parametric optimal control problems

• 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: