This paper describes recent results for controlling and stabilizing control systems of the form ú(t) = Au(t) + p(t) B(u(t)) where A is the infinitesimal generator C∞ semigroup

on a Banach space X, B' map from X into X, and p(t) is a real valued control. Application to a vibrating beam problem is given for illusstration of the theory

Ball, J. M.

Marsden, J. E.

Slemrod, M.

English

Caltech Authors - Main

J. M. Ball 
"U-WA/ U~\i-ll 
THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 
345 E 47 St., Naw York, N.Y_ 10017 
The Society shall not ~ ,espon$lb!cdor slatements Of OIII_a 4dvanCect .n paQe'S or 
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ilS pulllicaUons. Olscusslon is printed only it tllfl lJ/Jpq,IS tJUbllsh«l '" an ASME 
Journal Of ProclHICfmgS.RefeasiKS 10' g_aJ publicallOn upon presentatlon.FulI. 
credll Sltouldbe given 10 ASME.the TecIlnlcal DiviSion. aCICIllle Bulllor(S): 
Oe1Iartment 01 Matltematics. 
Heriot-wan University, 
Edinburgh, Scotland 
J. E. Marsden 
Ilepanment 01 Matltematics. 
University 01 California. 
Controllability and Stabilizability of 
Distributed Bilinear Systems: Recent 
Results and Open Problems 
Berkeley. Calif. 
M. Slemrod 
IlepaItment 01 Matltematical Sciences. 
This paper describes recent results for controlling and stabIlizing control systems of the 
fonn Il(t) = Au(t) + p(t) B(u(I)) where A is Iheinjinilesimalgenermoroja CO semigroup 
on a Banach space X. B' map from X into X, and p(l) is a real valued control. Applicalion 
10 a vibrating beam problem is given for iIIUstralion of the theory. 
Rensselaer' PoIytecItnic Institute, 
Troy.N. Y. 
1. INTRODllCTION 
The purpose of this paper is to provide a des-
cription of some problems and recent results in the 
theory of distributed bilinear control systems. The 
paper is in no way comprehensive and should be viewed 
only as an introduction to selected problems in con-
trol theory. Details may be found in Ball, Marsden, 
and Slemrod (1] and Ball and Slemrod [2]. 
As a motivation for our later analysis consider 
the following problem arising in the control of the 
transverse displacement w(x,t) of a vibrating beam of 
length ~ where we use axial force pet) as a control. 
A simple model of this situation is provided by the 
equation 
Wtt + wxxxx + pet) Wxx = o. 
Assume the beam has clamped ends so 
w ; w ~ 0 at x = O,~. 
x 
(l) 
If P is a constant two possibilities exist: either 
p is subcritica1 and the beam has undamped oscilla-
tory motion or p is supercritical and the equilibrium 
state {w,w } ; [O,O} is unstable. Hence if it is 
t 
our goal to stabilize the motion so the lW,Wt}~{O'O} 
as t~ (in some sense). a choice of constant p is 
doomed to failure. Similarly if we desire to control 
the motion to some full neighborhood of a given state, 
a constant p will allow us to reach only a highly 
Conlributed bv the Dvnamic SVSlems & Conlrol Division of The American 
Society of Mechinical Engineers Cor presentation at the Winter Annual ~te.:ting. 
Sovembcr 16·21. 1980. Chicago. illinOIS. Manuscript receIved 3t ASME 
Headquaners July 12. 1980. 
Copies will be available until August I. 1981. 
restricted set of states. For these reasons we are 
led to consider the following two mathematical prob-
lems where pet) is time varying. 
Controllability: 
Can we find a pet) such that on a finite time 
interval [O,T] we can bring w(x,t), wt(x,t) to a 
prescribed function values, i.e. 
w(X,T) = "'l(x), Wt(x,T) = w2 (x). 0 < x < ~? 
Stabilizability: 
Can we find pet) such that on the 
interval (O,~) pet) can be written as 
functional of w,wt in such a way that 
t~ (in some sense)? 
infinite time 
a (feedback) 
{w,w }+{O,O} as 
t 
We note that since {w,w } '" {O,O} is an €quili=--
t 
rium point of (1). (2) any reasonable uniqueness 
theorem will preclude us from being able to control 
to the state tw1 ,w2} = {O,O} in finite time. 
What methods should be used to attack the control-
lability and stabi1izability problems? A state space 
theory based on evolution equations in abstract inf~ 
ite dimensional vector spaces is one useful way to 
proceed. Such an approach provides both the structure 
to prove relevant theorems, yet the generality to 
cover many interesting examples. 
2. THE ABSTRACT EVOLl.iTION EQUATION 
Consider the abstract evolution equation 
dUCt) 
at Au(e) + pet) 6(u(t», 
u(O) u • 
o 
(3) 
(4) 
o Here A generates a C semigroup on a Banach space 
S ' 1 ' X, : X~X lS a C map and the control p lS real valued 
2 
f',mction of t. To see that (1), (2) is actually in 
the form (3) we set 
U' (:,)- Au .(~;: :)-. 
aod Su • (~;: : ) •• nd 1., 
? 2 
X be the Hilbert space H~IO,l)xL (0,1) 
with inner product 
.. * 11 * * * < (w1 ,w2 ), (w1 '''''2) >x : (wl WI ....... 2w2) dx. o XX xx 
Here H2(O,~) denotes the Sobolev space made up of 
o 
functions in L2 (O,2.) whose first and second general-
ized derivatives lie in L2 (O,2.) and satisfv (2). (A 
good reference for Sobolev spaces is the ~ok of 
Adams (3]: good reference on semigroups are the notes 
of Pazy (4] and the book of Balakrishnan (5).) 
Of course many other physical problems can be 
put in the form (3), (4). A list of such problems 
may be :o~~d in the papers of Ball and Slemrod [21, 
(61. 
3. Controllability: 
Having argued that (3), (4) is natural way to 
view distributed bilinear systems, we now proceed to 
the question of controllability of (4) with an eye 
to providing a taste for the methods and problems 
involved in developing an adequate theory. 
For (3), (4) the variation of constants formula 
gives 
t 
At f A(t-s) e u 0 + e p ( s) B ( u Is; p , uo» ds. 
o (5) 
At Here e denotes the s~~igroup generated by A and 
u(t;p,u
o
) means that the state vector u is evaluated 
at time t with control p and initial datum u . 
o 
One iteration of 
At ft 
u(t;p,u ) =e u + 
o 0 
o 
(5) yields the equation 
A(t-s) ( )s( As e p s e U
o 
+ 
A(S-T) 
e p(T)B(u(T;P,Uo»dTJds. 
(6) 
Continuing in this faShion we see that at least for-
mally we can write 
At ft A(t-s) As 
u(t:p,u ) = e u + e p(sIB(e u) ds 
000
o (7) 
+ higher order terms in p. 
(Actually this idea is due to Volterra (7]). 
for small p we could take 
t 
- At 1 A(t-s) As u(t;p,u ) - e u + e p(s)S(e u) 
o 0 0 
o 
Hence 
ds. (8) 
Thus a linear approximation to our control problem of 
finding a p such that 
(9) 
for h f X prescribed is the problem of finding a p 
such that 
AT iT e u
o 
+ 
o 
A(T-s) ( )B( As ) ds e p s e u
o 
If we set 
Lp i I 
(10) can be rewritten as 
AT Lp : h - e u 
o 
h. (10) 
(11) 
Here L is a linear operator acting on the space '~ere 
the p'S lie. If we were able to solve the linear 
approximating problem (11) for all h (X, then an 
application of the generalized inverse function 
theorem [a) would tell us that the original equation 
(9) could be solved for p as well if h - eATu
o 
is 
sufficiently small. 
All this sounds rather easy. However it is not 
usually simple to check the surjectivity of L except 
in rather special cases, e.g. A,B bounded linear 
operators a~~ X finite dimensional. The reason for 
this difficulty is that typically L will map the 
space of p's into but not onto X. In such situation~ 
X must be modified to a smaller space appropriate to 
the problem. We investigate this subject in detail 
in [1]. 
As mentioned above, the surjectivity of L is 
relatively easy to check if L has closed range R(L) 
and the Fredholm alternative holds. For instance, 
if X is finite dimensional X '" R (L) €I N (L*) (and not 
just X : RIL) <3 N(L*)). Hence if X is finite dimen-
sional, N (L") '" {O} implies R (L) '" X and surjecti'.'ity 
holds. But if the space of controls/is, say 
p (L2 (O,T], then it is easy to see L*q = 0 if and 
-As As > only if <q,e Be u = 0 for 0 < s < T. Expansion 
o 
of 
-AsS As s2 
e e U
o 
'" Suo - s [A,B] u
o 
+ ;! (A, [A,6]]u
o 
..... 
where [A,B] ~ AS - SA yieldS the well known control-
lability result {9]: 
Assume X = itn and that dim span 
{Su , (A, 5) u , [A, [A,Bll u , ... ! = n. There is an 
000
II AT '1 ( > 0 such that if e uo - hi < (then there is a 
2 PC; L [O,T) such that U(T;p,u
o
) - h. 
4. Stabilizability: 
The theory of stabilizability of (3), (4) is 
much more developed. It has been covered to a large 
extent in the papers of Ball and Slemrod [2), [6). 
We shall not go into very much detail here, only 
sketching the main ideas and open problems. 
Assume that A is dissipative, i.e. <Aw,'~ >X ~ 0 
for all ~ (D (A). For example in (1), (2) <A"" 1lJ > x=O). 
Let us now (at least formally) compute: 
1 d 2 
---llu(t)11 = <u,Au> + p(t)<u,B(u»x' 2 dt X X 
Since A is assumed to be dissipative we have 
2 d~j ju(t)il x ~ 2p(t) < u,8(u) >X· 
Hence if we choose 
then 
p (t) 
- <u,8(u» X 
2 2 ~I luCt) I Ix ~ -2 <u(t) ,8(u(t»>x 
2 
(12) 
and I luCt) I Ix is non-increasing. If we knew that 
u(t), 0 < t < =, belonged to a compact subset of X, 
then Hale's generalization [101 of LaSalle's Invari-
ance Principle [111 would tell us that ~, uCt) 
approached ~~e set of y such that 
<yCt),B(y(t»>x = 0 for all t> 0 (13) 
where yCt) is a solution of (3) with pet) = 
-<y(t),B(y(t»>. But this means that 
At At 
<e y ,B(e y l> = 0 for all t > 0 
o 0 X (14l 
where yeO) = Yo' Hence we may conclude that if u(t) 
belonged to a compact subset of X and the only solu-
:ion to (14) is Yo = 0, then u(t)~ as ~ in ~~e 
strong (norm) topology of X. Unfortunately it is 
not clear in general that u(t) will belong to a com-
pact subset of X. To overcome this difficulty Ball 
and Slemrod [2), [6] resorted to use of the weak 
topology on X. They were then able to sho\" under 
natural assumptions on A and B, that if the only 
solution to (14l is Yo '" 0 then u (t)+O as t" co 
weakly. Strong convergence is still an open question 
For example this weak topology approach leads 
for (1), C2l to the following result 
(Theorem 4.3 of [21): 
Set oCt) "'f~ w w ax. Then for all initial 
- 0 t xx 
data {w .w j (H2 (0,1) x L'(O,).), (ll. (2l possesses o 0 0 __ ~ __ --..-___ _ 
a unique weak solution ;.w'·"'tJ (C «(O,eo); H~ (0.1) x 
i:"2 (0,2.) and '~w'Wt} + ~~o,o1 weakly in 
H2(O.J.) x L2(0,;:'1 as t+"". 
Q 
Unfortunately this is not as nice as it looks. 
Even if we ignore the fact that we have only proven 
weak decay there is a more important difficulty, 
namely pet) as given (12) in general or 
~ 
r w w dx in our example depends on complete know-)0 t xx 
ledge of uCt), i.e. full state feedback. In practice 
we can only sense a finite dimensional projection of 
u. Thus it would be desirable to have a theory in 
which pet) could be chosen with less than a full 
knowledge of u. Based on ideas of Balas (12), [13] 
it seems unlikely that this can be done if the uncon-
trolled system has no damping since "spill-over" of 
energy from controlled to uncontrolled modes will 
occur. Nevertheless this problem and its understand-
ing may be crucial in real world applications of the 
abstract theory. 
5. Acknowledgement 
The work of J. M. Ball and J. E. Marsden was 
supported by grants from the U. S. Army and National 
Science Foundation at the University of California, 
Berkeley. 101. Slemrod was supported by a grant from 
the National science Foundation. 
REFERENCES 
1. Ball, J. M. Marsden, J. E., and Slemrod, M. 
in preparation. 
2. Ball, J. M. and Slemrod, M. "Nonharmonic 
Fourier Series and the Stabilization of Distributed 
Semi-linear Control Systems", Comm. on Pure and 
Apolied Math., Vol. 32, 1979, pp. 555-587. 
3. Adams, R. A., Sobolev Spaces. Academic Press, 
New York, 1975. 
4. Pazy, A., Semigroups of Linear Operators and 
Applications to Partial Differential Equations. Dept. 
of Mathematics, Univ. of Maryland, College Park, Md., 
Lecture Notes No. 10, 1974. 
5. Balakrishnan, A. V., Applied Functional Anal-
ysis, Aoolications of ~lathematics. Vol. 3, Springer, 
New York, 1976. 
6. Ball, J. M. and Slemrod. :'1., "Feedback Stabil-
ization of Distributed Semilinear Control Systems", 
Applied Mathematics and Optimization, Vol. 5. 1979, 
pp. 169-179. 
7. Volterra. V., Theory of Functionals and of 
Integral and !nteqrodifferential Eauations (Reprinted 
from 1931 :-lacMillan edition), Oover, New York. 1958. 
8. Luenberger, D. G •• Optimization by Vector 
Space Hethods, \'liley, New York, 1969, p. 240. 
9. Hermes, H., "On Local and Global €ontrol1a-
bility", S.I.A.t-I. J. Control, Vol. 12, 1974. pp. 252-
261. 
10. Hale, J. K., "Dynamical SysteMS and Stability', 
J. Math. Analysis and Aop1ications, Vol. 26, 1969, pp. 
39-59. 
11. LaSalle, J. P., "Stability Theor./ for Ordina.r:y 
Differential Equations", J. Differential Ecuations, 
Vol. 4. 1968, pp. 57-65. 
3 
4 
12. Balas, H., "Modal Control of Certain 
Flexible Dynamic Systems", S • I • A. lo!. J • Control and 
Optimization, Vol. 16, 1978, pp. 450-462. 
13. Balas, M., "Feedback Control of Flexible 
Systems", I::::E Trans. Automatic Control, Vol. AC-23, 
1978, pp. 673-679. 


Controllability and stabilizability of distributed bilinear systems: Recent results and open problems

Applied Functional Anal ysis,

Applied Functional Analysis,

Dynamical SysteMS and Stability',

Feedback Control of Flexible Systems&quot;,

Feedback Stabil  ization of Distributed Semilinear Control Systems&quot;,

Feedback Stabilization of Distributed Semilinear Control Systems&quot;,

G •• Optimization by Vector Space Hethods,

Modal Control of Certain Flexible Dynamic Systems&quot;,

Nonharmonic Fourier Series and the Stabilization of Distributed Semi-linear Control Systems&quot;,

On Local and Global €ontrol1a bility&quot;,

On Local and Global €ontrol1ability&quot;,

Semigroups of Linear Operators and Applications to Partial Differential Equations.

Sobolev Spaces.

Stability Theor./ for Ordina.r:y Differential Equations&quot;,

Theory of Functionals and of Integral and !nteqrodifferential Eauations (Reprinted from 1931 :-lacMillan edition), Oover,

http://authors.library.caltech.edu/20586/1/BaMaSl1980.pdf

Controllability and stabilizability of distributed bilinear systems: Recent results and open problems

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