Pointwise-in-Time Error Estimates for an Optimal Control Problem with Subdiffusion Constraint

Transcription

1 IMA Journal of Numerical Analysis (218) Page 1 of 22 doi:1.193/imanum/drnxxx Pointwise-in-Time Error Estimates for an Optimal Control Problem wit Subdiffusion Constraint BANGTI JIN Department of Computer Science, University College ondon Gower Street, ondon WC1E 6BT, UK. BUYANG I AND ZHI ZHOU Department of Applied Matematics, Te Hong Kong Polytecnic University, Hong Kong. [Received on December 22, 217; revised on June 4, 218; accepted on ] In tis work, we present numerical analysis for a distributed optimal control problem, wit box constraint on te control, governed by a subdiffusion equation wic involves a fractional derivative of order α (, 1) in time. Te fully discrete sceme is obtained by applying te conforming linear Galerkin finite element metod in space, 1 sceme/backward Euler convolution quadrature in time, and te control variable by a variational type discretization. Wit a space mes size and time stepsize τ, we establis te following order of convergence for te numerical solutions of te optimal control problem: O(τ min(1/2+α ε,1) + 2 ) in te discrete 2 (,T ; 2 (Ω)) norm and O(τ α ε + l 2 2 ) in te discrete (,T ; 2 (Ω)) norm, wit any small ε > and l = ln(2 + 1/). Te analysis relies essentially on te maximal p -regularity and its discrete analogue for te subdiffusion problem. Numerical experiments are provided to support te teoretical results. Keywords: optimal control, time-fractional diffusion, 1 sceme, convolution quadrature, pointwise-intime error estimate, maximal regularity. 1. Introduction et Ω R d (d = 1,2,3) be a convex polyedral domain wit a boundary Ω. Consider te distributed optimal control problem min q U ad J(u,q) = 1 2 u u d 2 2 (,T ; 2 (Ω)) + γ 2 q 2 2 (,T ; 2 (Ω)), (1.1) subject to te following time-fractional diffusion equation α t u u = f + q, < t T, wit u() =, (1.2) were T > is a fixed final time, γ > a fixed penalty parameter, : H 1 (Ω) H2 (Ω) 2 (Ω) te Diriclet aplacian, f : (,T ) 2 (Ω) a given source term, and u d : (,T ) 2 (Ω) te target function. Te admissible set U ad for te control q is defined by U ad = {q 2 (,T ; 2 (Ω)) : a q b a.e. in Ω (,T )}, wit a,b R and a < b. Te notation α t u in (1.2) denotes te left-sided Riemann-iouville fractional derivative in time t of order α (,1), defined by (Kilbas et al., 26, p. 7) α t u(t) = 1 d Γ (1 α) dt t (t s) α u(s)ds. (1.3) Corresponding autor. bangti.jin@gmail.com, b.jin@ucl.ac.uk bygli@polyu.edu.k, zizou@polyu.edu.k c Te autor 218. Publised by Oxford University Press on bealf of te Institute of Matematics and its Applications. All rigts reserved.

2 2 of 22 B. JIN, B. I AND Z. ZHOU Since u() =, te Riemann-iouville derivative t α u(t) is identical to te Caputo derivative (Kilbas et al., 26, p. 91). Furter, wen α = 1, t α u(t) coincides wit te first-order derivative u (t), and tus te model (1.2) recovers te standard parabolic problem. Te fractional derivative t α u in te model (1.2) is motivated by a growing list of practical applications related to subdiffusion processes, in wic te mean square particle displacement grows sublinearly wit time t, as opposed to linear growt for normal diffusion. Te list includes termal diffusion in fractal media, protein transport in plasma membrane and column experiments etc (see, e.g., Adams & Gelar (1992); Hatano & Hatano (1998); Nigmatulin (1986)). Te numerical analysis of te model (1.2) as received muc attention. However, te design and analysis of numerical metods for related optimal control problems only started to attract attention (see, e.g., Antil et al. (216); Du et al. (216); Ye & Xu (213, 215)). Te controllability of (1.2) was discussed in Fujisiro & Yamamoto (214) and ü & Zuazua (216). Ye & Xu (213, 215) proposed space-time spectral type metods for optimal control problems under a subdiffusion constraint, and derived error estimates by assuming sufficiently smoot state and control variables. Antil et al. (216) studied an optimal control problem wit space- and time-fractional models, and sowed te convergence of te discrete approximations via a compactness argument. However, no error estimate for te optimal control was given for te time-fractional case. Zou & Gong (216) proved te well-posedness of problem (1.1) (1.2) and derived 2 (,T ; 2 (Ω)) error estimates for te spatially semidiscrete finite element metod, and described a time discretization metod witout error estimate. To te best of our knowledge, tere is no error estimate for time discretizations of (1.1) (1.2). It is te main goal of tis work to fill tis gap. Tis work is devoted to te error analysis of bot time and space discretizations of (1.1)-(1.2). Te model (1.2) is discretized by te continuous piecewise linear Galerkin FEM in space and te 1 approximation (in & Xu (27)) or backward Euler convolution quadrature (ubic (1986)) in time, and te control q by a variational type discretization due to Hinze (25). Te analysis relies crucially on l p ( 2 (Ω)) error estimates for fully discrete solutions of te direct problem wit a nonsmoot source term. Suc results are still unavailable in te literature. We derive suc estimates in Teorems 2.2 and 2.3, and use tem to derive an O(τ min(1/2+α ε,1) + 2 ) error estimate in te discrete 2 (,T ; 2 (Ω)) norm for te numerical solutions of problem (1.1) (1.2), were and τ denote te mes size and time stepsize, respectively, and ε > is small, cf. Teorems 3.2 and 3.4. Te O(τ min(1/2+α ε,1) ) rate contrasts wit te O(τ) rate for te parabolic counterpart (see, e.g., Meidner & Vexler (28); Crysafinos & Karatzas (214); Gong et al. (214)). Te lower rate for α 1/2 is due to te limited smooting property of problem (1.2), cf. Teorem 2.1. Tis also constitutes te main tecnical callenge in te analysis. Based on te error estimate in te discrete 2 (,T ; 2 (Ω)) norm, we furter derive a pointwisein-time error estimate O(τ α ε +l 2 2 ) (wit l = log(2+1/), cf. Teorems 3.3 and 3.5). Our analysis relies essentially on te maximal p -regularity of fractional evolution equations (Bajlekova (21)) and its discrete analogue (Jin et al. (218a)). Te latter is te fractional extension of te discrete maximal p -regularity teory (Kovács et al. (216)), wic is a matematical tool for te numerical analysis of nonlinear parabolic equations (Akrivis et al. (217); Kunstmann et al. (218)). Numerical experiments in one- and two-dimensional spaces are provided to complement te teoretical analysis. Te rest of te paper is organized as follows. In Section 2, we discuss te solution regularity and numerical approximation for problem (1.2). In Section 3, we prove error bounds on spatially semidiscrete and fully discrete approximations to te optimal control problem (1.1) (1.2). Finally in Section 4, we provide one- and two-dimensional numerical experiments to support te teoretical results. Trougout, te notation c denotes a generic constant wic may differ at eac occurrence, but it is always independent of te mes size and time stepsize τ. 2. Regularity teory and numerical approximation of te direct problem In tis section, we recall preliminaries and present analysis for te direct problem α t u u = g, < t T, wit u() =, (2.1)

3 and its adjoint problem OPTIMA CONTRO PROBEM WITH SUBDIFFUSION CONSTRAINT 3 of 22 t α T z z = η, t < T, wit z(t ) =, (2.2) were te fractional derivative t T α z is defined in (2.5) below. Te adjoint equation follows from (2.6) below and standard integration by parts formula; see (Zou & Gong, 216, Section 3). In te case α (, 1/2], te initial / terminal condition sould be understood properly: for a roug source term g, te temporal trace at t = / t = T may not exist and te initial condition sould be interpreted in a weak sense (see, e.g., Gorenflo et al. (215)). We refrain from te case of a nonzero initial condition, and leave it to a future work. 2.1 Sobolev spaces of functions vanising at t = We sall use extensively Bocner-Sobolev spaces W s,p (,T ; 2 (Ω)). For any s and 1 p <, we denote by W s,p (,T ; 2 (Ω)) te space of functions v : (,T ) 2 (Ω), wit te norm defined by interpolation. Equivalently, te space is equipped wit te quotient norm v W s,p (,T ; 2 (Ω)) := inf ṽ ṽ W s,p (R; 2 (Ω)), (2.3) were te infimum is taken over all possible extensions ṽ tat extend v from (,T ) to R. For any < s < 1, one can define Sobolev Slobodeckiǐ seminorm W s,p (,T ; 2 (Ω)) by v p W s,p (,T ; 2 (Ω)) := T and te full norm W s,p (,T ; 2 (Ω)) by T v(t) v(ξ ) p 2 (Ω) t ξ 1+ps dtdξ, (2.4) v p W s,p (,T ; 2 (Ω)) = v p p (,T ; 2 (Ω)) + v p W s,p (,T ; 2 (Ω)). For s > 1, one can define similar seminorms and norms. et C (,T ; 2 (Ω)) := {v = w (,T ) : w C (R; 2 (Ω)) : supp(w) [, )}, and denote by W s,p (,T ;2 (Ω)) te closure of C (,T ;2 (Ω)) in W s,p (,T ; 2 (Ω)), and by W s,p R (,T ;2 (Ω)) te closure of CR (,T ;2 (Ω)) in W s,p (,T ; 2 (Ω)), wit C R (,T ; 2 (Ω)) := {v = w (,T ) : w C (R; 2 (Ω)) : supp(w) (,T ]}. By Sobolev embedding, for v W s,p being te integral part of s > ), and also v ( j) = if (s [s])p > 1. For v W s,p (,T ;2 (Ω)), tere olds v ( j) () = for j =,...,[s] 1 (wit [s] (,T ;2 (Ω)), te zero (,T ;2 (Ω)) = W s,p (,T ; 2 (Ω)), (,T ;2 (Ω)) as H s(,t ;2 (Ω)), and likewise HR s(,t ;2 (Ω)) for extension of v to te left belongs to W s,p (,T ; 2 (Ω)), and W s,p if s < 1/p. We abbreviate W s,2 W s,2 (,T ;2 (Ω)). R Similar to te left-sided fractional derivative t α u in (1.3), te rigt-sided Riemann-iouville fractional derivative t T α v(t) in (2.2) is defined by t T α 1 d v(t) := Γ (1 α) dt T t (s t) α v(s)ds. (2.5) et any p (1, ) and p (1, ) be conjugate to eac oter, i.e., 1/p + 1/p = 1. Since for u W α,p (,T ; 2 (Ω)),v W α,p R (,T ; 2 (Ω)), we ave t α u p (,T ; 2 (Ω)), t T αv p (,T ; 2 (Ω)). Tus, tere olds (Kilbas et al., 26, p. 76, emma 2.7): T ( α t u(t))v(t)dt = T u(t)( t T α v(t))dt, u W α,p (,T ; 2 (Ω)), v W α,p R (,T ; 2 (Ω)). (2.6)

4 4 of 22 B. JIN, B. I AND Z. ZHOU 2.2 Regularity of te direct problem Te next maximal p -regularity olds (cf. Bajlekova (21)), and an analogous result olds for (2.2). EMMA 2.1 If u = and g p (,T ; 2 (Ω)) wit 1 < p <, ten problem (2.1) as a unique solution u p (,T ;H 1 (Ω) H2 (Ω)) suc tat α t u p (,T ; 2 (Ω)) and u p (,T ;H 2 (Ω)) + α t u p (,T ; 2 (Ω)) c g p (,T ; 2 (Ω)), were te constant c is independent of g and T. Now we give a regularity result. Trougout, te notation denotes te Diriclet aplacian, wit its domain D( ) = H 1 (Ω) H2 (Ω). THEOREM 2.1 For g W s,p (,T ; 2 (Ω)), s [,1/p) and p (1, ), problem (2.1) as a unique solution u W α+s,p (,T ; 2 (Ω)) W s,p (,T ;D( )), wic satisfies u W α+s,p (,T ; 2 (Ω)) + u W s,p (,T ;H 2 (Ω)) c g W s,p (,T ; 2 (Ω)). Similarly, for η W s,p (,T ; 2 (Ω)), s [,1/p) and p (1, ), problem (2.2) as a unique solution z W α+s,p (,T ; 2 (Ω)) W s,p (,T ;D( )), wic satisfies z W α+s,p (,T ; 2 (Ω)) + z W s,p (,T ;H 2 (Ω)) c η W s,p (,T ; 2 (Ω)). Proof. For g W s,p (,T ; 2 (Ω)), s [,1/p) and p (1, ), extending g to be zero on Ω [(,) (T, )] yields g W s,p (R; 2 (Ω)) and g W s,p (R; 2 (Ω)) c g W s,p (,T ; 2 (Ω)). (2.7) Furter, we ave te identity t α g(t) = t α g(t) for t [,T ], and t α g = (iξ ) α ĝ(ξ ) (Kilbas et al., 26, p. 9), were denotes taking Fourier transform in t, and ĝ te Fourier transform of g. Ten, wit being te inverse Fourier transform in ξ, u = [((iξ ) α ) 1 ĝ(ξ )] is a solution of (2.1) and (1 + ξ 2 ) α+s 2 û(ξ ) = (1 + ξ 2 ) α 2 ((iξ ) α ) 1 (1 + ξ 2 ) s 2 ĝ(ξ ). Te self-adjoint operator : D( ) 2 (Ω) is invertible from 2 (Ω) to D( ), and generates a bounded analytic semigroup (Arendt et al., 211, Example 3.7.5). Tus te operator (1 + ξ 2 ) α 2 ((iξ ) α ) 1 (2.8) is bounded from 2 (Ω) to D( ) in a small neigborood N of ξ =. Furter, in N, te operator ξ d dξ (1 + ξ 2 ) α 2 ((iξ ) α ) 1 α ξ 2 = 1 + ξ 2 (1 + ξ 2 ) α 2 ((iξ ) α ) 1 is also bounded. If ξ is away from, ten + α(1 + ξ 2 ) α 2 ((iξ ) α ) 1 (iξ ) α ((iξ ) α ) 1 (2.9) (1 + ξ 2 ) α 2 ((iξ ) α ) 1 = (iξ ) α (1 + ξ 2 ) α 2 (iξ ) α ((iξ ) α ) 1. Now we sligtly abuse te notation for te operator norm on 2 (Ω). Recall also te resolvent estimate (z ) 1 c z 1 for any z Σ θ := {z C : z, arg(z) θ}, for all θ (π/2,π) (Arendt et al., 211, Example and Teorem ). Ten, te following inequalities (iξ ) α (1 + ξ 2 ) α 2 c and (iξ ) α ((iξ ) α ) 1 c

5 OPTIMA CONTRO PROBEM WITH SUBDIFFUSION CONSTRAINT 5 of 22 imply te boundedness of (2.8) and (2.9). Since boundedness of operators is equivalent to R-boundedness of operators in 2 (Ω) (see Kunstmann & Weis (24) for te concept of R-boundedness), te boundedness of (2.8) and (2.9) implies tat (2.8) is an operator-valued Fourier multiplier (Weis, 21, Teorem 3.4), and tus u W α+s,p (R; 2 (Ω)) [(1 + ξ 2 ) α+s 2 û(ξ )] p (R; 2 (Ω)) = [(1 + ξ 2 ) α 2 ((iξ ) α ) 1 (1 + ξ 2 ) s 2 ĝ(ξ )] p (R; 2 (Ω)) c [(1 + ξ 2 ) s 2 ĝ(ξ )] p (R; 2 (Ω)) c g W s,p (R; 2 (Ω)). Tis and (2.7) imply te desired bound on u W α+s,p (,T ; 2 (Ω)). Te estimate u W s,p (,T ;H 2 (Ω)) c g W s,p (,T ; 2 (Ω)) follows similarly by replacing (1 + ξ 2 ) α 2 ((iξ ) α ) 1 wit ((iξ ) α ) 1 in te proof. REMARK 2.1 Below we only use te cases p = 2, s = min(1/2 ε,α ε) and p > max(1/α,1/(1 α)), s = 1/p ε of Teorem 2.1. Bot cases satisfy te conditions of Teorem 2.1. A similar assertion olds for te more general case g W s,p (,T ;2 (Ω)), s > and 1 < p < : u W α+s,p (,T ; 2 (Ω)) + u W s,p (,T ;H 2 (Ω)) c g W s,p (,T ;2 (Ω)). In fact, for g W s,p (,T ;2 (Ω)), te zero extension of g to t belongs to W s,p (,T ; 2 (Ω)), wic can furter be boundedly extended to a function in W s,p (R; 2 (Ω)). Ten te argument in Teorem 2.1 gives te desired assertion. Tis also indicates a certain compatibility condition for regularity pickup. 2.3 Numerical sceme for problem (2.1) Now we describe numerical treatment of te forward problem (2.1), wic forms te basis for te fully discrete sceme of te control problem (1.1) (1.2) in Section 3. We denote by T a sape-regular and quasi-uniform triangulation of te domain Ω into d-dimensional simplexes, and let X = { v H 1 (Ω) : v K is a linear function, K T } be te finite element space consisting of continuous piecewise linear functions. Te 2 (Ω)-ortogonal projection P : 2 (Ω) X is defined by (P ϕ, χ ) = (ϕ, χ ), for all ϕ 2 (Ω), χ X, were (, ) denotes te 2 (Ω) inner product. Ten te spatially semidiscrete Galerkin FEM for problem (2.1) is to find u (t) X suc tat u () = and ( α t u (t), χ ) + ( u (t), χ ) = (g(t), χ ), χ X, t (,T ], (2.1) By introducing te discrete aplacian : X X, defined by ( ϕ, χ ) = ( ϕ, χ ), for all ϕ, χ X, problem (2.1) can be written as α t u (t) u (t) = P g(t), t (,T ], wit u () =. (2.11) Similar to Teorem 2.1, for s < 1/p tere olds u W α+s,p (,T ; 2 (Ω)) + u W s,p (,T ; 2 (Ω)) c g W s,p (,T ; 2 (Ω)), (2.12) were te constant c is independent of (following te proof of Teorem 2.1). emma 2.1 and Remark 2.1 remain valid for te semidiscrete solution u, e.g., u p (,T ; 2 (Ω)) + α t u p (,T ; 2 (Ω)) c g p (,T ; 2 (Ω)). (2.13)

6 6 of 22 B. JIN, B. I AND Z. ZHOU Tese assertions will be used extensively below witout explicitly referencing. To discretize (2.11) in time, we uniformly partition [,T ] wit grid points t n = nτ, n,=,1,2,...,n and a time stepsize τ = T /N 1, and approximate t α ϕ(t n ) by (wit ϕ j = ϕ(t j )): τ α ϕ n = τ α n β n j ϕ j, (2.14) j= were β j are suitable weigts. We consider two time stepping scemes: 1 sceme (in & Xu (27)) and backward Euler convolution quadrature (BE-CQ) (ubic (1986)), for wic β j are respectively given by (wit c α = 1/Γ (2 α)) 1 sceme: β = c α, and β j = c α (( j + 1) 1 α 2 j 1 α + ( j 1) 1 α ), j = 1,2,...,N, BE-CQ: β = 1, and β j = β j 1 (α j + 1)/ j, j = 1,2,...,N. Bot scemes extend te classical backward Euler sceme to te fractional case. Ten we discretize problem (2.1) by: wit g n = P g(t n ), find U n X suc tat τ α U n U n = gn, n = 1,2,...,N, wit U =. (2.15) By (Jin et al., 218b, Section 5) and (Jin et al., 216, Teorem 3.6), we ave te following error bound. EMMA 2.2 For g W 1,p (,T ; 2 (Ω)), 1 p, let u and U n be te semidiscrete solution and fully discrete solution, respectively, in (2.11) and (2.15). Ten tere olds U n u (t n ) 2 (Ω) cτtα 1 n tn g() 2 (Ω) + cτ (t n+1 s) α 1 g (s) 2 (Ω) ds. REMARK 2.2 emma 2.2 sligtly refines te estimates in Jin et al. (218b, 216), but can be proved in te same way using te following estimates in te proof of (Jin et al., 218b, Section 5): t α 1 n c(t + τ) α 1 and tn t s α 1 ds cτ(t + τ) α 1, for t [t n 1,t n ] and n = 1,...,N. For any Banac space X, we define ( N 1/p τ U (U n )N n= X) n p if 1 p <, l p (X) := n= max U n X if p =. n N Ten te maximal l p -regularity estimate olds for (2.15) (Jin et al., 218a, Teorems 5 and 7). EMMA 2.3 Te solutions (U n)n n=1 of (2.15) satisfy α ( τ U n )N n=1 l p ( 2 (Ω)) + ( U n )N n=1 l p ( 2 (Ω)) c p (g n )N n=1 l p ( 2 (Ω)), 1 < p <. 2.4 Error estimates Now we present l p ( 2 (Ω)) error estimates for g W s,p (,T ; 2 (Ω)), s 1, 1 p. Error analysis for suc g is still unavailable. We need an interpolation error estimate. Tis result seems standard, but we are unable to find a proof, and tus include a proof in Appendix A. EMMA 2.4 For v W s,p (,T ; 2 (Ω)), 1 < p < and s (1/p,1], let v n = τ 1 t n t n 1 v(t)dt. Ten tere olds (v(t n ) v n ) N n=1 l p ( 2 (Ω)) cτs v W s,p (,T ; 2 (Ω)).

7 OPTIMA CONTRO PROBEM WITH SUBDIFFUSION CONSTRAINT 7 of 22 Our first result is an error estimate for g W s,p (,T ;2 (Ω)) (i.e., compatible source). Since g may not be smoot enoug in time for pointwise evaluation, we define te averages ḡ n = τ 1 t n t n 1 P g(s)ds, and consider a variant of te sceme (2.15) for problem (2.1): find U n X suc tat α τ U n U n = ḡ n, n = 1,...,N, wit U =. (2.16) THEOREM 2.2 For g W s,p (,T ;2 (Ω)), 1 < p < and s [,1], let u and U n be te solutions of problems (2.11) and (2.16), respectively, and ū n := τ 1 t n t n 1 u (s)ds. Ten tere olds (U n ū n )N n=1 l p ( 2 (Ω)) cτs g W s,p (,T ;2 (Ω)). Proof. By Hölder s inequality and (2.12) (wit s = ), we ave N τ n=1 ū n N tn p 2 (Ω) = τ τ 1 u (s)ds p n=1 t 2 (Ω) n 1 T N ( tn τ1 p n=1 u (s) p 2 (Ω) ds c g p p (,T ; 2 (Ω)). Similarly, by applying emma 2.3 to (2.16) and te 2 (Ω) stability of P, we ave (U n ) N n=1 l p ( 2 (Ω)) c (ḡn )N n=1 l p ( 2 (Ω)) c g p (,T ; 2 (Ω)). t n 1 u (s) 2 (Ω) ds ) p Tis and te triangle inequality sow te assertion for s =. Next we consider g W 1,p (,T ;2 (Ω)), and resort to (2.15). Since g() =, by emma 2.2, Tis directly implies tn U n u (t n ) 2 (Ω) cτ (t n + τ s) α 1 g (s) 2 (Ω) ds. (U n u (t n )) N n=1 l ( 2 (Ω)) cτ g W 1, (,T ; 2 (Ω)). (2.17) Furter, let ψ(s) = τ N n=1 (t n + τ s) α 1 χ [,tn ], were χ S denotes te caracteristic function of a set S. Ten clearly, we ave Terefore, sup s [,T ] ψ(s) = ψ(t ) = τ N n=1 (U n u (t n )) N n=1 l 1 ( 2 (Ω)) cτ2 N = cτ T (nτ) α 1 tn n=1 T s α 1 ds = α 1 T α c T. (t n + τ s) α 1 g (s) 2 (Ω) ds ψ(s) g (s) 2 (Ω) ds c T τ g W 1,1 (,T ; 2 (Ω)). (2.18) Ten (2.17), (2.18) and Riesz-Torin interpolation teorem (Berg & ofstrom, 212, Teorem 1.1.1), yield for any 1 < p < (U n u (t n )) N n=1 l p ( 2 (Ω)) cτ g W 1,p (,T ;2 (Ω)). Since U n Un satisfies te discrete sceme, cf. (2.15) and (2.16), emmas 2.3 and 2.4 imply (U n Un ) N n=1 l p ( 2 (Ω)) c (ḡn P g(t n )) N n=1 l p ( 2 (Ω)) cτ g W 1,p (,T ; 2 (Ω)).

8 8 of 22 B. JIN, B. I AND Z. ZHOU Furter, by emma 2.4 and Remark 2.1, we ave (u (t n ) ū n )N n=1 l p ( 2 (Ω)) cτ u W 1,p (,T ; 2 (Ω)) cτ g W 1,p (,T ;2 (Ω)). Te last tree estimates sow te assertion for s = 1. Te case < s < 1 follows by interpolation. In Teorem 2.2, we compare te numerical solution U n to (2.16) wit te time-averaged solution ū n, instead of u (t n ). Tis is due to possible insufficient temporal regularity of u : it is unclear ow to define u (t n ) for t n (,T ] for g W s,p (,T ;2 (Ω)) wit s+α < 1/p. For s (1/p,1], W s,p (,T ;2 (Ω)) C([,T ]; 2 (Ω)) and so Teorem 2.2 requires te condition g() =. Suc a compatibility condition at t = is not necessarily satisfied by problem (1.1) (1.2). Hence, we state an error estimate below for a smoot but incompatible source g W s,p (,T ; 2 (Ω)). THEOREM 2.3 For g W s,p (,T ; 2 (Ω)) wit p (1, ) and s (1/p,1), let u and U n be te solutions of problems (2.11) and (2.15), respectively. Ten tere olds (U n u (t n )) N n=1 l p ( 2 (Ω)) cτmin(1/p+α,s) g W s,p (,T ; 2 (Ω)). (2.19) Moreover, if p > 1/α is so large tat α (,1/p ), ten (U n u (t n )) N n=1 l ( 2 (Ω)) cτα g W 1/p+α,p (,T ; 2 (Ω)). (2.2) Proof. For g(x,t) g(x), wic belongs to W 1,p (,T ; 2 (Ω)), by emma 2.2 we ave (U n u (t n )) N N n=1 p l p ( 2 (Ω)) = τ U n u (t n ) p 2 (Ω) cτ p+1 g p 2 (Ω) n=1 cτ pα+1 g p 2 (Ω) + cτ p g p 2 (Ω) T τ N n=1 t p(α 1) dt. t p(α 1) n Now for s < 1, tere olds cτ T pα+1 if α (,1/p ) cτ p(1/p+α) if α (,1/p ) τ p t p(α 1) dt cτ p if α (1/p,1) τ cτ p( ) cτ ps if α (1/p,1) 1 + ln(t /τ) if α = 1/p cτ ps if α = 1/p wic togeter wit te preceding estimate implies (U n u (t n )) N n=1 l p ( 2 (Ω)) cτmin(1/p+α,s) g W s,p (,T ; 2 (Ω)). For g W s,p (,T ; 2 (Ω)), by Sobolev embedding, g() exists, and in te splitting g(t) = g()+(g(t) g()), tere olds g g() s,p W (,T ;2 (Ω)) c g W s,p (,T ; 2 (Ω)). et v be te semidiscrete solution for te source g(t) g(), and v n = t n t n 1 v (t)dt. Since g(t) g() W s,p (,T ;2 (Ω)), by Teorem 2.2, te corresponding fully discrete solution V n by (2.16) satisfies Furter, by emma 2.4 and (2.12), we ave (V n v n )N n=1 l p ( 2 (Ω)) cτs g g() W s,p (,T ;2 (Ω)). (2.21) (v (t n ) v n )N n=1 l p ( 2 (Ω)) cτs v W s,p (,T ; 2 (Ω)) cτs g g() W s,p (,T ;2 (Ω)). Similarly, for te fully discrete solution V n emmas 2.3 and 2.4, we deduce corresponding to te source g(t) g() by (2.15), from (V n V n ) N n=1 l p ( 2 (Ω)) c (P g(t n ) ḡ n )N n=1 l p ( 2 (Ω)) cτ s g W s,p (,T ; 2 (Ω)). (2.22)

9 OPTIMA CONTRO PROBEM WITH SUBDIFFUSION CONSTRAINT 9 of 22 Tese estimates togeter wit te triangle inequality give (2.19). Finally, (2.2) follows by te inverse inequality in time and (2.19) wit s = 1/p + α, i.e., (U n u (t n )) N n=1 l ( 2 (Ω)) cτ 1/p (U n u (t n )) N n=1 l p ( 2 (Ω)) Tis completes te proof of te teorem. cτ α g W 1/p+α,p (,T ; 2 (Ω)). REMARK 2.3 Te estimate (2.19) is not sarp since, according to te proof, te restriction s < 1 is only needed for α = 1/p. Noneteless, it is sufficient for te error analysis in Section Te optimal control problem and its numerical approximation In tis section, we develop a numerical sceme for problem (1.1) (1.2), and derive error bounds for te spatial and temporal discretizations. 3.1 Te continuous problem Te first-order optimality condition of (1.1)-(1.2) was given in (Zou & Gong, 216, Teorem 3.4). THEOREM 3.1 Problem (1.1)-(1.2) admits a unique solution q U ad. Tere exist a state u 2 (,T ;D( )) H α (,T ;2 (Ω)) and an adjoint z 2 (,T ;D( )) H α R (,T ;2 (Ω)) suc tat (u,z,q) satisfies α t u u = f + q, < t T, wit u() =, (3.1) t α T z z = u u d, t < T, wit z(t ) =, (3.2) (γq + z,v q) 2 (,T ; 2 (Ω)), v U ad. (3.3) were (, ) 2 (,T ; 2 (Ω)) denotes te 2 (,T ; 2 (Ω)) inner product. et P Uad be te pointwise projection operator defined by It is bounded on W s,p (,T ; 2 (Ω)) for s 1 and 1 p P Uad (q) = max(a,min(q,b)). (3.4) P Uad u W s,p (,T ; 2 (Ω)) c u W s,p (,T ; 2 (Ω)). (3.5) Tis estimate olds trivially for s = and s = 1 (Ziemer, 1989, Corollary 2.1.8), and te case < s < 1 follows by interpolation. Ten (3.3) is equivalent to te complementarity condition q = P Uad ( γ 1 z ). (3.6) Now we give iger temporal regularity of te triple (u,z,q). Note tat te constant c depends on te scalar γ (see also te proof of emma 3.4) and te constraint bounds a and b. EMMA 3.1 For any s (,1/2), let u d H s (,T ; 2 (Ω)) and f H s (,T ; 2 (Ω)). Ten te solution (u,z,q) of problem (3.1) (3.3) satisfies te following estimate q H min(1,α+s) (,T ; 2 (Ω)) + u H α+s (,T ; 2 (Ω)) + z H α+s (,T ; 2 (Ω)) c. Proof. et r = min(1,α + s). By (3.6) and (3.5), we ave q H r (,T ; 2 (Ω)) c z H r (,T ; 2 (Ω)) c z H α+s (,T ; 2 (Ω)).

10 1 of 22 B. JIN, B. I AND Z. ZHOU Applying Teorem 2.1 to (3.2) yields z H α+s (,T ; 2 (Ω)) c( u H s (,T ; 2 (Ω)) + u d H s (,T ; 2 (Ω)) ) c u H s (,T ; 2 (Ω)) + c. Similarly, applying Teorem 2.1 to (3.1) gives u H α+s (,T ; 2 (Ω)) c( f H s (,T ; 2 (Ω)) + q H s (,T ; 2 (Ω)) ) c + c q H s (,T ; 2 (Ω)). (3.7) Te last tree estimates togeter imply q H r (,T ; 2 (Ω)) c + c q H s (,T ; 2 (Ω)) c + c ε q 2 (,T ; 2 (Ω)) + ε q H r (,T ; 2 (Ω)), were te last step is due to te interpolation inequality (Tartar, 26, emma 24.1) q H s (,T ; 2 (Ω)) c ε q 2 (,T ; 2 (Ω)) + ε q H r (,T ; 2 (Ω)). By coosing a small ε > and te bounded of q, cf. (3.6), we obtain q H r (,T ; 2 (Ω)) c + c ε q 2 (,T ; 2 (Ω)) c. (3.8) Tis sows te bound on q. (3.8) and (3.7) give te bound on u, and tat of z follows similarly. Next, we give an improved stability estimate on q. EMMA 3.2 et p > 1/α be sufficiently large so tat α (,1/p ), u d W α,p (,T ; 2 (Ω)) and f p (,T ; 2 (Ω)). Ten te optimal control q satisfies: q W 1/p+α ε,p (,T ; 2 (Ω)) c, were te constant c depends on u d W α,p (,T ; 2 (Ω)) and f p (,T ; 2 (Ω)). Proof. Te condition α (,1/p ) implies r := 1/p + α ε < 1. Tus (3.5) and Teorem 2.1 (wit s = r α) imply q W r,p (,T ; 2 (Ω)) c z W r,p (,T ; 2 (Ω)) c u u d W r α,p (,T ; 2 (Ω)), Since p > 1/α, r α = 1/p ε < α and tus Teorem 2.1 (wit s = ) and (3.6) give u u d W r α,p (,T ; 2 (Ω)) c u u d W α,p (,T ; 2 (Ω)) c f + q p (,T ; 2 (Ω)) + c c, were te second last inequality is due to Teorem 2.1 wit s =, and te last inequality due to te pointwise boundedness of q, cf. (3.6). Te last two estimates togeter imply te desired result. 3.2 Spatially semidiscrete sceme Now we give a spatially semidiscrete sceme for problem (1.1) (1.2): find q U ad suc tat subject to te semidiscrete problem min q U ad J(u,q ) = 1 2 u u d 2 2 (,T ; 2 (Ω)) + γ 2 q 2 2 (,T ; 2 (Ω)), (3.9) α t u u = P ( f + q ), < t T, wit u () =. (3.1) Note tat in te sceme (3.9) (3.1), te control variable q is not discretized directly, but instead in a variational sense due to Hinze (25). Tis coice greatly facilitates te analysis, wile also leads to optimal order convergence rates. In passing, note tat tere are oter possible strategies to discretize te control variable q, e.g., cellwise constant/linear approximation and postprocessing; see Meidner &

11 OPTIMA CONTRO PROBEM WITH SUBDIFFUSION CONSTRAINT 11 of 22 Vexler (28) for te standard parabolic counterparts. It would be interesting to analyze tese discretization strategies, wic, owever, is beyond te scope of te present work. Similar to Teorem 3.1, problem (3.9)-(3.1) admits a unique solution q U ad. Te first-order optimality system reads: α t u u = P ( f + q ), < t T, wit u () =, (3.11) t α T z z = P (u u d ), t < T, wit z (T ) =, (3.12) (γq + z,v q ) 2 (,T ; 2 (Ω)), v U ad. (3.13) Te variational inequality (3.13) is equivalent to q = P Uad ( γ 1 z ). (3.14) For te sceme (3.11) (3.13), we ave te next error bound (Zou & Gong, 216, Teorem 4.6). THEOREM 3.2 For f,u d 2 (,T ; 2 (Ω)), let (u,z,q) and (u,z,q ) be te solutions of problems (3.1) (3.3) and (3.11) (3.13), respectively. Ten tere old u u 2 (,T ; 2 (Ω)) + z z 2 (,T ; 2 (Ω)) + q q 2 (,T ; 2 (Ω)) c2, (u u ) 2 (,T ; 2 (Ω)) + (z z ) 2 (,T ; 2 (Ω)) c. Next, we present te regularity of te semidiscrete solution (u,z,q ). Te proof is similar to te continuous case in emmas 3.1 and 3.2 and ence omitted. EMMA 3.3 et s (,1/2), u d H s (,T ; 2 (Ω)) and f H s (,T ; 2 (Ω)). Ten te solution (u,z,q ) of problem (3.11) (3.13) satisfies q H min(1,α+s) (,T ; 2 (Ω)) + u H α+s (,T ; 2 (Ω)) + z H α+s (,T ; 2 (Ω)) c. Furter, for u d W α,p (,T ; 2 (Ω)), f p (,T ; 2 (Ω)), wit p > 1/α and α (,1/p ), tere olds q W 1/p+α ε,p (,T ; 2 (Ω)) c. ast, we derive a pointwise-in-time error estimate. THEOREM 3.3 For f, u d H 1 (,T ; 2 (Ω)), let (u,z,q) and (u,z,q ) be te solutions of problems (3.1) (3.3) and (3.11) (3.13), respectively. Ten wit l = log(2 + 1/), tere olds u u (,T ; 2 (Ω)) + z z (,T ; 2 (Ω)) + q q (,T ; 2 (Ω)) cl2 2, were te constant c depends on f H 1 (,T ; 2 (Ω)) and u d H 1 (,T ; 2 (Ω)). Proof. We employ te splitting u u = (u u (q)) + (u (q) u ) := ρ + ϑ, were u (q) X solves α t u (q) u (q) = P ( f + q), < t T, wit u (q)() =. Ten u (q) is te semidiscrete solution of (2.1) wit g = f + q, and ρ is te FEM error for te direct problem. By (Jin et al., 215, Teorem 3.7) and emma 3.2, we ave ρ (,T ; 2 (Ω)) cl2 2 f + q (,T ; 2 (Ω)) cl2 2. (3.15) Since ϑ satisfies α t ϑ ϑ = P (q q ), for < t T wit ϑ() =, (2.13), 2 (Ω)-stability of P, te conditions (3.6) and (3.14), and te pointwise contractivity of P Uad imply α t ϑ p (,T ; 2 (Ω)) P (q q ) p (,T ; 2 (Ω)) c q q p (,T ; 2 (Ω)) c z z p (,T ; 2 (Ω)). (3.16)

12 12 of 22 B. JIN, B. I AND Z. ZHOU Next, it follows from (3.2), (3.12) and te identity P = R (wit R : H 1 (Ω) X being Ritz projection) tat w := P z z satisfies w (T ) = and and tus by applying 1 t α T w w = P u u (P z R z), t < T, on bot sides leads to t α T 1 w 1 w = 1 (P u u ) (P z R z). (3.17) Te maximal p regularity (2.13) and triangle inequality imply w p (,T ; 2 (Ω)) Te 2 (Ω)-stability of P and triangle inequality yield 1 c (P u u ) (P z R z) p (,T ; 2 (Ω)) c P u u p (,T ; 2 (Ω)) + c P z R z p (,T ; 2 (Ω)). P u u p (,T ; 2 (Ω)) c u u p (,T ; 2 (Ω)) c( ϑ p (,T ; 2 (Ω)) + ρ p (,T ; 2 (Ω)) ), and Teorem 2.1 (wit s = ) and lemma 3.3 give P z R z p (,T ; 2 (Ω)) c z p (,T ;H 2 (Ω)) 2 c u u d p (,T ; 2 (Ω)) 2 c 2. Te last tree estimates and (3.15) yield Tus repeating te preceding argument yields w p (,T ; 2 (Ω)) c ϑ p (,T ; 2 (Ω)) + cl2 2. z z p (,T ; 2 (Ω)) z P z p (,T ; 2 (Ω)) + w p (,T ; 2 (Ω)) c ϑ p (,T ; 2 (Ω)) + cl2 2. Substituting tis estimate into (3.16) and by Sobolev embedding W α,p (,T ; 2 (Ω)) p α (,T ; 2 (Ω)), wit te critical exponent p α = p/(1 pα) if pα < 1, and p α = if pα > 1: ϑ pα (,T ; 2 (Ω)) c α t ϑ p (,T ; 2 (Ω)) c ϑ p (,T ; 2 (Ω)) + cl2 2. (3.18) A finite number of repeated applications of tis inequality yields ϑ (,T ; 2 (Ω)) c ϑ 2 (,T ; 2 (Ω)) + cl2 2 cl 2 2, (3.19) were we ave used te fact tat, by maximal p regularity (2.13) and Teorem 3.2, ϑ 2 (,T ; 2 (Ω)) = u (q) u 2 (,T ; 2 (Ω)) c q q 2 (,T ; 2 (Ω)) c2. Tis gives te desired bound on u u (,T ; 2 (Ω)). Te bounds on z z (,T ; 2 (Ω)) and q q (,T ; 2 (Ω)) follow similarly (and also by te contraction property of P U ad ). REMARK 3.1 By te best approximation properties for piecewise linear finite element spaces, te error estimates in Teorem 3.3 are of optimal order, up to te logaritmic factor l 2. Te latter factor is also present for te direct problem wit a source f (,T ; 2 (Ω)) (Jin et al., 215, Teorem 3.7).

13 3.3 Fully discrete sceme OPTIMA CONTRO PROBEM WITH SUBDIFFUSION CONSTRAINT 13 of 22 Now we turn to te fully discrete approximation of (1.1) (1.2), wit 1 sceme or BE-CQ time stepping (and variational discretization for te control variable). First, we define a time-discrete admissible set Uad τ = { Q = (Q n 1 ) N n=1 2 (Ω) N : a Q n 1 b, n = 1,2,...,N}, and consider te following fully discrete problem: τ min Q Uad τ 2 subject to te fully discrete problem N n=1 ( U n un d 2 2 (Ω) + γ Qn (Ω) α τ U n U n = f n + P Q n 1, n = 1,2,...,N, wit U =, wit u n d = u d(t n ) and f n = P f (t n ). Note tat like te semidiscrete case, te admissible set U ad is not directly discretized in space, but only in a variational sense. et α τ ϕ n be te 1/BE-CQ approximation of te rigt-sided Riemann-iouville fractional derivative t α T ϕ(t n): α τ ϕ N n = τ α n β n j ϕ N j. j= Ten te fully discrete problem is to find (U n,zn,qn ) suc tat (Zou & Gong, 216, Section 5) (γq n 1 τ α U n U n = f n + P Q n 1, n = 1,2,...,N, wit U =, (3.2) τ α Z n 1 Z n 1 = U n P u n d, n = 1,2,...,N, wit ZN =, (3.21) + Z n 1,v Q n 1 ), v 2 (Ω) s.t. a v b. (3.22) Similar to (3.13), (3.22) can be rewritten as Q n 1 = P Uad ( γ 1 Z n 1 To simplify te notation, we denote te l 2 ( 2 (Ω)) inner product by [v,w] τ = τ N n=1 ), ), n = 1,2,...,N. (3.23) (v n,w n ) v = (v n ) N n=1,w = (w n) N n=1 2 (Ω) N, and we sall identify vectors wit sequences below. et α τ v = ( α τ v n )N n=1 2 (Ω) N and α τ w = ( α τ w n 1 ) N n=1 2 (Ω) N. Ten te following identity olds (Zou & Gong, 216, Section 5.2) Tus, α τ is te adjoint to α τ wit respect to [, ] τ. et [ α τ v,w] τ = [v, α τ w] τ v,w 2 (Ω) N. (3.24) U = (U n )N n=1, Z = (Z n 1 ) N n=1, Q = (Q n 1 ) N n=1, u = (u (t n )) N n=1, z = (z (t n 1 )) N n=1, q = (q (t n 1 )) N n=1. Next we introduce two auxiliary problems. et U (q ) = (U n (q ))) N n=1 X N solve α τ U n (q ) U n (q ) = f n + q (t n 1 ), n = 1,...,N, wit U (q ) =. (3.25) By emma 3.3, te pointwise evaluation q (t n ) does make sense, and tus problem (3.25) is well defined. For any v = (v n )N n=1 X N, let Z (v ) = (Z n 1 (v )) N n=1 X N solve τ α Z n 1 (v ) Z n 1 (v ) = v n P u n d, n = 1,2,...,N, wit ZN (v ) =. (3.26) Te rest of tis part is devoted to error analysis. First, we bound q Q l 2 ( 2 (Ω)).

14 14 of 22 B. JIN, B. I AND Z. ZHOU EMMA 3.4 For Q, q, Z and Z (U (q )) defined as above, tere olds γ Q q 2 l 2 ( 2 (Ω)) [q Q,Z (U (q )) z ] τ. Proof. It follows from (3.2) and (3.25), similarly from (3.21) and (3.26), tat α ( τ )(U (q ) U ) = q Q and ( τ α )(Z (v ) Z ) = v U. Togeter wit (3.24), tese identities imply [q Q,Z Z (U (q ))] τ = [ ( α τ )(U (q ) U ),Z Z (U (q )) ] τ = [ U (q ) U,( τ α )(Z Z (U (q ))) ] τ = U (q ) U 2 l 2 ( 2 (Ω)). (3.27) Next, since (3.14) olds pointwise in time, i.e., q (t n 1 ) = P Uad ( γ 1 z (t n 1 )), we ave (q (t n 1 ) + γ 1 z (t n 1 ), χ q (t n 1 )), χ 2 (Ω) s.t. a χ b. (3.28) Upon setting v = q (t n 1 ) in (3.22) and χ = Q n 1 in (3.28), we deduce γ Q q 2 l 2 ( 2 (Ω)) = γ[q q,q ] τ γ[q q,q ] τ [q Q,Z ] τ [q Q,z ] τ = [q Q,Z Z (U (q ))] τ + [q Q,Z (U (q )) z ] τ. Now invoking (3.27) completes te proof of te lemma. Te next result gives an error estimate for te approximate state U (q ). EMMA 3.5 et f,u d H 1 (,T ; 2 (Ω)). For any ε (,min(1/2,α)), tere olds U (q ) u l 2 ( 2 (Ω)) cτ1/2+min(1/2,α ε). Proof. By te triangle inequality, we ave U (q ) u l 2 ( 2 (Ω)) U (q ) Ũ (q ) l 2 ( 2 (Ω)) + Ũ (q ) u l 2 ( 2 (Ω)), were Ũ (q ) = (Ũ n(q )) N n=1 is te solution to α τ Ũ n (q ) n Ũn (q ) = f n + qn, n = 1,2,...,N wit Ũ (q ) =, (3.29) wit q n = P q (t n ) (and q = ( q n )N n=1 ). Tat is, Ũ n (q ) is te fully discrete solution of problem (2.1) wit g = f + q. By emmas 2.3 and 2.4, we ave U (q ) Ũ (q ) l 2 ( 2 (Ω)) c q q l 2 ( 2 (Ω)) cτmin(1/2+α ε,1) q H 1/2+α ε (,T ; 2 (Ω)). (3.3) Furter, Teorem 2.3 (wit s = min(1, 1/2 + α ε) (1/2, 1)) implies Ũ (q ) u l 2 ( 2 (Ω)) c P f + q H s (,T ; 2 (Ω)) τs c( P f H s (,T ; 2 (Ω)) + q H s (,T ; 2 (Ω)) )τs. Te last two estimates and emma 3.3 (wit s = 1/2 ε) yield te desired assertion. Now we can give an l 2 ( 2 (Ω)) error estimate for te approximation (U n,zn,qn ).

15 OPTIMA CONTRO PROBEM WITH SUBDIFFUSION CONSTRAINT 15 of 22 THEOREM 3.4 For f H 1 (,T ; 2 (Ω)) and u d H 1 (,T ; 2 (Ω)), (u,z,q ) and (U n,zn,qn ) be te solutions of problems (3.11)-(3.13) and (3.2)-(3.22), respectively. Ten tere olds for any small ε > u U l 2 ( 2 (Ω)) + z Z l 2 ( 2 (Ω)) + q Q l 2 ( 2 (Ω)) cτ1/2+min(1/2,α ε), were te constant c depends on f H 1 (,T ; 2 (Ω)) and u d H 1 (,T ; 2 (Ω)). Proof. By emma 3.4 and te triangle inequality, we deduce Q q l 2 ( 2 (Ω)) c Z (U (q )) Z (u ) l 2 ( 2 (Ω)) + c Z (u ) z l 2 ( 2 (Ω)). It suffices to bound te two terms on te rigt and side. emmas 2.3 and 3.5 imply Z (U (q )) Z (u ) l 2 ( 2 (Ω)) c U (q ) u l 2 ( 2 (Ω)) cτr. wit r = 1/2 + min(1/2,α ε). Furter, since Z (u ) is a fully discrete approximation to z (u ), by Teorem 2.3 (wit s = r) and emma 3.3, we ave Z (u ) z l 2 ( 2 (Ω)) c u P u d H r (,T ; 2 (Ω)) τr cτ r. Tus, we obtain te estimate Q q l 2 ( 2 (Ω)) cτr. Next, by emmas 2.3 and 3.5, we deduce U u l 2 ( 2 (Ω)) U U (q ) l 2 ( 2 (Ω)) + U (q ) u l 2 ( 2 (Ω)) c Q q l 2 ( 2 (Ω)) + U (q ) u l 2 ( 2 (Ω)) cτr. Similarly, Z z l 2 ( 2 (Ω)) can be bounded by Z z l 2 ( 2 (Ω)) Z Z (u ) l 2 ( 2 (Ω)) + Z (u ) z l 2 ( 2 (Ω)) c U u l 2 ( 2 (Ω)) + Z (u ) z l 2 ( 2 (Ω)) cτr. Tis completes te proof of Teorem 3.4. ast, we give a pointwise-in-time error estimate for te approximation (U n,qn,zn ). THEOREM 3.5 For f,u d W 1,p (,T ; 2 (Ω)) H 1 (,T ; 2 (Ω)), p > 1/α wit α (,1/p ), let (u,z,q ) and (U n,zn,qn ) be te solutions of problems (3.11)-(3.13) and (3.2)-(3.22), respectively. Ten tere olds for any small ε > ( u n U n 2 (Ω) + zn 1 Z n 1 2 (Ω) + qn 1 max 1 n N Q n 1 2 (Ω)) cτ α ε, were te constant c depends on f W 1,min(p,2) (,T ; 2 (Ω)) and u d W 1,min(p,2) (,T ; 2 (Ω)). Proof. It follows from (3.2) and (3.25) tat U U (q ) = and α τ (U n Un (q )) (U n Un (q )) = Q n 1 q (t n 1 ), n = 1,...,N. By emma 2.3 and te inverse inequality (in time), we obtain for any 1/α < p 1 < α τ (U n Un (q )) N n=1 l p 1 ( 2 (Ω)) Tis and Teorem 3.4 imply c (Qn 1 q (t n 1 )) N n=1 l p 1 ( 2 (Ω)) cτ min(,1/p 1 1/2) (Q n 1 q (t n 1 )) N n=1 l 2 ( 2 (Ω)). α τ (U n Un (q )) N n=1 l p 1 ( 2 (Ω)) cτmin(1/p 1,1/2)+min(1/2,α ε).

16 16 of 22 B. JIN, B. I AND Z. ZHOU By coosing p 1 > 1/α sufficiently close to 1/α and discrete embedding (Jin et al. (218b)), (U n Un (q )) N n=1 l ( 2 (Ω)) c α τ (U n Un (q )) N n=1 l p 1 ( 2 (Ω)) cτ min(1/p 1,1/2)+min(1/2,α ε) cτ α ε, were te last inequality follows from te inequality min(1/p 1,1/2) + min(1/2,α ε) α ε, due to te coice of p 1. Furter, by te definition of Ũ n(q ) in (3.29), coosing p 2 > 1/α sufficiently large so tat α (,1/p 2 ) and applying (2.2) and emma 3.3, we get u (t n ) Ũ n (q ) 2 (Ω) cτα ε f + q W 1/p 2 +α ε,p 2 (,T ; 2 (Ω)) cτ α ε ( q W 1/p 2 +α ε,p 2 (,T ; 2 (Ω)) + c) cτα ε. ast, by coosing p 3 > 1/α sufficiently close to 1/α, emmas 2.3, 3.3, and 2.4, and discrete embedding (Jin et al. (218b)), we obtain (Ũ n (q ) U n (q )) N n=1 l ( 2 (Ω)) c α τ (Ũ n (q ) U n (q )) N n=1 l p 3 ( 2 (Ω)) c (q (t n 1 ) q (t n )) N n=1 l p 3 ( 2 (Ω)) cτα ε. Te last tree estimates yield te desired bound on u (t n ) U n 2 (Ω). Te bound on z (t n 1 ) 2 (Ω) follows similarly, and tat on q (t n 1 ) Q n 1 2 (Ω) by te contraction property of P U ad. Z n 1 REMARK 3.2 Teorem 3.5 gives an O(τ α ε ) convergence rate in te l ( 2 (Ω))-norm for te time discretization errors of te control, state and adjoint variables. Tis estimate agrees wit bot te numerical experiments in Section 4 and te regularity result in emma 3.3. Indeed, emma 3.3 implies u C α ε ([,T ]; 2 (Ω)) + z C α ε ([,T ]; 2 (Ω)) + q C α ε ([,T ]; 2 (Ω)) c( u H α+1/2 ε (,T ; 2 (Ω)) + z H α+1/2 ε (,T ; 2 (Ω)) + q W α+1/p ε,p (,T ; 2 (Ω))) c. Since te O(τ α ) convergence rate (uniform in time) is optimal for te direct problem (Jin et al., 218b, emma 4.2), te error estimate in Teorem 3.5 is optimal up to an ε order. REMARK 3.3 So far our discussion focuses on te case of a zero initial condition, i.e., u() =. Te analysis can be extended to te case of smoot initial data: α t u u = f, wit u() = u. were u D( ) and α t u denotes te Caputo fractional derivative. Ten te function w := u u satisfies (1.2) wit a source F = f + u, for wic our approac applies. Te case of a nonsmoot u, e.g., u 2 (Ω), requires new tecniques, due to a lack of regularity of te state variable u (near t = ). 4. Numerical results and discussions Now we present numerical experiments to illustrate te teoretical findings. 4.1 One-dimensional examples We perform experiments on te unit interval Ω = (,1). Te domain Ω is divided into M equally spaced subintervals wit a mes size = 1/M. To discretize te fractional derivatives t α u and t T α z, we fix te time stepsize τ = T /N. We present numerical results only for te fully discrete sceme by te Galerkin FEM in space and te 1 sceme in time, since BE-CQ gives nearly identical results. We consider te following two examples to illustrate te analysis. (a) f and u d (x,t) = e t x(1 x).

17 OPTIMA CONTRO PROBEM WITH SUBDIFFUSION CONSTRAINT 17 of 22 (b) f = (1 + cos(t))χ (1/2,1) (x) and u d (x,t) = 5e t x(1 x). Trougout, te penalty parameter γ is set to γ = 1, and te lower and upper bounds a and b in te admissible set U ad to a = and b =.5. Te final time T is fixed at T =.1. Te conditions from Teorems 3.3, 3.4 and 3.5 are satisfied for bot examples, and tus te error estimates terein old. In Tables 1 and 4, we present te spatial error e (u) in te (,T ; 2 (Ω))-norm for te semidiscrete solution u, defined by e (u) = max 1 n N u (t n ) u(t n ) 2 (Ω), and similarly for te approximations z and q. Te numbers in te bracket in te last column denote te teoretical rates. Since te exact solution to problem (1.2) is unavailable, we compute reference solutions on a finer mes, i.e., te continuous solution u(t n ) wit a fixed time step τ = T /1 and mes size = 1/128. Te empirical rate for te spatial error e is of order O( 2 ), wic is consistent wit te teoretical result in Teorem 3.3. For case (a), te box constraint is inactive, and tus te errors for te control q and adjoint z are identical (since γ = 1). Table 1: Spatial errors for example (a) wit N = 1 4. α M rate e (u) 4.57e e e e e e-8 2. (2.).4 e (q) 3.38e e e e e e-8 2. (2.) e (z) 3.38e e e e e e-8 2. (2.) e (u) 2.44e-6 6.7e e e e e-9 2. (2.).6 e (q) 3.62e-5 9.4e e e e e-8 2. (2.) e (z) 3.62e-5 9.4e e e e e-8 2. (2.) e (u) 8.93e e e-8 1.4e e e-1 2. (2.).8 e (q) 3.92e e e e e e-8 2. (2.) e (z) 3.92e e e e e e-8 2. (2.) Table 2: Temporal errors for example (a) wit M = 5. α N rate e τ,2 (u) 1.7e e e e e-7 1.6e-7.83 (.9).4 e τ,2 (q) 2.2e-5 1.2e-5 7.6e-6 4.9e e e-6.82 (.9) e τ,2 (z) 2.2e-5 1.2e-5 7.6e-6 4.9e e e-6.82 (.9) e τ,2 (u) 6.58e e e e-8 4.9e e-8.96 (1.).6 e τ,2 (q) 8.25e e e-6 1.2e e e-7.95 (1.) e τ,2 (z) 8.25e e e-6 1.2e e e-7.95 (1.) e τ,2 (u) 2.68e e-7 7.7e e e e-9.97 (1.).8 e τ,2 (q) 3.8e e-6 1.e e e e-7.98 (1.) e τ,2 (z) 3.8e e-6 1.e e e e-7.98 (1.) Next, to examine te convergence in time, we compute te l 2 ( 2 (Ω)) and l ( 2 (Ω)) temporal errors e τ,2 (u) and e τ, (u) for te fully discrete solutions U n, respectively, defined by e τ,2 (u) = (U n u (t n )) N n=1 l 2 ( 2 (Ω)) and e τ, (u) = (U n u (t n )) N n=1 l ( 2 (Ω)), and similarly for te approximations Z n and Qn. Te reference semidiscrete solutions are computed wit = 1/5 and τ = 1/( ). Numerical experiments sow tat te empirical rate for te temporal discretization error is of order O(τ min( 1 2 +α,1) ) and O(τ α ) in te l 2 ( 2 (Ω)) and l ( 2 (Ω))- norms, respectively, cf. Tables 2 3 and 5 6, for cases (a) and (b). Tese results agree well wit te teoretical predictions from Teorems 3.4 and 3.5, and tus fully support te error analysis in Section 3. In Fig. 1, we plot te optimal control q, te state u and te adjoint z. One clearly observes te

18 18 of 22 B. JIN, B. I AND Z. ZHOU weak solution singularity at t = for te state u and at t = T for te adjoint z. Te latter is especially pronounced for case (b). Te weak solution singularity is due to te incompatibility of te source term wit te zero initial/terminal data. Table 3: Pointwise-in-time temporal errors for example (a) wit M = 5. α N rate e τ, (u) 3.47e e-5 2.2e e e-5 1.3e-5.37 (.4).4 e τ, (q) 4.47e e e e e e-4.37 (.4) e τ, (z) 4.47e e e e e e-4.37 (.4) e τ, (u) 5.72e e e e-6 1.9e e-7.6 (.6).6 e τ, (q) 7.64e-5 5.6e e e e e-6.6 (.6) e τ, (z) 7.64e-5 5.6e e e e e-6.6 (.6) e τ, (u) 6.93e e e e-7 7.5e e-8.8 (.8).8 e τ, (q) 9.85e e e e-6 1.7e e-7.8 (.8) e τ, (z) 9.85e e e e-6 1.7e e-7.8 (.8) x.5 t x.5 t x.5 t x.5 t x.5 t x.5 t.1 U n Q n Z n FIG. 1: Plot of U n, Qn and Zn for example (a) (top) and (b) (bottom). 4.2 Two-dimensional example Now, we present numerical results of a two-dimensional example. Te domain Ω is taken to be te unit square Ω = (,1) 2. To discretize te problem, we first divide te unit interval (,1) into M equally spaced subintervals so tat Ω is divided into M 2 small squares, and ten obtain a uniform triangulation for te domain Ω by connecting te diagonal of eac small square. We consider te following data: (c) f, u d (x,t) = 5e t x(1 x)y 2 sin(2πy) and u(x,) = x(1 x)y(1 y). In te experiment, we set an active admissible set U ad wit lower bound a =.1 and upper bound b =.1. We take T =.1 and γ = 1, for evaluating bot te spatial and temporal errors (i.e., e, e τ,2 and e τ, ). Note tat in tis example, te initial data v is nonzero, but it is smoot and compatible wit

19 OPTIMA CONTRO PROBEM WITH SUBDIFFUSION CONSTRAINT 19 of 22 Table 4: Spatial errors for example (b) wit N = 1 4. α N rate e (u) 1.86e e e e e e-7 2. (2.).4 e (q) 1.59e e e e e e-7 2. (2.) e (z) 1.78e e e e e e-7 2. (2.) e (u) 1.99e e-5 1.2e e e e-7 2. (2.).6 e (q) 1.66e e-5 1.4e-5 2.6e-6 6.5e e-7 2. (2.) e (z) 1.86e e e e e e-7 2. (2.) e (u) 2.19e e e e e-7 2.1e-7 2. (2.).8 e (q) 1.71e e-5 1.7e e-6 6.7e e-7 2. (2.) e (z) 1.96e e e-5 3.7e e e-7 2. (2.) Table 5: Temporal errors for example (b) wit M = 5. α N rate e τ,2 (u) 1.5e e e e e e-6.79 (.9).4 e τ,2 (q) 9.e e e e-5 1.7e-5 6.1e-6.81 (.9) e τ,2 (z) 9.36e e e e-5 1.8e e-6.82 (.9) e τ,2 (u) 4.67e-5 2.5e e e e-6 1.9e-6.94 (1.).6 e τ,2 (q) 3.66e e-5 1.3e-5 5.4e e e-6.95 (1.) e τ,2 (z) 3.83e-5 2.3e-5 1.7e e e e-6.95 (1.) e τ,2 (u) 2.23e e e e e e-7.98 (1.).8 e τ,2 (q) 1.58e e e e e e-7.97 (1.) e τ,2 (z) 1.78e e-6 4.7e-6 2.4e e e-7.98 (1.) Table 6: Pointwise-in-time temporal errors for example (b) wit M = 5. α N rate e τ, (u) 2.4e e e e-3 1.4e e-4.34 (.4).4 e τ, (q) 2.7e e e-3 1.2e e e-4.37 (.4) e τ, (z) 2.7e e e-3 1.2e e e-4.37 (.4) e τ, (u) 5.11e e e e-4 1.3e e-5.59 (.6).6 e τ, (q) 3.54e e e-4 1.3e e e-5.6 (.6) e τ, (z) 3.54e e e-4 1.3e e e-5.6 (.6) e τ, (u) 6.96e-5 4.3e e e e e-6.8 (.8).8 e τ, (q) 4.6e e e e e e-6.8 (.8) e τ, (z) 4.6e e e e e e-6.8 (.8) te zero Diriclet boundary condition. Tus, one may reformulate te control problem according to Remark 3.3, and te analysis still applies. Te numerical results are given in Tables 7, 8 and 9, wic indicate tat te empirical convergence rates for bot spatial and temporal errors agree well wit te teoretical ones in Teorems 3.3, 3.4 and Conclusions In tis work, we ave developed a complete numerical analysis of a fully discrete sceme for a distributed optimal control problem governed by a subdiffusion equation, wit box constraint on te control variable, and derived nearly sarp pointwise-in-time error estimates for bot space and time discretizations. Tese estimates agree well wit te empirical rates observed in te numerical experiments. Te teoretical and numerical results sow te adverse influence of te fractional derivatives on te convergence rate wen te fractional order α is small.

20 2 of 22 B. JIN, B. I AND Z. ZHOU Table 7: Spatial errors for example (c) wit N = 5. α M rate e (u) 1.67e e-4 1.8e e e (2.).4 e (q) 6.11e e e e e (2.) e (z) 1.92e-3 5.8e e e e (2.) e (u) 1.82e e e e e (2.).6 e (q) 5.51e e e e e (2.) e (z) 1.98e e e e e (2.) e (u) 1.85e e e e e (2.).8 e (q) 4.51e e e e-6 1.8e (2.) e (z) 2.2e e e e e (2.) Table 8: Temporal errors for example (c) wit M = 5. α N rate e τ,2 (u) 3.83e e e e e-6 3.1e-6.78 (.9).4 e τ,2 (q) 3.14e-5 1.8e-5 1.4e-5 6.3e e-6 2.8e-6.77 (.9) e τ,2 (z) 3.14e e-5 1.2e e e e-6.78 (.9) e τ,2 (u) 1.76e e-6 5.6e e-6 1.4e e-7.94 (1.).6 e τ,2 (q) 1.34e e e-6 2.2e-6 1.6e e-7.94 (1.) e τ,2 (z) 1.44e e e e e-6 6.6e-7.94 (1.) e τ,2 (u) 6.49e e e e e e-7.96 (1.).8 e τ,2 (q) 4.85e e e e e e-7.97 (1.) e τ,2 (z) 5.27e e e e e e-7.96 (1.) Table 9: Pointwise-in-time temporal errors for example (c) wit M = 5. α N rate e τ, (u) 3.e e-4 2.8e e e-4 1.6e-4.34 (.4).4 e τ, (q) 1.95e e e e e e-5.28 (.4) e τ, (z) 2.47e-4 2.9e e e e e-5.34 (.4) e τ, (u) 6.47e e e e e e-6.59 (.6).6 e τ, (q) 3.65e e e e e e-6.58 (.6) e τ, (z) 5.45e e e e-5 1.8e e-5.59 (.6) e τ, (u) 8.64e e e e e e-7.8 (.8).8 e τ, (q) 8.7e e e e e e-7.82 (.8) e τ, (z) 7.23e e e e e e-7.8 (.8) A. Proof of emma 2.4 Proof. By Sobolev embedding, W s,p (,1; 2 (Ω)) C([,1]; 2 (Ω)) for s (1/p,1], and tus we can define an interpolation operator Π by Πv(ˆt) = v(1), for ˆt (,1), for any v W s,p (,1; 2 (Ω)). Te operator E = I Π is bounded from W s,p (,1; 2 (Ω)) to 2 (,1; 2 (Ω)): Ev 2 (,1; 2 (Ω)) = (I Π)v 2 (,1; 2 (Ω)) c v W s,p (,1; 2 (Ω)). By te fractional Poincaré inequality (cf. Hurri-Syrjänen & Vääkangas (213)), we ave Ev p (,1; 2 (Ω)) = inf E(v p) p R p (,1; 2 (Ω)) c inf v p W p R s,p (,1; 2 (Ω)) c v W s,p (,1; 2 (Ω)), (A.1)