Maximum principle for optimal control problems with delay-differential system equation

Banks, H. T.

English

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A MAXIMUM PRINCIPLJ3 FOR OPTIM4L CONTROL 
PROBLENS WITH FUNCTIONAL D I F F E R E W T U L  SYSTEMS 
H. T. Banks* 
Center for Dynarnical Systems 
Brown University 
Providence, Rhode Island 
_I- - - 
I 
- -  
* 
This research was supported in part by the Air Force Office of 
Scientific Research, under Grant No. AF-RFOSR 693-67, in part by 
the National Aeronautics and Space Administration, under G r a n t  No. 
NGR 40-002-013 and in part by the N a t i o n a l  Science Foundation, 
under Grant Wo. GP 9024, 
https://ntrs.nasa.gov/search.jsp?R=19680022022 2020-03-12T10:11:25+00:00Z
A MAXIMUM PRINCIPLF: FOR OPTIMAL CONTROL 
PR0BI;EPIIS WITH FUNCTIONAL DIFFERENTIAL SYSTEMS 
by 
H. T. Banks 
In  t h i s  note we present a maxitrmm principle i n  in tegra l  form 
for  optimal control  problems with delay-different ia l  system equa- 
t ions which also contain delays i n  the control. Recent re la ted  
resu l t s  for  par t icu lar  cases of the systems discussed below may be 
-.found i n  [l], [ 5 ] ,  and [ 6 ] .  Vector matrix notation w i l l  be used and 
we s h a l l  not dist inguish between a vector and i t s  transpose. 
L e t  a, and to be fixed i n  R1 with -W < a. < to, I = 
[a,, a) be a bounded in te rva l  containing [ a  t 3, and put Is = 
(to,a). For x continuous on I and t i n  I?, the notation 
F (x (* ) , t )  w i l l  mean t h a t  F i s  a functional i n  x, depending on 
any or  a l l  of the values x(T), a. 5 7 S t. 
c lass  of absolutely continuous n - 1 vector functions on [ a  o, to] 
Let be a given convex subset of the  c lass  of all bounded Bore1 
0' 0 
- 
Q, w i l l  denote the 
Q 
measurable f'unctions u defined on I in to  Rr, and 7 be a 
given C1 manifold i n  R2n-1. 
minimizing 
The problem considered i s  t h a t  of 
over 5 X il X C(I ,Rn- l )  X I' subject t o  
U 
2 
O I- P-l)  i s  an n-vector f = (f , f )  = (f  , f  ,..., 0 -  We assume t h a t  
functional of the form 
where the in tegra l  i s  a Lebesgue-Stieltjes integral .  Each 
h i (x(*) , t )  
gi(y,t) i s  c1 i n  (y,t) on R . me r x 1 vector function 
- 
is  assumed C1 i n  x and measurable i n  t, and each 
n 
q(t,s) i s  measurable i n  t,s, and of bounded var ia t ion i n  s on 
[ao,t]. It is also assumed tha t  the  var ia t ion of 7 i s  dominated 
by an L (Ir) function m. That is, v s(t,s) 5 m ( t )  f o r  
t E 1'. Finally, suppose t h a t  given x compact, $? C R , there  
ex is t s  an m i n  L1(Ir) satis- 
f i e s  
t 
s=a 
0 
1 
n-1 
Iv 
such t h a t  h = (h * ,h  ,..., hn-l) 
and 
. 
3 
1 
for any $ E C(I ,Rn-I)  
sup ( 1  T(s) I : s E [ao, t]] 
with respect t o  x. 
and E C(I ,%),  where 11T11, = 
and dh i s  the Frgchet derivative of h 
- 
( f A 1  denotes the Euclidean norm of A. ) 
-* * -* * 
If (9 ,u  ,x t ) i s  a solution of the above problem, we de- ’ 1  -* 
f ine  the n x n - 1 matrix function 7 for t E It, s E [ao,t] 
bY 
-2+ 
where T~ i s  such t h a t  
(The existence of T* i s  guaranteed by the Rie‘sz theorem.) Then we 1 
have the  following necessary conditions: 
4 
* *-* * 
Theorem: L e t  @ ,u ,x  ,tl) be a solution t o  the  problem under the 
* 
assumptions above. I n  addition, suppose t h a t  t is  a Lebesgue 
point of f ( x  (-),u (*),t). 
vector function h ( t )  = (ho(t) ,x( t ) )  of bounded var ia t ion on 
[t t*] continuous at  tl, satisfying: 
1 
A * Then there ex is t s  a non-tr ivial  n- 
* 
0' 1 ' 
* 
(a) ho(t)  = constant SO, A ( t l )  + o 
* 
t, 
-. -* 
%here 7 i s  defined by (2). 
for a l l  u E R .  
( e )  The 2n-1 vector 
* 
t. 
f (Z* ( ), u* ( ), t;, . 
The proof of t h i s  theorem involves showing t h a t  the c lass  of 
functions y =  {F(x(*), t) :  F(ji(*), t)  = f f i (*) ,u(*) ,%) ,  U E n) 
5 
i s  absolutely quasiconvex [2] and then using necessary conditions 
fo r  extremals given i n  [2]. Absolute quasiconvexity i s  a general- 
izat ion of ideas due t o  Ganikrelidze [4], who first  obtained an 
in tegra l  maximum principle fo r  control problems with ordinary d i f -  
f e r en t i a l  system equations. 
principle i n  in tegra l  form for  the above described optimal control 
The inequality i n  (b) i s  a maximum 
problem. 
In  many par t icular  cases of the systems defined by (l), one 
can show tha t  the multipliers x are actually absolutely continuous 
;and sa t i s fy  (a) i n  different ia ted form. 
becomes the usual known multiplier equation fo r  systems with simple 
This different ia ted form 
time lags i n  the s t a t e  variables (see [l]). The t ransversal i ty  
conditions given i n  (C)  a lso can be reduced t o  a simpler form for  
many special  cases of (1). 
Included i n  (1) are many integro-differential  systems and 
time lag systems which appear i n  physical problems. For example, 
i f  one modifies s l i gh t ly  the biological population model formulated 
by Cooke i n  [ 3 ] ,  one obtains the system equation 
;r(t) = u( t - r )x( t - r )  + i3 (t)u(t-r-e (t))x(t-2-0 (t)) 
where x ( t )  i s  the  number i n  the population at  time t, u( t )  i s  
the b i r t h  rate a t  time t, and z i s  the gestation period. Systems 
with 
6 
which ar i se  i n  the study of reactor dynamics [TI, and with 
where 'iE = x ( t + 8 ) ,  0 e [-T,O], are a l so  spec,al cases of (1). t 
I .  
b 
References 
H. T. Banks, "Necessary Conditions for  Control Problems with 
Variable Time Lags", J. SIAM Control 6 (1968), 9-47. 
, "Variational Problems Involving Functional Differ- 
e n t i a l  Equations", ( to  appear). 
[3] K.L. Cooke, "Functional-Differential Equations: Some Models 
and Perturbat ion Problems", Differential  Equations and 
Dynamical Systems, Academic Press, New York, 1967. 
*[4] R.V. Gamkrelidze, "On Some Extremal Problems i n  the Theory 
of Different ia l  Equations with Appplications t o  the Theory 
of Optimal Control", J. SLAM Control 3(1965), 106-128. 
[5] E.B. Lee, "Geometric Theory of Linear Controlled Systems", 
( to  appear). 
[6] , "Variational Problems for  Systems Having Delay i n  
the Control Action", ( t o  appear). 
[7] J. J. Levin and J.A. Nohel, "A System of Nonlinear Integro- 
Different ia l  Equations", Mich. Math. J. 13 (1966), 257-270. 


A maximum principle for optimal control problems with functional differential systems

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