shahlar meherrem department of mathematics yasar university – İzmir....
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Shahlar MeherremDepartment of Mathematics
Yasar University – İzmir.
shahlar.meherrem@yasar.edu.tr
Shahlar Meherrem
Shahlar Meherrem
Presentation Contents
Introduction
Some Definitions
Methodology
Necessary optimality condition
Results
Conclusions
Shahlar Meherrem
A switching system consists of a number of subsystems and a switching law.The switching law is to define of the subsystem to be activated at certain specified switching instants during the planing horizon.Switching system arise in many real world aplications,such as the control of mechanical systems,the automative industry,aircraft and air traffic control,and switching power converts. . In the present presentation, the author’s main aim is to formulate necessary optimality conditions for nonsmooth case (switching cost functional has Frechet superdiferential) by using nonsmooth analysis and the method which was suggested and formalized by Dubovitskii and Milyutin.The Dubovitskii and Milyutin formalizm contains the followings three major components:a) To treat local minima via the empty intersection of certain sets in the primal space builtupon the inital cost and constraints datab) To approximate the above sets by convex cones with no intersections.c) To arrive at dual necessary optimality conditions in the form of an abstract Euler eqation by employing convex separation. The main result in this article is a new optimality condition for the nonsmooth switching control system (Frechet subdifferential form).
Shahlar Meherrem
Problem formulation and tools of nonsmooth AnalysisExample :A car moves according the law ,yx ),(
1yugy Uu on the time
interval 101
, tt , and under the law ,yx ),(2
yugy 1
tyu at the time interval
Tt ,12
The inital and final time moments Tt ,0
a re fixed,the moment 1
t no to fixed , while the
set 1,0U , the functions ,,21
gg are positive and differentiable in 1R .The car starts from the
point 0,0, 00 yx and the s tate variables x and y are assumed to be continuous on the whole
interval T,0 . İt is requ i red to maximize Tx .To find the necessary optimality conditions we
have to build Hamilton - Pontryagin functions o each steps . A fter this by using increment formula and
conjugate systems we can get necessary condition for this step system. As a second example ,
consider a rocket with two types of engines that work consecutively. With work of the second
engine depends on the first one. Morerover, the rocket moves from one controlling area to a second
one that changes all the structure (controls, functions, conditions, etc.).
Shahlar Meherrem
Given a nonempty set nR , consider the associated distance function:
wxxdistu
inf;
and define the Euclidean projector of x to by:
;(|:; xdistwxwx .
If the set is closed, then the set ;x is nonempty for every nRx .
This nonconvex cone to closed sets and corresponding subdifferential of lower semicontinuous extended–real –valued functions satisfying these requirements were introduced by Mordukhovich in the beginning of 1975
the original normal cone definition was given in finite dimensional spaces by:
( ; ) : sup ;x x
N x Lim cone x x
, (1.1)
via the Euclidean projector, while the basic subdifferential )(x was defined geometrically via the normal cone to the epigraph of . Here it is assumed that is a real valued finite function and the basic subdifferential is defined as:
epixxNxRxx n ;,1,: . (1.2)
Here xRxepi n 1,: and is called the epigraph of a given extended real valued
function. Note that this cone is nonconvex and for the locally Lipschitzian functions, the convex hull of a subdifferential has a Clarke generalized gradient, 00 xcoxk .
If k is lower semicontinuous around x, then its basic subdifferential can be shown by:
0
0 ˆsupx x
x Lim x
.
Here,
0
0 0
0
0
,ˆ : | min 0n
u x
u x x u xx x R Li f
u x
(1.3)
is the Frechet subdifferential. By using plus-minus symmetric constructions, we can write
0 0: ,x x 00 ˆ:ˆ xx (1.4)
where denotes a basic superdifferential and ˆ denotes a Frechet superdifferential. Here
. 0
0 0
0
0
,ˆ : | sup 0n
u x
u x x u xx x R Li f
u x
(1.5)
For a Locally Lipschitzian function, the subdifferential and superdifferential may be different. For
example, if we take xx on R , then 1,10 , while 1,10ˆ .
If is Lipschitz continuous around point 0x then, the strictly differentiability of the function at 0x are equivalent to. Symmetrically, we can give upper regularity of the function at the point by
using definitions of superdifferential and Frechet superdifferential. Also if the extended-real-valued function is Lipschitz continuous around the given point and upper regular at this point then the Freshet superdifferential is not empty.
Definition. (Mordukhovich) is upper regular at x if x x
, If 00 ˆ xx
then, this function lower regular at 0x .
Shahlar Meherrem
Shahlar Meherrem
ttutxftx kkkk ),(),()( , kk ttt 1 , k=1,2,..,N (2.1)
, )( 001 xtx 0 1 1... N Nt t t t
0, NNNi ttxF i=1,2,...,l (2.2
0, NNNi ttxF i=l+1,l+2,..., m (2.3)
)),(()(1 kkkkkk ttxMtx , k=1,...,N-1 ( 121 ,...,, Nttt are unknowns and Nt not fixed) (2.4)
S({1
1 1
}, ) ( ( )) ( , , )k
k
tN N
k k k k k kk k t
u T x t L x u t dt
min (2.5)
Here,T=( ),,...,, 121 NN tttt , 1 2{ } ( , ,..., )k Nu u u u
In this problem, nrn
iRRRRf : ,
k ,
kM and
kF are given continuous, at least
continuously partially differentiable vector - valued functions with respect to it’s coordinates ,
RRRM n
i: and : , are given functions which satisfying
Frechet superdi fferential i
u (t): R r
iRU are controls . The set
iU , are assumed to be
nonempty and bounded
Shahlar Meherrem
Theorem1: Let 00 , kk xtu be an optimal solution to the control problem (2.1)-(2.5) under the standing assumptions
made.Then for every collectiion of Frechet supergradients kx 0ˆ 0
kk tx ,m=1,2,...N, there are multipliers the
1 2, ,..., 0, 0, 1, 2,...,l k k l and vector functions kp which
a) ttpuxHttptuxH kkkkkkkkUu kk
,,,),,,(max 000
, Nk ,...,2,1 , kk ttt ,1 , maximum condition
hold with the corresponding trajectory )(tpk
,)()K
KK x
Htpb
N
iNN
N
iN tp
x
Fx
1
0)( ,(transvers. condition)
c)Necessary conditions at the switching points
1
,( ) ( ) ,k k k k
k k k k kk
M x t tp t x p t
x
k=1,...,N-1
N
k
N
K k
kkkkkk
N
NNNk
t
ttxMtp
t
ttxF
1
1
11
,)(
,0
d) , 0, 1,2,...,k k N N NF x t t k l ,(complementarity slackness) hold.
here , , , , , , . ( , , , )T
k k k k k k k k k k k k kH x u p t L x u p t p f x u p t
Corollary(smooth version). Let 00 , xtu be an optimal process to problem (2.1)-(2.5),where
RR ni : is assumed to be differentiable at ii tx 0
.Then above conditions a)-d) satysfying with
ix itx0ˆ = ik tx 0
Shahlar Meherrem
Lemma. Let : nR R be Lipschitz continuous around x and upper regular at this point.Then
0 x x
Theorem2. Let 00 , kk xtu be an optimal process to the problem (2.1)-(2.5),where k is assumed to be
Lipschitz continuous around kk tx0 and upper regular at this point.Then for any element of Clarke
generalized gradient,the conditions a)-d) in the theorem1 satisfying
i.e. theorem1 satisfying for any 0k k kx x t
Shahlar Meherrem
In this final section of the presentation,we present nonsmooth versions of the maximum prinsiple for the above problem ,(2.1-2.5),with transversality conditions expressed in term of basic subgradient defined in the section tools of nonsmooth analysis by using metric approximations method.
Shahlar Meherrem
Theorem
Let 00 , kk xtu be an optimal process to the problem (2.1)-(2.5)
Let k be Lipschitz continuous around 0kx .Then there is 0
kx x such that
theorem 1 holds with the conditions a)-d)
It is difference between Frechet superdifferential and Mordukhovich subgradient for this problem.
Shahlar Meherrem
Theorem. Let *E is lower exhauster of a p.h. function : nh R R .Then
*C E
C
(0 )F nh
Theorem. : Let 00 , kk xtu be an optimal solution to the control problem (2.1)-(2.5) under the standing
assumptions made and the function .k is positevely homogeneous, Hadamard differentiable at the
point 0x .Then for every elements of intersection of lower exhauster, kx
*kC E
C ,k=1,2,...N, there are
multipliers the 1 2, ,..., 0, 0, 1,2,...,l k k l and vector functions kp which
a) ttpuxHttptuxH kkkkkkkkUu kk
,,,),,,(max 000
, Nk ,...,2,1 , kk ttt ,1 ,maximum
condition
hold with the corresponding trajectory )(tpk
,)()K
KK x
Htpb
N
iNN
N
iN tp
x
Fx
1
0)( ,(transvers. condition)
Exhausters in Switching Optimal Control Problem. Let : nf R R is lower semicontinuous and positevely homogeneous, Hadamard upper differentable function. Then
*( ) inf max( , )
v CC Ef g v g
for any ng R ( Demyanov and Rubinov ,ref.25).
The E * is called lower exhausters of the function f . Lemma(Demyanov and Roshchina, ref 25)
Let 0nx R ve fixed. Put 0( ) ( , )Hh g f x g Then 0( ) (0 )F F nf x h
Shahlar Meherrem
1
,( ) ( ) , 1, 2,..., 1k k k k
k k k k kk
M x t tp t x p t k N
x
1
11 1
, ,( ) 0
N Nk N N N k k k k
k kk KN k
F x t t M x t tp t
t t
c)Necessary conditions at the switching points
d) Complementarity slackness condition
, 0, 1, 2,..., ,k k N N NF x t t k l
hold.
Shahlar MeherremDepartment of Mathematics Yasar University – İZMİR
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