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r
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ON THE CONTROL OF A LINEAR
FUN%TIONAL-DIFFERENTIAL EQUATION
WITH QUADRATIC COST
1.
Tt21S ca .;,
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c
rel en Se and
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AFOSR
69-1287
N
'
HAROLD
I.
KUSHNER
AND
C
DANIEL 1. BARNEA
^J
MARCH, 1989
CENTER FOR DYNAMICAL SYSTEMS
Reproduced by the
C L E A R I N G H 0 U 5 E
for Federal Scientific & Technical
Information Springfield Va. 22151
OC)
ON THE CONTROL OF A LINEAR FUNCTIONAL
-
DIFFERENTIAL
EQUATION WITH QUADRATIC COST
by
Harold J. Kushner*
Division of Applied Mathematics and Engineering
Brown University
Providence, Rhode Island
F
s
1
and
Daniel I. Barnea**
Division of Engineering
Brown University
Providence, Rhode Island
This research was supported in part by the National Aeronautics and
Space Administration, under Grant No. NGR 40-002-015,
in
part by the
Air Force Office of Scientific Research, under Grant No. AF-AFOSR
693-67
and in part by the National Science Foundation, under Grant No. GK 2788.
This research was supported by the National Science Foundation, under
Grant No. GK 2788.
1. Tbis document has been approved for public
release and salo ; l.ts distribution is unlimited.
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ON THE CONTROL OF A LINPAR DIFFE'ONCE-DIFFERENTIAL
EQUATION WITH QUADRATIC COST
rHarold
J. Kushner
and
r
Daniel I. Barnea,
r
1.
Introduction.
Let
H
be the space of n-vector valued functions
y0p)
on the real finite interval
f -r,0],
r > 0, whose com-
Ponents are continuous on
[-r, 0].
Suppose
x(t)
is an n-vector valued
r
function defined on the real interval
[ -r, T], T > 0.
Fix
t
e .
[0, T].
Let
x
denote the element of
H
with values
x(")
at
cp, cp e [—r.0].
Let
x(•)
be'the solution of the delay equationtt
o
(1)
x(t) = A(t)x(t) + B(t)x(t-r) +
f
c(t,cp)x(t4<p)dcp + D(t)u(t)
-r
where
A(t), B(t), C(t,(p), D(t),
and the derivatives of
B(t)
and
C(t,cp)
for
(t,cp) a [0,T] x [-r,0], and the 'initial condition', x
o
, is
in
H.
This paper is concerned with finding the control
u(-)
which
rminimizes
the quadratic functional
m
1
(2)
vu(xt)t) =
f
[x'
(s)M(s)x(s) +
u' (s)N(s)u(s)]ds,
t
i
where
M(s)
and
DI(s)
are continuous ttt. M(s) ? 0, and
N(s) > 0
for
'The prime ' denotes transpose.
tt
(1)
is
treated for simplicity;
it will be obvious that replacing the
term
Bx(t
-r)
by
EB
i
x(t-
r
i
)
demands few changes in the development.
tttM z 0, N > 0
denote that
M
is non-negative definite and
N
is
positive definite.