Optimal control for stochastic heat equation with memory

In this paper, we investigate the existence and uniqueness of solutions for a class of evolutionary integral equations perturbed by a noise arising in the theory of heat conduction. As a motivation of our results, we study an optimal control problem when the control enters the system together with the noise.


Introduction
Our main goal in this paper is to analyse a class of stochastic integro-differential equation arising in the theory of heat conduction for materials with memory and to present an application to an optimal control problem where the control enters the system together with the noise. Needless to say that many physical phenomena are better described if one considers in the equation of the model some terms which take into consideration the past hystory of the system. Further, it is sensible to assume that the modeld of certain phenomena from the real world are more realistic if some kind of uncertainity, for instance, some randomness or enviromental noise, is also considered in the formulation.
We wish to mention that applications to optimal control problems naturally arise in the study of heating processes, for example in modeling heating with radiation boundary condition, simplified superconductivity, control of stationary flows, glueing in polymeric materials (for a thorough introduction to these problems we refer to the standard monograph by Lions [16] or Fredi [23].
Here we are concerned with the following semilinear heat equation in the bounded domain O ⊂ R d with Dirichlet boundary condition v |∂O (t, x) = 0, t ∈ R, x ∈ ∂O (2) and initial condition given by Notice that v 0 (·) represents the past history of the system and should satisfy suitable smoothness properties (as we will see later on). Moreover, the function k(t) = k 0 + t 0 k 1 (s)ds is called the convolution kernel of the system and k 1 is assumed to be 3-monotone (see Hypothesis 2.1 for the precise definition of this term).
We are interested in the analysis of the system (1) when the function g: i. is given by a Lipschitz continuous term f and an additive gaussian noise W with covariance Q, i.e.
ii. depends on a further parameter γ which introduces a control process in the system; this means that g = g(t, x, v, γ) = f (t, v(t, x)) + Q(r(t, v(t, x), γ(t, x)) + ∂ t W (t)), where r is a function with appropriate regularity.
The main question arising around the first case (which we refer to as uncontrolled problem) is to determine existence and uniqueness of the solution. This problem can be handled by reducing equation (1) to an abstract Cauchy problem on an appropriate product space, which contains the whole history of the solution. Within this framework the system can be represented with the following evolutionary equation where A is the generator of a C 0 -semigroup, F is a Lipschitz continuous function Q a linear operator and W a vector defined in term of the Wiener process (W (t)) t≥0 . Similar approaches are widely used in literature, see for example Miller [18] and Dafermos [10] for the deterministic counterpart, Caraballo and Chueshov [3,4] and the more recent work [2] for stochastic models. Anyway, differently for them, we are able to treat more general kernels k.
We stress that our approach may has the advantage that it naturally links the solution of a Volterra equation to a Markov process; this has the important development in view of the application of the analytic machinery to Volterra equations to solve the optimal control problems.
In the case g contains a control parameter, the natural problem is to determine a solution of the Volterra equation and a control process γ, within a set of admissible controls, in such a way that they minimize a cost functional. In particular in this paper we consider a cost of the form: where ℓ and φ are given real functions.
In the same way as the uncontrolled problem, the model can be translated into an abstract setting. In particular, it can be rewritten in the form where A, F, Q, W, X 0 are as above and R is given in terms of the function r. Notice here the special structure of the control term, which is clearly a restriction; however it arises from concrete models. Due to the special structure of the control term we are able to perform the synthesis of the optimal control, by solving in the weak sense the closed loop equation. Thus, we can characterize optimal controls by a feedback law.
The paper is organized as follows: in the next subsection we give the physical motivation of our work; in Section 2 we introduce the main assumptions on the coefficients of the problem while in Section 3 we reformulate the uncontrolled problem into a semilinear abstract evolution equation and we study the properties of leading operator, while in Section 4 we are concerned with the stochastic convolution of the rewritten equation. To this end, we study the so-called resolvent family (see Subsection 4.1) and the scalar resolvent family (see Subsection 4.2) associated with our problem. In Section 5 we prove the first main result of the paper: we determine existence and uniqueness of the solution of the uncontrolled Volterra equation 1. Finally, in Section 6 we perform the standard synthesis of the optimal control.

Motivation
Let us briefly explain one possible physical meaning of our model. Let O be a 3-dimensional homogeneous and isotropic rigid body (see Prüss [21, p.125] for more details on the physical terminology) which is represented by an open set O ⊂ R d (d = 1, 2, 3) with boundary ∂O of class C 1 . Points in O (i.e. material points) will be denoted by x, y, . . . . Suppose that the body O is subject to temperature changes. We denote by v = v(t, x) the temperature at time t ∈ R + , q(t, x) the heat flux vector field, e(t, x) the temperatue and f (t, x, v) the heat supply (possibly depending on the solution itself).
We denote by v = v(t, x) the temperature at time t ∈ R + , q(t, x) the heat flux vector field, e(t, x) the temperatue and f (t, x) the external heat supply. Balance of energy then reads as: with the boundary conditions basically either prescribed temperature or prescribed heat flux through the boundary. In particular, one (natural) choice is represented by Dirichlet boundary conditions: For the relationship between e and v we shall use the following linear law (or, more formally, constitutive law ): where e ∞ is a suitable positive phenomenological constant. Analogously, for the constitutive law relating q and v we choose where m, k ∈ BV loc (R + ) are scalar functions. Rearranging equation (4), we arrive at the following non autonomous heatequation with memory where * denotes the symbol for the convolution product between two functions. Remark 1.1. From the literature one can infer that m is a creep function, i.e. it is nonnegative, nondecreasing and concave which is also bounded. The natural form of this kind of functions is given by From a physical point of view, m 0 corresponds to the istantaneous heat capacity, i.e. the ratio of the change in heat energy of a unit mass of a substance to the change in temperature of the substance. The function m 1 is called energytemperature relaxation function while m(∞) = m 0 + ∞ 0 m 1 (s)ds is termed equilibrium heat capacity. In accordance with several works concerning with same type of problems (see, for example, Clément and Nohel [7], Nunziato [20], Monnieaux and Prüss [19], Grasselli and Pata [13,14]), in this paper we choose for semplicity m(t) ≡ m 0 = 1.
Concerning the function k, the literature is somewhat controversial. From Gurtin and Pipkin [15] and Nunziato [20] one can expext that k is a bounded creep function as well, in particular k ∞ = 0, k 1 ∈ L 1 (R + ) and k(∞) = k 0 + ∞ 0 k 1 (s)ds > 0. The constant k 0 is termed istantaneous conductivity, k(∞) is called equilibrium conductivity while k 1 is called heat conduction relaxation function. On the other hand, Clément and Nohel [7], Clément and Prüss [8] and Lunardi [17] and write k(t) = k 0 − t 0 k 1 (s)ds > 0 with k 1 positive and nonincreasing; in this case k is 2-monotone (see Hypothesis 2.1 for the explanation of this term). Also Bonaccorsi and Da Prato and Tubaro [2] consider k as above but they require k 1 completely monotone. In this theory the equilibrium conductivity k(∞) = k 0 − ∞ 0 k 1 (s)ds is smaller then the istantaneous conductivity, in contrast with Nunziato. We stress that in the present paper, we assume that k has the same form as in Gurtin and Pipkin [15] and Nunziato [20].

General assumptions
In equation (1), we are given the kernel k : R → R, the non linear term f : R → R and the stochastic perturbation (W (t)) t≥0 .
We assume the following.
where k 0 > 0 and the function k 1 is 3-monotone, that is, it satisfies the following conditions: h2) k 1 is positive and nonincreasing; h3) −k ′ 1 is nonincreasing and convex; Remark 2.2. We stress that the above assumption allows k 1 (t) to have a singularity at t = 0, whose order is less than 1, since k(t) is a non-negative function in L 1 (R + ). For instance, we are able to consider a weakly singular kernel of the following type Concerning the nonlinear part of the system we have: 1. f is continuous and differentiable on R.
2. f is Lipschitz continuous with respect to x, uniformly on t, and has sublinear growth; this means that there exists a constant L > 0 such that The conditions on the stochastic perturbation are given in the following.
1. The process (W (t)) t≥0 is a cylindrical Wiener process defined on a complete probability space with values in L 2 (O). In particular W (t) is of the form where {β k } k∈N is a sequence of real, standard, independent Brownian motions on (Ω, F, (F t ) t≥0 , P).
2. Q is a linear bounded operator, symmetric and positive. With no loss of generality, we shall assume in the sequel that A and Q diagonalizes on the same basis of L 2 (O) (this is required only for semplicity); 3. If {µ j } j∈N and {λ j } j∈N are respectively the eigenvalues of A and Q then we require where δ is the quantity and θ is any real number in (0, 1) such that 1 + θ > δ.
i. We notice that the quantity δ introduced in (5) depends only on the behaviour of the Laplace transform of the kernel k. In Pruss and Monnieaux [19] it is proved that, for the class of kernels considered by us (i.e. for 3monotone kernels), the Laplace transformk satisfies the following bound: and, consequently, δ belongs to (1,2). Following the terminology in Prüss (see [21]) we say that the kernel k is θ-sectorial.
It can be proved that the sectoriality of the kernel plays a central role in the study of the Volterra equation 1. In particular, it allows to prove existence of the resolvent family corresponding with the problem, and consequently to investigate existence and uniqueness of the solution. For more details we refer to Section 4.1 and the monograph [21, Section 3].
3 Statement and Reformulation of the uncontrolled equation

The abstract setting
In this section we are concerned with the following (uncontrolled) class of integral Volterra equations perturbed by an additive Wiener noise Our first purpose is to rewrite equation (7) as an evolution equation defined on a suitable Hilbert space.
To this end we denote with L 2 (O) the space of square integrable, real valued functions defined on O with scalar product u, v = O u(ξ)v(ξ)dξ, for any u, v ∈ L 2 (O). Sobolev spaces H 1 (O) and H 2 (O) are the spaces of functions whose first (resp. first and second) derivative are in L 2 (O). We set moreover H 1 0 (O) the subspace of H 1 (O) of functions which vanish (a.e.) on the boundary ∂O.
We let X = H −1 (O) the topological dual of H 1 0 (O). We recall that the operator ∆ with domain H 1 0 (O) is the generator of a C 0semigroup of contractions; since ∆ is self adjoint, the semigroup is analytic: see for instance [24,22, Theorem 1.5.7, Corollary 1.5.8].
In order to control the unbounded delay interval, we shall consider L 2 weighted spaces. Let and y X the corresponding norm. On this space, we introduce the delay operator K with domain D(K) = X by setting Our aim is to reduce this problem to an abstract Cauchy problem on the product space H in such a way that the first component gives the evolution of the system while the second contains all the informations concerning the whole history of the solution. The state variable in the Hilbert space H will be denoted by X(t). Thus (X(t)) t≥0 is a process in H and the initial condition is assumed to belong to H and satisfies suitable properties to be precised.
We introduce the linear operator A defined as: In order to handle the contribution of temperature values taken in the past, we introduce the new variable Moreover, we introduce the non linear operator F : where f is the non linear term in Equation 7. Finally we introduce the linear operator Q and the stochastic perturbation W on H as With the above notation, problem (7) can be rewritten in the form where X(t) stands for the pair v(t) η t (·) and X 0 := v η(·) is the initial condition. In the following (see Section 3.2) we will see that the dynamics of the system is described in terms of the transition semigroup e tA generated by the linear operator A. As a consequence we will read the solution of the original Volterra equation in the first component of X.
Before prooceding, let us recall the definition of mild solution for the stochastic Cauchy problem (9). Definition 3.1. Given an F t -adapted cylindrical Wiener process on a probability space (Ω, F, (F t ) t≥0 , P), a process (X(t)) t≥0 is a mild solution of (9) if it belongs to L 2 (0, T ; L 2 (Ω; H)) and satisfies P-a.s. the following integral equation Condition 10 implies that the integrals on the right-member are well defined. In particular, the second integral, which we shall refer to as stochastic convolution, is a mean-square continuous gaussian process with values in H. For the analysis of the stochastic convolution and its properties, we refer Section 4.

Generation properties
In this section we are dealing with the generation properties of the leading (matrix) operator and prove that, in our setting, the operator is quasi-m-dissipative (see inequality (11) below) and that the range of µ − A is dense in H for some (and all) µ > 0. In this way we will able to apply the Lumer-Phillips theorem to conclude that A − µ, and hence A, generates a C 0 -semigroup.
We start by proving the dissipativity properties.
Proof. We proceed in the same spirit as Bonaccorsi and Da Prato and Tubaro [2, Theorem 3.1], but we include the proof for completeness. The difference, here, is that we have no conditions linking the constant k 0 and the function k 1 . In contrast with [2], this point doesn't allow to prove the pure dissipativity of A, but only quasi-m-dissipativity. We compute the scalar product and we get Next, we consider the properties of the resolvent R(µ, A). Theorem 3.3. For every µ > 0 the equation has a unique solution φ ∈ D(A). moreover, using the monotonicity property of ρ it is possible to prove that η ∈ W 1,2 ρ (R + ; H 1 0 (O)). Straightforward calculation, gives where c k,µ is a positive constant depending only on k and µ whileṽ is the functionṽ Obviously u ∈ D(∆ D ). Finally, we notice that hence, it turns out that φ = u η ∈ D(A). Taking into account the above results, we can deduce the generation properties for the operator A. Precisely, we have

The stochastic convolution
The main object of investigation of this section is the stochastic convolution corresponding with our problem, that is the process In particular, our purpose is to prove that (W A (t)) t≥0 is a well-defined meansquare continuous gaussian process with values in H. Following the approach of Da Prato and Clement [5], Bonaccorsi and Da Prato and Tubaro [2], we can give a meaning to the stochastic convolution through the study of the socalled resolvent family associated with an abstract homogeneous linear Volterra equation of type where k is a kernel satsfying Hypothesis 2.1 and where ∆ D denotes the Laplace operator onŌ with Dirichlet boundary conditions. The concept of the resolvent plays a central role for the theory of linear Volterra equations and can be applied to inhomogeneous problem to derive a variation of parameters formula. The main tools for the resolvent are described in detail in the monograph [21]. In the next subsection we recall a few basic concepts and results.

The resolvent family
Following [21, Section 1], we define the resolvent family for the equation (13) as Definition 4.1. A family (S(t)) t≥0 of bounded linear operators in X is called a resolvent for equation (13) if the following conditions are satisfied: (S1) S(0) = I and, for all x ∈ X, t → S(t)x is continuous on R + ; (S2) S(t) commutes with ∆ D , that is for a.e. t ≥ 0, S(t)D(∆ D ) ⊂ D(∆ D ) and (S3) for anyv ∈ D(∆ D ), t → S(t)v is a strong solution of (7) on [0, T ], for any T > 0.
It turns out that if the kernel k satisfies Hypothesis 2.1 (or, more generally, if it is θ-sectorial for θ < π), then equation (13) admits a resolvent (S(t)) t≥0 which is uniformly bounded in L 2 (O) (see [21,Corollary 3.3]). Consequently (see [21, Proposition 1.1]), problem (13) is well-posed and its strong solution is given by the function v(t) = S(t)v. Besides, since k(t) belongs to BV loc (R + ), S(t) turns out to be differentiable and consequently (by differentiation of equation (13)) the function v is the mild solution of the homogeneous Cauchy problem Here the term dk * ∆ D v(t) denotes the function Analogously, it can be proved that if g is a function belonging to L 1 (0, T ; X), then the Cauchy problem is well-posed too and its (unique) mild solution can be represented through the variation of parameter formula as For a full discussion about the notion of well-posedness for equation (13), of mild solution for problems of type (14), (15) and their relationship between the resolvent family we refer to [21, Section 1].
Here we want to emphasize that the above arguments can be applied to the inhomogeneous Volterra equation (7) to obtain existence and uniqueness of a mild solution and its representation in terms of the resolvent family corresponding with the kernel k = 1 * dk. In fact, equation (7) is equivalent to v t (t, x) = k 0 ∆v(t, x) + t 0 k 1 (s)∆v(s, x)ds with boundary and initial conditions given by: In abstract form, we have Now, by the associativity property of the convolution product, the second term in the right member of (17) gives Hence equation (7) can be rewritten as follows: where the function h is given by

Now the variation of parameters formula implies that the function
is a mild solution of the Volterra equation (16), provided that h ∈ L 1 (0, T ; X).
We notice that the condition v 0 ∈ L 2 ρ (R + ; H 1 0 (O)) assures the requested regularity for the function h.

The scalar resolvent family
Suppose that (S(t)) t≥0 is the resolvent family for equation (13) and let {µ j } j∈N be the set of eigenvalues of ∆ D with respect to the basis {e j } j∈ bN . For any j ∈ N, we introduce the following one-dimensional Volterra equation Then (see [21, Section 1.3]) a unique solution to (19) exists and it satisfies S(t)e j = s j (t)e j , t ≥ 0.
In particular, the resolvent family S(t) admits a decomposition in the basis {e j } of L 2 (O) in terms of the solutions s j to (19).
In the sequel we state and prove some useful estimates on the scalar resolvent functions s j . They are crucial to study the stochastic convolution and descend immediately from the assumption on the kernel k. admits a solution s j (t) such that the following properties hold: Hence assertion 2 implies that the limit of s j (R) for R → ∞ exists; moreover, we have We observe that the last term in the above equality can be rewritten as Further, since s j satisfies equation (19), we get λŝ j (λ) = 1 λ + µ jk (λ) = λ λ 2 + k 0 +k 1 (λ) ; (20) in fact, we haveŝ We notice that, since k 1 belongs to L 1 (R + ), for any λ ≥ 0 it holds Taking into account the last inequality and equality (20) we see that the limit of s j (R) for R → ∞ satisfies For further use, we conclude this subsection with an estimate concerning the norm of s j in L 2 (R + ).
Now integrating both members of the previous inequality we obtain the thesis.

The representation of the semigroup
In the following we show that the semigroup corresponding with the linear operator e tA can be computed explicitly in terms of the resolvent family (S(t)) t≥0 . We recall that since the linear operator A generates a C 0 -semigroup, there exists a unique mild solution (X(t)) t≥0 for the deterministic equation The variation of parameter formula for abstract evolution equations applies to equation (21) and we can write: By construction, the first component of X satisfies the inhomegenous Volterra equation (23) and the variation of parameters formula for Volterra equations applied to (23) (see Subsection 4.1) yields where Comparing the first terms in equalities (22) and (24), we obtain Moreover, from the second part of (22) and (24), we have for s ≥ 0 Thus the semigroup e tA is completely described in terms of the resolvent family. As we will see in the next subsection, the above characterization allows to study the stochastic convolution process.

The stochastic convolution
We are now in the position to prove the main result of this section. We recall that (W (t)) t≥0 is a cylindrical Wiener process of the form where {β k } k∈N is a sequence of real, standard, independent Brownian motions on (Ω, F, (F t ) t≥0 , P). We have: Lemma 4.4. Under Hypothesis 2.5, for all T > 0 the process (W A (t)) 0≤t≤T defined as is a gaussian random variable with mean 0 and covariance operator Proof. It is well-known that the thesis follows provided that where C T is a positive constant depending only on T > 0. Recalling the repre-sentation of e tA given in (25) and (26), we have that We consider separately the two series in the previous formula. We recall that S(t)e j = s j (t) for any j ∈ N (see Subsection 4.2); hence we get where the last inequality follows from Lemma 4.2, point 1. Moreover, since it holds also that Concerning the second series in (28), applying Fubini's theorem, we get and, taking into account Lemma 4.3 and the definition of the function ρ (see (8)), By the above estimates and condition 3 in Hypothesis 2.5, we conclude that, for any θ ∈ (0, 1) such that 1 + θ > δ,

Existence and uniqueness
In this section we aim to prove existence and uniqueness of the solution for the uncontrolled equation where the coefficients k 0 , k 1 , f, Q satisfy the assumptions made in Section 2.
Recalling what has been showed in the previuos section, the above equation as an abstract equation on the space H : We recall that, from Proposition 3.4 A is the generator of a C 0 -semigroup, while from the assumption on the function f we get that F : H → H is Lipschitz continuous. Moreover, Q is a linear operator on H involving the covariance operator Q, X 0 = (v 0 (0, ·), (v 0 (−s, x)) s≥0 ) t and the stochastic convolution W A (t) introduced in (27)  Proof. The proof follows directly from Theorem 5.1. In fact, the mild solution of (29) is represented by the first component of the process (X(t)) t≥0 .

Synthesis of the optimal control
In this section we proceed with the study of the optimal control problem associated with the stochastic Volterra equation Here f is the nonlinear function introduced in Hypothesis 2.4 and γ = γ(ω, t, x) is the control variable, which is assumed to be a predictable real-valued process F t -adapted. The optimal control that we wish to treat consists in minimizing over all admissible controls a cost functional of the form where ℓ and φ are given real-valued functions. We will work under the following general assumptions. Concerning the function r, ℓ, φ we require: |r(t, x 1 , y) − r(t, x 2 , y)|+|ℓ(t, θ 1 , y) − ℓ(t, θ 2 , y)| ≤ C(1 + |θ 1 | + |θ 2 |) m |θ 1 − θ 2 |, |r(t, θ 1 , y)| + |ℓ(t, 0, y)| ≤ C.
2. φ ∈ C 1 (R) and there exist L > 0 and k ∈ N such that for every θ ∈ R In order to characterize the optimal control through a feedback law, we impose the following additional condition on the nonlinear term f : To handle the control problem, we first restate equation (31) in an evolution setting and we provide the synthesis of the optimal control by using the forwardbackward system approach.
Arguing as in Section 3, given a control process γ and any t ∈ [0, T ], v 0 ∈ L 2 ρ (R + ; H 1 0 (O)) we rewrite the problem (31) in the following abstract form where X 0 = (v 0 (0), v 0 (·)) and R : In this setting the cost functional will depend on X 0 and γ and is given by where L : for any t > 0, v η ∈ H, γ ∈ H and Φ : H → R is defined as There are different ways to give a precise meaning to the above problem; one of them is the so called weak formulation and will be specified below.
In the weak formulation the class of admissible control systems (a.c.s.) is given by the set U := (Ω,F, (F t ) t≥0 ,P,Ŵ ,γ), where (Ω,F,P) is a complete probability space; the filtration (F t ) t≥0 verifies the usual conditions, the procesŝ W is a Wiener process with respect to the filtration (F t ) t ≥ 0) and the control γ is an F t -predictable process taking value in some subset U of X with respect to the filtration (F t ) t≥0 ).
With an abuse of notation, for given X 0 ∈ H, we associate to every a.c.s. a cost functional J(x, U) given by the right side of (33). Altough formally the same, it is important to note that now the cost is a functional of the a.c.s. and not a functional ofγ alone. Any a.c.s. which minimizes J(x, ·), if it exists, is called optimal for the control problem starting from X 0 at time t in the weak formulation. The minimal value of the cost is then called the optimal cost. Finally we introduce the value function V : [0, T ] × H → R of the problem as: where the infimum is taken over all a.c.s. U.
At this moment it is convenient to list the relevant properties of the objects introduced so far in this section. Therefore we formulate the following proposition. Proposition 6.3. Under Hypothesis 2.1,2.4, 6.2, 2.5 and 6.1 the following properties hold: 1. The functions R and L are Borel measurable and there exist constants C, m, k ∈ N such that for any t > 0, X 1 , X 2 ∈ H and γ ∈ U 2. Φ is Gâteaux differentiable and there exist C Φ > 0 and k ∈ N such that for every X 1 , for every t ∈ [0, T ], X, X 1 , X 2 ∈ H. Moreover, for every t ∈ [0, T ], F(t, ·) has a Gâteaux derivative ∇F(t, X) at every point X ∈ H. Finally, the function (X, H) → ∇F(t, X)[H] is continuous as a map H × H → R.
Optimal control problems associated with equation (32) and the cost functional (33) when the coefficients has the properties listed in Proposition 6.3 has been exhaustively studied by Fuhrman and Tessitore in [12], compare Theorem 7. Within their approach the existence of an optimal control is related to the existence of the solution of a suitable forward backward system (FBSDE) that is a system in which the coefficients of the backward equation depend on the solution of the forward equation. Moreover, the optimal control can be selected using a feedback law given in terms of the solution to the corresponding FBSDE.
We introduce the hamiltonian function ψ : and we define the following set Γ(t, X, Z) = {γ ∈ U : L(t, X, γ) + Z, R(t, X, γ) = ψ(t, X, Z)} , For further use we require some additional properties of the function ψ: 1. For all t ∈ [0, T ], for all X, Z ∈ H there exists a unique Γ(t, X, Z) that realizes the minimum in (34). Namely: ψ(t, X, Z) = L(t, X, Γ(t, x, Z)) + Z, r(t, X, Γ(t, X, Z) with Γ ∈ C([0, T ] × H × H; U). Remark 6.5. It is easy to prove that combining the previous assumption with Proposition 6.3 we can deduce the following properties of ψ: 1. ψ is a measurable mapping and there exists a constant C such that for all X 1 , X 2 , Z ∈ H and t ∈ [0, T ].

Setting
whereW(t) = (W (t), 0) t . By Theorem 5.1 stated in Section 5, equation 35 is well-posed and the solution (X(t)) t≥0 : is a continuous process in H, adapted to the filtration (F t ) t≥0 . Moreover, the law of (W,X) is uniquely determined by X 0 , A, F and √ Q. We define the process W U (t) =W(t) − t 0 R(s,X(s),γ(s))ds, t ∈ [0, T ], and we note that, since R is bounded, by the Girsanov theorem there exists a probability measure P on (Ω, F) such that W U is a Wiener process under P.
Next we consider the backward stochastic differential equatioñ Y (t) + T tZ dW (σ) = Φ(X(T )) + T t ψ(σ,X(σ),Z(σ))dσ, t ∈ [0, T ], (37) where ψ is the hamiltonian function and Φ is the function defining the final cost. Under our assumptions , we can apply [12, Proposition 3.2 and Theorem 4.8] and state that there exists a solution (X,Ỹ ,Z) of the forward-backward system (35)-(37) on the interval [0, T ], whereỸ is unique up to indistinguishability and Z is unique up to modification. Moreover from the proof of Theorem 4.8 [12] it follows that the law of (Ỹ ,Z) is uniquely determined by the law of (W,X) and by Φ and Ψ. We note thatỸ (t), being measurable with respect to the degenerate σ-algebraF 0 , is deterministic; in particularỸ (t) = E(Ỹ (t)) only depends on the law ofỸ , and thus it is a functional of X 0 , A, F, √ Q, Φ, Ψ. To stress dependence on the initial datum X 0 , we will denote the solution of (35) and (37) by {(X X0 (t),Ỹ X0 (t),Z X0 (t)), t ∈ [0, T ]}.
The relevance of the solution of the Hamilton-Jacobi-Bellman equation to our control problem is explained in the following proposition.
Proof. The result follows immediately from the paper of Fuhrman and Tessitore [12,Theorem 7.2].