1 Introduction and main result
Stabilization of linear autonomous control systems is classically done in finite dimension by pole-shifting or by Riccati theory (see, e.g., [25, 29, 40, 43]). In infinite dimension, pole-shifting may be used for some appropriate classes of systems (see [5, 10, 11], see also [37, page 711] and [48, Chapter 3]), but such approaches rely on spectral considerations and in practice require the numerical computation of eigenelements, which may be hard in general. Riccati theory has also been much explored in infinite dimension (see, e.g., [12, 26, 27, 49] and provides a powerful way for stabilizing a linear control system. Anyway, in practice, computing an approximation of the Riccati operator requires to consider a numerical approximation scheme and to compute the solution of a high-dimensional algebraic Riccati equation (see, e.g., [4, 19, 30, 26, 27] for convergence results for space semi-discretizations of the Riccati procedure, see also the survey ), which raises also a number of numerical difficulties.
Given these facts, it appears interesting to use dimension reduction procedures. Indeed, model reduction can generate low-dimensional models for which one may expect reasonable performances for stabilization issues while keeping a computationally tractable numerical problem. Proper Orthogonal Decomposition (POD) is a popular reduction model approach and can be used to generate, from a finite number of snapshots, a reduced-order control system in dimension
, approximating in the least square sense the initial infinite-dimensional system. Such an approach is completely general and does not consist of computing eigenelements (POD does not see eigenvectors). It is then natural to expect that, ifis large enough, then a linear stabilizing feedback computed from the -dimensional reduced-order control system, stabilizes as well the whole infinite-dimensional control system. Proving that this assertion holds true under appropriate assumptions is the objective of this paper: we prove that a low-order feedback control obtained by the Riccati procedure applied to a POD reduced-order model suffices to stabilize the complete infinite-dimensional control system.
The idea of using POD as a way to efficiently stabilize infinite-dimensional control systems, such as controlled PDEs, by means of a low-order feedback control, has been implemented in [3, 23, 24], where a number of convincing numerical simulations have been provided, showing the relevance of that approach. Feasibility of this methodology is nicely illustrated in  for heat equations and in  for the Burgers equation. But, in these papers, the above theoretical issue has been let as an open problem. In this paper, we provide the first general theorem providing a positive answer.
The paper is structured as follows. In Section 1.1 we give all assumptions under which our general result will be established. We provide in Section 1.2 some elements on the POD approach. Our main result is stated in Section 1.3. An idea of the strategy of its proof is given in Section 1.4. Section 2 contains some reminders and useful results on POD, useful in the proof of the main result. Section 3 is devoted to proving the main theorem. In Section 4, we give a conclusion and some open problems and perspectives. Finally, in Appendix A, we establish an aymptotic result in Riccati theory, which is instrumental in the proof of our main result.
1.1 General setting and assumptions
Let and be real Hilbert spaces. Let be a densely defined, closed selfadjoint operator, such that there exists some for which is dissipative. By the Lumer-Phillips theorem (see [13, 34]), generates a quasicontraction semi-group on , i.e., satisfying for every . Let be a bounded control operator. Consider the control system
with controls .
In what follows, we denote by the norm in and by the corresponding scalar product. Throughout the paper, we make the following assumptions.
We assume that the Hilbert space can be written as the direct orthogonal sum
where is of dimension , is a closed subspace of such that is dense in (for the induced topology), satisfying and
(invariance under ).
We denote by the orthogonal projection of onto ; then is the orthogonal projection of onto . By a, we have
the (bounded) operator on (which can be identified with a matrix of size ) generates on the uniformly continuous semigroup , with for every ;
the operator on , of domain , generates the (quasicontraction) semigroup .
We make the two following assumptions on those semigroups:
We assume that the latter semigroup is exponentially stable, i.e., that there exists such that
The operator (restriction of to ) can be identified with a selfadjoint
matrix, which is therefore diagonalizable with real-valued eigenvalues. We assume that all eigenvalues ofare simple and have a positive real part. We define
In other words, we assume in particular that , that is the finite-dimensional instable part of the system and that is the exponentially stable part.
We assume that the pair satisfies the Kalman condition
This assumption is satisfied under the following much stronger assumption of unique continuation (which is equivalent, by duality, to approximate controllability for the system (1)): there exists such that, given any , if for every then .
where , is the Dirichlet-Laplacian on a bounded domain of , is a nonempty open subset of and
is its characteristic function. Taking, the operator on is selfadjoint and of compact inverse and thus is diagonalizable. We assume that and are such that the spectrum of is simple (this is true under generic assumptions, see ) and such that is not an eigenvalue. Then there exists a Hilbert basis of
consisting of real-valued eigenfunctions corresponding to the real eigenvalues
(with a slight abuse of notation because the number of instable modes may be equal to ). Taking and , Assumptions a, b and c are satisfied. Assumption d is satisfied because of unique continuation: indeed we have for .
Of course, when is such that all eigenvalues of are negative, any solution of (1) converges exponentially to . We are interested in the case where there are (a finite number of) positive eigenvalues, i.e., , and then stabilization is an issue.
More generally, the assumptions a, b, c are satisfied when is of compact inverse, with having a finite number of instable (positive) eigenvalues which are moreover simple. Our framework even allows for more general situations in which spectrum may not be discrete, but does not involve the case of wave-like equations for instance (for which is not selfadjoint). Assumption d follows from unique continuation but is much weaker and may be satisfied for finite-rank control operators .
Thanks to the assumptions a, b, c and d, to stabilize (1) it would suffice to focus on the finite-dimensional instable part of the infinite-dimensional system (1), as this was done for instance in [5, 10, 11] (see also [37, page 711] and [48, Chapter 3]). However, in practice eigenelements are not known in general or may be difficult to compute numerically. In particular, the integer is not known in general or may be difficult to compute although we know its existence.
Stabilizing the system from a finite-dimensional approximation of (1) that is not of a spectral nature but which is anyway, in some sense, compatible with the above spectral decomposition, is the main challenge that we address in this paper.
We address this issue by approximating the control system (1) thanks to the POD method, described hereafter, which generates a -dimensional reduced-order control system, with sufficiently large ( will be enough).
In what follows, we consider an arbitrary element
which, used as an initial condition, generates the trajectory , solution of (1) with . We will consider it to generate snapshots in the POD method as explained next.
1.2 Proper Orthogonal Decomposition (POD)
The main idea of POD is to design an orthogonal basis of reduced order (called a POD basis) from a given collection of data (called snapshots). In order to face with too costly computations of a too complex model, the rationale behind POD is to generate a reduced set of basis functions able to capture the essential information of the physical process under consideration. POD has been developed long time ago, and independently, by many authors in various contexts. POD is closely related to Karhunen-Loève decompositions and to principal component analysis (PCA) or factor analysis. It has been widely used in the context of fluid mechanics and in particular turbulence (see[6, 8, 15, 18, 22, 31]) of chemical reactions (see [32, 38, 42]) and it has become a classical approach for nonlinear model reduction (see [9, 16, 17, 21, 28, 33, 35, 39] and see [3, 20, 23, 24, 36, 45] for applications to control of PDEs). The POD method consists of designing an unstructured low-rank approximation of a matrix composed of snapshots of the state. It can roughly be thought of as a Galerkin approximation in the spatial variable, built from values of solutions of the physical system taken at prescribed times , for some . These values (assumed to be known) are called snapshots.
Here, we take snapshots
of the solution to (1) with initial condition and with the control , taken at times , for some . We set
Note that, since , we have .
Given some integer , the POD method consists of determining a subspace of , of dimension , such that the mean square discrepancy between all snapshots and their orthogonal projection onto is minimal, i.e., it consists of minimizing the functional
over all possible subspaces of of dimension (equivalently, of dimension equal to ). Here, is the orthogonal projection onto the orthogonal of in . This minimization problem has at least one solution (see, e.g., ) and we denote by the optimal value, but the optimal approximating subspace may not be unique.
The problem is often formulated as follows. Assume that and complete these orthonormal vectors into an orthonormal basis of ; write and . Then, the POD method consists of minimizing
over all possible orthonormal families in . A subspace , optimal solution of the minimization problem (7), i.e., such that , is then used as a best approximating subspace of of dimension . Any orthonormal basis of is called a POD basis of rank .
1.3 Main result
POD reduced-order control system.
Keeping the assumptions and notations of the previous subsections, we fix an arbitrary and we consider an optimal solution of the (POD) minimization problem (7), i.e., a best approximating -dimensional subspace of the space defined by (6).
Applying the orthogonal projection to the control system (1) yields
The last term at the right-hand side of the above equation is seen as a perturbation term, and we are thus led to consider the following POD reduced-order control system in the space
Note that is well defined because . The control system (8) is a linear autonomous control system in the -dimensional space with controls . The operator can be identified with a square matrix of size and with a matrix of size (with of finite or infinite dimension).
Stabilizing the reduced-order control system.
In the proof of our main result (Theorem 1 hereafter), we will prove that the POD reduced-order -dimensional control system (8) is stabilizable when and is large enough. To design an exponentially stabilizing linear feedback , we use the Riccati theory (see also Appendix A for some reminders on the Riccati theory).
Given any , since the pair is stabilizable, by [1, Corollary 2.3.7 page 55] there exists a (unique) maximal symmetric positive semidefinite solution (of size ) of the algebraic Riccati equation
Here and throughout,
denotes the identity matrix of size. Moreover, is semi-stabilizing, i.e., its eigenvalues have nonpositive real part. If then is positive definite, and is Hurwitz, i.e., its eigenvalues have a negative real part (these facts are established in Appendix A). Therefore, setting
the linear feedback
exponentially stabilizes the reduced-order control system (8) to the origin.
Given any and any , there exists a unique optimal control minimizing the functional
over all possible controls , where is the solution to (8) with control and with initial condition . If then the optimal control is exactly the stabilizing feedback .
We will prove that the closed-loop matrix , which is Hurwitz if , actually remains uniformly Hurwitz as (precise asymptotic results are established in Appendix A). In particular, the matrix decreases exponentially, with an exponential rate which remains uniformly bounded below by some positive constant as .
We now use the above feedback matrix in the original infinite-dimensional control system (1), by taking the feedback control
Since is bounded, the operator is defined on and generates a semigroup. Our main result establishes that, under appropriate assumptions, this semigroup is exponentially stable. In other words, the “finite-dimensional” feedback (10), which exponentially stabilizes the finite-dimensional control system (8), also exponentially stabilizes the infinite-dimensional control system (1) if the number of snapshots is large enough and if is small enough.
Let be arbitrary. There exist and such that, for every and every , the control system (1) in closed-loop with the feedback ,
is exponentially stable, meaning that any solution of (12) converges exponentially to in as .
Remark 2 (On Assumption (11)).
Under Assumption (11), we have that when . This is why we can take in the theorem. Since is selfadjoint, recalling that and noting that , we see that Assumption (11) is equivalent to the assumption that the pair satisfies the Kalman condition, i.e.,
This is rather this condition that we will use in the proof.
A second remark is the following. Let be an orthonormal basis of , consisting of eigenvectors of , corresponding to the (real-valued) eigenvalues , . Assumption (11) (equivalently, Assumption (13)) is satisfied if and only if all eigenvalues of are simple and
i.e., the component of in the direction is nonzero, for every . The condition (14) is generic in the sense that the set of of which one of the first spectral modes is zero has codimension (and thus has measure zero) in .
Let be the best decay rate of the exponentially stable quasicontraction semigroup (see Assumption b). Let be the best decay rate of the matrix ( is the spectral abscissa of ).
Then, in Theorem 1, the growth bound of the exponentially stable semigroup generated by satisfies
1.4 Strategy of the proof
Establishing Theorem 1 is easier under the additional assumption
and we first sketch the argument under this simplifying assumption. In this case, we write
By d, the pair satisfies the Kalman condition and thus the subsystem (18) is stabilizable. Besides, by b, the subsystem (19) is exponentially stable with control . It follows from Appendix A that the control system (18)-(19) (which is equivalent to (17)) is stabilizable by the Riccati procedure: the optimal feedback exponentially stabilizes the system (18)-(19), i.e., the closed-loop matrix
Now, plugging this finite-dimensional feedback into the initial control system (1), we obtain the closed-loop system (12), with that is written in the above decomposition as the infinite-dimensional matrix
Since as , the matrix (21) is approximately lower block triangular, with the first diagonal block being exponentially stable (because (20) is Hurwitz) and the second diagonal block being exponentially stable as well (because it is close to as is small enough). Therefore, (21) is exponentially stable and Theorem 1 follows, under the simplifying assumption (16).
In general, however, (16) is not true: there is indeed no reason that, when performing the POD reduction, the space contain the spectral subspace . Indeed, “POD does not see eigenmodes”.
Anyway, our complete analysis, done in Section 3, will reveal that this is almost the case: we will prove in particular that
when is large enough, which implies that the inclusion (16) is almost satisfied (because ). Establishing such a result will require a quite fine analysis. This shows that, in some sense, our problem is a small perturbation problem of (21) when is large. Theorem 1 will then be proved in Section 3.2, using an asymptotic result in Riccati theory developed in Appendix A, roughly stating that, when considering a linear system having an instable part and a stable part, the Riccati stabilization procedure with weight on the state and weight on the control yields feedbacks that essentially act on the instable part for small. The results established in Appendix A are a bit delicate and require a particular care, although all notions thereof remain quite elementary.
2 Some results on POD
2.1 Relationship with Singular Value Decomposition (SVD)
It is well known that optimal solutions of POD can be expressed thanks to SVD. Let be given by (5). We consider the -matrix
expressed in an arbitrary Hilbert basis of . We have and . Since the matrix is of finite rank , SVD works exactly as in finite dimension (because and are compact and selfadjoint, see ). According to the SVD theorem, we have
is an orthogonal matrix of infinite size (unitary operator in, consisting of eigenvectors of ), is an orthogonal matrix of size (consisting of eigenvectors of ) and is a matrix of size consisting of the diagonal (singular values of ), completed with zeros. The singular values of are nonnegative real numbers, with the first ones being positive and all others being zero. Denoting by and the columns of and , we have
Let be an integer. We define the -matrix as the submatrix of consisting of the first columns of , which are . Similarly, we define the -matrix as the submatrix of consisting of the first columns of , which are . Finally, we define the square diagonal matrix of size , consisting of the elements . It is then well known (see, e.g., [14, 33]) that the “best” projection of rank onto is and that the “best” approximation of rank of the matrix , over all matrices of rank , is the matrix
“Best” is understood here in the sense of the Frobenius norm as well as of the subordinate -norm (and actually, of any norm invariant under the orthogonal group), and the Frobenius norm of is
Recall that the square of the Frobenius norm of a matrix (of any size, possibly infinite) is equal to the sum of squares of all elements of . Moreover, when considering the Frobenius norm (also called Hilbert-Schmidt norm), we have uniqueness of the minimizer if and only if . Note that the range of is contained in the range of , and thus is also the best approximation of rank of over all possible matrices of rank whose range is contained in the range of .
By definition, the quantity defined by (7) is exactly the Frobenius norm of :
By the above remarks, since , the POD problem is exactly equivalent to searching the best approximation of rank of the matrix for the Frobenius norm. Therefore, we have
2.2 Boundedness of the optimal value
Recall that for every , where is fixed.
Since commutes with by Assumption a, we have , and hence
where we have used Assumption b to get the latter inequality, and similarly,
and the first item follows because .
For the second item, using the SVD interpretation of the POD, we have in particular (since )
and the lemma follows. ∎
It follows from the second item of Lemma 1 that there exists (which is the bound on ) such that, for all integers and satisfying , we have
3 Proof of Theorem 1
3.1 Several convergence results
If then there exists such that