Fractional diffusion models have recently received a tremendous amount of attention. They have shown a remarkable flexibility in capturing anomalous behavior. Recently in 
, the authors have derived the fractional Helmholtz equations using the first principle arguments combined with a constitutive relationship. This article also shows a qualitative match between the real data and the numerical experiments. It is natural to consider the optimal control or inverse problems with such partial differential equations (PDEs) as constraints and has also motivated the current study.
The main goal of this paper is to consider an optimal control problem with fractional parabolic PDEs as constraints. We denote the fractional exponent by which lies strictly between 0 and 1. The key novelty stems from the state constraints. Optimal control of fractional PDEs have recently received a significant amount of attention. We refer to [4, 2] for optimal control of fractional elliptic PDEs with state constraints. Moreover, see  for the optimal control of semilinear fractional PDEs with control constraints, see also  for the linear case. We also refer to [3, 5] for the exterior optimal control of fractional elliptic and parabolic PDEs with control constraints.
We emphasize that the optimal control of parabolic PDEs with state constraints is not new, see for instance  for the classical case . Similarly to our observation in the elliptic case , we emphasize that almost none of the existing works can be directly applied to our fractional setting. For instance, we need finer integrability conditions on the data to show the continuity of solution to the fractional PDEs, and we need to create the notion of very-weak solution to fractional parabolic PDEs with measure-valued datum.
Next we present the statements of our main results on fractional PDEs while delaying proofs to the body of the paper. We do this in order to show the novelties of this work and clearly distinguish it from previous similar, yet different, papers. The novelties for the optimal control problem are described after these results. For , a bounded open set with boundary , we wish to study the following fractional parabolic optimal control problem:
|subject to the fractional parabolic PDE: Find solving|
|with the state constraints|
|and control constraint imposed for|
Here, and subsequently, is the space of continuous functions in and . Furthermore, is a non-empty, closed and convex set, and the real numbers and fulfill
The precise definition of the function spaces and will be given in the forthcoming sections. We note that the control problem with nonlocal PDE constraint in (1.1b) may look similar to  at first glance, however, the two problems are significantly different. First of all, the control is taken as a source in this paper rather than in the exterior of the domain in . Secondly, in the current paper, we consider both the control and the state constraints rather than just the control constraints in .
Our first main result, stated below, involves the boundedness of .
Theorem 1.1 ( is bounded).
We remark that a similar result has been recently shown in [21, Theorem 29 and Corollary 3]. First, our proof (see Section 3) is much simpler, it follows directly by using semigroup arguments. Secondly, we were not able to follow the arguments given in the proof of [21, Theorem 29]. We think that the proof maybe contain some non obvious misprints.
From Theorem 1.1, we get the continuity of as a corollary.
Corollary 1.2 ( is continuous).
Our next result shows the well-posedness of the PDE when the right-hand side is a Radon measure. This result is needed because the Lagrange multiplier corresponding to the state constraints (1.1c) is a measure. Therefore, when we study the adjoint equation, we have a measure in the right-hand side of the equation. We also allow for the possibility of non-zero measure-valued initial condition as can be seen in the following: For let and , and consider
The above result is used to show the well-posedness of the adjoint equation, a fractional parabolic PDE with measure-valued right-hand-side. The adjoint equation arises when we derive the first order optimality conditions of the control problem (1.1). We again emphasize the novelty of (1.1) in that our PDE constraint is a fractional parabolic PDE with control taken in (space and time) and we impose additional control and state constraints. The Lagrange multiplier associated with the state constraints is a Radon measure, which necessitates the introduction of the notion of very-weak solutions to the adjoint equation (see Sections 3 and 4). Due to the difficulties in solving such an equation in practice, we introduce a regularized version of the optimal control problem using the Moreau-Yosida regularization (see [20, 16, 2], among others). This enables us to create an algorithm to solve the optimal control problem. While this approach is well-known, there are almost no works in the literature for fractional parabolic problems. For example, see , where the regularization is presented for parabolic problems in the classical case, but the optimality conditions of the original problem are not discussed. For fractional problems, the Moreau-Yosida regularization was used recently in  in the fractional elliptic case, and in this work we provide the details needed to extend those results to the current context.
The rest of the paper is organized as follows. In Section 2, we first introduce some notation and the relevant function spaces. The content of this section is well-known. The proofs of our main results stated above, as well as the precise definitions needed to state the results, are found in Section 3. In Section 4, we study our optimal control problem, establish its well-posedness, and derive the first order optimality conditions. We introduce the Moreau-Yosida regularized problem in Section 5 and show convergence (with rate) of the regularized solutions to the original solution. Our numerical examples in Section 6, clearly confirm all our theoretical findings.
2. Notation and preliminaries
The goal of this section is to introduce some notation and state some preliminary results that are needed in the proofs of our main results. Unless otherwise stated, is an open bounded set, . We define the space
which is a Sobolev space when we endow it with the norm
where we are taking to be the space of smooth functions with compact support in .
To study the Dirichlet problem (1.1b) we also need to consider the following fractional order Sobolev space
defines an equivalent norm on .
We recall the following result taken from [4, Remark 2.2].
are continuous, where we have set
Assume that that has a Lipschitz continuous boundary. If , then the spaces and coincide with equivalent norm. But if , then is a proper subspace of . We refer to [15, Chapter 1] for more details.
We shall denote by the dual of , i.e., , with respect to the pivot space so that we have the following continuous embeddings:
We are now ready to give a rigorous definition of the fractional Laplace operator . Consider the space
Then for a function in and , we let
Here the normalization constant is given by
provided that the limit exists for a.e. .
Next we define the operator in as follows:
The operator has a compact resolvent and generates a strongly continuous submarkovian semigroup on .
We refer to  for further qualitative properties of the semigroup . We conclude this section with the following observation.
The operator can be viewed as a bounded operator from into . In addition, we have that the operator generates a strongly continuous semigroup on the space . The semigroups and coincide on . Throughout the following, if there is no confusion we shall simply denote the semigroup by .
3. State equation and an equation with measure valued datum
For the remainder of this section, unless otherwise stated, we assume that () is a bounded open set. For each result we shall clarify if a regularity on is needed. Moreover, given a Banach space and its dual , we shall denote by their duality pairing.
The goal of this section is to establish the continuity of solutions to (1.1b) if the datum belongs to with fulfilling (1.2) and to study the well-posdendess of (1.1b) with . The latter space denotes the space of all Radon measures on such that
In addition, we have the following norm on this space:
Towards this end, we notice that (1.1b) can be rewritten as the following Cauchy problem:
Next we state the notion of weak solution to (3.1).
Definition 3.1 (Weak solution).
is said to be a weak solution to (1.1b) if a.e. in and the equality
holds, for every and almost every . Here
The existence of a weak solution to (1.1b) can be shown by using standard arguments. In addition this weak solution belongs to . From [5, Proposition 3.3] we also recall that this weak solution can be written as follows.
Proposition 3.2 (weak solution to (1.1b)).
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1.
Firstly, if , then the result is a direct consequence of the semigroup property, the representation (3.3) of solutions and the Sobolev embedding (2.1). Thus, we can assume without any restriction that . We give the proof for . The case follows similarly as the case .
Let then and fulfill (1.2) and assume that . We notice that it follows from the embedding (2.1) that the submarkovian semigroup is ultracontractive in the sense that it maps into . More precisely, following line by line the proof of [14, Theorem 2.16], or the proof of the abstract result in [23, Lemma 6.5] (see also [11, Chapter 2]), we get that for every , there exists a constant such that for every and
we have the estimate:
denotes the first eigenvalue of the operator.
Proof of Corollary 1.2.
We prove the result in two steps.
Step 1: Let be real number, and consider the following elliptic Dirichlet problem:
By a weak solution of (3.8) we mean a function such that the equality
holds for every .
The existence and uniqueness of weak solutions to the Dirichlet problem (3.8) are a direct consequence of the classical Lax-Milgram theorem.
In our recent work , we have shown that, if with (see also  for the case ), then every weak solution of the Dirichlet problem (3.8) belongs to . Thus, the resolvent operator maps into for every . In particular, this shows that for every , the operator maps () into the space , that is, the semigroup has the strong Feller property. In addition, we have the following result which is interesting in its own, independently of the application given in this proof.
Let be the part of the operator in , that is,
From the above properties of the resolvent operator and the semigroup, together with the fact that is dense in , we can deduce that the operator generates a strongly continuous semigroup on . Thus, for every we have that the function
belongs to . We can then deduce that for every , the unique weak solution to (1.1b) given by
belongs to .
Step 2: Now, let and satisfy (1.2). Since the space is dense in , we can construct a sequence such that
Let be the weak solution to (1.1b) with datum . It follows from Step 1 that . Whence, subtracting the equations satisfied by and and using the estimate (1.3) from Theorem 1.1 we obtain that there is a constant such that for every we have
As a result, we have that in as . Since is the uniform limit on of a sequence of continuous functions on , it follows that is also continuous on . The proof is finished. ∎
Now, throughout the rest of the paper, without any mention, we assume that () is a bounded open set with a Lipschitz continuous boundary and has the exterior ball condition.
Next we study the well-posedness of the state equation (1.1b) where is a Radon measure and the initial condition is equal to a Radon measure. However, prior to this result, we need to introduce the notion of such solutions. We call them very-weak solutions. We refer to  for a similar notion for the fractional elliptic problems and [3, 5] for fractional elliptic and parabolic problems with nonzero exterior conditions. To motivate the need to discuss such solutions, we introduce the adjoint equation, which we will derive in the next section when discussing the first-order optimality conditions. For , we can write , where , , and . We then consider the following problem with and
Note that at the final time the data for the adjoint variable is a measure on . Now, making a change of variables in (3.9) yields, for ,
This problem now resembles (1.5), and so we define the notion of very-weak solution in this context.
Definition 3.3 (very-weak solutions).
Now, we are ready to prove the existence and uniqueness of very-weak solutions as stated in Theorem 1.3.
Proof of Theorem 1.3.
We prove the theorem in three steps.
Step 1: Given where fulfill (1.2), we begin by considering the following “dual” problem
After using semigroup theory as in Proposition 3.2, we can deduce that (3.11) has a unique weak solution . In addition, from (3.11) we have that . It follows from Corollary 1.2 that . Thus is a valid “test function” according to Definition 3.3.
Step 2: Towards this end, we define the map
Due to Corollary 1.2, is linear and continuous.
for every . It follows from Step 1 that the mapping
is surjective. Thus, we can deduce from (3.13) that
for every . Exploiting the fundamental lemma of the calculus of variations we can conclude from the preceding identity that a.e. in and we have shown the uniqueness.
4. Optimal control problem
The main goal of this section is to establish well-posdeness of the optimal control problem (1.1) and to derive the first order necessary optimality conditions. We start by equivalently rewriting the optimal control problem (1.1) in terms of the constraints (3.1). Recall that is the realization of in with zero Dirichlet exterior conditions and it is a self-adjoint operator. In terms of , (1.1) becomes
We next define the appropriate function spaces. Let
Here, is a Banach space with the graph norm
We let to be a nonempty, closed, and convex set and as in (1.1c). We require the spaces and to be reflexive. This is needed to show the existence of solution to .
is well-defined, linear, and continuous. Due to the continuous embedding of in we can in fact consider the control-to-state map as
and we can define the admissible control set as
and thus the reduced minimization problem is given by
Let be a closed, convex, bounded subset of and a closed and convex subset of such that is nonempty. Moreover, let be weakly lower-semicontinuous. Then (4.2) has a solution.
The proof follows by using similar arguments as in the elliptic case, see [4, Theorem 4.1] and has been omitted for brevity. ∎
Next, we derive the first order necessary conditions under the following Slater condition.
There is some control such that the corresponding state fulfills the strict state constraints
Under this assumption, we have the following first order necessary optimality conditions.