parametric optimal control problems
DESCRIPTION
UNIVERSIDADE de AVEIRO Departamento de Matematica, 2005. PARAMETRIC OPTIMAL CONTROL PROBLEMS. Olga Kostyukova. Institute of Mathematics National Academy of Sciences of Belarus Surganov Str.11, Minsk, 220072 e’mail [email protected]. OUTLINE. Problem statement; - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/1.jpg)
![Page 2: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/2.jpg)
• 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
![Page 3: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/3.jpg)
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
![Page 4: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/4.jpg)
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
![Page 5: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/5.jpg)
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
![Page 6: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/6.jpg)
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
![Page 7: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/7.jpg)
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.
![Page 8: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/8.jpg)
{ ( ) 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:
![Page 9: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/9.jpg)
![Page 10: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/10.jpg)
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
![Page 11: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/11.jpg)
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
![Page 12: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/12.jpg)
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
![Page 13: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/13.jpg)
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
![Page 14: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/14.jpg)
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
![Page 15: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/15.jpg)
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
![Page 16: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/16.jpg)
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
![Page 17: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/17.jpg)
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
![Page 18: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/18.jpg)
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?
![Page 19: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/19.jpg)
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
\
\
![Page 20: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/20.jpg)
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
![Page 21: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/21.jpg)
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
![Page 22: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/22.jpg)
*For each , , a) or b) ?*j Rt j J \ J
![Page 23: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/23.jpg)
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
![Page 24: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/24.jpg)
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
![Page 25: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/25.jpg)
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
![Page 26: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/26.jpg)
0
0
situation ) if 0,
situation ) if 0.
j
j
a s
b s
Solution of problem B):
![Page 27: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/27.jpg)
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
![Page 28: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/28.jpg)
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
![Page 29: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/29.jpg)
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.
![Page 30: PARAMETRIC OPTIMAL CONTROL PROBLEMS](https://reader035.vdocument.in/reader035/viewer/2022062408/56813098550346895d9676f7/html5/thumbnails/30.jpg)
• 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: