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Temporally semidiscrete approximation of a Dirichlet boundary control for a fractional/normal evolution equation with a final observation

Optimal Dirichlet boundary control for a fractional/normal evolution with a final observation is considered. The unique existence of the solution and the first-order optimality condition of the optimal control problem are derived. The convergence of a temporally semidiscrete approximation is rigorously established, where the control is not explicitly discretized and the state equation is discretized by a discontinuous Galerkin method in time. Numerical results are provided to verify the theoretical results.


Introduction
There is an extensive literature on the numerical optimization with PDE constraints. So far, most of the literature focuses on the distributed control problems, and the works on the Dirichlet boundary control problems are rather limited. Compared with the distributed control problems, the Dirichlet boundary control problems are more challenging in the following senses. Firstly, the solution of the state equation of a Dirichlet boundary control problem is of significantly lower regularity than that of a distributed control problem, and this increases the difficulty in both theoretical and numerical analysis. Secondly, in the weak form of the solution of the state equation of a Dirichlet boundary control problem, the test function space is more regular than the trial function space, and hence the weak form is not appropriate for the discretization. Thirdly, since the normal derivative of the adjoint state occurs in the first-order optimality condition, the discrete first-order optimality condition will essentially involve the discrete normal derivative of the discrete adjoint state, and this increases the implementation difficulty.
We summarize the works on the parabolic Dirichlet boundary control problems briefly as follows. Using an integral representation formula derived by the semigroup theory (cf. [3,Section 4.12] and [30]), Lasiecka [29,31] analyzed spatial Galerkin approximations of an optimal Dirichlet boundary control problem and a time optimal Dirichlet boundary control problem for the parabolic equations. Kunisch and Vexler [28] analyzed constrained Dirichlet boundary control problems for a class of parabolic equations and derived the convergence of the PDAS strategy for two Dirichlet boundary control problems. Applying the Robin penalization method to a Dirichlet boundary control problem for a parabolic equation with a final observation, Belgacem et al. [4] obtained a penalized Robin boundary control problem. Gong et al. [16] analyzed a finite element approximation of a Dirichlet boundary control for a parabolic equation, where the variational discretization approach [19] was used and the state equation was discretized by the usual H 1 -conforming P 1-element in space and discretized by the dG(0) scheme in time. Recently, Gong and Li [17] improved the spatial accuracy derived in [16], using the maximal L p -regularity theory. We note that, for the state equation with rough Dirichlet boundary data, [29,31] used the semigroup theory to define the solution whereas [16,17,28] used the transposition technique to define the solution (called the very weak solution).
For the numerical analysis of parabolic Neumann/Robin boundary control problems, we refer the reader to [1,7,27,44]. For the numerical analysis of other optimal control problems for parabolic equations, we refer the reader to [8,15,34,35,47,48,49,50] and the references therein. Although the spatial discretization is not considered in this paper, we would like to refer the reader to [5,13,14,32] for the numerical analysis of elliptic and parabolic equations with rough Dirichlet boundary data.
To our best knowledge, no convergence result is available for the Galerkintype approximations of the Dirichlet boundary control problems governed by the parabolic equations with final observations. The fractional evolution equation is an extension of the normal evolution equation, widely used to describe the physical phenomena with memory effect [55]. Recently, Harbir et al. [2] studied an optimal distributed control problem for a space-time fractional diffusion equation. For the numerical analysis of the optimal distributed problems governed by the time fractional diffusion equations, we refer the reader to [18,26,38,61]. To our knowledge, no numerical analysis is available for the Dirichlet boundary control problems governed by the fractional evolution equations. Hence, this paper tries to analyze the Dirichlet boundary control problems for the fractional and normal evolution equations in a unified way.
In this paper, we establish the convergence of a temporally semidiscrete approximation of an abstract optimal control problem governed by a fractional/normal evolution equation with a final observation. This approximation uses the variational discretization concept [19] and uses a discontinuous Galerkin method to discretize the state equation in time. The discontinuous Galerkin method is the famous dG(0) scheme for the normal evolution equation, and is equivalent to the well-known L1 scheme [39] with uniform temporal grids for the fractional evolution equations. The derived numerical analysis is applied to a Dirichlet boundary control problem. We note that there are many works (see [20,21,22,23,25,36,41,43] and the references therein) devoted to the numerical analysis of the fractional diffusion equations with rough initial value and source term, but, to our knowledge, no numerical analysis is available for the fractional diffusion equation with rough Dirichlet boundary value. This paper also fills in this gap.
The rest of this paper is organized as follows. Section 2 establishes the convergence of a temporally semidiscrete approximation of an abstract optimal control problem. Section 3 applies the theory developed in the previous section to a Dirichlet boundary control problem. Section 4 performs three numerical experiments to confirm the theoretical results.
2 An abstract optimal control problem

Preliminaries
We will use the following conventions: for each linear vector space, the field of the scalars is C; for a Hilbert space X , we use (·, ·) X to denote its inner product; for a Banach space B, we use ·, · B to denote a duality paring between B * (the dual space of B) and B; for a linear operator A, ρ(A) denotes the resolvent set of A and R(z, A) denotes the inverse of z − A for each z ∈ ρ(A); for two Banach spaces B 1 and B 2 , L(B 1 , B 2 ) is the set of all bounded linear operators from B 1 to B 2 , and L(B 1 , B 1 ) is abbreviated to L(B 1 ); I denotes the identity map; for a Lebesgue measurable subset D ⊂ R l , 1 l 4, p, q D means the integral D pq, where q is the conjugate of q; for a function v defined on (0, T ), by v(t−), 0 < t T , we mean the limit lim s→t− v(s); the notation C × means a positive constant, depending only on its subscript(s), and its value may differ at each occurrence; for any 0 < θ < π, define where Γ θ and Υ θ are so oriented that the negative real axis is to their left.

Time fractional Sobolev spaces
Assume that −∞ < a < b < ∞ and B is a Banach space. Define and endow them with the two norms For each 0 < γ < 1, define where (·, ·) γ,2 means the interpolation space defined by the K-method (cf. [42]). For convenience, the spaces 0 H γ (a, b; C) and 0 H γ (a, b; C) will be abbreviated to

Riemann-Liouville fractional calculus operators
Assume that −∞ a < b ∞ and B is a Banach space. For any γ > 0, define for all v ∈ L 1 (a, b; B), where Γ(·) is the gamma function. In addition, let D 0 where D is the first-order differential operator in the distribution sense. We summarize some properties of the above fractional calculus operators, which will be used later in this paper.
where C 1 and C 2 are two positive constants depending only on γ.
If D γ For the proof of Lemma 2.1, we refer the reader to [11]. For the proof of Lemma 2.2, we refer the reader to [43]. Since the proof of Lemma 2.3 is a standard density argument by Lemmas 2.1 and 2.2, it is omitted here.

Definitions of A and A *
Assume that X and Y are two Hilbert spaces such that X is continuously embedded into Y and X is dense in Y . We will regard Y as a subspace of X * , the dual space of X, in the sense that v, w X := (v, w) Y for all v ∈ Y and w ∈ X.
Let A and A * be two bounded linear operators from X to Y satisfying that where π/2 < ω 0 < π, c 0 , c 1 and M 0 are four positive constants. By the transposition technique, A and A * can be extended as two bounded linear operators from Y to X * by that for all v ∈ Y and w ∈ X.
Also, we have (13)). Consequently, 2.1.4 Definitions of E α and E * α For any 0 < α 1 and t > 0, define For any v ∈ X * and w ∈ Y , by the definitions of A and A * and Lemma 2.4 we have that and hence from (19) and (20) we obtain Furthermore, by Lemma 2.4, a routine calculation (cf. [59,23]) yields the following lemma.

Solutions of the fractional evolution equations
Following the mild solution theory of fractional/normal evolution equations in [54,41,46,23], we introduce the following mild solutions. Assume that 0 < α 1 and 0 < T < ∞. For any the mild solution to the following fractional evolution equation: For the mild solution to (29) with g = vδ 0 , where δ 0 is the Dirac measure in time concentrated at t = 0. Symmetrically, for any the mild solution to the following backward fractional evolution equation: For the mild solution to equation (32) with g = vδ T , where δ T is the Dirac measure in time concentrated at t = T .
This proves (36) and hence this lemma.
Proof. Since (28) implies it suffices to prove that To this end, note that Then a straightforward calculation gives, by (19) and Lemma 2.4, that for all t > 0. It follows that Therefore, Lebesgue's dominated convergence theorem yields (38) and thus concludes the proof. and where 0 < ǫ < 1 and c is a positive constant independent of g.
Proof. Since a complete rigorous proof of this lemma is tedious but standard (cf. [54]), we only present briefly the key ingredients of the proof.
Step 1. Define A straightforward computation gives that, for any 0 < t T , where Γ ω0 is deformed so that the origin is to its left. Hence, we conclude from (10) and Lemma 2.4 the following properties: Step 2. By the theory of Laplace transform, from (28) we obtain that Hence, by the properties of η presented in Step 1, (19)) = g(t) + A(S α g)(t) (by (28)) for each 0 t T . This implies equality (39).
Step 3. For convenience, we will use c to denote a positive constant, whose value is independent of g but may differ at each occurrence. A routine calculation gives, by (26), (27) and (28), that so that (10) implies Since Combining (39), (43) and (44) proves (40) and thus concludes the proof.

Continuous problem
Let Z be a Hilbert space and let U ad ⊂ L ∞ (0, T ; Z) be a convex, bounded and closed subset of L 2 (0, T ; Z). We consider the following abstract optimal control problem: where 0 < α 1, y d ∈ Y , ν > 0 is a regularization parameter, and R θ0 : Z → [X * , Y ] θ0 is a bounded linear operator for some 0 < θ 0 1. (45) is a strictly convex functional on L q (0, T ; Z), and U ad is a convex, bounded and closed subset of L q (0, T ; Z). By (36), a routine argument (cf. [60, Theorems 2.14 and 2.21]) yields the following theorem.
Theorem 2.1. Problem (45) admits a unique solution u ∈ U ad , and the following first-order optimality condition holds:

Remark 2.3. Scheme
Remark 2.4. We note that the idea of using the Galerkin methods to discretize the time fractional calculus operators was firstly developed by McLean and Mustapha [45,52,53,51]. The L1 scheme [39,57] is widely used for the discretizations of the fractional diffusion equations. Jin et al. [24,Remark 3] discovered that the L1 scheme is equivalent to discretization (47) with uniform temporal grids. For the numerical analysis of discretization (47) with nonuniform temporal grids, we refer the reader to [36,37].
Using the variational discretization concept proposed in [19], we consider the following temporally discrete problem: Note that (57) implies that (S α,τ R θ0 ·)(T −) is a bounded linear operator from L 2 (0, T ; Z) to Y . In addition, U ad is a convex, bounded and closed subset of L 2 (0, T ; Z). Hence, applying [60, Theorems 2.14 and 2.21] to problem (51) yields the following theorem, by Lemma 2.9.
This completes the proof.
Together the above estimates of I 1 and I 2 yields (77). Since (78) can be proved analogously by (13) and (74), this completes the proof.
Finally, we are in a position to prove Lemma 2.10 as follows.

A Dirichlet boundary control problem
Assume that 0 < α 1, 0 < T < ∞, and Ω ⊂ R d (d = 2, 3) is a bounded convex polygonal domain with a boundary ∂Ω. Define where u * and u * are two given constants. For any y ∈ C((0, T ]; L 2 (Ω)) and u ∈ L 2 (0, T ; L 2 (∂Ω)), define where y d ∈ L 2 (Ω) and ν > 0 is a regularization parameter. We are concerned with the following optimal Dirichlet boundary control problem: Here, ∂ α 0+ , a fractional partial differential operator, is the scalar-valued version of D α 0+ with respect to the time variable t.
For α = 1 the above results are standard (cf. [12]). Hence, by the method of transposition (cf. [40]), we define the very weak solution y ∈ L 2 (0, T ; Y ) to the state equation of problem (84) with u ∈ L 2 (0, T ; Z) by that Then we will prove that, for any 0 < θ 0 < 1/4 and u ∈ L 2 (0, T ; Z), S α R θ0 u is identical to the very weak solution to the state equation of problem (84), and hence the application of the theory in the previous section to problem (84) is reasonable.
Also, (89) and (17) imply Consequently, by Lemma 2.3 we have and it is evident by (8) that Combining (90) and the above two equations gives The arbitrariness of ϕ ∈ 0 H α (0, T ; Y )∩L 2 (0, T ; X) proves that S α R θ0 u is indeed the very weak solution. The general case u ∈ L 2 (0, T ; Z) then follows from a standard density argument by S α R θ0 ∈ L(L 2 (0, T ; Z), L 2 (0, T ; Y )), which is a direct consequence of (34) and the fact R θ0 ∈ L(Z, [X * , Y ] θ0 ). This completes the proof.
To approximate S α (vδ 0 ), we use discretization (47)(0 < α < 1) or (49)(α = 1) in time and use the usual H 1 (Ω)-conforming P 1-element method in space. Let U M be the corresponding numerical approximation with time step τ = T /2 M and spatial mesh size h = 2 −9 . Table 2     To approximate problem (84), we will use the temporal discretization in Section 2.3 and the H 1 (Ω)-conforming P 1-element method to discretize the state equation in time and space, respectively; see [16] for the implementation details. Let U M be the corresponding numerical solution with time step τ = T /2 M and spatial mesh size h = 2 −8 . The numerical results in Table 3 show that U M − U 13 L 2 (0,T ;L 2 (∂Ω)) is close to O(τ 0.125 ), which agrees with error estimate (88).