H. T. BANKS
AND R. H. FABIANO y
Abstract. We consider computational aspects of using semidiscrete approximation schemes to solve problems with innite-dimensional dynamics. We survey theory and convergence results for the simulation or forward problem, the feedback control problem, and the parameter estimation problem, all for the case in which the underlying system dynamics is governed by a partial dierential equation. In particular we investigate the critical sucient conditions required for convergence of semidiscrete approximations of these problems. These sucient conditions require that the approximation scheme demonstrate system convergence, adjoint system convergence, and uniform preservation of system stability. By considering in detail several specic examples, we illustrate the diculties which may arise when these sucient conditions are not satised.
Keywords. approximation, optimal control, parameter estimation,C0-semigroup, preservation of stability under approximation.
1. Introduction.
In this paper we consider computational issues which might arise when one attempts to `solve' a problem involving innite-dimensional dynamics (ie. partial dierential equations, delay equations, etc.) by solving instead an ap-proximate problem involving nite-dimensional dynamics. The main point we wish to make is that a carefully constructed approximation scheme which performs well on one type of problem (a simulation problem, for example) may perform poorly on another type of problem (an inverse or optimization problem, for example), even when the underlying dynamics is the same for both problems.We restrict our consideration to problems in which the underlying dynamics are governed by the following Cauchy problem evolving in an innite dimensional Hilbert spaceH:
_
z(t) = A(q)z(t) +Bu(t) (1.1)
z(0) = z0:
HereA(q) is a parameter dependent linear operator (typically unbounded), andBu(t) represents a control input or an external force. We consider solution of three dierent types of problems associated with the dynamics modeled by (1.1). These can be described loosely as follows:
Simulation| givenz
0,q,B andu(t), determine the unique solutionz(t). Parameter Estimation | given z
0, B, u(t) and data consisting of (perhaps
partial) measurements ofz(t) at certain timesti, estimateq.
Optimal Control | given z
0, q, B and a cost functional J, determine u(t)
such thatJ is minimized.
Considering for now only the simulation problem, we know that one way to `solve' such a problem is to introduce a semidiscrete approximation scheme, whereby instead
Center for Research in Scientic Computation, North Carolina State University, Raleigh, NC 27695-8205. The research of this author was supported in part by the Air Force Oce of Scientic Research under grant AFOSR F4962095-1-0236. ([email protected] )
yDepartment of Mathematical Sciences, 340 Bryan Building, University of North Carolina at Greensboro, Greensboro, NC 27412. The research of this author was supported in part by the National Science Foundation under grant DMS-9696239 and UNCG New Faculty grant 97NF04. ([email protected])
of (1.1) one solves the nite dimensional system _
zN(t) = AN(q)zN(t) +BNu(t)
(1.2)
zN(0) = zN
0 :
The idea is that since (1.2) is theoretically a solved problem, we are nished `solving' the simulation problem once we have demonstrated appropriate convergence ofzN(t)
to z(t). In this context, when we say that (1.2) is a solved problem we mean that there are known, reasonably well-behaved computational algorithms with which one can solve (1.2) on a computer (actually one considers a matrix representation of (1.2) to be solved computationally). This usually turns out to be a highly satisfactory method for solving the simulation problem.
It seems reasonable, then, to attempt the same methodology for solving the parameter estimation and/or optimal control problems, especially since the nite-dimensional versions of the parameter estimation/optimal control problems are (the-oretically) solved. This is, in fact, a common approach to solving innite-dimensional optimization problems and is sometimes described heuristically as `rst discretize, then optimize.' It has been observed, however, and this is precisely the main theme of this paper, that caution must be exercised in choosing discretization schemes and optimization algorithms which are `compatible'. Indeed it turns out that an excellent discretization scheme (excellent for simulation, that is) and an excellent optimization algorithm may perform poorly when used together.
In the remainder of the paper we will expand on these ideas. More detailed def-initions and statements of convergence theorems will be given so that we may focus more precisely on the theoretical reasons for the behavior which has been described. In x2 we consider the simulation problem and recall some standard denitions and
convergence results. Inx3 we survey theoretical results for the linear-quadratic
regu-lator problem (an optimal control problem). We will focus on recent results for two specic partial dierential equations, one a model for a delay system and the other a model for an elastic structure with boundary damping. Finally in x4 we pursue
similar ideas for the parameter estimation problem.
2. The simulation problem.
In this section we recall some standard denitions and results for the simulation problem. Specically, consider a linear Cauchy problem_
z(t) = Az(t) +f(t) (2.1)
z(0) = z0
on an innite dimensional Hilbert spaceH. We assume that A: domAH !H is
the innitesimal generator of aC0-semigroupT(t) onH. This assumption guarantees
that (2.1) is well-posed (we take as a standing assumption throughout the paper that the underlying dynamics or `forward' problem is well-posed). The simulation problem consists of ndingz(t) when z0 andf(t) are known. Proceeding as described in the
introduction, we introduce a semidiscrete approximation scheme for (2.1), which will consist of a sequence of nite dimensional subspaces HN H, N = 1;2;:::, the
corresponding orthogonal projectionsPN :H !HN, and operatorsAN :HN!HN.
This denes a nite dimensional analog of (2.1) given by _
zN(t) = ANzN(t) +PNf(t)
(2.2)
z(0) = PNz
The operatorAN generates the semigroupTN(t) =etA onHN. The system (2.2) is
equivalent to a rst order matrix dierential equation which can be solved computa-tionally, and from this the solutionzN(t) of (2.2) can be recovered. Many standard
approximation methods can be formulated in this semigroup-theoretic framework, in-cluding Galerkin, nite element, spectral, and nite dierence methods. In order for this approach to be a useful method for solving the simulation problem, we need a re-sult guaranteeing convergence of solutions of (2.2) to solutions of (2.1). The following hypotheses are sucient.
(H1)
For eachz2H,PNz!z inH (that is,PN converges strongly toP).(H2)
For eachz2H,TN(t)PNz!T(t)z inH, uniformly on compactt-intervals.Recall that the mild solution of (2.1) can be written (using the variation of con-stants formula) as
z(t) =T(t)z0+ Z t
0
T(t?s)f(s)ds:
Similarly, the variation of constants solution of (2.2) is given by
zN(t) =TN(t)PNz
0+ Z t
0
TN(t?s)PNf(s)ds:
Thus, we see that if (H1) and (H2) hold, then zN(t) ! z(t) in H, uniformly on
compactt-intervals. In many applications,H is a function space (or cross product of function spaces),HNis the span of a nite number of basis functions, andN is related
to the number of basis functions and/or the `mesh size'. Typically, hypothesis (H1) follows from the approximation properties of the basis functions. A useful tool for verifying (H2) is the Trotter-Kato theorem, a semigroup theoretic version of the well known dictum [32] `stability plus consistency yields convergence'. For various versions and applications of the Trotter-Kato theorem, see [2], [7], [9], [26], [34]. In summary, in order to solve the simulation problem (2.1) via the semidiscrete approximation (2.2), the scheme must have the properties that the nite dimensional spaces HN
approximate well the space H (hypothesis (H1)), and that the nite dimensional operators AN are a stable and consistent approximation of the operator A (which
yields hypothesis (H2)).
3. The optimal control problem.
In this section we consider computational issues for optimal control problems, and in particular we focus on the linear-quadratic regulator (LQR) problem. To set up the problem, letH,U, andW be Hilbert spaces representing respectively the state space, the control space and the output space. Consider a cost functionalJ(z0;u) = Z
1
0
[hCz(t);Cz(t)iW +hRu(t);u(t)iU]dt
(3.1)
and the state equation
_
z(t) = Az(t) +Bu(t) (3.2)
z(0) = z0:
As in the simulation problem, we assume thatA: domAH !H is the innitesimal
in the cost functional satisfyC2L(H;W) andR2L(U), withR >0. (We use the
standard notation L(H
1;H2) to denote the space of bounded linear operators from
H1 toH2, and L(H
1) ifH1=H2). For simplicity of presentation we assume that the
control operator is bounded, that is,B 2L(U;H), although it should be noted that
many of the results summarized below are available for the case in whichBand/orC
is unbounded. (For a recent summary of some of these results, see Chapter 7 of [11].) With these denitions, we can state the LQR problem of interest, which is
min
u2L2(0;1;U)
J(z0;u)
(3.3)
subject to dynamics governed by (3.2). Among many references considering this problem, we cite [16] and [22].
As we did with the simulation problem, we will rst recall some conditions which guarantee that the LQR problem is `well-posed' (here this means that there exists a unique control given in feedback form). Then we will introduce a semidiscrete ap-proximation of the LQR problem, and consider necessary conditions for convergence. To proceed, let us recall the following denitions relevant to the LQR problem.
The C
0-semigroup T(t) is exponentially stable if there exists M
1 and ! > 0
such thatkT(t)kMe
?!t for allt 0.
The pair (A;B) is stabilizable if there existsK2L(H;U) such thatA?BKis the
innitesimal generator of an exponentially stable semigroup.
The pair (A;C) is detectable if there existsL2L(W;H) such that A?LC is the
innitesimal generator of an exponentially stable semigroup.
The control u 2 L 2(0;
1;U) is admissible for the initial condition z 0
2 H if
J(z0;u)< 1.
The system (3.1)-(3.2) is optimizable if there exists an admissible control u for
everyz0 2H.
Note that if (A;B) is stabilizable then the system (3.1)-(3.2) is optimizible, and ifA
generates an exponentially stable semigroup, then (A;B) is stabilizable and (A;C) is detectable. Stabilizability and detectability hypotheses are important sucient conditions for well-posedness of the LQR problem, as the following well-known result indicates.
Theorem 3.1. For the system (3.1)-(3.2), there exists a nonnegative self-adjoint solution 2L(H) of the algebraic Riccati equation (ARE)
A + A
?BR
?1B +CC= 0
(3.4)
if and only if (3.1)-(3.2) is optimizable. In this case, if is the minimal nonnegative self-adjoint solution of the ARE (3.4), then the unique optimal control for the LQR problem is given in feedback form by
e
u(t) =?R ?1B
e
z(t);
(3.5)
where the optimal trajectoryze(t) is the solution of the closed loop system
_
z(t) = (A?BR
?1B)z(t)
(3.6)
z(0) = z0:
That is, ez(t) = S(t)z
0, where S(t) is the closed loop semigroup generated by A ?
BR?1B. If in addition (A;C) is detectable, then there exists a unique nonnegative
An alternative useful hypothesis is the following.
(H3)
For eachz02H, there exists an admissible control, and any admissible control
drives the statez(t) to zero ast!1(that is, lim
t!1
kz(t)kH= 0).
If (H3) holds, then there exists a unique nonnegative self-adjoint solution of the ARE (3.4). IfCC >0, then (H3) holds andS(t) is exponentially stable.
From Theorem 3.1 we see that the optimal control is given in feedback form by u = Kx, where K = ?R
?1B is sometimes referred to as the feedback gain
operator. In solving the LQR problem, we may be interested in the closed loop solutionez(t), the closed loop semigroup S(t), the controlu(t), the solution of the
ARE, or perhaps just the gain K. To nd any of these computationally, we need a semidiscrete approximation scheme and appropriate convergence results.
Therefore, consider a semidiscrete approximation scheme consisting of a sequence of nite dimensional subspacesHN H,N = 1;2;:::, the corresponding orthogonal
projectionsPN:H !HN, and operatorsAN:HN!HN. Also letBN2L(U;HN)
andCN2L(HN;W). This allows one to dene an approximate cost functional
JN(PNz
0;u) = Z
1
0
[hCNz(t);CNz(t)iW +hRu(t);u(t)iU]dt
(3.7)
and nite dimensional state equation _
zN(t) = ANzN(t) +BNu(t)
(3.8)
zN(0) = PNz0:
The nite dimensional LQR problem is min
u2L 2
(0;1;U)
JN(PNz
0;u)
(3.9)
subject to dynamics governed by (3.8). As in the innite dimensional case, there are various hypotheses which guarantee well-posedness for the nite dimensional LQR problem. Whichever hypotheses are used, however, should be satised for allN. The following result is typical.
Theorem 3.2. Assume that (H3) holds for each N. Then there exists a unique nonnegative self-adjoint solutionN of the algebraic Riccati equation (ARE)
ANN+ NAN
?NBNR
?1BNN+CNCN= 0:
(3.10)
Moreover, the unique optimal control for the nite dimensional LQR problem is given in feedback form by
e
uN(t) =?R
?1BNN e
zN(t);
(3.11)
where the optimal trajectoryezN(t) is the solution of the closed loop system
_
zN(t) = (AN?BNR
?1BNN)zN(t)
(3.12)
zN(0) = PNz
0:
That is, zeN(t) = SN(t)PNz
0, where S
(t) = e(A N
?B
NR?1BN
N
)t is the closed loop
semigroup generated byAN?BNR
We are interested in conditions which will be sucient to guarantee appropriate convergence for these problems. In particular, we are interested in ensuring that N ! in an appropriate sense as N ! 1. Based on our discussions so far, it
seems reasonable to expect to need (H1)-(H3), as well as some sort of convergence of
BN to B andCN toC. These are stated as:
(H4)
(a) For eachv2U,BNv!Bv; (b) For eachz2H,BN PNz!B z.
(H5)
For eachz2H,CNPNz!Cz and for eachw2W,CN w!C
w.
If (H1)-(H5) were sucient for the convergence we desire, then there would be little point to this paper. Indeed, these hypotheses are not sucient, and most theoretical convergence results in the literature require what amounts to two additional hypothe-ses. One is an `adjoint convergence' condition, and the other is a `preservation of stability under approximation' condition. The adjoint condition can be given as fol-lows.
(H6)
For eachz2H,TN(t) PNz!T(t)
zinH, uniformly on compactt-intervals.
HereT(t) is the Hilbert space adjoint of the operatorT(t), and it is well known [34]
thatAis the innitesimal generator ofT(t). Condition (H6) is not unexpected once
we note that the controlu(t) (resp. uN(t)) is dened in terms of the operator (resp.
N), which is in turn a solution of an algebraic Riccati equation involvingA (resp.
AN).
It is important to consider (H6) when designing approximation schemes for the LQR problem. In particular, (H6) is independent of (H2), although it should be noted that (H2) implies weak operator convergence of TN(t)PN to T(t) ((H6) is strong
operator convergence). We shall see below an example of an approximation scheme which satises (H2) but not (H6), and which performs well for the simulation problem but poorly for the control problem. If one were not familiar with the results surveyed here, this type of behavior might be rather surprising.
The `preservation of stability under approximation' condition appears in various forms in the literature. The following result is typical and was given in [8].
Theorem 3.3. Suppose (H1), (H2), (H4), (H5), (H6) hold, and (H3) holds for eachN. (Note that this guarantees that the nite dimensional LQR problem is well-posed for each N). Let N denote the unique nonnegative self-adjoint solution of the ARE (3.10). Further assume that the ARE (3.4) has a unique nonnegative self-adjoint solution. LetS(t) andSN(t) denote the closed loop semigroups generated by
A?BR
?1B andAN ?BNR
?1BNN, respectively. IF there are positive constants
M1,M2 and! (independent ofN) satisfying
kSN(t)kHN M 1e
?!t fort 0
(3.13) and
kNkHN M 2;
(3.14)
for allN, THEN
NPNz!z for everyz2H;
(3.15)
SN(t)PNz!S(t)z for everyz2H;
where the convergence is uniform int on bounded intervals, and
kS(t)kM 1e
?!t fort 0:
(3.17)
In the statement of this theorem, one interprets (3.13) as a uniform stability condition. Unfortunately, for many approximation schemes it is not easy to verify (3.13) and (3.14). Assuming that the control space U is nite dimensional, we can sometimes instead use a related result, which makes use of the following denitions.
Given an approximation scheme as described above for the LQR problem, we say
that (AN;BN) is uniformly stabilizable if there existsKN2L(HN;U),M
1 and! >0 such that supkKNk<1andke (A
N ?B
NKN )tPN
kMe
?!t for
allt0 andN suciently large.
Given an approximation scheme as described above for the LQR problem, we say
that (AN;CN) is uniformly detectable if there existsLN 2L(W;HN),M1
and! >0 such that supkLNk<1andke (A
N ?L
NCN )tPN
kMe
?!t for all
t0 andN suciently large.
Theorem 3.4. (see [25]) Suppose (H1), (H2), (H4), (H5) and (H6) hold, and that (AN;BN) is uniformly stabilizable and (AN;CN) is uniformly detectable. Then for
eachN the nite dimensional ARE (3.10) has a unique nonnegative solutionN and there exists M1
1, ! >0 such that (3.13) holds. Moreover, (A;B) is stabilizable.
If in addition(A;C) is detectable, then the ARE (3.4) has a unique nonnegative self-adjoint solution, and (3.15) holds.
It is possible to obtain (3.15) under weaker versions of the `preservation of stability under approximation' condition. For example, in [30] the authors dene uniform out-put stabilityand uniform input-output stability for approximation schemes and show that these conditions (which are weaker than uniform stabilizability and detectability) together with (H1)-(H6) imply (3.15). Alternatively, in some applications it happens that the open-loop semigroupT(t) is exponentially stable, and if the approximation scheme preserves this property (as characterized in the following denition) a conver-gence result can be obtained for the LQR problem.
Assume that T(t) is exponentially stable. Given an approximation scheme as
described above for the LQR problem, we say that TN(t) = eANt
is uni-formly exponentially stable if there exists M 1 and ! > 0 such that
supkTN(t)kMe
?!t for allt
0 andN suciently large.
Assuming that T(t) is exponentially stable,TN(t) is uniformly exponentially stable,
and (H1)-(H6) hold, it can be inferred from the results in [23] and [8] that (3.15) and (3.16) hold.
We turn next to a couple of examples in which the `adjoint convergence' and `preservation of stability under approximation' conditions will be explored for certain specic approximation schemes. These examples illustrate in very specic ways why the theoretical concepts and issues raised in the preceeding discussions are indeed of computational importance. Diculties that arise when certain of the sucient conditions are not met will be rather apparent.
Example 1Consider a cost functional
J(u) =
Z 1
y(t;0)2+u(t)2
and state equation
@y
@t(t;s) = @y@s(t;s); s2(?1;0); t >0;
@y
@t(t;0) = y(t;0) +y(t;?1) +u(t);
y(0;s) = (s); ?1s0:
This system arises in the study of delay equations (in fact, by settingx(t+s) =y(t;s) the system is equivalent to the retarded delay equation _x(t) =x(t) +x(t?1) +u(t),
t >0). In order to formulate an abstract LQR problem such as we have considered, dene the state spaceH = 1R1
L
2(
?1;0), the control spaceU = 1R
1, and the state
z(t) = (y(t;0);y(t;)). Then the above system can be written as
_
z(t) = Az(t) +Bu(t); z(0) = z0= ((0););
whereB2L(U;H) is dened byBu= (u;0), andAis dened on the domain
domA=
(;)2Hj2H 1(
?1;0); (0) =
by A((0);) = ((0) +(?1); dds). If we take R = 1 and C 2 L(H) is given by
C(;) = (;0), then the cost functionalJ can be written as
J(u) =Z 1
0
hCz(t);Cz(t)iH+Rju(t)j 2 dt:
We can consider the simulation problem (2.1) and the LQR problem (3.3), both of which have received much attention in the literature. It is known (see [2] and [16]) that
Ais the innitesimal generator of aC0-semigroupT(t) onH, so that the simulation
problem is well-posed. Also (see [16], [5] and [23]) the LQR problem is well-posed and the optimal control is given in feedback form by (3.5). Because of the product space structure ofH, one can write the operator solution of the ARE (3.4) as
=
00 01
10 11
;
where 00 is a scalar (i.e., in L(1R
1)), 11
2 L(L 2(
?1;0)), 10
2 L(1R 1;L
2( ?1:0))
and 01 2 L(L
2(
?1;0);1R
1). It follows from the structure ofB, C and R that the
optimal feedback control has the form
e
u(t) =?
00 e
y(t;0) +
Z 0
?1
10(s) e
y(t;s)ds
;
whereze(t) = (ye(t;0);ye(t;)) is the solution of the closed loop system (3.6). In
partic-ular, the feedback gain kernel 10(s) is a function inL2( ?1;0).
Let us now consider approximation for the simulation and LQR problems. Recall that an approximation scheme for the simulation problem consists of constructing a sequence of nite dimensional spaces HN H and operators AN : HN ! HN.
it is natural to use this scheme for the LQR problem by simply deningBN=PNB
andCN=PNC.
For the particular system under consideration, constructingHN amounts to
dis-cretizing the function spaceL2(
?1;0)), and deningAN amounts to discretizing the
dierential operator dds. The paper [2] is one of the earliest to consider approximation for this problem within the context of semigroup-theoretic convergence results. In it Banks and Burns considered the so-called averaging approximation scheme (referred to here as the AVE scheme), using piecewise constant functions to discretize L2(
?1;0)
and nite dierencing to discretize the operator dds. Specically, let tNj = ?j=N,
j = 0;1;:::;N be a partition of the interval [?1;0], and let Nj = [t
N j ;t
N j?1
] be the
usual characteristic function for the interval [tNj;tNj?1]. Dene the basis elements
eN;0
AVE= (1;0) ande
N;j
AVE= (0;Nj),j= 1;2;:::;N, and set
HN
AVE= span feN;
0 AVE;e N;1 AVE;:::;e N;N AVE g:
ThenHN
AVE
H, andAN AVE :H
N
AVE !HN
AVE is dened by
ANAVE
N
X
j=0
jeN;jAVE= (
0+N)e
N;0 AVE+
N
X
j=1
N(j?1
?j)eN;j AVE:
Banks and Burns veried that (H1) and (H2) hold for the AVE scheme, and used it to solve some simulation and control problems (but not the LQR problem). They also obtained a convergence rate for the simulation problem, establishing that the scheme is order 1=N.
In an eort to improve on the convergence rate for the simulation problem, Banks and Kappel [7] considered a Galerkin type approximation scheme (denoted as the SPL scheme) based on using piecewise spline functions to discretizeL2(
?1;0). Specically,
dene the piecewise linear functions
bN
0(s) =
Nr(s?tN 1); ift
N
1
s0,
0; elsewhere,
bNN(s) =
?Nr(s?tNN
?1); iftNN
stNN ?1,
0; elsewhere, and, forj= 1;2;:::;N?1,
bNj(s) =
8 <
:
?Nr(s?tNj
?1); iftNj
stNj ?1,
Nr(s?tNj
+1); iftNj+1
stNj,
0; elsewhere.
These functions are the familiar and standard `hat' functions used in linear nite element schemes. Next dene the basis elements eN;0
SPL = (1;b
N
0), e
N;j
SPL = (0;bNj) for
j= 1;2;:::;N, and set
HN
SPL= span
eN;0 SPL;e
N;1 SPL;:::;e
N;N
SPL :
ThenHN
SPL
domAH, andAN SPL :H
N
SPL !HN
SPL can be dened byA
N
SPL=P
N
SPLA,
wherePN
SPL is the orthogonal projection ofH onto H
N
SPL. Banks and Kappel veried
1=N2. They used the scheme to solve some simulation problems, and presented
com-putational examples verifying convergence rates for the two schemes. Not surprisingly, the SPL scheme outperformed the AVE scheme for the simulation problem.
However, when researchers began investigating the possibility of using each of these schemes to solve the LQR problem ([5],[23]), the spline based scheme was seen to have some shortcomings. The authors of [5] applied both the AVE and SPL schemes to the LQR problem dened above, and generated approximations to the functional feedback gain 10. Recall that 10 is a function dened on (
?1;0), and because of
the structure of each approximation scheme, the AVE approximation of 10 will be
a piecewise constant function and the SPL approximation will be a piecewise linear function. Based on the superiority of the SPL scheme for the simulation problem, there was some expectation by the authors of [5] that the same might hold true for the LQR problem. However, Figure 1 indicates that this is not the case.
−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
−2 −1 0 1 2 3 4
spline
AVE
Figure 1 - Approximate feedback functional gain N
10,N = 16
The AVE scheme provides a more useful reconstruction of 10, and Banks et al
conjectured that the SPL reconstruction only obtains some type of weak convergence for this gain. Burns et al [12] subsequently showed that the SPL scheme does not satisfy the adjoint convergence condition (H6) (Gibson [23] showed that the AVE scheme satises (H6)). This lack of adjoint convergence is not too surprising when we observe thatHN
SPL is contained in domA but not in domA =
f(; )2H j 2
H1(
With this observation in mind, and in an eort to improve convergence rates for LQR approximations, Kappel and Salamon [28] dened a modied spline based scheme (referred to as the NEW scheme) with the adjoint convergence property (H6). Specically, using the piecewise linear functionsbNj from above, dene basis elements
eN;0
NEW= (1;0), e
N;j
NEW= (0;bNj
?1) forj= 1;2;:::;N+ 1, and set
HN
NEW= span
eN;0 NEW;e
N;1 NEW;:::;e
N;N+1
NEW :
Since HN
NEW
6domA, one cannot use PN
NEWA to dene an operator A
N
NEW onH
N
NEW
(herePN
NEW is the orthogonal projection ofH ontoH
N
NEW). Instead, extendA to the
operatorAedened on the domain domAe= 1R
H
1(
?1;0) by
e
A(;) = (+(?1); dds+(s)[?(0)]);
where(s) denotes the Dirac delta impulse ats= 0. ThusHN
NEW dom
e
A, andAN
NEW:
HN
NEW !HN
NEW can be dened byA
N
NEW=P
N
NEW e
A. Kappel and Salamon [28] showed that (H1)-(H6) hold for this scheme, and observed numerically that for the LQR problem, the scheme converged and outperformed both the AVE and SPL schemes. They did not prove convergence theoretically because, as they showed, the NEW scheme does not satisfy a uniform exponential stability condition. In two later papers [29] [30], Kappel and Salamon nally showed theoretical convergence of the NEW scheme for the LQR problem, by rst showing that the NEW scheme satises a weaker uniform stability condition and then showing that this is enough for convergence of the gains.
Let us pursue the issue of uniform stability behavior a bit further. To visualize this property numerically, we plot the eigenvalues ofAN
AVEandA
N
NEWfor several values
ofN in Figure 2 and Figure 3. We see that the AVE scheme exhibits uniform stability numerically, and this has been established theoretically in [36]. For the NEW scheme, it is the existence of eigenaluesN
NEW ofA
N
NEW with the property that
lim
N!1
ReN
NEW= 0 and limN !1
ImN
NEW= 1
that causes the lack of uniform exponential stability. The eigenvalues of the SPL scheme exhibit the same behavior. Thus even though uniform exponential stability was not necessary for convergence of the feedback gains in the LQR approximation problem, the lack of this property could still be considered a shortcoming of the NEW scheme. There is ongoing research activity involving the construction of approxima-tion schemes for such delay systems, and particularly on schemes which satisfy the adjoint convergence and uniform exponential stability conditions (see [27] and refer-ences therein).
We complete our considerations of this example with a discussion of one particu-lar recently developed method [20] for constructing Galerkin approximation schemes which satisfy a uniform stability condition. Recall that in our interpretation of the Galerkin method as used above, we constructed nite dimensional spacesHNdomA
and denedAN:HN !HN byAN=PNA, wherePN is the orthogonal projection
ofH ontoHN. A more standard interpretation of the Galerkin method is to consider
a sesquilinear form(;) :V V !C
j which is related toAby domA
V and
−100 −50 0 −60
−40 −20 0 20 40 60
AVE eigenvalues N=8
Real
Imaginary
−100 −50 0
−60 −40 −20 0 20 40 60
AVE eigenvalues N=16
Real
Imaginary
−100 −50 0
−60 −40 −20 0 20 40 60
AVE eigenvalues N=32
Real
Imaginary
−100 −50 0
−60 −40 −20 0 20 40 60
AVE eigenvalues N=64
Real
Imaginary
Figure 2 - Eigenvalues ofAN
AVE (AVE scheme)
One then constructs nite dimensional spacesHNV and denes AN :HN!HN
by
hANx;yi=(x;y) for allx;y 2HN:
(3.18)
(Both the SPL and NEW schemes can be dened according to this approach to the Galerkin method). The idea in [20] is to select a more appropriate equivalent inner producth;ie onH (two inner products are equivalent if their compatible norms are
equivalent) and corresponding sesquilinear forme such that
hAx;yie=e(x;y) for allx;y2V:
One can then use the same nite dimensional spacesHNV and deneANe:HN !
HN by
hANex;yie=e(x;y) for allx;y2HN:
(3.19)
For the example under consideration here, we have
h(;);(; )i=+ Z
0
−20 −10 0 −100
−50 0 50 100
NEW eigenvalues N=8
Real
Imaginary
−20 −10 0
−100 −50 0 50 100
NEW eigenvalues N=16
Real
Imaginary
−20 −10 0
−100 −50 0 50 100
NEW eigenvalues N=32
Real
Imaginary
−20 −10 0
−100 −50 0 50 100
NEW eigenvalues N=64
Real
Imaginary
Figure 3 - Eigenvalues ofAN
NEW (NEW scheme)
and
((;);(; )) = [+(?1)]+ Z
0
?1
d
ds(s) (s)ds:
Using the nite dimensional spacesHN
NEWand (3.18) leads to the NEW scheme.
Con-sider instead the inner product
h(;);(; )ie=+e ?
Z 0
?1
e?2s(s) (s)ds;
where is the unique real eigenvalue ofA, and also consider the sesquilinear form
e((;);(; )) = [+(?1)]+e ?
Z 0
?1
e?2s d
ds(s) (s)ds+e?[
?(0)] (0):
(see [20] for details). Then using the nite dimensional spacesHN
NEWand (3.19) denes
an operatorAbN NEW:H
N
NEW !HN
NEW which satises a uniform stability property (this
was shown in [20]). In Figure 4 we plot the eigenvalues ofAbN NEW.
−40 −30 −20 −10 0 −100
−50 0 50 100
N=8
Real
Imaginary
−40 −30 −20 −10 0 −100
−50 0 50 100
N=16
Real
Imaginary
−40 −30 −20 −10 0 −100
−50 0 50 100
N=32
Real
Imaginary
−40 −30 −20 −10 0 −100
−50 0 50 100
N=64
Real
Imaginary
Figure 4 - Eigenvalues ofAbN
NEW(equivalent inner product)
seems to be a computationally desirable property. That is, if an operatorA gener-ates an exponentially stable semigroupT(t) satisfyingkT(t)kMe
?!t, then it would
seem that an approximation scheme should consist of operatorsANwith the property
that TN(t) = eANt
is uniformly exponentially stable. However the present authors are unable to make a denitive statement concerning the precise negative implications for the case that TN(t) = eANt
is not uniformly exponentially stable. We consider this to be an open and unresolved issue. To complete this section we will consider the issue again as it arises in a model of a weakly damped elastic system.
Example 2Consider the following equation:
ytt(t;) =y(t;); 0< <1;
(3.20)
y(t;0) = 0; y(t;1) =?yt(t;1); >0;
y(0;) =y0(); yt(0;) =v0():
as a boundary controller and the problem can be treated as an LQR problem, but we will instead treat this energy dissipation term as part of the system dynamics and consider (3.20) as a simulation problem (in [24] the authors do consider the LQR problem and use a regularized algebraic Riccati equation to overcome the lack of uniform stability which arises for some approximation schemes for this problem).
By choosing a state z(t) = (y(t;);yt(t;)), we can write (3.20) as the Cauchy
problem
_
z(t) = Az(t) (3.21)
z(0) = (y0;v0)
on the Hilbert spaceH =H1
L(0;1)L
2(0;1), where
H1
L(0;1) =fu2H
1(0;1) : u(0) = 0 g:
The energy norm onH is given by
k(u;v)k 2
H =
Z 1
0
(ju 0()
j 2+
jv()j 2)d;
with the compatible energy inner producth;iH. The operatorAin (3.21) is dened
on the domain
domA=f(u;v)2H : u2H
2(0;1); v 2H
1
L(0;1); u0(1) =
?v(1)g
by
A(u;v) = (v;u00):
It is known (see [14], [31]) that A is the innitesimal generator of an exponentially stable semigroupT(t). In [6] the authors investigated the uniform stability behavior (or lack thereof) for a number of standard approximation schemes, including nite dierences, Galerkin nite elements, mixed methods, spectral methods, etc. Through numerical experiments (i.e., computing the eigenvalues of the nite dimensional op-erators AN arising for each of these schemes) they discovered that many of these
schemes do not uniformly preserve the exponential stability of the original system. (Subsequent rigorous theoretical verication of some of the disturbing numerical be-havior can be found in [35].) This is noteworthy and perhaps even alarming given that these are popular approximation schemes and this is a seemingly reasonable physical model of a damped elastic system. The reader is referred to [6] for more detailed discussion and examples. Here we will only illustrate typical behavior by considering a standard Galerkin nite element scheme.
Toward this end, dene the Hilbert space V = H1
L(0;1)H 1
L(0;1) H with
norm
k(u;v)k 2
V =
Z 1
0
(ju 0()
j 2+
jv 0()
j 2)d
and the sesquilinear form:V V !C j by
((u;v);(f;g)) =
Z 1
[v0()f 0()
?u 0()g
0()]d
?v(1)g(1):
Then the operatorAis related to by
hA(u;v);(f;g)iH =((u;v);(f;g)) for all (u;v);(f;g)2V:
Next, let Ni = i=N, i = 0;:::;N be a partition of the interval [0;1]. Dene linear splineshNi() by
hNi() =
8 <
:
N(?Ni
?1) ifNi?1
Ni, ?N(?Ni
+1) ifNi
Ni +1,
0 elsewhere, i= 1;:::;N?1
hNN() =
N(?NN
?1) ifNN?1
1,
0 elsewhere.
Set SN
1 = span fhNigNi
=1 and V
N = SN
1 SN
1 . Then V
N V and the operator
AN :VN!VN is dened by
hAN(u;v);(f;g)iH=((u;v);(f;g)) for all (u;v);(f;g)2VN:
(3.23)
This is a standard Galerkin nite element approximation scheme for (3.20). Even thoughT(t) is exponentially stable, this scheme is not uniformly exponentially sta-ble. This is seen convincingly in Figure 5, in which we plot (for the case = :5 in the boundary condition) the eigenvalues of AN for N = 8;16;32;64 (note: for
scaling purposes a few eigenvalues with large negative real part have been omitted from this gure). This is typical of the behavior observed by the authors of [6] for several dierent schemes. In particular, the popular nite dierence scheme suers a similar behavior for its eigenvalues. In [6] the authors used mixed element methods to construct some uniformly exponentially stable schemes.
It is also possible to dene Galerkin projections in a dierent inner product, and this has been pursued in [19]. In particular, dene the inner producth;ieby
h(u;v);(f;g)ie=h(u;v);(f;g)iH+ Z
1
0
a()[u0()g() +v()f 0()]d;
(3.24)
wherea() =e?1 and= ln (1+)
2 1+
2 . Also setVe=H 2(0;1)
\H 1
L(0;1)H 1
L(0;1)
with norm
k(u;v)k 2
Ve = Z
1
0
(ju 00 j 2+ ju 0 j 2+ jv 0 j 2)d;
and dene a sesquilinear forme:VeVe!C j by
e((u;v);(f;g)) =
Z 1
0
[v0()f 0()
?u 0()g
0()]d
?v(1)g(1)
+
Z 1
0
a()[u00()f 0() +v
0()g()]d
?a(1)[u
0(1) +v(1)]f 0(1):
Observe that domAVeand that
e(x;y) =hAx;yie for allx2 domA; y2Ve:
−4 −2 0 −300
−200 −100 0 100 200 300
N=8
Real
Imaginary
−4 −2 0
−300 −200 −100 0 100 200 300
N=16
Real
Imaginary
−4 −2 0
−300 −200 −100 0 100 200 300
N=32
Real
Imaginary
−4 −2 0
−300 −200 −100 0 100 200 300
N=64
Real
Imaginary
Figure 5 - Eigenvalues ofAN (nite element method)
Notice that we cannot deneANeby usingVe,eandVN in (3.23), becauseVN6Ve.
Instead, set
SN
2 = span f
Z
0
hNi(t)dtgNi =1;
and deneVeN =SN
2 SN
1 . Now,V
N
e Ve and we can deneANe:VeN !VeN by
hANex;yie=e(x;y) 8x;y2VeN:
It can be shown theoretically that the semigroupsTeN(t) = etAN
e are uniformly
ex-ponentially stable (see [19] for details). We can see this numerically in Figure 6, in which the eigenvalues ofANe are plotted forN = 8;16;32;64.
4. The parameter estimation problem.
In this section we continue the theme of investigating computational issues for semidiscrete approximations of in-nite dimensional problems, and in particular we focus on the parameter estimation problem. To begin, consider the following Cauchy problem evolving in a Hilbert spaceH:
_
z(t) = A(q)z(t) +f(t); 0< t < T;
(4.1)
−4 −2 0 −300
−200 −100 0 100 200 300
N=8
Real
Imaginary
−4 −2 0
−300 −200 −100 0 100 200 300
N=16
Real
Imaginary
−4 −2 0
−300 −200 −100 0 100 200 300
N=32
Real
Imaginary
−4 −2 0
−300 −200 −100 0 100 200 300
N=64
Real
Imaginary
Figure 6 - Eigenvalues ofANe (Galerkin method in equivalent inner product)
Here the operatorA(q) depends on a parameterq2Q1R
M. We assume that that
f 2 L
1(0;T;H) and that for eachq
2Q, the operator A(q) satises domA(q)H
and thatA(q) is the innitesimal generator of aC0-semigroupT(t;q) onH. We can
write (4.1) in the equivalent variation of constants form
z(t;q) =T(t;q)z0+ Z t
0
T(t?s;q)f(s)ds
(4.2)
for eachq2Q. Roughly speaking, a parameter estimation problem involves
estimat-ing the parameterqusing data consisting of observations of the system (4.2). There are many ways to formulate such a problem, and we restrict ourselves to consideration of a least squares t-to-data formulation, with data consisting of observations of the system at discrete timesti and discrete spatial locationsxj. That is, we are assuming
thatH =L2() where is some spatial domain, so thatz(ti;xj;q) makes sense and
represents system output which is to be t to data dij which represents actual
sys-tem observations. We refer the reader to [9] and the references therein for discussion of more general formulations of the parameter estimation problem (for example, one could allow a parameter dependent forcing termf(t;q) in (4.1), parameter dependent Hilbert spaces H(q), innite dimensional parameter space Q, dierent data forms,
We consider the least squares problem of minimizing the functional
J(q) =X
i;j
jz(ti;xj;q)?dijj 2
(4.3)
overq 2Q, subject to (4.2). This is a nonlinear optimization problem with innite
dimensional constraints (4.2). As before, to solve this problem computationally we need a semidiscrete approximation scheme and appropriate convergence results. Thus, let us introduce a semidiscrete approximation scheme consisting of a sequence of nite dimensional subspaces HN H, N = 1;2;:::, the corresponding orthogonal
projections PN : H ! HN, and operatorsAN(q) : HN ! HN which generateC 0
-semigroupsTN(t;q) =eAN
(q)tonHN. This denes a nite dimensional system
_
zN(t) = AN(q)zN(t) +PNf(t); 0< t < T;
(4.4)
z(0) = PNz
0;
or, equivalently,
zN(t;q) =TN(t;q)PNz
0+ Z t
0
TN(t?s;q)PNf(s)ds:
(4.5)
The approximate least squares problem is to minimize the functional
JN(q) =X
i;j
jzN(ti;xj;q)?dijj 2
(4.6)
overq2Q, subject to (4.5).
This is a nonlinear optimization problem with nite dimensional constraints, and there are many reasonable computational algorithms available for its solution (for example, see [9] and [21]). Such an algorithm will produce an optimal least squares solutionqN for eachN, and we desire a convergence result along the lines ofqN !q,
whereq is an optimal least squares solution (not necessarily unique) for the original innite dimensional problem. Assuming that Q is compact, one possibility for
de-veloping a satisfactory convergence theory ([4],[9],[10]) is based on establishing the following `forward' convergence result:
For arbitraryfqNginQwithqN!qwe havezN(t;qN)!z(t;q) in an appropriate
norm (frequently stronger than theH norm).
From our analysis of the simulation problem, we know that for each xedq, (H1) and (H2) implyzN(t;qN)!z(t;q) in theHnorm. In view of the parameter dependence of
the semigroupT(t;q) and the form of the functionalsJ andJN(pointwise evaluations
of the state), we consider the following hypotheses:
(H7)
For eachq2Qandz2H,TN(t;q)PNz!T(t;q)zinH, uniformly on compactt-intervals.
(H8)
The convergencezN(t;q)!z(t;q) obtained using (4.2), (4.5), (H1) and (H7)is suciently strong so thatzN(t;x;q)!z(t;x;q) fort2(0;T),x2.
There are results available to show that (H1), (H7) and (H8) imply the conver-genceqN !q. This indicates that the strategy `rst discretize, then optimize' may
However, recent computational results (see below) indicate that this is not always the case, and, motivated by our discussion for the optimal control problem, we shall try to provide a possible explanation for this.
Many optimization algorithms (e.g., see [17], [9, p. 173-175] and the references in each) require the computation of gradients (i.e., @J=@qm, m = 1;:::;M). One
popular method for computing gradients in optimization problems with dierential equation constraints is the so-called costate or adjoint method (see [9],[13]). In this method the gradient is dened in terms of a solution of a costate equation which involves the adjoint operator. In particular (we refer the reader to [9, p. 175-179] for details), one can formally rewrite (4.3) as
J(q) =X
i;j
Z T
0
jz(t;xj;q)?dj(t)j 2(t
?ti)dt;
(4.7)
wherez is dened by (4.1) anddj(t) =dij forti?1< t
ti. One then nds formally
that
@J
@qm(q) =? Z T
0
(t); @A@qm(q)z(t;q)
H dt
(4.8)
with the costate variabledened by _
(t) = ?A
(q)(t) + ?(t;q); 0< t < T;
(4.9)
(T) = 0;
and
?(t;x;q) = 2X
i;j
?
[z(t;x;q)?dj(t)](t?ti)(x?xj)
:
Once an approximation scheme is given for the original least squares problem, it de-nes a natural approximation of the costate equation and the formal costate depen-dent gradients. More specically, we have (tacitly assuming that our approximation scheme is such thatHNdomA)
@JN
@qm(q) =? Z T
0
N(t); @A
@qm(q)zN(t;q)
H dt
(4.10)
with the approximate costate variableN dened by
_
N(t) = ?A
N(q)N(t) +PN?N(t;q); 0< t < T;
(4.11)
N(T) = 0;
and
?N(t;q) = 2X
i;j
?
[zN(t;q)?dj(t)](t?ti)(x?xj)
:
as discussed above. However, if the gradients are calculated with the costate variable, then we could reasonably argue that such an optimization algorithm denes the op-timal solution qN in terms of the costate variable N, which is in turn dened as a
solution to an equation involving the adjoint operator. (See [1] for a precise statement of a convergence result for such an approach.) But this is precisely the same scenario that arises in the LQR problem (where the optimal control ueN is dened in terms
of the operator N, which is in turn dened as a solution to an algebraic Riccati
equation involving the adjoint operator), and we have seen there that an additional hypothesis of adjoint semigroup convergence (H6) is needed to establish theoretical convergence. We have also seen in the LQR problem that approximation schemes (e.g. the SPL scheme) that don't satisfy (H6) may converge for the simulation problem but not for the LQR problem. Our contention is that this is a possible explanation for the poor performance of some parameter estimation approximation schemes which utilize costate-based gradient calculations. Clearly the theoretical issue is still un-resolved, but there is numerical evidence to support this convention. For example, in [3] Banks et al used methods such as those described here to estimate material parameters in models of elastic beams. They found that for models in which the system operatorA(q) is skew-adjoint (i.e., A(q) =
?A(q), as occurs in beams with
no damping), the costate based gradient methods were highly competitive with other methods such as Levenberg-Marquardt. But in models which included damping (so that A(q) is not skew-adjoint), Banks et al experienced extreme diculties (tting the data) with the costate based gradient methods and abandoned them in favor of the Levenberg-Marquardt algorithm. Of course, for skew-adjoint system operators, strong semigroup convergence guarantees strong adjoint semigroup convergence, but this is not necessarily true for non skew-adjoint system operators.
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