case of H = 12 (i.e., the case when (B H (t), t ≥ 0) is a V -valued Wiener process), that is, by the independent increments of (B H (t), t ≥ 0) it follows that "Z # T 1/2 ψt = E[ϕ(t)|Ft ] = E UP (s, t)P (s)CdB (s) = 0 t

so the optimal control is u ¯(t) = −R−1 B ∗ P (t)X(t). To demonstrate the analogous correspondence for the optimal cost, it is necessary to verify the suitable expectation which for H = 12 is ∞ Z T Z s X E hλij (s, t)dβ i (t) ◦ dβ j (t) = (19) i,j=1

0

0

∞ Z T X

hλii (s, s)ds

0

i=1

T

Z

˜ T r[C ∗ R∗ (λ)P (s)R(λ)C Q]ds

= 0

This family of integrals can be shown to converge to Z T ˜ T r[C ∗ P (s)C Q]ds. 0

as λ → ∞. Summarizing, the well-known expression for the optimal cost for a V -valued Wiener process is obtained 1 ˜ J(x) = J(x, u ¯) = < P (0)x, x >H (20) 2 Z T ˜ + T r[C ∗ P (s)C Q]ds. 0

4. SOME EXAMPLES Consider the stochastic controlled heat equation ∂y (t, ξ) = ∆y(t, ξ) + u(t, ξ) + η(t, ξ) (21) ∂t for (t, ξ) ∈ R+ × D with the following initial condition and Dirichlet boundary conditions u(0, ξ) = x(ξ) (22) for ξ ∈ D and u|R+ ×∂D = 0 (23) where D ⊂ Rd is a bounded domain with a smooth boundary, u is the control and η is a noise process that is the formal time derivative of a space dependent fractional Brownian motion. To provide a precise meaning to (21)– (23), the parabolic system is rewritten as an infinite dimensional stochastic differential equation dX(t) = AX(t)dt + u(t)dt + dB H (t) (24) for t ≥ 0 in the space H = L2 (D), where A = ∆|Dom(A) generates an analytic semigroup (S(t), t ≥ 0) on H with Dom(A) = H 2 (D) ∩ H01 (D), U = V = H and the noise η is modeled as the formal derivative (dB H /dt)(t), where (B H (t), t ≥ 0) is a cylindrical fractional Brownian motion ˜ ∈ L(V ). The assumptions in in V with covariance Q Theorem 3 for an optimal control can be verified. Consider a stochastic wave equation formally described as ∂2y ˜ (t, ξ) = ∆u(t, ξ) + B(u(t))(ξ) + η(t, ξ) (25) ∂t2 for (t, ξ) ∈ R+ × D where D and η satisfy the conditions in ˜ belongs to the previous example and the control operator B L(U, L2 (D)), U being an arbitrary control (Hilbert) space. The initial and boundary conditions are ∂y (0, ξ) = x1 (ξ) (26) ∂t y(0, ξ) = x2 (ξ) (27) y(t, ξ) = 0 (28) for ξ ∈ D and (t, ξ) ∈ R+ × ∂D, respectively. The corresponding infinite dimensional stochastic differential equation is dX(t) = AX(t)dt + Bu(t)dt + dB H (t) (29) X(0) = x = (x1 , x2 ). with the following choice of operators and spaces: Let Λ = ∆|Dom(Λ) , Dom(Λ) = H01 (D) ∩ H 1 (D), Dom(A) = Dom(Λ) × Dom((−Λ)1/2 ) and   0 I A= (30) Λ 0 It is well known that A generates a strongly continuous semigroup in the space H = Dom(−Λ)1/2 × L2 (D). Let (B H (t), t ≥ 0) be a fractional Brownian motion on H with the covariance   0 0 ˜ Q= ˜2 0 Q 1/2

˜ where Q is a Hilbert-Schmidt operator on L2 (D). The 2 control operator B is defined as   0 0 B= ˜ 0 B

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Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011

and clearly B ∈ L(U, H). The space of controls is chosen as U = L2F (U ). The assumptions in Theorem 3 for an optimal control can be verified. Consider the heat equation with boundary noise and control ∂y (t, ξ) = ∆y(t, ξ) (31) ∂t for (t, ξ) ∈ R+ × D with the initial condition y(0, ξ) = x(ξ) for ξ ∈ D and the boundary conditions that are either of Dirichlet type y(t, ξ)|R+ ×∂D = u(t, ξ) + η(t, ξ) (32) or of Neumann type ∂y (t, ξ) = u(t, ξ) + η(t, ξ) (33) ∂ν for (t, ξ) ∈ R+ × ∂D where ∂/∂ν is a normal derivative, D is a bounded domain in Rd with smooth boundary ∂D, u is the control and η is a noise process on ∂D. For suitable choice of α the assumptions in Theorem 3 for an optimal control can be verified. The authors are not aware of any numerical work on the control of stochastic partial differential equations with fractional Brownian motions. 5. CONCLUDING REMARKS An optimal control has been given explicitly for a linear stochastic equation in a Hilbert space driven by a fractional Brownian motion with the Hurst parameter H ∈ ( 21 , 1) and a cost functional that is quadratic in the state and the control. The method is also applicable to the case of a Brownian motion, that is, H = 21 . These stochastic equations can model linear stochastic partial differential equations and linear stochastic hereditary (delay) equations. For future work it is desirable to extend the optimal control results to H ∈ (0, 12 ) and thereby to include all fractional Brownian motions. However in this case it is known (Duncan et al. [2002], Pasik-Duncan et al. [2006]) that the conditions for solutions of the linear equations differ from the case for H ∈ ( 12 , 1). Furthermore it is also important to consider the associated ergodic control problems with H ∈ ( 12 , 1) and H ∈ (0, 12 ). REFERENCES T. E. Duncan, Prediction for some processes related to a fractional Brownian motion. Stat. Prob. Lett., 76: 128-134, 2006. T. E. Duncan, J. Jakubowski, and B. Pasik-Duncan, Stochastic integration for fractional Brownian motions in a Hilbert space. Stoch. Dyn., 6: 53-75, 2006.

T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stochastic Process. Appl., 115: 1357-1383, 2005. T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Linearquadratic control for stochastic equations in a Hilbert space with fractional Brownian motions, preprint. T. E. Duncan and B. Pasik-Duncan, Control of some linear systems with a fractional Brownian motion. Proc. 48th IEEE Conference Decision and Control, 8518-8522, 2009. T. E. Duncan and B. Pasik-Duncan, Stochastic linearquadratic control for systems with a fractional Brownian motion. Proc.49th IEEE Conference on Decision and Control, 6163-6168, 2010. T. E. Duncan, B. Pasik-Duncan and B. Maslowski, Fractional Brownian motions and stochastic equations in Hilbert spaces. Stoch. Dyn., 2: 225-250, 2002. F. Flandoli, Direct solutions of a Riccati equation arising in a stochastic control problem with control and observation on the boundary. Appl. Math. Optim., 14: 107-129, 1986. W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control. Springer-Verlag, New York, 1975. W. Grecksch and V. V. Anh, A parabolic stochastic differential equation with fractional Brownian motion input. Statist. Probab. Lett., 41: 337-346, 1999. H. E. Hurst, Long-term storage capacity in reservoirs. Trans. Amer. Soc. Civil Eng., 116: 400-410, 1951. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, New York, 1988. M. L. Kleptsyna, A. Le Breton and M. Viot, About the linear quadratic regulator problem under a fractional Brownian perturbation. ESAIM Probab. Stat., 9: 161170, 2003. A. N. Kolmogorov, Wienersche spiralen und einige andere interessante kurven in Hilbertschen Raum. C.R. (Doklady) Acad. USSS (N.S.), 26: 115-118, 1940. B. Pasik-Duncan, T. E. Duncan and B. Maslowski, Linear stochastic equations in a Hilbert space with a fractional Brownian motion. In H. Yan, G. Yin and Q. Zhang, Editors, Stochastic Processes, Optimization, and control Theory: Applications in Financial Engineering, Queueing Networks, and Manufacturing Systems Springer-Verlag, New York, 2006. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983. S. G. Samko, A. A. Kilbas, and O. I Marichev, Fractional Integrals and Derivatives Gordon and Breach, Yverdon, 1993. S. Tindel, C. A. Tudor, and F. Viens, Stochastic evolution equations with fractional Brownian motion. Probab. Theory Related Fields, 127: 186-204, 2003.

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