INTERPRETATION OF PROPER ORTHOGONAL DECOMPOSITION AS SINGULAR VALUE DECOMPOSITION AND HJB-BASED FEEDBACK DESIGN PAPER MSC-506

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1 INTERPRETATION OF PROPER ORTHOGONAL DECOMPOSITION AS SINGULAR VALUE DECOMPOSITION AND HJB-BASED FEEDBACK DESIGN PAPER MSC-506 SIXTEENTH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL THEORY OF NETWORKS AND SYSTEMS (MTNS2004), KATHOLIEKE UNIVERSITEIT LEUVEN, BELGIUM, JULY 5-9, 2004 S. VOLKWEIN Abstract. The proper orthogonal decomposition (POD) is a method to compute a reduced basis for a dynamical system. In this work the close connection between the singular value decomposition (SVD) and POD is studied in complex Hilbert spaces. For practical computation of the POD basis functions perturbation error bounds are presented. Moreover the combination of model reduction techniques based on POD with the numerical treatment of the Hamilton-Jacobi-Bellman (HJB) equation for infinite horizon optimal control problems by a modification of an algorithm originated by Gonzales-Rofman and further developed by Falcone-Ferretti is addressed. 1. Introduction. 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. We concern with the proper orthogonal decomposition which was used as an efficient procedure to compute low-order models, see for instance [1, 16] and the references given therein. The paper is organized in the following manner: In Section 2 the close connection between POD and SVD is investigated in complex finite-dimensional Hilbert spaces. For practical issues we consider the computation of the POD-basis and present error bounds in Section 3, which are based on the perturbation theory for the eigenvalues and the eigenvectors of Hermitean matrices. The feedback-design based on the Hamilton-Jacobi-Belman (HJB) equation is briefly mentioned in Section POD and SVD. First we formulate the singular value decomposition for linear mappings from a finite dimensional complex Hilbert space V into another finite dimensional complex Hilbert space W. Proposition 2.1. Let F : V W be a linear operator, where V and W denote two finite dimensional complex Hilbert spaces with inner products (, ) V and (, ) W, Date: June 8, Mathematics Subject Classification. 35Kxx, 49Lxx, 65Kxx. Key words and phrases. Proper orthogonal decomposition, singular value decomposition, matrix perturbation theory, Hamilton-Jacobi-Bellman equation, closed loop control. 1

2 2 S. VOLKWEIN respectively, and with dim V = m and dim W = n, where n m. Then there exist real numbers σ 1 σ 2... σ m 0 and orthonormal bases {v k } m of V and {w k } n of W such that (2.1) F (v k ) = σ k w k and F (w k ) = σ k v k for k = 1,..., m. Proof. Consider the linear mapping F F : V V, where F : W V denotes the Hilbert space adjoint of F defined by (F (v), w) W = (v, F (w)) V for all v V and w W. Then it is obvious that F F is self-adjoint. Hence, there exist an orthonormal basis {v k } m of V and real eigenvalues λ 1 λ 2... λ m such that (2.2) F F (v k ) = λ k v k for k = 1,..., m. Since F F is non-negative, the eigenvalues λ k of F F are non-negative. Let l {1,..., m} such that λ l > 0, but λ l+1 = 0. Then we set σ k = λ k for k = 1,..., m and w k = F (v k )/σ k for k = 1,..., l. It follows that F (v k ) = σ k w k for k = 1,..., l and (w k, w l ) W = 1 (F (v k ), F (v l )) σ k σ W = 1 (F F (v k ), v l ) l σ k σ V = σ kδ kl l σ l for k, l {1,..., l}. Thus, w 1,..., w l are orthonormal in W. We choose orthonormal w l+1,..., w n in such a way that {w k } n is an orthonormal basis of W. Since σ k = 0 for k > l, we obtain that the first identity of (2.1) holds for k = 1,..., m, whereas the second one follows from (2.2). Example 2.2. For V = C m and W = C n Proposition 2.1 is well-known as the singular value decomposition of matrices, see, e.g., [13]. Proposition 2.1 and its proof imply directly the next corollary. Corollary 2.3. Under the hypotheses of Proposition 2.1 we have F F (w k ) = σ 2 k w k and F F (v k ) = σ 2 k v k for k = 1,..., m n. By L(V, W ) we denote the normed linear space of all bounded linear operators from V into W. Let V = {v 1,..., v m } and W = {w 1,..., w n } be bases of V and W, respectively. The uniquely determined representation matrix MW V (F ) = ((a ij)) C n m of a given F L(V, W ) with respect to V and W is defined by n F (v j ) = a ij w i for j = 1,..., m. i=1 It is well-known that the mapping M V W : L(V, W ) Cn m is linear and bijective. Remark 2.4. Let F L(V, W ). Due to Proposition 2.1 there exist a diagonal matrix Σ = diag (σ 1,..., σ m ) C m m and orthonormal bases V and W of V and W, respectively, so that MW V (F ) = (Σ 0)T, where 0 is an (n m) m matrix of zeros. The previous remark motivates the next definition. Definition 2.5. Let F and Σ be given as in Proposition 2.1 and Remark 2.4, respectively. We define a norm on L(V, W ) by ( m 1/2 F = Σ F = σk) 2 for F L(V, W ), where F denotes the Frobenius norm. Furthermore, the rank of F is given by rank Σ.

3 INTERPRETATION OF POD AS SVD 3 Remark 2.6. Since F is a norm, the mapping is a norm on L(V, W ). Furthermore, the properties of the Frobenius norm imply that Σ F = MW V (F ) F for any orthonormal bases V and W of V and W, respectively. Now we formulate POD as a constrained minimization problem. For that purpose let X be a complex Hilbert space endowed with the inner product (, ) X and norm X. Let y 1,..., y m X be given elements. We assume that dim X m and that y 1,..., y m are linearly independent in X. Otherwise we decrease m. We set endowed with the inner product (, ) X. W = span {y 1,..., y m } X Example 2.7. The ensemble can be determined from a dynamical system. For this purpose let us consider the following semi-linear initial value problem: { dy(t) (2.3) + Ay(t) = f(t, y(t)) for t (0, T ), dt y(0) = φ, where A is the infinitesimal generator of a C 0 -semigroup S(t), t > 0, on X, φ X and f : [0, T ] X X is continuous in t and uniformly Lipschitz-continuous on X for every t. Problem (2.3) has a unique mild solution y C([0, T ]; X) given by the integral representation y(t) = S(t)φ + t 0 S(t s)f(s, y(s)) ds for t [0, T ], see, for instance, [12]. Then for given time instances 0 t 1 <... < t m T the members of the ensemble can be given by the mild solution to (2.3): y j = y(t j ) for j = 1,..., m. Let {w k } m denote any orthonormal basis for W. Then we have (2.4) y j = (y j, w k ) X w k for j = 1,..., m. For any l N, l < m, we want to find that orthonormal basis of W, which minimizes the mean square error between the members of the ensembles and their corresponding l-th partial sum of (2.4): l 2 (P l min J(w 1,..., w l ) = y j (y j, w k ) ) X w k X subject to (w i, w j ) X = δ ij for 1 i, j l. We call the solution {w k } l to (Pl ) the POD-basis of rank l. To solve (P l ) we shall apply Proposition 2.1 with V = C m. Let us introduce the linear mapping (2.5) Y L(C m, W ) with Y ( e k ) = y k for k = 1,..., m, where { e k } m denotes the canonical basis in Cm. Obviously, we obtain that Y ( v) = m ( v, e k) C my k for v C m holds. From the definition of the adjoint Y L(W, C m ) of Y we derive that Y (w) = m (w, y k) X e k for w W. This imply the identities (2.6) Y Y = (, y k ) X y k and Y Y = K,

4 4 S. VOLKWEIN where (2.7) K C m m with K ij = (y j, y i ) X denotes the correlation matrix. Example 2.8. Let X = C n endowed with the Euclidean norm and the matrix Y C n m consist in the members y j C n of the ensemble. Then Y satisfies (2.5) and Y = Y F. Theorem 2.9 (POD and SVD). Let σ 1 σ 2... σ m denote the singular values of Y and let V = { v 1,..., v m } and W = {w 1,..., w m } be the corresponding orthonormal bases of C m and W, respectively, determined by Proposition 2.1 such that Y ( v k ) = σ k w k for k = 1,..., m. If we define Y l L(C m, W ) by (2.8) Y l = Y on span { v 1,..., v l } and Y l = 0 otherwise, problem (P l ) is solved by Y l, and the solution to (P l ) is given by w k for k = 1,..., l. Moreover, for all F L(C m, W ) with rank F = l we have (2.9) Y F 2 Y Y l 2 = σk. 2 Proof. Definition 2.5 yield ( m 1/2 ( l 1/2. Y = σk) 2 and Y l = σk) 2 From y j = m (y j, w k ) X w k for j = 1,..., m and from (2.6) we infer that Y Y l = σk 2 = (Y Y 2 w k, w k ) X = (y j, w k ) X w k = J(w 1,..., w l ). Let F l be an arbitrary linear mapping with rank F = l. Then the matrix M V W (F l ) has rank l. Due to Remark 2.6 and Theorem 6.7 in [13] we obtain that Y F l = Σ F = MW(Y V F l ) F = MW(Y V ) MW(F V l ) F ( m 1/2, MW(Y V ) MW(Y V l ) F = MW(Y V Y l ) F = σk) 2 where Σ is a diagonal matrix containing the singular values of Y F l. Hence, the solution of (P l ) is given by {w k } l and argmin (Pl ) = m σ2 k. Remark Theorem 2.9 extends the results in [8], where Y and F were complex matrices. Example Let us consider the Hilbert space X = L 2 (Ω; C), where Ω denotes a bounded domain in R d, d N. We define the kernel (2.10) r(x, x) = y j (x)y j ( x) for x, x Ω, X

5 INTERPRETATION OF POD AS SVD 5 which is often called the averaged autocorrelation function. Then Y Y is the integral operator Y Y (w) = w( x)r(, x) d x for w W, Ω whereas Y Y is the correlation matrix K defined in (2.7). If we extend Y Y as a mapping from L 2 (Ω; C) into itself by (2.11) R(v)(x) = v( x)r(x, x) d x for v L 2 (Ω; C) Ω then the existence of the non-negative eigenvalues σ σ 2 m 0 follows also from the Hilbert-Schmidt theorem if X is separable. Obviously, we have r(x, ) W for every x Ω. Hence, r(x, x) = for every x, x Ω. = (r(x, ), w k ) L 2w k ( x) = σkw 2 k (x)w k ( x) Y Y (w k )(x)w k ( x) To determine the POD-basis of rank l we have to compute the singular value decomposition of the mapping Y. If m dim X holds, one may compute the PODbasis as stated in Proposition 2.12 below, which follows directly from Corollary 2.3, (2.7) and Theorem 2.9. Corollary The POD-basis {w k } l 1) Solve the eigenvalue problem of rank l can be determined as follows: (EVP) K v k = λ k v k for k = 1,..., l to determine the positive eigenvalues {λ k } l and eigenvectors { v k} l. 2) Set w k = Y ( v k )/ λ k for k = 1,..., l. Remark The eigenvalue problem (EVP) has dimension m m and can be solved by a numerical algorithm. If X is an infinite dimensional Hilbert space, the elements (y j, y i ) X, 1 i, j m, of the correlation matrix K has to be computed by a numerical quadrature formula. 3. Approximation of (EVP). In many applications X is an infinite dimensional Hilbert space, which has to be discretized for numerical realization. For that purpose we suppose that we are given a family {h}, h > 0, with accumulation point zero and a family {X h } h of Hilbert spaces with finite dimension. In practice, the parameter h is called mesh-size of the finite dimensional space and h varies over a sequence. Furthermore, we introduce a family {Π h } h>0 of bounded linear and surjective restrictions from X onto X h. Example 3.1. An important class of discretization methods is provided by an Galerkin approach, where for n N the finite dimensional space is given by X h = span {ϕ 1,..., ϕ n },

6 6 S. VOLKWEIN and the set {ϕ i } n i=1 is linearly independent in X. Then we define the family {Πh } h as follows: n ( n (3.1) Π h v = (Mij) h 1 (v, ϕ j ) X )ϕ i for any v X, i=1 where M h denotes the mass matrix (3.2) M h = (( Mij h )) C n n with M h ij = (ϕ j, ϕ i ) X. If {ϕ i } n i=1 is an orthonormal set in X, the matrix M h is the identity in C n n and (3.1) reduces to the n-th partial sum of the Fourier expansion of the element v. Now we prove that Π h is the orthogonal bounded projection from X onto X h : For every h the operator Π h belongs to L(X, X h ), the normed linear space containing all bounded and linear operators mapping from X into X h : From (3.1) it follows directly that Π h is linear for each h. The identity (3.3) (Π h v, v h ) X = (v, v h ) X for any v X and v h X h. implies that Π h v v (X h ) for all v X and for each h. Thus, Π h v 2 X Πh v 2 X + v Πh v 2 X = v 2 X for all v X, so that the family {Π h } h is bounded for each h. Π h is an orthogonal projection: Since (Π h ) 2 = Π h holds for each h, the operators Π h are projections. From (3.3) it follows that these projections are orthogonal. Each member of the ensemble is approximated by the projection of itself onto the space X h yj h = Π h y j X h for j = 1,..., m, and together with (P l ) we investigate the finite dimensional discrete problems (P l h ) min J h (w1 h,..., wl h l ) = yj h (yj h, wk) h X w h k 2 X subject to (wi h, wh j ) = δ X ij for 1 i, j l. From Corollary 2.12 we directly infer the next result. Proposition 3.2. Assume that for each h the set span {y h 1,..., y h m} has dimension m h with l m h m. Analogously to (2.5) we introduce the family of mappings Y h L(C m, W ) with Y h ( e k ) = y h k for k = 1,..., m. Let {wk h}l be determined as follows: 1) Solve the eigenvalue problem (EVP h ) K h v h k = λ k v h k for k = 1,..., l, where K h C m m with K h i,j = (yh j, yh i ) X, to determine the positive eigenvalues {λ h k }l and eigenvectors { vh k }l. 2) Set w h k = Y h ( v h k )/ λ h k for k = 1,..., l. Then (w h 1,..., w h l ) solves (Pl h ) satisfying argmin (Pl h ) = m λ h k.

7 INTERPRETATION OF POD AS SVD 7 Remark 3.3. Clearly, K h is positive semi-definite. Since the set {Π h y j } m is not necessarily linear independent, K h need not to be positive definite. Example 3.4. As in Example 3.1 we are given n linear independent functions {ϕ i } n i=1 in X and set X h = span {ϕ 1,..., ϕ n }. Assume that the coefficient matrix C h C n m arises from discretization of the ensemble {y j } m such that n yj h = Cijϕ h i. i=1 Then the eigenvalue problem (EVP h ) leads to (3.4) (C h ) H M h C h v h k = λ h k v h k for k = 1,..., l, where the mass matrix M h was already introduced in (3.2). Let us introduce the family {K(h)} h of matrices by { K h for h > 0, (3.5) K(h) = K for h = 0. Since K(h) is positive semi-definite, there exist a number m h m depending on h and m real eigenvalues such that λ 1 (h) λ 2 (h)... λ m h(h) > λ mh +1(h) =... = λ m (h) = 0 for h 0. In the next proposition we give a sufficient condition for the right-continuity of K(h) at h = 0. Proposition 3.5. Let K(h) be given by (3.5). pointwise convergent in X, i.e. (3.6) lim h 0 Π h u = u for any u X If the family of restrictions is then K(h) is right-continuous at h = 0. If in addition there exists ε > 0 such that (3.7) max 1 j m Πh y j y j X = O(h ε ) for h 0, then K K(h) 2 = O(h ε ) for h 0, where 2 denotes the spectral norm for Hermitean matrices. Proof. Let k {1,..., m} such that y k X = max 1 i m y i X. Then we estimate for h > 0 K K(h) 2 K K h = max K ij Kij h max 1 i m 1 i m y i X y j Π h y j X + Π h y j X y i Π h y i X m y k X y j Π h y j X +m y k X Π h L(X,Xh ) y i Π h y i X.

8 8 S. VOLKWEIN With (3.6) holding Π h u X is bounded for all u X. According to the principle of uniform boundedness (see, e.g., [18]) there exists a constant C > 0 independent of h such that Π h L(X,Xh ) C for all h. Hence, we have lim K K(h) 2 = 0. h 0 If in addition (3.7) is valid, the claim follows directly. Remark 3.6. Notice that K is positive definite, but that K(h) is only positive semi-definite for h > 0. If (3.6) holds, K(h) converge to K as h tends to zero. Thus, K(h) is positive definite for sufficiently small h, i.e. the family {Π h y j } m is linear independent for sufficiently small h. In particular, the hypothesis l dim(span {y1 h,..., yh m}) m in Proposition 3.2 holds for sufficiently small h. Example 3.7. Let us turn back to Example 2.7. If in addition φ belongs to the domain D(A) of A and f is also Lipschitz-continuous in the first variable then the mild solution of (2.3) is a classical one, i.e., y(t) D(A) for 0 < t < T, y(t) is differentiable on (0, T ) and (2.3) holds on [0, T ), see, e.g., [12]. Hence the members of the ensemble y j = y(t j ) for j = 1,..., m are more regular than the elements of X and there exists a constant C > 0 such that max y j 1 j m D(A) C is satisfied. For appropriate approximation schemes we have estimates of the following form max 1 j m Πh y j y j X max y j 1 j m D(A) h ε for some ε > 0 so that assumption (3.7) can be fulfilled. In the next theorem a perturbation bound for the eigenvalues is presented, which was proved, e.g., in [5]. Theorem 3.8. For any i {1,..., m} we have λ i λ i (h) K K(h) 2, i.e. the condition number of an eigenvalue of K is one. Remark 3.9. From Theorem 3.8 we infer that the condition number of the eigenvalues is equal to one. An analogous perturbation bound also holds for the singular values of the mapping Y, see [5]. With (3.6) and (3.7) holding we conclude that i = 1,..., m. λ i λ i (h) = O(h ε ) for h 0, The sensitivity or condition of an eigenvector depends on the gap of its corresponding eigenvalue: a small gap implies a sensitive eigenvector. This fact is formulated in the next theorem proved in [5]. Theorem For i {1,..., m} let v i and v i (h) denote the i-th eigenvector of K and K(h), respectively. Then 1 2 sin(2θ) K K(h) 2 min j i λ j λ i if min j i λ j λ i > 0 and 1 2 sin(2θ) K K(h) 2 if min min j i λ j (h) λ i (h) λ j(h) λ i (h) > 0, j i where θ i denote the acute angle between v i and v i (h).

9 INTERPRETATION OF POD AS SVD 9 Remark Note that sin(2θ)/2 sin θ θ if θ 1. Let us mention that frequently we know only the eigenvalues of K(h), since they are typically the output of the eigenvalue algorithm that we have used. In this case we can only estimate min j i λ j λ i. Finally, we establish a result for sums of eigenvalues and then specialize it to a single eigenvalue. For a proof of the next proposition we refer the reader to [14]. Proposition For any h > 0 and i = 1,..., m Remark λ i (h) [λ i + λ m (h), λ i + λ 1 (h)]. a) Note that m [λ i + λ m (h), λ i + λ 1 (h)] i=1 contains all eigenvalues of K(h). However, we know just which eigenvalue to look for in each interval. Moreover it is impossible for an eigenvalue corresponding to one of a cluster of overlapping intervals to migrate outside its own interval. b) The intervals are not symmetric about the eigenvalues λ i. In fact if λ m is positive, the i-th interval will not contain the eigenvalue λ i. This occurs if K K(h) is positive definite. Hence: If K K(h) is positive definite, its eigenvalues must increase. 4. HJB-POD based feedback design. In many applications the discretization of optimal control problems for time dependent partial differential equations, e.g., for the unsteady Navier-Stokes equations, require the solution of nonlinear systems with a large number of degrees of freedom. In particular, to compute closed loop controls in state feedback form we have to solve the Hamilton-Jacobi-Bellman (HJB) equation, which is numerically infeasible for parabolic differential equations on a standard workstation equipment till today if classical approximations like finite elements or finite differences are used. In [9] model reduction is applied to reduce the number of unknowns significantly. The obtained low-dimensional models should guarantee a reasonable performance of the controlled plant while being computationally tractable. Proper orthogonal decomposition (POD) provides a method for deriving appropriate low order models. It can be thought of as a Galerkin approximation in the spatial variable, built from functions corresponding to the solution of the physical system at pre-specified time instances. These are called the snapshots. Due to possible linear dependence or almost linear dependence a singular value decomposition of the snapshots is carried out (see Corollary 2.12) and the leading generalized eigenfunctions are chosen as a basis, referred to as the POD basis. Once a low order model of the dynamical system is available feedback synthesis based on approximate solutions to the stationary HJB equation becomes feasible. The feasibility of the proposed approach is demonstrated by means of an optimal boundary control problem for the Burgers equation. Compared to open loop control much less attention has been paid to the important problem of closed loop control. We mention the work by Byrnes-Gilliam-Shubov [3], where a fixed feedback-operator is used and analyzed and Burns-Kang [2] where the feedback synthesis is based on Riccati operators for the linearized equations. In [6] instantaneous control was applied to construct a feedback law which matches a desired state, but at considerable

10 10 S. VOLKWEIN control costs. In [8] the authors utilized model reduction with POD to construct a suboptimal feedback synthesis, and an optimal output feedback reduced-order control law was designed by POD discretization in [10]. In [9] the HJB-based approach is utilized numerically for an optimal boundary control problem for the viscous Burgers equation. Let u a u b. We set U ad = {u R : u a u u b } and define the set of admissible controls (4.1) U ad = {u L 2 loc(0, ) : u(t) U ad for almost all t (0, )}. For a control u U ad we consider the viscous Burgers equation (4.2a) (4.2b) (4.2c) (4.2d) y t νy xx + yy x = 0 in Q = (0, ) Ω, νy x (, 0) + σ 0 y(, 0) = u νy x (, 1) + σ 1 y(, 1) = g in (0, ), in (0, ), y(0, ) = y in Ω = (0, 1), where y L 2 (Ω) is a given initial condition, and σ 0, σ 1, g are real numbers. Henceforth we consider weak solutions y W loc (0, ; V ) of (4.2) satisfying (4.2d) and (4.3) y t (t), ϕ V,V + σ 1y(t, 1)ϕ(1) σ 0 y(t, 0)ϕ(0) y x (t), ϕ + y(t)y (t)ϕ dx = gϕ(1) uϕ(0), Ω for all ϕ H 1 (Ω) and t (0, ) a.e. For the functional analytic treatment of (4.2) we refer to [15, 17], for example. We shall consider the cost functional ( 1 J(y, u) = y(t, x) z(x) 2 dx + β 2 2 u(t) 2) e λt dt, 0 Ω where z L 2 (Ω) is a given desired state, and λ, β > 0 are positive constants. The optimal control problem is given by (OC) min J(y, u) such that (y, u) W loc (0, ) U ad satisfies (4.3). It is straightforward to argue the existence of an optimal control for (OC). Using the l POD basis functions for the Galerkin approximation of the optimal control problem (OC) we obtain min J l (w l, u) (OC l { ) s.t. u U ad and ẇl (t) = F (w l (t), u(t)) for t > 0, w l (0) = w with the state variable w l : [0, T ] R l, w R l and with an appropriate nonlinear F : R l R R l. For (OC l ) an HJB-based feedback law F : H 1 loc (0, ; Rl ) satisfying u l = F(w l ) is computed numerically in [9]. References [1] G. Berkooz, P. Holmes, and J. L. Lumley. Turbulence, Coherent Structuress, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, [2] J. A. Burns and S. Kang. A control problem for Burgers equation with bounded input/output ICASE Report 90-45, [3] C. I. Byrnes, D. S. Gilliam, and V. I. Shubov. On the global dynamics of a controlled viscous Burgers equation. J. Dyn. Control Syst., 4: , 1995.

11 INTERPRETATION OF POD AS SVD 11 [4] H. Choi, R. Temam, P. Moin, and J. Kim. Feedback control for unsteady flow and its application to the stochastic Burgers equation. J. Fluid Mech., 253: , [5] J. W. Demmel. Applied Numerical Linear Algebra. SIAM Marcel Dekkar, Inc., [6] M. Hinze and S. Volkwein. Analysis of instantaneous control for the Burgers equation. Nonlinear Analysis Theory and Methods and Applications, 50:1-26, [7] S. Kang, K. Ito, and J. A. Burns. Unbounded observation and boundary control problems for Burgers equation. In 30th IEEE Conf. on Decision and Control, , [8] K. Kunisch and S. Volkwein. Control of Burgers equation by a reduced-order approach using Proper Orthogonal decomposition. J. Optim. Theory and Appl., 102: , [9] K. Kunisch, S. Volkwein, and L. Xie. HJB-POD based feedback design for the optimal control of evolution problems. Submitted, [10] F. Leibfritz and S. Volkwein. Reduced order output feedback control design for PDE systems using proper orthogonal decomposition and nonlinear semidefinite programming. Technical Report No. 233, Special Research Center F 003 Optimization and Control, Project area Continuous Optimization and Control, University of Graz & Technical University of Graz, February 2002, submitted. [11] H. V. Ly, K. D. Mease, and E. S. Titi. Distributed and boundary control of the viscous Burgers equation. Numerical Functional Analysis and Optimization, 18: , [12] A. Pazy. Semigoups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, [13] G. W. Stewart. Introduction to Matrix Computations. Academic Press, New York, [14] G. W. Stewart. Matrix Perturbation Theory. Academic Press, New York, [15] R. Temam. Infinite-Dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, [16] S. Volkwein. Optimal and suboptimal control of partial differential equations: augmented Lagrange-SQP methods and reduced order modelling with proper orthogonal decomposition. Grazer Math. Berichte, Bericht Nr. 343, [17] S. Volkwein. Lagrange-SQP techniques for the control constrained optimal control problems for the Burgers equation. Computational Optimization and Applications, 26: , [18] A. Wouk. A Course of Applied Functional Analysis. Wiley-Interscience, New York, S. Volkwein, Karl-Franzens-Universität Graz, Institut für Mathematik, Heinrichstrasse 36, A-8010 Graz, Austria address: stefan.volkwein@uni-graz.at