Editors: M. Thoma and W. Wyner
R.E Curtain (Editor) A. Bensoussan, J.L. Lions (Honorary Eds.)
Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite- Dimensional Systems Proceedings of the 10th Intemational Conference Sophia-Antipolis, France, June 9-12, 1992
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest
Advisory Board
L.D. Davisson • A.GJ. MacFarlane. H. Kwakemaak J.L. Massey-Ya Z. Tsypkin "A.J. Viterbi
Editor
Honorary Editors
A. Bensoussan INRIA - Universit~ Paris IX Dauphine
L Lions Collage de France - CNES, Paris
INRIA Institut National de Recherche en Informatique et en Automatique Domaine de Voluceau, Rocquencourt, B.P. 105 78153 Le Chesnay, France
ISBN 3-540-56155-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-56155-2 Springer-Verlag NewYork Berlin Heidelberg
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F O R E W O R D
The 10th Conference on "Analysis and Optimisation of Systems" organised by INRIA is marked by a change in form : instead of covering a wide range of topics, as in the past, this conference is devoted to a specific domain of the theory of infinite- dimensional systems.We plan to proceed in the same way in the future, and we will cover the whole field of Analysis and Optimisation of Systems by a series of specialized conferences.
Among the advantages, this allows the possibility of covering a specific area in more depth and it gives the participants more opportunity for fruitful interaction.
We would like to express our thanks to the Organizations which have sponsored this Conference: DRET and ONREUR
We also would like to extend our gratitude to :
- the authors who have shown an active participation in this Conference - the many reviewers who have accepted the difficult task of selecting
papers - the chairpersons for having run with efficiency all the sessions
of the Conference - all the members of the Organization Committee, and particularly
Ruth Curtain who did a wonderful job. - The Public Relations Departement of INRIA and particulary C. Genest
and F. Tapissier for their excellent work in making this Conference really happen.
- Professor M. THOMA and the Editor SPRINGER VERLAG who have accepted to publish this series in the Lecture Notes in Control and Information Sciences.
A. Bensoussan J.L. Lions
This conference is under the sponsorship of:
DRET Direction des Recherches Etudes et Techniques ONREUR Office of US Naval Research in Europe
INTERNATIONAL PROGRAMME COMMI'I"rEE
Chairperson R. CURTAIN University of Groningen / INRIA-Rocquencourt
J. BALL J. BARAS L. BARATCHART J. BLUM F. CALLIER G. Da PRATO M. DELFOUR P. GAHINET W. KRABS A.J. PRITCHARD M. SORINE J.P. YVON
J. ZABCZYK
LOCAL ORGANIZING COMI'I"rEE
L. BARATCHART J. BLUM P. GAHINET M. SORINE J.P. YVON
INRIA-Sophia, France Universitd de Grenoble, France INRIA-Roequencourt, France INRIA-Roequencourt, France INRIA-Rocquencourt / Universitd de Technologic de Compi~gne, France
CONFERENCE SECRETARIAT
INTRODUCTION
The aim of the conference was to bring together engineers and mathematicians working in the field of infinite-dimensional systems who are specialists in one or more of the following categories of mathematical approaches :
_ semi-group approaches _ partial differencial approaches _ frequency domain approaches _ synthesis of state and frequency domain approaches
Since these approaches use very different sophisticated mathematical techniques, it is unusual for a scientist to be expert in all of them. On the other hand, these different approaches all purport to address the same control problems for the same classes of linear infinite-dimensional systems. It is therefore important to compare and discuss the advantages and disadvantages of these different mathematical techniques. To help ameliorate the communication gap a series of introductory tutorial lectures were given by specialists in above-mentionned fields. The writter accounts of these introductory lectures, together with their references, form an extremely useful source of background information on a very wide class of problems and approaches in linear infinite-dimensional sys tems .
For example, the semi-group theme is represented by an article on general background, an up-to*date survey on the Linear Quadratic Control Problem and Riceati
equations and an account of the state-space approach to HOO-optimal control problem. The results on this last very recent topic are less maticre than those on the Linear Quadratic Control Problem, but since the techniques involved are very similar a strong interaction between these two domains of activity can be expected in the future. Indeed,. it has already begun.
The second theme concerns input-output descriptions of systems in terms of transfer functions.After an introductory article on transfer functions, an outline of the coprime factorisation approach to control synthesis for different classes of transfer functions follows. The final paper on this theme concerns the application of these techniques to the important problem of robust controller design and in particular, finite-dimensional controller design in these articles, the connections with the first theme (semi-group approach) arc emphasized.
The third theme on partial differential equations is also introduced with a background article on the Lions approach, followed by recent results on Exact Controllability and Stabilisation using both the Hilbert Uniqueress Method and high frequency asymptotic methods.
The last theme on frequency domain approaches is also based on a transfer function description, but the connections with a state-space representation are not relevant.The first article describes the utilization of classical ideas in Harmonic Analysis, such as Hank¢l operators and Nehari's Theorem, to the systems theory context. Robust controller design is treated in a more general and deeper context in the second article and the final
contribution to this theme surveys recent results on the same HOO-domain viewpoint using techniques from Harmonic Analysis.
In addition, the proceedings contains several key survey papers on fundamental topics such as geometric theory, robust stability radii and relationships between input- output stability and exponential stability. Last, but not least, the proceedings contains many shorter contributions on recent original research covering all four of the above- mentionncd themes.
VII I
Of course, bringing together scientist from four very different approaches with the aim of having them interact with each other is an ambitious one. In this we succeeded, especially during the final round table discussion on "Different Approaches to Control of Infinite-Dimensional Linear Systems". The aim of these proceedings is to help to promote further interaction and fruiful collaboration in the future between scientists working on different approaches to the challenging problems in the field of infinite-dimensional linear systems.
T A B L E O F C O N T E N T S
1. T U T O R I A L L E C T U R E S
S E M I - G R O U P A P P R O A C H
Introduct ion to Semigroup Theory A. J. PRITCHARD ...................................................................................................
Riccati Equations Arising from Boundary and Point Control Problems I. LASIECKA ............................................................................................................
A State-Space Approach to the H**-Control Problems for I n f i n i t e - D i m e n s i o n a l Sys t ems B. VAN KEULEN .....................................................................................................
23
46
SYNTHESIS OF STATE AND F R E Q U E N C Y DOMAIN A P P R O A C H E S
Infini te Dimensional System Transfer Funct ions F. M. CALLIER, J . WINKIN ............................................................................... 72
Stabi l izat ion and Regulat ion of Inf in i te-Dimensional Systems Using Copr ime Factor izat ions H. LOGEMANN ......................................................................................................... 102
Robust Control lers for Inf in i te -Dimensional Systems R. F. CURTAIN ......................................................................................................... 140
P A R T I A L D I F F E R E N T I A L E Q U A T I O N A P P R O A C H E S
Control for Hyperbol ic Equations G. LEBEAU ................................................................................................................. 160
An Introduction to the Hilbert Uniqueness Method A. BENSOUSSAN ........................................................................................................ 184
F R E Q U E N C Y DOMAIN A P P R O A C H E S
The Nehari Problem and Optimal Hankel Norm Approximation N. J. YOUNG ............................................................................................................. 199
Topological Approaches to Robutness T.T. G E O R G I O U , M. C. SMITH ............................................................................. 222
Frequency Domain Methods for the H**-Optimization of Distr ibuted Systems A. T A N N E N B A U M .................................................................................................... 242
2. CONTRIBUTED PAPERS
Disturbance Decoupling Problem for Infinite-Dimensional Systems H. J. ZWART .............................................................................................................. 279
Simultaneous Triangular-Decoupling, Disturbance-Rejection and Stabilization Problem for Infinite-Dimensional Systems N. OTSUKA, H. INABA, K. TORAICHI ............................................................. 290
Robust Stability Radii for Distributed Parameter Systems : A Survey S. TOWNLEY ............................................................................................................... 302
Solutions of the ARE in terms of the Hamiltonian for Riesz-spectral Systems C. R. KUIPER, H. J. ZWART ............................................................................... 314
Regional Controllability of Distributed Systems A. EL JAI, A. J. PRITCHARD ............................................................................. 326
On Filtering of the Hilbert Space-valued Stochastic Process over Discrete-continuous Observations Y. V. ORLOV, M. V. BASIN ................................................................................... 336
Boundary Stabilization of Rotating Flexible Systems C. Z. XU, G. SALLET ................................................................................................ 347
SYNTHESIS OF STATE AND FREQUENCY DOMAIN APPROACHES
Frequency Domain Methods for Proving the Uniform Stability of Vibrating Systems R. REBARBER .......................................................................................................... 366
The Well-Posedness of Acceleromctcr Control Systems K. A. MORRIS .......................................................................................................... 378
Comparison of Robustly Stabilizing Controllers for a Flexible Beam Model with Additive, Multiplicative and Stable Factor Perturbations J.B O N T S E M A , R . F . C U R T A I N , C . R . K U I P E R , H . M . O S I N G A ............................ 388
On the Stability Uniformity of Infinite-Dimensional Systems H. ZWART, Y. YAMAMOTO, Y. GOTOH ............................................................ 401
×l
Modelling and Controllability of Plate-Beam Systems J. E. LAGNESE .......................................................................................................... 423
A Simple Viscoelastic Damper Model - Application to a Vibrating String G. MONTSENY, J. AUDOUNET, B. MBODJE .................................................... 436
Controllability of a Rotating Beam W. KRABS .................................................................................................................. 447
Min-Max Game Theory for a Class of Boundary Control Problems C. MCMILLAN, R. TRIGGIANI ......................................................................... 459
Microlocal Methods in the Analysis of the Boundary Element Method M. PEDERSEN ........................................................................................................... 467
Stochastic Control Approach to the Control of a Forward Parabolic Equation, Reciprocal Process and Minimum Entropy A. BLAQUIERE, M. SIGAL-PAUCHARD ......................................................... 476
Observability of Hyperbolic Systems with Interior Moving Sensors A. KHAPALOV .......................................................................................................... 489
Controllability of a Multi-Dimensional System of Schr0dinger Equations : Application to a System of Plate and Beam Equations J. P. PUEL, E. Z U A Z U A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
Decay of Solutions of the Wave Equation with Nonlinear Boundary Feedback F. CONRAD, B. RAO ................................................................................................ 512
Boundary Approximate Controllability for Semilinear Heat Equations C. FABRE, J. P. PUEL, E. ZUAZUA ................................................................... 524
On the Stabilization of the Wave Equation O. MORGUL ................................................................................................................ 531
FREQUENCY DOMAIN APPROACHES
The Hankel Singular Values of a Distributed Delay Line A Fredholm Equation Approach L. PANDOLFI ............................................................................................................ 543
Rational Approximation of the Transfer Function of a Viscoelastic Rod K. B. HANNSGEN, R. L. WHEELER, O. J. STAFFANS ................................ 551
Some Extremal Problems linked with Identification from Partial Frequency Data D. ALPAY, L. BARATCHART, J. LEBLOND ................................................... 563
XII
Approximat ion of Inf in i te-Dimensional Discrete Time Linear Systems via Balanced Realizations and an Application to Fractional Filters C. BONNET .................................................................................................................. 574
A "Relaxation" Approach for the Hankel Approximation of some Vibra t ing St ructures N. M A I Z I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
Numerical Methods for H** Control of Distributed Parameter Systems D. S. FLAMM, K. K L I P E C .................................................................................... 598
Robust Controller Design for Uncertain Time Delay Systems Z. Q. WANG, S. SKOGESTAD ................................................................................. 610
Parameter Ident i f icat ion of Large Spacecraf t Systems Based on Frequency Character is t ics D. R. AUGESTEIN, J. S. BARAS, S. M. F ISHER ........................................... 624
On the Optimal Minimax Tuning of Controllers for Distributed Paramete r Sys tems S. P O H J O L A I N E N , M. LAAKSONEN .................................................................. 636
Introduction to Semigroup
Coventry CV4 7AL, UK
1 A b s t r a c t
In this paper some basic sytem theoretic concepts will be introduced for abstract systems of the form
e(t) = Ax(t) + Bu(0, x(0) = ~0, y(t) = C.(t). (1)
Here A is the infinitesimal generator of a strongly continuous semigroup S(t) on a Banach space Z and necessary and sufficient conditions for this to be the case are given by the Hille-Yosida theorem. For U another Banach space B E £(U, Z) and x ° • Z, u(.) • L2(0, co; U) a mild solution is defined to be
j~0 t ~(t) = s(t)~ ° + s(t- s)Z~,(s)d~ (2)
and z(.) E C(0; co; Z). Various definitions of controllablity, observability, stabi- lizability, detectability, identifiability and realizability will be given and theorems which characterize them will be stated. Throughout the paper examples will be given (albeit trivial ones) which illustrate the way the abstract definitions and re- sults can be applied to concrete problems defined via pa~-tiaI differential equations and delay equations. In preparing this introduction I have made considerable use of the following book by Ruth Curtain and Hans Zwart An Introduction to Infinite Dimensional Linear Systems Theory which is to be published soon.
2 F o r m u l a t i o n o f t h e s y s t e m
Let z0 be the state, at time zero, of a linear, time-invariant, uncontrolled dynamical system on a Banach space Z and z(t) denote the state at Lime t. Then we may define a linear operator S(t) : Z --* Z, such that
S(O) = I (identity on Z) (1) *(*) = S(*)~0. (2)
2
If the solution is unique, then z(t + a) is the same point in Z as the point reached by allowing the dynarnics to evolve from z(a) for .a time t, thus
z(t+~)=S(t+s)zo=S(t)z(s)=S(t)S(s)Zo, t>8>O, WoEZ
whence, the semigroup property
S(~+8)=S(OS(~), ~>_ ,>_0. (3)
If, for each t, the state z(t) varies continuously with variations in the initial state, then S(t) is a bounded map. Finally, we impose some smoothness with respect to time on the solution or trajectory z(.). In fact, we only require z(t) --* zo as * --~ 0 +, since this will imply that the trajectory is continuous for all time. This motivates the following:
Defini t ion 2.1 (Strongly continuous semigroup) A strongly continuous semigroup is a map S(t) from R + to £(Z) which satisfies
s(o) = z (4)
s(~ + 8) = s(os( , ) vt,8 >_ o (~) I IS (O=o- =oll -~ o as ~ -~ o + , V=o e z . (6)
It follows immediately from this definition that there exists constants M and w such that
I IS(~) l l - - M e ~' . (7)
However, since we have only assumed S(~)zo is continuous in ~, it is not possible in general to differentiate S(f.)zo. Nevertheless, we define:
Defini t ion 2.2 (Infinitesimal generator of a strongly continuous semigroup) The infinitesimal generator A of a strongly continuous semigroup S(t) on a Banach space Z is defined by
1
= lira ~ [ s ( o - z]~ (8) Az t-*O+
where the domain, D(A), of A is the set of z for which the above limit exists.
Then, it can be shown that A is a closed, densely defined, linear operator, and
d(S(Ozo ) = AS(Ozo = S(OAzo, zo e D(A) . (9)
Thus, z(~) = S(t)Zo is the solution of the abstract differential equation
~. = A z , z(9) = Zo E D(A) . (10)
The Hille-Yosida Theorem gives a complete characterization of those operators which generate strongly continuous semigroups.
T h e o r e m 2.3 (Hille-Yosida) A necessary and sufficient condition for a closed linear operator A, with dense do- main in a Banach space Z, to generate a strongly continuous semigroup S(t) is ~hat there exist real numbers M and w such that for all real A > w, A is in the resolvent set of A, and
M II(,~i - A ) - ' I I < (A _ ~) - - - - - - - ; , r = 1 , 2 , . . . (11)
i,~ ~,hich c~e IIS(t)ll <__ Me".
E x a m p l e 2.4 Let A E £(Z) , then
S(O = e ~' = ~ A"~"/n! n=0
is a strongly continuous semigroup on Z with infinitesimal generator A.
(12)
E x a m p l e 2.5 Let {~o,},>1 be an orthonormal basis in a sepazable Hilbert space H and (,~.)n_>l be a sequence of real numbers. If sup A. < eo then
n
n=l
{h e n : Z ~=.(h, ~.>~ < ~}. n = l
We are part icularly interested in control problems which can be wri t ten formally as
~(t) = Az(t) + Bu( t ) , z(O) = Zo (13)
where we assume B 6 £.(U, Z), U is a Banach space, and u 6 L2(0, T; U). The first p roblem encountered is the concept of a solution of (13). If Z is finite- dimensional, then (13) has the solution
f0 L ~(~) = s(t):o + s ( t - s)B~(s)ds. (14)
However, if Z is infinite-dimensional, we cannot in general differentiate (14) to obtain (13). If this were the case, we would say that (14) is a str /ct solution. Nevertheless, z(.) is well defined by (14) and, in fact, z E C(0, T; Z). We call (14) the mild solution of (13). And although our problems are mot ivated by differential equations of the form (13), we will actually define our system via the integral equation (14).
4
Finally, we will assume that the output from the system is related to the state by means of the equation
y(~) = CzCt) (15)
where we assume C 6 £ ( Z , Y ) and Y is a Banach space. Then, from (14),
f y(t) = cs(t) o + cs( - (16)
If zo = 0, note that
( s I - A)-~z = e - a S ( t ) z d t for Re s > w (17)
then, by taking the Laplace transform of (16), we obtain
9(s) = C(sI - A ) - ' Bfi(s) . (18)
Thus, the transfer function of the system is given by
G(s) = C(sl - A ) - I B . (19)
For many physical systems it is possible to obtain a different representation which is important in the application of numerical methods and approximation theory. In order to see how this can be carried out, we specialize and assume that the operator A is defined on a Hilbert space H, and has a distinct set of eigenvalues {Ai}, with corresponding eigenvectors ~i such that {~Pl} span H. Then,
A%ai = Ai~i. (20)
If lbi axe eigenvectors of the adjoint operator A', then
A'~bi = Ai¢i (21)
where Ai is the complex conjugate of Ai. Moreover,
(~bi ,~j)=0 if i # j (22)
and we may normalize ¢i, qal, so that
(~/,i,~ol) = i V{. (23)
Now, let us assume that there exists a solution of the form
z( t ) = (24) i = l
Substituting in (13), we find ~ = x d,(t)~, = Z~=I Aia,(t)~pi + Bu(t ) . Taking the inner product with ~b, gives
a,Ct) = A,a, Ct) + (¢, , Bu( t ) ) , ,, = 1 ,2 , . . . (25)
We also have
oo
ai(O) = aio. (27)
sCt)z = f i e~'%(¢. ~>. (28) i = l
So we have seen that there are a number of differen~ ways of describing our system. The differential equation may be given in the form (13) or, alternatively, its mild so- lution may be formulated in terms of the semigroup. A transfer function description may be given or we may describe the system in terms of an infinite set of ordinary differential equations. In order to see how these different approaches are realized in practice, we consider the following simple example.
E x a m p l e 2.6 Consider a metal bar in a furnace. The bar is heated by jets whose spatial distribution b(z) is fixed but whose magnitude (intensity) can be varied by a control u. For one-dimensional flow, heat balance considerations over a small region about the point z yield
= ~ ~ . - + b (z ) . ( t )d2
where 8(x, t) is the temperature in the bar at time t and distance x from one end, a is the thermometric conductivity. Differentiation yields
08 0~8" t) + b(x)u(t) (29) 0-7 = a0~z2 ~2'
Let us assume that the temperature at each end is kept at a constant value T, and that the initial temperature distribution in the bar is 8o, so that 8(0,t) = 8(l , t) = and 8(2,0) = 8o(x), where I is the length of the bar. Set 8(2, t) = T ( z , t ) + "~ and 0o(2) = T o ( 2 ) + ¢ , then
OT O~T" t) + b(2)~(t) (30)
T ( O , t ) = r ( l , t ) = O (31) T(z,O) = To(z). (32)
Finally, assume that it is possible to sense the temperature at a given point, and that this is accomplished by taking some weighted average over nearby points. Then the output of
I the sensor can be modelled by h(t) = fo c(z)8(x, t)dx. Writing h(t) = y(t) -F T f t o c(z)dx, then
f y(t) = ~ ( ~ ) T ( ~ , t ) e ~ . (33)
Equations (3O)-(33) represent a single-input, single-output distributed parameter system. Let us define an operator A by
d2T (AT) ( z ) = a-~Tx2 (z ) (34)
D(A) = H=(O,l) n HI(O,O (35)
then we may abstract the differential equations (30/-(33) to obtain
9.(t) = AzCt)+ buCt), z(O) = zo (36 / ~(~) = (~ ,K~ ) ) (37)
where z(t) = T( . , t ) , and the inner product is taken on L2(O,l). It is easy to show that the operator A generates a strongly continuous semigroup S(t), where
s(~)z = ~] ~-,--~-'~.(~.. ~) (38) n = l
with ~ . ( z / = (V/~7Osin nTrz/l, so that we can obtain the mild solution (14/. In order to construct the transfer function, we have to find (sI - A) -1, i.e. we have to solve the problem (sI - A)z = ~. for z E D(A) and any Y. E La(O, l). It is readily shown that the solution is
1 [s inh(~ / ,~ ) ' /~= ['sinhC~l~l,/~Cl_ol~(~ld~_/o'sinhC.l,~ll/~C=_,.l~C,.ld,.] Z(Z) - (,,,)'/= ' ~ .,o
so that the transfer function is given by
G(~) = (c, (~I - A)-Ib)
f0' c(~) [sinh(d~)~/2x [ sinh(8/c~)l/2(l a)b(~)da (~)w~ t sinh(d~)ml J0
- /o" s~,~(~/,,)'/'(= - ,,)b(,,le,,]e=. (391
equivalent infinite-dimensional system, we assume that b(z) =
(40)
(41)
To obtain the oo ~"~'n=l b , . ~ . ( ) , c(x) = ~ = 1 c.¢2.(z), and (z(t))(x) = ~ = 1 a.(t)~on(x), then
a . ( t ) = - l---~-a.(t) + b . ~ ( t ) , ~ = 1 , 2 , . . .
yCt) = ~ c . a . ( t ) .
3 Linear feedback control
In many appl icat ions , we wish to consider s ta te or output feedback control, in which c a s e
~(~) = F z ( ~ ) (1)
o r
~(~) = ~yCt) = ~c~(t) (2) where we assume F E S(Z, U), ~' E £(Y, U), Y is a Banach space. Note that (2) is just a special case of (I) for which we have the following result:
Propos i t i on 3.1 Let A generate a strongly continuous semigroup S(t) on a Banach space Z, B E E( U, Z) and F E ~.( Z, U), then A + B F generates a strongly continuous semigroup S F( t ) which satisfies
Z' s~ ( t ) z = s ( t ) z + s ( t - s ) B F S ~ ( s ) z e s
and g Ils(t)ll < M e ~'` then
IIS~(t)ll < M e (~+Mlla~H)' • (3)
A direct consequence of the above proposition is that z(t) = SF(t)zo is the mild solution of the differential equation ~(t) = (A + BF)z( t ) , z(O) = zo. In some cases, we wish to consider time-varying feedback operators of the form
,,(t) = F(t)z(~) . (4)
Obviously, in such cases the operator A + BF( t ) will not generate a strongly con- tinuous semigroup; consequently, we introduce the following generalization:
Defini t ion 3.2 (Mild evolution operator) Let A(T) = {(s,~) : 0 < s < t < T} then U(. , . ) : A(T) ~ / : ( Z ) is a mild evolution operator if:
(a) U(t, t) = I Vt e [O,T], (b) U ( t , r ) U ( r , s ) = U( t ,~) O < s < r < t < T , (c) U(., s) is strongly continuous on Is, T], (d) U(t, . ) is strongly continuous on [0, ~].
A consequence of the above proprty is that
ess sup IIUCt, s)ll < ~ . C 5) a(T)
We have not assumed any differentiability properties of the mild evolution operator; however, the following concept is useful.
Defini t ion 3.3 (Weak evolution operator) Suppose that for all t > 0 A(t) is a linear operator on Z, its domain D(A) is dense in Z, independent of t and A(.)z is continuous for every z E D(A). If
or(t, ~)z - z = u(~, e )A (p )~ep ~, ~ ~ ~ ( T )
then U(f., s) is said to be a weak evolution operator with the generator .4(-).
Note that
O u ( ~ , s ) z , = - U ( t , s)A(s)z0 s e [0,t], Zo e D(A) .
Now suppose that F E C(0, oo; £(Z, U)) then we have the following:
P ropos i t i on 3.4 Let A generate a strongly continuous semigroup S(t) on a Banach space Z, then
u(~, ~)z = s ( t - ~ ) . + s ( t - e ) ~ F ( p ) c r ( e , s )~ep (6)
defines uniquely a mild evolution operator U(t, s). Moreover, U(t, s) is a weak evo- lution operator with weak generator A + B F.
4 Controllability
Note that if we require a strict solution of (2.13), then necessarily z(t) E D(A) , t > 0. So if A is unbounded (i.e. D(A) ~ Z) it will be impossible to steer a strict solution to every point in Z. For mild solutions we have the following:
Defini t ion 4.1 (Exact controllability on [0, T]) System (2.14) is said to be exactly controllable on [0, T] if, given any two points zo, zl E Z, there exists a control u E L2(0,T; U) such that
zl = S(T)zo + S (T - (1)
It can be shown that, if Z and U are reflexive Banach spaces, then (2.14) is exactly controllable on [0, T] if and only if there exists 7 > 0 such that
> 71{='llz.. (2)
E x a m p l e 4.2 We consider the controlled conduction problem of Example 2.6 and ask whether or not it is possible to steer the system, via controls u E L2(O,T), from the zero temperature distribution to a prescribed temperature distribution T1 E L2(0, I) in time T. Let TI(X ) = ~ . ° ° = l YnWn(X), where ~o,(x) = x / (2~s in nrz / l . Then {vn} e l 2 and solving (2.40), we require
Z r n = e;~"(T-')b,u(s)ds, n = 1,2,. . .
where .k, = -an2~r2/l 2. Using the Schwaxtz inequality,
I r . I 2 _< Ib . l 2 e2~"(T-°)ds u2(8)ds, n = 1,2, . . .
But, if the control energy is to be finite, then
I([b, I ~'lb, I 1/2
for some constants K and 2~'. So we see that the system cannot be exactly controllable and only those states with "sufficiently smooth" Fourier coefficients axe likely to be achievable.
o
The above example indicates that the concept of exact controllability may be too strong for infinite dimensional systems. In fact, it can be shown that systems gov- erned by parabolic partial differential equations are never exactly controllable with control action which occurs in practice (although this is not the case for hyperbolic systems). Frequently, however, it may be possible to control the system exactly to a linear subspace of a Bane.ch space Z.
Defini t ion 4.3 (Exact W-subspace controllability on [0, T]) System (2.14) is said to be exactly W-subspace controllable if, given any two points z0, zl, with z0 and zl E W C Z, there exists a control u E L2(0, T; U) such that
/o T zt = S(T)zo + S ( T - s )Bu(s )ds .
If W, Z and U are reflexive Banach spaces, W C Z with continuous, dense injection, S ( T ) W C W then (2.14) is exactly W-subspace controllable on [0,T] if and only if there exists 7 > 0 such that
IlB*S'(.)z*llL~to,r;v.) >__ 71lz'llw.. (3)
E x a m p l e 4.4 Consider the hyperbolic system
~ 2 " W . . at 2 (z, t) = (A)(x, t) + u(z, t) (z, t) e 12 x (0, T)
Ow "x t" w ( x , O ) = - ~ ( , ) = 0 x e 1 2 (4)
~(~, t) = 0 (~,t) ~ 0n x (0, T)
where D C R a is an open bounded set. It is assumed that (¢nj) is an orthonormal basis (in L2(D)) of eigenfunctions of the Laplacian A with Dirichlet boundary conditions. The associated eigenvalues A,~ have multiplicities r~. We may write (4) in the state space form (2.13) [°l on the Banach space Z = Hol(a) x L2(a), with
Then A generates a strongly continuous semigroup S(t) where
r n
~ [/~,, ,.;> cos ~ -~om~ + ~_~o~1,~,,,,,~> ~o ~_~o~,~] ,o~
Z2 ~__j~_.~[__(__)~n)l/2(Zl,¢nj)Sin((__~n)l]2~.}_~(Z2,¢nj)gos{(--~n)l/2~.}](/)nj n j = l
We assume u e L2(0,T; L2(~)). Now
IlD*S'(t) [ zl ] 2 z~ ILL,( . ) =
Hence
+(z2, ¢-i) cos {(-;~.)z/2t}] 2.
i ~ ~ { _ ~o<z,.~o~>, [~ - sin ~ ~.~1,,T~1 . j=l 2(-'X")x/2 J
+2(_~.),/~(~, ¢.j)(~, ¢.j)[1 - cos (2(-~.)~/~T}]
+(z2, ¢.j)2 [T + sin ~{~(-~")l/2T} ]j}.
r n
11~l12z = ~ ~ [ - A.(~, 4,.j) 2 + (~2, ¢.s)2]. n j=l
10
So the system is exactly controllable if there exists 7 such that
1 ~ "~ { sin {2(-A.)II2T} ]2(_A.) t12 J IT +sin {2(-~A.)112T} ]j - ~.<z,, ¢.j)~ [T - + <z2, ¢.i) 2
c~ {2(-;,.)InT} +2( - A")m(z~' ¢"#)(z:z' '/""J) 1 - 2(-A.)~n l
r n
. j=l
This will certainly be the case if there exists 7 > 0 such that
I. r r _ sin { 2 ( - ~ , , ) t / ~ r } _ 2 ~ 1 j + 2 X Y 1 - cos {2 ( - .X , , )~ /~r} X 2 2(-A.) I I 2 2(--A.)I/2
+Y~[r + ~in {2(-~.)~I~r} _ ~;~] > 0 2(- ,x . )m
for X, Y E R. Such a 7 will exist if
T > sin {2(-An)l/2T}2(_An)II 2 II
and T2 [sin {2(-A.)II~T}] 2 > [i - cos {2(-A.)II2T}] 2
- - 4 . x . - ~ ( ~ I •
T > [sin {(-,X.) ' /~T} ( - ,~ . )m
which is valid for any T > 0. This result shows that for all T > 0 the system is exactly controllable, m
Since stability is very often an important consideration, the origin plays a distin- guished role.
Def ini t ion 4.5 (Exact null controllability on [0, T]) System (2.14) is said to be exactly null controllable on [0,T] if, given z0 6 Z, there exists u 6 L2(0, T; U) such that
- S(T)zo = S(T - s)Bu(s)ds. (5)
If U and Z are reflexive Banach spaces, then (2.14) is exactly null controllable on [0, T] if and only if there exists 7 > 0 such that
IIB'S*(.)z'IIL2(o,T;U. ) >_ 7}IS*(T)z*]Iz.. (6)
A weaker concept of controllability is the following:
11
Definition 4.6 (Approximate controllability on [0, T]) System (2.14) is said to bc approximately controllable on [0, T] if, given Zo, zl E Z, and s > 0, there exists a control u e L2(0,T; U) with IIz(T) - zlU <_ s.
Although the assumptions of reflexivity of the Baaach spaces U, Z axe not essential here, we will assume that they hold, then (2.14) is approximately controllable on [0, T] if and only if
B'S*(~)z* = 0 V~ e [0,T] =~ z" = 0 . (7)
E x a m p l e 4.7 For Example 4.2, it is easy to show that S(t) = S*($), where
S*(t)z* = f i e'X"t~2n(~p,,z*) and B'z = (b,z) n = l
and where An, £a,~ are defined in Example 4.2. Hence
B's'c)~* = ~ b,,e~"(~., ~').
Obviously, if b, = 0 for some n, then we cannot make any conclusion about (£a., z ' ) and certainly cannot conclude that B*S*(t)z* = 0 implies z* = 0. So we require b, ~ 0. In fact, it can be shown that this is a neccssary and sufficient condition for approximate controllability, r~
D e f i n i t i o n 4.8 (Approximate null controllability on [0, T]) System (2.14) is said to be approximately null controllable on [0,T] if, given any z0 E Z and e > 0, there exists a control u E L2(0, T; U) such that Ilz(T)]] < e.
It can be shown that a necessary and sufficient condition for approximate null con- trollability on [0,T] is that the sct of points for which
B'S ' ( t ) z* = 0 Vt e [0,T] (8)
is contained in the set of points for which
S ' (T ) z" = O. (9)
5 Observability
The input-output map of the system is given by (2.16), namely,
f y(~) = cs ( t )~o + CS( t - . ) B ~ ( s ) d ~ .
If we assume that the control is known on the interval [0, T], then so is the function
f ~(~) = y(f,) - CS(~ - s)Bu(s)ds (1)
12
and we have ~(0 = csCt)zo. (~)
Questions of observability are concerned with the problem of determining the state of the system z(.) given the output y(.) or, equivalently, .~(.). Clearly, we are able to compute z(t) for all t >_ t if z(t) is known. Here, we consider two possible values of t, namely 0 and T. Although it may seem from (2) that # E C([0, T]; Y), since we want to consider the possibility of perturbations to the output we will take ~ E L2(0, T; Y) with Y a reflexive Banach space.
Definit ion 5.1 (Initial observability on [0, T]) System (2) is said to be initially observable on [0, T] if
c s ( O z o = o for a.~. t e [0,T] ~ zo = 0 . (3)
We see that if the system is initially observable then there is a one-to-one relationship between the output and the initial state, i.e. given g(.) and z0 satisfies (2) then z0 is unique. In many physical problems this concept is not strong enough since, if the output y is slightly perturbed, then the corresponding initial state could vary considerably. We therefore introduce a continuity hypothesis.
Definit ion 5.2 (Continuous initial observability on [0, T]) System (2) is said to be continuously initially observable on [0,T] if there is a continuous map between the output and the initial state, i.e. if there exists 7 > 0 such that
I I~(Olla,(oz;~) = I lCS( ' )~ol l~=(oz~Y)>-~l lzo l lz- (4)
We note immediately that the conditions (3) aad (4) are similar to the control- lability conditions (4.7) and (4.2), respectively. In fact, the concepts are dual to each other in the following sense. Suppose Z is reflexive. Let us define our orig- inal system by the sextuplet {Z, U, Y, S(t), B, C} and denote the dual system by {Z*, Y', U*, S'(t), C*, B'}. Then, the original system is approximately controllable on [0,T] if and only if the dual system is initially observable on [0,T], and the original system is exactly controllable on [0, T] if and only if the dual system is con- tinuously initially observable on [0, T]. But we have seen that exact controllability may be too strong a requirement, so the same must be said of continuous initial observability. We have two possible ways of preserving the continuity of the map fi'om output to initial state. One way is to assume that the output is smoother thaal that which we have previously considered, the other is to seek a space larger than Z for which the continuity holds.
Example 5.3 Suppose Z = L2(0, co), to > 0
(sco~) (= ) o O < z < t
{ z ( z ) = __. to (C~) (z ) = 0 0 < = < to
13
Then
= LT fm~,~(to_t,o)lZ(P)12dP dr.
If T < to and z(x) = O, x >_ l0 - T, then f [ HCS(t)zH2dt = 0 and so the system cannot be continuously initially observable. However if T > to, then
: Z '°/,o=_ > (m- to) --L ~ Iz(p)2dp
= (T - to)llz(-)ll~.,(o.~). Hence the system is continuously initially observable.
Defini t ion 5.4 (Continuous initial V-large space observability on [0, T]) System (2) is said to be continuously V-large space observable on [0, T] if there exists a Banach space V such that Z C V and the output to initial state map is continuous between L2(0, T; Y) and V, i.e.
IlCS(.)~ollL=(oz;~,) > ;II;o11,,, (s)
for some 7 > 0.
Clearly, this is the dual concept to exact V*-subspace controllability. Very often we require knowledge of the state z(.) in order to construct controls. If we observe over an interval [0, T], then the construction of controls may require z(t) for ~ >_ T, this leads to the following definitions:
Defini t ion 5.5 (Final observability on [0, T]) System (2) is said to be finally observable on [0, T] if
CS(t)zo = 0 for a.e. ~ 6 [0,T] (6)
implies S(T)zo = O. (7)
Defini t ion 5.6 (Continuous final observability on [0, T]) System (2) is said to be continuously finally observable on [0, T] if there exists 7 > 0 such that
HCS(')ZollL~(o,r;r) >_ "flIS(T)zollz. (S)
14
6 Control labi l i ty and observabil ity in Hi lbert space
In many examples, U, Y, Z will be Hllbert spaces. Then if the system is exactly controllable to the subspace W on [0, T], we require
B * * 2 II s (')~lbco,r,~) -> ~11~11~. (1) Let us further assume that there exists a 2 such that
a~ll~ll~. > lib S (')zllL, co,r;u) (2)
then
If we define the operator G by
f Gu = S(T - s)Bu(s)ds (4)
then G e £(L2(O,T; U), W), and
(G" z)Ct) = B*S*(T - t)z (5)
with G" 6 £(W*, L2(O, T; U)). Moreover,
/0 f C a " z = S ( T - s ) Z ~ B ' S ' ( T - s )~d~ = S ( t ) B B ' S ' ( t ) z a
and from (3), GG* • ECW', IV) and (GG')-' • £(W, W'). ~Ve define the operator GG* to be the controllability operator W(T), i.e.
W(T) = GG*. (6)
Now we axe in a position to construct a control which steers z0 • Z to zl • W. Specifically, we require
zl = S(T)zo + S(T - s)Bu(s)ds (7)
with zl - S(T)zo • W by hypothesis. Let
u(t) = B*S'(T - t)W-'(T)[z~ - S(T)zo] (8)
then ll,,llt.,coz,u) <- IlV'll~c~.,~.,(o,~w))llW-'ll,-cw.w.)llz~ - S(T)zo[iw.
Thus, the control is well defined and it is easy to show that (7) is satisfied. Similar arguments lead to a dual result on continuous initial V-laxge space observ- ability. If we construct an operator O(T) defined by
f O(T)z = S*(t)C'CS(t)zdt (9)
15
] : s'(t)C'y(t)e =
whence
Note
fi £ ( V , V ' ) . So, if ~(t) = CS(t)Zo, then operating by S ' ( t ) C " and
j~0 T O - I ( T ) S*( t )C'9( t )d t = zo. (10)
~00 T lizol[v <_ llO-1(T)Hr.(v.,v)Jl S*(t)C'dt[[~(L,(O,T;Y),V.)I{~I[L,(o,T;y).
We have therefore obtained explicit formulae (8) and (10) for the control steering Zo to zl, and for the estimation of the initial state. In the first case, we require exact subspace controllability and zl E W; in the second case, we do not necessarily expect the initial state to be a Z-valued function. This may seem unsatisfactory but the alternative is to assume that the output data is smooth. However, if we are only interested in obtMning z (T) , then we require continuous final observability and we have to adjust (10) to give
S ( T ) O - I ( T ) S ' ( t )C '~ ( t )d t = z ( T ) . (11)
Similarly, if we require exact null controllability, then the control is
u(t) = - B * S ' ( T - t)~Y -1 ( T ) S ( T ) z o . (12)
The advantage of such considerations is that, for parabolic systems, S ( T ) is a smoothing operator so it may be that z (T ) as in (11) lies in Z not V; similarly, S(T)zo may lie in W. On the other hand, for hyperbolic systems, where the semi- group is not smooting, it may be possible to have W = V = Z (see Example 4.4).
7 Stability, stabilizability and detectability We have seen that, for any strongly continuous scmigroup S(t) , there exist constants M and w such that [[S(t)l [ ~ M e ~'t, t >_ O. We say that the semigroup is exponentially stable if w can be chosen to be negative. In this case, the solution z(.) = S(')Zo of the uncontrolled system
~(t) = Az ( t ) , z(O) = zo
is such that I[z(t)ll <_ Me~'tllZoll and so decays exponentially to zero. An important criterion for exponentiM stability is the following:
P ropos i t i on 7.1 The strongly continuous semigroup S i t ) on a Banach space Z is ezponentiaUy stable i f and only i f there exists 7 > 0 such that
fo llS(t)zll2d _< (Izll', Z. Z E
16
If Z is a Hilbert space the above result can be reformulated as a Liapunov equation.
P ropos i t i on 7.2 Suppose that A is the generator of a strongly continuous semi- group on a Hilbert space H. Then S(t) is exponentially stable if and only if there exists a positive operator P = P* E L(H) such that
(Az, Pz)H + (Pz, Az)H =--IIzi]2H, z e D(A) .
For finite dimensional systems a matrix A generates an exponentially stable semi- group e A t if and only in sup{Re A : A E a(A)} < 0. This is not necessarily the case in infinite dimensions since the growth rate w is not necessarily determined via the spectrum.
Defini t ion 7.3 We say that the semigroup S(~) satisfies the spectrum determined growth assumption if
sup{Re A: A • a(A)} = l im{ln IIS( )ll/t} = ~0. (1)
(1) holds for a large class of semigroups and when this is the case for every w > w0 there exists M~ such that lIS(l)ll < M~e ~L, t > O. If S(t) is not exponentially stable it may be possible to stabilize it by state feedback.
Def ini t ion 7.4 (A, B) is said to be stabilizable if there exists F • L(Z, U) such that the semigroup SF(I) generated by A + B F is exponentially stable.
One might hope that if the pair (A, B) is approximately controllable then (A, B) is stabilizable. The following exaraple shows that this is not necessarily the case.
E x a m p l e 7.5 Let Z = 12, U = C
Az = (z,,~212,z313,...,z,l~,...) ZU = (hi u , b 2 u , . . , , b n u , , . . )
where b, ~ O, ~ = 1 Inb,] 2 < co. Then B e Z(C, Z) and it is easy to show that (A, B) is approximately controllable. For any F e ~(Z,C) there exists f e 12 such that Fz = {z, f). Consider the solutions of
Then
or
x . = z . / n + b . ( z , / )
z . = n x . - n b . ( z , f ) .
Now (nb,) E 12, but there exists x E 12 such that (nx,) ¢. 12. Thus (A + BF) is not boundedly invertible and this implies 0 E a(A + BF). m
One criterion which links controllability and stabilizahility can be obtained via linear quadratic optimal control.
Propos i t ion 7.6 Suppose (A, B) is exactly null controllable on a Hilbert space H, then ( A, B) is stabilizable.
17
In order to develop a more satisfactory theory of stabilizability and also to ask converse questions like "what conditions on the pair (A, B) are implied by the as- sumption of stabilizability" we will assume the following:
Def ini t ion 7.7 If there exists a rectifiable, simple curve r enclosing an open set a+(A) of a(A) in its interior and a(A) \a+(A) in its exterior we say that the operator A satisfies the spectrum decomposition assumption.
P r o p o s i t i o n 7.8 Suppose (A, B) is stabiIizable and B is of finite rank then there exists 6 > 0 such that
a,(A) = a(A) N {A E C; Re k > 6}
consists of only parts of the point spectrum of A. Furthermore, for evry )t E an(A) and every u > 0 dim k e r ( A I - A) v < oo.
We see therefore that under the conditions of the above proposition that A satisfies the spectrum decomposition assumption. If we set
Pz = ~ / (AI - A)- 'zdA
Corresponding to this decomposition we sct
Z_ := (I- P ) Z .
a [A+ 0 ] 0 ] [B+] 0 A_ ' 0 S_(t) , B = B_ "
T h e o r e m 7.9 I f A generates a strongly continuous semigroup on a Banach space Z and B is of finite rank, then the following assertions are equivalent
(i) (A, B) is stabilizable, (ii) A satisfies the spectrum decomposition assumption and there exists a r such
that Z+ is finite dimensional, S_( t ) is exponentially stable and ( A+, B+) is control- lable.
As a consequence of this theorem we recover the usual finite dimensional character- isation of stabilizability "A is stable on the uncontrollable subspace".
Def ini t ion 7.10 (A, C) is said to be detectable on a Banach space Z if there exists K E £(Y, Z) such that the strongly continuous semigroup SK(t) generated by A + KG is exponentially stable.
If Z is reflexive then (A, C) is detectable if and only if (A*, C*) is stabilizable.
18
8 Identif iabil ity
In this section we will be concerned only with parameter identifiability, although it is possible to combine the ideas introduced here with those of observability to develop the basic structure for parameter and state estimation. We suppose that the operators A, B, C, or equivalently S(t), B, C depend on a parameter a; we denote this dependence by S°(~), B(o), C(o) and A(4). For simplicity, we assume that o 6 £t C R", and define an ezperiment as a pair [zo, u], denoting the collection of experiments by E = [z0, u : u E /d]. Heze, we mean to imply that the initial state z0 is fixed and a variety of experiments are conducted by varying the controls in a sct/2. Of course, we could easily extend the arguments to include a variety of initial states. The relationship between the output and input is given by
io t
y(t,a) = C(o)S~(t)Zo + C(a)S~(t - s)B(4)u(s)ds. (i)
D e f i n i t i o n 8.1 (Indistinguishabili ty on [0, T]) The pair of parameters 4, (7 6 £t are said to bc indistinguishable if y(t, a) = y(t, 8) for a.e. t 6 [0, T] and for all experiments in E. If this is not the case, then the pair is said to be distinguishable.
Defini t ion 8.2 (Identifiablity on [0, T]) The parameter set gt is identifiable at 8 if (8, 0) is a distiguishable pair for all 8,06~,4#8.
Although this is a desirable requirement, the mathematical analysis associated with this concept is extremely difficult because the map V(t, ") : f~ ~ Y, t 6 [0,T], is a highly nonlinear one.
D e f n i t i o n 8.3 (Local identifiability on [0, T]) A parameter set ~t is said to be locally identifiable at 8 if there exists e > 0 such that 8 ,4 is distinguishable for all 4 with I[0 - a]]R, < ~.
Let us assume that Zo = 0, then clearly the set ~ is not identifiable at 8 if for a # 8
o' [c(°)sa(* - ~)B(o) - cco)so(, - ~)BCo)NC,)a~ = 0 (2)
for all u E Zd and almost all t 6 [0,T]. Moreover, if the class of control experiments is sufficiently large, (2) will hold if and only if
c(4)sa(t)s(o) = c ( e ) s q O s ( o ) (3)
for almost all t e [0, T]. In general, it is di•cult to check condition (3). However, if we set
19
where F(. , a) 6 £(U, L2(0, T; Y)) we may appeal to the imphcit function theorem to check that the set f~ is locally identifiable on [0, T] at 8. This requires that the operators depend in some smooth way on the parameter a and that the Frechet derivative of F(. , a) at 8 = a is invertible. The most usual way of tackling the problem of identifying a is to construct a cost functional
J(~) = IIv(t,,~) - v.(t)ll~.e (4)
where y~ is the measured outcome of a single experiment. Then assuming the initial state z0 is known the parameter a is chosen to minimmize J ( a ) . Since the problem is highly nonlinear, mathematical analysis is usually confined to imposing conditions which ensure that the minimum exists and the major effort is concentrated on ob- taining efficient computational algorithms. Clearly, if y(., a) depends continuously on a E f14 a compact metric space, then the minimum will be achieved. In order to compute the minimum it is usually necessary to calculate the Frechet differential of J ( a ) so extra connditions are assumed which guarantee this derivative exists. There are at least two problems with the above approach. First of all the initial state z0 is often not known and secondly the purpose of the identification is to obtain a good model for a variety of inputs and not just those used in the identification experiment. Below we suggest methods by which these difficulties may be overcome. Suppose the initial state is unknown and for simplicity the input function is zero, then
Y(~, a) = CCa)Sa(OZo = (CoZo)(0 •
Assuming that for all a 6 f~ the system is large V-space observable, we choose
Then (4) becomes
Then
~0 't v ( 0 = w ( t - , ) = ( , ) d , .
~0 T 7(~) _< I I C ( ~ ) s " ( t ) B ( ~ ) - ,.,,(OIl~(v,~.)dt
Y(~) = -<yo, ao(a:,a,,)-la~,v°) + Ily, ll= (6)
and a is chosen to minimize (6). Suppose now that Zo = 0 and we wish to choose a E F/ so that the model (1) is reasonable for a large class of inputs. One way of doing this is to choose a 6 ~ so that
7(~) = sup j(~); II~(')II~,(o,~,~)= i (7) is minimized. But note that we do not know in advance yc(~) for all inputs with Ilu(')llL,(o,r;v) = 1. However it may be that from one input experiment it is possible to obtain the input-output map in the form
20
Y(,~) = sup IlC(,~)(/, , ,z - A(a))-XB(a) - wCi,,,)ll,~c~,,Y) • (8) W
The RHS of this expression may be taken as a performance index for the choice of a. We see that a variety of performance idices can be associated with the identification problem. Just which or which combination is the most appropriate depends very much on the application.
9 R e a l i s a t i o n t h e o r y
There are two aspects to building a model of a system. One can assume by using physical laws that the structure of the system is given and the problem is then re- duced to one of finding the values of parameters in the partial differential equations. For complex systems it may not be possible to use physical laws to model all of the system and a black box approach must be used. Here controlled experiments are carried out which result in a knowledge of the input-output map of the form
yCt) = ~oCt) + ~Ct- s)~Cs)~s. (i)
The question then arrises as to whether it is possible to construct a state space model of the form
k(t) = Az(t) + Bu( t ) , z(O) = Zo e Z (2)
u(O = cz(o
and in what sense such a model is unique. This is the subject of realisation theory. Of course we must interpret (2) in the mild form
i' u(~) = cs(Ozo + c s ( t - ~)B~(s)d~. (3)
So that we require cs(Ozo = wo(O, cscoB = ~ ( 0 . (4)
In this section we will restrict our attention to the Hilbert space c u e and Zo = 0, so that given w(t) we need to derive methods of obtaining C, B and S(t) and exarnine the uniqueness of the realisation of these operators. Clearly a necessary condition on w(0 is
~,(~ + ~) = c ( O / - / ( s ) , tl s _> o (5)
where G(t) E £(Z , Y) and H(s) 6 £(U, Z). Given such a factorisation we construct a canonical factorisation.
Definition 9.1 (Canonical factorisation) The operators G, H are said to be a canonical factorisation of w if
AkerC(0 = {0}, nkerH'(s)= {0}. (6) t>0 °>_0
21
w(t + s) = CS(t)S(s)B, G(t) = CS(t), H(s) = S(s)B (7)
then a canonical factorisation will yield a realisation which is approximately con- trollable and initially observable on [0, oo). To construct a canonical realisation we set
M = ~'] ker G(t) the unobservable space (8) t>o
N = ( ' ] ker H ' ( s ) the uncontrollable space (9) a_>0
and set Z' = M ± Cl N j" which is a Hilbert space with the topology induced by Z. We then denote by Pz, the orthogonal projection of Z onto Z' and by ~r = P~, the injection of Z' into Z. Then we define G'(t) =- G(O~r , H'(s) -= Pz, H(s) and it is easy to show that
w(t + s) = G'(t)H'(s), s , t > O
is a canonical factorisation. Since we have a canonical realisation, the operator
Oz = e-"'G'*(p)G'(p)zdp (10)
is well defined for some w > 0 and is 1-1 and selfadjoint. If we now assume
Z o s ( t ) ~ = o -~ e-~"c"(e)a'(p + t)~eo (11)
defines a strongly continuous semigroup on Z', then there exists a realisation with
C = a ' (0) , B = H'(0). (12)
In order to study the relationships between two canonical realisations of the same w(.) it is necessary to strengthen the definition of canonical. In fact we will assume that a realisation C, B, S(t) on Z is exactly controllable to the subspace W and continuously observable to the larger space V, with the corresponding assumptions on C, B, S(t) on 2 which is another realisation of w(t). Then if
C S ( t ) B = ~(t)~ (13) ^ A
it follows that there exist operators M, N, M, N such that
C = CN B = M B S(t) = M S ( t ) N (14) ¢ = c]~ b = PZB $(t) = ~ s c t ) ~
N E £(W,I~.), M E £(V, V), IV E #(IV, W), 1(¢ E £(V, (/), M N is the injection W ~ V, M N is the injection W --* V and
S(t)]V -= MS(t ) $ ( t )N = ~IS(t) . (15)
22
The essential idea behind the above is illustrated as follows
o * e-~,( t+~)S,( t )C,CS(t)S(p)S(7)BB.S,(7)zdtd7
= foz*e- 'Kt+~)S'( t )C*dS(t)sCp)SCT)~B'S'(7)zdtd7
by (13). Thus
W~z = fo~"e-~"S(t)BB'S'(t)zdt.
Then M = 0 - 1 0 1 , N = YVIYV -1 and similar definitions for the other terms. Note that the operators M, N etc. are not bounded from Z to Z unless, of course, V = W = Z, V = 1~ = Z which is the case for exact controllability to Z and continuous initial observability. The converse result is also valid, namely if (14) hold then the realisations (C, S(t), B), (C, S(t), B) give the same input-output map. In most applications w(t) will not be known for t > 0 and we have to extend w(.) on say [0, T] either by analytic continuation or periodicity.
Riccati equations arising from boundary and point control problems
Irena Lasiecka Department of Applied Mathematics
University of Virginia Charlottesville, VA 22903
Abstract
We present a survey of results on differential and algebraic Riccati equations which include the cases
that arise from boundary/point control problems for partial differential equations (P.D.E.'s). As the
Riccati theory rests on dynamical properties of the underlying P.D.E.'s (such as regularity, exact
controllability, stabilization, etc.), particular emphasis will be paid to these. To this end, P.D.E.
methods (including pseudo-differential techniques and mierolocal analysis) will be emphasized. The
paper will highlight an interplay between semigroups or operator methods and P.D.E. techniques. This
survey is an update of the recent Springer-Vedag volume [L-T.13].
1. Introduction
1.1 Classical theory: B bounded.
Let Y, U, Z and W be Hilben spaces. Let A: Y ~ D (A) ---> Y be the generator of a s.c. sernigroup eat
on Y: t > 0 and B: U -.> Y be a given linear bounded (for now) operator. The classical linear quadratic
optimal control problem consists in finding u ° e I.~(0,T; U)) and y0 e La(0, T; Y) such that
(1.I) J (u 0, y°(u°)) = rain J (u, y(u)) u ~ I.~(O.T; U)
where the quadratic functional J(u, y) is defined by
T 2+lu(t)12 dt+lGy(T)12 w (1.2) J ( u , y ) = ~ [IRy(t)l z u ]
o
and y (u) is the solution due to u of the dynamical system
(1.3) y t = A y + B u ; y ( 0 ) = y o ~ Y .
Here R (resp. G) are bounded linear operators from Y --> Z (resp. Y ---> W). In (1.2) T > 0 may be finite
or infinite. I f T = ~o we shall then drop the last term with G in (1.2).
It is well known that there exists a unique optimal solution (u 0, y0) to the minimization problem (I.I),
under an additional finite cost/stabilizability condition if T = ~,. The problem of interest in control
theory is to find a pointwise feedback representation of the optimal control. This amounts to finding the
operator say, C (t); Y ---> U, independent on Yo, and time independent in the case T = ~, such that
(1.4) u°(t; Y0) = C (t) y°(t; Y0) a.e. in 0 < t ~ T .
24
The advantage of having a "closed loop" feedback control is well known and documented, with
motivation coming from engineering problems. Indeed, the fundamental reason for using a feedback
representation is to accomplish performance objectives in the presence of uncertainty. In many
situations, knowledge of the system is only part~; or else, the available model is based on many
simplifying assumptions which question its accuracy. An effective feedback reduces the effects of
uncertainties, because it tends to compensate for all errors, regardless of their origin. It is well known,
at least since the work of R. Kalman in the early sixties in the finite dimensional case, that the existence
of the feedback operator C (t) is closely related to the solvability of the following Riecati Equations, for
the two cases T < ,,o and T = ,~, respectively.
Case T < o=: the Differential Riccati Equation in the unknown P(t) E £, (Y), for all x, y e D (A)
d (DRE) ('~'t P(t) x, y)y = (A*P(t) x, y)y + (P(t) Ax, y)y + (Rx, Ry) z - (B* P(t) x, B*P(t)Y)u; P(T) = G*G.
Case T = oo: the Algebraic Riccati Equation in the unknown P e L C/), for all x, y E ~ (A)
(ARE) (A*P x, y)y + (PA x, y)y + (Rx, Ry)y = (B* Px, B* PY)u "
It is well known that for the classical linear quadratic control problem (i.e. when all the operators B, R,
and G are bounded), a unique solution P(t) e L (Y) to the DRE equation for T < oo [respect. a unique
solution P e £ (Y) to the ARE equation for T = ~,] exists in the class of nonnegative selfadjoint
operators [under the appropriate finite cost (stabilizability) and detectability conditions, if T = oo], in
which case the operator C (O in (1.4) has the specific form
(l.Sa) C( t )=-B*P(t ) , henceu°( t ;yo)=-B*P(Oy°( t ;yo) a.e. 0 < t < T
(l.5b) [C=-B*P, henceu°(t;yo)=-B*Py*(t;yo) a.e. 0:~t<oo].
Moreover, in the infinite horizon case (i.e. T = oo), the feedback -B*P in (1.5b) constructed via the
Riccati operator yields a closed loop dynamics yt ° (t; 3'o) = (A - BB*P) 3,0 (% Yo) which is exponentially
stable; i.e. there exist constants M, 03 > 0 such that
(1.6) le (A-BB'p)' Iz(y) <Me-t°t; t > 0 .
Thus, in the case T = 0% when the original dynamics e At is unstable, the Riccad feedback in (1.Sb) has
the additional attractive property of inducing uniform stability of the closed loop optimal dynamics. All
the results mentioned above are suitable extensions of finite dimensional theories (going back to R.
Kalman), combined with basic theory of linear semigroups (see [B.1]; [C-P]; [B-P]; [Lio.1]).
1.2 More recent theories
In more recent years considerable attention has been paid to control problems with data, i.e. operators R,
G and B, which do not satisfy the usual regularity assumptions. Here, the main motivation comes from
2 5
boundary/point control problem for partial differential equations with boundary/point observations.
Indeed, in some systems, for physical and technical reasons, only the boundary of the spatial domain, or
else some selected points in the interior, may be aecessible to external manipulations by actuators and
for sensing. From a theoretical standpoint, boundary/point control problems pose much more difficult
questions than those with "distributed" controls. The main mathematical feature that distinguishes
boundary/point control models from distributed models is the fact that in the first case the control
operator B: U ~ Y is unbounded. In fact, in this case, the control operator B is defined only in a
"larger" space i.e.:
(1.7) B e L(U;[~D(A*)]'), equivalently ( 'Ao-A) -1 B e £,(U;'Y); Xo~ p(A)
where [D (A*)]' is the dual (pivotal) to D (A*) with respect to Y-inner product; p (A) is the resolvent
set. A consequence of (1.7) is that the operator mapping the con~ol into the state space may be
unbounded. Moreover, and more importantly, the quadratic term in the Riccati Equation involving the
gain operator B*P may become unbounded (see below). This, of course, complicates substantially the
mathematical analysis of the problem where standard methods of proving existence of solutions to
Riccati Equations arc no longer applicable[ The main goal of this paper is to present a survey of recent
results, thus updating [L-T.13] on the solvability of Riccati Equations which arise in "nouregnlar"
control problems i.e. when the operator B is unbounded and subject only to (1.7), as in [L-T.4]. In an
analogous Way, we can also treat the nonregular problems where the observations R and G arc
unbounded, or where both control and observations are unbounded. For lack of space, since the results
with nonregular observations arc somewhat simpler than those with nonregnlar controls (and can
certainly be obtained by the techniques dealing with nonregular inputs), we shall concentrate only on
the latter. Thus, in the sequel, we shall consider explicitly only the case where the control operator B is
unbounded as in (1.7), while the observation operators R, G axe bounded. Then, in order to ¢xu'act "best
possible" results in boundary/point control problems, it is n~essary to distinguish different (not
necessarily mutually exclusive) classes of dynamics. One class includes free dynamics which generate
s.c. analytic semigroups ¢ At on Y: t > 0. Canonical examples include not only heat/diffusion
equations, but also damped wave/beam/plate equations with a sufficiently strong degree of damping.
This class will be analyzed in section 2 below (theory) and in section 3 (examples). It will be seen,
among the wealth of results available, that the gain operator B*P is always bounded on Y in this case, a
result which reflects and contains the property that in this analytic case P is a smoothing operator.
Another class to be treated in section 4.1 (theory) and in section 5 (examples) is motivated by, and
includes, wave, plate, and Schrodinger equations. It is characterized by the property (in addition to
(1.7)) that the input-solution operator is wall-defined in Y. This property is equivalent, by duality or
transposition, to assumption (H.2) in section 4.1, which is an "abstract trace property." It is precisely in
the latter dual form (H.2) that this property has been established, over the past ten years, for a large
variety of partial differential equations by purely p.d.e methods (differential and pseudodifferential).
For this second class (H.2) of section 4.1, very different techniques arc needed in the study of Riecati
equations, as compared to those used for the analytic class (H-l) in section 3. In particular, the operator
2t~
P is now typically an isomorphism on Y (unlike the analytic class (H.1)), and moreover B*P is now then
inherently unbounded in the most representative dynamics of this class (H.2), such as conservative
problems. We shall see in section 4.1 that there is a link between the desirable property of the feedback
B* P to yield the exponential decay (1.6) of the (optimal) feedback semigronp and the unboundedness of
B*P.
FinaLly, in subsection 4.2, with motivations coming again from wave and plate boundary control
problems (illustrated in section 6), we shall consider a very general class where the continuity of the
input-solution map is not fulfilled, (i.e. (H.2) is violated). In this latter class, P has generally unbounded
inverse and moreover, in the full generality of the present assumptions, new pathological features make
now their appearance, such as the optimal feedback dynamics, which is now claimed to be only a one-
time integrated semigroup, rather than a bonafid¢ s.c. scmigroup. F'mally, section 7 deals with more
general "nonstandard" Riccati equations where the quadratic term is nonpositive and unbounded. These
types of equations arise in the context of game theory and in particular of H" theory.
2. Analytic semigroups.
The main hypothesis assumed throughout this section is
(H-l) thes.c, semigroup e^t is analytic on Y, t > 0 a n d A ~ B E f. (U; H) for some 0 < y <l
where A = (7,.0I - A); 7,0 e p (A).
Examples of dynamics complying with this hypothesis are heat equation with Dirichlet or Neumann
boundary control, strongly damped plate equation with point or boundary controls, etc. (see sect. 3).
It should be noted that the main technical difficulties in the analytic case arise when the constant 3¢ in the
hypothesis (H-l) is greater than ~ . Indeed, if ~ < ~,,~, then the solution operator u --* y(t) is bounded:
L2(0,T; U) ~ C ([(3, "17; Y). This fact together with the analytic estimates of the semigroup yields a
priori bounds for the gain operator B*P. The situation is more demanding if, instead, T > ½, in which
case the solution operator is not bounded (even for 1-dimensional problems) in the above sense. In
order to obtain a meaningful definition and eventually a priori bounds of the gain operator B*P (hence
of the quadratic term in the Riccati Equation), much more refined arguments involving the theory of
singular integrals are used.
2.1. Differential Riccati Equation
The results and the regularity of the Riccati operator depend on the degree of smoothness of the
"terminal" operator G ~ f~ (Y; W). We first deal with the case when the "terminal cost" operator
satisfies the following smoothing assumption
(2.t) A"I'G*GE L 0 0 , 'g as in (I-I.1) .
27
Theorem 2.1. (see [L-T.I], [FAD
Assume (H-I) and (2.1). Then there exists a nonnegative selfadjoint operator P(t) = P*(t) > 0 such that
(2.2) P(t) ¢ L (Y; C ([0, T]; Y)) ;
in fact, even more for any 0 < 0 < I
"*0 CT (2.3) I A P(t) l £ (Y) < (T - t) ° ---'---~ "
(2.4) B*P(-)¢ L(Y;C[0, T];U) and the synthesis in (1.4) holds for aU t ¢ [0,T].
(2.5) For 0 < t < T; P(t) satisfies the (DRE) equation for all x,y ¢ D (,~); V 8 > 0.
(2.6) lira P(t)x=G*Gx; x¢ Y. t.--~T
(2.7) The solution P(t) is unique within the class of positive selfadjoim operators such that
(2.4) holds.
(2.8) (regularity of optimal solutions): u 0 ~ C ([0,T]; l.D; yO ¢ C ([0, T; U). Moreover u ° and y0 are
infinitely many times differentiable on (0,T).
Notice that in the case when the operator G is subject to the "smoothing" hypothesis (2.1), the optimal
control, the optimal trajectory, and the feedback operator B*P(t) are regular (in fact, continuous) for
0 < t < T. Instead, in the absence of the smoothing condition (2.1), the gain operators C(t) -- - B 'P( t ) as
well as the optimal control and trajectories develop singularities at the terminal point t -- T. This is not
surprising, since the benefit of analyticity of the original dynamics can not be relied on at the end point t
-- T. The next theorem provides the results pertinent to the general "nonsmoothing" observation G. We
shall let I11.: L2(0,T; U) ---r Y be the (unbounded) operator
T L T u = t eAfr- t) B u(t) dt
with densely defined domain ~9 (LT) = {u ~ L2(0,T; U); I.: r u ¢ Y}. Clearly, L T is closable.
Theorem 2.2. [L-T.1]:
(2.9) GLT: L2(0 ,T;U) ~ W isc losab le .
28
Then there exists a non negative selfadjoint operator P(t) = P* (t) such that
(2.10) P (.) • : (Y; C ([0,T); Y)
and the (DRE) equation holds as in (2.5). Moreover
cT (2.11) l(,~')°P(t)l m <--~- ~ , o<e<l,
CT (2.12) IB*P(t)IL(y;u) < (T-t)'f ' 0<t<T,
(2.13)
(2.14)
where
0<t<T,
lira (P(t) x, y)y = (G*G x, y), V x, y • Y, z...cT
U ° • C.f ([0, "r]; U); y0 • C2r-i+, ([0, T]; Y),
CT ([0,T]; ~0 "" [f(t) e C ([0,T); Y); t f l C,(10.T]; ",9 = teS~ (T-t)r I f(0 t Y } .
Moreover, the optimal control and trajectory arc differcndablc functions of t for t e (0, T). []
Remark 2.1. It is shown in [L-T.I] by examining further an example in IF.3] that without assurapdon
(2.9). the optimal control does not exist.
Remark 2.2. A sufficient condition for (2.9) to hold is
(2.15) (,~*)It~ O °
bc densely defined as an operator W ~ Y for sorac ~ > 2 T- I. Notice that if Y <: lh then condition (2.15)
is automatically satisfied. For other related results on this topic see [L-T. I] and [F.2].
2.2. Algebraic Riccati Equations
In this subsection wc shall then discuss the solvability of the Algebraic Riccati Equation (ARE). A
necessary and sufficient condition for the existence of the optimal control corresponding to the
rainiraizadon praoblcra (I. I) with T -- oo is the following
(F.C.C.) Finite Cost Condition: For oven/ Y0 • Y there exists u • L2(0,0o ; y) such that the
corresponding value of the functional in (1.2) satisfies J (u,y(u)) < oo.
29
I. Existence
Assume hypotheses (H-l) and (F.C.C.). Then there exists a selfadjoint, nonnegative definite solution
P = P* • L (Y) of the (ARE) equation such that
(2.16) (~k*) t - t p • r .(Y), V e > 0 , inpardcular
(2.17) B*Pe L ( Y ; U )
(2.18) For each fixed Y0 • Y, we have y0 (t; Y0) = e(A-BB'P)t Y0, where the s.c. semigroup e (A-sB" p)t
is analytic on Y.
II. Uniqueness. In addition to the assumption of part I, we assume the following "detectability
condition."
(DC) There exists K • L (Z; Y) such that the S, C. semigroup e (^+zR)t generated by A + KR is
exponentially stable.
(2.19) Then, the solution P to (ARE) is unique within the class of non-negative selfadjoint operators in
L (Y) which satisfy (2.17).
(2.20) The s.c. analytic semigroup e Art generated by Ap = A - BB*P is exponentially stable on Y. •
Remark 2.3. Notice that in the analytic case the Riccati operator P has a "smoothing" effect (inherited
from the analyticity of eat). Indeed, (2.16) asserts that P is bounded from Y into ~)(/~*l-r). If the
resolvent of A is compact, this implies compactness of P in L (Y).
3. Examples illustrating the results ofsection 2
3.1. Heat equation with Dirichlet boundary control [L-T.13, p. 51].
Let ~ c R n be an open bounded domain with sufficiently smooth boundary F. In ~ , we consider the
Dirichlet mixed problem for the heat equation in the unknown y(t,x):
Yt = Ay + c2y in (0,T] × f~ -= Q ;
(3.1) Jy(0, • ) = yo in f~ ; I
[ylz =u in (0,T] x r-- Z;
with boundary control u e L2(~) and Yo • L2(~). The cost functional which we wish to minimize is
then
T
i fT < oo ,o r in the case T = oo
30
(3.3) J(u'Y)= i [Y(t)12ta + lu(t)l~]dt
where we denote I y I n = l y I In(n) and I u I r -= I u I Lz(r)" To put problem (3.1) into the abstract setting
of section 2, we introduce the operator
(3.4) A h = A h + c 2 h ; a ) ( A ) f H 2 ( ~ ) m H 1 ( f 2 ) '
next extend isomorphicalIy A as L2(f2) -+ [~D (A*)]' and select the spaces
(3.5) Z = Y = W = L2(f2); U = I..2(1"3,
and finally define the operators Bu - - ADu; R = I; G = I where D (Diriehlet map) is defined by
h = D g i f f ( A + e 2 ) h = 0 i n ~ a n d h l r f g .
It can be shown (see [L-T.13 p. 52]) that hypothesis (H-l ) is satisfied with 7 = ¾ + e where e > 0.
Moreover, since the condition (2.15) holds trivially with G = I, hypothesis (2.9) is satisfied as well.
Hence, the conclusions of Theorem 2.2 apply. In the ease of infinite horizon problem (T = **), the
arguments of IT.l], [L-T.13 p. 52] provide a construction of stabilizing feedbacks, which in turn,
implies the Finite Cost Condition (F.C.CT. The detectability condition holds automatically true since R
-- I. Thus, all the hypotheses of Theorem 2.3 are satisfied and the statements (2.16)-(2.207 are valid for
problem (3.17 - (3.3).
3.2. Structurally damped plate equation with point control [L-T.13, p. 57].
Consider the following model of a plate equation in the deflection w (t, x), where 13 > 0 is any constant
wtt + A2w - pAwt = 8(x - x 0) u(t); in (0,T] × f2 - Q
(3.6) ~w(O, • ) = wo; wt(O, • ) = wl in f2 /
[wly =,Xwly =o in Z
where x 0 is an (interior) point of ft , dim ~ = n. The cost functional associated with (3.6) is
(consistently with IT.2])
T (3.7) J ( u , w ) = t [IAw(t) l ~ + l w t ( t ) l ~ + l u ( t ) l ~ . ] d t + l A w m l ~ -
To put problcm (3.6) (3.77 into the abstract setting, we introduce the strictly positive definite operator
. ~ h = A 2 h ; ~ ( . ~ ) = { h E H 4 ( ~ ) ; h l r = 0 ; A h l r = 0 } ;
and select the spaces and operators
Y = D (~'~) x L2 (~ ) = [H2(f2) n n~(f~)] × L2(t2) ; W = Z = Y ; U = R t ;
31
I -°, 'I ,, I -° I I°I • o I °l A = -pA ½ ' 8(x x0)u ' ' "
n It can be verified see (sea [L-T.13, section 6.3] that the hypothesis (H-l) is sadsfiexi with "/= ~-+8,
which then requires n ~; 3. As in the previous case, one verifies that the condition (2.15), hence (2.9),
holds true. Thus Theorem 2.2 applies. Similar analysis applies to the infinite horizon case (see [L-T]
scct. 6.3) where now Theorem 2.3 applies.
3.3. Structurally damped plate equation with boundary control. [L-T.13, p. 64]
Consider
(3.8) w(0 , ' )=wo;wt (0 , ' )=wl in £2;
wlz=0; AwIZ=ue L2(0, T; I.a(ID).
T with (3.8)we associa,e the functional ,(u.y>= I t,~w~,), ~ + ,u~0 ,~ ld ,+ ,w~ ,~ To put
problem 0.8) into the abstract setting, we introduce Y, W, Z, A as in example 3.2: U = I.,2(1") and the
operator B I°l BU ~ - ~ Du
It can b~ shown (sea [L-T.13, p. 65]) that hypothesis (H-I) holds with 1:ffi ¾ + ~ Since condition (2.1)
is satisfied, the conclusion of Theorem 2.1 applies as well.
4. Hyperbolic and "hyperbolic-like" dynamics
Here we shall consider "unbounded control" dynamics governed by general C0-semigroups, with
particular emphasis on hyperbolic and "hyperbolic-like" dynamics (waves and plates). Because of space
limitations, we shall focus only on infinite horizon problems and corresponding Algebraic Riccati
Equations. The results on finite horizon problem (Differential Riccati Equations) can be found in [D-L-
T], [L-T.2], [L-T.3], [L-T.13] and references therein. A main reason why Algebraic Riccati Equations
are particularly interesting is that the existence of solutions to these equations is closely related to the
problem of stabilizability of the original, usually unstable dynamics. On the other hand, questions of
stabilizability/controllability for hyperbolic (hyperbolic-like) equations present many challenging
mathematical problems and have atwacted in recent years a lot of attention. We shall make an effort to
enlighten this interplay between Riccafi theory and smbilizability pmpe~es of the underlying dynamics.
32
4.1 Unbounded control operators subject to (H-2) hypothesis In this subsection we present results on
solvability of Algebraic Riccati Equations in the case when (1.7) holds and the unbounded control B
operator satisfies the following hypothesis
T (H-2) [ tB* e A't y 12 d t<CT lyl 2" u y, ye ~D(A*); extended to all y ~ Y
where B" e r.(~)(A*);U) and (B* v, U)u =(v, BU)y; ve ~D(A*); ue U. As we shall see later,
hypothesis (H-2) in the case of hyperbolic p.d.e.'s, expresses certain "wace regularity" properties of the
homogeneous problems (see section 5).
Theorem 4.1. [L-T.2] [F-L-T]
I. Existence Assume that, in addition to the (F.C.C) and (1.7), the regularity hypothesis (H-2) holds. Then there
exists a nonnegative solution P = P* e L ('Y) of the (ARE) such that
(4.1) B*Pe LCD(A); U);
(4.2) uO(t;yo)=-B*Py°(t;yo)e I~(0, oo;U); y0(t;y0)=e(A-BB'P)ty0;
(4.3) j(u 0, y0) = (pyO, y0)y, YO e Y.
when the operator Ap mA-BB*P is closed, densely defined with ~9(Ap)cY and it generates a
semigroup on Y.
II. Uniqueness
In addition to the hypotheses of part I, we assume that the detectability condition (D.C.) holds true with
an operator K satisfying
(D-CA) IK*xl Z < C[IB*xl U + {Xiy].
Then the solution to (ARE) is unique within the class of selfadjoint, positive operator satisfying (4.1).
Finally, e Apt is exponentially stable on Y Ill
As mentioned in the introduction, it is important to notice that, in contrast with the analytic case
described in section 2, the gain operator B*P is now generally unbounded. Indeed, this property follows
from the next results.
Theorem 4.2 [F-L-T] In addition to the hypotheses of part I of Theorem 4.1, we assume the following exact controllability
condition:
33
(E.C.) The equation Yt = A* y + R*v is exacdy controllable from the origin over some [0, T], T < **,
within the class of L2(0, T; Z) controls v.
Then the solution operator P to the (ARE) guaranteed by Theorem 4.1 is an isomorphism on Y. •
Corollary 4.1
Under the assumptions of Theorem 4.2 the operator B 'P: Y --~ U is bounded iff B: U ~ Y is bounded.
Corollary 4.2 [L-T.13 p. 43]
Assume the hypotheses of Theorem 4.1. In addition assume that the free dynamics e At is a s.c group
uniformly bounded for negative times. Then the conclusion of Corollary 4.1 applies.
As Corollary 4.2 shows, the property that the gain operator B°P is unbounded is intrinsic to time
reversible dynamics under the hypothesis (I-I-2). We shall see later that one "loses" this property for
classes of control problems (i.e. such that they do not satisfy (H-2)) control operators which are treated
in the next section.
A different treatment of control problems with unbounded control operators (subject to (H-2)) is via the
so called Dual Riccati Equations, as proposed by F. Flandoli. This is to say that instead of looking at
the solution to (ARE) one considers the "Dual Riccati Operator" say Q ~ .6 ~ which solves an
appropriate "Dual Riceati Equation." If the original dynamics is represented by a group e At, then the
Dual Riccati Equation takes the form
(ARE-l) (AQ x,y)y + (QA*x,y)y - (B°x, B*y)u + (R*RQx, Qy)y = 0 x,y ~ ~9 (A*) c Y
The Dual Riccati Equation is simpler than (ARE) since the quadratic term in (ARE-l) is now bounded.
The relation between ARE and ARE-1 in the case of group dynamics is given by the following
Theorem.
Theorem 4.3 [F-L-T]
In addition to hypotheses of Theorem 4.1 we assume that A generates a s.c. group and that both pairs
{A, B} and {A °, R*} are exactly controllable. Then there exists unique solution Q E L (Y) to (ARE-l)
and
p'q = Q . •
Some extensions of the results of Theorem 4.3 to more general dynamics are provided in [B.2] and [B- D].
Remark 4.1. Under the additional "smoothness" assumption imposed on the observation R
T t IR, ReAtBulydt<CT [ul U
34
one obtains the additional regularity of the Riccati operator: B* P e L (Y, U) (see [D-L-T], [L-T.13 p.
44]). However, the above assumption imposed on the observation R, in the most interesting cases of
conservative hyperbolic dynamics, is in conflict with the detectability condition and, ultimately, with the
stability of the feedback semigroup (see the results of Theorem 4.2 and Corollary 4.2). For this reason
the results leading to the bounded gain operator B*P (under (H-2)-hypothesis) are of very limited
interest and will not be further pursued.
4.2. Fully unbounded control operators
In this section we shall dispense with regularity hypothesis (H-2). Our motivation comes from several
hyperbolic and plate-like problems (see sec. S) where this hypothesis is violated. Some results on the
wellposedness of Algebraic Riccad equations for the case of fully unbounded control operators are
given by the following Theorem
Theorem 4.4 see [L-T.4]
In add/t/on to (1.7) and to the (F.C.C), assume that
(4.4) R*R is strictly positive
(4.5) The operator A* R*R A -1 ~ I"- CO-
Then there exists a positive selfadjoim solution P ~ £ (Y) of the (ARE) such that
(4.6) B*P e f~CD(A); U)
(4.7) u°Ct; YO) = -]3* Py°(t; YO) e I..2(0, -- ; U)
(4.8) J (u °, yO) = (PYo, YO)y.
Moreover, P is the unique solution of (ARE) within the class of selfadjoint operators P subject to
regularity requirements (4.6) •
In the present case the Riccati theory displays new features, or pathologies, over the more regular case
when hypothesis (H-2) is in place. They include the following ones (i) While in the case of assumption (H-2), the opt/real trajectory y°(t) defines a s.c. semigroup no such
claim is now made. Indeed, in general y°(t) may be only a l-firne integrated semigroup. No claim is made that the domain ~D (Ap) of the generator Ap of 1-t/me integrated semigroup is dense in Y.
(ii) While in the case of hypothesis (H-2), the s.c. semigroup e Apt under detectability conditions (weaker then (4.4)) is exponentially stable on Y, no such claim is now made in full generality. Indeed, the original closed loop vrajectory y°(t) is exponentially stable but for initial data Y0 contained in a subspace which is strictly contained in Y see [I.,-T.4].
Off) In the case of hypothesis (H-2), the Riccati operator is an isomorphism on Y when exact controllability (E.C.) holds (which in our case (see (4.4)) holds for groups). Now, however P is bounded and injective, but with possible unbounded inverse p--1 (sea [L-T.4] for examples where not only is p--t unbounded but P changes drastically the feedback dynamics from hyperbolic of e At
to parabolic, o f eAt=t).
3 5
As mentioned above, in general, the feedback semigroup need not generate a s.c. semigroup on Y.
Ho~vever, for several specific dynamics one can show that this is indeed the case i.e. the feedback
trajectory y0 is strongly continuous in Y. If this happens, uniform decay as t --~ ** of the feedback
semigroup can be proved. This result is stated in the following Theorem
Theorem 4.5 [L-T.4]
In addition to hypotheses of Theorem 4.4 we assume that for some T > 0
(4.9) y°(t, 0; Y0) e C ([0,T]; Y).
Then the operator Ap = A - BB* P generates a strongly continuous semigroup on Y and, moreover,
there exist constants C, co > 0 such that
l e ( A - B B ' P ) t l r ( y ) ~ C e " ~ t ; t > 0 . • (4.10)
The cnax of the matter is, of course, to verify hypothesis (4.9). In concrete example this is rather
technical p.d.e, question. In fact, as we shall see in the next section, for several hyperbolic-like
problems, a positive answer is provided by applying P.D.E. arguments based on rnicrolocal analysis and
pseudodifferential techniques.
5. Examples il lustrating the results of sub-section 4.1 (case of H-2 hypothesis)
Here we shall provide several examples illustrating Theorems 4.1 - 4.3 which assume the validity of
hypothesis (H-2). In fact formulation of hypothesis (H-2) (see [L-T.13]) was precisely motivated by the
discovery that it expresses sharp trace regularity of underlying p.d.e, problems which hold true. The
abstract formulation of these trace regularity properties is precisely (H-2).
5.1. Wave equation with boundary control [L-T.13, p. 71].
We consider the following problem
Wtt = AW ; in Q ;
(5.1) ~ w ( 0 , ' ) = w 0 ; w t ( 0 ; ' ) = w z ; i n f l ; / Lwlz=u onE.
where we take the boundary control u e I.q(~). With (5.1) we associate the cost functional (which is
motivated by regularity properties [L-T.9], [L-T. 10], [Lie.l], [L-L-T]):
(5.2) J ( u , w ) = f [Iw(t)l 2 + I u(t) I ta2(r)] dt. b la(n)
To put problem (5.1), (5.2) into abstract setting we introduce the positive selfadjoim operator gh = -Ah;
~9 (.~) = HI(f2) ~ H2(~) and define the operators
36
A= I:D,I IA 01 and the space Z = Y = I.,2(f2) x H -I (f2) ; U = (L2(F)). It is well known that A generates group on Y and
A-1Bu = [Du I is bounded: L2(I") --d, Y.
The crux of the matter is hypothesis (H-2). Indee..d, it can be shown (see [L-T.2]) that (I-{-2) is
equivalent to the following inequality
(5.3) ~rf dE < CT [ l~bO [ 2 (t"2) + I ~b I 12L2(t.I)]
where ~b satisfies the homogene