A functional analytic approach to infinite dimensional stochastic linear systems
AA FUNCTIONAL ANALYTIC APPROACH TO INFINITEDIMENSIONAL STOCHASTIC LINEAR SYSTEMS
F.-Z. LAHBIRI AND S. HADD
Abstract.
In this paper, we are going to present a functional analytic approach tothe well-posedness of input-output infinite dimensional stochastic linear systems. Thisis in fact a stochastic version of the known Salamon-Weiss linear systems. We also provecontrollability and observability properties of such stochastic systems. On the otherhand, we use the obtained results to prove a new variation of constants formula forperturbed partial stochastic differential equations. Several applications and examplesare given. Introduction
In the last decades, the abstract theory of deterministic linear systems has been oneof the main themes in infinite-dimensional systems theory. In this area one studies theproperties of abstract differential equations˙ x ( t ) = Ax ( t ) + Bu ( t ) , x (0) = x, t (cid:62) ,y ( t ) = Cx ( t ) + Du ( t ) . (1.1)Here the standard assumptions were that A is the infinitesimal generator of a stronglycontinuous semigroup, and that the other operators are linear and bounded. Early con-tributions (1970-1980) were made by Butkovskii, Lions, Bensoussan, Curtain, Pritchard,and others. The widely–cited references [1] and [4] treats this in details.Starting from the early 80’s researchers have studied the above abstract differentialequation under weaker assumptions. Important contributions have been made by Salamon[29], Staffans [30] and Weiss [36] (see also [31]). The motivation for relaxing the conditionscame from applications. Namely, an important model class which can be written as anabstract differential equation (1.1) are partial differential equations (PDE’s). A PDE withboundary control and point observation can be written as an abstract linear system (1.1),but B and C are unbounded operators.Actually, when control and observation operators have a certain degree of unbounded-ness, one could still define system properties like observability, controllability, and sta-bilizability, and furthermore, one could solve control problems, such as optimal control,see e.g. Lions [21], Lasiecka–Triggiani [19]. These applications have been the motivatingforce behind many Theorems for abstract linear systems. However, since the state space, Key words and phrases.
Stochastic systems, white noise, boundary control, admissible operators,Hilbert spaces, controllability and observability.Submission date to SICON 18/07/2019 Version. a r X i v : . [ m a t h . O C ] M a y .e., the space in which x ( t ) has its values, of PDE’s is normally some energy space with anatural inner product, the main part of the theory has been developed for Hilbert spaces.In this paper, we are concerned with the stochastic version of infinite dimensionalsystems theory. To the best of our knowledge, this theory is not well investigated for sto-chastic systems unless a few tentative, see e.g. [7], [25] which treat a class of input-outputstochastic systems in Hilbert spaces. Our main objective is to show that stochastic sys-tems can inherit properties from infinite dimensional linear systems like well-posedness,observability, controllability, etc. To be more precise, let (Ω , T , F , P ) be a filtered prob-ability space with filtration F t = ( F t ) t (cid:62) and the filtration is generated by the standardone-dimensional Brownian motion ( W t ) t (cid:62) . Let A : D ( A ) ⊂ H → H be a generator ofa C -semigroup ( T ( t )) t (cid:62) on a Hilbert space H, and F : H → H be a linear boundedoperator. In Section 2, we first investigate and define, in a rigourous way, the concept ofwell-posedness and exact observability of the following stochastic observed linear system( C, A, W ) (cid:40) dX ( t ) = AX ( t ) dt + F ( X ( t )) dW t , X (0) = ξ, t (cid:62) ,Y ( t ) = CX ( t ) , t (cid:62) , where the initial process ξ is F -measurable with ξ ∈ L (Ω , H ), and C : D ( A ) → U isa linear operator (generally not closed), called the observation operator, where U is aHilbert space. It is known (see e.g. [6]) that the mild solution of the evolution stochasticequation in ( C, A, W ) is given by X ( t ) = T ( t ) ξ + (cid:90) t T ( t − s ) F ( X ( s )) dW s := T ( t ) ξ + W A ( t ) , (1.2)for all t (cid:62) ξ ∈ L (Ω , H ). Basically, in systems theory one looks for well definedobservation process Y = CX with some L -regularity. For the system ( C, A, W ), theseproperties are not obvious as the operator C is only defined on a ”small” domain D ( C ) = D ( A ). It is then convenient to work not with C but with an appropriate extension of C . We say that the stochastic system ( C, A, W ) is well-posed if there exists an extension˜ C : D ( ˜ C ) ⊂ H → U of the operator C such that X ( t ) ∈ D ( ˜ C ) for a.e. t > P -a.s, and the map ( t (cid:55)→ ˜ Y ( t ) := ˜ CX ( t )) ∈ L ([0 , α ] × Ω , U ) for any α >
0. To provethe well-posedness of (
C, A, W ) we will assume that (
C, A ) is an admissible pair in thedeterministic sense (see Definition 2.1). In this case, we show (see Proposition 2.6) thatthe stochastic convolution W A ( t ) satisfies W A ( t ) ∈ D ( C Λ ) for a.e. t (cid:62) P -a.s, inaddition E (cid:90) α (cid:107) C Λ W A ( t ) (cid:107) dt (cid:54) c E (cid:107) ξ (cid:107) , for any ξ ∈ L (Ω , H ), and for a constant c := c α >
0, where C Λ is the Yosida extensionof C with respect to A (see (2.3)). On the other hand, we use Weiss’s representationtheorem [34] to conclude that the system ( C, A, W ) is well-posed, see Theorem 2.7. Ifin addition, if we assume that the deterministic system (
C, A ) is exactly observable, it is o for the stochastic system ( C, A, W ), whenever the norm (cid:107) F (cid:107) L ( H ) is small enough, seeTheorem 2.11.In Section 3, we consider a perturbation system of ( C, A, W ),(
C, A + L, W ) (cid:40) dX ( t ) = ( A + L ) X ( t ) dt + F ( X ( t )) dW t , X (0) = ξ, t (cid:62) ,Y ( t ) = CX ( t ) , t (cid:62) , where L : D ( L ) ⊂ H → H is a linear operator. We show that if ( C, A ) and (
L, A ) areadmissible in the deterministic sense, then the stochastic system (
C, A + L, W ) is well-posed. Moreover, we prove that the mild solution of (
C, A + L, W ) satisfies X ( t ) ∈ D ( L Λ )for a.e. t > P -a.s, and X ( t ) = T ( t ) ξ + (cid:90) t T ( t − s ) L Λ X ( s ) ds + (cid:90) t T ( t − s ) F ( X ( s )) dW s , for any t (cid:62) ξ ∈ L (Ω , H ), where L Λ : D ( L Λ ) ⊂ H → H is the Yosida extension of L for A . Inspiriting from this formula, we introduce a new representation of solutions ofstate delay stochastic linear systems, see Theorem 3.2.Section 4 is devoted to the well-posedness and exact controllability of boundary controlstochastic linear system(BCSS) (cid:40) dX ( t ) = A m X ( t ) dt + F ( X ( t )) dW t , X (0) = ξ, t (cid:62) ,GX ( t ) = u ( t ) , t (cid:62) , where ξ ∈ L F (Ω; H ), A m : Z ⊂ H → H is a closed linear operator (differential operator)on H, and Z a densely and continuously embedded Hilbert space in H . The boundaryoperator G : Z → U is linear and surjective such that A := A m with domain D ( A ) = ker G is a generator of a C -semigroup ( T ( t )) t (cid:62) on H . The control function u : [0 , + ∞ ) × Ω → U is a square integrable function, F and ( W t ) t (cid:62) are as above. Under suitable conditions,we prove that the mild solution of (BCSS) is given by X ( t ) = T ( t ) ξ + Φ stoct u, for any t (cid:62) u ∈ L ( R + , L (Ω , U )), whereΦ stoct : L ( R + , L (Ω , U )) → L (Ω , H ) , t (cid:62) , are linear and bounded operators. We mention that in the proof of the existence of themild solution we have based on the work [8]. As an application we proved the well-posedness of a class of stochastic systems with input delays. In addition, we have inves-tigated the concept of exact controllability for the system (BCSS) in the case (cid:107) F (cid:107) L ( H ) when is small enough. An example of exact controllability of an initial-boundary valueproblem for the string equation is given.In the last section, we will be based on the concept of well-posed infinite dimensionallinear systems in the sense of Salamon-Weiss for deterministic systems to give sense toinput-output stochastic linear systems in Hilbert spaces. We recall that a linear systemdefined by a generator A : D ( A ) ⊂ H → H , a control operator B : U → H − and anobservation operator C : D ( A ) ⊂ H → U is well-posed (in the Salamon-Weiss sense) if C, A ) and (
A, B ) are admissible and the application F that relate the control function u to the observation function y define a linear bounded operator on L ([0 , α ] , U ) for any α >
0. We also mention that Weiss [36] have introduced an important small class ofwell-posed systems called regular linear systems (or compatibe systems), see Section 5 fordetails. We prove that for stochastic systems these two concepts coincide. We give anexample of a tranport system to ullistarte the result.
Notation.
Let ( H, (cid:107) · (cid:107) ) be a Hilbert space and A : D ( A ) ⊂ H → H be the generatorof a C –semigroup T := ( T ( t )) t (cid:62) on H . The graph norm associated with A is definedby (cid:107) x (cid:107) A := (cid:107) x (cid:107) + (cid:107) Ax (cid:107) for x ∈ D ( A ). We know that H A := ( D ( A ) , (cid:107) · (cid:107) A ) is a Hilbertspace and we have a dense embedding H A ⊂ H . On the other hand, we denote by ρ ( A ) the resolvent set of A and we denote the resolvent operator of A by R ( λ, A ) :=( λI − A ) − , λ ∈ ρ ( A ) . We define a new norm on H by setting (cid:107) x (cid:107) − := (cid:107) R ( β, A ) x (cid:107) for x ∈ H and β in the resolvent set ρ ( A ). The completion of H with respect to this norm isa Hilbert space denoted by H − and satisfies D ( A ) (cid:44) → H (cid:44) → H − . The extension of the semigroup T to H − is again a strongly continuous semigroup T − :=( T − ( t )) t (cid:62) on H − whose generator A − : D ( A − ) = H → H − is the extension of A to H . L ( E , E ) is the Banach space of linear bounded operators from a Banach space E to another Banach space E . Admissibility and observability of stochastic process
In this section we shall use the concept of admissible observation operator developedin [34] to establish the well-posedness and observability of the stochastic observed linearsystem (
C, A, W ) defined in Section 1.Let ( U, (cid:107)·(cid:107) ) be a Hilbert space (we use the same notation for the norm), and C : H A → U be a linear operator (not necessarily closed). Consider the (deterministic) observed linearsystem ˙ x ( t ) = Ax ( t ) , x (0) = x , y ( t ) = Cx ( t ) , t (cid:62) , (2.1)for initial states x ∈ H . From semigroup theory, we know that the mild solution ofthe Cauchy problem in (2.1) is given by x ( t ) = T ( t ) x for any t (cid:62) x ∈ H . Insystems theory observation functions y are needed to be 2-locally integrable functions witha certain continuity with respect to the initial states x ∈ H . For instance, we can onlywrite y ( t ) = CT ( t ) x for t (cid:62) x ∈ D ( A ) (because C is an unbounded operator andit is only defined on the domain D ( A )). It is then important to look for an extension of y .To solve this problem, Weiss [34] introduced the following class of observation operators: Definition 2.1.
An operator C ∈ L ( H A , U ) is called an admissible observation operatorfor A (or simply ( C, A ) is admissible) if for some (hence all) α > := γ ( α ) > (cid:90) α (cid:107) CT ( t ) x (cid:107) dt (cid:54) γ (cid:107) x (cid:107) , ∀ x ∈ D ( A ) . (2.2)It is to be noted that the constant γ in the above definition depends only on α . Theestimation (2.2) shows that the following operatorΨ : D ( A ) → L loc ( R + , U ) , x (cid:55)→ CT ( · ) x is extended to a bounded linear operator on H (noted by the same symbol). The extendedoutput function associated with y is then given by˜ y ( t ) = (Ψ x )( t )for a.e. t (cid:62) x ∈ H . Define the following extension of C, called the Yosidaextension of C w.r.t. A, as D ( C Λ ) := (cid:26) x ∈ H : lim σ → + ∞ CσR ( σ, A ) x exists in U (cid:27) ,C Λ x := lim σ → + ∞ CσR ( σ, A ) x. (2.3)Clearly, D ( A ) ⊂ D ( C Λ ) and C Λ x = Cx for all x ∈ D ( A ). It is show in [34] that if ( C, A )is admissible then Range( T ( t )) ⊂ D ( C Λ ) and (Ψ x )( t ) = C Λ T ( t ) x (2.4)for a.e. t > x ∈ H . In what follows we assume that C ∈ L ( D ( A ) , H ) is anadmissible observation operator for A. Definition 2.2. [31] Let C ∈ L ( D ( A ) , H ) such that ( C, A ) is admissible. This pair (
C, A )(or the system (2.1)) is called exactly observable in time τ (or on [0 , τ ]) if there exists a κ τ > (cid:107) Ψ x (cid:107) L ([0 ,τ ] ,U ) (cid:62) κ τ (cid:107) x (cid:107) , ∀ x ∈ H. (2.5)In the rest of this section, we keep the same assumptions on A, F, C and ( W t ) t (cid:62) . Weshall investigate well-posedness and observability of the stochastic linear system ( C, A, W ). Definition 2.3. An H -valued stochastic process ( X ( t )) t (cid:62) is called a mild solution of thestochastic differential equation dX ( t ) = AX ( t ) dt + F ( X ( t )) dW t , t > , X (0) = ξ, if ( X ( t )) t (cid:62) is F -adapted, mean-square continuous and for all t (cid:62) , for all ξ ∈ L F (Ω , H ) X ( t ) = T ( t ) ξ + (cid:90) t T ( t − s ) F ( X ( s )) dW s . (2.6)The mild solution is proved to be continuous, in most cases, due to the continuity ofstochastic convolution term (see [6]).Next, we will give sense to the observation process Y ( t ) associated to the stochasticlinear system ( C, A, W ). We fist introduce the following well-posedness concept for a suchsystem. efinition 2.4. The stochastic system (
C, A, W ) is well-posed if there exists an extension˜ C : D ( ˜ C ) ⊂ H → U of the operator C such that X ( t ) ∈ D ( ˜ C ) f or a.e. t > , P − a.s, and for any α >
0, the map( t (cid:55)→ ˜ Y ( t ) := ˜ CX ( t )) ∈ L ([0 , α ] × Ω , U ) . The following result can easily obtained by using the Weiss’s representation theorem(see [34, Theorem 4.5]).
Lemma 2.5.
Assume that ( C, A ) is admissible, then (1) For a.e t (cid:62) and P -a.s ω ∈ Ω we have T ( t ) x ( ω ) ∈ D ( C Λ ) . (2) Moreover for ξ ∈ L F (Ω; H ) , there exists γ > , such that E (cid:90) τ (cid:107) C Λ T ( s ) ξ (cid:107) U ds (cid:54) γ E (cid:107) ξ (cid:107) H . (2.7)We have the following localisation of the stochastic convolution. Proposition 2.6.
For any admissible pair ( C, A ) and f ∈ L loc ( R + , L (Ω , H )) , we have ( T (cid:5) f )( t ) := (cid:90) t T ( t − s ) f ( s ) dW s ∈ D ( C Λ ) , a.e.t (cid:62) , P − a.s. In addition E (cid:90) α (cid:107) C Λ ( T (cid:5) f )( t ) (cid:107) U dt (cid:54) C E (cid:90) α (cid:107) f ( s ) (cid:107) H ds, for any α > and a constant C := C ( α ) > independent of f . roof. First we denote C λ := CλR ( λ, A ) for λ ∈ ρ ( A ) and let α >
0. Using Fubini’stheorem, Itˆo’s isomerty and the admissibility of (
C, A ) , we obtain E (cid:90) α (cid:107) C λ (cid:90) t T ( t − s ) f ( s ) dW s (cid:107) U dt = E (cid:90) α (cid:107) (cid:90) t C λ T ( t − s ) f ( s ) dW s (cid:107) U dt = (cid:90) α E (cid:107) (cid:90) t C λ T ( t − s ) f ( s ) dW s (cid:107) U dt = (cid:90) α E (cid:90) t (cid:107) C λ T ( t − s ) f ( s ) (cid:107) U dsdt = (cid:90) α E (cid:90) t (cid:107) CT ( t − s ) λR ( λ, A ) f ( s ) (cid:107) U dsdt (cid:54) E (cid:90) α (cid:90) αs (cid:107) CT ( t − s ) λR ( λ, A ) f ( s ) (cid:107) U dtds (cid:54) E (cid:90) α (cid:90) α − s (cid:107) CT ( τ ) λR ( λ, A ) f ( s ) (cid:107) U dτ ds (cid:54) E (cid:90) α (cid:90) α (cid:107) CT ( τ ) λR ( λ, A ) f ( s ) (cid:107) U dτ ds (cid:54) γ E (cid:90) α (cid:107) λR ( λ, A ) f ( s ) (cid:107) H ds where γ := γ ( α ) (cid:62)
0. Then for µ and λ large enough we get that E (cid:90) α (cid:107) ( C λ − C µ )( T (cid:5) f )( t ) (cid:107) U dt (cid:54) γ E (cid:90) α (cid:107) ( λR ( λ, A ) − µR ( µ, A )) f ( s ) (cid:107) H ds. Since E (cid:90) α (cid:107) ( λR ( λ, A ) − µR ( µ, A )) f ( s ) (cid:107) H ds → λ,µ (cid:55)→ + ∞ , then ( C n ( T (cid:5) f )) n ∈ N ∗ is a Cauchy sequence on L loc ( R + × Ω , U ), thuslim k (cid:55)→ + ∞ C n k ( T (cid:5) f )( t ) exists a.e. t (cid:62) , P − a.s . By Yosida extension we obtain( T (cid:5) f )( t ) ∈ D ( C Λ ) , a.e.t (cid:62) , P − a.s. Moreover E (cid:90) α (cid:107) C λ ( T (cid:5) f )( t ) (cid:107) U dt (cid:54) γ E (cid:90) α (cid:107) λR ( λ, A ) f ( s ) (cid:107) H ds (cid:54) M γ E (cid:90) α (cid:107) f ( s ) (cid:107) H ds = C ( α ) E (cid:90) α (cid:107) f ( s ) (cid:107) H ds. (2.8) y letting λ → + ∞ we obtain E (cid:90) α (cid:107) C Λ ( T (cid:5) f )( t ) (cid:107) U dt (cid:54) C ( α ) E (cid:90) α (cid:107) f ( s ) (cid:107) H ds. (2.9) (cid:3) We now able to state the well-posedness result for the stochastic system (
C, A, W ). Theorem 2.7.
Assume that the observation operator C is admissible for A . Then X ( t ) := T ( t ) ξ + W A ( t ) ∈ D ( C Λ ) , a.e t (cid:62) , P − a.s, (2.10) and E (cid:90) α (cid:107) C Λ X ( t ) (cid:107) U dt (cid:54) ˜ C ( α ) E (cid:107) ξ (cid:107) H , ∀ ξ ∈ L F (Ω; H ) . (2.11) In particular the stochastic observed system ( C, A, W ) is well-posed.Proof. Let ( X ( t )) t (cid:62) be the mild solution of the stochastic differential equation in( C, A, W ) and set f ( t, ω ) := F ( X ( t, ω )) , t (cid:62) , ω ∈ Ω . We have (cid:90) α E (cid:107) f ( t, · ) (cid:107) dt (cid:54) (cid:107) F (cid:107) L ( H ) (cid:90) α E (cid:107) X ( t ) (cid:107) H dt. On the other hand, we have for t ∈ [0 , α ] , E (cid:107) X ( t ) (cid:107) H (cid:54) M e | ω | t (cid:18) E (cid:107) ξ (cid:107) H + (cid:107) F (cid:107) L ( H ) E (cid:90) t (cid:107) X ( σ ) (cid:107) H dσ (cid:19) (cid:54) M e | ω | α (cid:18) E (cid:107) ξ (cid:107) H + (cid:107) F (cid:107) L ( H ) (cid:90) t E (cid:107) X ( σ ) (cid:107) H dσ (cid:19) . By applying Gronwall’s inequality to the function t → E (cid:107) X ( t ) (cid:107) H , we obtain E (cid:107) X ( t ) (cid:107) H ds (cid:54) κ α E (cid:107) ξ (cid:107) H , where the constant κ α := 2 M e | ω | α exp(2 α (cid:107) F (cid:107) L ( H ) M e | ω | α ). Thus (cid:90) α E (cid:107) f ( t, · ) (cid:107) dt (cid:54) ακ α (cid:107) F (cid:107) L ( H ) E (cid:107) ξ (cid:107) H . Now (2.10) and (2.11) immediately follow from Lemma 2.5 and Proposition 2.6. (cid:3)
We give now the definition of an extend of the output function.
Definition 2.8.
If (
C, A, W ) is well-posed, we call˜ Y ( t ) := C Λ X ( t ) , a.e t (cid:62) , P − a.s, the generalized observation process which satisfies˜ Y ∈ L loc ([0 , + ∞ ) × Ω , U ) . emark . We can obtain the same results if we assume that F : H → H is a non-linearand globally lipschitz operator, and F (0) = 0. In fact, this is due to the following estimate (cid:107) F ( x ) (cid:107) (cid:54) K (cid:107) x (cid:107) for all x ∈ H and a constant K >
C, A ) is admissible which impliesthat (
C, A, W ) is well-posed and investigate the exact observability of this later. To thatpurpose, we define the following operatorΨ stoc : L (Ω , H ) → L loc ( R + , L (Ω , U )) , ξ (cid:55)→ Ψ stoc ξ = Ψ ξ + C Λ W A ( · ) , where W A ( · ) and Ψ are respectively defined by (1.2) and (2.4). From Theorem 2 . stoc is well defined. Moreover the estimation(2.11) shows that Ψ stoc is linear bounded from L (Ω , H ) to L loc ( R + , L (Ω , U )) . Then wecan set the following definition.
Definition 2.10.
The system (
C, A, W ) is called exactly observable in time τ if thereexists a κ > (cid:107) Ψ stoc ξ (cid:107) L ([0 ,τ ] ,L (Ω ,U )) (cid:62) κ (cid:107) ξ (cid:107) L (Ω ,H ) , ∀ ξ ∈ L (Ω , H ) . (2.12)To show that the system ( C, A, W ) is exactly observable, we shall use Theorem 2 . . Theorem 2.11.
Assume that the pair ( C, A ) is exactly observable in time τ in the de-terministic sense. Then there exists a constant θ τ > such that the stochastic observedsystem ( C, A, W ) is exactly observable whenever (cid:107) F (cid:107) L ( H ) < θ τ . Proof.
Since (
C, A ) is exactly observable in time τ , so there exists a constant γ τ > ξ ∈ H we have (cid:107) Ψ ξ (cid:107) L ([0 ,τ ] ,U ) (cid:62) γ τ (cid:107) ξ (cid:107) , P − a.s. It follows from estimation (2.11) that (cid:107) Ψ stoc ξ (cid:107) L ([0 ,τ ] ,L (Ω ,U )) (cid:62) (cid:107) Ψ ξ (cid:107) L ([0 ,τ ] ,L (Ω ,U )) − (cid:107) Ψ stoc ξ − Ψ ξ (cid:107) L ([0 ,τ ] ,L (Ω ,U )) (cid:62) (cid:107) Ψ ξ (cid:107) L ([0 ,τ ] ,L (Ω ,U )) − (cid:107) C Λ W A (cid:107) L ([0 ,τ ] ,L (Ω ,U )) (cid:62) γ τ (cid:107) ξ (cid:107) L (Ω ,H ) − C τ (cid:107) ξ (cid:107) L (Ω ,H ) = ( γ τ − C τ ) (cid:107) ξ (cid:107) L (Ω ,H ) . Let us now define the function h : R + → R + , (cid:107) F (cid:107) (cid:55)→ h ( (cid:107) F (cid:107) ) = (cid:107) F (cid:107) exp( a (cid:107) F (cid:107) ) , where a = τ M exp(2 | ω | τ ) . e claim that h is bijective on R + , then the stochastic observed system ( C, A, W ) isexactly observable whenever (cid:107) F (cid:107) L ( H ) < h − (cid:18) γM (cid:114) γ τ τ exp( −| ω | τ ) (cid:19) = θ τ . The proof is therefore completed. (cid:3)
Example 2.12.
Consider the following clamped Euler-Bernoulli with torque observationat an endpoint ∂ ∂t z ( x, t ) + ∂ ∂x z ( x, t ) − b ( x ) z ( x, t ) ∂∂t W t = 0 , ( x, t ) ∈ (0 , × [0 , ∞ ) ,z (0 , t ) = z (1 , t ) = 0 , t (cid:62) ∂∂x z (0 , t ) = ∂∂x z (1 , t ) = 0 , t (cid:62) z ( x,
0) = z ( x ) , z ( x,
1) = z ( x ) , x ∈ [0 , , (2.13)where b ∈ L ∞ (0 , . The output function is given by y ( t ) = ∂ ∂x z (0 , t ) t (cid:62) . Let H = L [0 ,
1] and A ,m : D ( A ,m ) → H be the operator described by A ,m f = ∂ f∂x , D ( A ,m ) = H (0 , , and the boundary operators G f = ∂f∂x (0) , G f = ∂f∂x (1) , f ∈ H (0 , . We define also the following Hilbert spaces H = H (0 , ∩ H (0 , , H = H (0 , , where H = (cid:26) f ∈ H (0 , | f (0) = f (1) = ∂f∂x (0) = ∂f∂x (1) = 0 (cid:27) . We denote H = H × H, H = H × H . We introduce the space Z ⊂ H as follows Z = H (0 , × H , and we select the operators A ,m = (cid:18) I − ∆ (cid:19) : Z → H , G := δ δ G G : Z → C . Let B be the following bounded operator B = (cid:18) b (cid:19) , b = b ( x ) . y using these notation, and introducing the new state ρ ( t ) = (cid:18) z ( t )˙ z ( t ) (cid:19) , t (cid:62) , the equation (2.13) can be rewritten as follows dρ ( t ) = A ,m ρ ( t ) dt + B ρ ( t ) dW t , t (cid:62) G ρ ( t ) = 0 , t (cid:62) ρ (0) = (cid:0) z z (cid:1) , (2.14)We now check that the operators A ,m and G satisfy the conditions of Greiner [10]. Tothat purpose we define the operator A = A ,m , D ( A ) = ker G . Then A = (cid:18) I − A (cid:19) , D ( A ) = H , where A is the strictly positive fourth derivative operator defined on H . From [31,Proposition 3.7.6] we know that the operator A is skew-adjoint, so it generates a unitarygroup ( T ( t )) t (cid:62) on H . Moreover, the operator G is onto. Then the system (2.13) can bereformulated in the following abstract form dρ ( t ) = A ρ ( t ) dt + B ρ ( t ) dW t , t (cid:62) ρ (0) = (cid:0) z z (cid:1) ,y ( t ) = C ρ ( t ) , t (cid:62)
0, (2.15)where C : H → C is a torque observation operator at an endpoint given by C (cid:18) gh (cid:19) = ∂ g∂ x (0) , ∀ (cid:18) gh (cid:19) ∈ H . We know by [31, Proposition 6.10.1] that ( C , A ) is admissible and exactly observable inthe deterministic sense in any τ > . Moreover for all (cid:0) gh (cid:1) ∈ H , we obtain (cid:13)(cid:13)(cid:13)(cid:13) B (cid:18) gh (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) H = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) bh (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) H (cid:54) (cid:107) b (cid:107) ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) gh (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) H . We denote α τ > C , A ) . Then the stochasticobserved Euler equation (2.13) is exactly observable, whenever (cid:107) b (cid:107) ∞ < h − (cid:18) α τ √ τ γM exp( −| ω | τ ) (cid:19) , where γ > C for A . . Application to perturbed stochastic evolution equations
In this section, we consider the perturbed stochastic linear system (
C, A + L, W ) definedin the introductory section, where we assume that the operators
A, C, F, and the process( W ( t )) t (cid:62) are as in the previous sections, and L : D ( A ) → H is a linear operator. We shallapply results obtained in Section 2 to show that the system ( C, A + L, W ) is well-posedin the sense of Definition 2.4.If we assume that (
L, A ) is admissible, then by H¨older inequality one easily see that L is a Miyadera-Voigt perturbation for A . This implies that the operator A L := ( A + L )with domain D ( A L ) := D ( A ) generates a strongly continuous semigroup ( T L ( t )) t (cid:62) on H (see [9, p.196]). According to [11], this semigroup satisfies the following variation ofconstants formula on the hull space H,T L ( t ) x = T ( t ) x + (cid:90) t T ( t − s ) L Λ T L ( s ) xds, = T ( t ) x + (cid:90) t T L ( t − s ) L Λ T ( s ) xds, (3.1)for any t (cid:62) x ∈ H , where L Λ is the Yosida extension of L for A . Henceforth, themild solution of the stochastic equation ( C, A + L, W ) is given by X ( t ) = T L ( t ) ξ + (cid:90) t T L ( t − s ) F ( X ( s )) dW s , t (cid:62) . (3.2)The question now is how can we reformulate this mild solution using the first semigroup( T ( t )) t (cid:62) . This problem have been thoroughly investigated for deterministic perturbedevolution equations, see for example [11, Theorem 2.1]. We recall from Proposition 2.6that for any f ∈ L loc ( R + , L (Ω , H )) , we have (cid:90) t T ( t − s ) f ( s ) dW s ∈ D ( L Λ ) , a.e. t (cid:62) , P − a.s. (3.3)Moreover, we have the following technical lemma. Lemma 3.1.
Assume that ( L, A ) is admissible and let f ∈ L loc ( R + , L (Ω , H )) . Then Ω × [0 , t ] → H ( ω, s ) → (Ψ L f ( s ))( t − s ) (3.4) is measurable and L Λ (cid:90) t T ( t − s ) f ( s ) dW s = (cid:90) t (Ψ L f ( s ))( t − s ) dW s , (3.5) for a.e t ∈ [0 , α ] ( α > ) and P -a.s, where Ψ L is the extended output map associated with L and A, defined as in (2.4) . roof. It is clear that ( ω, s ) → (Ψ L f ( s ))( t − s ) is measurable if f belongs to the densespace D L = span { E φ ( · ) T L ( · − r ) x : E ∈ F , x ∈ H,r (cid:62) , φ ∈ C C ( R + ) , φ ( t ) = 0 , (cid:54) t < r } , and then by the density, it will be measurable for f ∈ L loc ( R + , L (Ω , H )). Now insteadof f we will work with λR ( λ, A ) f for large λ > . (cid:90) t (Ψ L λR ( λ, A ) f ( s ))( t − s ) dW s = (cid:90) t L Λ T ( t − s ) λR ( λ, A ) f ( s ) dW s = (cid:90) t LλR ( λ, A ) T ( t − s ) f ( s ) dW s = LλR ( λ, A ) (cid:90) t T ( t − s ) f ( s ) dW s According to Proposition 2.6,lim λ (cid:55)→ + ∞ (cid:90) t (Ψ L λR ( λ, A ) f ( s ))( t − s ) dW s = L Λ (cid:90) t T ( t − s ) f ( s ) dW s . (3.6)Now it suffices to prove thatlim λ (cid:55)→ + ∞ (cid:90) t (Ψ L λR ( λ, A ) f ( s ))( t − s ) dW s = (cid:90) t (Ψ L f ( s ))( t − s ) dW s . To this end, let us define the following function g λ ( t ) := (cid:90) t (Ψ L λR ( λ, A ) f ( s ) − f ( s ))( t − s ) dW s , t > . For α > . E (cid:90) α (cid:107) g λ ( t ) (cid:107) dt = (cid:90) α E (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) t (Ψ L λR ( λ, A ) f ( s ) − f ( s ))( t − s ) dW s (cid:13)(cid:13)(cid:13)(cid:13) dt = (cid:90) α E (cid:90) t (cid:107) (Ψ L λR ( λ, A ) f ( s ) − f ( s ))( t − s ) (cid:107) dsdt = E (cid:90) α (cid:90) t (cid:107) (Ψ L λR ( λ, A ) f ( s ) − f ( s ))( t − s ) (cid:107) dsdt = E (cid:90) α (cid:90) αs (cid:107) (Ψ L λR ( λ, A ) f ( s ) − f ( s ))( t − s ) (cid:107) dtds (cid:54) E (cid:90) α (cid:90) α (cid:107) (Ψ L λR ( λ, A ) f ( s ) − f ( s ))( τ ) (cid:107) dτ ds (cid:54) E (cid:90) α γ (cid:107) λR ( λ, A ) f ( s ) − f ( s ) (cid:107) ds → λ → ∞ Thus we may conclude from inequality (3.6) the desired result. (cid:3) heorem 3.2. Assume that ( L, A ) is admissible and let X : [0 , + ∞ ) × Ω → H be theprocess defined by (3.2) . Then the perturbed stochastic system ( C, A + L, W ) is well-posedand X ( t ) ∈ D ( L Λ ) , a.e. t > , P − a.s., E (cid:90) α (cid:107) L Λ X ( t ) (cid:107) U dt (cid:54) C ( α ) E (cid:107) ξ (cid:107) H , for any ξ ∈ L F (Ω; H ) and some constants α > and C ( α ) > . Moreover, X ( t ) = T ( t ) ξ + (cid:90) t T ( t − s ) L Λ X ( s ) ds + (cid:90) t T ( t − s ) F ( X ( s )) dW s (3.7) for any t (cid:62) .Proof. We recall from [13] that if L is an admissible observation operator for A then L is also an admissible observation operator for A L . Moreover, the Yosida extensions of L with respect to A and A L denoted respectively by L Λ and ˜ L Λ coincide, that is,˜ L Λ ≡ L Λ . Now, according to Theorem 2.7, we have X ( t ) = T L ( t ) ξ + W A L ( t ) ∈ D ( ˜ L Λ ) = D ( L Λ ) , a.e t > , P − a.s. and E (cid:90) α (cid:107) L Λ X ( t ) (cid:107) U dt = E (cid:90) α (cid:107) ˜ L Λ X ( t ) (cid:107) U dt (cid:54) C ( α ) E (cid:107) ξ (cid:107) H , for any ξ ∈ L F (Ω; H ). Let us now prove the variation of constants formula (3.7). Usingthe formula in (3.1), we obtain X ( t ) = T ( t ) ξ + (cid:90) t T ( t − s ) F ( X ( s )) dW s + (cid:90) t T ( t − s ) L Λ T L ( s ) ξds + (cid:90) t (cid:90) t − s T ( t − s − τ ) L Λ T L ( τ ) F ( X ( s )) dτ dW s . y using Fubini’s stochastic theorem, Lemma 3 . L Λ ≡ L Λ , the lastterm of the solution X ( t ) becomes (cid:90) t (cid:90) t − s T ( t − s − τ ) L Λ T L ( τ ) F ( X ( s )) dτ dW s = (cid:90) t (cid:90) t − s T ( t − s − τ )( ˜Ψ L F ( X ( s )))( τ ) dτ dW s = (cid:90) t (cid:90) t − s T ( t − s − τ )(Ψ L F ( X ( s )))( τ ) dτ dW s = (cid:90) t (cid:90) σ T ( t − σ )(Ψ L F ( X ( s )))( σ − s ) dW s dσ = (cid:90) t T ( t − σ ) L Λ W A L ( σ ) dσ where ˜Ψ L is the extended output map associated with L and A L and W A L ( σ ) := (cid:90) σ T L ( σ − s ) F ( X ( s )) dW s it follows that X ( t ) = T ( t ) ξ + (cid:90) t T ( t − s ) F ( X ( s )) dW s + (cid:90) t T ( t − s ) L Λ ( S ( s ) ξ + W A L ( s )) ds. This ends the proof. (cid:3)
Remark . It is shown in [13] that if (
C, A ) is admissible then (
C, A + L ) is also admissibleand if we denote by C Λ and ˜ C Λ the Yosida extensions for A and A L respectively, then C Λ ≡ ˜ C Λ on D ( C Λ ) ∩ D ( L Λ ) . (3.8)Theorem 2.7 shows that X ( t ) ∈ D ( ˜ C Λ ) for a.e. t > P -a.s. On the other hand, usingLemma 2.5, Proposition 2.6, and [11, Prop.3.3], we have also X ( t ) ∈ D ( C Λ ) for a.e. t > P -a.s. Thus, according to (3.8),˜ C Λ X ( t ) = C Λ X ( t ) . In the rest of this section, we shall apply Theorem 3.2 to introduce a new variation ofconstants formula for the solutions of the following stochastic delay equation dX ( t ) = ( AX ( t ) + RX ( t + · )) dt + F ( X ( t )) dW t , t (cid:62) X (0) = ξ ∈ L F (Ω , H ) ,X ( θ, ω ) = ϕ ( θ, ω ) , θ ∈ [ − r, ω ∈ Ω , (3.9)where X : [ − r, + ∞ ) × Ω → H and its history function X ( t + · ) : [ − r, × Ω → H isdefined for any t (cid:62) θ (cid:55)→ X ( t + θ ). The operators A, F and the process ( W t ) t (cid:62) areas above, and the delay operator R : W , ([ − r, , H ) → H is defined by Rg = (cid:90) − r dµ ( θ ) g ( θ ) , here µ : [ − r, → L ( H ) is a function of bounded variations, continuous and µ (0) = 0.We assume that the initial process history ϕ ∈ L ([ − r, × Ω; H ).We mention that the solution of the delay equation (3.9) is a vector formed by X ( t )and the history X ( t + · ) at any time t (cid:62)
0. We thus introduce the following new statespace H := H × L ([ − r, H ) , (cid:107) ( xφ ) (cid:107) := (cid:107) x (cid:107) H + (cid:107) φ (cid:107) . Now we select the process solution Z ( t ) := (cid:18) X ( t ) X ( t + . ) (cid:19) ∈ L (Ω , H ) (cid:39) L (Ω , H ) × L ([ − r, × Ω; H ) . We will see that t (cid:55)→ Z ( t ) satisfies a free-delay stochastic equation in L (Ω , H ). To thatpurpose, we define the matrices operators A = (cid:18) A ddθ (cid:19) , D ( A ) = (cid:26)(cid:18) xg (cid:19) ∈ D ( A ) × W , ([ − r, , H ) : g (0) = x (cid:27) , and L = (cid:18) R (cid:19) : D ( A ) → H . In addition define the application˜ F : H → H (cid:18) xϕ (cid:19) → ˜ F ( xφ ) = (cid:18) F ( x )0 (cid:19) . The equation (3.9) is now reformulated as (cid:40) dZ ( t ) = ( A Z ( t ) + L Z ( t )) dt + ˜ F ( Z ( t )) dW t , t (cid:62) Z (0) = (cid:0) ξϕ (cid:1) , (3.10)It is well known that A generates the following C -semigroup on HT ( t ) = (cid:18) T ( t ) 0 T t S ( t ) (cid:19) , t (cid:62) , (3.11)where ( S ( t )) t (cid:62) is the left shift semigroup on L ([ − r, , H ) given by( S ( t ) ϕ )( θ ) = (cid:26) if t + θ (cid:62) ,ϕ ( t + θ ) if t + θ (cid:54) , for ϕ ∈ L ([ − r, , H ) , and the maps T t : H → L ([ − r, , H ) are given by( T t x )( θ ) = (cid:26) T ( t + θ ) x if t + θ (cid:62) , if t + θ (cid:54) , See [11] for more details. We also recall that the generator of ( S ( t )) t (cid:62) is given by Qϕ = ϕ (cid:48) , D ( Q ) = { ϕ ∈ W , ([ − r, , H ) : ϕ (0) = 0 } . roposition 3.4. The stochastic differential equation (3.10) (and hence the delay sto-chastic equation (3.9) ) admits a unique mild solution Z ( t ) = (cid:18) X ( t ) X ( t + · ) (cid:19) , t (cid:62) , such that X ( s + . ) ∈ D ( R Λ ) , a.e. s > X ( t ) = T ( t ) ξ + (cid:90) t T ( t − s ) R Λ X ( s + . ) ds + (cid:90) t T ( t − s ) F ( X ( s )) dW s ,X ( t + . ) = T ( t ) ξ + S ( t ) ϕ + (cid:90) t T t − s R Λ X ( s + . ) ds + (cid:90) t T t − s F ( X ( s )) dW s for any t (cid:62) and P -a.s., where R Λ is the Yosida extension of R respect to Q .Proof. To apply Theorem 3.2 it suffices to show that ( L , A ) is admissible. In fact, from[11], for α > , we have (cid:90) α (cid:107) RS ( t ) ϕ (cid:107) dt (cid:54) ( | µ | ([ − r, (cid:107) ϕ (cid:107) , ∀ ϕ ∈ D ( Q ) , where | µ | is the total variation of µ . This implies that ( L , A ) is admissible. Moreover (see[11]) the Yosida extension of L for A satisfies D ( L Λ ) = H × D ( R Λ ) and L Λ = (cid:18) R Λ (cid:19) . Now Theorem 3.2 implies that Z ( t ) ∈ D ( L Λ ) (hence X ( t + · ) ∈ D ( R Λ )) for a.e. t > P -a.s., and the mild solution of equation (3.10) is given by Z ( t ) = T ( t ) (cid:18) ξϕ (cid:19) + (cid:90) t T ( t − s ) L Λ Z ( s ) ds + (cid:90) t T ( t − s ) ˜ F ( Z ( s )) dW s , (3.12)for any t (cid:62) ξϕ ) ∈ L (Ω , H ). The rest of the proof follows by replacing in (3.12) T , L Λ and ˜ F by their explicit expressions. (cid:3) Controllability of stochastic boundary control linear systems
In this section we will discuss the concept of the exact controllability for infinite di-mensional stochastic boundary control system (BCSS) defined in Section 1. In order toreformulate the boundary system (BCSS) as a distributed stochastic evolution equation,we need some extra conditions. We then assume:( A A = A m , with D ( A ) = ker G generates a C -semigroup T := ( T ( t )) t (cid:62) on H. ( A G is surjective.According to Greiner ([10, Lem.1.2, Lem.1.3]), assumptions ( A
1) and ( A
2) imply thatfor each λ ∈ ρ ( A ) , the operator G | ker( λ − A m ) is invertible with inverse D λ := (cid:0) G | ker( λ − A m ) (cid:1) − : U → ker( λ − A m ) . he operator D λ is called the Direchlet operator and satisfies D λ ∈ L ( U, H ) , λ ∈ ρ ( A ) . Let H − be the extrapolation space associated with A and H and ( T − ( t )) t (cid:62) be theextrapolation semigroup on H − , extension of the initial semigroup ( T ( t )) t (cid:62) (see notationin Section 1). Define the following control operator B := ( λ − A − ) D λ ∈ L ( U, H − ) , λ ∈ ρ ( A ) . (4.1)Due to the resolvent equation, the operator B is independent of λ . We recall that BG | Z = ( A m − A − ) | Z . (4.2)Using (4.2), the system (BCSS) is reformulated as (cid:40) dX ( t ) = A − X ( t ) dt + Bu ( t ) dt + F ( X ( t )) dW t , t (cid:62) ,X (0) = ξ, (4.3)Remark that this equation is defined in H − because Range ( B ) (cid:40) H − .We introduce the following definition Definition 4.1.
We say that the system (BCSS) is well posed if(1) X ( t ) ∈ H, ∀ t (cid:62) , P − a.s. (2) For any t >
0, there exist C > C > E (cid:107) X ( t ) (cid:107) H (cid:54) C E (cid:107) ξ (cid:107) H + C E (cid:107) u (cid:107) L ( R + ; U ) . The convolution Φ t u := (cid:90) t T − ( t − s ) Bu ( s ) ds (4.4)takes value in H − . Hence we may have X ( t ) ∈ H − for any t (cid:62) P -a.s. To prove thewell-posedness of the control stochastic system (BCSS), we need the concept of admissiblecontrol operator which can be found in [31, Chap.4] or [35]. Definition 4.2.
The operator B ∈ L ( U, H − ) is called admissible control operator for A (or the pair ( A, B ) is admissible) if there is a t > t ) ⊂ H. By using the closed graph theorem it is shown in [35] that if (
A, B ) is admissible, thenRange(Φ t ) ⊂ H for any t (cid:62) (cid:107) Φ t v (cid:107) H (cid:54) c (cid:107) v (cid:107) L (0 ,t ; U ) (4.5)for any t (cid:62) v ∈ L ( R + , U ) and a constant c := c ( t ) > t .Moreover, we have (cid:107) Φ t (cid:107) (cid:54) (cid:107) Φ α (cid:107) (4.6)for all 0 (cid:54) t (cid:54) α (the operator norm). or each t (cid:62) , we define linear operators ˜Φ t : L ( R + × Ω , U ) → L (Ω; H − ) by˜Φ t u := (cid:90) t T − ( t − s ) Bu ( s, . ) ds, ∀ u ∈ L ( R + × Ω , U ) . (4.7)We will use the following assumption( A
3) Assume that B ∈ L ( U, H − ) is admissible for A. Lemma 4.3.
Let assumptions ( A , ( A and ( A be satisfied, then there exists aconstant C t > such that E (cid:107) ˜Φ t u (cid:107) H (cid:54) C t (cid:107) u (cid:107) L ([0; t ) × Ω ,U ) . Proof.
The proof follows immediately from (4.5) and the fact that E ( · ) is monotone. (cid:3) The well-posedness of (BCSS) is given in the following:
Theorem 4.4.
Let assumptions ( A , ( A and ( A be satisfied. Then for every ξ ∈ L F (Ω; H ) and u ∈ L ( R + × Ω , U ) there exists a unique F -adapted mild solution of thecontrol stochastic system (BCSS) , X : R + × Ω → H, given by X ( t ; ξ, u ) = T ( t ) ξ + (cid:90) t T ( t − s ) F ( X ( s ; ξ, u )) dW s + ˜Φ t u (4.8) for any t (cid:62) and P -a.s.Proof. Let t > W , r ([0 , t ] , U ) := { g ∈ W , ([0 , t ] , U ) : g ( t ) = 0 } , and the matrix operator A := (cid:18) A − Bδ dds (cid:19) , D ( A ) := (cid:26)(cid:18) xg (cid:19) ∈ H × W , r ([0 , t ] , U ); A − x + Bg (0) ∈ H (cid:27) . According to [8, Theorem 2], the condition ( A
3) implies that the operator ( A , D ( A ))generates on the product space H × L ([0 , t ] , U ) , the following C -semigroup T ( t ) = (cid:18) T ( t ) Φ t S U ( t ) (cid:19) , t (cid:62) . where ( S U ( t )) t (cid:62) is the right shift semigroup on L ([0 , t ] , U ) . By introducing the newstate ρ ( t ) = (cid:18) X ( t ) S U ( t ) u (cid:19) , t (cid:62) . The problem (4.3) becomes (cid:40) dρ ( t ) = A ρ ( t ) dt + F ( ρ ( t )) dW t , t (cid:62) ρ (0) = (cid:0) ξu (cid:1) , (4.9) here F ( ρ ( t )) = (cid:0) F ( X ( t ))0 (cid:1) . It is well known in the literature that the standard system(4.9) is well-posed. Moreover the first projection of ρ ( t ) satisfies for any t > P -a.sthe formula in (4.8). (cid:3) Notation.
Under assumptions of Theorem 4.4, we denote the control part of the mildsolution of (BCSS) byΦ stoct u := X ( t, , u ) = (cid:90) t T ( t − s ) F ( X ( s, , u )) dW s + ˜Φ t u. Using this notation, we have the following result.
Theorem 4.5.
Let assumptions ( A , ( A and ( A be satisfied. Then for each α (cid:62) the operator Φ stocα is linear bounded from L ([0 , α ] , L (Ω; U )) to L (Ω; H ) , and the processsolution of (BCSS) can be rewritten as X ( t ) = T ( t ) ξ + Φ stoct u (4.10) for any t (cid:62) , P -a.s., ξ ∈ L F (Ω , H ) and u ∈ L ( R + , L (Ω , H )) . Moreover, it has acontinuous modification.Proof. Let α > , M = sup s ∈ [0 , (cid:107) T ( s ) (cid:107) , ω > ω ( A ) , t ∈ [0 , α ] and denote M α := M e | ω | α (cid:107) F (cid:107) . By using Itˆo’s isometry, the admissibility of the operator B and the estimate(4.6), we obtain E (cid:107) X ( t, , u ) (cid:107) H = E (cid:107) Φ stoct u (cid:107) H = E (cid:107) (cid:90) t T ( t − s ) F ( X ( s, , u )) dW s + ˜Φ t u (cid:107) H (cid:54) E (cid:107) (cid:90) t T ( t − s ) F ( X ( s, , u )) dW s (cid:107) H + 2 (cid:90) Ω (cid:107) Φ t u ( · , ω ) (cid:107) H d P ( ω ) (cid:54) M α E (cid:90) t (cid:107) X ( s, , u ) (cid:107) H ds + 2 (cid:90) Ω (cid:107) Φ t (cid:107) (cid:107) u ( · , ω ) (cid:107) L ([0 ,t ] ,U ) d P ( ω ) (cid:54) M α E (cid:90) t (cid:107) X ( s, , u ) (cid:107) H ds + 2 (cid:107) Φ α (cid:107) (cid:90) Ω (cid:107)(cid:107) u ( · , ω ) (cid:107) L ([0 ,α ] ,U ) d P ( ω ) (cid:54) M α (cid:90) t E (cid:107) X ( s, , u ) (cid:107) H ds + 2 (cid:107) Φ α (cid:107) E (cid:107) u (cid:107) L ([0 ,α ] ,U ) )The Granwal’s inequality implies that E (cid:107) X ( t, , u ) (cid:107) H (cid:54) c E (cid:107) u (cid:107) L ([0 ,α ] ,U ) , for a constant c := c ( α ) >
0. In particular, E (cid:107) Φ stocα u (cid:107) H = E (cid:107) X ( α, , u ) (cid:107) H (cid:54) c E (cid:107) u (cid:107) L ([0 ,α ] ,U ) . (4.11)For the linearity, let u, v ∈ L ( R + × Ω , U ) and λ ∈ R . Consider the evolution equation dX u ( t ) = AX u ( t ) dt + Bu ( t ) dt + F ( X u ( t )) dW t , X u (0) = 0 , t (cid:62) . he mild solution of this equation is given by X u ( t ) = Φ stoct u. In a similar way, the mild solution of the following stochastic system dX λv ( t ) = ( AX λv ( t ) + λBv ( t )) dt + F ( X λv ( t )) dW t , X λv (0) = 0 , t (cid:62) ,X λv ( t ) = Φ stoct ( λv ) . On the other hand, the solution of the following equation (cid:40) dX u + λv ( t ) = AX u + λv ( t ) dt + B ( u + λv )( t ) dt + F ( X u + λv ( t )) dW t , t (cid:62) ,X u + λv (0) = 0 . (4.12)is given by X u + λv ( t ) = Φ stoct ( u + λv ) . To prove that Φ stoct is a linear operator, it suffices to show that X u + λv ( t ) = X u ( t ) + X λv ( t ) , We then define Z ( t ) = X u ( t ) + X λv ( t ) . Clearly Z is the mild solution of (4.12). The result now follows by the uniqueness of themild solution of (4.12). (cid:3) Example 4.6.
We consider the following input-delay stochastic equation dX ( t ) = AX ( t ) dt + J u ( t + · ) dt + F ( X ( t )) dW t , t (cid:62) ,X (0) = ξ,u ( t, ω ) = ϕ ( t, ω ) , ( t, ω ) ∈ [ − r, × Ω , λ ⊗ P , (4.13)where A , F , and ( W t ) t (cid:62) are as in the previous sections, the control process u : [ − r, ∞ ) × Ω → U , the history control process u ( t + · ) : [ − r, × Ω → H is defined by ( θ, ω ) (cid:55)→ u ( t + θ, ω ), and the input delay operator J : W , ([ − r, , U ) → H is given by J ψ = (cid:90) − r dν ( θ ) ψ ( θ ) , where ν : [ − r, → L ( U, H ) is a function of bounded variations continuous at 0 and ν (0) = 0. The initial conditions ξ ∈ L F (Ω , H ) and ϕ ∈ L ([ − r, × Ω , U ).In order to reformulate the delay system (4.13) to a distributed one, we consider thefollowing process v ( t, θ ) = u ( t + θ ) , t (cid:62) , θ ∈ [ − r, , which is the solution of the following equation in L ([ − r, , L (Ω , U )): ˙ v ( t, . ) = Q m v ( t, . ) , t (cid:62) v (0 , . ) = ϕ,v ( t,
0) = u ( t ) , t (cid:62)
0, (4.14) ombining systems (4.13) and (4.14), we obtain a new equivalent system of (4.13): dX ( t ) = AX ( t ) dt + J v ( t, . ) dt + F ( X ( t )) dW t , t (cid:62) dv ( t, . ) = Q m v ( t, . ) dt, t (cid:62) v ( t,
0) = u ( t ) , t (cid:62) X (0) = ξ,v (0 , . ) = ϕ, (4.15)We define the Hilbert space H := H × L ([ − r, , U ), and we denote ρ ( t ) the columnvector with components ( X ( t ) , v ( t, . )). On the product space H we define the matrixoperators A m = (cid:18) A J Q m (cid:19) , D ( A m ) = D ( A ) × W , ([ − r, , U ) , G = (cid:0) δ (cid:1) : D ( A m ) → U, Γ : H → H , Γ( xf ) = (cid:0) F ( x )0 (cid:1) . Under the above notation, our problem (4.13) can be reformulated as dρ ( t ) = A m ρ ( t ) dt + Γ( ρ ( t )) dW t , t (cid:62) G ρ ( t ) = u ( t ) , t (cid:62) ρ (0) = (cid:0) ξϕ (cid:1) , (4.16)Next, we shall verify the assumptions of Theorem 4.5 in order to give a new expressionto process ρ ( t ) . Then we consider the operator A on the product space H , given by A = A m , D ( A ) := ker G . Then A = (cid:18) A J Q (cid:19) , D ( A ) = D ( A ) × D ( Q ) , where Qϕ = ϕ (cid:48) , D ( Q ) := { ϕ ∈ W , ([ − r, , U ) , ϕ (0) = 0 } , generates the left shift semigroup ( S ( t )) t (cid:62) on L ([ − r, , U ). As shown in [12], ( A , D ( A ))generates the following C -semigroup on H , T ( t ) = (cid:18) T ( t ) N ( t )0 S ( t ) (cid:19) , t (cid:62) , where N ( t ) : L ([ − r, , U ) → H are defined by N ( t ) ψ = (cid:90) t T ( t − s ) J Λ S ( s ) ψds, t (cid:62) , here J Λ is the Yosida extension of J for Q (we recall ( J, Q ) is admissible). Clearly, G isalso surjective. It follows that G | ker( λ −A m ) is invertible and we denote his inverse by D λ := (cid:0) G | ker( λ −A m ) (cid:1) − , λ ∈ ρ ( A ) . Using standard argument, we can see that the Direchlet operator associated with A m and G is given by D λ a = (cid:18) R ( λ, A ) J e λ ae λ a (cid:19) , λ ∈ ρ ( A ) , for any a ∈ U, where e λ : U → L ([ − r, , U ) defined by ( e λ a )( θ ) = e λθ a for a ∈ U and θ ∈ [ − r, B := ( λ − A − ) D λ ∈ L ( U, H − , A ) . Then the system (4.13) can be written in the following abstract form (cid:40) dρ ( t ) = A − ρ ( t ) dt + B u ( t ) dt + Γ( ρ ( t ) dW t , t (cid:62) ,ρ (0) = (cid:0) ξϕ (cid:1) . (4.17)Next, we show that the pair ( A , B ) is admissible. We defineΦ A , B t u := (cid:90) t T − ( t − s ) B u ( s ) ds for u ∈ L ( R + , U ). The Laplace transform of is given by (cid:92) Φ A , B· u ( λ ) = R ( λ, A − ) B (cid:98) u ( λ )= (cid:18) R ( λ, A ) J e λ (cid:98) u ( λ ) e λ (cid:98) u ( λ ) (cid:19) On the other hand, we select β := ( λ − Q − ) e λ , λ ∈ C , We now that β ∈ L ( U, L ([ − r, , U ) − ) is an admissible control operator for Q (see [14]).We define Φ Q,βt u = (cid:90) t S − ( t − s ) βu ( s ) ds, t (cid:62) . We recall also from [14] that the triple (
Q, β, J ) is compatible in the sense of Definition5.2. This means that Φ
Q,βt u ∈ D ( J Λ ) for a.e. t > u ∈ L loc ( R + , U ). Moreover, theoperator ( F u )( t ) := J Λ Φ Q,βt u is linear bounded from L ([0 , α ] , U ) to L ([0 , α ] , H ) for any α > (cid:99) F u ( λ ) = J Λ R ( λ, Q − ) β (cid:98) u ( λ ) = J e λ (cid:98) u ( λ ) . If we denote by Φ F t u = (cid:90) t T ( t − s )( F u )( s ) ds Φ Q,βt u , hen (cid:92) Φ A , B· u ( λ ) = (cid:100) Φ F · u ( λ ) . By the injectivity of Laplace transform we deduce thatΦ A , B t u = Φ F t u ∈ H . This shows that ( A , B ) is admissible. Then by Theorem 4.5 there exists a family˜Φ stoct : L loc ( R + , L (Ω; U )) → L (Ω; H ) , linear bounded such that the solution of the system (4.13) can be expressed as ρ ( t ) = T ( t ) (cid:18) ξϕ (cid:19) + ˜Φ stoct u, t (cid:62) . Next we investigate the exact controllability for stochastic boundary control system(BCSS). Before doing so, let us first recall the exact controllability in the deterministiccase of the following control system (cid:40) dz ( t ) = Az ( t ) dt + Bu ( t ) dt, t (cid:62) ,z (0) = x ∈ H, (4.18)where A and B are defined as in the previous sections. The following definition can befound in [31]. Definition 4.7.
The system (4.18) (or (
A, B ) is exactly controllable) in time τ > , if Range (Φ τ ) = H. (4.19)The results presented in this Theorem can be found in the book [31, p.357]. Theorem 4.8.
The pair ( A, B ) is exactly controllable in time τ if and only if there exists κ t > , such that (cid:90) t (cid:107) B ∗ T ∗ ( s ) v (cid:107) U ds (cid:62) κ t (cid:107) v (cid:107) , ∀ v ∈ D ( A ∗ ) . (4.20)Now in order to give a characterization of the exact controllability, we need the followingimportant result on surjective linear operators, which can be found in [18, p.227]. Lemma 4.9.
We denote by S ( O, E ) := { G ∈ L ( O, E ) : G surjective } . Then S ( O, E ) is an open set of L ( O, E ) with respect to the uniform topology. This meansthat for any G ∈ S ( O, E ) there exists r > such that for any G ∈ L ( O, E ) such that (cid:107) G − G (cid:107) < r we have G ∈ S ( O, E ) . We call r the radius of surjectivity of G . In the deterministic case, the exact controllability of infinite dimensional system is wellinvestigated, see e.g. [31]. In this section, we will translate this theory to stochastic volution equations. Specially we are interested in the stochastic version of the model(4.18) given by (cid:40) dX ( t ) = ( AX ( t ) + Bu ( t )) dt + F ( X ( t )) dW t , t (cid:62) ,X (0) = ξ. (4.21)We mention that the operator F is defined as in the previous sections. If one want toextend the concept of controllability to stochastic systems, it is important to deals withthe following questions: • What is the definition of the exact controllability for stochastic systems ? • For what conditions on the operator F, the stochastic system (BCSS) is exactlycontrollable ?According to Theorem 4 .
5, we can set the following definition.
Definition 4.10.
The system (BCSS) is called controllable in time τ if Range (Φ stocτ ) = L (Ω , H ) . To prove that the system (BCSS) is exactly controllable, we shall use Theorem 4 . . . Theorem 4.11.
Assume that the pair ( A, B ) is exactly controllable on [0 , τ ] in the de-terministic sense. Then there exists a constant γ τ > such that the stochastic controlsystem (BCSS) is exactly controllable whenever (cid:107) F (cid:107) < γ τ . Proof. If B ∈ L ( U, H − ), then the adjoint of ˜Φ τ defined in (4.7) which is in L ( L (Ω , H ); L (Ω , L ([0 , τ ] , U ))) , can be expressed using B ∗ . Let us denote by ˜Ψ dτ theoutput maps corresponding to the semigroup T ∗ with the observation operator B ∗ , thenfor every t (cid:62) ξ ∈ H we get E (cid:104) ˜Φ t u, ξ (cid:105) = (cid:90) Ω (cid:10) (cid:90) t T − ( t − s ) Bu ( s, ω ) ds, ξ ( ω ) (cid:11) d P ( ω )= (cid:90) Ω (cid:90) t (cid:104) u ( s, ω ) , B ∗ T ∗ ( t − s ) ξ ( ω ) (cid:105) dsd P ( ω )= (cid:90) Ω (cid:90) t (cid:104) u ( s, ω ) , ( ˜Ψ dt ξ ( ω ))( s ) (cid:105) dsd P ( ω )= (cid:104) u, ˜Ψ dt ξ (cid:105) L (Ω × R + ,U ) . Then for every τ > ξ ∈ H ( ˜Φ ∗ τ ξ )( s ) = (cid:26) if , s > τB ∗ T ∗ ( τ − s ) ξ if s ∈ [0 , τ ] , -a.s. According to Theorem 4 .
8, the pair ( B ∗ , A ∗ ) is exactly observable on [0 , τ ], andthen there exists a constant κ τ >
0, such that for every ξ ∈ L (Ω , H ) we obtain E (cid:90) τ (cid:107) ( ˜Φ ∗ τ ξ )( s ) (cid:107) U ds = E (cid:90) τ (cid:107) B ∗ T ∗ ( τ − s ) ξ (cid:107) U ds = E (cid:90) τ (cid:107) B ∗ T ∗ ( σ ) ξ (cid:107) U dσ (cid:62) κ τ E (cid:107) ξ (cid:107) H . (4.22)This implies that the operator ˜Φ t is surjective (see H.Brezis [2]). By Itˆo’s formula, weobtain E (cid:107) Φ stocτ u − ˜Φ τ u (cid:107) H = E (cid:107) W A ( τ ) (cid:107) H = E (cid:107) (cid:90) τ T ( τ − σ ) F ( X ( σ, , u )) dW σ (cid:107) H = E (cid:90) τ (cid:107) T ( τ − σ ) F ( X ( σ, , u )) (cid:107) H dσ (cid:54) ( M e | ω | τ ) E (cid:90) τ (cid:107) F ( X ( σ, , u )) (cid:107) H dσ (cid:54) ( M e | ω | τ ) (cid:107) F (cid:107) L ( H ) E (cid:90) τ (cid:107) X ( σ, , u ) (cid:107) H dσ (cid:54) ˜ C ( M e | ω | τ ) (cid:107) F (cid:107) L ( H ) (cid:107) u (cid:107) L ([0 ,τ ] × Ω; U ) . It follow that E (cid:107) Φ stocτ − ˜Φ τ (cid:107) H (cid:54) ˜ C ( M e | ω | τ ) (cid:107) F (cid:107) L ( H ) . If (cid:36) denotes the radius of surjectivity of ˜Φ τ , then the stochastic system (4.21) is exactlycontrollable, for all (cid:107) F (cid:107) L ( H ) < (cid:36)M e | ω | τ ( ˜ C ) = γ τ . (cid:3) Example 4.12.
We consider the following initial-boundary value problem for the stringequation ∂ ∂t v ( x, t ) = ∂ ∂x v ( x, t ) + q ( x ) v ( x, t ) ∂∂t W t , ( x, t ) ∈ (0 , π ) × [0 , ∞ ) ,v ( π, t ) = 0 , ∂∂x v (0 , t ) = u ( t ) , t (cid:62) v ( x,
0) = f ( x ) , ∂∂t v ( x,
0) = g ( x ) , x ∈ (0 , π ) , (4.23)Let Σ = (0 , π ) , we assume in this text that q ∈ L ∞ (Σ) . In order to reformulate (4.23)as a distributed stochastic control system, we introduce the solution space and the statespace respectively by Z = (cid:0) H (Σ) ∩ H π (Σ) (cid:1) × H π (Σ) , H = H π (Σ) × H, here H π (Σ) := { ϕ ∈ H (Σ) | ϕ ( π ) = 0 } , H := L (Σ) . Also, the control space U = C . Define the following operators Lψ = ∂ ∂x ψ, Gψ = ∂ψ∂x (0) , ψ ∈ H (Σ) ∩ H π (Σ) . Moreover, we consider the operators matrix A m : Z → H and G : Z → U which are givenby A m = (cid:18) IL (cid:19) , G = (cid:0) G (cid:1) . Let K be the bounded operator, where q = q ( x ) , K = (cid:18) q (cid:19) . If we put ω ( x, t ) = (cid:18) v ( x, t ) ∂v∂t ( x, t ) (cid:19) , t (cid:62) , x ∈ (0 , π ) , the equation (4.23) can be reformulated as dω ( x, t ) = A m ω ( x, t ) dt + K ω ( x, t ) dW t , t (cid:62) x ∈ (0 , π ) , G ω ( x, t ) = u ( t ) , t (cid:62) x ∈ (0 , π ) ,ω ( x,
0) = (cid:0) fg (cid:1) , x ∈ (0 , π ) ,. (4.24)We shall prove some generation results for the operator matrix A = (cid:18) IA (cid:19) , D ( A ) = { (cid:18) hf (cid:19) ∈ Z| ∂h∂x (0) = 0 } , where A : D ( A ) → H is the operator defined as A = L, D ( A ) = { φ ∈ H (Σ) ∩ H π (Σ) | ∂φ∂x (0) = 0 } , it is well known that the operator A generates a positive C -semigroup ( T ( t )) t (cid:62) on H, so that is A generates a semigroup ( T ( t )) t (cid:62) on H for more details see [31, Example2.7.15]. Moreover G is surjective, then the assumptions of Greiner are satisfied. We thendefine B := ( λ − A − ) D λ ∈ L ( U, D ( A ∗ ) (cid:48) ) for λ ∈ ρ ( A ) where D λ is the Dirichlet operatorassociated with A and G . Then the problem (4.23) becomes (cid:40) dω ( t ) = A ω ( t ) dt + B u ( t ) dt + K ω ( t ) dW t , t (cid:62) ω (0) = (cid:0) fg (cid:1) , . (4.25)Next we denote by C = B ∗ , it follows from [31, Proposition 6.2.25] that C is admissibleobservation operator, hence ( A , B ) admissible. So we have indeed assumptions of Theorem4 . K t : L loc ( R + , L (Ω; U )) → L (Ω; H ) , inear bounded such that the solution of the system (4.23) can be expressed as ω ( t ) = T ( t ) (cid:18) fg (cid:19) + Φ K t u, t (cid:62) . According to [31, Proposition 6.2.5], the pair ( A , C ) is exactly observable in any time τ > π, it follows from Theorem 4 . A , B ) is exactly controllable in any time τ > π. Moreover for all (cid:0) fh (cid:1) ∈ H , we obtain (cid:13)(cid:13)(cid:13)(cid:13) K (cid:18) fh (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) H = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) qh (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) H (cid:54) (cid:107) q (cid:107) ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) fh (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) H , if we denote by ε the radius of surjectivity of Φ st (where Φ st u := (cid:82) t T − ( t − s ) B u ( s ) ds ),there is a γ > (cid:107) q (cid:107) ∞ < εγM e | ω | τ . Well-posedness of input–output stochastic linear systems
In this section we investigate the well-posedness of the following input–output stochasticlinear system dX ( t ) = A m X ( t ) dt + F ( X ( t )) dW t , t (cid:62) ,X (0) = ξ,GX ( t ) = u ( t ) , t (cid:62) ,Y ( t ) = M X ( t ) , t (cid:62) , (5.1)where the operators, A m , F and G are defined as in previous sections and M : Z ⊂ H → H is a linear operator. Throughout this section we assume that the conditions ( A1 ) and( A2 ) (defined in Section 4) hold. Let B ∈ L ( U, H − ) the operator defined by (4.1).Moreover, we define C : D ( A ) → U, C := M | D ( A ) . The operator C will plays the same role as in Section 2.According to the transformation introduced in Section 4, the system (5.1) can be re-formulated as the following distributed linear system( A, B, C, W ) dX ( t ) = ( AX ( t ) + Bu ( t )) dt + F ( X ( t )) dW t , t (cid:62) ,X (0) = ξ,Y ( t ) = CX ( t ) , t (cid:62) . We introduce the following definition.
Definition 5.1.
We say that the system (
A, B, C, W ) (hence system (5.1)) is well-posedif (
A, B ) and (
C, A ) are admissible, there exists an extension ˜ C : D ( ˜ C ) ⊂ H → U of C uch that X ( t ) ∈ D ( ˜ C ) for a.e t > P − a.s , and ( t (cid:55)→ ˜ CX ( t )) ∈ L ([0 , α ] × Ω , U )for some α > (cid:107) ˜ CX ( · ) (cid:107) L ([0 ,α ] × Ω ,U ) (cid:54) c (cid:0) (cid:107) ξ (cid:107) L (Ω ,H ) + (cid:107) u (cid:107) L ([0 ,α ] × Ω ,U ) (cid:1) for any ξ ∈ L (Ω , H ) and u ∈ L ([0 , α ] × Ω , U ) , where c := c α > Definition 5.2.
We say that the triple (
A, B, C ) is called a compatible triple (in thedeterministic sense) if the following hold: • ( A, B ) and (
C, A ) are admissible • for some µ ∈ ρ ( A ) (hence all µ ∈ ρ ( A )) we have Range( R ( µ, A − ) B ) ⊂ D ( C Λ ),where C Λ is the Yosida extension of C for A .In the terminology of Weiss (see [36]), compatible triples are also called regular triples.For u ∈ W , ,loc ( R + , L (Ω , U )), the space of functions u ∈ W , loc ( R + , L (Ω , U )) such that u (0) = 0, we haveΦ t u = (cid:90) t T − ( t − s )( − A − D ) u ( s ) ds = R (0 , A − ) Bu ( t ) − (cid:90) t T ( t − s ) D ˙ u ( s ) ds for t (cid:62) , using an integration by parts (where we have assumed that 0 ∈ ρ ( A ) withoutloosing generality). If we assume that ( A, B, C ) is compatible, then according to [36] andDefinition 5.2, we have Φ t u ∈ D ( C Λ ) for a.e. t > Theorem 5.3.
Assume that the triple ( A, B, C ) is compatible. Then the stochastic system ( A, B, C, W ) is well-posed.Proof. The stochastic maps are given byΦ stoct u = ˜Φ t u + (cid:90) t T ( t − s ) F (cid:0) Φ stocks u (cid:1) dW s . For almost every ω ∈ Ω, we have (cid:16) ˜Φ t u (cid:17) ( ω ) = (cid:90) t T − ( t − s ) Bu ( s, ω ) ds = Φ t u ( · , ω ) ∈ D ( C Λ )almost every t > , due to the fact that ( A, B, C ) is compatible. In addition, (cid:90) α (cid:107) C Λ Φ t u ( · , ω ) (cid:107) U dt (cid:54) κ (cid:90) α (cid:107) u ( t, ω ) (cid:107) U dt This shows that E (cid:90) α (cid:107) C Λ ˜Φ t u (cid:107) U dt (cid:54) κ (cid:107) u (cid:107) L ([0 ,α ] ,L (Ω ,U )) According to estimate (4.11) and Proposition 2.6, we have E (cid:90) α (cid:13)(cid:13)(cid:13)(cid:13) C Λ (cid:90) t T ( t − s ) F (cid:0) Φ stocs u (cid:1) dW s (cid:13)(cid:13)(cid:13)(cid:13) dt (cid:54) C α E (cid:90) α (cid:107) F (Φ stocs u ) (cid:107) ds (cid:54) C α C (cid:48) α (cid:107) F (cid:107)(cid:107) u (cid:107) L ([0 ,α ] ,L (Ω ,U )) . oreover, according to Lemma 2.5 and equality (4.10), the mild solution of the stochasticdifferential equation in ( A, B, C, W ) satisfies X ( t ) ∈ D ( C Λ ) for a.e t > P -a.s, and (cid:107) C Λ X ( · ) (cid:107) L ([0 ,α ] × Ω ,U ) (cid:54) c (cid:0) (cid:107) ξ (cid:107) L (Ω ,H ) + (cid:107) u (cid:107) L ([0 ,α ] × Ω ,U ) (cid:1) for any ξ ∈ L (Ω , H ) and u ∈ L ([0 , α ] × Ω , U ) , where c := c α > (cid:3) We end this section by the following transport stochastic equation which illustrate theabove concept of well-posedness for stochastic systems.
Example 5.4.
Let U be a Hilbert space and r > H = L ([ − r, , U ) we consider the following input-output transport stochastic system dX ( t, θ ) = ∂∂θ X ( t, θ ) dt + K ( θ ) X ( t, θ ) dW t , t (cid:62) , θ ∈ [ − r, ,X (0 , θ ) = ϕ ( θ ) ,X ( t,
0) = u ( t ) , t (cid:62) ,Y ( t ) = (cid:90) − r dµ ( θ ) X ( t + θ ) , t (cid:62) , (5.2)where K : [ − r, → L ( U ) is a continuous function, ( W t ) t (cid:62) is a standard one-dimensionalBrownian motion, and µ : [ − r, → L ( U, Y ) is a function of bounded variations. In orderto apply our abstract result, we select the following operators A m ϕ = ϕ (cid:48) , D ( A m ) = W , ([ − r, , U ) ,G : W , ([ − r, , U ) → U, Gf = f (0) ,F : H → H, F g := K ( · ) g ( · ) ,M : W , ([ − r, , U ) → Y, M ϕ = (cid:90) − r dµ ( θ ) ϕ ( θ ) . As K ( · ) is continuous then for any f ∈ H, (cid:107) F f (cid:107) H (cid:54) (cid:107) K ( · ) (cid:107) ∞ (cid:107) f (cid:107) H . The dirac operator is surjective, so G is surjective. Moreover, the operator Aϕ = ϕ (cid:48) , D ( A ) = { ϕ ∈ W , ([ − r, , U ) : ϕ (0) = 0 } generates the left shift semigroup ( S ( t )) t (cid:62) on H . In addition the control operator asso-ciated with (5.4) is given by B = ( − A − ) e ( e u )( θ ) = u, u ∈ U. We also set C := M | D ( A ) . It is shown in [14] that the triple ( A, B, C ) is compatible (seedefinition 5.2). Now according to Theorem 5.3, the transport stochastic system (5.2) iswell-posed. eferences [1] Bensoussan, A., Da Prato, G., Delfour, M.C. and Mitter, S.K., Representation and Control of Infinite-Dimensional Systems. Birkh¨auser, Boston, Basel, Berlin, 2007.[2] H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer Science andBusiness Media, 2010.[3] R.F. Curtain, A.J. Pritchard. Infinite Dimensional Linear Systems, in: Lecture Notes in Control andInformation Sciences, Vol. 8, Springer-Verlag, Berlin, 1978.[4] R.F. Curtain and H. Zwart. Introduction to Infinite–Dimensional Linear Systems. TMA 21, Springer–Verlag, New York, 1995.[5] G. Da Prato and J. Zabczyk. A note on stochastic convolution, Stochastic Analysis and Applications,10(2), 143–153, 1992.[6] G. Da Prato and J. Zabczyk. Stochastic Equations in infinite Dimensions. Cambridge UniversityPress, 2014.[7] G. Da Prato and J. Zabczyk. Evolution equations with white-noise boundary conditions, Stochasticsand Sochastics Reports, 42: 167–182, 1993.[8] K.-J. Engel, On the characterization of admissible control- and observation operators, Systems ControlLetters 34(4): 225–227, 1998.[9] K.-J. Engel, R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag,New York, Berlin, Heidelberg, 2000.[10] G. Greiner. Perturbing the boundary conditions of a generator, Houston J. Math. 18: 405–425, 2001.[11] S. Hadd, Unbounded perturbations of C -semigroups on Banach spaces and applications, SemigroupForum 70: 451–465, 2005.[12] S. Hadd, A. Idrissi, Regular linear systems governed by systems with state, input and output delays,IMA J. of Math. Control Information 22 (4), pp. 423-439.[13] S. Hadd, A. Idrissi. On the admissibility of observation for perturbed C –semigroups on Banachspaces, Systems Control Letters. 55: 1–7, 2006.[14] Hadd, S., Idrissi, A., Rhandi, A. (2006). The regular linear systems associated with the shift semi-groups and application to control linear systems with delay. Mathematics of Control, Signals and Sys-tems, 18(3), 272-291.[15] S. Hadd, R. Manzo, A. Rhandi. Unbounded perturbations of the generator domain, Discrete Con-tinuous Dynamical Systems A, 35: 703–723 ,2015.[16] B. Jacob, J.R. Partington. Admissibility of control and observation operators for semigroups: asurvey, in: J.A. Ball, J.W. Helton, M.Klaus, L. Rodman (Eds.), Current Trends in Operator Theory andits Applications, Proceedings of IWOTA 2002, Operator Theory: Advances and Applications, vol.149,Birkhuser, Basel, pp.199-221, 2004.[17] T. Kato, Perturbation Theory for Linear Operators. Springer-Verlag, 1980.[18] Kolmogorov, Andrej N., and Sergej V. Fomin. Elementi di teoria delle funzioni e di analisi funzionale.1980.[19] I. Lasiecka and R. Triggiani, Control theory for partial differential equations: continuous and ap-proximation theories. II. Abstract hyperbolic–like systems over a finite time horizon, Encyclopedia ofMathematics and its Applications, 75 Cambridge University Press, Cambridge, England, 2000.[20] Lee, E.B. and Markus, L., Foundations of Optimal Control Theory, Wiley, New York, 1967.[21] J.L. Lions, Contrˆole Optimal des Syst`emes Gouvern´es par des Equations de Deriv´ees Partielles,Dunod, Paris, 1968.[22] Lu, Q. Exact controllability for stochastic Schr¨odinger equations. Journal of Differential Equations,255(8), (2013) 2484-2504.[23] Lu, Q. Observability estimate for stochastic Schrodinger equations and its applications. SIAM Jour-nal on Control and Optimization, 51(1), (2013) 121-144.