Sparse optimal control for fractional diffusion^{†}^{†}thanks: EO has been supported in part by CONICYT through FONDECYT project 3160201. AJS has been supported in part by NSF grant DMS1418784.
Abstract
We consider an optimal control problem that entails the minimization of a nondifferentiable cost functional, fractional diffusion as state equation and constraints on the control variable. We provide existence, uniqueness and regularity results together with first order optimality conditions. In order to propose a solution technique, we realize fractional diffusion as the DirichlettoNeumann map for a nonuniformly elliptic operator and consider an equivalent optimal control problem with a nonuniformly elliptic equation as state equation. The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation. We propose a fully discrete scheme: piecewise constant functions for the control variable and first–degree tensor product finite elements for the state variable. We derive a priori error estimates for the control and state variables which are quasi–optimal with respect to degrees of freedom.
remarkRemark \headersSparse controlE. Otárola, A.J. Salgado
ptimal control problem, nondifferentiable objective, sparse controls, fractional diffusion, weighted Sobolev spaces, finite elements, stability, anisotropic estimates.
26A33, 35J70, 49K20, 49M25, 65M12, 65M15, 65M60.
1 Introduction
In this work we shall be interested in the design and analysis of a numerical technique to approximate the solution to a nondifferentiable optimal control problem involving the fractional powers of a uniformly elliptic second order operator; control constraints are also considered. To make matters precise, let be an open and bounded polytopal domain of with . Given and a desired state , we define the nondifferentiable cost functional
(1) 
where and are positive parameters. We shall thus be concerned with the following nondifferentiable optimal control problem: Find
(2) 
subject to the fractional state equation
(3) 
and the control constraints
(4) 
The operator , with , is a fractional power of the second order, symmetric, and uniformly elliptic operator
(5) 
supplemented with homogeneous Dirichlet boundary conditions; and is symmetric and positive definite. The control bounds and, since we are interested in the nondifferentiable scenario, we assume that [CHW:12, Remark 2.1].
The design of numerical techniques for the optimal control problem (2)–(4) is mainly motivated by the following considerations:

Fractional diffusion has recently become of great interest in the applied sciences and engineering: practitioners claim that it seems to better describe many processes. For instance, mechanics [atanackovic2014fractional], biophysics [bio], turbulence [wow], image processing [GH:14], nonlocal electrostatics [ICH] and finance [MR2064019]. It is then natural the interest in efficient approximation schemes for problems that arise in these areas and their control.

The objective functional contains an –control cost term that leads to sparsely supported optimal controls; a desirable feature, for instance, in the optimal placement of discrete actuators [MR2556849]. This term is also relevant in settings where the control cost is a linear function of its magnitude [MR2283487].
One of the main difficulties in the study and discretization of the state equation (3) is the nonlocality of the fractional operator [CS:07, CDDS:11, ST:10]. A possible approach to this issue is given by a result of Caffarelli and Silvestre in [CS:07] and its extensions to bounded domains [CDDS:11, ST:10]: Fractional powers of can be realized as an operator that maps a Dirichlet boundary condition to a Neumann condition via an extension problem on the semi–infinite cylinder . Therefore, we shall use the Caffarelli–Silvestre extension to rewrite the fractional state equation (3) as follows:
(6) 
where is the lateral boundary of , , and the conormal exterior derivative of at is
(7) 
the limit being understood in the distributional sense [CS:07, CDDS:11, ST:10]. Finally, the matrix is defined by . We will call the extended variable and the dimension in the extended dimension of problem (6). As noted in [CS:07, CDDS:11, ST:10], and the DirichlettoNeumann operator of (6) are related by
The analysis of optimal control problems involving a functional that contains an –control cost term has been previously considered in a number of works. The article [MR2556849] appears to be the first to provide an analysis when the state equation is a linear elliptic PDE: the author utilizes a regularization technique that involves an –control cost term, analyze optimality conditions, and study the convergence properties of a proposed semismooth Newton method. These results were later extended in [MR2826983], where the authors obtain rates of convergence with respect to a regularization parameter. Subsequently, in [CHW:12], the authors consider a semilinear elliptic PDE as state equation and analyze second order optimality conditions. Simultaneously, the numerical analysis based on finite element techniques has also been developed in the literature. We refer the reader to [MR2826983], where the state equation is a linear elliptic PDE and to [CHW:12again, CHW:12] for extensions to the semilinear case. The common feature in these references, is that, in contrast to (3), the state equation is local. To the best of our knowledge, this is the first work addressing the analisys and numerical approximation of (2)–(4).
The main contribution of this work is the design and analysis of a solution technique for the fractional optimal control problem (2)–(4). We overcome the nonlocality of by using the Caffarelli–Silvestre extension: we realize the state equation (3) by (6), so that our problem can be equivalently written as: Minimize subject to the extended state equation (6) and the control constraints (4); the extended optimal control problem. We thus follow [MR3429730, MR3504977] and propose the following strategy to solve our original control problem (2)–(4): given a desired state , employ the finite element techniques of [NOS] and solve the equivalent optimal control problem. This yields an optimal control and an optimal extended state . Setting for all , we obtain the optimal pair that solves (2)–(4).
The outline of this paper is as follows. In section 2 we introduce notation, define fractional powers of elliptic operators via spectral theory, introduce the functional framework that is suitable to analyze problems (3) and (6) and recall elements from convex analysis. In section 3, we study the fractional optimal control problem. We derive existence and uniqueness results together with first order necessary and sufficient optimality conditions. In addition, we study the regularity properties of the optimal variables. In section 4 we analyze the extended optimal control problem. We begin with the numerical analysis for our optimal control problem in section 5, where we introduce a truncated problem and derive approximation properties of its solution. Section 6 is devoted to the design and analysis of a numerical scheme to approximate the solution to the control problem (2)–(4): we derive a priori error estimates for the optimal control variable and the state.
2 Notation and Preliminaries
In this work is a bounded and open convex polytopal subset of () with boundary . The difficulties inherent to curved boundaries could be handled with the arguments developed in [Otarola_controlp1] but this would only introduce unnecessary complications of a technical nature.
We follow the notation of [MR3429730, NOS] and define the semi–infinite cylinder with base and its lateral boundary, respectively, by and . For , we define the truncated cylinder and accordingly.
Throughout this manuscript we will be dealing with objects defined on and . It will thus be important to distinguish the extended –dimension, which will play a special role in the analysis. We denote a vector by with and .
In what follows the relation means that for a nonessential constant whose value might change at each occurrence.
2.1 Fractional powers of second order elliptic operators
We proceed to briefly review the spectral definition of the fractional powers of the second order elliptic operator , defined in (5). To accomplish this task we invoke the spectral theory for , which yields the existence of a countable collection of eigenpairs such that
In addition, is an orthonormal basis of and an orthogonal basis of . Fractional powers of , are thus defined by
(8) 
Invoking a density argument, the previous definition can be extended to
(9) 
This space corresponds to [Lions, Chapter 1]. Consequently, if , can be characterized by
and, if , then . If , the space corresponds to the socalled Lions–Magenes space [Tartar, Lecture 33]. When deriving regularity results for the optimal variables of problem (2)–(4), it will be important to characterize the space for . In fact, we have that, for such a range of values of , ; see [ShinChan].
For we denote by the dual of . With this notation, is an isomorphism.
2.2 Weighted Sobolev spaces
The localization results by Caffarelli and Silvestre [CS:07, CDDS:11, ST:10] require to deal with a nonuniformly elliptic equation posed on the semi–infinite cylinder . To analyze such an equation, it is instrumental to consider weighted Sobolev spaces with the weight ( and ). We thus define
(10) 
For we have that the weight belongs to the so–called Muckenhoupt class , see [Muckenhoupt, Turesson]. Consequently, , endowed with the norm
(11) 
is a Hilbert space [Turesson, Proposition 2.1.2] and smooth functions are dense [Turesson, Corollary 2.1.6]; see also [GU, Theorem 1]. We recall the following weighted Poincaré inequality:
(12) 
[NOS, ineq. (2.21)]. We thus have that is equivalent to (11) in . For , we denote by its trace onto , and we recall ([NOS, Prop. 2.5])
(13) 
2.3 Convex functions and subdifferentials
Let be a real normed vector space. Let be convex and proper, and let with . By convexity of and the fact that we conclude that the graph of can always be minorized by a hyperplane. If is not differentiable at , then a useful substitute for the derivative is a subgradient, which is nothing but the slope of a hyperplane that minorizes the graph of and is exact at . In other words, a subgradient of at is a continuous linear functional on that satisfies
(14) 
where denotes the duality pairing between and . We immediately remark that a function may admit many subgradients at a point of nondifferentiability. The set of all subgradients of at is called subdifferential of at and is denoted by . Moreover, by convexity, the subdifferential for all points in the interior of the effective domain of . Finally, we mention that the subdifferential is monotone, i.e.,
(15) 
We refer the reader to [MR1058436, MR2330778] for a thorough discussion on convex analysis.
3 The fractional optimal control problem
In this section we analyze the fractional optimal control problem (2)–(4). We derive existence and uniqueness results together with first order necessary and sufficient optimality conditions. In addition, in section 3.2, we derive regularity results for the optimal variables that will be essential for deriving error estimates for the scheme proposed in section 6.
For defined as in (2), the fractional optimal control problem reads: Find subject to (3) and (4). The set of admissible controls is defined by
(16) 
which is a nonempty, bounded, closed, and convex subset of Â·. Since we are interested in the nondifferentiable scenario, we assume that and are real constants that satisfy the property [CHW:12, Remark 2.1]. The desired state while and are both real and positive parameters.
As it is customary in optimal control theory [Lions, Tbook], to analyze (2)–(4), we introduce the so–called control to state operator.
[fractional control to state map] The map , where solves (3), is called the fractional control to state map.
This operator is linear and bounded from into [CDDS:11, Lemma 2.2]. In addition, since , we may also consider acting from into itself. With this operator at hand, we define the optimal fractional state–control pair.
[optimal fractional statecontrol pair] A state–control pair is called optimal for (2)–(4) if and
for all such that .
With these elements at hand, we present an existence and uniqueness result.
[existence and uniqueness] The fractional optimal control problem (2)–(4) has a unique optimal solution . {proof} Define the reduced cost functional
(17) 
In view of the fact that is injective and continuous, it is immediate that is strictly convex and weakly lower semicontinuous. The fact that is weakly sequentially compact allows us to conclude [Tbook, Theorem 2.14].
3.1 First order optimality conditions
The reduced cost functional is a proper strictly convex function. However, it contains the –norm of the control variable and therefore it is not nondifferentiable at . This leads to some difficulties in the analysis and discretization of (2)–(4), that can be overcome by using some elementary convex analysis [MR1058436, MR2330778]. With this we shall obtain explicit optimality conditions for problem (2)–(4). We begin with the following classical result; see, for instance, [MR2330778, Chapter 4].
Let be defined as in (17). The element is a minimizer of over if and only if there exists a subgradient such that
for all .
In order to explore the previous optimality condition, we introduce the following ingredients.
[fractional adjoint state] For a given control , the fractional adjoint state , associated to , is defined as , where denotes the –adjoint of .
We also define the convex and Lipschitz function by — the nondifferentiable component of the cost functional — and
(18) 
the differentiable component of . Standard arguments yield that is Fréchet differentiable with [Tbook, Theorem 2.20]. Now, invoking Definition 3.1, we obtain that, for , we have
(19) 
It is rather standard to see that if and only if the relations
(20) 
hold for a.e. . With these ingredients at hand, we obtain the following necessary and sufficient optimality conditions for our optimal control problem; see also [CHW:12, Theorem 3.1] and [MR2826983, Lemma 2.2].
[optimality conditions] The pair is optimal for problem (2)–(4) if and only if and satisfies the variational inequality
(21) 
where and . {proof} Since the convex function is Fréchet differentiable we immediately have that [MR2330778, Proposition 4.1.8]. We thus apply the sum rule [MR2330778, Proposition 4.5.1] to conclude, in view of the fact that is convex, that . This, combined with Lemma 3.1 and (19) imply the desired variational inequality (21).
To present the following result we introduce, for , the projection formula
[projection formulas] Let , , and be as in Theorem 3.1. Then, we have that
(22)  
(23)  
(24) 
See [CHW:12, Corollary 3.2].
[sparsity] We comment that property (23) implies the sparsity of the optimal control . We refer the reader to [MR2556849, Section 2] for a thorough discussion on this matter.
3.2 Regularity estimates
Having obtained conditions that guarantee the existence and uniqueness for problem (2)–(4), we now study the regularity properties of its optimal variables. This is important since, as it is well known, smoothness and rate of approximation go hand in hand. Consequently, any rigorous study of an approximation scheme must be concerned with the regularity of the optimal variables. Here, on the the basis of a bootstraping argument inspired by [MR3429730, MR3504977], we obtain such regularity results.
[regularity results for and ] If , then the optimal control for problem (2)–(4) satisfies that . In addition, the subgradient , given by (24), satisfies that . {proof} We begin the proof by invoking the convexity of , the fact that is a pseudodifferential operator of order and that to conclude that
(25) 
the space , for , was characterized in Section 2.1. We now consider the following cases:
Case 1, : We immediately obtain that . This, in view of the projection formula (24) and [KSbook, Theorem A.1] implies that ; notice that formula (24) preserves boundary values. Now, since both functions and belong to , an application, again, of [KSbook, Theorem A.1] and the projection formula (22), for , implies that . We remark that, in view of the assumption , the formula (22) also preserves boundary values.
Case 2, : We now begin the bootstrapping argument like that in [MR3429730, Lemma 3.5]. In this case, (25) implies that . This, on the basis of a nonlinear operator interpolation result as in [MR3429730, Lemma 3.5], that follows from [Tartar, Lemma 28.1], guarantees that . We notice, once again, that formula (24) preserves boundary values. Similar arguments allow us to derive that .
Case 2.1, : Since , we conclude that and that , where . We now invoke that to deduce that . This, in view of (24), implies that , which in turns, and as a consequence of (22), allows us to derive that .
Case 2.2, : As in Case 2.1 we have that . We now invoke, again, a nonlinear operator interpolation argument to conclude that and then that . These regularity results imply that and then that , where .
Case 2.2.1, : We immediately obtain that . This implies that , and thus that .
Case 2.2.2, : We proceed as before.
After a finite number of steps we can thus conclude that, for any , and belong to . This concludes the proof.
As a byproduct of the proof of the previous theorem, we obtain the following regularity result for the optimal state and optimal adjoint state.
[regularity results for and ] If , then , where and , where .
4 The extended optimal control problem
In this section we invoke the localization results of Caffarelli and Silvestre [CS:07] and their extensions [CDDS:11, ST:10] to circumvent the nonlocality of the operator in the state equation (3). We follow [MR3429730] and consider the equivalent extended optimal control problem: Find subject to the extended state equation:
(26) 
where, for all , the bilinear form is defined by
(27) 
5 The truncated optimal control problem
The state equation (26) of the extended optimal control problem is posed on the infinite domain and thus it cannot be directly approximated with finite element–like techniques. However, the result of Proposition 5 below shows that the optimal extended state decays exponentially in the extended variable . This suggests to truncate to , for a suitable truncation parameter , and seek solutions in this bounded domain.
[exponential decay] For every , the optimal state , solution to problem (26), satisfies
(30) 
where denotes the first eigenvalue of the operator . {proof} See [NOS, Proposition 3.1].
This motivates the truncated optimal control problem: Find subject to the truncated state equation:
(31) 
where
and for all , the bilinear form is defined by
(32) 
To formulate optimality conditions we introduce the truncated adjoint problem:
(33) 
With this adjoint problem at hand, we present necessary and sufficient optimality conditions for the truncated optimal control problem: the pair is optimal if and only if solves (31) and
(34) 
where solves (33) and .
We now introduce the following auxiliary problem:
(35) 
The next result follows from [MR3429730, Lemma 4.6] and shows how approximates .
[exponential convergence] If and are the optimal pairs for the extended and truncated optimal control problems, respectively, then
(36) 
and
(37) 
Set and in (29) and (34), respectively. Adding the obtained inequalities we arrive at the estimate
As a first step to control the right hand side of the previous expression, we recall that and so that, by (15),
Consequently,
(38) 
To control the right hand side of the previous expression, we add and subtract the adjoint state as follows:
Let us now bound I. Notice that solves
On the other hand, we also observe that solves
Setting and we immediately conclude that .
To control the term II we write , where solves (35). The first term is controlled in view of the trace estimate (13), the well–posedness of problem (35) and an application of the estimate [NOS, Theorem 3.5]:
Similar arguments yield: In view of (38), a collection of these estimates allow us to obtain (36).
The estimate (37) follows from similar arguments upon writing This concludes the proof.
We now state projection formulas and regularity results for the optimal variables and , together with a sparsity property for .
[projection formulas] Let the variables , , and be as in the variational inequality (34). Then, we have that
(39)  
(40)  
(41) 
See [CHW:12, Corollary 3.2].
[regularity results for and ] If , then the truncated optimal control . In addition, the subgradient , given by (41), satisfies that . {proof} The proof is an adaption of the techniques elaborated in the proof of [Otarola_controlp1, Proposition 4.1] and the bootstrapping argument of Theorem