Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control

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1 Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control Michael Hinze 1 and Stefan Volkwein 1 Institut für Numerische Mathematik, TU Dresden, D-169 Dresden, Germany hinze@math.tu-dresden.de Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens Universität Graz, Heinrichstrasse 36, A-81 Graz, Austria stefan.volkwein@uni-graz.at 1 Motivation Optimal control problems for nonlinear partial differential equations are often hard to tackle numerically so that the need for developing novel techniques emerges. One such technique is given by reduced order methods. Recently the application of reduced-order models to optimal control problems for partial differential equations has received an increasing amount of attention. The reduced-order approach is based on projecting the dynamical system 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 elements of the subspaces are uncorrelated to the physical properties of the system that they approximate. The reduced basis method as developed, e.g., in [IR98] is one such reduced-order method with the basis elements corresponding to the dynamics of expected control regimes. Proper orthogonal decomposition (POD) provides a method for deriving low order models of dynamical systems. It was successfully used in a variety of fields including signal analysis and pattern recognition (see [Fuk9]), fluid dynamics and coherent structures (see [AHLS88, HLB96, NAMTT3, RF94, Sir87]) and more recently in control theory (see [AH1, AFS, LT1, SK98, TGP99]) and inverse problems (see [BJWW]). Moreover, in [ABK1] POD was successfully utilized to compute reduced-order controllers. The relationship between POD and balancing was considered in [LMG, Row4, WP1]. Error analysis for nonlinear dynamical systems in finite dimensions were carried out in [RP]. In our application we apply POD to derive a Galerkin approximation in the spatial variable, with basis functions corresponding to the solution of the physical system at pre-specified time instances. These are called the snap-

2 Michael Hinze and Stefan Volkwein shots. Due to possible linear dependence or almost linear dependence, the snapshots themselves are not appropriate as a basis. Rather a singular value decomposition (SVD) is carried out and the leading generalized eigenfunctions are chosen as a basis, referred to as the POD basis. The paper is organized as follows. In Section the POD method and its relation to SVD is described. Furthermore, the snapshot form of POD for abstract parabolic equations is illustrated. Section 3 deals with reduced order modeling of nonlinear dynamical systems. Among other things, error estimates for reduced order models of a general equation in fluid mechanics obtained by the snapshot POD method are presented. Section 4 deals with suboptimal control strategies based on POD. For optimal open-loop control problems an adaptive optimization algorithm is presented which in every iteration uses a surrogate model obtained by the POD method instead of the full dynamics. In particular, in Section 4. first steps towards error estimation for optimal control problems are presented whose discretization is based on POD. The practical behavior of the proposed adaptive optimization algorithm is illustrated for two applications involving the time-dependent Navier- Stokes system in Section 5. For closed-loop control we refer the reader to [Gom, KV99, KVX4, LV3], for instance. Finally, we draw some conclusions and discuss future research perspectives in Section 6. The POD method In this section we propose the POD method and its numerical realization. In particular, we consider both POD in C n (finite-dimensional case) and POD in Hilbert spaces; see Sections.1 and., respectively. For more details we refer to, e.g., [HLB96, KV99, Vol1a]..1 Finite-dimensional POD In this subsection we concentrate on POD in the finite dimensional setting and emphasize the close connection between POD and the singular value decomposition (SVD) of rectangular matrices; see [KV99]. Furthermore, the numerical realization of POD is explained. POD and SVD Let Y be a possibly complex valued n m matrix of rank d. In the context of POD it will be useful to think of the columns {Y,j } m j=1 of Y as the spatial coordinate vector of a dynamical system at time t j. Similarly we consider the rows {Y i, } n of Y as the time-trajectories of the dynamical system evaluated at the locations x i. From SVD (see, e.g., [Nob69]) the existence of real numbers σ 1 σ... σ d > and unitary matrices U C n n with columns {u i } n and V C m m with columns {v i } m such that

3 POD: Error Estimates and Suboptimal Control 3 ( ) U H D Y V = =: Σ C n m, (1) where D = diag (σ 1,..., σ d ) IR d d, the zeros in (1) denote matrices of appropriate dimensions, and the superindex H stands for complex conjugation. Moreover, the vectors {u i } d and {v i} d satisfy Y v i = σ i u i and Y H u i = σ i v i for i = 1,..., d. () They are eigenvectors of Y Y H and Y H Y with eigenvalues σi, i = 1,..., d. The vectors {u i } m i=d+1 and {v i} m i=d+1 (if d < n respectively d < m) are eigenvectors of Y Y H and Y H Y, respectively, with eigenvalue. If Y IR n m then U and V can be chosen to be real-valued. From () we deduce that Y = UΣV H. It follows that Y can also be expressed as Y = U d D(V d ) H, (3) where U d C n d and V d C m d are given by It will be convenient to express (3) as U d i,j = U i,j for 1 i n, 1 j d, V d i,j = V i,j for 1 i m, 1 j d. Y = U d B with B = D(V d ) H C d m. Thus the column space of Y can be represented in terms of the d linearly independent columns of U d. The coefficients in the expansion for the columns Y,j, j = 1,..., m, in the basis {U d,i }d are given by the B,j. Since U is Hermitian we easily find that Y,j = d B i,j U d,i = d U,i, Y,j C nu d,i, where, C n denotes the canonical inner product in C n. In terms of the columns y j of Y we express the last equality as y j = d B i,j u i = d u i, y j C nu i, j = 1,..., m. Let us now interpret singular value decomposition in terms of POD. One of the central issues of POD is the reduction of data expressing their essential information by means of a few basis vectors. The problem of approximating all spatial coordinate vectors y j of Y simultaneously by a single, normalized vector as well as possible can be expressed as

4 4 Michael Hinze and Stefan Volkwein m max y j, u C n subject to (s.t.) u C n = 1. j=1 (P) Here, C n denotes the Euclidean norm in C n. Utilizing a Lagrangian framework a necessary optimality condition for (P) is given by the eigenvalue problem Y Y H u = σ u. (4) Due to singular value analysis u 1 solves (P) and argmax (P) = σ1. If we were to determine a second vector, orthogonal to u 1 that again describes the data set {y i } m as well as possible then we need to solve max m yj, u C n s.t. u C n = 1 and u, u 1 C n =. (P ) j=1 Rayleigh s principle and singular value decomposition imply that u is a solution to (P ) and argmax (P ) = σ. Clearly this procedure can be continued by finite induction so that u k, 1 k d, solves max m y j, u C n s.t. u C n = 1 and u, u i C n =, 1 i k 1. (P k ) j=1 The following result which states that for every l k the approximation of the columns of Y by the first l singular vectors {u i } l is optimal in the mean among all rank l approximations to the columns of Y is now quite natural. More precisely, let Û Cn d denote a matrix with pairwise orthonormal vectors û i and let the expansion of the columns of Y in the basis {û i } d be given by Y = Û ˆB, where ˆB i,j = û i, y j C n for 1 i d, 1 j m. Then for every l k we have Y Û l ˆBl F Y U l B l F. (5) Here, F denotes the Frobenius norm, U l denotes the first l columns of U, B l the first l rows of B and similarly for Û l and ˆB l. Note that the j-th column of U l B l represents the Fourier expansion of order l of the j-th column y j of Y in the orthonormal basis {u i } l. Utilizing the fact that Û ˆB l has rank l and recalling that B l = (D(V k ) H ) l estimate (5) follows directly from singular value analysis [Nob69]. We refer to U l as the POD-basis of rank l. Then we have d d ( m ) d ( m σi = B i,j ˆB i,j ). (6) and i=l+1 i=l+1 j=1 i=l+1 j=1

5 l σi = POD: Error Estimates and Suboptimal Control 5 l ( m ) B i,j j=1 l ( m ˆB i,j ). (7) Inequalities (6) and (7) establish that for every 1 l d the POD-basis of rank l is optimal in the sense of representing in the mean the columns of Y as a linear combination by a basis of rank l. Adopting the interpretation of the Y i,j as the velocity of a fluid at location x i and at time t j, inequality (7) expresses the fact that the first l POD-basis functions capture more energy on average than the first l functions of any other basis. The POD-expansion Y l of rank l is given by j=1 Y l = U l B l = U l ( D(V d ) H) l, and hence the t-average of the coefficients satisfies B l i,, B l j, C m = σ i δ ij for 1 i, j l. This property is referred to as the fact that the POD-coefficients are uncorrelated. Computational issues Concerning the practical computation of a POD-basis of rank l let us note that if m < n then one can choose to determine m eigenvectors v i corresponding to the largest eigenvalues of Y H Y C m m and by () determine the POD-basis from u i = 1 σ i Y v i, i = 1,..., l. (8) Note that the square matrix Y H Y has the dimension of number of timeinstances t j. For historical reasons [Sir87] this method of determine the PODbasis is sometimes called the method of snapshots. For the application of POD to concrete problems the choice of l is certainly of central importance, as is also the number and location of snapshots. It appears that no general a-priori rules are available. Rather the choice of l is based on heuristic considerations combined with observing the ratio of the modeled to the total information content contained in the system Y, which is expressed by E(l) = l σ i d σ i for l {1,..., d}. (9) For a further discussion, also of adaptive strategies based e.g. on this term we refer to [MM3] and the literature cited there.

6 6 Michael Hinze and Stefan Volkwein Application to discrete solutions to dynamical systems Let us now assume that Y IR n m, n m, arises from discretization of a dynamical system, where a finite element approach has been utilized to discretize the state variable y = y(x, t), i.e., y h (x, t j ) = n Y i,j ϕ i (x) for x Ω, with ϕ i, 1 i n, denoting the finite element functions and Ω being a bounded domain in IR or IR 3. The goal is to describe the ensemble {y h (, t j )} m j=1 of L -functions simultaneously by a single normalized L - function ψ as well as possible: max m yh (, t j ), ψ L (Ω) s.t. ψ L (Ω) = 1, ( P) j=1 where, L (Ω) is the canonical inner product in L (Ω). Since y h (, t j ) span {ϕ 1,..., ϕ n } holds for 1 j n, we have ψ span {ϕ 1,..., ϕ n }. Let v be the vector containing the components v i such that ψ(x) = n v i ϕ i (x) and let S IR n n denote the positive definite mass matrix with the elements ϕ i, ϕ j L (Ω). Instead of (4) we obtain that Y Y T Sv = σ v. (1) The eigenvalue problem (1) can be solved by utilizing singular value analysis. Multiplying (1) by the positive square root S 1/ of S from the left and setting u = S 1/ v we obtain the n n eigenvalue problem Ỹ Ỹ T u = σ u, (11) where Ỹ = S1/ Y IR n m. We mention that (11) coincides with (4) when {ϕ i } n is an orthonormal set in L (Ω). Note that if Y has rank k the matrix Ỹ has also rank d. Applying the singular value decomposition to the rectangular matrix Ỹ we have Ỹ = UΣV T (see (1)). Analogous to (3) it follows that Ỹ = U d D(V d ) T, (1) where again U d and V d contain the first k columns of the matrices U and V, respectively. Using (1) we determine the coefficient matrix Ψ = S 1/ U d IR n d, so that the first k POD-basis functions are given by

7 ψ j (x) = POD: Error Estimates and Suboptimal Control 7 n Ψ i,j ϕ i (x), j = 1,..., d. Due to (11) and Ψ,j = S 1/ U d,j, 1 j d, the vectors Ψ,j are eigenvectors of problem (1) with corresponding eigenvalues σ j : Y Y T SΨ,j = Y Y T SS 1/ U k,j = S 1/ Ỹ Ỹ T U k,j = σ j S 1/ U k,j = σ j Ψ,j. Therefore, the function ψ 1 solves ( P) with argmax ( P) = σ 1 and, by finite induction, the function ψ k, k {,..., d}, solves max m y h (, t j ), ψ L (Ω) s.t. ψ L (Ω) = 1, ψ, ψ i L (Ω) =, i < d, ( P k ) j=1 with argmax ( P k ) = σk. Since we have Ψ,j = S 1/ U d,j, the functions ψ 1,..., ψ d are orthonormal with respect to the L -inner product: ψ i, ψ j L (Ω) = Ψ,i, SΨ,j C n = u i, u j C n = δ ij, 1 i, j d. Note that the coefficient matrix Ψ can also be computed by using generalized singular value analysis. If we multiply (1) with S from the left we obtain the generalized eigenvalue problem SY Y T Su = σ Su. From generalized SVD [GL89] there exist orthogonal V IR m m and U IR n n and an invertible R IR n n such that ( ) V (Y T E S)R = =: Σ 1 IR m n, (13a) US 1/ R = Σ IR n n, (13b) where E = diag (e 1,..., e d ) with e i > and Σ = diag (s 1,..., s n ) with s i >. From (13b) we infer that Inserting (14) into (13a) we obtain that R = S 1/ U T Σ. (14) Σ 1 ΣT 1 = Σ 1 RT SY V T = US 1/ Y V T, which is the singular value decomposition of the matrix S 1/ Y with σ i = e i /s i > for i = 1,..., d. Hence, Ψ is again equal to the first k columns of S 1/ U. If m n we proceed to determine the matrix Ψ as follows. From u j = (1/σ j ) S 1/ Y v j for 1 j d we infer that

8 8 Michael Hinze and Stefan Volkwein Ψ,j = 1 σ j Y v j, where v j solves the m m eigenvalue problem Y T SY v j = σ i v j, 1 j d. Note that the elements of the matrix Y T SY are given by the integrals y(, t i ), y(, t j ) L (Ω), 1 i, j n, (15) so that the matrix Y T SY is often called a correlation matrix.. POD for parabolic systems Whereas in the last subsection POD has been motivated by rectangular matrices and SVD, we concentrate on POD for dynamical (non-linear) systems in this subsection. Abstract nonlinear dynamical system Let V and H be real separable Hilbert spaces and suppose that V is dense in H with compact embedding. By, H we denote the inner product in H. The inner product in V is given by a symmetric bounded, coercive, bilinear form a : V V IR: ϕ, ψ V = a(ϕ, ψ) for all ϕ, ψ V (16) with associated norm given by V = a(, ). Since V is continuously injected into H, there exists a constant c V > such that We associate with a the linear operator A: ϕ H c V ϕ V for all ϕ V. (17) Aϕ, ψ V,V = a(ϕ, ψ) for all ϕ, ψ V, where, V,V denotes the duality pairing between V and its dual. Then, by the Lax-Milgram lemma, A is an isomorphism from V onto V. Alternatively, A can be considered as a linear unbounded self-adjoint operator in H with domain D(A) = {ϕ V : Aϕ H}. By identifying H and its dual H it follows that D(A) V H = H V, each embedding being continuous and dense, when D(A) is endowed with the graph norm of A.

9 POD: Error Estimates and Suboptimal Control 9 Moreover, let F : V V V be a bilinear continuous operator mapping D(A) D(A) into H. To simplify the notation we set F (ϕ) = F (ϕ, ϕ) for ϕ V. For given f C([, T ]; H) and y V we consider the nonlinear evolution problem d dt y(t), ϕ H + a(y(t), ϕ) + F (y(t)), ϕ V,V = f(t), ϕ H (18a) for all ϕ V and t (, T ] a.e. and y() = y in H. (18b) Assumption (A1). For every f C([, T ]; H) and y V there exists a unique solution of (18) satisfying y C([, T ]; V ) L (, T ; D(A)) H 1 (, T ; H). (19) Computation of the POD basis Throughout we assume that Assumption (A1) holds and we denote by y the unique solution to (18) satisfying (19). For given n IN let = t < t 1 <... < t n T () denote a grid in the interval [, T ] and set δt j = t j t j 1, j = 1,..., n. Define t = max (δt 1,..., δt n ) and δt = min (δt 1,..., δt n ). (1) Suppose that the snapshots y(t j ) of (18) at the given time instances t j, j =,..., n, are known. We set V = span {y,..., y n }, where y j = y(t j ) for j =,..., n, y j = t y(t j n ) for j = n + 1,..., n with t y(t j ) = (y(t j ) y(t j 1 ))/δt j, and refer to V as the ensemble consisting of the snapshots {y j } n j=, at least one of which is assumed to be nonzero. Furthermore, we call {t j } n j= the snapshot grid. Notice that V V by construction. Throughout the remainder of this section we let X denote either the space V or H. Remark 1 (compare [KV1, Remark 1]). It may come as a surprise at first that the finite difference quotients t y(t j ) are included into the set V of snapshots. To motivate this choice let us point out that while the finite difference quotients are contained in the span of {y j } n j=, the POD bases differ depending on whether { t y(t j )} n j=1 are included or not. The linear dependence does

10 1 Michael Hinze and Stefan Volkwein not constitute a difficulty for the singular value decomposition which is required to compute the POD basis. In fact, the snapshots themselves can be linearly dependent. The resulting POD basis is, in any case, maximally linearly independent in the sense expressed in (P l ) and Proposition 1. Secondly, in anticipation of the rate of convergence results that will be presented in Section 3.3 we note that the time derivative of y in (18) must be approximated by the Galerkin POD based scheme. In case the terms { t y(t j )} n j=1 are included in the snapshot ensemble, we are able to utilize the estimate n t α j y(t j ) j=1 l t y(t j ), ψ i X ψ i X d i=l+1 λ i. () Otherwise, if only the snapshots y j = y(t j ) for j =,..., n, are used, we obtain instead of (37) the error formula n y(tj α j ) j= and () must be replaced by n t α j y(t j ) j=1 l y(t j ), ψ i X ψ i X = d i=l+1 l t y(t j ), ψ i X ψ i X (δt) λ i, d i=l+1 λ i, (3) which in contrast to () contains the factor (δt) on the right-hand side. In [HV3] this fact was observed numerically. Moreover, in [LV3] it turns out that the inclusion of the difference quotients improves the stability properties of the computed feedback control laws. Let us mention the article [AG3], where the time derivatives were also included in the snapshot ensemble to get a better approximation of the dynamical system. Let {ψ i } d denote an orthonormal basis for V with d = dim V. Then each member of the ensemble can be expressed as y j = d y j, ψ i X ψ i for j =,..., n. (4) The method of POD consists in choosing an orthonormal basis such that for every l {1,..., d} the mean square error between the elements y j, j n, and the corresponding l-th partial sum of (4) is minimized on average: min J(ψ 1,..., ψ l ) = n α j yj j= l y j, ψ i X ψ i s.t. ψ i, ψ j X = δ ij for 1 i l, 1 j i. X (P l )

11 POD: Error Estimates and Suboptimal Control 11 Here {α j } n j= are positive weights, which for our purposes are chosen to be α = δt 1, α j = δt j + δt j+1 and α j = α j n for j = n + 1,..., n. for j = 1,..., n 1, α n = δt n Remark. 1) Note that I n (y) = J(ψ 1,..., ψ l ) can be interpreted as a trapezoidal approximation for the integral I(y) = T y(t) l y(t), ψ i X ψ i X + yt (t) l y t (t), ψ i X ψ i X dt. For all y C 1 ([, T ]; X) it follows that lim n I n (y) = I(y). In Section 4. we will address the continuous version of POD (see, in particular, Theorem 4). ) Notice that (P l ) is equivalent with l max n j= α j yj, ψ i X s.t. ψ i, ψ j X = δ ij, 1 j i l. (5) For X = C n, l = 1 and α j = 1 for 1 j n and α j = otherwise, (5) is equivalent with (P). A solution {ψ i } l to (P l) is called POD basis of rank l. The subspace spanned by the first l POD basis functions is denoted by V l, i.e., V l = span {ψ 1,..., ψ l }. (6) The solution of (P l ) is characterized by necessary optimality conditions, which can be written as an eigenvalue problem; compare Section.1. For that purpose we endow IR n+1 with the weighted inner product v, w α = n j= α j v j w j (7) for v = (v,..., v n ) T, w = (w,..., w n ) T IR n+1 and the induced norm. Remark 3. Due to the choices for the weights α j s the weighted inner product, α can be interpreted as the trapezoidal approximation for the H 1 -inner product T v, w H 1 (,T ) = vw + v t w t dt for v, w H 1 (, T ) so that (7) is a discrete H 1 -inner product (compare Section 4.).

12 1 Michael Hinze and Stefan Volkwein Let us introduce the bounded linear operator Y n : IR n+1 X by Y n v = n j= Then the adjoint Y n : X IRn+1 is given by α j v j y j for v IR n+1. (8) Y nz = ( z, y X,..., z, y n X ) T for z X. (9) It follows that R n = Y n Y n L(X) and K n = Y n Y n IR (n+1) (n+1) are given by R n z = n j= α j z, y j X y j for z X and ( K n )ij = α j y j, y i X (3) respectively. By L(X) we denote the Banach space of all linear and bounded operators from X into itself and the matrix K n is again called a correlation matrix; compare (15). Using a Lagrangian framework we derive the following optimality conditions for the optimization problem (P l ): R n ψ = λψ, (31) compare e.g. [HLB96, pp ] and [Vol1a, Section ]. Thus, it turns out that analogous to finite-dimensional POD, we obtain an eigenvalue problem; see (4). Note that R n is a bounded, self-adjoint and nonnegative operator. Moreover, since the image of R n has finite dimension, R n is also compact. By Hilbert-Schmidt theory (see e.g. [RS8, p. 3]) there exist an orthonormal basis {ψ i } i IN for X and a sequence {λ i } i IN of nonnegative real numbers so that R n ψ i = λ i ψ i, λ 1... λ d > and λ i = for i > d. (3) Moreover, V = span {ψ i } d. Note that {λ i} i IN as well as {ψ i } i IN depend on n. Remark 4. a) Setting σ i = λ i, i = 1,..., d, and we find v i = 1 σ i Y n ψ i for i = 1,..., d (33) K n v i = λ i v i and v i, v j α = δ ij, 1 i, j d. (34) Thus, {v i } d is an orthonormal basis of eigenvectors of K n for the image of K n. Conversely, if {v i } d is a given orthonormal basis for the image of

13 POD: Error Estimates and Suboptimal Control 13 K n, then it follows that the first d eigenfunctions of R n can be determined by ψ i = 1 σ i Y n v i for i = 1,..., d, (35) see (8). Hence, we can determine the POD basis by solving either the eigenvalue problem for R n or the one for K n. The relationship between the eigenfunctions of R n and the eigenvectors for K n is given by (33) and (35), which corresponds to SVD for the finite-dimensional POD. b) Let us introduce the matrices D = diag (α,..., α n ) IR (n+1) (n+1), K n = (( y j, y i X )) i,j n IR(n+1) (n+1). Note that the matrix K n is symmetric and positive semi-definite with rank K n = d. Then the eigenvalue problem (34) can be written in matrixvector-notation as follows: K n Dv i = λ i v i and v T i Dv j = δ ij, 1 i, j d. (36) Multiplying the first equation in (36) with D 1/ from the left and setting w i = D 1/ v i, 1 i d, we derive D 1/ Kn D 1/ w i = λ i w i and w T i w j = δ ij, 1 i, j d. where the matrix K n = D 1/ Kn D 1/ is symmetric and positive semidefinite with rank K n = d. Therefore, it turns out that (34) can be expressed as a symmetric eigenvalue problem. The sequence {ψ i } l solves the optimization problem (P l). This fact as well as the error formula below were proved in [HLB96, Section 3], for example. Proposition 1. Let λ 1... λ d > denote the positive eigenvalues of R n with the associated eigenvectors ψ 1,..., ψ d X. Then, {ψi n}l is a POD basis of rank l d, and we have the error formula J(ψ 1,..., ψ l ) = n j= α j yj l y j, ψ i X ψ i X = d i=l+1 λ i. (37) 3 Reduced-order modeling for dynamical systems In the previous section we have described how to compute a POD basis. In this section we focus on the Galerkin projection of dynamical systems utilizing the POD basis functions. We obtain reduced-order models and present error estimates for the POD solution compared to the solution of the dynamical system.

14 14 Michael Hinze and Stefan Volkwein 3.1 A general equation in fluid dynamics In this subsection we specify the abstract nonlinear evolution problem that will be considered in this section and present an existence and uniqueness result, which ensures Assumption (A1) introduced in Section.. We introduce the continuous operator R : V V, which maps D(A) into H and satisfies Rϕ H c R ϕ 1 δ1 V Aϕ δ1 H Rϕ, ϕ V,V c R ϕ 1+δ V ϕ 1 δ H for all ϕ D(A), for all ϕ V for a constant c R > and for δ 1, δ [, 1). We also assume that A + R is coercive on V, i.e., there exists a constant η > such that a(ϕ, ϕ) + Rϕ, ϕ V,V η ϕ V for all ϕ V. (38) Moreover, let B : V V V be a bilinear continuous operator mapping D(A) D(A) into H such that there exist constants c B > and δ 3, δ 4, δ 5 [, 1) satisfying B(ϕ, ψ), ψ V,V =, B(ϕ, ψ), φ V,V cb ϕ δ3 H ϕ 1 δ3 V B(ϕ, χ) H + B(χ, ϕ) H c B ϕ V χ 1 δ4 V Aχ δ4 H, B(ϕ, χ) H c B ϕ δ5 H ϕ 1 δ5 V ψ V φ δ3 V φ 1 δ3 H, χ 1 δ5 V Aχ δ5 H, for all ϕ, ψ, φ V, for all χ D(A). In the context of Section. we set F = B + R. Thus, for given f C(, T ; H) and y V we consider the nonlinear evolution problem d dt y(t), ϕ H + a(y(t), ϕ) + F (y(t)), ϕ V,V = f(t), ϕ H for all ϕ V and almost all t (, T ] and (39a) The following theorem guarantees (A1). y() = y in H. (39b) Theorem 1. Suppose that the operators R and B satisfy the assumptions stated above. Then, for every f C(, T ; H) and y V there exists a unique solution of (39) satisfying y C([, T ]; V ) L (, T ; D(A)) H 1 (, T ; H). (4) Proof. The proof is analogous to that of Theorem.1 in [Tem88, p. 111], where the case with time-independent f was treated.

15 POD: Error Estimates and Suboptimal Control 15 Example 1. Let Ω denote a bounded domain in IR with boundary Γ and let T >. The two-dimensional Navier-Stokes equations are given by ϱ ( u t + (u )u ) ν u + p = f in Q = (, T ) Ω, (41a) div u = in Q, (41b) where ϱ > is the density of the fluid, ν > is the kinematic viscosity, f represents volume forces and (u )u = ( u 1 u 1 x 1 + u u 1 x, u 1 u x 1 + u u x ) T. The unknowns are the velocity field u = (u 1, u ) and the pressure p. Together with (41) we consider nonslip boundary conditions and the initial condition u = u d on Σ = (, T ) Γ (41c) u(, ) = u in Ω. (41d) In [Tem88, pp , ] it was proved that (41) can be written in the form (18) and that (A1) holds provided the boundary Γ is sufficiently smooth. 3. POD Galerkin projection of dynamical systems Given a snapshot grid {t j } n j= and associated snapshots y,..., y n the space V l is constructed as described in Section.. We obtain the POD-Galerkin surrogate of (39) by replacing the space of test functions V by V l = span {ψ 1,..., ψ l }, and by using the ansatz Y (t) = l α i (t)ψ i (4) for its solution. The result is a l-dimensional nonlinear dynamical system of ordinary differential equations for the functions α i (i = 1,..., l) of the form M α + Aα + n(α) = F, Mα() = ( y, ψ j H ) l j=1, (43) where M = ( ψ i, ψ j H ) l i,j=1 denotes the POD mass matrix, A = (a(ψ i, ψ j )) l i,j=1 the POD stiffness matrix, n(α) = ( F (Y ), ψ j V,V ) l j=1 the nonlinearity, and F = ( f, ψ j H ) l j=1. We note that M is the identity matrix if in (P l) X = H is chosen. For the time discretization we choose m IN and introduce the time grid = τ < τ 1 <... < τ m = T, δτ j = τ j τ j 1 for j = 1,..., m,

16 16 Michael Hinze and Stefan Volkwein and set δτ = min{δτ j : 1 j m} and τ = max{δτ j : 1 j m}. Notice that the snapshot grid and the time grid usually does not coincide. Throughout we assume that τ/δτ is bounded uniformly with respect to m. To relate the snapshot grid {t j } n j= and the time grid {τ j} m j= we set for every τ k, k m, an associated index k = argmin { τ k t j : j n} and define σ n {1,..., n} as the maximum of the occurrence of the same value t k as k ranges over k m. The problem consists in finding a sequence {Y k } m k= in V l satisfying Y, ψ H = y, ψ H for all ψ V l (44a) and τ Y k, ψ H + a(y k, ψ) + F (Y k ), ψ V,V = f(τ k), ψ H (44b) for all ψ V l and k = 1,..., m, where we have set τ Y k = (Y k Y k 1 )/δτ k. Note that (44) is a backward Euler scheme for (39). For every k = 1,..., m there exists at least one solution Y k of (44). If τ is sufficiently small, the sequence {Y k } m k=1 is uniquely determined. A proof was given in [KV, Theorem 4.]. 3.3 Error estimates Our next goal is to present an error estimate for the expression m β k Y k y(τ k ) H, k= where y(τ k ) is the solution of (39) at the time instances t = τ k, k = 1,..., m, and the positive weights β j are given by β = δτ 1, β j = δτ j + δτ j+1 for j = 1,..., m 1, and β m = δτ m. Let us introduce the orthogonal projection P l n of X onto V l by P l n ϕ = l ϕ, ψ i X ψ i for ϕ X. (45) In the context of finite element discretizations, Pn l is called the Ritz projection.

17 Estimate for the choice X = V POD: Error Estimates and Suboptimal Control 17 Let us choose X = V in the context of Section.. Since the Hilbert space V is endowed with the inner product (16), the Ritz-projection P l n is the orthogonal projection of V on V l. We make use of the following assumptions: (H1) y W, (, T ; V ), where W, (, T ; V ) = {ϕ L (, T ; V ) : ϕ t, ϕ tt L (, T ; V )} is a Hilbert space endowed with its canonical inner product. (H) There exists a normed linear space W continuously embedded in V and a constant c a > such that y C([, T ]; W ) and a(ϕ, ψ) c a ϕ H ψ W for all ϕ V and ψ W. (46) Example. For V = H 1(Ω), H = L (Ω), with Ω a bounded domain in IR l and a(ϕ, ψ) = ϕ ψ dx for all ϕ, ψ H 1 (Ω), Ω choosing W = H (Ω) H 1(Ω) implies a(ϕ, ψ) ϕ W ψ H for all ϕ W, ψ V, and (46) holds with c a = 1. Remark 5. In the case X = V we infer from (16) that a(p l n ϕ, ψ) = a(ϕ, ψ) for all ψ V l, where ϕ V. In particular, we have P l n L(V ) = 1. Moreover, (H) yields P l n L(H) c P for all 1 l d where c P = cl/λ l (see [KV, Remark 4.4]) and c > depends on y, c a, and T, but is independent of l and of the eigenvalues λ i. The next theorem was proved in [KV, Theorem 4.7 and Corollary 4.11]. Theorem. Assume that (H1), (H) hold and that τ is sufficiently small. Then there exists a constant C depending on T, but independent of the grids {t j } n j= and {τ j} m j=, such that m k= β k Y k y(τ k ) H Cσ n τ( τ + t) y tt L (,T ;V ) ( d + C i=l+1 ( ψi, y σ n τ ) ) V + λ i + σ n τ t y t L δt (,T ;V ). (47) Remark 6. a) If we take the snapshot set Ṽ = span {y(t ),..., y(t n )}

18 18 Michael Hinze and Stefan Volkwein instead of V, we obtain instead of (47) the following estimate: m β k Y k y(τ k ) H k= C d i=l+1 ( ψi, y σ ( n 1 ) ) V + δt δτ + τ λ i + Cσ n τ t y t L (,T ;V ) + Cσ n τ( τ + t) y tt L (,T ;H) (compare [KV, Theorem 4.7]). As we mentioned in Remark 1 the factor (δt δτ) 1 arises on the right-hand of the estimate. While computations for many concrete situations show that d i=l+1 λ i is small compared to τ, the question nevertheless arises whether the term 1/(δτ δt) can be avoided in the estimates. However, we refer the reader to [HV3, Section 4], where significantly better numerical results were obtained using the snapshot set V instead of Ṽ. We refer also to [LV4], where the computed feedback gain was more stabilizing providing information about the time derivatives was included. b) If the number of POD elements for the Galerkin scheme coincides with the dimension of V then the first additive term on the right-hand side disappears. Asymptotic estimate Note that the terms {λ i } d, {ψ i} d and σ n depend on the time discretization of [, T ] for the snapshots as well as the numerical integration. We address this dependence next. To obtain an estimate that is independent of the spectral values of a specific snapshot set {y(t j )} n j= we assume that y W, (, T ; V ), so that in particular (H1) holds, and introduce the operator R L(V ) by Rz = T z, y(t) V y(t) + z, y t (t) V y t (t) dt for z V. (48) Since y W, (, T ; V ) holds, it follows that R is compact, see, e.g., [KV, Section 4]. From the Hilbert-Schmidt theorem it follows that there exists a complete orthonormal basis {ψ i } i IN for X and a sequence {λ i } i IN of nonnegative real numbers so that Rψ i = λ i ψ i, λ 1 λ..., and λ i as i. The spectrum of R is a pure point spectra except for possibly. Each non-zero eigenvalue of R has finite multiplicity and is the only possible accumulation point of the spectrum of R, see [Kat8, p. 185]. Let us note that T y(t) X dt = λ i and y X = y, ψ i X.

19 Due to the assumption y W, (, T ; V ) we have POD: Error Estimates and Suboptimal Control 19 lim R n R t L(V ) =, where the operator R n was introduced in (3). The following theorem was proved in [KV, Corollary 4.1]. Theorem 3. Let all hypothesis of Theorem be satisfied. Let us choose and fix l such that λ l λ l+1. If t = O(δτ) and τ = O(δt) hold, then there exists a constant C >, independent of l and the grids {t j } n j= and {τ j} m j=, and a t >, depending on l, such that m β k Y k y(τ k ) H C ( y ), ψi V + λ i ( i=l+1 ) + C τ t y t L (,T ;V ) + τ( τ + t) y tt L (,T ;V ) k= for all t t. (49) Remark 7. In case of X = H the spectral norm of the POD stiffness matrix with the elements ψ j, ψ i V, 1 i, j, d, arises on the right-hand side of the estimate (47); see [KV, Theorem 4.16]. For this reason, no asymptotic analysis can be done for X = H. 4 Suboptimal control of evolution problems In this section we propose a reduced-order approach based on POD for optimal control problems governed by evolution problems. For linear-quadratic optimal control problems we among other things present error estimates for the suboptimal POD solutions. 4.1 The abstract optimal control problem For T > the space W (, T ) is defined as W (, T ) = { ϕ L (, T ; V ) : ϕ t L (, T ; V ) }, which is a Hilbert space endowed with the common inner product (see, for example, in [DL9, p. 473]). It is well-known that W (, T ) is continuously embedded into C([, T ]; H), the space of continuous functions from [, T ] to H, i.e., there exists an embedding constant c e > such that ϕ C([;T ];H) c e ϕ W (,T ) for all ϕ W (, T ). (5) We consider the abstract problem introduced in Section.. Let U be a Hilbert space which we identify with its dual U, and let U ad U a closed and convex

20 Michael Hinze and Stefan Volkwein subset. For y H and u U ad we consider the nonlinear evolution problem on [, T ] d dt y(t), ϕ H + a(y(t), ϕ) + F (y(t)), ϕ V,V = (Bu)(t)), ϕ V,V for all ϕ V and (51a) y() = y in H, (51b) where B : U L (, T ; V ) is a continuous linear operator. We suppose that for every u U ad and y H there exists a unique solution y of (51) in W (, T ). This is satisfied for many practical situations, including, e.g., the controlled viscous Burgers and two-dimensional incompressible Navier-Stokes equations, see, e.g., [Tem88, Vol1b]. Next we introduce the cost functional J : W (, T ) U IR by J(y, u) = α 1 Cy z 1 W 1 + α Dy(T ) z W + σ u U, (5) where W 1, W are Hilbert spaces and C : L (, T ; H) W 1 and D : H W are bounded linear operators, z 1 W 1 and z W are given desired states and α 1, α, σ >. The optimal control problem is given by min J(y, u) s.t. (y, u) W (, T ) U ad solves (51). (CP) In view of Example 1 a standard discretization (based on, e.g., finite elements) of (CP) may lead to a large-scale optimization problem which can not be solved with the currently available computer power. Here we propose a suboptimal solution approach that utilizes POD. The associated suboptimal control problem is obtained by replacing the dynamical system (51) in (CP) through the POD surrogate model (43), using the Ansatz (4) for the state. With F replaced by ( (Bu)(t), ψ j H ) l j=1 it reads min J(α, u) s.t. (α, u) H 1 (, T ) l U ad solves (43). (SCP) At this stage the question arises which snapshots to use for the POD surrogate model, since it is by no means clear that the POD model computed with snapshots related to a control u 1 is also able to resolve the presumably completely different dynamics related to a control u u 1. To cope with this difficulty we present the following adaptive pseudo-optimization algorithm which is proposed in [AH, AH1]. It successively updates the snapshot samples on which the the POD surrogate model is to be based upon. Related ideas are presented in [AFS, Rav]. Choose a sequence of increasing numbers N j. Algorithm 4.1 (POD-based adaptive control) 1. Let a set of snapshots y i, i = 1,..., N be given and set j=.

21 POD: Error Estimates and Suboptimal Control 1. Set (or determine) l, and compute the POD modes and the space V l. 3. Solve the reduced optimization problem (SCP) for u j. 4. Compute the state y j corresponding to the current control u j and add the snapshots y j+1 i, i = N j +1,..., N j+1 to the snapshot set y j i, i = 1,..., N j. 5. If u j+1 u j is not sufficiently small, set j = j+1 and goto. We note that the term snapshot here may also refer to difference quotients of snapshots, compare Remark 1. We note further that it is also possible to replace its step 4. by 4. Compute the state y j corresponding to the current control u j and store the snapshots y j+1 i, i = N j + 1,..., N j+1 while the snapshot set y j i, i = 1,..., N j is neglected. Many numerical investigations on the basis of Algorithm 4.1 with step 4 can be found in [Afa]. This reference also contains a numerical comparison of POD to other model reduction techniques, including their applications to optimal open-loop control. To anticipate discussion we note that the number N j of snapshots to be taken in the j-th iteration ideally should be determined during the adaptive optimization process. We further note that the choice of l in step might be based on the information content E defined in (9), compare Section 5.. We will pick up these items again in Section 6. Remark 8. It is numerically infeasible to compute an optimal closed-loop feedback control strategy based on a finite element discretization of (51), since the resulting nonlinear dynamical system in general has large dimension and numerical solution of the related Hamilton-Jacobi-Bellman (HJB) equation is infeasible. In [KVX4] model reduction techniques involving POD are used to numerically construct suboptimal closed-loop controllers using the HJB equations of the reduced order model, which in this case only is low dimensional. 4. Error estimates for linear-quadratic optimal control problems It is still an open problem to estimate the error between solutions of (CP) and the related suboptimal control problem (SCP), and also to prove convergence of Algorithm 4.1. As a first step towards we now present error estimates for discrete solutions of linear-quadratic optimal control problems with a POD model as surrogate. For this purpose we combine techniques of [KV1, KV] and [DH, DH4, Hin4]. We consider the abstract control problem (CP) with F and U ad U. We note that J from (5) is twice continuously Fréchet-differentiable. In particular, the second Fréchet-derivative of J at a given point x = (y, u) W (, T ) U in a direction δx = (δy, δu) W (, T ) U is given by J(x)(δx, δx) = α 1 Cδy W 1 + α Dδy(T ) W + σ δu U.

22 Michael Hinze and Stefan Volkwein Thus, J(x) is a non-negative operator. The goal is to minimize the cost J subject to (y, u) solves the linear evolution problem y t (t), ϕ H + a(y(t), ϕ) = (Bu)(t), ϕ H (53a) for all ϕ V and almost all t (, T ) and y() = y in H. (53b) Here, y H is a given initial condition. It is well-known that for every u U problem (53) admits a unique solution y W (, T ) satisfying y W (,T ) C ( y H + u U ) for a constant C > ; see, e.g., [DL9, pp. 51-5]. If, in addition, y V and if there exist two constants c 1, c > with Aϕ, ϕ H c 1 ϕ D(A) c ϕ H for all ϕ D(A) V, then we have y L (, T ; D(A) V ) H 1 (, T ; H), (54) compare [DL9, p. 53]. From (54) we infer that y is almost everywhere equal to an element of C([, T ]; V ). The minimization problem, which is under consideration, can be written as a linear-quadratic optimal control problem min J(y, u) s.t. (y, u) W (, T ) U solves (53). (LQ) Applying standard arguments one can prove that there exists a unique optimal solution x = (ȳ, ū) to (LQ). There exists a unique Lagrange-multiplier p W (, T ) satisfying together with x = (ȳ, ū) the first-order necessary optimality conditions, which consist in the state equations (53), in the adjoint equations p t (t), ϕ H + a( p(t), ϕ) = α 1 C (Cȳ(t) z 1 (t)), ϕ H (55a) for all ϕ V and almost all t (, T ) and and in the optimality condition p(t ) = α D (Dȳ(T ) z ) in H, (55b) σū B p = in U. (56) Here, the linear and bounded operators C : W 1 L (, T ; H), D : W H, and B : L (, T ; H) U stand for the Hilbert space adjoints of C, D, and B, respectively.

23 Introducing the reduced cost functional POD: Error Estimates and Suboptimal Control 3 Ĵ(u) = J(y(u), u), where y(u) solves (53) for the control u U, we can express (LQ) as the reduced problem min Ĵ(u) s.t. u U. (ˆP) From (56) it follows that the gradient of Ĵ at ū is given by Let us define the operator G : U U by Ĵ (ū) = σū B p. (57) G(u) = σu B p, (58) where y = y(u) solves the state equations with the control u U and p = p(y(u)) satisfies the adjoint equations for the state y. As a consequence of (56) it follows that the first-order necessary optimality conditions for ( ˆP) are G(u) = in U. (59) In the POD context the operator G will be replaced by an operator G l : U U which then represents the optimality condition of the optimal control problem (SCP). The construction of G l is described in the following. Computation of the POD basis Let u U be a given control for (LQ) and y = y(u) the associated state satisfying y C 1 ([, T ]; V ). To keep the notation simple we apply only a spatial discretization with POD basis functions, but no time integration by, e.g., an implicit Euler method. Therefore, we apply a continuous POD, where we choose X = V in the context of Section.. Let us mention the work [HY], where estimates for POD Galerkin approximations were derived utilizing also a continuous version of POD. We define the bounded linear Y : H 1 (, T ; IR) V by Yϕ = T ϕ(t)y(t) + ϕ t (t)y t (t) dt for ϕ H 1 (, T ; IR). Notice that the operator Y is the continuous variant of the discrete operator Y n introduced in (8). The adjoint Y : V H 1 (, T ; IR) is given by ( Y z ) (t) = z, y(t) + y t (t) V for z V. (compare (9)). The operator R = YY L(V ) is already introduced in (48).

24 4 Michael Hinze and Stefan Volkwein Remark 9. Analogous to the theory of singular value decomposition for matrices, we find that the operator K = Y Y L(H 1 (, T ; IR)) given by ( ) T Kϕ (t) = y(s), y(t) V ϕ(s)+ y t (s), y t (t) V ϕ t (s) ds has the eigenvalues {λ i v i (t) = 1 λ i } ( Y ψ i and the eigenfunctions ) (t) = 1 λ i ψ i, y(t) + y t (t) V for ϕ H 1 (, T ; IR) for i {j IN : λ j > } and almost all t [, T ]. In the following theorem we formulate properties of the eigenvalues and eigenfunctions of R. For a proof we refer to [HLB96], for instance. Theorem 4. For every l N the eigenfunctions ψ1,..., ψ l minimization problem V solve the min J(ψ 1,..., ψ l ) s.t. ψ j, ψ i X = δ ij for 1 i, j l, (6) where the cost functional J is given by J(ψ,..., ψ l ) = T y(t) l y(t), ψ i V ψ i X + yt (t) l y t (t), ψ i V ψ i V dt. Moreover, the eigenfunctions {λ i } i IN and eigenfunctions {ψi } i IN of R satisfy the formula J(ψ1,..., ψl ) = λ i. (61) i=l+1 Proof. The proof of the theorem relies on the fact that the eigenvalue problem Rψ i = λ i ψ i for i = 1,..., l is the first-order necessary optimality condition for (6). For more details we refer the reader to [HLB96]. Galerkin POD approximation Let us introduce the set V l = span {ψ1,..., ψl } V. To study the POD approximation of the operator G we introduce the orthogonal projection P l of V onto V l by P l ϕ = l ϕ, ψi V ψi for ϕ V. (6)

25 (compare (45)). Note that POD: Error Estimates and Suboptimal Control 5 J(ψ,..., ψ l ) = T y(t) P l y(t) + yt (t) P l y t (t) dt = V V i=l+1 λ i. (63) From (16) it follows directly that a(p l ϕ, ψ) = a(ϕ, ψ) for all ψ V l, where ϕ V. Clearly, we have P l L(V ) = 1. Next we define the approximation G l : U U of the operator G by where p l W (, T ) is the solution to G l (u) = σu B p l, (64) p l t (t), ψ H + a(pl (t), ψ) = α 1 C (Cy l z 1 ), ψ H for all ψ V l and t (, T ) a.e. and p l (T ) = α P l( D (Dy l (T ) z ) ) (65a) (65b) and y l W (, T ), which solves y l t(t), ψ H + a(y l (t), ψ) = (Bu)(t), ψ H (66a) for all ψ V l and almost all t (, T ) and y l () = P l y (66b) Notice that G l (u) = are the first-order optimality conditions for the optimal control problem min Ĵ l (u) s.t. u U, where Ĵ l (u) = J(y l (u), u) and y l (u) denotes the solution to (66). It follows from standard arguments (Lax-Milgram lemma) that the operator G l is well-defined. Furthermore we have Theorem 5. The equation G l (u) = in U (67) admits a unique solution u l U which together with the unique solution u of (59) satisfies the estimate u u l U 1 ) ( B (P P l )Bu σ U + B (S Sl )C z 1 U. (68) Here, P := S C CS, P l := S l C CS l, with S, S l denoting the solution operators in (53) and (66), respectively.

26 6 Michael Hinze and Stefan Volkwein A proof of this theorem immediately follows from the fact, that u l is a admissible test function in (59), and u in (67). Details will be given in [HV5]. Remark 1. We note, that Theorem 5 remains also valid in the situation where admissible controls are taken from a closed convex subset U ad U. The solutions u, u l in this case satisfy the variational inequalities and G(u), v u U for all v U ad, G l (u l ), v u l U for all v U ad, so that adding the first inequality with v = u l and the second with v = u and straightforward estimation finally give (68) also in the present case. The crucial point here is that the set of admissible controls is not discretized a- priori. The discretization of the optimal control u l is determined by that of the corresponding Lagrange multiplier p l. For details of this discrete concept we refer to [Hin4]. It follows from the structure of estimate (68), that error estimates for y y l and p p l directly lead to an error estimate for u u l. Proposition. Let l IN with λ l > be fixed, u U and y = y(u) and p = p(y(u)) the corresponding solutions of the state equations (53) and adjoint equations (55) respectively. Suppose that the POD basis of rank l is computed by using the snapshots {y(t j )} n j= and its difference quotients. Then there exist constants c y, c p > such that and y l y L (,T ;H) + yl y L (,T ;V ) c y i=l+1 λ i (69) p l p L (,T ;V ) ( c p i=l+1 ) λ i + P l p p L (,T ;V ) + Pl p t p t L (,T ;V ), (7) where y l and p l solve (66) and (65), respectively, for the chosen u inserted in (66a). Proof. Let y l (t) y(t) = y l (t) P l y(t) + P l y(t) y(t) = ϑ(t) + ϱ(t), where ϑ = y l P l y and ϱ = P l y y. From (16), (6), (63) and the continuous embedding H 1 (, T ; V ) L (, T ; H) we find

27 POD: Error Estimates and Suboptimal Control 7 ϱ L (,T ;H) + ϱ L (,T ;V ) c E λ i (71) i=l+1 with an embedding constant c E >. Utilizing (53) and (66) we obtain ϑ t (t), ψ H + a(ϑ(t), ψ) = y t (t) P l y t (t), ψ H for all ψ V l and almost all t (, T ). From (16), (17) and Young s inequality it follows that d dt ϑ(t) H + ϑ(t) V c V y t (t) P l y t (t) V. (7) Due to (66b) we have ϑ() =. Integrating (7) over the interval (, t), t (, T ], and utilizing (37), (45) and (63) we arrive at for almost all t (, T ). Thus, t ϑ(t) H + ϑ(s) V ds c V i=l+1 λ i T esssup ϑ(t) H + ϑ(s) V ds c V t [,T ] i=l+1 λ i. (73) Estimates (71) and (73) imply the existence of a constant c y > such that (69) holds. We proceed by estimating the error arising from the discretization of the adjoint equations and write p l (t) p(t) = p l (t) P l p(t) + P l p(t) p(t) = θ(t) + ρ(t), where θ = p l P l p and ρ = P l p p. From (16), (5), and (65b) we get θ(t ) H α D L(H,W 1) yl (T ) y(t ) H α D L(H,W 1) yl y C([,T ];H). Thus, applying (5), (69) and the techniques used above for the state equations, we obtain T esssup θ(t) H + θ(s) V ds t [,T ] ( c V c V c ec y D 4 L(H,W 1) i=l+1 Hence, there exists a constant c p > satisfying (7). λ i + p t P l p t L (,T ;V )).

28 8 Michael Hinze and Stefan Volkwein Remark 11. a) The error in the discretization of the state variable is only bounded by the sum over the not modeled eigenvalues λ i for i > l. Since the POD basis is not computed utilizing adjoint information, the term P l p p in the H 1 (, T ; V )-norm arises in the error estimate for the adjoint variables. For POD based approximation of partial differential equations one cannot rely on results clarifying the approximation properties of the POD-subspaces to elements in function spaces as e.g. L p or C. Such results are an essential building block for e.g. finite element approximations to partial differential equations. b) If we have already computed a second POD basis of rank l IN for the adjoint variable, then we can express the term involving the difference P lp p by the sum over the eigenvalues corresponding to eigenfunctions, which are not used as POD basis functions in the discretization. c) Recall that {ψi } i IN is a basis of V. Thus we have T p(t) P l p(t) V dt T i=l+1 a(p(t), ψ i ) dt. The sum on the right-hand side converges to zero as l tends to. However, usually we do not have a rate of convergence result available. In numerical applications we can evaluate p P l p L (,T ;V ). If the term is large then we should increase l and include more eigenfunctions in our POD basis. d) For the choice X = H we have instead of (71) the estimate ϱ L (,T ;H) + ϱ L (,T ;V ) C S i=l+1 λ i, where C is a positive constant, S denotes the stiffness matrix with the elements S ij = ψj, ψ i V, 1 i, j l, and stands the spectral norm for symmetric matrices, see [KV, Lemma 4.15]. Applying (58), (64), and Proposition we obtain for every u U G l (u) G(u) U c G ( λ i i=l+1 + P l p p L (,T ;H) + Pl p t p t L (,T ;H) ) (74) for a constant c G > depending on c λ and B. Suppose that u 1, u U are given and that y l 1 = y l 1(u 1 ) and y l = y l (u ) are the corresponding solutions of (66). Utilizing Young s inequality it follows that there exists a constant c V > such that

29 POD: Error Estimates and Suboptimal Control 9 y l 1 y l L (,T ;H) + yl 1 y l L (,T ;V ) c V B L(U,L (,T ;H)) u 1 u U. (75) Hence, we conclude from (65) and (75) that p l 1 pl L (,T ;H) + pl 1 pl L (,T ;V ) max ( α 1 c V C L(L (,T ;H),W, α 1) D L(H,W ( ) )) y1 l yl L (,T ;H) + yl 1 yl L (,T ;V ) C u 1 u U. (76) where C = c4 V B L(U,L (,T ;H)) max ( α 1c V C L(L (,T ;H),W, α 1) D ) L(H,W ). If the POD basis of rank l is computed for the control u 1, then (64), (74) and (76) lead to the existence of a constant Ĉ > satisfying G l (u ) G(u 1 ) U G l(u ) G l (u 1 ) U + G l(u 1 ) G(u 1 ) U Ĉ u u 1 U ( + Ĉ ) λ i + P l p 1 p 1 L (,T ;V ) + Pl (p 1 ) t (p 1 ) t L (,T ;V ). i=l+1 Hence, G l (u ) is close to G(u 1 ) in the U-norm provided the terms u 1 u U and i=l+1 λ i are small and provided the l POD basis functions ψ1,..., ψl leads to a good approximation of the adjoint variable p 1 in the H 1 (, T ; V )- norm. In particular, G l (u) in this case is small, if u denotes the unique optimal control of the continuous control problem, i.e., the solution of G(u) =. We further have that both, G and G l are Fréchet differentiable with constant derivatives G σid B p and G l σid B (p l ). Moreover, since B p and B p are selfadjoint positive operators, G and G l are invertible, satisfying (G ) 1 L(U), (G l ) 1 L(U) 1 σ. Since G l also is Lipschitz continuous with some positive constant K we now may argue with a Newton-Kantorovich argument [D85, Theorem 15.6] that the equation G l (v) = in U admits a unique solution in u l B ɛ (u), provided (G l ) 1 G l (u) U ɛ and Kɛ σ < 1.