• Problem statement;• Solution structure and defining elements;• Solution properties in a neighborhood of regular point;• Solution properties in a neighborhood of irregular point: • construction of new Lagrange vector;
• construction of new structure and defining elements;
• Generalizations.
OUTLINE
Family of parametric optimal control problems:
0
0
0
( ( ) ) ( ( ) ( ) ( ) ( ) ( ) ( )) min
( ) ( ) ( ) ( ) ( )OC( )
( ) ( ) ( ) ( ) ( ) [0 ]
(0) ( ) ( ( )
(1)
(2)
(3)
(4)
(
) 0, {1 }
( ) 1 5)
t
i
f x t x t D x t u t R u t dt
x t A x t B u t
d x t g u t t t T t
x x f x t
h h h
h h
i M m
u t t T
hh h h
h h
0, , ( ) ( ) 0, ( ) ( ) 0, ( ) ( ) ( )n r h h h h h h hx R u R D D R R A B x
( ) 0,..., ; ( ) [ ]ni h hf x i m t x R t hT h h
are given functions,
[ ]h hh is a parameter.
Problem statement
Optimal control and trajectory for problem ( )OC h
( ) ( ( ) ) ( ) ( ( ) )u u t t T xh h h x t t Th
The aims of the talk are
• to investigate dependence of the performance index and
( ) ( )h hu x on the parameter h;
• to describe rules for constructing solutions to ( )OC h
for all [ ]h h h
Terminal control problem OC(h)
( ), ( )hu xh
0 ( ( )) min
( ) ( ) ( ) (0) ( )OC( )
( ( )) 0 {1 }
( ) 1 [0
1
]
( )i
f x t
x t Ax t bu t x z
f x t i M … m
u t t T t
hh
[ ],n hx R u R h h is solution to the problem OC(h),
functions ( ) 0 , are convex.if x i M
Maximum Principle
my R
0 ( ) ( )() 3)(
f x f xA t y
x x
( )u h to be optimal in ОС (h) In order for admissible control
it is necessary and sufficient that a vector
exists such that the following conditions are fulfilled
0 ' ( ( )) (20 )y y f x ht
1( ( )) ( ) max ( ( ))
ut y x t bu th h t y x t bu th T
Here ( ), , , ,m nt y x t T y R x R is a solution to system
Denote by ( ) mY h R the set of all vectors y, satisfying (2), (3)
(4( ) [ ] ).Y hh h h h
• The set ( )Y h is not empty and is bounded for [ ]hh h
and consider the mapping
• The mapping (4) is upper semi-continuous.
Let ( ) ( ( ) ) ( ).ih hy i M hy Y Denote by
( ( ) ) ( ( ) ( ))h h h ht y t y x t b t T
the corresponding switching function.
{ ( ) 1,..., ( )} { ( ( ) ) 0}j h ht j p ht T t hy
1( ) ( ) 1,..., ( ) 1j jt t jh h hp
( ( ) ( ) )( ) { {1 ( )} 0}j h h hy
L p ht
jt
h
1 1( ) 1 if ( ) 0 ( ) 0 if ( ) 0l t lh h h t h
( ) ( )( ) 1 if ( ) ( ) 0 if ( )
+0 ±1,
hp p hh hl t t l t t
k( )= u(
h h
=h | )h
0 ( ) { ( ) ( ) 0}.a ih hM i M hy
( ) { ( ( | )) 0},a ihM i M f t hx
Zeroes of the switching function:
Active index sets:
Double zeroes:
Solution structure:
0( ) { ( ) ( ) ( ) ( ) ( ) ( ) ( )}aS p k M l lh h h h h h h hM L
Defining elements:
( ) ( ( ) 1,..., ( ) ( ))jh h hj p hQ t y
Regularity conditions for solution ( )u h (for parameter h)
0( ) ( ) ( ) ( ) ( ) 0h h h hl L hM l
Lemma 1. Property of regularity (or irregularity) for control ( )u h
does not depend on a choice of a vector ( ) ( )y Yh h
Suppose for a given 0 [ [h hh we know
• solution 0( )u h to problem 0( ),OC h
• a vector 0 0( ) ( )y Yh h
• corresponding structure 0( )S h and defining elements 0( ).Q h
The question is how to find
0( ) ( ) ( ) for ( )?h h hu Q hhS E
is a sufficiently small right-side neighborhood of 0( )E h
the point 0.h
Here
Solution Properties in a Neighborhood of
Regular Point
0( ) : { ( ) ( ) ( ) ( ) ( ) ( ) ( )}aS p k M lh h h h l Mh Lh h h
Solution structure does not change:
0 0 0 0 0 0 0 00{ ( ) ( ) ( ) ( ), ( ) ( ) ( )} : ( )ap k M l lh h M L Sh h h h h h
Defining elements
with initial conditions
0 0 0 0( 0) ( ) 1,..., ( 0) ( )j jt t j p yh h h hy
0 0 0ap k( ) ( ), M ( )ah hk hp M
are uniquely found from defining equations
( ) ( ( ), 1,... ( ))jQ t jh p hyh
ap( ( ), | , k M, ) 0Q h h
a( , | ) ,
( , 1,
p,k
..., ; ) ;
,M p m
p mj
Q R
Q t j
h
p y R
where
Optimal control ( )u h for ОС(h):
1( ) ( 1) [ ( ) ( )[
0,... ,
jj jh hu t k t t ht
j p
0 1( ) 0 ( )ph ht t t
Construction of solutions in neighborhood of irregular point
The set consists of more than one vector.0( )Y h
0 0 0 0 0 0( 0) ( ) ( 0) ( ) ( 0) ( )y y S S Qh h h Qh h h
The first Problem: How to find 0( 0) ?y h
The second Problem: How to find 0 0( 0), ( 0)?S Qh h
0h
0Costruction of vector ( 0)hy
Theorem 1. The vector 0( 0)y h is a solution to the problem
0 0 0min(0 ( )) ( 0) (SI)( )y x t Yh hz yh
The problem (SI) is linear semi-infinite programming problem.
The set 0( )Y h is not empty and is bounded
the problem (SI) has a solution.
Suppose that the problem (SI) has a unique solution y
0( 0)hy y
0 0New Lagrange vector ( 0) ( ) is foundh hy y y
00
0( ) ( ( ) )t t y h h t T
New switching function 0( )( ) t y ht t T Old switching function
0 0Construction of new ( 0) ).and ( 0S h hQ
A) What indices i M are in the new set of active
0( 0)?a hM
B) How many switching points 0( 0)p h will new
0( 0)hu optimal control
indices
have?
0( 0)?a hM Form the index sets
0{ ( ( )) 0},a iM i M f x ht
0 { 0} { 0}.a a i a a iM i M y M i M y
It is true that 0 0( ) ( 0) ( 0)a a a aM M M M Mh h ‚
0
0
0
( 0)
( 0)
a
a
a
M
i M
M M
h
h
‚
?
A): How to determine
\
\
0( 0)?p h
0
Let 1,..., be zeroes of new unperturbed switching function
) ( )(
jt j p
t y tt h T
0 0{1,2,..., }, { : ( 0 | ) ( 0 | )}.R j jhJ p J j J u t u t h
7, {2,4,5,7}Rp J
B): How to determine
0
1,..., ( ) are zeroes of perturbed switching functi( )
( ( ) )
on
, , 0.
jt
t
j p
t T hhh h
h
hy
h
h
0 0*
1,... are zeroes of unperturbed switching function
( ) , with ( 0)) ,(
jt
t h h
j p
t y t T y y
7, ( ) 8p p h
*For each , , a) or b) ?*j Rt j J \ J
Using known vector 0( 0) ,z h
and sets 0, , ,a a RM M J J
form quadratic programming problem (QP):
min( ) ( ) ( ) 2S
I s g s Dg s s Ds
00 0( ( )
( ) 00
aR
a
ij
MJ
if x tg Js s
h
Mj
x i
‚
0where ( ) ( ) ( 0) .js s j gJ s hF sz B
Theorem 2. Suppose that there exist finite derivatives
0 0( 0) ( 0)
1,... (, )jdt
h h
dyj
d
h hp
d
Then the problem (QP) has a solution which can be uniquely found using derivatives
0 0( )js s j J 0 0( ).i ai M
Then derivatives are uniquely calculated by 0 0, .s
( .)
( )
Suppose the problem (QP) has a unique optimal solution:
primal and dual
Let (QP) have unique optimal plans 0 0 0 0( ), ( ).j i as s j J i M
0 ?A): ai M , \ ?B): j Rt j J J
We had problems:
Solution of problem A):
0 00Index belongs to ( 0) if 0.a a ii M M h
0 00Index does not belong to ( 0) if 0.a a ihi M M
0
0
situation ) if 0,
situation ) if 0.
j
j
a s
b s
Solution of problem B):
0 a0 0
Consequently
( 0) { 0}: Ma a a iM M ih M
( ) )0 (0( p0) :hp J J
( ) (0)
where
{ 0} { 0 0 }R R j R j jJ J j J J s J j J J s t t ‚ ‚
( ) (0)Put { 1 } { },j j jt j … p t j J t j J
1, 1 1,j jt t j … p
1 10 0( 0 ) if 0 ( 0 ) ifk 0k hu u tht
0 0 0 0 a( 0) { ( 0)
New structur
p k( 0 M) ( 0) }
e
ah h h hS p k M
0(
New defining element
0) ( 1,..., )p
s
jhQ Q t j y
Theorem 3. Let h0 be an irregular point and the problem (QP) have a unique solution. 0 0( )Then for E h hh \
problems ОС(h) have regular solutions with constant structure
0 a( ) ( 0) p, ;M }: ,{ khShS defining elements Q(h) are uniquely found from
0( 0) ,with initial condition Q Qhs
p pa( , | ) , ( , 1,..., ; )p, k, p ;M m m
jwhere Q R Q t j Rh y
optimal control ( )u h is constructed by the rules
1( ) ( 1) [ ( ) ( )[ 0,. , p.k . ,jj ju t th hth t j
p0 1( ) 0 ( )t t th h
ap, k,( ( ), | ) 0Mdefining equations h hQ
On the base of these results the following problems are investigated and solved
differentiability of performance index and solutions to problems
( ), [ ] with respect to parameter ;h hhO h hC
path-following (continuation) methods for constructing solutions to a family of optimal control problems;
fast algorithms for corrections of solutions to perturbed problems
0 0
0
( ), [ ] with respect to small variations of a
parameter ;
OC h hhh h h
h
construction of feedback control.
• Kostyukova O.I. Properties of solutions to a parametric linear-quadratic optimal control problem in neighborhood of an irregular point. // Comp. Math. and Math. Physics, Vol. 43, No 9, 1310-1319 (2003).• Kostyukova O.I. Parametric optimal control problems with a variable index. Comp. Math. and Math. Physics, Vol. 43, No 1, 24-39 (2003).• Kostyukova, Olga; Kostina, Ekaterina. Analysis of properties of the solutions to parametric time-optimal problems. // Comput. Optim. Appl. 26, No.3, 285-326 (2003).• Kostyukova, O.I. A parametric convex optimal control problem for a linear system. // J. Appl. Math. Mech. 66, No.2, 187-199 (2002).• Kostyukova, O.I. An algorithm for solving optimal control problems. // Comput. Math. and Math. Phys. 39, No.4, 545-559 (1999).• Kostyukova, O.I. Investigation of solutions of a family of linear optimal control problems depending on a parameter. // Differ. Equations 34, No.2, 200-207 (1998).
Results of these investigations are presented in the papers: