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.

Consider a stochastic controlled problem governed by the following stochastic evolution equation (SEE):

This system is driven mainly by a possibly unbounded linear operator

We will derive the maximum principle for this control problem. More precisely, we will concentrate on providing necessary conditions for optimality for this optimal control problem, which gives this minimization. For this purpose we will apply the theory of

In our work here, we allow

On the other hand, we recall that control problems governed by SPDEs that are driven by martingales are studied in [

In the present work, we will show how to handle this open problem in great success, and as we stated earlier, we can and will allow all coefficients in (

Let

For a separable Hilbert space

This space is Hilbert with respect to the norm

Moreover, if

Let us assume that

Suppose that

Let

For example, one can take

The optimal control problem of system (

If this happens, the corresponding solution

We close this section by the following theorem.

Assume that

The proof of this theorem can be derived in a similar way to those in [

From here on, we will assume that

It is known from the literature that BSDEs play a fundamental role in deriving the maximum principle for SDEs. In this section, we will search for such a role for SEEs like (

Then, we consider the following BSEE on

As in the previous section, a

Assume that

Then, there exists a unique (mild) solution

The proof of this theorem can be found in [

Our main result is the following.

Suppose that the following two conditions hold.

If

The proof of this theorem will be given in Section

Let

The Hamiltonian is then given by the formula:

From the construction of the solution of (

On the other hand, for fixed

It is easy to see that with these choices all the requirements of Theorem

Finally, the value function attains the formula

A concrete example in the setting of Example

The computations in this case of

Let

We note that the convexity of

Let

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.

Assume condition (i) of Theorem

The solution of (

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

Assuming condition (i) of Theorem

Observe first from (

Hence,

Secondly, from condition (i), we get

Similarly,

Finally, by applying (

Keeping the notations

Let

From the corresponding equations (

But (i), (

Similarly, we have

On the other hand, as done for (

Finally applying (

Hence, from (

The following theorem contains our main variational equation, which is one of the main tools needed for deriving the maximum principle stated in Theorem

Suppose that (i) and (ii) in Theorem

We can write

Note that with the help of our assumptions and by making use of Lemmas

On the other hand, applying Lemmas

As a result, the theorem follows from (

Let us next introduce an important variational inequality.

Let hypotheses (i), (ii) in Theorem

Since

The following duality relation between (

Under hypothesis (i) in Theorem

The proof is done by using Yosida approximation of the operator

We are now ready to establish (or complete in particular) the proof of Theorem

Recall the BSEE (

Applying (

Now, by applying (

Finally, (

The author would like to thank the associate editor and anonymous referee(s) for their remarks and also for pointing out the recent work of Fuhrman et al., [