Stabilization of Flexible Structures: Third Working Conference Montpellier, France, January 1989

332 Pages·1990·4.331 MB·English

by  Walter Littman (auth.), J. P. Zoléesio (eds.)

Lecture Notes ni
Control and
noitamrofnI Sciences
detidE yb amohT.M dna renyW.A
147
.J .P Zolesio (Editor)
Stabilization
of Flexible Structures
drihT gnikroW ecnerefnoC
,reilleptnoM ,ecnarF yraunaJ 9891
Springer-Verlag
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DRO~LROF
This volume contains the papers presented during the third
working conference "Stabilization of Flexibles Structures " held
in Montpellier,January 1989.The three conferences,Nice June 1987,
Montpellier January 1988,Montpellier January 1989, were sponsored
by the following French institutions working on control and stabili-
zation~
Centre National de la Recherche Scientifique (CNRS) ,
Centre de Math4matiques Appliqu~es (CMA),which a department of
l'Ecole Nationale Sup4rieure des Mines de Paris,located in Sophia-
Antipolis
A4rospatiale , Cannes - la - Bocca
The collaboration of these three institutions was initiated in
1986.As a result there exist now several joint works and a regular
seminar "Stabilization of Flexible Structures "
This volume is divided in three parts:
Examples of Flexibles Structures
Mathematics concerning Stability,wave Equation,Non Cylindrical
Domain
Shape Variation in Hyperbolic Problems
J would like to express my thanks to Professor M.THOMA who has
accepted again to publish these proceedings in the Lecture Notes in
Control and Information Sciences.
J.P. ZOLESIO
CONTENTS
Recent Work on the Scole Model
W. LFFFMAN
Mathematical Study of Large Space Structures.
D.CIORANESCU, J.SAINT JEAN PAULIN 6
Symbolic Formulation of Dynamic Equation for Interconnected Flexibles Bodies:
The GEMMES Software.
C.GARNIER 71
Adaptive Optics, Shape Control of an Adaptive Mirror.
C.TRUCHI 28
Energy Decay Estimates for a Beam with Non Linear Boundary Feedback.
F.CONRAD, J.LEBLOND, J.P.MARMORAT 46
Uniform Stabilization of the Wave Equawtiitohn Dirichlet Feedback
Control Without Geometrical Conditions.
I. LASIECKA, R. TRIGGIANI 62
Actuators and Controllability of Distributed Systems.
A. ELJ AI 109
Linear Quadratic Control Without Stabilizability.
G.DA PRATO, M.C. DELFOUR 126
Riccati Equation in Non Cylindrical Domains.
ECANNARSA, G. DA PRATO, J.P. ZOLESIO 148
Boundary Control Problems for Non Autonomous Parabolic Systems.
P. ACQUISTAPACE, B. TERRENI 156
Existence and OptiCmoanlt rol for the Wave Equation in Moving Domain.
G. DA PRATO, J.P. ZOLESIO 761
V
Galerkin Approximation for Wave Equation in Moving Domain.
LP. ZOLESIO 191
Further Results on Exact Controllability of the Euler-BernouUi
Equation With Controls on the Dirichlet and Netnnann Boundary conditions.
I. LASIECKA, R. TRIGGIANI 226
Some Properties of the ValFuuen ction of a non Linear Control
Problem in Infinite Dimension.
P. CANNARSA, G. DA PRATO 235
Identification of Coefficients with Bounded Variation in the Wave Equation.
J.P. ZOLESIO 248
Shape Hessian by the VeloMceitthyo d: a Lagrangian Approach.
M.C. DELFOUR, J.P. ZOLESIO 280
Differential Stability of Pertubed Optimization with Application to
Parameter estimation.
J. SOKOLOWSKI, B. RAO 298
A Numerical Method For Drag Minimization Via the Suction and Injection of
Mass Through the Boundary.
M. GUNZBURGER, L. HOU, T.P. SVOBODNY 312
Using the Physical Properties of Systems for Control.
H.J.C. HUIJBERT 322
RECENT WORK ON THE SCOLE MODEL
Walter Littman
University of Minnesota
Minneapolis, MN 55455~ USA
1. Introduction.
In a number of papers (see for example [BT D Balakrishnan and Taylor introduced the
"SCOLE ~ (Spacecraft Control Laboratory Experiment) model for a vibrating flexible mast,
which at one end is attached to a spaceship, and at the other end to an antenna reflector.
Mathematically, the system consists, essentially, of three uncoupled partial differential
equations, two of which are the Euler Bernoulli beam equation while the third is the
one dimensional wave equation. At the "left" end "clamped" boundary conditions are
imposed. At the "right ~ end control forces and torques are imposed, yielding complicated
non homogeneous boundary conditions which are nonlinear and in which the unknown
functions - representing beam deflections and the torsion angle about the beam axis - are
coupled.
Two problems present themselves: one is the "open loop" exact controllability of
the system: can an initial disturbance - in an appropriated function space - be exactly
controlled to rest in a finite time by applying the forces and torques at the right end in
an appropriate fashion? The second question is one of closed loop stabilization: Can the
(inhomogeneous) control forces and torques at the "right" end be chosen as functions of
the velocities and angular velocities at that end in such a way that the energy of the
system approaches zero asymptotically as t -~ co. In that case can this decay be made to
exponential? In thls note we shall discuss some recent work dealing with the first question.
2. The open loop problem.
In [LM1] the "reduced SCOLE" system is considered, consisting of a single Euler
Bernoulli beam equation, arising from the plane motion of a beam.
Consider the mixed problem:
0<z<l, t>0
w(0,t) = w~(t,0) = o
w~ - #lw... = fl(t) w~tt +/~zw~. = fz(t) at z = 1
~(~, )0 = ~o(~) ~(~,0) = ~(~) < 0 < ~ .I
(Here/~ > ,0 ~2 > .)0
The control problem: Given laitini conditions wo(z) and w1(z) (possibly gniyfsitas some
fx(t) f2(t)
compatibility conditions at z = 0), can we find functions and such that the
resulting solution of the mixed problem vanishes for t __> T?
An answer was given in [LM1]: (Here the H's refer to Sobolev spaces)
Given initial data in H 6 x H 4 on 0 < z < 1, with compatibility conditions
w0(0) = w~(0) = ,0 wx(0) = wi(0) = 0
o~)0(~'~
= ~o )0(~'c = o;
then rof each positive duration T, exist there two srellortnoc fx )t( and )t(zf continuous ni
]T,0[ and C °~ on ,0( T] such that the corresponding ,noitulos w(z, )t to the mixed problem
vanishes from t > T. Furthermore the functions )t(tf and )t(~f are given by ticilpxe
formulas.
Note: In the proof ti actually seciffus rof the laitini data to be ni HS½ × H3½.
3. Improvements.
There are several directions in which the method of [LM1] can be extended and im-
proved.
First of all, although the original three dimensional SCOL]~ system seems much more
complicated, the methods of [LM1] encompass essentially all mathematical difficulties,
and the exact controllability of the three dimensional model can be achieved by a minor
modification of the method. The only difference is that since one of the equations in the
full system is a wave equation, the time T is not arbitrarily small, but is governed by the
time it takes to control the one dimensional wave equation.
Secondly, to what extent is the high degree of smoothness of the initial data really
necessary? It follows from a result of Triggiani that an initial disturbance assumed only to
3
have finite energy can not be controlled by locally integrable fl and fa (see the discussion
in [LM2]). The method described above yields fl and f2 which are C ~¢ for positive t but
which may have singularities at t = 0, in the case where the initial data is assumed to
be merely L 2, or have finite energy. It tan eb shown however that if we merely require fl
and f2 Ot eb in L2[O,T], rather than eb continuous at zero, it su~ees to take the initial
data merely in H4½ X H2½. This has the advantage that the compatibility conditions
)'(oW )0( = 0 dna )O()S(w = 0 nac be now desnepsid with.
Finally it is of interest to consider the esac where the material properties of the beam
vary from point to point. This problem has been recently solved by Steven Taylor, a Ph.D.
student at Minnesota, who in the course of providing the solution has obtained a number
of related results of independent interest. We describe Taylor's work in the next section.
4. The work of Steven Taylor.
The main ingredient in the work is the establishment of a certain degree ofr egularity
in t of solutions of a class of equations with variable coefficients on the semi infinite interval
0 < x < oo. Consider the equation
02w w~gc
Lw=-g-~+ E b/(z)-wT..=O O<z<oo, t>O.
0</<__4 dx- -- --
assume 4b > 0,b4 E C4,b8 E C3,b2,b, ob each is C 2, and that these coefficients are
We
constant outside of a bounded set. If w is a solution to the initial value problem consisting
of the above equations and boundary condition
w(o,t) = o w (o,t) = o
and initial conditions (satisfying appropriate compatibility conditions at x = 0)
w(x,O) -- wo(x) e H2([0, oo)) with compact support
wt(x,O) = wx(x) e L2([0,¢o)) with compact support,
then the solution w(x, t) belongs to the Gevrey class q(2) in the variable t. Loosely speak-
ing, a ~ O function f(t) is in the Gevrey class )6(/~ (6 > 1) if for each 0 > 0 one can find a
4
constant Co > 0 such that [)t()"{f[ < GoO'n s" rof lla tr = ,O 1,2,... coafIs ne functions
)t,x(w ti si understood that the stated estimates hold uniformly in compact subsets of
,O__>x t>O.
Once the Gevrey regularity result is established, the controllability result follows as
in [LM1]. We briefly outline the procedure. Suppose the basic interval is 0 < x < 1.
We write L = T+A, where T = a-~" We extend the initial data as smoothly as possible
to have compact support in the larger interval 0 < z < + 1 ¢ and solve the resulting mixed
problem for the half line x >_ 0. By the Gevrey regularity result the resulting solution
W(x,t} will be of a class if(2) for t > 0. Letting T1 be an arbitrary positive number one
can find a function h(t) of class "7(2) such that h(t) = 1 for t < T1/2 and h(t) = 0 for
t > T1. Then h(t)W(x,t) is of class "/{~) in t and satisfies (T + A)h(t)W(x,t) = F(x,t),
where F(x,t) is of class 7" )2( in t, vanishes for t >_ T1 and 0 _< t < 2'1/2. Then one
constructs a solution g of (T+ A)Z = F vanishing for t _> T1,0 < t < 1"i/2. The solution
is given by the formula
z(., t) =
where A -~ is an appropriately defined inverse of operator A. It can be shown that the series
for W converges because F(x,t) is of class if(2) in t. One then sets w(x,t) -=- W(x,t)h(t) -
Z(x, t). Then w(z, t) satisfies the equation Lw = o, the correct initial conditions in 0 < z <
1, and the correct boundary conditions at x = 0, and furthermore vanishes for t > T1. The
boundary controllers are then read off from the traces of the correct boundary operators
at t=l.
This method has recently been extended by Taylor to equations of the form
a2w a~w w~O
--=)z(~b
+ r~ + _r,= -_-_-_-a_~(~-) _-_-_-_-_-_ = o
where b4 (x) > 0 and a2 (x) < .O
5
YHPARGOILBIB
TB[ 1 Balakrishnan, A. V., and Taylor, .L W., A Mathematlcal Problem and a Spacecraft
Control Laboratory Experiment (SCOLE) Used to Evaluate Control Laws for Flexible
Spacecraft, Proceedings of NASA SCOLE workshop, Dec. .4891
]1ML[ Littman, W., Markus, .L Exact Boundary Controllability of a HybridS ystem of Elas-
ticity, Archive for Rational Mechanics and Analysis, 301 ~3 1988, .632-391
]2MLI Littman, W., Markus, L., Remarks on Exact Controllability dam Stabilization of a
Hybrid System in Elasticity Through Boundary Damping, Proceedings of Symposium
in Santiago Spain Springer (1987), Lecture Notes on Control and Information Sciences
~114, .9891
Acknowledgement: This research was in part sponsored by NSF Grant .2042278