Finite element method for parabolic optimal control problems with a bilinear state equation

Journal of Computational and Applied Mathematics 367 (2020) 112431

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Finite element method for parabolic optimal control problems with a bilinear state equation Pratibha Shakya, Rajen Kumar Sinha

∗

Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India

article

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Article history: Received 2 May 2018 Received in revised form 14 August 2019 Keywords: Parabolic optimal control problem Bilinear state equation Finite element method Spatially discrete scheme Linearized backward-Euler scheme A priori error estimates

This article studies a finite element discretization of the optimal control problem governed by a parabolic equation in a convex polygonal domain. The control variable enters the state equation as a coefficient and is subject to the pointwise inequality constraints. We adopt optimize-then-discretize strategy to approximate the control problem. Both spatial and temporal discretizations of the state equations are considered and analyzed. The space discretization uses continuous piecewise linear finite elements for the approximation of the state variable and piecewise constant functions for the control variable. A linearized backward Euler scheme is used for the time discretization. We derive a priori error estimate in the L2 (0, T ; L2 (Ω )) norm for the state and control variables for both the spatially discrete and fully-discrete schemes. Numerical experiment is performed to illustrate our theoretical results. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In this paper we derive a priori error estimates for the finite element approximation of parabolic optimal control problem (POCP), where the control variable enters the state equation as a coefficient. We consider the following POCP for the state variable u and the control variable q involving pointwise control constraints: min J(q, u) =

∫ T( 1

q∈Qad

0

2

∥u − ud ∥2L2 (Ω ) +

α 2

) ∥q − qd ∥2L2 (Ω ) ds

(1.1)

subject to the state equation ut − ∆u + qu = f

in Ω × (0, T ]

(1.2)

and the initial and boundary conditions u(x, 0) = u0 (x) u = 0

in Ω ,

(1.3)

on ∂ Ω × (0, T ],

(1.4) ∂u ∂t

where Ω ⊂ R (d = 2 or d = 3) is a convex polygonal domain with boundary ∂ Ω , ut = and 0 ≤ T < ∞. The initial function u0 (x) and the forcing function f (x, t) are assumed to be smooth in their respective domain of definition, α > 0 and Qad denotes the set of admissible control variable defined by d

Qad = {q ∈ L∞ (0, T ; L∞ (Ω )) : qa ≤ q(x, t) ≤ qb a.e. in Ω × (0, T ]}, ∗ Corresponding author. E-mail addresses: [email protected] (P. Shakya), [email protected] (R.K. Sinha). https://doi.org/10.1016/j.cam.2019.112431 0377-0427/© 2019 Elsevier B.V. All rights reserved.

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P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

where the control bounds qa , qb ∈ R fulfill 0 < qa < qb . Further, we assume that ud ∈ H 1 (0, T ; L2 (Ω )), f ∈ H 1 (0, T ; L2 (Ω )), u0 ∈ H01 (Ω ) ∩ H 2 (Ω ) and qd ∈ L2 (0, T ; H 1 (Ω )). Optimal control problems play an important role in many aspects of modern life such as social, economic, scientific and engineering numerical simulations. Efficient numerical methods are critical for successful application of optimal control problems. Numerous researchers have made various contributions for optimal control problems. A systematic introduction of finite element method for optimal control problems can be found in [1,2]. A good number of literature are available concerning a priori error estimates for optimal control problems, see [3–7] for the elliptic case and [8–13] for the parabolic case. In these papers, the control variable enters the state equation either on the source term or on the boundary condition. For the elliptic optimal control problem, the error estimate of order O(h) is shown for the control variable 3 using cellwise constant discretizations of the control variable. An error estimate of order O(h 2 ) is established in [14] when the control variable is discretized by (bi-/tri-) linear H 1 -conforming element. Further, Hinze [15] has introduced a variational discretization concept for optimal control problems. The key feature is not to discretize the control but to implicitly utilize the optimality conditions and the discretization of the state and co-state for the discretization of the control, which improves the order of convergence of the control variable. Recently, Braack et al. [16] have studied the well-posedness of a nonlinear parabolic diffusion–reaction equation and derived the first and second order optimality conditions. The purpose of this article is to investigate finite element approximations of POCP with bilinear state equation. In general, there are two approaches to solve the optimal control problems, namely discretize-then-optimize and optimizethen-discretize. In the discretize-then-optimize approach the state equation is discretized and the optimality system for the finite dimensional optimization problems is derived. In this approach, the discrete gradient is the right direction of descent but it is not clear whether the discrete co-state equation is an appropriate discretization of the continuous co-state equation. On the other hand, in the optimize-then-discretize strategy first necessary optimality conditions are established on the continuous level consisting of the state, co-state and the optimality equations. These equations are then discretized using finite element method. It is well known that these two approaches lead to the same discretization scheme provided the pure Galerkin discretization is used [14]. The optimize-then-discretize approach leads to better asymptotic convergence properties. In this paper, we have used optimize-then-discretize approach to derive the error between the control variable in the continuous case and the discretized case in appropriate norm. As the solution of the state equation depends nonlinearly on the control variable, we cannot guarantee the unique solution of the optimization problem globally. Therefore, we study the errors in the control variable locally. More precisely, this paper generalizes the work of Kröner and Vexler [17] to time dependent parabolic optimal control problem with bilinear state equation with control constraints. While the state variable is approximated by continuous piecewise linear element and the control variable by piecewise constant functions, a linearized fully discrete backward-Euler scheme is used for the temporal discretization. The convergence properties for the state and control variables in L2 (0, T ; L2 (Ω )) norm are established. Finally, numerical experiment is performed to support our theoretical results. The paper is organized as follows. Section 2 presents the precise formulation of the optimization problem (1.1)–(1.4) and discusses optimality conditions and the regularity of optimal solutions. The finite element approximation and the discrete optimality condition are discussed in Section 3. Section 4 is devoted to the error analysis of the spatial discrete scheme for the state and control variables. In Section 5, the temporal discretization scheme based on linearized backward Euler scheme is discussed and error bounds for the state and control variables are obtained. Finally, numerical results are presented in Section 6 to support our theoretical findings. { } We adopt the standard notation W m,p (Ω ) for Sobolev space Ω with ∥·∥W m,p (Ω ) . We set H01 (Ω ) ≡ v ∈ H 1 (Ω ) : v|∂ Ω = 0 and denote W m,2 (Ω ) by H m (Ω ). We denote Lr (0, T ; W m,p (Ω )) as the Banach space of all Lr integrable functions from [0, T ] into W m,p (Ω ) with norm

) 1r

T

(∫ ∥v∥Lr (0,T ;W m,p (Ω )) =

r W m,p (Ω )

∥v∥ 0

ds

for r ∈ [1, ∞) and the standard modification for r = ∞. We shall denote the L2 -inner product by (·, ·) and its norm by ∥·∥, L2 (ΩT ) := L2 (0, T ; L2 (Ω )), L∞ (ΩT ) := L∞ (0, T ; L∞ (Ω )) and X (0, T ) := L2 (0, T ; H01 (Ω )) ∩ H 1 (0, T ; L2 (Ω )). The details can be found in [18]. Throughout this paper C denotes a positive generic constant independent of discretizations parameters. 2. Abstract formulation of POCP In this section we describe the precise setting of the optimization problem under consideration. Further, we recall some basic results on existence, uniqueness, and regularity of optimal solutions as well as optimality conditions. To formulate the parabolic optimal control problem with the cost functional J : Qad × X (0, T ) → R+ 0, the weak form of the parabolic optimal control problem may be stated as follows: min J(q, u) =

q∈Qad

∫ T( 1 0

2

∥u − ud ∥2L2 (Ω ) +

α 2

) ∥q − qd ∥2L2 (Ω ) ds,

(2.1)

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

3

subject to

{

(ut , φ ) + (∇ u, ∇φ ) + (qu, φ ) = (f , φ ), ∀φ ∈ H01 (Ω ), t > 0,

u(·, 0) = u0 , in Ω .

(2.2)

The standard energy arguments yield the following a priori estimates for u. Lemma 2.1. For every q ∈ Qad and u0 ∈ L2 (Ω ), the state equation (2.2) admits a unique solution u ∈ X (0, T ). Then, for 0 < t ≤ T , we have t

∫

2

[

2 1

2

t

∫

∥u(s)∥ ds ≤ C ∥u0 ∥ +

∥u(t)∥ +

] ∥f (s)∥2 ds .

(2.3)

0

0

Moreover, with f = pg in (2.2), where p ∈ L2 (ΩT ) and g ∈ L2 (0, T ; H01 (Ω )), we have

∥u(t)∥2 +

t

∫

∫ t [ ] ∥u(s)∥21 ds ≤ C ∥u0 ∥2 + ∥p(s)∥2 ∥∇ g(s)∥2 ds .

0

(2.4)

0

Proof. Choose φ = u in (2.2) and use the fact (qu, u) =

∫ Ω

qu2 dx ≥ qa

∫ Ω

u2 dx = qa ∥u∥2L2 (Ω ) ≥ 0

to obtain 1 d 2 dt

∥u∥2 + ∥∇ u∥2 ≤ (f , u),

an application of Cauchy–Schwarz inequality and Poincaré inequality yields d dt

∥u∥2 + ∥u∥21 ≤ C ∥f ∥∥u∥1 .

Integrating the above from 0 to t and using standard kick-back argument, the estimate (2.3) follows. A similar argument as above together with Poincaré inequality ∥g ∥ ≤ C ∥∇ g ∥ proves the second estimate (2.4). This completes the proof. □ Lemma 2.1 ensures the existence of a unique control-to-state mapping S : L∞ (ΩT ) → X (0, T ), q ↦ → S(q),

(2.5)

where S(q) is the solution of (2.2). Using the above mapping we introduce the reduced cost functional j : L∞ ( ΩT ) → R + 0, q ↦ → j(q) := J(q, S(q)). Hence, the optimal control problem (2.1)–(2.2) can be equivalently reformulated as min j(q).

q∈Qad

(2.6)

The following lemma guarantees the existence of a solution to (2.1)–(2.2). For a proof, we refer to [19, Page. 236]. Lemma 2.2 (Existence). There exists a solution (q¯ , u¯ ) ∈ Qad × X (0, T ) of problem (2.1)–(2.2). Since we cannot guarantee global unique solution of (2.1)–(2.2), we consider local optimal solutions. Therefore, we use the following standard definition. Definition 2.1 (Local Solution). A control q¯ ∈ Qad is called a local solution of (2.1)–(2.2) if for each fixed t ∈ [0, T ], there exists ϵ > 0 such that for all q ∈ Qad with ∥q − q¯ ∥L2 (ΩT ) < ϵ , we have j(q) ≥ j(q¯ ). The existence of a local solution of the optimal control problem (2.1)–(2.2) is guaranteed by Lemma 2.1. We now need certain differentiation properties of the mappings S and j. Therefore, we recall the following definition of Q-differentiability [17]. Definition 2.2 (Q-differentiability). Let X and Y be Banach spaces, and let Q ⊂ X be a convex set. The mapping F : Q → Y is said to be Q-differentiable in q ∈ Q with respect to Q , if there exists a mapping FQ′ (q) ∈ L(X , Y ) such that for all p ∈ X

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P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

with q + p ∈ Q ,

∥F (q + p) − F (q) − FQ′ (q)(∥p∥X )∥Y ∥p∥X

→ 0 as ∥p∥X → 0.

We shall omit the index Q and write F ′ = FQ′ . The following lemma will prove to be convenient. Lemma 2.3. Let the control to the state operator S : L∞ (ΩT ) → X (0, T ) be defined by (2.5). Then, its derivatives have the following properties for all directions p1 , p2 ∈ L∞ (ΩT ):

• S ′ (q)(p1 ) ∈ X (0, T ) is the solution v of (vt , φ ) + (∇v, ∇φ ) + (qv, φ ) = −(p1 S(q), φ ),

∀φ ∈ H01 (Ω ),

v (·, 0) = 0. • S (q)(p1 , p2 ) ∈ X (0, T ) is the solution w of ′′

(wt , φ ) + (∇w, ∇φ ) + (qw, φ ) = −(p2 S ′ (q)(p1 ), φ ) − (p1 S ′ (q)(p2 ), φ ),

∀φ ∈ H01 (Ω ),

w(·, 0) = 0. Proof. The proof of the lemma follows from the definition of Q -differentiability and simple calculations. □ Using directional derivative of j in q¯ ∈ Qad , we can formulate the necessary optimality condition for a local solution. For a proof, see [19]. Proposition 2.4. Let q¯ be a local solution of (2.1)–(2.2). Then the following inequality j′ (q¯ )(q∗ − q¯ ) ≥ 0,

∀q∗ ∈ Qad

(2.7)

holds. We now introduce the co-state parabolic equation. Find z ∈ X (0, T ) such that

{ −(φ, zt ) + (∇φ, ∇ z) + (φ, qz) = (φ, u − ud ), z(·, T ) = 0,

∀φ ∈ H01 (Ω ), t < T ,

(2.8)

where q ∈ Qad and u = S(q). Then the first directional derivative at q¯ ∈ Qad of the reduced cost functional can be expressed as j′ (q¯ )(q∗ − q¯ ) =

T

∫

{α (q¯ − qd , q∗ − q¯ ) − ((q∗ − q¯ )u, z)} ds, ∀q∗ ∈ Qad .

(2.9)

0

For the solution of the co-state equation (2.8), we can also introduce a control to co-state operator for a given q ∈ Qad . Lemma 2.5. There exists a unique operator Z : L∞ (ΩT ) → X (0, T ), q ↦ → Z (q), where Z (q) is the solution of (2.8). Proof. The proof follows by the same arguments as used in Lemma 2.1.

□

The optimality condition (2.7) may be read as follows: Every local solution q¯ ∈ Qad satisfying (2.7) fulfills q¯ = PQad

(1 α

)

S(q¯ )Z (q¯ ) + qd ,

(2.10)

where PQad denotes the pointwise projection on the admissible set Qad , i.e., PQad : L∞ (ΩT ) → Qad defined as PQad (χ ) := min{qb , max{qa , χ}}, which can be verified by standard arguments, see [13,19,20]. For f˜ ∈ L2 (ΩT ), we assume that ζ and ψ are the solutions of the following forward and backward in time parabolic problems:

⎧ ⎨ψt − ∆ψ + qψ = f˜ in Ω × (0, T ], ψ = 0 on ∂ Ω × (0, T ], ⎩ ψ (·, 0) = 0 in Ω ,

(2.11)

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

5

and

⎧ ⎨−ζt − ∆ζ + qζ = f˜ in Ω × [0, T ), ζ = 0 on ∂ Ω × [0, T ), ⎩ ζ (·, T ) = 0 in Ω .

(2.12)

2.1. Regularity results for optimal solutions This section provides some a priori bounds for the state, co-state and control variables. Lemma 2.6. solution of

{

Assume that f ∈ L2 (ΩT ), u0 ∈ H01 (Ω ) and q ∈ Qad . Let u ∈ L2 (0, T ; H 2 (Ω ) ∩ H01 (Ω )) ∩ H 1 (0, T ; L2 (Ω )) be the

(ut , φ ) + (∇ u, ∇φ ) + (qu, φ ) = (f , φ )

∀φ ∈ H01 (Ω ),

u(·, 0) = u0 .

(2.13)

Then, the following estimate holds:

∥u(t)∥21 +

∫ t{

∫ t ∫ t )] )( } [( ∥f (s)∥2 ds . ∥q(s)∥2 ds ∥u0 ∥21 + ∥us (s)∥2 + ∥u(s)∥22 ds ≤ C 1 +

0

0

0

Proof. Set φ = ut in (2.13) and apply Cauchy–Schwarz inequality to have

∥ut ∥2 +

1 d 2 dt

( ) ∥u∥21 ≤ C ∥f ∥2 + ∥qu∥2 .

Integrate the above from 0 to t to obtain t

∫

∫ t ∫ t ] [ ∥qu∥2 ds . ∥f ∥2 ds + ∥us (s)∥2 ds + ∥u(t)∥21 ≤ C ∥u0 ∥21 +

(2.14)

0

0

0

Use of (2.3) in (2.14) leads to t

∫

∥us (s)∥2 ds + ∥u(t)∥21 ≤ C

[(

t

∫

∥q∥2 ds

1+

0

)(

∥u0 ∥21 +

t

∫

∥f ∥2 ds

)]

.

(2.15)

0

0

Since ∥u∥2 ≤ C ∥∆u∥ = C ∥ut − f + qu∥, an integration from 0 to t together with (2.15) completes the rest of the proof. □ Lemma 2.7. Let q ∈ Qad and f˜ ∈ L2 (ΩT ). For t ≤ T , let ζ (t) ∈ L2 (0, T ; H 2 (Ω ) ∩ H01 (Ω )) ∩ H 1 (0, T ; L2 (Ω )) be the solution of the backward problem

{

−(φ, ζt ) + (∇φ, ∇ζ ) + (φ, qζ ) = (φ, f˜ ), ζ (·, T ) = 0.

∀φ ∈ H01 (Ω ),

t < T,

Then, we have

∥ζ (t)∥2 +

T

∫

∥ζ (s)∥21 ds ≤ C t

T

∫

∥f˜ (s)∥2 ds, t

∥ζ (·, 0)∥ ≤ C ∥f˜ ∥L2 (ΩT ) , and

∥ζ (t)∥21 +

∫ T{

T

∫ } ( ∥ζs (s)∥2 + ∥ζ (s)∥22 ds ≤ C 1 +

t

T

)∫ ∥q(s)∥2 ds

t

∥f˜ (s)∥2 ds. t

Proof. We proceed as in Lemma 2.6 and standard estimation techniques to complete the proof. □ Remark 2.8.

Let q ∈ Qad and p ∈ L∞ (ΩT ). Then, for t ∈ (0, T ], we deduce from Lemmas 2.1 and 2.3 that

∥S ′ (q)(p)∥2L∞ (0,T ;L2 (Ω )) ≤ C ∥p∥2L2 (Ω

T)

∥S ′′ (q)(p, p)∥2L∞ (0,T ;L2 (Ω ))

∫

t

∥S(q)∥2 ds ∫ t ( ) 2 ∥ u ∥ + ∥f (s)∥2 ds , 0 )

0

≤ C ∥p∥2L2 (Ω T 0 ∫ t 2 ′ 2 ≤ C ∥p∥L2 (Ω ) ∥S (q)(p)∥ ds. T

0

6

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

Using formulation (2.10), we obtain the following regularity result. Lemma 2.9 (Regularity). Let (q¯ , u¯ ) be the solution of the optimization problem (2.1)–(2.2) and z¯ = z(q¯ ) be the corresponding co-state. Then there holds u¯ , z¯ ∈ L2 (0, T ; H 2 (Ω ) ∩ H01 (Ω )) ∩ H 1 (0, T ; L2 (Ω )), q¯ ∈ L2 (0, T ; W 1,p (Ω )) ∩ H 1 (0, T ; L2 (Ω )) ∩ L∞ (ΩT ) for any p < ∞ when d = 2 and p ≤ 6 when d = 3. Proof. The regularity of the parabolic equation implies u¯ , z¯ ∈ L2 (0, T ; H 2 (Ω ) ∩ H01 (Ω )) ∩ H 1 (0, T ; L2 (Ω )). The embedding H 2 (Ω ) ↪→ W 1,p (Ω ) and the property (2.10) imply the desired result for q¯ . □ We now formulate explicit representation of some a priori bounds for a local solution and its spatial derivatives which we will be useful for the error estimates. Remark 2.10. Applying Cauchy–Schwarz inequality and using Eq. (2.10) and Remark 2.8, we can estimate a local solution and its derivatives by the data:

∥∇ q¯ ∥L2 (ΩT )

1(

) ∥S(q¯ )∥L2 (ΩT ) ∥Z (q¯ )∥L2 (ΩT ) + ∥qd ∥L2 (ΩT ) , α ) 1( ∥∇ S(q¯ )∥L2 (ΩT ) ∥Z (q¯ )∥L2 (ΩT ) + ∥S(q¯ )∥L2 (ΩT ) ∥∇ Z (q¯ )∥L2 (ΩT ) ≤ α + ∥∇ qd ∥L2 (ΩT ) .

∥¯q∥L2 (ΩT ) ≤

We assume the following second-order sufficient optimality condition. Second-order sufficient optimality condition. Let q¯ fulfill the necessary optimality condition (2.7). Then there exists a constant γ > 0 such that j′′ (q¯ )(p, p) ≥ γ ∥p∥2L2 (Ω ) , T

∀p ∈ L2 (ΩT ).

(2.16)

The next lemma discusses the Lipschitz continuity of S , S ′ and S ′′ with respect to the L2 (ΩT )-norm (see. [16]). Lemma 2.11. Let q, q˜ ∈ Qad and p ∈ L∞ (ΩT ). Then the solution operator S, the directional derivative of the solution operator S ′ and the second order directional derivative of the solution operator S ′′ satisfy the following estimates

∥S(q) − S(q˜ )∥L∞ (0,T ;L2 (Ω )) ≤ C ∥q − q˜ ∥L2 (ΩT ) ,

(2.17)

∥S (q)(p) − S (q˜ )(p)∥L∞ (0,T ;L2 (Ω )) ≤ C ∥p∥L2 (ΩT ) ∥q − q˜ ∥L2 (ΩT ) ,

(2.18)

∥S ′′ (q)(p, p) − S ′′ (q˜ )(p, p)∥L∞ (0,T ;L2 (Ω )) ≤ C ∥p∥2L2 (Ω ) ∥q − q˜ ∥L2 (ΩT )

(2.19)

′

′

and T

with a positive constant C independent of q, q˜ and p. Proof. In order to prove the Lipschitz continuity of the operator S, we note that w := S(q) − S(q˜ ) satisfies the following (wt , φ ) + (∇w, ∇φ ) + (qw, φ ) = ((q˜ − q)S(q˜ ), φ ), ∀φ ∈ H01 (Ω ),

{

w(·, 0) = 0,

An application of Lemma 2.1 yields

∥w∥L∞ (0,T ;L2 (Ω )) ≤ C ∥˜q − q∥L2 (ΩT ) ∥S(q˜ )∥L2 (ΩT ) . By Lemma 2.3, v := S ′ (q)(p) − S ′ (q˜ )(p) satisfies (vt , φ ) + (∇v, ∇φ ) + (qv, φ ) = (p(S(q˜ ) − S(q)), φ ) + ((q˜ − q)S ′ (q˜ )(p), φ ),

{

v (·, 0) = 0. We use Lemma 2.1, Remark 2.8 and (2.17) to obtain (2.18). Similarly, we find that v˜ := S ′′ (q)(p, p) − S ′′ (q˜ )(p, p) satisfies the equation

{

(v˜ t , φ ) + (∇ v˜ , ∇φ ) + (qv˜ , φ ) = −(2pS ′ (q)(p), φ ) + (2pS ′ (q˜ )(p), φ ) + ((q˜ − q)S ′′ (q˜ )(p, p), φ ),

v˜ (·, 0) = 0.

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

7

By Lemma 2.1, we have

( ) ∥˜v ∥L∞ (0,T ;L2 (Ω )) ≤ C ∥p∥L2 (ΩT ) ∥S ′ (q)(p) − S ′ (q˜ )(p)∥L2 (ΩT ) + C ∥˜q − q∥L2 (ΩT ) ∥S ′′ (q˜ )(p, p)∥L2 (ΩT ) , which together with Remark 2.8 and (2.18) leads to the desired estimate (2.19). This completes the proof. □ To prove the coercive property of reduced cost functional in a neighborhood of local solution, we need the following lemma. Lemma ( 2.12.

The second derivative ) of the reduced cost functional fulfills a Lipschitz-condition, i.e., there exists a constant C˜ = C˜ ∥u0 ∥, ∥f ∥L2 (ΩT ) , ∥ud ∥L2 (ΩT ) > 0 such that, for all q, q˜ ∈ Qad and p ∈ L∞ (ΩT ), the following estimate

|j′′ (q)(p, p) − j′′ (q˜ )(p, p)| ≤ C˜ ∥q − q˜ ∥L2 (ΩT ) ∥p∥2L2 (Ω

T)

holds. Proof. Using Eq. (2.9), we obtain j′′ (q)(p, p) =

T

∫

(S ′ (q)(p), S ′ (q)(p)) + (S(q) − ud , S ′′ (q)(p, p)) + α (p, p) ds,

{

}

(2.20)

0

and from which we have j′′ (q)(p, p) − j′′ (q˜ )(p, p) =

∫ T{

(S ′ (q)(p), S ′ (q)(p) − S ′ (q˜ )(p))

0

+ (S ′ (q)(p) − S ′ (q˜ )(p), S ′ (q˜ )(p)) + (S(q) − ud , S ′′ (q)(p, p) − S ′′ (q˜ )(p, p)) } + (S(q) − S(q˜ ), S ′′ (q˜ )(p, p)) ds, and hence,

|j′′ (q)(p, p) − j′′ (q˜ )(p, p)| ≤ ∥S ′ (q)(p)∥L2 (ΩT ) ∥S ′ (q)(p) − S ′ (q˜ )(p)∥L2 (ΩT ) + ∥S ′ (q)(p) − S ′ (q˜ )(p)∥L2 (ΩT ) ∥S ′ (q˜ )(p)∥L2 (ΩT ) + ∥S(q) − ud ∥L2 (ΩT ) ∥S ′′ (q)(p, p) − S ′′ (q˜ )(p, p)∥L2 (ΩT ) + ∥S(q) − S(q˜ )∥L2 (ΩT ) ∥S ′′ (q˜ )(p, p)∥L2 (ΩT ) . The desired assertion follows from the Cauchy–Schwarz inequality, Remark 2.8, Lemmas 2.11 and 2.1. □ Lemma 2.13.

Let q¯ be a local solution of (2.6) and let the sufficient optimality condition (2.16) be true. Then there exists

ϵ > 0 such that j′′ (q)(p, p) ≥

γ 2

∥p∥2L2 (Ω

(2.21)

T)

for all p ∈ L∞ (ΩT ) and q ∈ Qad with ∥q − q¯ ∥L2 (ΩT ) < ϵ . Proof. Using (2.16) and Lemma 2.12, we have j′′ (q)(p, p) = j′′ (q¯ )(p, p) + (j′′ (q)(p, p) − j′′ (q¯ )(p, p))

≥ γ ∥p∥2L2 (Ω ) − C ϵ∥p∥2L2 (Ω ) T T γ ≥ ∥p∥2L2 (Ω ) 2

for ϵ sufficiently small.

T

□

Theorem 2.14. If a control satisfies the necessary and sufficient optimality conditions (2.7) and (2.16), then there are constants ϵ, σ > 0 such that j(q) ≥ j(q¯ ) + σ ∥q − q¯ ∥2L2 (Ω ) , T

for q ∈ Qad and ∥q − q¯ ∥L2 (ΩT ) < ϵ . Proof. Using Q-differentiability, the proof follows by standard arguments, see [19]. □

8

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

3. Spatially discrete optimization problem In this section, we spatially discretize the optimization problem. ¯ = ∪K ∈Th K¯ with h = Let {Th }h>0 be a partition of Ω into a finite number of elements called simplexes, i.e., Ω max{diam(K ) : K ∈ Th }. We assume that {Th }h>0 is quasi-uniform (cf. [21]). Let Vh be the finite dimensional subspace of H01 (Ω ) be given by

¯ ) : vh |K ∈ P1 (K ), ∀K ∈ Th , vh |∂ Ω = 0}, Vh = {vh ∈ C (Ω where P1 (K ) is the space of all polynomial of degree≤ 1. The space Vh satisfies the following approximation property: inf {∥v − vh ∥ + h∥v − vh ∥1 } ≤ Ch2 ∥v∥2 , ∀v ∈ H 2 (Ω ) ∩ H01 (Ω ).

vh ∈Vh

(3.1)

Let Qh be a finite dimensional approximate subspace of Qad consisting of piecewise constant functions and be given by Qh = {ˆqh ∈ Qad : qˆ h |K = constant , ∀K ∈ Th , t ∈ (0, T ]}. The spatially discrete optimization problem is formulated as follows: min J(qh , uh ) =

∫ T( 1

qh ∈Qh

0

2

∥uh − ud ∥2 +

α 2

) ∥qh − qd ∥2 ds

(3.2)

subject to (uh,t , φh ) + (∇ uh , ∇φh ) + (qh uh , φh ) = (f , φh ),

{

∀φh ∈ Vh , t > 0,

uh (·, 0) = uh,0 , in Ω ,

(3.3)

where uh,0 be the suitable approximation of u0 in Vh . Concerning existence, uniqueness and stability results, we have the following lemma. Lemma 3.1.

For every q ∈ Qad and uh,0 ∈ Vh , the state equation

(uh,t (q), φh ) + (∇ uh (q), ∇φh ) + (quh (q), φh ) = (f , φh ),

{

∀φh ∈ Vh , t > 0,

uh (q)(·, 0) = uh,0 , in Ω ,

(3.4)

admits a unique solution uh (q) ∈ H 1 (0, T ; Vh ) and the following stability estimate holds:

∥uh (q)(t)∥2 +

t

∫

∫ t ) ( ∥f (s)∥2 ds . ∥uh (q)∥21 ds ≤ C ∥uh,0 ∥2 + 0

0

Moreover, for p ∈ L2 (ΩT ), g ∈ L2 (0, T ; Vh ) and uh,0 ∈ Vh , let uh (q) ∈ H 1 (0, T ; Vh ) be the unique solution of (uh,t (q), φh ) + (∇ uh (q), ∇φh ) + (quh (q), φh ) = (pgh , φh ),

{

∀φh ∈ Vh , t > 0,

uh (q)(·, 0) = uh,0 , in Ω .

Then the following estimate

∥uh (q)(t)∥2 +

t

∫

∫ t ( ) ∥uh (q)∥21 ds ≤ C ∥uh,0 ∥2 + ∥p(s)∥2 ∥∇ g(s)∥2 ds

0

0

holds. Like continuous case, we introduce the discrete control to state operator: Sh : L∞ (ΩT ) → H 1 (0, T ; Vh ) q ↦ → Sh (q),

(3.5)

where Sh (q) is the solution of (3.4). The operator Sh is well defined by Lemma 3.1. Using this operator we introduce the discrete reduced cost functional jh : L∞ (ΩT ) → R+ 0, q ↦ → jh (q) := J(q, Sh (q)), and reformulate the discrete optimal control problem (3.2)–(3.3) as min jh (qh ).

qh ∈Qh

We now define a discrete local solution as follows.

(3.6)

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

9

Definition 3.1 (Discrete Local Solution). A control q¯ h ∈ Qh is called a discrete local solution of (3.2)–(3.3) if for each fixed t ∈ [0, T ], there exists ϵ > 0 such that for all qh ∈ Qh with ∥q − q¯ h ∥L2 (ΩT ) < ϵ , jh (qh ) ≥ jh (q¯ h ) holds. Analogous to the continuous case, we obtain the existence of a local solution. In the following lemma, we now give the differentiability properties of the operators Sh and jh which can be defined in a similar way as in the continuous case. Lemma 3.2. Let the operator Sh : L∞ (ΩT ) → H 1 (0, T ; Vh ) be defined by (3.5). Then its derivatives have the following properties for all directions p1 , p2 ∈ L∞ (ΩT ): (a) Sh′ (q)(p1 ) ∈ H 1 (0, T ; Vh ) is the solution vh of (vh,t , φh ) + (∇vh , ∇φh ) + (qvh , φh ) = −(p1 Sh (q), φh ), ∀φh ∈ Vh ,

vh (·, 0) = 0. (b) Sh (q)(p1 , p2 ) ∈ H (0, T ; Vh ) is the solution wh of 1

′′

(wh,t , φh ) + (∇wh , ∇φh ) + (qwh , φh ) = −(p2 Sh′ (q)(p1 ), φh ) − (p1 Sh′ (q)(p2 ), φh ), ∀φh ∈ Vh ,

wh (·, 0) = 0. Proof. The proof follows by the definition of Q -differentiability and simple calculations. □ Now, we can formulate the discrete necessary optimality condition for q¯ h ∈ Qh as j′h (q¯ h )(q∗h − q¯ h ) ≥ 0, ∀q∗h ∈ Qh ,

(3.7)

where j′h (q¯ h )(q∗h − q¯ h ) =

∫ T{

} α (q¯ h − qd , q∗h − q¯ h ) − ((q∗h − q¯ h )u¯ h , z¯h ) ds.

(3.8)

0

With u¯ h = Sh (q¯ h ), let z¯h ∈ H 1 (0, T ; Vh ) be the solution of discrete co-state equation given by

{ −(φh , z¯h,t ) + (∇φh , ∇ z¯h ) + (φh , q¯ h z¯h ) = (φh , u¯ h − ud ) ∀φh ∈ Vh , t ≤ T , z¯h (·, T ) = 0.

(3.9)

For the solution of the discrete co-state equation (3.9), we can also introduce a discrete control-to-costate operator as follows: There exists a unique operator Zh : L∞ (ΩT ) → H 1 (0, T ; Vh ), q ↦ → Zh (q), where Zh (q) is the solution of (3.9). 4. Error analysis for the spatially discrete problem This section considers error analysis for the spatially discrete optimization problem and derives error estimates for the state and control variables. We now recall the elliptic projection Rh : H01 (Ω ) ↦ → Vh defined by (∇ (Rh u − u), ∇φh ) = 0, ∀φh ∈ Vh , for u ∈ H01 (Ω ).

(4.1)

Set ρ = Rh u − u. Note that elliptic projection is well defined, and as a consequence of standard elliptic error estimates [22], we have the following error estimates for ρ . Lemma 4.1.

Assume that (3.1) holds. Then, with Rh defined by (4.1), we have

∥ρ∥ + h∥∇ρ∥ ≤ Ch2 ∥u∥2 . We have the following approximation results associated with the L2 -projection, see [21]. Lemma 4.2.

Let Lh : L2 (Ω ) → Vh be the L2 -projection operator defined by

(Lh u − u, φh ) = 0

∀φh ∈ Vh .

Then,

∥Lh u − u∥−1 + h∥Lh u − u∥ ≤ Ch2 ∥u∥1 .

10

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

Now, we estimate the error between the solutions of continuous problem and semi-discrete problem. To achieve this we need to use the duality argument. Thus, we introduce the semi-discrete finite element approximation of the forward and backward problems (2.11) and (2.12) for q ∈ Qad : (ψh,t (q), φh ) + (∇ψh (q), ∇φh ) + (qψh (q), φh ) = (f˜ , φh )

∀φh ∈ Vh ,

{ −(ζh,t (q), φh ) + (∇ζh (q), ∇φh ) + (qζh (q), φh ) = (f˜ , φh ) ζh (q)(·, T ) = 0,

∀φh ∈ Vh ,

{

(4.2)

ψh (q)(·, 0) = 0,

and (4.3)

where ζh (q), ψh (q) ∈ H 1 (0, T ; Vh ). At first, we need to derive the error estimates for ∥ζ − ζh (q)∥L∞ (0,T ;L2 (Ω )) and ∥ψ − ψh (q)∥L∞ (0,T ;L2 (Ω )) , which will play a crucial role in the derivation of main results. Lemma 4.3. estimates:

Let ψ and ψh (q) be the solutions of problem (2.11) and (4.2), respectively. Then, we have the following error

( ) ∥ψ − ψh (q)∥L∞ (0,T ;L2 (Ω )) + ∥ψ − ψh (q)∥L2 (0,T ;H 1 (Ω )) ≤ Ch ∥ψ∥L2 (0,T ;H 2 (Ω )) + ∥ψt ∥L2 (ΩT ) .

(4.4)

Similarly, ζ and ζh (q) be the solutions of (2.12) and (4.3), respectively. Then, we have

( ) ∥ζ − ζh (q)∥L∞ (0,T ;L2 (Ω )) + ∥ζ − ζh (q)∥L2 (0,T ;H 1 (Ω )) ≤ Ch ∥ζ ∥L2 (0,T ;H 2 (Ω )) + ∥ζt ∥L2 (ΩT ) ,

(4.5)

where the constant C depends on the upper and lower bounds of q. Proof. The proof techniques are different from the standard semi-discrete error analysis, which require higher regularity, e.g. [22]. We follow the proof techniques of [23] to derive the error estimate in ∥ψ − ψh (q)∥L∞ (0,T ;L2 (Ω )) -norm. By subtracting (4.2) from the weak form of (2.11), we obtain the orthogonality condition (ψt − ψh,t (q), φh ) + (∇ (ψ − ψh (q)), ∇φh ) + (q(ψ − ψh (q)), φh ) = 0,

∀φh ∈ Vh .

This leads to the relation 1 d

∥ψ − ψh (q)∥2 + ∥∇ (ψ − ψh (q))∥2 + (q(ψ − ψh (q)), ψ − ψh (q)) 2 dt = (ψt − ψh,t (q), ψ − ψh (q)) + (∇ (ψ − ψh (q)), ∇ (ψ − ψh (q))) + (q(ψ − ψh (q)), ψ − ψh (q)) = (ψt − ψh,t (q), ψ − φh ) + (∇ (ψ − ψh (q)), ∇ (ψ − φh )) + (q(ψ − ψh (q)), ψ − φh ),

(4.6)

for every φh ∈ H 1 (0, T ; Vh ). Let Lh ψ be the L2 -projection operator defined in Lemma 4.2. Then, Lh ψ ∈ L2 (0, T ; H01 (Ω )) ∩ H 1 (0, T ; L2 (Ω )). Using repeatedly the definition of L2 -projection, we obtain (ψt − ψh,t , ψ − Lh ψ ) = (ψt , ψ − Lh ψ ) = (ψt − Lh ψt , ψ − Lh ψ ) 1 d = ∥ψ − Lh ψ∥2 . 2 dt

(4.7)

Set φh = Lh ψ in (4.6) and with the help of (4.7) we are led to 1 d

∥ψ − ψh (q)∥2 + ∥∇ (ψ − ψh (q))∥2 + (q(ψ − ψh (q)), ψ − ψh (q))

2 dt 1 d

=

2 dt

∥ψ − Lh ψ∥2 + (∇ (ψ − ψh (q)), ∇ (ψ − Lh ψ )) + (q(ψ − ψh (q)), ψ − Lh ψ ).

Using the Cauchy–Schwarz inequality and the Young’s inequality, we obtain 1 d

∥ψ − ψh (q)∥2 + ∥∇ (ψ − ψh (q))∥2 + (q(ψ − ψh (q)), ψ − ψh (q))

2 dt 1 d

=

2 dt 1

1

1

1

2

2

2

∥ψ − Lh ψ∥2 + ∥∇ (ψ − ψh (q))∥2 + ∥∇ (ψ − Lh ψ )∥2 + ∥q(ψ − ψh (q))∥2

+ ∥ψ − Lh ψ∥2 . 2

Since (q(ψ −ψh (q)), ψ −ψh (q)) ≥ qa ∥ψ −ψh (q)∥2 ≥ 0 and ∥q(ψ −ψh (q))∥2 ≤ q2b ∥ψ −ψh (q)∥2 , a use of Poincaré inequality leads to d dt

∥ψ − ψh (q)∥2 + ∥∇ (ψ − ψh (q))∥2 = C

(d dt

) ∥ψ − Lh ψ∥2 + ∥∇ (ψ − Lh ψ )∥2 + ∥ψ − Lh ψ∥2 .

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

11

Integrate the above with respect to time and using ψ (·, 0) = 0, we obtain

∥ψ (t) − ψh (t)∥ + 2

∫

t

( ∥∇ (ψ (s) − ψh (q)(s))∥2 ds ≤ C ∥ψ (t) − Lh ψ (t)∥2 0 ∫ t ) + {∥∇ (ψ (s) − Lh ψ (s))∥2 + ∥ψ (s) − Lh ψ (s)∥2 }ds . 0

Using ∥∇ (Lh ψ − ψ )∥ ≤ ∥∇ (Rh ψ − ψ )∥ (cf. [23]) and Lemmas 4.1 and 4.2, the desired estimate (4.4) follows. Similarly, estimate (4.5) for the backward problem (4.3) is obtained, and this completes the proof of the lemma. □ 4.1. Some auxiliary error estimates This section is concerned with some auxiliary error estimates for the state and co-state variables. Theorem 4.4. Let q ∈ Qad and u0 ∈ H01 (Ω ). Further, let u ∈ L2 (0, T ; H 2 (Ω ) ∩ H01 (Ω )) ∩ H 1 (0, T ; L2 (Ω )) and uh (q) ∈ H 1 (0, T ; Vh ) be the solutions of

{

(ut , φ ) + (∇ u, ∇φ ) + (qu, φ ) = (f , φ ), ∀φ ∈ H01 (Ω ),

(4.8)

u(·, 0) = u0 , and

{

(uh,t (q), φh ) + (∇ uh (q), ∇φh ) + (quh (q), φh ) = (f , φh ), ∀φh ∈ Vh ,

(4.9)

uh (q)(·, 0) = uh,0 ,

respectively, where uh,0 is a suitable finite dimensional approximation of u0 . Then, we have

( )( ) ∥u − uh (q)∥L2 (ΩT ) ≤ C ∥u0 − uh,0 ∥ + Ch 1 + ∥q∥L2 (ΩT ) ∥u0 ∥ + ∥f ∥L2 (ΩT ) . Proof. Let ζ be the solution of (2.12) with f˜ ∈ L2 (ΩT ). Then, from (2.2) and (4.9), we have

∫ ΩT

(u − uh (q))f˜ dxds =

T

∫ 0

∫ Ω

(u − uh (q))(−ζt − ∆ζ + qζ ) dxds

= −(u, ζt ) + (∇ u, ∇ζ ) + (qu, ζ ) + (uh (q), ζt ) − (∇ uh (q), ∇ζ ) − (uh (q), ζ ) = (ut , ζ ) + (u0 , ζ (0)) + (∇ u, ∇ζ ) + (qu, ζ ) + (uh (q), ζt ) − (∇ uh (q), ∇ζ ) − (uh (q), qζ ) = (u0 , ζ (0)) + (f , ζ ) + (uh (q), ζt ) − (∇ uh (q), ∇ζ ) − (uh (q), qζ ).

(4.10)

Eqs. (4.3) and (2.12) lead to the orthogonality condition

−(ζt − ζh,t (q), φh ) + (∇ (ζ − ζh (q)), ∇φh ) + (q(ζ − ζh (q)), φh ) = 0,

∀φh ∈ Vh .

Use (4.9), (4.11) in (4.10) and ζh (q)(·, T ) = 0 to obtain

∫ ΩT

(u − uh (q))f˜ dxds = (u0 , ζ (0)) + (f , ζ ) + (uh (q), ζh,t (q))

− (∇ uh (q), ∇ζh (q)) − (uh (q), qζh (q)) = (u0 − uh,0 , ζ (0)) + (uh,0 , ζ (0) − ζh (q)(0)) + (f , ζ − ζh (q)) ≤ ∥u0 − uh,0 ∥∥ζ (0)∥ + ∥uh,0 ∥∥ζ − ζh (q)∥L∞ (0,T ;L2 (Ω )) + ∥f ∥L2 (ΩT ) ∥ζ − ζh (q)∥L2 (ΩT ) .

We apply Lemma 4.3 to obtain

∫ ΩT

(u − uh (q))f˜ dxds ≤ C ∥u0 − uh,0 ∥∥f˜ ∥L2 (ΩT )

( )( ) + Ch ∥u0 ∥ + ∥f ∥L2 (ΩT ) ∥ζ ∥L2 (0,T ;H 2 (Ω )) + ∥ζt ∥L2 (ΩT ) .

(4.11)

12

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

Then, Lemma 2.1 and the definition of L2 (ΩT )-norm yield

∥u − uh (q)∥L2 (ΩT ) =

sup

(u − uh (q), f˜ )

∥f˜ ∥L2 (ΩT ) ( )( ) ≤ C ∥u0 − uh,0 ∥ + Ch 1 + ∥q∥L2 (ΩT ) ∥u0 ∥ + ∥f ∥L2 (ΩT ) . f˜ ∈L2 (ΩT ),f˜ ̸ =0

This completes the proof of the theorem. □ Theorem 4.5. Let q ∈ Qad , p ∈ L2 (ΩT ), g1 ∈ L2 (0, T ; H01 (Ω )), g2 ∈ L2 (0, T ; H01 (Ω )) and u0 ∈ H01 (Ω ). Assume that u ∈ L2 (0, T ; H 2 (Ω ) ∩ H01 (Ω )) ∩ H 1 (0, T ; L2 (Ω )) and uh (q) ∈ H 1 (0, T ; Vh ) are the solutions of (ut , φ ) + (∇ u, ∇φ ) + (qu, φ ) = (pg1 , φ ), ∀φ ∈ H01 (Ω ),

{

(4.12)

u(·, 0) = u0 ,

and (uh,t (q), φh ) + (∇ uh (q), ∇φh ) + (quh (q), φh ) = (pg2 , φh ), ∀φh ∈ Vh ,

{

(4.13)

uh (q)(·, 0) = uh,0 ,

respectively. Here, uh,0 is a suitable approximation of u0 onto Vh . Then the following estimates hold true:

∥u − uh (q)∥L2 (ΩT ) + ∥u − uh (q)∥L2 (0,T ;H 1 (Ω )) ≤ ∥u0 − uh,0 ∥ ) 21 ( )( ) (∫ T ∥p∥2 ∥∇ (g2 − g1 )∥2 ds . + Ch 1 + ∥q∥L2 (ΩT ) ∥u0 ∥ + ∥f ∥L2 (ΩT ) + C 0

Proof. Let uˆ h (q) be the solution of (uˆ h,t (q), φh ) + (∇ uˆ h (q), ∇φh ) + (quˆ h (q), φh ) = (pg1 , φh ), ∀φh ∈ Vh ,

{

(4.14)

uˆ h (q)(·, 0) = uh,0 .

From (4.13) and (4.14) with φh = uh (q) − uˆ h (q), we obtain (uh,t (q) − uˆ h,t (q), uh (q) − uˆ h (q)) + (∇ (uh (q) − uˆ h (q)), ∇ (uh (q) − uˆ h (q)))

+ (q(uh (q) − uˆ h (q)), uh (q) − uˆ h (q)) = (p(g2 − g1 ), uh (q) − uˆ h (q)).

(4.15)

Since (q(uh (q) − uˆ h (q)), uh (q) − uˆ h (q)) =

∫ Ω

q(uh (q) − uˆ h (q))2 dx ≥ qa ∥uh (q) − uˆ h (q)∥2 ≥ 0,

we obtain from (4.15) and Poincaré inequality, 1 d 2 dt

∥uh (q) − uˆ h (q)∥2 + ∥∇ (uh (q) − uˆ h (q))∥2 ≤

1 2

1

∥p∥2 ∥∇ (g2 − g1 )∥2 + ∥∇ (uh (q) − uˆ h (q))∥2 . 2

Integration from 0 to t yields

∥uh (q) − uˆ h (q)∥2 +

∫

t

∥∇ (uh (q) − uˆ h (q))∥2 ds ≤ C

t

∫

0

∥p∥2 ∥∇ (g2 − g1 )∥2 ds. 0

The desired estimate follows from the triangle inequality

∥u − uh (q)∥L2 (ΩT ) ≤ ∥u − uˆ h (q)∥L2 (ΩT ) + ∥ˆuh (q) − uh (q)∥L2 (ΩT ) , and Theorem 4.4. This completes the proof. □ Now, we summarize some estimates for the operators S , Sh and their derivatives. Lemma 4.6.

Let q, q˜ ∈ Qad and p ∈ L∞ (ΩT ). Then the following estimates hold:

( ) ∥S ′ (q)(p) − Sh′ (q)(p)∥L2 (ΩT ) ≤ Ch 1 + ∥q∥L2 (ΩT ) ∥S(q)∥L2 (ΩT ) ∥p∥L2 (ΩT ) (∫ T ) 12 +C ∥p∥2 ∥∇ (S(q) − Sh (q))∥2 ds , 0

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

∥S ′′ (q)(p, p) − Sh′′ (q)(p, p)∥L2 (ΩT )

13

(

) ≤ Ch 1 + ∥q∥L2 (ΩT ) ∥S ′ (q)(p)∥L2 (ΩT ) ∥p∥L2 (ΩT ) ) 12 (∫ T +C ∥p∥2 ∥∇ (S ′ (q)(p) − Sh′ (q)(p))∥2 ds , 0

and

∥Sh (q) − Sh (q˜ )∥L∞ (0,T ;L2 (Ω )) ≤ Cˆ ∥q − q˜ ∥L2 (ΩT ) ( ) with Cˆ = C ∥uh,0 ∥, ∥f ∥L2 (ΩT ) . Proof. The proof follows from a simple calculation and Theorem 4.4. □ Similarly, for the operators Z and Zh , we have the following result. Theorem 4.7.

Let q, q˜ ∈ Qad . Then, we have

( )( ) ∥Z (q) − Zh (q)∥L2 (ΩT ) ≤ Ch 1 + ∥q∥L2 (ΩT ) ∥S(q)∥L2 (ΩT ) + ∥ud ∥L2 (ΩT ) + ∥S(q) − Sh (q)∥L2 (ΩT ) . Moreover, the operator Zh satisfies

ˆ ∥Zh (q) − Zh (q˜ )∥L2 (ΩT ) ≤ Cˆ ∥q − q˜ ∥L2 (ΩT ) ( ) ˆ where Cˆ = C ∥uh,0 ∥, ∥f ∥L2 (ΩT ) , ∥ud ∥L2 (ΩT ) .

(4.16)

Proof. Let ψ be the solution of (2.11) with f˜ ∈ L2 (ΩT ). We now proceed as in Theorem 4.4, and use Lemma 4.3 to obtain the desired estimate. This completes the proof. □ 4.2. Discrete coercivity In this section, we derive some auxiliary estimates and we further verify that the second derivative of the discrete reduced cost functional is coercive in a neighborhood of a local solution if the second order optimality condition (2.16) is fulfilled for the continuous problem. Lemma 4.8.

Let q ∈ Qad and p ∈ L∞ (ΩT ). Then

|j′ (q)(p) − j′h (q)(p)| ≤ Cˆ h∥p∥L2 (ΩT ) , ( ) where Cˆ = C ∥ud ∥L2 (ΩT ) , ∥f ∥L2 (ΩT ) , ∥u0 ∥, ∥uh,0 ∥, ∥q∥L2 (ΩT ) . Further, j′h fulfills a Lipschitz condition, i.e., there exists a ( ) constant C˜ = C ∥u0 ∥, ∥f ∥L2 (ΩT ) , ∥ud ∥L2 (ΩT ) > 0, such that for all q, q˜ ∈ Qad and all p ∈ L∞ (ΩT ), we have |j′h (q)(p) − j′h (q˜ )(p)| ≤ C˜ ∥q − q˜ ∥L2 (ΩT ) ∥p∥L2 (ΩT ) . Proof. From (2.9) and (3.8), we have

|j′ (q)(p) − j′h (q)(p)| ≤

∫ T( ) |(p(S(q) − Sh (q)), Z (q))| + |pSh (q), Z (q) − Zh (q)| ds 0

≤ ∥p(S(q) − Sh (q))∥L2 (ΩT ) ∥Z (q)∥L2 (ΩT ) + ∥pSh (q)∥L2 (ΩT ) ∥Z (q) − Zh (q)∥L2 (ΩT ) ( ) ≤ ∥S(q) − Sh (q)∥L2 (ΩT ) ∥Z (q)∥L2 (ΩT ) + ∥Sh (q)∥L2 (ΩT ) ∥Z (q) − Zh (q)∥L2 (ΩT ) ∥p∥L2 (ΩT ) . Using Lemmas 2.7, 3.1, Theorems 4.4 and 4.7, we obtain

|j′ (q)(p) − j′h (q)(p)| ≤ Cˆ h2 ∥p∥L2 (ΩT ) , ( ) where Cˆ = C ∥ud ∥L2 (ΩT ) , ∥f ∥L2 (ΩT ) , ∥u0 ∥, ∥uh,0 ∥, ∥q∥L2 (ΩT ) . This proves the first assertion. The second inequality can be treated as in the proof of Lemma 2.12.

□

The following result shows the coercive property of second derivative of the discrete reduced cost functional in a neighborhood of a local solution.

14

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

Lemma 4.9. Let q¯ be a local solution of (2.6) and Eq. (2.16) be valid. Then there exists an ϵ > 0 such that, for all q ∈ Qad with ∥q − q¯ ∥L2 (ΩT ) < ϵ and all p ∈ L∞ (ΩT ),

γ

j′′h (q)(p, p) ≥

∥p∥2L2 (Ω ) T 4 holds for h sufficiently small.

(4.17)

Proof. From (2.9), we have T

∫

j′ (q)(p) =

(S ′ (q)(p), S(q) − ud ) ds + α 0

(q − qd , p) ds, 0

∫ T(

j′′ (q)(p, p) =

T

∫

)

(S ′ (q)(p), S ′ (q)(p)) + (S ′′ (q)(p, p), S(q) − ud ) + α (p, p) ds.

0

Similarly, for the discrete case, we have jh (q)(p, p) = ′′

∫ T(

)

(Sh′ (q)(p), Sh′ (q)(p)) + (Sh′′ (q)(p, p), Sh (q) − ud ) + α (p, p) ds.

0

Note that

|j′′ (q)(p, p) − j′′h (q)(p, p)| ⏐∫ T{ ⏐ (S ′ (q)(p) − Sh′ (q)(p), S ′ (q)(p)) + (Sh′ (q)(p), S ′ (q)(p) − Sh′ (q)(p)) =⏐ 0 } ⏐ ⏐ + (S ′′ (q)(p, p) − Sh′′ (q)(p, p), S(q) − ud ) + (Sh′′ (q)(p, p), S(q) − Sh (q)) ds⏐ ≤ ∥S ′ (q)(p) − Sh′ (q)(p)∥L2 (ΩT ) ∥S ′ (q)(p)∥L2 (ΩT ) + ∥Sh′ (q)(p)∥L2 (ΩT ) ∥S ′ (q)(p) − Sh′ (q)(p)∥L2 (ΩT ) + ∥S ′′ (q)(p, p) − Sh′′ (q)(p, p)∥L2 (ΩT ) ∥S(q) − ud ∥L2 (ΩT ) + ∥Sh′′ (q)(p, p)∥L2 (ΩT ) ∥S(q) − Sh (q)∥L2 (ΩT ) . Using Lemmas 4.6 and 3.1 together with Theorem 4.4 leads to

γ

∥p∥2L2 (Ω ) , T 4 for h sufficiently small. Therefore, the assertion follows from Lemma 2.13. |j′′ (q)(p, p) − j′′h (q)(p, p)| ≤

□

4.3. Error estimates for the control and state variables This section is devoted to derive the errors in the control and state variables for the spatially-discrete optimization problem. We begin with the formulation of an auxiliary problem: For ϵ > 0, h > 0, min jh (qh ),

(4.18)

qh ∈Uϵh (q¯ )

where Uϵh (q¯ ) is defined by Uϵh (q¯ ) = {qh ∈ Qh : ∥qh − q¯ ∥L2 (ΩT ) ≤ ϵ} ⊂ Qad . In order to prove the existence of the solution of auxiliary problem (4.18) for sufficiently small h, we need the following lemma. For a proof, we refer to [24]. Lemma 4.10.

πh v =

Let πh : L2 (ΩT ) → Qh denote the L2 -projection operator defined by

∫

1

|K |

v (x)dx, x ∈ K , K

for all K ∈ Th and v ∈ L2 (ΩT ). Then πh Qad ⊂ Qh and the estimate

∥πh v − v∥ ≤ ch∥∇v∥ holds for all v ∈

H01 (Ω ).

(4.19)

In addition, for all qh ∈ Qh , v ∈ L (ΩT ), we have 2

(qh , πh v − v ) = 0. Using the above lemma, we have the following existence, uniqueness result. Lemma 4.11. For all ϵ > 0 and h > 0 sufficiently small, the auxiliary problem (4.18) has a solution. Moreover, j′′h satisfies (4.17) for q ∈ Uϵh (q¯ ), p ∈ L∞ (ΩT ) and the auxiliary problem (4.18) admits a unique solution.

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

15

Proof. Let qˆ h = πh q¯ ∈ Qh and for h small enough we have ∥¯q − qˆh ∥L2 (ΩT ) < ϵ . Therefore, qˆh ∈ Uϵh (q¯ ). Hence, Uϵh (q¯ ) is not empty. For the rest of the argument we refer to the standard techniques as in [19]. For uniqueness, let q¯ h , q˜ h ∈ Uϵh (q¯ ) be the global minima of jh on Uϵh (q¯ ) with q˜ h ̸ = q¯ h and jh (q˜ h ) = jh (q¯ h ). Utilizing the necessary optimality condition and the coercivity we obtain for some λ ∈ [0, 1], jh (q˜ h ) = jh (q¯ h ) + j′h (q˜ h − q¯ h ) +

≥ jh (q¯ h ) +

γ 8

1 2

j′′h (λ˜qh + (1 − λ)q¯ h )(q˜ h − q¯ h , q˜ h − q¯ h )

∥˜qh − q¯ h ∥2L2 (Ω ) > 0 T

for h sufficiently small. As a result, we get 0 = jh (q˜ h ) − jh (q¯ h ) ≥

γ 8

∥˜qh − q¯ h ∥2L2 (Ω ) > 0, T

for h sufficiently small and this leads to a contradiction.

□

Lemma 4.12. Let ϵ > 0 be small enough, such that j′′h is coercive on Uϵh for h sufficiently small. Moreover, let q¯ ϵh be a solution of (4.18) with q¯ ϵh → q¯ in L2 (ΩT )-norm. Then q¯ ϵh is a local solution of (3.6) for h sufficiently small. Proof. To prove that q¯ ϵh is a local solution of (3.6) we have to verify that jh (qh ) ≥ jh (q¯ ϵh )

(4.20)

¯ϵ

holds for all qh ∈ Qh with ∥qh − qh ∥L2 (ΩT ) <

ϵ

ϵ

. By the definition of q¯h we know (4.20) only for those qh ∈ Qh with 2 ϵ ¯ ¯ ∥qh − q∥L2 (ΩT ) ≤ ϵ . Let qh ∈ Qh satisfy ∥qh − qh ∥L2 (ΩT ) < 2ϵ . Then we have for h sufficiently small

∥qh − q¯ ∥L2 (ΩT ) ≤ ∥qh − q¯ ϵh ∥L2 (ΩT ) + ∥¯qϵh − q¯ ∥L2 (ΩT ) ≤

ϵ 2

+

ϵ 2

≤ ϵ.

This completes the proof. □ Now, we are in position to show convergence with respect to space discretization parameter h of the error ∥¯qh − q¯ ∥L2 (ΩT ) of solutions of the discretized problem (3.6) and the continuous problem (2.6). Theorem 4.13. Let q¯ be a local solution of (2.6), and let condition (2.16) hold true. Then, for ϵ > 0 and h > 0 small enough, the problem (4.18) has a unique solution denoted by q¯ ϵh and, the following estimate holds

α C Cˆ ∥¯q − q¯ ϵh ∥L2 (ΩT ) ≤ C √ h∥∇ qd ∥L2 (ΩT ) + √ h + h, γ γ γ where Cˆ is defined in Lemma 4.8 and

(

)

C = C α, ∥u0 ∥, ∥f ∥L2 (ΩT ) , ∥ud ∥L2 (ΩT ) .

(4.21)

Proof. Let ϵ > 0 be small enough such that j′′ (q)(p, p) ≥

γ 2

∥p∥2L2 (Ω ) , T

(4.22)

for all q ∈ Qad with ∥q − q¯ ∥L2 (ΩT ) ≤ ϵ and, for sufficiently small h j′′h (qh )(p, p) ≥

γ 4

∥p∥2L2 (Ω ) , T

(4.23)

for all qh ∈ Uϵh = {qh ∈ Qh : ∥qh − q¯ ∥L2 (ΩT ) ≤ ϵ} and p ∈ L∞ (ΩT ), which is possible in view of Lemmas 2.13 and 4.9. With this ϵ , we consider (4.18) and formulate another auxiliary problem min j(qh ),

qh ∈Uϵh (q¯ )

(4.24)

where we only discretize the control variable. For h sufficiently small, (4.18) and (4.24) have unique solutions. This is a consequence of Lemma 4.11. Let q¯ ϵh and qˆ ϵh denote the solutions of (4.18) and (4.24), respectively. To derive an error estimate, we split the error as q¯ (t) − q¯ ϵh (t) = (q¯ (t) − qˆ ϵh (t)) + (qˆ ϵh (t) − q¯ ϵh (t)).

(4.25)

16

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

With ξ = λ¯q + (1 − λ)qˆ ϵh , λ ∈ [0, 1] and h sufficiently small, we have by (4.22)

γ

2

∥¯q − qˆ ϵh ∥2L2 (Ω

T)

≤ j′′ (ξ )(q¯ − qˆ ϵh , q¯ − qˆ ϵh ) = j′ (q¯ )(q¯ − qˆ ϵh ) − j′ (qˆ ϵh )(q¯ − qˆ ϵh ) = j′ (q¯ )(q¯ − qˆ ϵh ) − j′ (qˆ ϵh )(q¯ − πh q¯ ) − j′ (qˆ ϵh )(πh q¯ − qˆ ϵh ).

The necessary optimality conditions imply for h sufficiently small, j′ (q¯ )(q¯ − qˆ ϵh ) ≤ 0 and

− j′ (qˆ ϵh )(πh q¯ − qˆ ϵh ) ≤ 0,

and hence, using the properties of πh and the Young’s inequality, we obtain

γ

∥¯q − qˆ ϵh ∥2L2 (Ω ) ≤ −j′ (qˆ ϵh )(q¯ − πh q¯ ) T ∫ T =− (α (qˆ ϵh − qd ) − S(qˆ ϵh )Z (qˆ ϵh ), q¯ − πh q¯ ) ds 0 ∫ T{ } =− (α (πh qd − qd ), q¯ − πh q¯ ) + (S(qˆ ϵh )Z (qˆ ϵh ) − πh S(qˆ ϵh )Z (qˆ ϵh ), q¯ − πh q¯ ) ds 0 ∫ T{ 2 } α 1 ≤ ∥qd − πh qd ∥2 + ∥S(qˆ ϵh )Z (qˆ ϵh ) − πh S(qˆ ϵh )Z (qˆ ϵh )∥2 + ∥¯q − πh q¯ ∥2 ds. 2

0

2

2

Thus, we have

∥¯q − qˆ ϵh ∥2L2 (Ω ) ≤ T

∫ T{ 2 } α C C C ∥qd − πh qd ∥2 + ∥S(qˆ ϵh )Z (qˆ ϵh ) − πh S(qˆ ϵh )Z (qˆ ϵh )∥2 + ∥¯q − πh q¯ ∥2 ds, γ γ γ 0

an application of (4.19) leads to

∥¯q − qˆ ϵh ∥2L2 (Ω ) ≤ T

∫ T{ 2 } C C α 2 h ∥∇ qd ∥2 + h2 ∥∇ (S(qˆ ϵh )Z (qˆ ϵh ))∥2 + h2 ∥∇ q¯ ∥2 ds. C γ γ γ 0

Now using (2.10), Remark 2.10 together with Lemmas 2.1, 2.6 and 2.7, we estimate the first term of (4.25). To estimate the second term in (4.25), we use the necessary optimality conditions leading to the following relation for all q˜ h ∈ Uϵh (q¯ ) j′h (q¯ ϵh )(q¯ ϵh − q˜ h ) ≤ 0 ≤ j′ (qˆ ϵh )(q˜ h − qˆ ϵh ). Again, with ξ = λ¯qϵh + (1 − λ)qˆ ϵh , λ ∈ [0, 1], and h sufficiently small, we obtain using (4.23)

γ

4

∥¯qϵh − qˆ ϵh ∥2L2 (Ω

T)

≤ j′′h (ξ )(q¯ ϵh − qˆ ϵh , q¯ ϵh − qˆ ϵh ) = j′h (q¯ ϵh )(q¯ ϵh − qˆ ϵh ) − j′h (qˆ ϵh )(q¯ ϵh − qˆ ϵh ) ≤ j′ (qˆ ϵh )(q¯ ϵh − qˆ ϵh ) − j′h (qˆ ϵh )(q¯ ϵh − qˆ ϵh ) ≤ Cˆ h∥¯qϵh − qˆ ϵh ∥L2 (ΩT ) ,

where the last step follows from Lemma 4.8, and this completes the rest of proof.

□

Corollary 4.14. Let q¯ be a local solution of (2.6) and the condition (2.16) holds true. Then, for a h0 > 0, there exists a discrete solution (q¯ h )0 0 and h > 0 small enough, the solution q¯h of (4.18) is a local solution of (3.6). Hence, the assertion follows from Theorem 4.13. □ We are now in a position to estimate the error between the continuous and the discrete state variables. Theorem 4.15. Let u and uh be the solutions of (2.2) and (3.3), respectively. Then, we have

( )( ) ∥u − uh ∥L2 (ΩT ) ≤ ∥u0 − uh,0 ∥ + Ch 1 + ∥q∥L2 (ΩT ) ∥u0 ∥ + ∥f ∥L2 (ΩT ) ( ) + ∥q − qh ∥L2 (ΩT ) ∥uh,0 ∥ + ∥f ∥L2 (ΩT ) .

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

17

Proof. To find the error for state variable, we split the state error as u − uh = (u − uh (q)) + (uh (q) − uh ). Then, by the triangle inequality, we have

∥u − uh ∥L2 (ΩT ) ≤ ∥u − uh (q)∥L2 (ΩT ) + ∥uh (q) − uh ∥L2 (ΩT ) .

(4.26)

The first term of (4.26) is estimated in Theorem 4.4. Now, we need to estimate the second term. From (3.3) and (4.9), we have (uh,t − uh,t (q), φh ) + (∇ (uh − uh (q)), ∇φh ) − ((q − qh )uh , φh )

+ (q(uh − uh (q)), φh ) = 0,

∀φh ∈ Vh .

(4.27)

Choose φh = uh − uh (q) in (4.27) and using (q(uh − uh (q)), uh − uh (q)) ≥ qa ∥uh − uh (q)∥2 ≥ 0, we obtain 1 d 2 dt

∥uh − uh (q)∥2 + ∥∇ (uh − uh (q))∥2 ≤ |((q − qh )uh , uh − uh (q))| 1

≤

2

1

∥(q − qh )uh ∥2 + ∥uh − uh (q)∥2 . 2

Use of Poincaré inequality and ∥∇ (uh − uh (q))∥2 ≥ 0 yields d dt

∥uh − uh (q)∥2 ≤ C ∥(q − qh )uh ∥2 ,

integrating the above and using the stability estimate ∥uh ∥ ≤ C (∥uh,0 ∥ + ∥f ∥L2 (ΩT ) ), we obtain

( ) ∥uh (t) − uh (q)(t)∥ ≤ C ∥q − qh ∥L2 (ΩT ) ∥uh,0 ∥ + ∥f ∥L2 (ΩT ) . Estimate (4.28) together with Theorem 4.4 completes the rest of the proof.

(4.28) □

5. Fully discrete optimization problem This section considers the backward Euler fully discrete approximations of the optimal control problem (1.1)–(1.4) and derives the related error estimates for the control and state variables. Let 0 = t0 < t1 < · · · < tN −1 < tN = T be a partition of [0, T ], and let In := (tn−1 , tn ] with k := tn − tn−1 for n = 1, 2, . . . , N. Now, set ψ n := ψ (x, tn ) and n n−1 ¯ n := ψ − ψ , ∂ψ

k

n = 1, 2, . . . , N .

Let Thn denote the triangulation at time level t = tn . Let Vhn ⊂ H01 (Ω ) and the finite element space for the discrete control is defined by Qd := {q ∈ Qad : q|In ×K = constant , n = 1, 2, . . . , N , K ∈ Th }. The backward Euler fully discrete optimization problem is stated as follows: min

qnh ∈Qd

N ∫ ∑ n=1

tn

J(qnh , unh ) ds = min

qnh ∈Qd

tn−1

N ∫ ∑ n=1

tn

{

} ∥unh − und ∥2 + α∥qnh − qnd ∥2 ds

(5.1)

tn−1

subject to (∂¯ unh , φh ) + (∇ unh , ∇φh ) + (qnh unh , φh ) = (f n , φh ), n ≥ 1, ∀φh ∈ Vhn ,

{

u0h = uh,0 .

(5.2)

The problem (5.1)–(5.2) has a solution if and only if there exists a co-state zhn−1 such that the triplet (unh , qnh , zhn−1 ) satisfies the following optimality conditions: For all φh ∈ Vhn , (∂¯ unh , φh ) + (∇ unh , ∇φh ) + (qnh unh , φh ) = (f n , φh ), n ≥ 1, u0h

= uh,0

−(∂¯ zhn , φh ) + (∇ zhn−1 , ∇φh ) + (qnh−1 zhn−1 , φh ) = (unh − und , φh ), zhN = 0, (α

(qnh

−

qnd )

−

unh zhn−1

,

qnh

− q˜ h ) ≥ 0, ∀˜qh ∈ Qd .

(5.3) (5.4) (5.5) (5.6) (5.7)

Eqs. (5.3)–(5.7) are nonlinear. The state and co-state equations (5.3) and (5.5) admit a unique solution if h and the mesh ratio k/h, are sufficiently small, [22]. The above procedure has a disadvantage that a nonlinear system of algebraic equations has to be solve at each time step due to the presence of qnh unh in (5.3), qnh−1 zhn−1 in (5.5) and unh zhn−1 in (5.7).

18

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

Therefore, we consider a linearized modification of the method by replacing qnh unh by qnh unh−1 in (5.3), qnh−1 zhn−1 by qnh zhn−1

in (5.5) and unh zhn−1 by unh−1 zhn−1 in (5.7). Thus, we obtain the following linearized optimality conditions: For all φh ∈ Vhn , (∂¯ unh , φh ) + (∇ unh , ∇φh ) + (qnh unh−1 , φh ) = (f n , φh ), n ≥ 1,

(5.8)

u0h = uh,0 ,

−(∂¯ , φh ) + (∇ zhn

zhn−1

, ∇φh ) +

(qnh zhn−1

, φh ) = zhN

(unh

−

(5.9) und

, φh ), n ≤ N ,

(5.10)

= 0,

(5.11)

(α (qnh − qnd ) − unh−1 zhn−1 , qnh − q˜ h ) ≥ 0, ∀˜qh ∈ Qd .

(5.12)

Similar to the spatially discrete case we can reformulate the fully discrete optimal control problem as: min jnh (qnh ) := J(qnh , unh ),

(5.13)

qnh ∈Qd

and the expression for (jnh )′ (q)(p) is given by (jnh )′ (q)(p)

=

N ∫ ∑ n=1

(α (q(s) − qd (s)) − unh−1 (q)zhn−1 (q), p) ds,

∀ p ∈ L2 ( ΩT ) .

(5.14)

In

Now, we prove a priori error estimates for the fully discrete scheme. First, we shall derive some intermediate estimates which will be used to prove the main result of this section. For this purpose, we now decompose the errors as follows: unh − u(tn ) = (unh − R¯ h u(tn )) + (R¯ h u(tn ) − u(tn )) =: θ n + ρ n ,

(5.15)

− R¯ h z(tn )) + (R¯ h z(tn ) − z(tn )) =: Θ n + ηn ,

(5.16)

zhn

− z(tn ) =

(zhn

where R¯ h u ∈ Vhn is defined on In as ∫ 1 Rh u(·, s) ds, R¯ h u = R¯ h un = k

n>0

In

with R¯ h u0 = Rh u(·, 0). In the following theorem, we shall investigate the error between the continuous solutions (u, z) and the intermediate solutions (unh (q), zhn−1 (q)) of (∂¯ unh (q), φh ) + (∇ unh (q), ∇φh ) + (q(tn )unh−1 (q), φh ) = (f n , φh ),

(5.17)

·, 0) = uh,0 ,

−(∂¯ zhn (q), φh ) + (∇ zhn−1 (q), ∇φh ) +

unh (q)( (q(tn )zhn−1 (q) zhN (q)(

, φh ) =

(unh (q)

(5.18)

−

und

, φh ),

·, T ) = 0,

(5.19) (5.20)

for all φh ∈ Vhn . Theorem 5.1. Let (u, z) and (unh (q), zhn−1 (q)) be the solutions of (2.2), (2.8) and (5.17)–(5.20), respectively. Then, for sufficiently small h, the mesh ratio k/h, we have for n = 1, 2, . . . , N, N ∫ (∑ n=1

∥unh (q) − u(tn )∥2 ds

) 21

≤ ∥u0 − uh,0 ∥ + C˜˜ (h2 + k),

In

and N ∫ (∑ n=1

∥ In

zhn−1 (q)

2

− z(tn−1 )∥ ds

) 12

≤C

N ∫ (∑ n=1

∥unh (q) − u(tn )∥2 ds

) 21

ˆ + Cˆ (h2 + k),

In

( ) ˜ where C˜ = C ∥u0 ∥2 , ∥u∥L2 (0,T ;H 2 (Ω )) , ∥ut ∥L2 (ΩT ) , ∥ft ∥L2 (ΩT ) , qa , qb and ( ) ˆ Cˆ = C ∥z ∥L2 (0,T ;H 2 (Ω )) , ∥zt ∥L2 (0,T ;H 1 (Ω )) , ∥ud,t ∥L2 (ΩT ) , qa , qb . Proof. In analogy with (5.15), we write unh (q) − u(tn ) = (unh (q) − R¯ h u(tn )) + (R¯ h u(tn ) − u(tn )) =: θ n (q) + ρ n . In view of Lemma 4.1, ρ n is bounded as desired. It remains to bound θ n (q). To estimate θ n (q), we follow the proof techniques from [25]. Let ζ n (q) ∈ Vhn satisfy k−1 (ζ n (q) − ζ n−1 (q), φh ) − (∇ζ n−1 (q), ∇φh ) − (qζ n−1 (q), φh ) = (θ n (q), φh ), ∀φh ∈ Vhn

(5.21)

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

with ζ N (q) = 0. Set φh = θ n (q) in (5.21). Then integrating over In and taking summation over n, we have N ∫ ∑

n

2

∥θ (q)∥ ds = In

n=1

N ∫ { ∑

k−1 (ζ n (q) − ζ n−1 (q), θ n (q)) − (∇ζ n−1 (q), ∇θ n (q))

In

n=1

} − (qζ n−1 (q), θ n (q)) ds N ∫ { ∑ k−1 (ζ n (q) − ζ n−1 (q), unh (q)) − (∇ζ n−1 (q), ∇ unh (q)) = In

n=1

N { } ∑ − (qζ n−1 (q), unh (q)) ds − k−1 (ζ n (q) − ζ n−1 (q), R¯ h u(tn )) n=1 } − (∇ζ n−1 (q), ∇ R¯ h u(tn )) − (qζ n−1 (q), R¯ h u(tn )) ds.

A summation by parts yields N ∫ ∑

∥θ n (q)∥2 ds = −

In

n=1

N ∫ { ∑ n=1

k−1 (ζ n−1 (q), unh (q) − unh−1 (q)) + (∇ζ n−1 (q), ∇ unh (q))

In

N ∫ } [ ] ∑ + (qζ n−1 (q), unh (q)) ds + k−1 (ζ n (q), unh (q)) − (ζ n−1 (q), unh−1 (q)) ds In

n=1

−

N ∫ { ∑

k−1 (ζ n (q) − ζ n−1 (q), R¯ h u(tn )) − (∇ζ n−1 (q), ∇ R¯ h u(tn ))

In

n=1 } − (qζ n−1 (q), R¯ h u(tn )) ds.

Using (5.17), we obtain N ∫ ∑ In

n=1

∥θ n (q)∥2 ds = −(ζ 0 (q), uh,0 ) −

N ∫ ∑

(ζ n−1 (q), f n ) ds −

N ∫ { ∑

In

k−1 (ζ n (q) − ζ n−1 (q), R¯ h u(tn ))

In

n=1 n=1 } − (∇ζ n−1 (q), ∇ R¯ h u(tn )) − (qζ n−1 (q), R¯ h u(tn )) ds N ∫ ∑ = −(ζ 0 (q), uh,0 − R¯ h u0 ) − (ζ n−1 (q), f n ) ds n=1

+

N ∫ { ∑ n=1

k−1 (ζ n−1 (q), R¯ h u(tn ) − R¯ h u(tn−1 )) + (∇ζ n−1 (q), ∇ R¯ h u(tn ))

In

+ (qζ n−1 (q), R¯ h u(tn ))

}

= −(ζ (q), uh,0 − R¯ h u0 ) − 0

ds N ∫ ∑ n=1

+

N ∫ { ∑ n=1

In

(ζ n−1 (q), f n − f ) ds In

k−1 (ζ n−1 (q), R¯ h u(tn ) − u(tn ) − (R¯ h u(tn−1 ) − u(tn−1 )))

In

+ (∇ζ n−1 (q), ∇ (R¯ h u(tn ) − u)) + (qζ n−1 (q), R¯ h u(tn ) − u) where we have subtracted the equation

∫ {

}

k−1 (u(tn ) − u(tn−1 ), φ ) + (∇ u, ∇φ ) + (qu, φ ) − (f , φ ) ds = 0

In

for φ = ζ n−1 (q). Using the fact N ∫ ∑ n=1

∫ In

(∇ζ n−1 (q), ∇ (R¯ h u(tn ) − u)) ds = 0 yields

∥θ n (q)∥2 ds = −(ζ 0 (q), uh,0 − R¯ h u0 ) − In

N ∫ ∑ n=1

+

N ∫ { ∑ n=1

In

(ζ n−1 (q), f n − f ) ds In

k−1 (ζ n−1 (q), ρ n − ρ n−1 ) + (qζ n−1 (q), ρ n )

}

ds,

19

20

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

} + (qζ n−1 (q), u(tn ) − u) ds = −(ζ 0 (q), uh,0 − u0 − (R¯ h u0 − u0 )) −

N ∫ ∑ n=1

N ∫ { ∑

−

n=1

(ζ n−1 (q), f n − f ) ds In

k−1 (ζ n (q) − ζ n−1 (q), ρ n ) − (qζ n−1 (q), ρ n )

In

} − (qζ n−1 (q), u(tn ) − u) ds. Apply the Cauchy–Schwarz inequality to have N ∫ ) 12 ) (∑ ¯ ∥ζ n−1 ∥2 ds ∥θ (q)∥ ds ≤ ∥ζ (q)∥ ∥uh,0 − u0 ∥ + ∥Rh u0 − u0 ∥ +

N ∫ ∑

n

2

(

0

In

n=1

In

n=1

×

N ∫ (∑

∥f n − f ∥2 ds

) 12

In

n=1

(∑ ∫

N

1 2

N ∫ (∑

N ∫ ) 12 (∑

In

n=1

In

n=1

∥qζ n−1 (q)∥2 ds

n=1

) 12

) 21 ∥ρ n ∥2 ds

) }(∑ ∫

In

n=1

+

∥qζ n−1 (q)∥2 ds

k−2 ∥ζ n (q) − ζ n−1 (q)∥2 ds In

n=1

N

+

+

N ∫ {(∑

∥u(tn ) − u∥2 ds

) 12

.

(5.22)

In

Choose φh = ζ n (q) − ζ n−1 (q) in (5.21) and using the Cauchy–Schwarz inequality and the Young’s inequality to obtain k−1 ∥ζ n (q) − ζ n−1 (q)∥2 + ∥∇ζ n−1 (q)∥2 + (qζ n−1 (q), ζ n−1 (q)) ≤ ∥∇ζ n (q)∥2 + k∥θ n (q)∥2 1 + ∥qζ n−1 (q)∥2 . 2 Since qa ∥ζ n−1 (q)∥2 ≥ 0 and ∥qζ n−1 ∥2 ≤ q2b ∥ζ n−1 ∥2 . Use of Poincaré inequality leads to k−1 ∥ζ n (q) − ζ n−1 (q)∥2 + ∥∇ζ n−1 (q)∥2 ≤ ∥∇ζ n (q)∥2 + Ck∥θ n (q)∥2 . Taking summation over n from n = 1 to N and after cancellation the terms to have N ∑

k−1 ∥ζ n (q) − ζ n−1 (q)∥2 + ∥∇ζ 0 (q)∥2 ≤ C

n=1

N ∑

k∥θ n (q)∥2 = C

n=1

N ∫ ∑ n=1

∥θ n (q)∥2 ds.

(5.23)

In

Similarly, set φh = kζ n−1 (q) in (5.21). An application of the Cauchy–Schwarz inequality, the Young’s inequality and taking summation over n from n = 1 to N to have

∥ζ 0 (q)∥2 +

N ∑

k∥∇ζ n−1 (q)∥2 ≤ C

N ∑

n=1

k∥θ n (q)∥2 = C

n=1

N ∫ ∑ n=1

∥θ n (q)∥2 ds.

(5.24)

In

Now, using (5.23) and (5.24) in (5.22) together with Poincaré inequality leads to N ∫ ∑ n=1

×

[ N ( ) (∑ ) 12 ∥θ (q)∥ ds ≤ C ∥ζ 0 (q)∥ ∥uh,0 − u0 ∥ + ∥R¯ h u0 − u0 ∥ + k∥∇ζ n−1 ∥2 n

2

In

n=1

N ∫ (∑ n=1

) 21

∥f n − f ∥2 ds In

(∑ ∫

k−1 ∥ζ n (q) − ζ n−1 (q)∥2

) 21

+

n=1

N

×

+

N {(∑

∥ρ n ∥2 ds

) 12

+

N ∫ (∑

In

N (∑

kq2b ∥∇ζ n−1 (q)∥2

) 12 }

n=1

q2b ∥∇ζ n−1 (q)∥2 ds

N ∫ ) 21 (∑

In

∥u(tn ) − u∥2 ds

) 12 ]

In

n=1 n=1 [(n=1 ) ≤ C ∥uh,0 − u0 ∥ + ∥R¯ h u0 − u0 ∥ + k∥ft ∥L2 (ΩT ) + qb k∥ut ∥L2 (ΩT )

+ (1 + qb )

N ∫ (∑ n=1

∥ρ n ∥2 ds In

N ∫ ) 12 ](∑ n=1

) 21 ∥θ n (q)∥2 ds . In

Now, Lemma 4.1 together with (5.25) yields the first estimate of the theorem.

(5.25)

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

21

To estimate error in the co-state variable, we split the error zhn (q) − z(tn ) = Θ n (q) + ηn . Since the estimate of ηn is

bounded in Lemma 4.1. It remains to bound Θ n (q). Let ψ n (q) ∈ Vhn with ψ 0 (q) = 0 satisfy

k−1 (ψ n (q) − ψ n−1 (q), φh ) + (∇ψ n (q), ∇φh ) + (qψ n−1 (q), φh ) = (Θ n−1 (q), φh ), ∀φh ∈ Vhn .

(5.26)

Set φh = Θ n−1 (q) in (5.26) and proceed as in the case of forward problem to obtain the second estimate. This completes the proof. □ Let j′ (q)(p) and (jnh )′ (q)(p) be given by (2.9) and (5.14), respectively. Then

Lemma 5.2.

|j′ (q)(p) − (jnh )′ (q)(p)| ≤ Cˆ˜ (h2 + k)∥p∥L2 (ΩT ) , where

(

ˆ

C˜ = C ∥u0 − uh,0 ∥, ∥u0 ∥2 , ∥u∥L2 (0,T ;H 2 (Ω )) , ∥ut ∥L2 (0,T ;H 1 (Ω )) , ∥z ∥L2 (0,T ;H 2 (Ω )) ,

) ∥zt ∥L2 (0,T ;H 1 (Ω )) , ∥ud,t ∥L2 (ΩT ) , qa , qb . Proof. From (2.9) and (5.14), we have N ∫ ⏐∑ ⏐ |j′ (q)(p) − (jnh )′ (q)(p)| = ⏐

tn

tn − 1

n=1 N

−

( ) α (q(s) − qd (s)) − u(s)z(s), p ds tn

∑∫

) ⏐ ( ⏐ α (q(s) − qd (s)) − unh−1 (q)(s)zhn−1 (q)(s), p ds⏐

tn−1

n=1 N ∫ ⏐∑ ⏐ =⏐ n=1

tn

) ⏐ ⏐

(

unh−1 (q)zhn−1 (q) − u(s)z(s), p ds⏐.

tn−1

Adding and subtracting the terms we obtain N ∫ ⏐∑ ⏐ |j′ (q)(p) − (jnh )′ (q)(p)| ≤ ⏐ n=1

≤

tn

(

tn−1

[ (∑ N ∫ n=1

N ∫ ) 21 (∑ ) 12 ∥unh−1 (q) − u∥2 ds ∥zhn−1 (q)∥2 ds In

N

+

(∑ ∫ n=1

) ⏐ ⏐

(unh−1 (q) − u(s))zhn−1 (q) + (zhn−1 (q) − z(s))u(s), p ds⏐

In

n=1

In

] ) 21 ∥zhn−1 (q) − z ∥2 ds ∥u∥L2 (ΩT ) ∥p∥L2 (ΩT )

≤ Cˆ˜ (k + h2 )∥p∥L2 (ΩT ) . The last step follows from Lemma 2.1, Theorem 5.1 and the stability estimate N ∫ ∑ n=1

∥zhn−1 (q)∥2 ds ≤ C In

N ∫ ∑ n=1

In

( ) ∥∇ zhn−1 (q)∥2 ds ≤ C ∥unh (q)∥L2 (ΩT ) + ∥und ∥L2 (ΩT ) .

This completes the proof. □ We now proceed to estimate the error between a local solution q¯ of the continuous parabolic optimal control problem (2.6) and a fully-discrete solution q¯ nh of problem (5.13). As in spatially-discrete case, we start with the formulation of an auxiliary problem for ϵ > 0, h > 0: For a given local solution q¯ , associate the following discrete problem: min jnh (qnh ),

qnh ∈Uϵh (q¯ )

(5.27)

where Uϵh (q¯ ) is defined by Uϵh (q¯ ) = {qnh ∈ Qd : ∥qnh − q¯ ∥ ≤ ϵ} ⊂ Qad . Now, we introduce the L2 -projection πd : L2 (ΩT ) → Qd and note that due to piecewise constant discretization, the following property holds true:

πd Qad ⊂ Qd . The existence and uniqueness of the problem (5.27) follows from Lemma 4.11.

22

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

Theorem 5.3. Let q¯ be a local solution of (2.6) and the second order optimality condition (2.16) is valid. Then, we can choose ϵ > 0 and h > 0 small enough, such that (5.27) has a unique solution denoted by q¯ nh,ϵ and the following estimate holds: N ∫ (∑ n=1

) 21 α C Cˆ ∥¯q − q¯ nh,ϵ ∥2 ds ≤ C √ h∥∇ qd ∥L2 (ΩT ) + √ h + h + γ γ γ In

ˆ

C˜

γ

(h2 + k)

ˆ

for h and mesh ratio k/h sufficiently small, where C , Cˆ and C˜ are given by (4.21), Lemmas 4.8 and 5.2. Proof. Let ϵ > 0 be small enough such that j′′ (q)(p, p) ≥

γ 2

∥p∥2L2 (Ω ) ,

(5.28)

T

for all q ∈ Qad with ∥q − q¯ ∥L2 (ΩT ) ≤ ϵ and for h sufficiently small, (jnh )′′ (qnh )(p, p) ≥

N ∫ γ ∑

4

n=1

∥p∥2 ds,

(5.29)

In

for all qnh ∈ Uϵh = {qnh ∈ Qd : ∥qnh − q¯ ∥L2 (ΩT ) ≤ ϵ} and p ∈ L∞ (ΩT ), which is possible in view of Lemmas 2.13 and 4.9. With this ϵ , we consider (5.27) and formulate another auxiliary problem min j(qnh ),

(5.30)

qnh ∈Uϵh (q¯ )

where we only discretize the time and the control variable. For h sufficiently small, (5.27) and (5.30) have unique solutions. This is a consequence of Lemma 4.11 . Let q¯ nh,ϵ and qˆ nh,ϵ be the solutions of (5.27) and (5.30), respectively. To derive an error estimate, we split the error q¯ − q¯ nh,ϵ = (q¯ − qˆ nh,ϵ ) + (qˆ nh,ϵ − q¯ nh,ϵ ). With ξ = λ¯q + (1 − λ)qˆ nh,ϵ , λ ∈ [0, 1] and h sufficiently small, by (5.28), we have n ∫ γ ∑

2

In

n=1

∥¯q − qˆ nh,ϵ ∥2 ds ≤ j′′ (ξ )(q¯ − qˆ nh,ϵ , q¯ − qˆ nh,ϵ ) = j′ (q¯ )(q¯ − qˆ nh,ϵ ) − j′ (qˆ nh,ϵ )(q¯ − qˆ nh,ϵ ) = j′ (q¯ )(q¯ − qˆ nh,ϵ ) − j′ (qˆ nh,ϵ )(q¯ − πd q¯ ) − j′ (qˆ nh,ϵ )(πd q¯ − qˆ nh,ϵ ).

The necessary optimality conditions imply for h sufficiently small j′ (q¯ )(q¯ − qˆ nh,ϵ ) ≤ 0, and − j′ (qˆ nh,ϵ )(πd q¯ − qˆ nh,ϵ ) ≤ 0, and hence, we get with the properties of πd and the Young’s inequality n ∫ γ ∑

2

n=1

In

∥¯q − qˆ nh,ϵ ∥2 ds ≤ −j′ (qˆ nh,ϵ )(q¯ − πd q¯ )

∫ T( ) α (qˆ nh,ϵ − qnd ) − S(qˆ nh,ϵ )Z (qˆ nh,ϵ ), q¯ − πd q¯ ds 0 ∫ T( ) =− (α (πd qnd − qnd ), q¯ − πd q¯ ) + (S(qˆ nh,ϵ )Z (qˆ nh,ϵ ) − πd (S(qˆ nh,ϵ )Z (qˆ nh,ϵ )), q¯ − πd q¯ ) ds 0 ∫ T( 2 ) α 1 ≤ ∥qnd − πd qnd ∥2 + ∥S(qˆ nh,ϵ )Z (qˆ nh,ϵ ) − πd (S(qˆ nh,ϵ )Z (qˆ nh,ϵ ))∥2 + ∥¯q − πd q¯ ∥2 ds. =−

2

0

2

Therefore, we have N ∫ ∑ n=1

In

∥¯q − qˆ nh,ϵ ∥2 ds ≤

∫ T( 2 α C ∥qnd − πd qnd ∥2 + ∥S(qˆ nh,ϵ )Z (qˆ nh,ϵ ) − πd (S(qˆ nh,ϵ )Z (qˆ nh,ϵ ))∥2 C γ γ 0 ) C + ∥¯q − πd q¯ ∥2 ds. γ

(5.31)

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

23

Applying (4.19) we obtain N ∫ ∑ n=1

∫ T( 2 ) α 2 C C h ∥∇ qd ∥2 + h2 ∥∇ (S(qˆ nh,ϵ )Z (qˆ nh,ϵ ))∥2 + h2 ∥∇ q¯ ∥2 ds. ∥ ds ≤ C γ γ γ 0

qnh,ϵ 2

∥¯q − ˆ In

Now altogether (2.10), Remark 2.10, Lemmas 2.1, 2.6 and 2.7 yield the bound for the first term of (5.31). To estimate the second term in (5.31), we use the necessary optimality conditions leading to the following relation for all q˜ nh ∈ Uϵh (q¯ ), (jnh )′ (q¯ nh,ϵ )(q¯ nh,ϵ − q˜ nh ) ≤ 0 ≤ j′ (qˆ nh,ϵ )(q˜ nh − qˆ nh,ϵ ). Hence, we obtain with (4.22) the following estimate for ξ = λ¯qnh,ϵ + (1 − λ)qˆ nh,ϵ with λ ∈ [0, 1] and h sufficiently small: n ∫ γ ∑

4

In

n=1

∥¯qnh,ϵ − qˆ nh,ϵ ∥2 ds ≤ (jnh )′′ (ξ )(q¯ nh,ϵ − qˆ nh,ϵ , q¯ nh,ϵ − qˆ nh,ϵ ) = (jnh )′ (q¯ nh,ϵ )(q¯ nh,ϵ − qˆ nh,ϵ ) − (jnh )′ (qˆ nh,ϵ )(q¯ nh,ϵ − qˆ nh,ϵ ) ≤ j′ (qˆ nh,ϵ )(q¯ nh,ϵ − qˆ nh,ϵ ) − (jnh )′ (qˆ nh,ϵ )(q¯ nh,ϵ − qˆ nh,ϵ ) N ∫ (∑ ) 21 ˆ 2 ˜ ≤ C (h + k) ∥¯qnh,ϵ − qˆ nh,ϵ ∥2 ds , n=1

In

the last step follows from Lemma 5.2 . This completes the proof.

□

Using Lemmas 4.11 and 4.12, we can choose ϵ > 0 small enough such that, for h > 0 sufficiently small, the solution q¯ nh of (5.27) is a local solution of (5.13). As a consequence and Theorem 5.3, we have the following assertion. Corollary 5.4. Let q¯ be a local solution of (2.6) and condition (2.16) holds true. Then for a h0 > 0 there exists a sequence (q¯ nh )0
ˆ ) 21 α C Cˆ C˜ ∥¯qnh − q¯ ∥2 ds ≤ C √ h∥∇ qd ∥L2 (ΩT ) + √ h + h + (h2 + k) γ γ γ γ In

N ∫ (∑ n=1

ˆ for h and mesh ratio k/h sufficiently small, where C , Cˆ and C˜ are given by (4.21), Lemmas 4.8 and 5.2. We now derive the main result for the fully discrete backward-Euler method. Theorem 5.5. Let u and unh be the solutions of (2.2) and (5.8), respectively. Then, for sufficiently small h and the mesh ratio k/h, we have for n = 1, 2, . . . , N, N ∫ (∑

∥unh − u(tn )∥2 ds

) 21

≤ ∥u0 − uh,0 ∥ + C˜˜ (h2 + k)

In

n=1

N ∫ ( )(∑ ) 12 + C ∥u0 ∥ + ∥f ∥L2 (ΩT ) ∥qnh − q∥2 ds . n=1

In

˜

where C˜ is defined in Theorem 5.1. Proof. As before we decompose the error unh − u(tn ) =: θ n + ρ n . In view of Lemma 4.1, ρ n is bounded as desired. To estimate θ n , let ζ ∈ Vhn with ζ N = 0 satisfy k−1 (ζ n − ζ n−1 , φh ) − (∇ζ n−1 , ∇φh ) − (qnh ζ n−1 , φh ) = (θ n , φh ),

∀φh ∈ Vhn .

(5.32)

Set φh = θ n in (5.32) and proceed as in Theorem 5.1, and obtain N ∫ ∑ n=1

∥θ n ∥2 ds = −(ζ 0 , uh,0 − R¯ h u0 ) − In

N ∫ ∑ n=1

+

N ∫ { ∑ n=1

In

(f n − f , ζ n−1 ) ds − In

N ∫ ∑ n=1

k−1 (ζ n − ζ n−1 , ρ n ) ds In

}

(qnh ζ n−1 , ρ n ) + ((qnh − q)u(tn ), ζ n−1 ) + (qζ n−1 , u(tn ) − u) ds.

24

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

Apply the Cauchy–Schwarz inequality to obtain N ∫ ∑ n=1

∥θ n ∥2 ds ≤ ∥ζ 0 ∥∥u0h − R¯ h u0 ∥ +

N ∫ (∑

In

+

n=1

+

n=1 N ∫ ) 21 (∑

k−2 ∥ζ n − ζ n−1 ∥2 ds In

N ∫ (∑ n=1

∥qnh ζ n−1 ∥2 ds

∥ρ n ∥2 ds In

(∑ ∫

n=1

n=1

+

n=1

To estimate the term N ∫ ∑ n=1

∑N

∫

n=1

In

N ∫ ) 12 (∑

In

(∑ ∫

n=1

∥qζ n−1 ∥2 ds

) 21

In

In

) 12

) 21

In

∥(qnh − q)u(tn )∥2 ds

N

∥ρ n ∥2 ds

) 12 ∥ζ n−1 ∥2 ds

In

n=1 N ∫ ) 12 (∑

N

+

N ∫ ) 21 (∑

In

n=1 N ∫ (∑

∥f n − f ∥2 ds

) 12 ∥ζ n−1 ∥2 ds In

k∥ut ∥L2 (ΩT ) .

(5.33)

∥(qnh − q)u(tn )∥2 ds, we use Lemma 2.1 to have

∥(qnh − q)u(tn )∥2 ds ≤ max ∥u(t)∥2 t ∈[0,T ]

In

N ∫ ∑ n=1

∥qnh − q∥2 ds In

N ∫ ( )∑ 2 2 ≤ C ∥u0 ∥ + ∥f ∥L2 (Ω ) ∥qnh − q∥2 ds. T

n=1

In

Now, Lemma 4.1 together with Theorem 5.1, (5.23) and (5.33) completes the rest of the proof. □ Remark 5.6. Observe that Theorem 5.3 provides the error estimate of order O(h + k) for the control variable, where piecewise constant functions are used to approximate the control variable. Since the error in the state variable (see Theorem 5.5) depends on the error in the control variable, we obtain an error estimate of order O(h + k) for the state variable. 6. Numerical experiment In this section, we consider an example problem to validate theoretical results. For computations, we develop computer code using FreeFem++ [26] software. To solve the optimal control problem numerically we have used the projection gradient algorithm [1]. Consider min j(q),

q∈Qad

where j(q) is a convex functional on L∞ (ΩT ) and Qad is a closed subset of L∞ (ΩT ). The iterative scheme reads (i = 0, 1, 2, . . .) : b(qi+ 1 , vh ) = b(qi , vh ) − α˜ i (j′ (qi ), vh ),

{

2

∀vh ∈ Qh ,

qi+1 = PQad (qi+ 1 ),

(6.1)

2

∫T

where b(vh , wh ) = 0 (vh , wh ) ds is a symmetric and positive definite bilinear form, α˜ i is a step size of iteration. The bilinear form b(·, ·) provides a suitable precondition for the projection algorithm. For an acceptable error tolerance Tol, by applying (6.1) to the discretized parabolic optimal control problem (5.1)–(5.2), we have the following algorithm. Algorithm 6.1. Step 1. Solve the following equations:

⎧ ∫ T ⎪ ⎪ b(q , v ) = b(q , v ) − α ˜ ({α (qi − qd ) − zi ui } ds, vh ), qi+ 1 , qi ∈ Qh , ∀vh ∈ Qh , ⎪ i+ 1 h i h i ⎪ 2 2 ⎪ 0 ⎪ ⎪ ⎨1 n n−1 n n n−1 n (ui − ui , φh ) + (∇ ui , ∇φh ) + (qi ui , φh ) = (f , φh ), ∀φh ∈ Vh , uni , uni −1 ∈ Vh , k ⎪ 1 ⎪ ⎪ ⎪ − (zin − zin−1 , φh ) + (∇ zin−1 , ∇φh ) + (qni −1 zin−1 , φh ) = (uni − und , φh ), ∀φh ∈ Vh , ⎪ ⎪ ⎪ ⎩q k = P (q 1 ). i+1 Qad i+ 2

Step 2. Calculate the iterative error : Ei+1 = ∥qi+1 − qi ∥L2 (ΩT )

(6.2)

P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

25

Table 1 The numerical error for state, control and co-state variables. h

∥u − uh ∥L2 (ΩT )

1/4 1/8 1/16 1/32 1/64 1/128

3.09031e−3 9.51835e−4 2.86666e−4 7.92353e−5 2.19264e−5 5.9878e−6

Rate

∥q − qh ∥L2 (ΩT )

1.70 1.73 1.86 1.86 1.87

7.56581e−3 3.89067e−3 1.99197e−3 1.02322e−3 5.1843e−4 2.6125e−4

Rate

∥z − zh ∥L2 (ΩT )

Rate

0.96 0.97 0.96 0.98 0.99

1.00472e−1 3.09797e−2 8.9604e−3 2.6142e−3 7.5532e−4 2.1902e−4

1.70 1.78 1.78 1.79 1.79

Fig. 1. The profiles of the exact(left) and approximation(right) controls at T = 0.1.

Step 3. If Ei+1 ≤ Tol, stop, else go to Step 1. We consider the following parabolic optimal control problem: min

q∈Qad

1

T

∫

2

( ) ∥u − ud ∥2 + α∥q − qd ∥2 ds 0

subject to the state equation ut − ∆u + qu = f , x ∈ Ω , t ∈ (0, T ], with the initial and boundary condition u(·, 0) = u0 (x), x ∈ Ω , u(x, t) = 0,

x ∈ ∂ Ω , t ∈ [0, T ].

Next, we introduce the co-state parabolic equation

−zt − ∆z + qz = u − ud , x ∈ Ω , t ∈ [0, T ), subject to initial and boundary condition z(·, T ) = 0, x ∈ Ω , z(x, t) = 0, x ∈ ∂ Ω , t ∈ [0, T ]. We choose the domain Ω = [0, 1] × [0, 1], T = 0.1 and α = 1. We adopt the same mesh partition for the state and the control variables. The solutions are computed on a series of uniformly triangular meshes with time step size k = .001. Example 6.1.

For this example, the data under testing are as follows:

u = sin(π x)sin(π y)t , z = 2sin(π x)sin(π y)(1 − t), f = sin(π x)sin(π y) + 2π 2 u + qu, ud = −2sin(π x)sin(π y) − 2π 2 z − qz + u, q = max(qa , min(qb , uz)), qd = 0, qa = 0.005 and qb = 0.3. The errors ∥u − uh ∥L2 (ΩT ) , ∥q − qh ∥L2 (ΩT ) and ∥z − zh ∥L2 (ΩT ) obtained on a sequence of uniformly refined triangular meshes are presented in Table 1. To discretize the control problem we use piecewise linear and continuous finite elements for the state and co-state variables whereas the control variable is approximated by piecewise constant functions. To find

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P. Shakya and R.K. Sinha / Journal of Computational and Applied Mathematics 367 (2020) 112431

the errors we use different discretization of the mesh parameter h at the final time T = 0.1. We observe that the errors in the state, co-state and control variables decrease with the decrease of the mesh parameter h (see Table 1). Further, the rate of convergence for the state, control and co-state variables is calculated using the formula Rate ≃

log(Ei∗ /Ei∗+1 ) log(hi /hi+1 )

,

where i corresponds to the spatial partition, and Ei∗ denotes the error norm. The profiles for the exact and approximate controls are shown in Fig. 1. Acknowledgments The authors thank the anonymous referees for their helpful comments which greatly improved the quality of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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