Necessary conditions for optimality for stochastic evolution equations

This paper is concerned with providing the maximum principle for a control problem governed by a stochastic evolution system on a separable Hilbert space. In particular, necessary conditions for optimality for this stochastic optimal control problem are derived by using the adjoint backward stochastic evolution equation. Moreover, all coefficients appearing in this system are allowed to depend on the control variable. We achieve our results through the semigroup approach.


Introduction
Consider a stochastic controlled problem governed by the following stochastic evolution equation (SEE): dX(t) = (AX(t) + b(X(t), ν(t)))dt + σ(X(t), ν(t))dW (t), t ∈ (0, T ], X(0) = x 0 . (1.1) We shall be interested in trying to minimize the cost functional, which is given by equation (2.2) below, over a set of admissible controls. This system is driven mainly by a possibly unbounded linear operator A on a separable Hilbert space H and a cylindrical Wiener process W on H. Here ν(·) denotes a control process.
We shall derive the maximum principle for this control problem. More precisely, we shall concentrate on providing necessary conditions for optimality for 0 * This work is supported by the Science College Research Center at Qassim University, project no. SR-D-012-1610. this optimal control problem, which gives this minimization. For this purpose we shall apply the theory of backward stochastic evolution equations (BSEEs shortly) as in equation (3.2) in Section 3. These equations together with backward stochastic differential equations (BSDEs) have become of great importance in a number of fields. For example in [4], [6], [13], [15], [16], [17] and [19] one can find applications of BSDEs to stochastic optimal control problems. Some of these references have also studied the maximum principle to find either necessary or sufficient conditions for optimality for stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs). Necessary conditions for optimality of the control process ν(·) and its corresponding solution X ν(·) but for the case when the noise term σ does not depend on ν(t) can be found in [13].
In our work here we allow σ to depend on the control variable and study a stochastic control problem associated with the former SEE. This control problem is explained in details in Section 2, and the main theorem is stated in Section 3 and is proved together with all necessary estimates in Section 4. Sufficient conditions for optimality for this optimal control problem can be found in [6]. We refer the reader also to [4].
On the other hand, we recall that control problems governed by SPDEs that are driven by martingales are studied in [5]. In fact in [5] we derived the maximum principle (necessary conditions) for optimality of stochastic systems governed by SPDEs. The technique used there relies heavily on the variational approach. The reason beyond that is that the only known way until now to find solutions to the resulting adjoint BSPDEs is achieved through the same variational approach, and is established in details in [3]. Thus the semigroup approach to get mild solutions (as done here in Theorem 3.1 below and in Section 3) cannot be used to study such adjoint BSPDEs considered in [5]. Moreover, it is not obvious how one can allow the control variable ν(t) to enter in the noise term and in particular in the mapping G in equation (1.1) of [5] and obtain a result like Theorem 3.2 below. This problem is still open and is also pointed out in [5,Remark 6.4].
In the present work, we shall show how to handle this open problem in great success, and as we stated earlier, we can and will allow all coefficients in (1.1) and especially in the diffusion term to depend on the control variable ν(t). We emphasize that our work here does not need go through the technique of Hamilton-Jacobi-Bellman equations nor the technique of viscosity solutions. We refer the reader to [10] for this business and to [8] and some of the related references therein for the semi-group technique. Thus our results here are new. In this respect we thank the anonymous referee for pointing out the recent and relevant work of Fuhrman et al. in [11].

Statement of the problem
Let (Ω, F , P) be a complete probability space and denote by N the collection of Pnull sets of F . Let {W (t), 0 ≤ t ≤ T } be a cylindrical Wiener process on H with its completed natural filtration F t = σ{ℓ • W (s) , 0 ≤ s ≤ t , ℓ ∈ H * } ∨ N , t ≥ 0; see [1] for more details.
For a separable Hilbert space E denote by L 2 This space is Hilbert with respect to the norm is the space of all Hilbert-Schmidt operators on H, the stochastic integral f (t)dW (t) can be defined and is a continuous stochastic martingale in H. The norm and inner product on L 2 (H) will be denoted respectively by || · || 2 and ·, · 2 .
Let us assume that O is a separable Hilbert space equipped with an inner product ·, · O , and U is a convex subset of O. We say that ν(·) and ν(t) ∈ U a.e., a.s. The set of admissible controls will be denoted by U ad .
Suppose that b : H × O → H and σ : H × O → L 2 (H) are two continuous mappings, and consider the following controlled SEE: where ν(·) ∈ U ad . A solution (in the sense of the following theorem) of (2.1) will be denoted by X ν(·) to indicate the presence of the control process ν(·). Let ℓ : H × O → R and φ : H → R be two measurable mappings such that the following cost functional is defined: For example one can take ℓ and φ to satisfy the assumptions of Theorem 3.2 in Section 3.
The optimal control problem of the system (2.1) is to find the value function J * := inf{J(ν(·)) : ν(·) ∈ U ad } and an optimal control ν * (·) ∈ U ad such that If this happens, the corresponding solution X ν * (·) is called an optimal solution of the stochastic control problem (2.1)-(2.3) and (X ν * (·) , ν * (·)) is called an optimal pair. We close this section by the following theorem.
Theorem 2.1 Assume that A is an unbounded linear operator on H that generates a C 0 -semigroup {S(t), t ≥ 0} on H, and b, σ are continuously Fréchet differentiable with respect to x and their derivatives b x , σ x are uniformly bounded. Then for every ν(·) ∈ U ad there exists a unique mild solution X ν(·) on [0, T ] to (2.1). That is X ν(·) is a progressively measurable stochastic process such that The proof of this theorem can be derived in a similar way to those in [9,Chapter 7] or [14].
From here on we shall assume that A is the infinitesimal generator of a C 0semigroup {S(t), t ≥ 0} on H. Its adjoint operator A * :

Stochastic maximum principle
It is known from the literature that BSDEs play a fundamental role in deriving the maximum principle for SDEs. In this section we shall search for such a role for SEEs like (2.1). To prepare for this business let us first define the Hamiltonian by the following formula: Then we consider the following BSEE on H: where ∇φ denotes the gradient of φ, which is defined, by using the directional derivative . This equation is the adjoint equation of (2.1).
As in the previous section a mild solution (or a solution) of (3.2) is a pair Theorem 3.1 Assume that b, σ, ℓ, φ are continuously Fréchet differentiable with respect to x, the derivatives b x , σ x , σ ν , ℓ x are uniformly bounded, and The proof of this theorem can be found in [2] or [12]. An alternative proof by using finite dimensional framework through the Yosida approximation of A can be found in [18].
Our main result is the following.
Theorem 3.2 Suppose that the following two conditions hold. (i) b, σ, ℓ are continuously Fréchet differentiable with respect to x, ν, φ is continuously Fréchet differentiable with respect to x, the derivatives b x , b ν , σ x , σ ν , ℓ x , ℓ ν are uniformly bounded, and If (X ν * (·) , ν * (·)) is an optimal pair for the control problem (2.1)-(2.3), then there exists a unique solution (Y ν * (·) , Z ν * (·) ) to the corresponding BSEE (3.2) s.t. the following inequality holds: The proof of this theorem will be given in Section 4 below. Now to illustrate this theorem let us present an example.

Example 3.3
Let H and O be two separable Hilbert spaces as considered earlier, and let U = O. We shall study in this example a special case of the control problem (2.1)-(2.3). In particular, given φ as in Theorem 3.2, we would like to minimize the cost functional: subject to:

5)
where B is a bounded linear operator from O into H and D is another bounded linear operator from O into L 2 (H).
The Hamiltonian is then given by the formula: where (x, ν, y, z) ∈ H × O × H × L 2 (H), and the adjoint BSEE is From the construction of the solution of (3.6), as e.g. in [2, Lemma 3.1], this BSEE attains an explicit solution: On the other hand, for fixed (x, y, z), we note that the function ν → H(x, ν, y, z) attains its minimum at ν = as a candidate optimal control. It is easy to see that with these choices all the requirements of Theorem 3.2 are verified. Hence this candidate ν * (·) given in (3.7) is an optimal control for the problem (3.4)-(3.5), and its corresponding optimal solution X ν * (·) is the solution of the following SEE: Finally, the value function attains the formula The computations in this case of H, Y * , Z * , ν * , X * become direct from the corresponding equations in Example 3.3.
Let p be the solution of the following linear equation: The following three lemmas contain estimates that will play a vital role in deriving the desired variational equation and the maximum principle for our control problem. Proof. The solution of (4.1) is given by the formula By using Minkowski's inequality (triangle inequality), Holder's inequality, Burkholder's inequality for stochastic convolution together with assumption (i) and Gronwall's inequality we obtain easily for some constant C > 0.
Finally, by applying (4.6), (4.8) in (4.4) and then using Gronwall's inequality we find that for some constant C 5 > 0 that depends in particular on C i , i = 1, . . . , 4, and M. Hence the proof is complete.
Keeping the notationsb x and δ ε b used in the preceding proof let us state the following lemma.
Hence from (4.11) and Gronwall's inequality we obtain The following theorem contains our main variational equation, which is one of the main tools needed for deriving the maximum principle stated in Theorem 3.2.
Let us next introduce an important variational inequality.
The following duality relation between (4.1) and (3.2) is also needed in order to establish of proof of Theorem 3.2. Proof. The proof is done by using Yosida approximation of the operator A and Itô's formula for the resulting SDEs, and can be gleaned directly from the proof of Theorem 2.1 in [18].
We are now ready to establish (or complete in particular) the proof of Theorem 3.2.
From Theorem 3.1 there exists a unique solution (Y * , Z * ) to it. Thereby it remains to prove (3.4).