Uniform convergence of the POD method and applications to optimal control

We consider proper orthogonal decomposition (POD) based Galerkin approximations to parabolic systems and establish uniform convergence with respect to forcing functions. The result is used to prove convergence of POD approximations to optimal control problems that automatically update the POD basis in order to avoid problems due to unmodeled dynamics in the POD reduced order system. A numerical example illustrates the results.

(Communicated by the associate editor name) 1. Introduction. Proper orthogonal decomposition (POD) is a powerful Galerkin type technique for model reduction of evolution systems. A POD basis presents an optimal representation of "snapshots" of the dynamical system and it is used to derive reduced-order models (ROM) of the system. From the rich literature on POD, we can only select a few contributions that are in some way related to the present work [1,2,4,5,15,18].
In this work, we focus on the convergence of POD approximations as the number of basis elements tends to infinity. For the case of fixed forcing functions, such error estimates were first obtained in [6]. In the present paper, we derive error estimates for linear abstract parabolic evolution problems that establish the uniform convergence of the POD ROM solution with respect to forcing functions and also give a convergence rate.
Using the POD method in the context of optimal control, the problem of "unmodeled dynamics" may arise: The POD basis elements are computed from a reference trajectory which may contain features that are quite different from those of the optimally controlled trajectory. In Optimality Systems POD (OS-POD), this problem is avoided by augmenting the optimality system of the control problem with the POD basis generation criteria. Here, for a linear quadratic control problem, we apply the uniform POD convergence result in order to prove the (weak) convergence of the optimal "OS-POD control" to the full-order optimal control as the dimension of the POD-ROM system goes to infinity.
Finally, we give numerical examples which illustrate that even under bad starting conditions that cause the classical POD method to fail, the OS-POD basis update still leads to satisfactory optimal control results. Moreover, we show that convergence of the OS-POD controls as the dimension of the ROM tends to infinity is also obtained numerically.
The structure of the paper is the following. In Section 2, necessary prerequisites are summarized. Section 3 contains the uniform POD convergence result, which is applied to OS-POD in Section 4. The numerical tests are given in Section 5. The proof of a POD convergence result is deferred to an Appendix.
The paper is based on the second author's thesis [10], which is available under http://media.obvsg.at/p-AC08836134-2001.

Parabolic State Equation.
Here, we gather some facts on parabolic equations that will be used in the remainder of the paper.
Let V and H be real separable Hilbert spaces and let (V, H, V * ) be a Gelfand triple. In particular, there exists a constant C V > 0 such that Denote the V * -V -duality pairing by (·, ·) V * ,V and let a : V ×V → R be a symmetric bilinear form such that a is bounded and coercive in the following sense: There exist constants β, κ > 0 such that for all v, w ∈ V , we have The condition κ > β 2 will not be used before Proposition 3.5. For f ∈ L 2 (0, T ; V * ), g ∈ H and 0 < T < ∞, we consider We recall an existence and regularity result which will be used in Section 3.2. For an initial value g ∈ H, the solution operator S g : L 2 (0, T ; V * ) → W (0, T ), y = S g (f ) is bounded. If further g ∈ V and f ∈ H k (0, T ; V * ) for some k ∈ N and they are chosen such that the compatibility conditions ∂ j t y(0) ∈ V , j = 0, . . . , k − 1 and ∂ k t y(0) ∈ H hold, problem (2) admits a solution y ∈ H k (0, T ; V ) ∩ H k+1 (0, T ; V * ) and there exists a constant C > 0 such that In particular, assumption a) therein is naturally fulfilled since we have assumed a to be constant in time. Furthermore, b) and a special case of c) are given by (1). The additional regularity (k > 0) is stated in [17, Satz 27.2]. (Note that assumption d) therein is fulfilled since in our case, a is assumed to be time independent.) The final estimate is given in [9,Satz 8.7], for instance.
As a special case of the evolution problem above, we consider a parabolic initialvalue problem with homogeneous Dirichlet boundary conditions and a right-hand side (RHS) that is space-time-separable. Different types of boundary conditions as for instance inhomogeneities in (5b) or Neumann-or Robin-type conditions can be treated provided that regularity and a-priori estimates as in Proposition 2.3 can be established (see [17, Theorem 27.5], for instance).
Problem 2.2 (Initial Boundary Value Problem). Let Ω ⊂ R d be a bounded open set with sufficiently smooth boundary ∂Ω, choose q ∈ N even and define the control operator Forû ∈ L 2 (0, T ; R M ), y 0 ∈ H q+1 (Ω) and L a second-order strongly elliptic operator, define the initial boundary value problem (Ω)). We will need additional regularity in space as provided by the following proposition, for which a proof can be found in [ satisfying the a-priori estimate with a constant C(T ) > 0. . Choose a target state z ∈ L 2 (0, T ; H) and a control operator B ∈ L(R M , V * ). Set U := L 2 (0, T ; R M ) and let U ad ⊂ U be closed, convex and non-empty. Findû * ∈ U ad that solves min u∈U ad such that (û, y) satisfies (3) with RHS f = Bû and initial condition g ∈ V fixed, i.e., there holds In order to characterize an optimal solution, let us quote first order necessary optimality conditions; cf. [ (7) and, with the unique Lagrange-multiplier p ∈ W (0, T ), satisfies the following adjoint equation as well as the optimality condition where the operator G : U → U is defined by Corollary 2.6 (Optimality Conditions in Unconstrained Case). Set U ad := U. Let the pairz = (y * , p) ∈ W (0, T ) × W (0, T ) satisfy the following system of evolution problems Then, the pair x * = (y * ,û * ) is the (unique) solution of Problem 2.4 if and only if where the series converges w.r.t. the operator norm. Furthermore, (σ 2 n ) n is the monotonically decreasing sequence of eigenvalues of the operator A * A.
The existence of the Schmidt representation is proved in [11,Satz 16.3]. The second assertion follows since The sequence (σ n ) n of Proposition 2.7 is called the sequence of singular values of the respective compact operator A, which we shall denote by (σ n (A)) n . Further, we quote [11,Lemma 16.5] to show that in our context, the notions of "'singular values" and "approximation numbers" coincide: where α n (A) is called the n-th approximation number of A.
The following inequality for singular values will play a crucial role in the proofs of the theorems in Subsection 3.2. A proof can be found in [11,Lemma 16.6 (6)], for instance. The uniform convergence theorems will also require the following result on Sobolev embeddings: is compact. Furthermore, there exists a constant C Ω > 0 such that we have for all k ∈ N that the k-th singular value σ k of J satisfies σ k (J) ≤ C Ω 1 k r/d .

KARL KUNISCH AND MARKUS MÜLLER
In particular, for all s ∈ N such that sr/d > 1, we have the following convergence result: Proof. The compactness of the embedding map as well as the first assertion on its singular values are given in [14,Theorem 1.107]. According to Remark 1.108 therein, this includes the case of Sobolev embeddings. For the second assertion, let be ∈ N. For all k ∈ N, there holds σ k (J) ≥ 0 and due to the first assertion, we have for all m ∈ N with m > that Thus, both limits exist for m → ∞ since sr/d > 1. Then, the integral test for convergence yields Note that the integrand 1 x sr/d is monotonically decreasing and positive. 3. Uniform Convergence for Continuous POD.
3.1. Continuous POD Theory. In the continuous POD setting that we follow, we consider the trajectories of y u and ∂ t y u as our snapshot set. In order to ensure the existence of a POD basis, we construct a suitable POD operator, show the existence of eigenvalues and eigenvectors and then give a definition of a POD basis. Finally, we comment on optimality and convergence properties of the basis. Henceforth, y u := y(u) stands for the solution of (2) with f := u ∈ L 2 (0, T ; V * ). We further assume that y u ∈ H 1 (0, T ; V ), (9) which is satisfied for u ∈ H 1 (0, T ; V * ) by Proposition 2.1. We next define the continuous POD problem for the snapshot set Problem 3.1 (Continuous POD Problem). Choose X ∈ {V, H}. Find an orthonormal basis B = {ψ k } k=1 that fulfills POD Operator. We define the operator Y u : (L 2 (0, T )) 2 → X, Let us show that the adjoint of Y u is given by UNIFORM CONVERGENCE OF THE POD METHOD 7 For this purpose, let be W := L 2 (0, T ), v ∈ X as well as w ∈ W 2 and observe Finally, we set the POD operator to be K u := Y u Y * u : X → X, and define an auxiliary operator Existence of a POD Basis. We now wish to define a POD basis by means of the operator introduced above. Beforehand, we need an auxiliary result on the spectrum of the POD operators such that a POD basis is well-defined. Proposition 3.2 (Spectra of POD Operators). The operator K u possesses at most countably many eigenvalues {λ u i } i . All these eigenvalues are non-negative and can be ordered, taking into account their multiplicities. Furthermore, there exists an orthonormal system of eigenvectors Additionally, ψ is an eigenvector of K u if and only if Y * u ψ is an eigenvector ofK u .
Proof. • Spectrum ofK u : Let us observe thatK u is self-adjoint and set W := (L 2 (0, T )) 2 to see thatK u is non-negative: Furthermore, k presents an L 2 -kernel ofK u . Thus, by [11,Satz 16.12], we infer that K u is Hilbert-Schmidt and in particular compact (due to [12, Théorème (10,2), 1]). Hence, by [13, Subsection 4.2.6, Theorem 1], there exist at most countably many eigenvalues. These eigenvalues may be ordered, taking into account their multiplicities, to a non-increasing sequence (λ u k ) k . Since the operator is non-negative, all its eigenvalues are non-negative as well. Furthermore, there exists an orthonormal system of corresponding eigenvectors V u = (v u k ) k .
• Spectrum of K u : Finally, note thatK u possesses the same eigenvalues (with identical multiplicities) as K u , except for possibly zero. This fact is shown in [7, Proposition 2.1], together with the last assertion of the proposition.
In Proposition 3.2, we numbered the eigenvalues of the POD operator according to their multiplicity. A POD basis of order is given by the eigenvectors of K u corresponding to the first eigenvalues, i.e., by Ψ u = (ψ u k ) k=1 . In other words, a POD basis criterion reads: Note that if among the first eigenvalues, an eigenvalue appears k times, any orthonormal basis of the corresponding k-dimensional subspace can be used. Furthermore, we define a projection onto V u := span(Ψ u ) by Remark 3.3 (Optimality and Convergence of POD). It is a typical feature for a POD basis that it approximates the trajectories that it is calculated from better than any other orthonormal basis. Also, this approximation is arbitrarily good for increasing dimension of the POD subspace V u . For continuous POD, we refer to [3, Proposition 2.2.3], for instance. In particular, for X ∈ {V, H} and a POD basis {ψ u k } k=1 based on the trajectories of (y(u), ∂ t y(u)), for any integer ≥ 1, there holds for all orthonormal basis (ϕ k ) k of X that Continuous POD ROM Problem. For a RHS f , determine a POD basis Ψ f based on the snapshots set (10). Using V f := span(Ψ f ) as a test space in problem (2) and choosing a RHS u ∈ L 2 (0, T ; V * ) for the reduced problem, we arrive at: the POD reduced-order model for problem (2), such that for a.e. t ∈ (0, T ]: In the case that the RHS f for the determination of the basis is equal to the RHS u of the low-order system, we abbreviate the notation of the low-order solution by As the eigenvectors of the POD operators are not uniquely determined necessarily, a POD basis Ψ f need not be unique. However, for ∈ N with λ f +1 = λ f , a "POD ROM problem" is uniquely setup by the snapshot trajectory since the span V f of the POD basis is uniquely determined. Otherwise, uniqueness does not hold necessarily. Convergence Estimate for POD ROM for Fixed Data. The main part of the following result (estimate (18)) was already derived in [5,Theorem 10]; see also [3,Proposition 3.3.2]. For the sake of completeness and to verify the remaining part, an independent proof is given in the Appendix. Note also that there is an implicit requirement on the regularity of the RHS u by assuming y(u) ∈ H 1 (0, T ; V ) such that a POD basis can be constructed. According to Proposition 2.1, it suffices to choose u ∈ H 1 (0, T ; V * ), for instance. Due to (1), there exists an > 0 such that Proposition 3.5 (Convergence of POD ROM for Fixed Data). Fix u and let y(u) ∈ H 1 (0, T ; V ) be the solution to problem (2) with RHS f := u. For X := V , construct a POD basis Ψ u from the trajectories of (y(u), ∂ t y(u)) and let y (u) := y Ψ u (u) be a solution to the -ROM approximation (17) of problem (2) with the same RHS u and ansatz space V u = span(Ψ u ).
Then, for c : > 0, independent of u, the following estimates hold: as well as where C H is the constant of the continuous embedding of Remark 3.6 (Why to use X = V ?). For X = H, an error estimate as in Proposition 3.5 cannot be derived in a similar fashion. In fact, note that for X = H, a POD representation estimate as in Remark 3.3 does not hold in V , but only in H. Thus, we try to estimate all terms involving the V -norm in (45) and infer which we may rearrange to However, the term ỹ 2 L 2 (0,T ;V ) still cannot be "POD estimated" in the case X = H. Yet, this term is unavoidable since a(·, ·) can only be bounded in the V -Norm. Note that an analogue phenomenon was observed in the discrete context as well (cf. the occurrence of the stiffness matrix in [6, Lemma 3, Theorem 7]).
Remark 3.7 (Why to use POD on (y(u), ∂ t y(u))?). Using a POD basis based on solely the trajectory of y(u), we may derive an analogue of estimation (18) -c.f. [5,Theorem 9]. Yet according to (45), the RHS of this estimate includes the term ∂ tỹ (t) H , which in this case, cannot be estimated by ∞ k= +1 λ u k . In [1, Corollary 1], an alternative error estimate is presented which avoids the addition of a time derivative term. However, the estimate includes a constant depending on the ROM dimension and is hence not suitable for our purpose of deriving a uniform convergence estimate.
3.2. Uniform POD Convergence. Assuming more regularity for the RHS u, and hence obtaining more regularity for the solution y, we show that the convergence of the POD low-order solution is actually uniform w.r.t. the RHS u in bounded subsets of a suitable set of functions. Also, we now find a rate of the convergence w.r.t. the ROM dimension.
First, we will assume additional regularity in time and afterwards, we will require additional spatial regularity in an abstract sense. Finally, we consider a more specific problem with space-time-separable RHS.
The additional regularity allows to estimate the eigenvalues of the POD operator by the Courant Fischer inequality. In this way, the sum of eigenvalues in the RHS of Proposition 3.5 can be estimated by the known decay of singular values of Sobolev embeddings and the a-priori estimate of the solution.
Additional Regularity in Time. The first way for attaining the desired uniform convergence uses additional regularity of the solution in time: Theorem 3.8 (Uniform POD Convergence, Temporal Regularity). Let W and U g be such that for each (u, g) ∈ W × U g , problem (2) admits a solution y(u) ∈ H 2 (0, T ; V ). Moreover, assume that there exists a constant C a > 0 such that the a-priori estimate holds for all (u, g) ∈ W × U g .
For X := V , construct a POD basis Ψ u from the trajectories of (y(u), ∂ t y(u)), depending on u. Let y (u) := y Ψ u (u) be the solution to the -ROM approximation (17) of problem (2) with the same RHS u and ansatz space V u = span(Ψ u ).
Then, for each g ∈ U g , the sequence (y (u)) converges strongly to y(u) in L 2 (0, T ; V ), uniformly w.r.t. u in bounded subsets of W for → ∞. In particular, for a constant C = C(T, α, β) > 0, the following estimate holds true for each ≥ 1: Proof. • Introduction of Embedding: Due to the representation of Y * u in (12), there holds Y * u v ∈ (L 2 (0, T )) 2 for all v ∈ X := V . Since we assumed y(u) ∈ H 2 (0, T ; V ), we see that there even holds Y * u v ∈ (H 1 (0, T )) 2 for all v ∈ V . We can thus introduce the operator such that there holds for J : (H 1 (0, T )) 2 → (L 2 (0, T )) 2 being the compact embedding, • Estimation of λ u k : Let σ k (Y * u ) and σ k (J) denote the k-th singular values of Y * u and J, respectively. Setting A := Y * u in Proposition 2.7, we have for the eigenvalues Since J is compact andỸ * u is bounded, we may use Lemma 2.9 and together with (21) and (22), we have for all integers k ≥ 1: . (23) • Estimation of Thus, we are allowed to sum over (23) and since the norm term is independent of the summation index k, we arrive at . (24) • Estimation of ||Ỹ * u ||: By the definition of · (H 1 (0,T )) 2 , the Cauchy Schwarz inequality and finally by assumption (20), we have for a constant C a > 0, independent of u: (25) • Finalization: Note that our assumptions are stronger than those of Proposition 3.5. Thus, we may in particular use (19), estimate the sum term therein by (24) and use (25) to arrive at which clearly converges to zero for → ∞, completing the proof.
Note that in the context of optimal control, the estimate of Theorem 3.8 is restrictive due to the regularity requirements for the controls: For the compatibility condition for the initial value g in Proposition 2.1 with k = 2, it is sufficient to require W = H 1 (0, T ; V ) in order to obtain Ag + u(0) ∈ V and for the second temporal derivative, g ∈ D(A 2 ), u(0) = u t (0) = 0 suffices, for instance. Together with the requirement for the RHS that u ∈ H 2 (0, T ; V * ), a sufficient choice for the control space would be W Additional Regularity in Space. In a second approach, additional regularity of the solution in space is required. We can then use the same idea of estimation, but make use of a spatial Sobolev embedding. (The spaceŨ is introduced in order to take care of boundary conditions imposed on y u , for instanceŨ := H 2 (Ω) ∩ H 1 0 (Ω).) and letŨ be a function space continuously embedded into U . Let W and U g be such that for each (u, g) ∈ W × U g , problem (2) admits a solution y(u) ∈ H 1 (0, T ;Ũ ). Moreover, assume that there exists a constant C a > 0 such that the a-priori estimate holds for all (u, g) ∈ W × U g . For X := V , construct a POD basis Ψ u from the trajectories of (y(u), ∂ t y(u)), depending on u. Let y (u) := y Ψ u (u) be a solution to the -ROM approximation (17) of problem (2) with the same RHS u and ansatz space V u = span(Ψ u ).
Then, for each g ∈ U g , the sequence (y (u)) converges strongly to y(u) in L 2 (0, T ; V ), uniformly w.r.t. u in bounded subsets of W. In particular, there exists a constant C > 0, independent of , such that for each ≥ 1, we have Proof. • Introduction of Embedding: Due to the definition of Y u in (11), there holds Y u v ∈ X = V for all v ∈ (L 2 (0, T )) 2 . Since we assumed y(u) ∈ H 1 (0, T ;Ũ ), there even holds Y u v ∈Ũ for all v ∈ (L 2 (0, T )) 2 . We can thus introduce the operator Y u : (L 2 (0, T )) 2 →Ũ , and set for the continuous embedding E U :Ũ → U , By the definition of U and V , there exists a compact embedding E : U → V for which we have • Estimation of λ u k : Let σ k (Y u ) and σ k (E) denote the k-th singular values of Y u and E, respectively. According to Proposition 3.2, we have (λ u k ) k = (λ u k ) k , where (λ u k ) k and (λ u k ) k denote all non-zero eigenvalues of K u andK u = Y * u Y u in (13), respectively. Setting A := Y u in Proposition 2.7, we haveK u = A * A and hence for all integers k ≥ 1, there holds Since E is compact andỸ u is bounded, we may use Lemma 2.9 and together with (27) and (28), we have for all integers k ≥ 1: Note that estimate (29) is trivially true for all λ u k = 0 since the RHS is non-negative. Hence, the estimate is true for all eigenvalues (λ u k ) k of K u .
since by the assumption on r, there holds 2r/d > 1. Thus, we may sum over (29), in which the norm term is independent of the summation index k. Hence, we arrive at • Estimation of ||Ỹ u ||: Set W := L 2 (0, T ). Due to the continuous embedding E U with constant C U , by definition of · L(W 2 ,U ) and · W 2 , the Cauchy Schwarz inequality and the fact that for all where the last step is due to the triangular inequality for integrals and changing the order of integration. Hence, by assumption (26), we have for a constant C a > 0, independent of u, Ug . (31) • Finalization: Note that the assumptions of Proposition 3.5 are fulfilled (in particular, we have y ∈ H 1 (0, T ;Ũ ) → H 1 (0, T ; V )). Thus, we can use (19), estimate the sum term therein by (30) and use (31) to arrive at which clearly approaches zero for → ∞, completing the proof.
Space-time-separable RHS. The previous results have given sufficient conditions for the POD convergence to be uniform. Now, we give a concrete example in which this uniformity holds. (Ω)). For X := V , construct a POD basis Ψ u from the trajectories of (y(u), ∂ t y(u)), depending on u. Let y (u) := y Ψ u (u) be a solution to the -ROM approximation of Problem 2.2 with the same RHS u and ansatz space V u = span(Ψ u ).

Convergence of Optimality Systems POD.
In this section, we use the uniform convergence result of Theorem 3.8 to establish convergence of Optimality Systems POD (OS-POD), which was first proposed in [7] and subsequently extended in [16].

OS-POD.
The starting point is the suboptimal control based on POD reduction with respect to a reference trajectory with input f : such that (u, y Ψ f ) satisfies the corresponding POD-reduced order model (17).
Here one is confronted with the problem that the POD basis is computed from a reference trajectory which is not the optimal one. Moreover, during the course of iterative strategies to solve (32), the trajectory changes and the controlled dynamics may not be sufficiently well represented in the POD modes computed from the state corresponding to f . OS-POD was introduced to overcome this problem of "unmodeled dynamics". In the OS-POD procedure, the basis, with fixed dimension , is updated with the goal that in the asymptotic limit of the iteration, the PODbasis corresponds to the optimal trajectory at discretization level . Here we shall further justify the OS-POD procedure by proving that as the dimension → ∞, the optimal controls of the finite dimensional OS-POD problems converge weakly to the solution u * of the infinite dimensional Problem 2.4. In the OS-POD algorithm, this issue is taken care of by augmenting the optimization problem with the POD basis construction criteria. For that matter, we need to include the full system (7) as well as the POD basis condition (14). Altogether, this then reads: For J in (32), solve min J (û, y Ψ u ) such that • (û, y(u)) fulfills the full problem (7): d dt y(u)(t) + Ay(u)(t) = u = B(û(t)) for a.e. t ∈ (0, T ], (33a) • A POD basis Ψ u is constructed according to: Compact Statement. Let us gather the conditions into a more compact form and recall the following notation: • u * = B(û * ) denotes the optimal solution of the full-order optimization problem and u * = B(û * ) is the optimal solution of the OS-POD problem. • y Ψ u (u * ) denotes the solution of the -th-order ROM with RHS u * and ansatz space span(Ψ u ) where Ψ u is determined based on the trajectory of y(u ). Then, the OS-POD problem reads where z u := P Ψ u z, and y Ψ u satisfies (17) with RHS = u , and the basis Ψ u is determined from the trajectories y(u ), ∂ t y(u ) of the full solution y(u ) of (33) with RHS u as well, i.e., f = u .
In the remainder of this section, we restrict our attention to the special case of the initial boundary value Problem 2.2 with controls which act as temporal coefficients of finitely many spatial ansatz functions as stated in (4).
For this situation, Theorem 2.2 in [7] guarantees the existence of a solution u * to the OS-POD problem, provided that for eachû ∈ L 2 (0, T ; R M ), we have min λ(K u (y)) > 0, where y solves (33): Further, assume that for the eigenvalues λ of the POD operator K u , there holds min λ(K u (y)) y solves (33) withû ∈ L 2 (0, T ; R M ) > 0.
Then, for each ∈ N, the OS-POD Problem 4.2 admits a (global) solution , where y * denotes the full-order optimal solution. (Note that y * depends on as the RHS u * does.) The OS-POD problem is certainly computationally demanding. For an efficient numerical treatment, see [10, Chapter 6], [7] and [16], for instance.

4.2.
Convergence of OS-POD. In this subsection, we show that the solutions u * of the OS-POD problem converge weakly to the full-order optimal control u * as goes to infinity. First, we consider the case z = 0. Then, we allow a state z = 0.
Theorem 4.4 (OS-POD Convergence for z = 0). Set X = V , choose the target state z = 0 and let B q be as in Problem 2.2 with q even such that q > d/2 + 1, where d is the dimension of Ω. Letû * ∈ L 2 (0, T ; R M ) be the optimal control of Problem 2.4 and set u * := B qû * . Further, letû * ∈ L 2 (0, T ; R M ) be a solution to the OS-POD Problem 4.2 and set u * := B qû * . Then, (û * ) converges weakly toû * in L 2 (0, T ; R M ). Furthermore, the sequence of corresponding OS-POD states (y Ψ u * (u * )) converges to y(u * ) strongly in L 2 (0, T ; H).
In order to simplify the notation, let us re-index the sub-subsequence (û * kp ) p to (û * k ) k since the former also converges toũ ∈ L 2 (0, T ; R M ) due to the uniqueness of weak limits. We can then say that (y k ) k = (y(u * k )) k converges strongly to y ∈ H 1 (0, T ; V ) in the sense of L 2 (0, T ; H).
• Taking the Limit in the Equation: Let us now show that (y k ) k actually converges strongly to y(ũ) in the sense of L 2 (0, T ; H), i.e., that in L 2 (0, T ; H), we have y = y(ũ).
• Convergence of OS-POD State: We have that the sequence (û * k ) k is bounded in L 2 (0, T ; R M ). Making use of Corollary 3.10 and zero-adding the term y(u * k ), we infer, for a constant C > 0, using Young's inequality and V → H, where s := 2q−2 d − 1 > 0 due to the assumption on q. The previous step implies that the first summand of (34) approaches zero for k → ∞ since (y(u * k )) k converges strongly to y(ũ) in L 2 (0, T ; H). The second summand approaches zero for k → ∞ since û * k L 2 (0,T ;R M ) is bounded and s > 0. Thus, we have that y Ψ u * k k (u * k ) approaches y(ũ) strongly in L 2 (0, T ; H) for k → ∞, which gives the second assertion of the theorem as soon as we proofũ =û * .
• Optimality ofũ: Note that since z = 0, the cost functional J does not explicitly depend on . By weakly lower semi continuity of J and the previous step, we have for allû ∈ L 2 (0, T ; R M ), setting u := B qû andū := B qũ , that where estimation (35) holds for allû ∈ L 2 (0, T ; R M ) since (û * k , y Ψ u * k k ) is minimal for J for each integer k ≥ 1. In (36), we used that the subsequence (y Ψ u k (u)) k converges strongly to y(u) in L 2 (0, T ; V ) → L 2 (0, T ; H) for k → ∞ which is justified by Corollary 3.10.
Altogether, we infer thatũ is a solution of the optimal control Problem 2.4. Hence, by the uniqueness of the solution to this problem, we haveũ =û * .
Since y = w + z satisfies Problem 2.2, we infer w t + z t + L(w + z) = y t + Ly = B qû and hence, w fulfills Since we assumed y 0 ∈ H q+1 (Ω) and there holds z(0) ∈ H q+3 0 (Ω), we have g := w(0) = y 0 − z(0) ∈ H q+1 (Ω). Together with w = y − z = 0 on Γ, we infer that w satisfies Problem 2.2 with initial value g and RHS u := B qû − z t − Lz. k=1ũ kbk and we can apply Corollary 3.10 withũ in place ofû andb k in place of b k : For the low-order approximation w of w and s = 2q−2 d − 1 > 0, we find for a constant C > 0, which ensures the convergence of the corresponding OS-POD state as in (34): We can thus repeat the proof of Theorem 4.4 with w in place of y. Since we have J z=0 (u, w) = J z (u, y), this completes the proof.
Remark 4.6 (Non-separable Target State). The OS-POD convergence can also be obtained for a target state z = 0 that is not space-time-separable (in contrast to the assumption in Corollary 4.5). In this case, it is sufficient to require f := z t + Lz ∈ H 2 (0, T ; V * ) as well as the compatibility conditions (implicitly) assumed for f in Proposition 2.1: In fact, for w in the proof of Corollary 4.5, due to linearity, we can make the ansatz w = w 1 + w 2 , where w 1 and w 2 solve Problem 2.2 with RHS u 1 := B qû and u 2 := −z t − Lz, respectively. For u 1 , we can apply Corollary 3.10 in order to obtain the uniform convergence of the low-order solution corresponding to w 1 . For u 2 , we can apply Theorem 3.8 due to the additional regularity assumed for z and obtain the uniform convergence of the low-order solution corresponding to w 2 .
5. Numerical Investigation. The purpose of this section is to illustrate the convergence result of Theorem 4.4 and to show that the OS-POD basis update leads to satisfactory control results when classical POD suboptimal control does not perform well.
5.1. Numerical Strategy. In order to efficiently solve the OS-POD problem, we use a splitting algorithm that alternatingly considers the constraints given by the partial differential equation and the eigenvalue problems (cf. [7]). This approach involves the OS-POD optimality condition derived in [7, Theorem 2.3]: Let be X ∈ {V, H} and let I : X * → X denote the canonical Riesz isomorphism. We assume here that the eigenvalues λ (K u (y)) of the POD operator K u are distinct.
Note that if this is not the case, we have to keep the orthonormality condition on the subspace corresponding to a multiple eigenvalue as explicit constraints.
In order to simplify the notation, we denote the POD basis Ψ u by Ψ . Using a suitable operator G : [7, p. 9]), there holds: denote a solution to the OS-POD Problem 4.2. Furthermore, assume that Then, there exist (p , q , µ , η ) ∈ L 2 (0, T ; R ) × L 2 (0, T ; V ) × X × R such that the following optimality system holds: where for X := i=1 X, we have used the notation Algorithm 5.2 (OS-POD Algorithm). Choose an initial POD basis Ψ 0 = {ψ 0 k } k=1 of rank , a tolerance δ > 0 and set n = 0.
For the sake of comparison, Problem (41) is solved by an FE approach. The FE solution is obtained by discretizing the optimality system (8a)-(8d) of Corollary 2.6 by piece-wise linear finite elements on an equidistant grid and using backward Euler for the time discretization. The discrete system is solved by a direct solver. The POD reduction is based on (32) and is utilized within OS-POD as presented in Proposition 5.1 and Algorithm 5.2. The OS-POD optimality systems were also solved by an exact solver. In all cases, the initial snapshots are taken at every 8th time step, based on the trajectory of the solution corresponding to u = v = 0. All POD bases are used for the state as well as the adjoint state. All computations were carried out using a Matlab 2008a implementation on an Intel Quad Core CPU with 2.83 GHz and 3.25 GB RAM. The following parameter settings were used. For the model problem, we set: T = 1, σ = 10 6 , β = 10 −3 , c = 10 and g(x) = sin(2πx). For the discretization, we used a spatial as well as a temporal grid size of h = τ = 1/256. (Note that the solutions are plotted on an eight times coarser mesh than the computational grid.) For this example, POD suboptimal control fails for = 6; see column "Iter 0" of Table 1. However, Table 1 also shows that OS-POD yields satisfactory results after three basis updates. Figure 1 depicts the target state, the FE optimal control and the FE optimal state. In the first/second row of Figure 2, we see the OS-POD optimal states/controls for zero to five basis updates. Figure 3 finally depicts the first two corresponding POD basis functions.    Table 2. Control error uFE − u OSPOD L 2 (R 2 ) for OS-POD basis updates for different choices of .
In Table 2, we show the dependence of the error in the OS-POD control u OSPOD on the size of the low-order system . Note that this presents a numerical justification for the convergence of the OS-POD algorithm that we established in Theorem 4.4: For sufficiently many basis updates, the error in the control approaches zero for being increased. On the other hand, we see in the first column that increasing without updating the basis does not significantly reduce the error, i.e., the classical POD method fails for our choices of . Also, we observe that for < 6, the error in the control settles at a rather high value. Furthermore, the higher the choice of , the less basis updates are necessary to achieve a satisfactory result.
We report that for this example, updating the POD basis on the basis of the current suboptimal state without invoking OS-POD does not improve the POD suboptimal control result.
The OS-POD update also performs well with higher convection c = 20. However, we have to set ≥ 7 and carry out a minimum of two OS-POD basis updates to obtain satisfactory results.
Proof. Since u is fixed, we shall simplify the notation by omitting the u-dependencies. Also, we sometimes denote d dt by ∂ t . • Splitting the Error: Let P X : X → V denote the POD projection defined in (15). Note that for the low-order solution y ∈ V = span(Ψ ) of problem (17), there holds for all t ∈ [0, T ], y (t) = P X y (t) = k=1 (y (t), ψ k ) X ψ k .
By means of that, we split the approximation error into two components: y − y 2 L 2 (0,T,X) = P X y +ỹ − y where the second equation is true since P X y − y andỹ are orthogonal in X.