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