Trust-Region POD using A-Posteriori Error Estimation for Semilinear Parabolic Optimal Control Problems

Universität Konstanz

Trust-Region POD using A-Posteriori Error Estimation for Semilinear Parabolic Optimal

Control Problems

Sabrina Rogg Stefan Trenz Stefan Volkwein

Konstanzer Schriften in Mathematik Nr. 359, März 2017

ISSN 1430-3558

© Fachbereich Mathematik und Statistik Universität Konstanz

Fach D 197, 78457 Konstanz, Germany

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-401106

Trust-Region POD using A-Posteriori Error Estimation for Semilinear Parabolic Optimal Control Problems

Sabrina Rogg, Stefan Trenz, and Stefan Volkwein

AbstractAn optimal control problem governed by a semilinear heat equation is solved using a globalized inexact Newton method. To reduce the computational ef- fort a model order reduction approach based on proper orthogonal decomposition (POD) is applied. Within a trust region framework we guarantee that the reduced- order models are sufficiently accurate by ensuring gradient accuracy. The gradient error is successfully monitored by an a-posteriori error estimate. Numerical results are presented and discussed.

1 Introduction

Optimal control problems for partial differential equations are often hard to tackle numerically because their discretization leads to very large scale optimization prob- lems. Therefore different techniques of model reduction were developed to approx- imate these problems by smaller ones that are tractable with less effort. Recently the application ofreduced-order modelsto linear time varying and nonlinear sys- tems, in particular to nonlinear control systems, has received an increasing amount of attention. The reduced-order approach is based on projecting the dynamical sys- tem onto subspaces consisting of basis elements that contain characteristics of the expected solution. This is in contrast to, e.g., finite element techniques, where the basis elements of the subspaces do not relate to the physical properties of the system that they approximate. Thereduced basis(RB) method, as developed in [26] and [35, 17], is one such reduced-order method, where the basis elements correspond to the dynamics of expected control regimes. Let us refer to the [11, 23, 32, 34] for the successful use of reduced basis method in PDE constrained optimization problems.

CurrentlyProper orthogonal decomposition(POD) is probably the mostly used and Sabrina Rogg, Stefan Trenz and Stefan Volkwein

Universit¨at Konstanz, Fachbereich Mathematik und Statistik, Universit¨atsstraße 10, 78457 Kon- stanz, Germany, e-mail:{Sabrina.Rogg, Stefan.Trenz, Stefan.Volkwein}@uni-konstanz.de

1

most successful model reduction technique for nonlinear optimal control problems, where the basis functions contain information from the solutions of the dynamical system at pre-specified time-instances, so-called snapshots. Due to a possible linear dependence or almost linear dependence the snapshots themselves are not appropri- ate as a basis. Hence a singular value decomposition is carried out and the leading generalized eigenfunctions are chosen as a basis, referred to as the POD basis. POD is successfully used in a variety of fields including fluid dynamics, coherent struc- tures [2, 4] and inverse problems [7]. Moreover in [6] POD is successfully applied to compute reduced-order controllers. The relationship between POD and balancing was considered in [30, 39, 48]. An error analysis for non-linear dynamical systems in finite dimensions were carried out in [37] and a missing point estimation in mod- els described by POD was studied in [5]. Let us also mention that POD and the reduced basis method are successfully combined by variants of the POD greedy algorithm; see [21, 22], for instance.

In this paper we consider an optimal control problem governed by a semilin- ear parabolic equation together with control constraints. For the numerical solution we apply a POD Galerkin approximation. However, to obtain the state data under- lying the POD model, it is necessary to solve once the full state system using a reference control. Consequently, the POD approximations depend on the chosen reference control, so that the choice of a reference control turned out to be essential for the computation of a POD basis for the optimal control problem. To overcome this problem we investigate theTrust-Region POD method[4, 13, 38, 40, 41] as a basis update strategies for improving the POD basis. In this strategy the POD basis is changed in the optimization method in order to ensure convergence and a certain accuracy for the obtained controls. For that reason we apply an a-posteriori error estimation of the POD approximation. Let us also refer to the papers [49] and [36], where the trust-region optimization is efficiently combined with the reduced-basis method for linear elliptic and parabolic problems without control constraints. More- over,optimality system PODas a further basis update strategy has been introduced in [29] and numerically tested in [15, 18, 46].

The paper is organized as follows: In Section 2 the optimal control problem is introduced. First and second-order necessary optimality conditions are studied in Section 3. The finite element Galerkin approximation is explained in Section 4. In Section 5 the reduced-order modeling is described. The a-posteriori error analysis for the POD approximation is presented and illustrated by numerical tests in Sec- tion 6. Finally, the TR-POD method is studied in Section 6, where also numerical experiments are shown.

2 The optimal control problem

Suppose thatΩ⊂R^d,d∈ {1,2,3}, is a bounded domain with Lipschitz-continuous boundary Γ =∂ Ω. For T >0 we set Q= (0,T)×Ω andΣ = (0,T)×Γ. Let b₁, . . . ,b_Nu ∈L^∞(Ω) be given nonnegative shape functions with N_u∈N. We set

V =H¹(Ω)andH=L²(Ω), whereV is endowed with the standard inner product hϕ,ϕi˜ _V =

Z

Ω

ϕϕ˜+∇ϕ·∇ϕ˜dx forϕ,ϕ˜∈V.

For the definition of Sobolev spaces we refer, e.g., to [1, 12]. The Bochner space L²(0,T;V)denotes the space of (equivalence classes) of measurable abstract func- tionsϕ:[0,T]→V, which are square integrable, i.e.,

Z _T 0

kϕ(t)k_V²dt<∞.

Whentis fixed, the expressionϕ(t)stands for the functionϕ(t,·)considered as a function inΩ only. Recall that

W(0,T) =

ϕ∈L²(0,T;V):|ϕt∈L²(0,T;V⁰)

is a Hilbert space supplied with its common inner product. In particular, we have Z T

0

hϕ_t(t),χ(t)i_V0,V+hχ_t(t),ϕ(t)i_V0,Vdt= Z T

0

d

dthϕ(t),χ(t)i_Hdt

=hϕ(T),χ(T)i_H− hϕ(0),χ(0)i_H

for everyϕ,χ∈W(0,T), whereh·,·i_V0,Vdenotes the dual pairing betweenV and its dual spaceV⁰. For the details we refer the reader to [10, pp. 472-479], for instance.

Let us consider the following optimal control problem:

min

(y,u)J(y,u) =1

2ky(T)−y_Ωk²_H+κ

2kuk²_U (1a)

subject to the semilinear evolution problem c_py_t(t,x)−∆y(t,x) +c_ny(t,x)³=

Nu

k=1

∑

u_k(t)b_k(x) +f(t,x), (t,x)∈Q,

∂y

∂n(t,s) =0, (t,s)∈Σ, y(0,x) =y◦(x), x∈Ω,

(1b)

and to bilateral control constraints u∈U_ad=

˜

u∈U:u_a≤u˜≤u_balmost everywhere (a.e.) in[0,T] (1c) withU=L²(0,T;R^Nu). In (1a) we assume that the desired states satisfyy_Ω∈L^∞(Ω) andκ>0. In (1b) let f ∈L^r(Q)withr>d/2+1,y◦∈L^∞(Ω)andc_p>0,c_n>0.

In (1c),u_a= (u_ai)1≤i≤N_u,u_b= (u_bi)1≤i≤N_u∈L^∞(0,T;R^Nu)satisfyu_a(t)≤u_b(t)for almost all (f.a.a.)t∈[0,T]and ’≤’ is interpreted componentwise. In particular, we haveU_ad⊂L^∞(0,T;R^Nu).

The functiony∈Y =W(0,T)∩L^∞(Q)is called a weak solution to (1b) provided y(0) =y◦holds inHand

c_p d

dthy(t),ϕi_H+ Z

Ω

∇y(t)·∇ϕ+c_ny(t)³ϕdx

= Z

Ω

f(t) +

Nu

∑

i=1

u_i(t)b_i

ϕdx for allϕ∈V and f.a.a.t∈(0,T].

(2)

It follows from [43, Theorem 5.5] that for any controlu∈U_adthere exists a unique statey=y(u)∈Y solving (2). Hence, we can introduce thereduced cost functional by

J(u) =ˆ J(y(u),u) foru∈U_ad.

Instead of (1) we consider now thereduced optimal control problem

min ˆJ(u) subject to (s.t.) u∈U_ad. (P)ˆ The next result is proved in [43, Theorem 5.7], for instance.

Theorem 1.Problem(P)ˆ admits a (global) solution.

Sincec_n>0 holds, (P) is a nonconvex programming problem. Therefore, differentˆ local minima might occur. Numerical methods will deliver a local minimum close to their starting point. Throughout this paper we will assume that a fixed reference solution ¯u∈U_ad is given satisfying first-and second-order optimality conditions (ensuring local optimality of the solution).

3 First- and second-order optimality conditions

Using well-known arguments as in [43, Sections 5.5 and 5.7] it can be shown that the mappingU_ad3u7→J(u)ˆ is twice continuously Fr´echet-differentiable and the second derivative is locally Lipschitz-continuous. Moreover, for any u∈U_ad the reduced gradient∇J(u)ˆ ∈Ucan be expressed by

∇J(u) =ˆ κu(·) +

Z

Ω

p(·,x)b_i(x)dx

1≤i≤N_u a.e. in[0,T], where the adjoint variablep=p(t,x)solvesp(T) =y(T)−yΩ inHand

−c_p d

dthp(t),ϕi_H+ Z

Ω

∇p(t)·∇ϕ+3c_ny(t)²p(t)ϕdx=0 for allϕ∈Vand f.a.a.t∈[0,T).

(3)

Further,y=y(u)denotes the solution to (2). For the second derivative at a point u∈Uwe derive that

∇²J(u)uˆ ^δ =

κu^δ(·) + Z

Ω

p^δ(·,x)b_i(x)dx

1≤i≤Nu

a.e. in[0,T] for everyu^δ = (u^δ₁, . . . ,u^δ_N

u)∈U. Here the statey=y(u)is the solution to (2), the dualp=p(u)solves (3), the linearized dualp^δ =p^δ(u^δ)satisfies

−c_p d

dthp^δ(t),ϕi_H+ Z

Ω

∇p^δ(t)·∇ϕ+3c_ny(t)²p^δ(t)ϕdx

= Z

Ω

1−6c_ny(t)p(t)

y^δ(t)ϕdt for allϕ∈Vand f.a.a.t∈[0,T) together withp^δ(T) =y^δ(T)inHand the linearized statey^δ =y^δ(u^δ)solves

c_p d

dthy^δ(t),ϕi_H+ Z

Ω

∇y^δ(t)·∇ϕ+3c_ny(t)²y^δ(t)ϕdx

= Z

Ω

f(t) +

Nu

∑

i=1

u^δ_i(t)b_i

ϕdx for allϕ∈Vand f.a.a.t∈(0,T] withy^δ(0) =0 inH.

Theorem 2 (First-order conditions). Suppose thatu¯∈U_ad is a local solution to (P)ˆ and y¯=y(u)¯ denotes the associated state solving (2) for u=u. Then, there¯ exists a Lagrange multiplier p¯∈Y satisfying(3)for y=y. Moreover, we have the¯ variational inequality

Nu

∑

i=1 Z T

0

κu¯_i(t) +

Z

Ω

p(t,¯ x)b_i(x)dx

u_i(t)−u¯_i(t)

dt≥0 for all u∈U_ad. (4)

Proof.The claim is shown in [43, Theorem 5.12].

Suppose that we have computed a solution to the variational inequality (4). The next theorem gives sufficient conditions that we have found a local minimum to (2). For a proof we refer to [43, Theorem 5.17].

Theorem 3 (Second-order conditions).Letu¯∈U_adbe an admissible control solv- ing together with the associated statey¯6=0and dualp¯6=0the variational inequality (4). Assume that there exist positive constantsγandτsuch that the hessian∇²J(ˆu)¯ satisfies the second-order sufficient optimality condition

h∇²J(ˆu)u¯ ^δ,u^δi_U≥γku^δk_U² (5) for every u^δ = (u^δ₁, . . . ,u^δ_N

u)∈U belonging to theτ-critical cone, i.e., u^δ satisfies

u^δ_i(t)







=0 if t∈A^τi,

≥0 ifu¯i(t) =uai(t)and t6∈A^τ_i,

≤0 ifu¯_i(t) =u_bi(t)and t6∈A^τi







for i=1, . . . ,N_u,

where now for i=1, . . . ,N_uthe sets of strongly active constraintsA^τ_i are given by A^τ_i =n

t∈[0,T]: κu¯_i(t) +

Z

Ω

¯

p(t,x)b_i(x)dx >τ

o .

Then, there are positive constantsεandσsuch that the quadratic growth condition J(u)ˆ ≥J(ˆu) +¯ σku−uk¯ _U² for all u∈U_adwithku−uk¯ _L∞(0,T;R^Nu)≤ε holds. In particular,u is a strict local minimum of¯ (P).ˆ

Remark 1.Condition (5) can be ensured provided the function Q3(t,x)7→1− 6c_ny(t,¯ x)p(t,¯ x)is nonnegative f.a.a.(t,x)∈Q; see [43, Sections 5.5 and 5.7] and [14, Section 2.3], for instance. This can be ensured if the residual termsαQky¯− y_Qk_Lr(Q)andα_Ωky(T¯ )−y_Ωk_L∞(Ω)are small enough; cf. [45, Proposition 9]. In this case we are able to boundkpk¯ _L∞(Q)by 1/(6c_nkyk¯ _L∞(Q)). This implies that 1−6 ¯yp¯≥

0 holds inQa.e. ♦

4 The finite element (FE) Galerkin approximation

The FE space.ForN_x∈Nthe functionsϕ1, . . . ,ϕNxdenoteN_xlinearly independent nodal piecewise linear finite element (FE) ansatz functions. Then, we define the N_x-dimensional subspace

V^h=span

ϕ1, . . . ,ϕNx ⊂V

endowed with the topology inV. Moreover, we introduce the FE projectionP^h: H→V^h as follows: For anyw∈H the elementw^h=P^hw∈V^his given as the solution to the linear system

hw^h,ϕ_i^hi_H=hw,ϕ_i^hi_H fori=1, . . . ,Nx. (6) Letw^h=∑^N_j=1^x w^h_jϕ^h_j ∈V^h. Then, the coefficient vector w^h= (w^h_i)1≤i≤N_x∈R^Nx is uniquely determined as the solution to the linear system

M^hw^h=b^h

with the mass matrix M^h= ((hϕ^h_j,ϕ_i^hi_H))∈R^Nx×Nx and the right-hand side b^h= (hw,ϕ_i^hi_H)1≤i≤N_x∈R^Nx. From (6) we infer that

hP^hy◦,ϕ^hi_H=hy◦,ϕ^hi_H for allϕ^h∈V^h. Thus,

kP^hy◦k²_H=hP^hy◦,P^hy◦i_H=hP^hy◦,y◦i_H≤ kP^hy◦k_Hky◦k_H

which implieskP^hy◦k ≤ ky◦k_H.

FE approximation for(P).ˆ First we apply a standard Galerkin scheme for the state equation. Thus, we look for a function

y^h(t) =

Nx

∑

y^h_i(t)ϕ_i∈V^h f.a.a.t∈[0,T]

satisfying the initial conditiony^h(0) =P^hy◦∈V^hand the variational equation c_p d

dthy^h(t),ϕ^hi_H+ Z

Ω

∇y^h(t)·∇ϕ^h+c_ny^h(t)³ϕ^hdx

= Z

Ω

f(t) +

Nu

i=1

∑

u_i(t)b_i

ϕ^hdx for allϕ^h∈V^hand f.a.a.t∈(0,T].

(7)

Lemma 1.Let y◦∈H, f ∈L²(0,T;H), u∈U and b₁, . . . ,b_Nu ∈L^∞(Ω). Then a solution y^hto(7)satisfies

ky^hk²_L∞(0,T;H)+ky^hk²_L2(0,T;V)≤C ky◦k²_H+kfk²_L2(0,T;H)+kuk_U²

(8) with a constant C>0.

Proof.Choosingϕ^h=y^h(t)∈V^hf.a.a.t∈[0,T]and using c_p

Z

Ω

y^h(t)⁴dx≥0 f.a.a.t∈[0,T], we obtain from (7)

c_p 2

d

dtky^h(t)k²_H+ Z

Ω

|∇y^h(t)|²dx≤ kf(t)k_H+√

c_b|u(t)|₂

ky^h(t)k_H

≤ kf(t)k²_H+c_b|u(t)|²₂+1

2ky^h(t)k²_H, (9)

where| · |₂stands for the Euclidean norm (here inR^Nu) andc_b=∑^N_i=1ⁿ kb_ik²_H. By the Gronwall lemma [10, p. 559] it follows that

ky^h(t)k²_H≤ e^t c_p

ky^h(0)k²_H+ Z _t

0

kf(s)k²_H+c_b|u(s)|²₂ds

≤ e^t

c_p kP^hy◦k²_H+kfk²_L2(0,T;H)+c_bkuk_U²

≤c_H ky◦k²_H+kfk²_L2(0,T;H)+kuk²_U

(10)

for the time-independent constantc_H=e^Tmax{1,c_b}/c_p. Integrating (9) over[0,T] and using (10) we find that

ky^hk²_L2(0,T;V)≤cp

2ky^h(0)k²_H+kfk²_L2(0,T;H)+c_bkuk_U²+3

2kyk²_L2(0,T;H)

≤c_V ky◦k²_H+kfk²_L2(0,T;H)+kuk²_U

withcV =max{c_p/2,1,c_b}+3c_HT/2. Now the claim follows with the constant

C=c_H+c_V.

Remark 2.By using Lemma 1 it can be shown by standard arguments that (7) admits a unique solution which even belongs toH¹(0,T;V^h),→W(0,T). Throughout we

also assume thaty^h∈L^∞(Q)holds. ♦

We suppose that (7) admits a unique FE solutiony^h=y^h(u)∈Y for anyu∈U_ad. Thus, we define the FE discretization of the reduced functional ˆJby

Jˆ^h(u) =J(y^h(u),u) foru∈U_ad. The FE approximation for (P) reads as follows:ˆ

min ˆJ^h(u) s.t. u∈U_ad. (Pˆ^h) The FE optimality conditions.To characterize a local solution to (Pˆ^h) we introduce the FE adjoint statep^h(t) =∑^N_i=1^x p_i(t)ϕ_i∈V^hwhich solvesp^h(T) =y^h(T)−P^hy_Ω together with

−c_p d

dthp^h(t),ϕi_H+ Z

Ω

∇p^h(t)·∇ϕ+3c_ny^h(t)²p^h(t)ϕdx

= Z

Ω

y^h(t)−y_Q(t)

ϕdx for allϕ∈V^hand f.a.a.t∈[0,T).

(11)

Hence, the FE reduced gradient is given by

∇Jˆ^h(u) = κu(·) +

Z

Ω

p^h(·,x)b_i(x)dx

1≤i≤N_u a.e. in[0,T].

Consequently, the FE approximation for the variational inequality (4) is given as

Nu

∑

i=1 Z T

0

κu¯^h_i(t) + Z

Ω

¯

p^h(t,x)b_i(x)dx

u_i(t)−u¯^h_i(t)

dt≥0 (12)

for all u∈U_ad, where ¯u^h= (u¯^h_i)1≤i≤N_u ∈U_ad is a local solution to (Pˆ^h) and ¯p^h solves (11) for the optimal FE state ¯y^h=y^h(u¯^h). Analogously to Theorem 3 the second-order sufficient optimality condition reads: There exist positive constantsγ andτsuch that the hessian∇²Jˆ^h(u¯^h)satisfies the second-order sufficient optimality condition

h∇²Jˆ^h(u¯^h)u^δ,u^δi_U≥γku^δk²_U for everyu^δ = (u^δ₁, . . . ,u^δ_N

u)∈Ubelonging to theτ-critical cone, i.e.,u^δ satisfies

u^δ_i(t)











=0 ift∈A^h,τ_i ,

≥0 if ¯u^h_i(t) =u_ai(t)andt6∈A^h,τ_i ,

≤0 if ¯u^h_i(t) =u_bi(t)andt6∈A^h,τ_i











fori=1, . . . ,N_u,

where fori=1, . . . ,N_uthe sets of strongly active constraintsA^h,τ_i are given by A^h,τ_i =n

t∈[0,T]:

κu¯^h_i(t) + Z

Ω

¯

p^h(t,x)b_i(x)dx >τ

o .

5 Reduced order modelling

Computation of the POD basis.Suppose that for an admissibleu∈U_adthe trajec- toriesy^h(t),p^h(t)∈V^h,t∈[0,T], are the FE solutions to (7) and (11), respectively.

We introduce the snapshot space V^h=span

y^h(t),p^h(t):t∈[0,T] ⊂V^h⊂V

andd^h=dimV^h≤N_x. Notice that we do not include the time derivatives into the snapshot space; cf. [20, 28]. For any`∈ {1, . . . ,d^h}we construct a low-dimensional orthonormal basis by solving the optimization problem

min Z T

0

y^h(t)−

`

∑

i=1

hy^h(t),ψ_i^hi_Vψ_i^h

2 V+

p^h(t)−

`

∑

i=1

hp^h(t),ψ_i^hi_Vψ_i^h

2 Vdt s.t.{ψ_i^h}^`_i=1⊂V^handhψ_i^h,ψ^h_ji

X=δ_{i j}for 1≤i,j≤`.

(13)

The solution to (13) is presented in [20, 24], for instance. Let us define the linear, bounded, finite-rank, nonnegative and selfadjoint operatorR^h:V →V^hby

R^hψ= Z T

0

hy^h(t),ψi_Vy^h(t) +hp^h(t),ψi_Vp^h(t)dt forψ∈V.

Now the solution to (13) is given by the eigenvectors corresponding to the`largest (positive) eigenvaluesλ₁^h≥. . .≥λ^h

d^h>0 solving the symmetric eigenvalue problem R^hψ_i^h=λ_i^hψ_i^h fori=1, . . . ,d^h.

We can quantify the POD approximation error as follows Z T

0

y^h(t)−

` i=1

∑

hy^h(t),ψ_i^hi_Vψ_i^h

2 V

+ p^h(t)−

` i=1

∑

hp^h(t),ψ_i^hi_Vψ_i^h

2 Vdt=

d^h

∑

i=`+1

λ_i^h.

Next, we introduce a POD Galerkin scheme for (1b). Suppose that we have deter- mined the POD basis{ψ_i^h}<sup>`</sup><sub>i=1</sub>⊂V<sup>h</sup>of rank`N_x. We define

V^h,`=span

ψ₁^h, . . . ,ψ_`^h ⊂V^h⊂V

endowed with the topology inV. We introduce the projection operator P^{h,`</sup>:H→V<sup>h,`}, v^{h,`</sup>=P<sup>h,`}ϕsolves min

w^{h,`</sup>∈V<sup>h,`}

kϕ−w^{h,`</sup>k<sub>H</sub>forϕ∈H. (14) Lemma 2.The projectionP<sup>h,`}is given as

P^h,`ϕ=

` i=1

∑

ϕ^{h,`</sup><sub>i</sub> ψi forϕ∈H, (15) where the vectorϕ<sup>h,`}= (ϕ^{h,`</sup><sub>1</sub> , . . . ,ϕ<sup>h,`}<sub>`</sub> )<sup>></sup>∈R<sup>`</sup>solves the linear system

`

∑

hψ_j,ψii_Hϕ<sup>h,`</sup><sub>j</sub> =hϕ,ψ<sub>i</sub>i<sub>H</sub> for1≤i≤`. (16)

In particular,P^h,`is linear, continuous and orthogonal in H.

Proof.The first-order optimality conditions for (14) reads: for anyϕ∈Hthe element ϕ^{h,`</sup>=P<sup>h,`}ϕ∈V^h,`solves

hϕ^{h,`</sup>,ψ<sub>i</sub><sup>h</sup>i<sub>H</sub>=hϕ,ψ<sub>i</sub><sup>h</sup>i<sub>H</sub> for 1≤i≤`. (17) Writing ϕ^{h,`</sup>in the form ϕ<sup>h,`}=∑^{`</sup><sub>j=1</sub>ϕ<sup>h,`}_j ψ^h_j we derive from (17) that the vector ϕ^{h,`</sup>= (ϕ<sup>h,`}_i )<sub>1≤i≤`</sub>satisfies (16). Thus, we have shown (15). Utilizing (16) and (17) the continuity ofP<sup>h,`}follows from

kP^h,`ϕk²_H=

` i=1

∑

ϕ^h,`_i ψ_i^h

2

H

=

` i=1

∑

`

∑

ϕ^{h,`</sup><sub>i</sub> ϕ<sup>h,`}_j hψ_i^h,ψ^h_ji

H

=

`

∑

j=1

ϕ<sup>h,`</sup><sub>j</sub> `

∑

i=1

ϕ^h,`_i hψ_i^h,ψ^h_ji

H

=

`

∑

j=1

ϕ^{h,`</sup><sub>j</sub> hϕ<sup>h,`},ψ^h_ji

H

=

`

∑

j=1

ϕ^h,`_j hϕ,ψ^h_ji

H=hϕ,P^{h,`</sup>ϕi<sub>H</sub>≤ kϕk<sub>H</sub>kP<sup>h,`}ϕk_H

for everyϕ∈H. For the specific choiceϕ=∑^{`</sup><sub>i=1</sub>ϕ<sup>h,`}_i ψi∈Hwith arbitrarily chosen ϕ^{h,`</sup><sub>1</sub> , . . . ,ϕ<sup>h,`}<sub>`</sub> we even find thatP<sup>h,`</sup>ϕ=ϕ. Thus, the operator norm ofP^h,`is equal

to one.

POD approximations for(Pˆ^h).Next, we look for a reduced-order state solution y^h,`(t) =

`

∑

i=1

y^{h,`</sup><sub>i</sub> (t)ψ<sub>i</sub>∈V<sup>h,`} f.a.a.t∈[0,T]

satisfyingy^{h,`</sup>(0) =P<sup>h,`}y◦∈V^h,`and c_p d

dthy^h,`(t),ψ^hi_H+ Z

Ω

∇y^{h,`</sup>(t)·∇ψ<sup>h</sup>+c<sub>n</sub>y<sup>h,`}(t)³ψ^hdx

= Z

Ω

f(t) +

Nu

∑

i=1

u_i(t)b_i

ψ^hdx for allψ^h∈V^h,`and a.e. in(0,T].

(18)

Utilizing similar arguments as in [43, Theorem 5.5] it can be shown that (18) admits a unique solutiony^{h,`</sup>=y<sup>h,`}(u)∈Y. Thus, we define the POD discretization of the reduced functional ˆJby

Jˆ^{h,`</sup>(u) =J(y<sup>h,`}(u),u) foru∈U_ad. Thus, POD approximation for (P) reads as follows:ˆ

min ˆJ^{h,`</sup>(u) s.t. u∈U<sub>ad</sub>. (Pˆ<sup>h,`}) Remark 3 (A-priori analysis).For a givenu∈U_adsuppose thaty^h=y^h(u)andy^{h,`</sup>= y<sup>h,`}(u)denote the solutions to (7) and (18), respectivly. To derive an a-priori error estimate for the term

Z T 0

ky^h(t)−y^h,`(t)k_V²dt one makes use of the decomposition

y^h(t)−y^{h,`</sup>(t) =y<sup>h</sup>(t)−P<sup>h,`}y^h(t)−P^{h,`</sup>y<sup>h</sup>(t)−y<sup>h,`}(t) =ρ^{h,`</sup>(t) +ϑ<sup>h,`}(t) withρ^{h,`</sup>(t) =y<sup>h</sup>(t)−P<sup>h,`}y^h(t)∈V^handϑ^{h,`</sup>(t) =P<sup>h,`}y^h(t)−y^{h,`</sup>(t)∈V<sup>h,`}f.a.a.

t∈[0,T]. From [42, Theorem 5.2] we have Z T

0

kρ^h,`(t)k_V²dt= Z T

0

ky^h(t)−P^h,`y^h(t)k²_Vdt

=

d^h

∑

i=`+1

λ_i^hkψ_i^h−P^h,`ψ_i^hk_V².

(19)

In [47, Section 3.4] an estimate forϑ^h,`is derived for a linear evolution problem.

Since our nonlinearity is monotone, we can apply similar arguments here. It follows that there exists a constantC>0 satisfying

Z _T 0

kϑ^h,`(t)k²_Vdt≤C

kϑ^h,`(0)k²_H+

d^h i=`+1

∑

λ_i^hkψ_i^h−P^h,`ψ_i^hk_V²

. (20)

From (17),ψ_i^h∈V^hfor 1≤i≤`and (6) we infer that

hP<sup>h,`</sup>y◦,ψ<sub>i</sub><sup>h</sup>i<sub>H</sub>=hy<sub>◦</sub>,ψ<sub>i</sub><sup>h</sup>i<sub>H</sub>=hP<sup>h</sup>y◦,ψ<sub>i</sub><sup>h</sup>i<sub>H</sub> for 1≤i≤`.

Let Id denotes the identity operator. Sinceϑ^{h,`</sup>(0) =P<sup>h,`}(P^h−Id)y◦∈V^h,`holds, we have

kϑ^{h,`</sup>(0)k<sup>2</sup><sub>H</sub>=kP<sup>h,`}(P^h−Id)y_◦k²_H=hP^{h,`</sup>(P<sup>h</sup>−Id)y<sub>◦</sub>,ϑ<sup>h,`}(0)i_H

=hP^h(P^h−Id)y_◦,ϑ^{h,`</sup>(0)i<sub>H</sub>=h(P<sup>h</sup>−P<sup>h</sup>)y<sub>◦</sub>,ϑ<sup>h,`}(0)i_H=0, i,e,ϑ^h,`(0) =0 is valid inH. Combining (19) and (20) we have

ky^h−y^h,`k²_L2(0,T;V)≤2 Z _T

0

kρ^h,`(t)k²_Hdt+2 Z _T

0

kϑ^h,`(t)k²_Hdt

≤C˜

d^h

∑

i=`+1

λ_i^hkψ_i^h−P^h,`ψ_i^hk²_V

for the positive constant ˜C=2(1+C). ♦

The POD optimality conditions.Next we define the reduced-order dual variable p^h,`=

`

∑

i=1

p^{h,`</sup><sub>i</sub> (t)ψ<sub>i</sub><sup>h</sup>∈V<sup>h,`} f.a.a.t∈[0,T].

We suppose that the terminal condition p^{h,`</sup>(T) =y<sup>h,`}(T)−P^h,`yΩ and the varia- tional equation

− hc_pp^h,`_t (t),ψi_V0,V+ Z

Ω

∇p^{h,`</sup>(t)·∇ψ+3c<sub>n</sub>y<sup>h,`}(t)²p^h,`(t)ψdx

= Z

Ω

y^h,`(t)−y_Q(t)

ψdx for allψ∈V^h,`and a.e.[0,T)

(21)

hold. Finally, the POD Galerkin approximation of the reduced gradient is given by

∇Jˆ^h,`(u) = κu(·) +

Z

Ω

p^h,`(·,x)b_i(x)dx

1≤i≤N_u a.e. in[0,T].

Proceeding formally as in Section 4 we obtain that a local solution ¯u^{h,`</sup>∈U<sub>ad</sub> to (Pˆ<sup>h,`}) is characterized by the following reduced-order variational inequality

Nu

∑

i=1 Z T

0

κu¯^h,`_i (t) + Z

Ω

¯

p^h,`(t,x)b_i(x)dx

u_i(t)−u¯^h,`_i (t)

dt≥0 (22) for allu∈U_ad.

6 A-posteriori error estimation for the reduced gradient

A-posteriori error estimation for the state equation. For an arbitrarily chosen controlu∈Uadlety^h=y^h(t)∈V^handy^{h,`</sup>=y<sup>h,`}(t)∈V^h,`be the unique solutions to (7) and (18), respectively. The goal is to estimate the error

e^pr(t) =y^h(t)−y^h,`(t)∈V^h fort∈[0,T] (23) between the FE and the POD solutions. We infer from (7), (18) and (23) that

cp

d

dthe^pr(t),ϕi_H+ Z

Ω

∇e^pr(t)·∇ϕ+cn y^h(t)³−y^h,`(t)³ ϕdx

=D

c_p y^h_t −y^h,`_t (t),ϕ

E

V⁰,V+ Z

Ω

∇ y^h−y^h,`

(t)·∇ϕ+c_n y^h(t)³−y^h,`(t)³ ϕdx

=−hc_py^h,`_t (t),ϕi_V0,V− Z

Ω

∇y^h,`(t)·∇ϕ−

cny^h,`(t)³−f(t)−

Nu

i=1

∑

ui(t)b_i

ϕdx

=−hR^pr(t),ϕi_V0,V for allϕ∈V^hand f.a.a.t∈(0,T] with the primal residual

R^pr(t) =c_py^{h,`</sup><sub>t</sub> (t)−∆y<sup>h,`}(t) +c_ny^h,`(t)³−f(t)−

m

∑

i=1

u_i(t)b_i∈(V^h)⁰. Thus, we conclude that

cp

d

dthe^pr(t),ϕi_H+ Z

Ω

∇e^pr(t)·∇ϕ+c_n y^h(t)³−y^h,`(t)³ ϕdx

=−hR^pr(t),ϕi_V0,V for allϕ∈V^hand f.a.a.t∈(0,T].

(24)

For fixedt∈[0,T]we define the function Ψ(s;t) =c_n y^h,`(t) +se^pr(t)3

=c_n y^{h,`</sup>(t) +s(y<sup>h</sup>(t)−y<sup>h,`}(t))3

fors∈[0,1], so that

Ψ(0;t) =c_ny^{h,`</sup>(t)<sup>3</sup>,Ψ(1;t) =c<sub>n</sub>y<sup>h</sup>(t)<sup>3</sup>andΨ<sup>0</sup>(s;t) =3c<sub>n</sub> y<sup>h,`}(t) +se^pr(t)2

e^pr(t) holds f.a.a.t∈[0,T]. This implies

c_n y^h(t)³−y^h,`(t)³

=Ψ(1;t)−Ψ(0;t) = Z 1

0

Ψ⁰(s;t)ds

=3c_n Z ₁

0

y^h,`(t) +se^pr(t)2

ds

e^pr(t)

(25)

f.a.a.t∈[0,T]. Choosingϕ=e^pr(t)∈V^hin (24) and using (25) we obtain

cp

2 d

dtke^pr(t)k²_H+ Z

Ω

∇e^pr(t)

2dx+

Z

Ω

c_n y^h(t)³−y^h,`(t)³ e^pr(t)dx

=c_p 2

d

dtke^pr(t)k²_H+ke^pr(t)k²_V− ke^pr(t)k²_H +3c_n

Z ₁ 0

Z

Ω

y^h,`(t) +se^pr(t)2 e^pr(t)

2dxds f.a.a.t∈[0,T]. From

Z 1 0

Z

Ω

y^h,`(t) +se^pr(t)2 e^pr(t)

2dxds≥0 for allt∈[0,T]we derive that

cp

2 d

dtke^pr(t)k²_H+ Z

Ω

∇e^pr(t)

2dx+

Z

Ω

c_n y^h(t)³−y^h,`(t)³ e^pr(t)dx

≥cp

2 d

dtke^pr(t)k²_H+ke^pr(t)k²_V− ke^pr(t)k²_H

f.a.a.t∈[0,T]. . Utilizing (24) and Young’s inequality [3, p. 52]

ab= √ εa

b

√ ε

≤εa² 2 +b²

2ε for alla,b∈Randε>0 witha=kR^pr(t)k_(Vh)⁰,b=ke^pr(t)k_V, andε=1 we find that

c_p 2

d

dtke^pr(t)k²_H+ke^pr(t)k²_V≤ ke^pr(t)k²_H+kR^pr(t)k_(Vh)⁰ke^pr(t)k_V

≤ ke^pr(t)k²_H+1

2kR^pr(t)k²_(Vh)⁰+1

2ke^pr(t)k²_V, which implies

d

dtke^pr(t)k²_H+ke^pr(t)k_V² ≤ 2 cp

ke^pr(t)k²_H+ 1 cp

kR^pr(t)k²_(Vh)⁰ f.a.a.t∈[0,T]. (26) Recall thate^pr(0) =y^h(0)−y^{h,`</sup>(0) = (P<sup>h</sup>−P<sup>h,`})y_◦holds. Thus, from Gronwall’s lemma [10, p. 559] it follows that

ke^pr(t)k²_H≤e^2t/cp

k(P^h−P^h,`)y◦k²_H+ 1 c_p

Z t 0

kR^pr(s)k²_(Vh)⁰ds

(27) f.a.a.t∈[0,T]. Integrating (26) over[0,t]⊂[0,T]and applying (27) we derive that

ke^pr(t)k²_H+ Z t

0

ke^pr(s)k²_Vds

≤ ke^pr(0)k²_H+ 2 c_p

Z _t 0

ke^pr(s)k²_Hds+ 1 c_p

Z _t 0

kR^pr(s)k²_(Vh)⁰ds

≤ ke^pr(0)k²_H+ 2 c_p

Z t 0

e^2s/cp

ke^pr(0)k²_H+ 1 c_p

Z s 0

kR^pr(τ)k²_(Vh)⁰dτ

ds + 1

cp Z _t

0

kR^pr(s)k²_(Vh)⁰ds Consequently,

ke^pr(t)k²_H+ Z t

0

ke^pr(s)k_V²ds

≤ ke^pr(0)k²_H+ e^2t/cp−1

ke^pr(0)k²_H+ 1 c_p

Z _t 0

kR^pr(τ)k²_(Vh)⁰dτ

+ 1 c_p

Z t 0

kR^pr(s)k²_(Vh)⁰ds

≤e^2t/cpke^pr(0)k²_H+

e^2t/cp+c_p+1 cp

^Z t 0

kR^pr(s)k²_(Vh)⁰ds

Proposition 1.For an arbitrarily chosen control u∈U_ad let y^h=y^h(t)∈V^hand y^{h,`</sup>=y<sup>h,`}(t)∈V^h,`be the unique solutions to(7)and(18), respectively. Then,

ke^pr(t)k²_H=ky^h(t)−y^h,`(t)k²_H

≤e^2t/cp

k(P^h−P^h,`)y_◦k²_H+ 1 c_p

Z t 0

kR^pr(s)k²_(Vh)⁰ds

f.a.a. t∈[0,T] and

Z _t 0

ky^h(s)−y^{h,`</sup>(s)k<sub>V</sub><sup>2</sup>ds≤e<sup>2t/cp</sup>k(P<sup>h</sup>−P<sup>h,`})y◦k²_H + e^2t/cp+c˜_p^{Z t}

0

kR^pr(s)k²_(Vh)⁰ds

f.a.a. t∈[0,T] withc˜_p= (1+c_p)/c_p.

A-posteriori error estimation for the dual equation.Suppose thatu∈U_ad is an arbitrary admissible control. Lety^handy^{h,`</sup>denote the associated unique solutions to (7) and (18), respectively. By p<sup>h</sup>and p<sup>h,`} we denote the unique corresponding solutions to (11) and (21), respectively. We estimate the dual errore^du(t) =p^h(t)− p^h,`(t)∈V^h. From (11) and (21) we infer for allϕ∈V^hand f.a.a.t∈[0,T)

−c_p d

dthe^du(t),ϕi_H+ Z

Ω

∇e^du(t)·∇ϕ+3c_n y^h(t)²p^h(t)−y^{h,`</sup>(t)<sup>2</sup>p<sup>h,`}(t) ϕdx

=−c_p d

dthp^h(t),ϕi_H+ Z

Ω

∇p^h(t)·∇ϕ+3c_ny^h(t)²p^h(t)ϕdx +c_p d

dthp^h,`(t),ϕi_H− Z

Ω

∇p^{h,`</sup>(t)·∇ϕ−3c<sub>n</sub>y<sup>h,`}(t)²p^h,`(t)ϕdx

= Z

Ω

y^h(t)−y_Q(t)

ϕdx− hR^du(t),ϕi_V0,V+ Z

Ω

y^h,`(t)−y_Q(t) ϕdx

=−hR^du(t),ϕi_V0,V− he^pr(t),ϕi_H,

where we define the dual residual as

R^du(t;u) =−c_pp^{h,`</sup><sub>t</sub> (t)−∆p<sup>h,`}+3c_ny^{h,`</sup>(t)<sup>2</sup>p<sup>h,`}(t)−y^h,`(t) +y_Q(t)∈(V^h)⁰. Thus, we have

−cp

d

dthe^du(t),ϕi_H+ Z

Ω

∇e^du(t)·∇ϕ+3cn y^h(t)²p^h(t)−y^{h,`</sup>(t)<sup>2</sup>p<sup>h,`}(t) ϕdx

≤ kR^du(t)k_(Vh)⁰kϕk_V+ke^pr(t)k_Hkϕk_H

(28) for allϕ∈V^hand f.a.a.t∈[0,T]. Notice that

Z

Ω

y^h(t)²p^h(t)−y^{h,`</sup>(t)<sup>2</sup>p<sup>h,`}(t)

e^du(t)dx

= Z

Ω

y^h(t)²e^du(t)²+ y^h(t)²−y^h,`(t)²

p^h,`(t)e^du(t)dx

≥ Z

Ω

y^h(t)²−y^h,`(t)²

p^h,`(t)e^du(t)dx f.a.a.t∈[0,T]. Thus, choosingϕ=e^du(t)∈V^hwe have

−c_p d

dtke^du(t)k²_H+ Z

Ω

∇e^du(t)

2+3c_n y^h(t)²p^h(t)−y^{h,`</sup>(t)<sup>2</sup>p<sup>h,`}(t)

e^du(t)dx

≥ −cp

2 d

dtke^du(t)k²_H+ke^du(t)k²_V− ke^du(t)k²_H +3c_n

Z

Ω

y^h(t)²e^du(t)²+ y^h(t)²−y^h,`(t)²

p^h,`(t)dx

≥ −c_p 2

d

dtke^du(t)k²_H+ke^du(t)k²_V− ke^du(t)k²_H +3c_n

Z

Ω

y^h(t)²−y^h,`(t)²

p^h,`(t)e^du(t)dx f.a.a.t∈[0,T]. It follows that