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Optimal Control
Prof. Daniela Iacoviello
Lecture 4
31/10/2018 Pagina 2Prof.Daniela Iacoviello- Optimal Control
THESE SLIDES ARE NOT SUFFICIENT
FOR THE EXAM:YOU MUST STUDY ON THE BOOKS
Part of the slides has been taken from the References indicated below
Pagina 231/10/2018
31/10/2018 Pagina 3Prof.Daniela Iacoviello- Optimal Control
Course outline
• Introduction to optimal control
• Nonlinear optimization
• Dynamic programming
• Calculus of variations
• Calculus of variations and optimal control
• LQ problem
• Minimum time problem
Pagina 331/10/2018
R.F.Hartl, S.P.Sethi. R.G.Vickson, A Survey of the
Maximum Principles for Optimal Control Problems with
State Constraints, SIAM Review, Vol.37, No.2,
pp.181-218, 1995
Prof.Daniela Iacoviello- Optimal Control Pagina 431/10/2018
Pontryagin
Lev Semenovich Pontryagin (3 September 1908 – 3
May 1988) was a Soviet Russian mathematician. He was
born in Moscow and
lost his eyesight in a stove explosion when he was 14.
Despite his blindness he was able to
become a mathematician due
to the help of his mother who read
mathematical books and papers to him.
He made major discoveries in a number of fields of
mathematics, including the geometric parts of topology.
Prof.Daniela Iacoviello- Optimal Control Pagina 531/10/2018
The Pontryagin principle
Problem 1: Consider the dynamical system:
with:
Assume fixed the initial control instant and the initial and final
values :
Define the performance index :
with
( )uxfx ,=
( ) niURCx
ffRUtuRtx n
i
pn ,...,2,1,,)(,)( 0 =
Pagina 631/10/2018Prof.Daniela Iacoviello- Optimal Control
ff
ii xtxxtx == )()(
( ) ( ) ))(()(),(,, f
t
t
f txGduxLtuxJ
f
i
+=
( ) 20 ,,...,2,1,, CGniURCx
LL n
i
=
Determine:
▪ the value ,
• the control
• the state
that satisfy:
✓ the dynamical system,
✓ the constraint on the control,
✓ the initial and final conditions
✓ and minimize the cost index
( ) ,if tt
)(0 RCuo
)(1 RCxo
Pagina 731/10/2018Prof.Daniela Iacoviello- Optimal Control
Hamiltonian function
( ) ( ) ( )uxftuxLuxH T ,)(,,,, 00 +=
Pagina 831/10/2018Prof.Daniela Iacoviello- Optimal Control
The Pontryagin principle
Theorem 1 (necessary condition):
Assume the admissible solution is a minimum
there exist a constant
and a n-dimensional vector
not simultaneously null such that :
( )*** ,, ftux
00
*1* , fi ttC
T
x
H*
*
−=
( ) ( )U
ttutxHttxH
,)(,),(),()(,,),( **0
****0
*
Pagina 931/10/2018Prof.Daniela Iacoviello- Optimal Control
0*=H
The Pontryagin principle
Remark- If is fixed, the condition
Is substituted by
Pagina 1031/10/2018Prof.Daniela Iacoviello- Optimal Control
0*=H
ft
fi tttkH ,,*
=
The Pontryagin principle
Problem 2: Consider the dynamical system:
with:
( )uxfx ,=
( ) niURCx
ffRUtuRtx n
i
pn ,...,2,1,,)(,)( 0 =
Pagina 1131/10/2018Prof.Daniela Iacoviello- Optimal Control
Assume fixed the initial control instant and the initial state
while for final values assume:
where is a function of dimension of C1 class.
Define the performance index :
with
ii xtx =)(
( ) 0)( = ftx
nf
( ) ( ) ))(()(),(,, f
t
t
f txGduxLtuxJ
f
i
+=
( ) 20 ,,...,2,1,, CGniURCx
LL n
i
=
Determine:
▪ the value
▪ the control
▪ and the state
that satisfy
✓ the dynamical system,
✓ the constraint on the control,
✓ the initial and final conditions
✓ and minimize the cost index .
( ) ,if tt
)(0 RCuo
)(1 RCxo
Pagina 1231/10/2018Prof.Daniela Iacoviello- Optimal Control
Theorem 2 (necessary condition):
Consider an admissible solution such that
If it is a minimum
there exist a constant and an n-dimensional vector
not simultaneously null such that :
Moreover there exists a vector such that:
( )*** ,, ftux
00
*1* , fi ttC
fftdx
drank =
*
)(
Pagina 1331/10/2018Prof.Daniela Iacoviello- Optimal Control
T
x
H*
*
−=
( ) ( )U
ttutxHttxH
,)(,),(),()(,,),( **0
****0
*
T
ff
tdx
dt
*
)()(
=
fR
0*=H
The Pontryagin principle
Remark- If is fixed condition
Is substituted by
Pagina 1431/10/2018Prof.Daniela Iacoviello- Optimal Control
0*=H
ft
fi tttkH ,,*
=
The Pontryagin principle
Problem 3: Consider the dynamical system:
with:
( )tuxfx ,,=
( ) niURCx
ffRUtuRtx n
i
pn ,...,2,1,,)(,)( 0 =
Pagina 1531/10/2018Prof.Daniela Iacoviello- Optimal Control
Assume fixed the initial control instant and the initial state
while for final values assume: where
is a function of dimension of C1 class.
Define the performance index :
with
ii xtx =)(
( ) 0),( = ff ttx
1+ nf
( ) ( )=f
i
t
t
f duxLtuxJ ),(),(,,
( ) niRURCt
L
x
LL n
i
,...,2,1,,, 0 =
Determine:
▪ the value
▪ the control
▪ and the state
that satisfy
the dynamical system,
the constraint on the control,
the initial and final conditions
and minimize the cost index .
( ) ,if tt
)(0 RCuo
)(1 RCxo
Pagina 1631/10/2018Prof.Daniela Iacoviello- Optimal Control
Theorem 3: Consider an admissible solution such that
IF it is a minimum there exist a constant and an
n-dimensional vector not simultaneously null
such that :
( )*** ,, ftux
00
*1* , fi ttC
( ) fff ttx
rank =
*
),(
Pagina 1731/10/2018Prof.Daniela Iacoviello- Optimal Control
,
**
T
x
H
−= Rkkd
HH
ft
t
=
+ ,
*
**
( ) ( ) UttutxHttxH ,)(,),(),()(,,),( **0
****0
*
Moreover there exists a vector such that: fR
T
ft
T
f
ft
Htx
tf
*
*
*
**
)()(
−=
=
The Pontryagin principle
Problem 4: Consider the dynamical system:
with:
and initial instant and state fixed
( )tuxfx ,,=
( )RURCt
f
x
ffRUtuRtx npn
0,,)(,)(
ii xtx =)(
Pagina 1831/10/2018Prof.Daniela Iacoviello- Optimal Control
For the final values assume: where is a
function of dimension of C1 class.
Assume the constraint with
Define the performance index :
with
( ) 0),( = ff ttx
1+ nf
( ) kduxh
f
i
t
t
= ),(),(
( ) niRURCt
h
tx
hh n ,...,2,1,,
)(, 0 =
( ) ( )=f
i
t
t
f duxLtuxJ ),(),(,,( )RURC
t
L
x
LL n
0,,
Determine
• the value
• the control
• and the state
that satisfy
✓ the dynamical system,
✓ the constraint on the control,
✓ the initial and final conditions
✓ and minimize the cost index .
( ) ,if tt
)(0 RCuo
)(1 RCxo
Pagina 1931/10/2018Prof.Daniela Iacoviello- Optimal Control
Hamiltonian function
( ) ( ) ( ) )),(),((,)(,,,, 00 ttutxhuxftuxLuxH TT ++=
Pagina 2031/10/2018Prof.Daniela Iacoviello- Optimal Control
Theorem 4 (necessary condition):
Consider an admissible solution such that
IF it is a local minimum
there exist a constant , ,
not simultaneously null such that :
( )*** ,, ftux
00 *1* , fi ttC
( ) f
ff ttxrank =
*
),(
Pagina 2131/10/2018Prof.Daniela Iacoviello- Optimal Control
,
**
T
x
H
−=
( ) ( )U
ttutxHttxH
,)(,),(),()(,,),( **
0
****
0
*
Moreover there exists a vectorsuch that:fR
T
ft
T
f tH
txT
f
*
*
*
**
)()(
−=
=
RkkdH
H
ft
t
=
+ ,
*
**
R*
The discontinuities of may occur only in the instants
in which u has a discontinuity and in these instants the
Hamiltonian is continuous
*
Pagina 2231/10/2018Prof.Daniela Iacoviello- Optimal Control
Remark
If the set U coincides with Rp the minimum
condition reduces to :
0=
u
H
Pagina 2331/10/2018Prof.Daniela Iacoviello- Optimal Control
The Pontryagin principle - convex case
Problem 5: Consider the dynamical linear system:
with A and B of function of C1 class; assume fixed the initial
and final instants and the initial state, and
Assume
where U is a convex set.
utBxtAx )()( +=
nfff Rtxorfixedxtx = )()(
fip tttRUtu ,)(
Pagina 2431/10/2018Prof.Daniela Iacoviello- Optimal Control
Define the performance index :
with
L convex function with respect to x(t), u(t) in
per ogni
G is a scalar function of C2 class and convex with
respect to x(tf).
( ) ( ) ( ))(),(),(, f
t
t
txGduxLuxJ
f
i
+=
( ) nittURCt
L
x
LL fi
n
i
,...,2,1,,,, 0 =
URn
fi ttt ,
Pagina 2531/10/2018Prof.Daniela Iacoviello- Optimal Control
Determine:
the control
and the state
that satisfy
the dynamical system,
the constraint on the control,
the initial and final conditions
and minimize the cost index .
fio ttCu ,0
fio ttCx ,1
Pagina 2631/10/2018Prof.Daniela Iacoviello- Optimal Control
Theorem 5 (necessary and sufficient condition):
Consider an admissible solution such that
It is a minimum normal (i.e.λ0 =1)
if and only if there exists an n-dimensional vector
such that :
( )oo ux ,
fio ttC ,1
( ) f
o
ff ttxrank =
),(
Pagina 2731/10/2018Prof.Daniela Iacoviello- Optimal Control
oTo
x
tuxH
−=
),,,(
( ) ( ) UttutxHttxH ooooo ,)(),(),()(,),(
nf Rtx )(
oT
f
fo
tdx
dGt
)()( =Moreover, if
Example 3 (from L.C.Evans)
Control of production and consumption
Consider a factory whose output can be controlled.
Let’s set:
x(t) the amount of output produced at time t , 0 ≤ t.
Assume we consume some fraction of the output at each time
and likewise reinvest the remaining fraction u(t) .
It is our control, subject to the constraint
0 ≤ u(t) ≤ 1
Pagina 2831/10/2018Prof.Daniela Iacoviello- Optimal Control
The corresponding dynamics are:
The positive constant k represents the growth rate of our
reinvestment. We will chose K=1.
Assume as cost index the function:
The aim is to maximize the total consumption of the
output
0)0(
0),()()(
xx
ktxtkutx
=
=
( ) −=
ft
dttxtuuJ
0
)()(1))((
Pagina 2931/10/2018Prof.Daniela Iacoviello- Optimal Control
We apply the Pontryagin Principle; the Hamiltonian is:
The necessary conditions are:
( ) ( ) xuxuuxH +−= 1,,
( )
( ) ( ) 1)()()()(max)(),(),(
)()()(
0)(1)()(1)(
10−+=
=
=−−−=
ttxtutxttutxH
txtutx
tttut
u
f
Pagina 3031/10/2018Prof.Daniela Iacoviello- Optimal Control
1,0),()()()()()()()()()()()( −+−+ txtttxttxtxtuttxtutx
1,0,1)()(1)()( −− ttttu since x(t)>0
Pagina 3131/10/2018Prof.Daniela Iacoviello- Optimal Control
=
1)(0
1)(1)(
tif
tiftu
1,0,1)()(1)()( −− ttttu
From the equation of the costate, since ,
by continuity we deduce for t<tf, t close to tf, that
thus for such values of t.
Therefore and consequently:
More precisely so long as
and this holds for:
0)( =ft
0)( =tu
1)( t
1)( −=t ttt f −=)(
1)( t
ff ttt −1
ttt f −=)(
Pagina 3231/10/2018Prof.Daniela Iacoviello- Optimal Control
For times with t near tf we have
Therefore the costate equation yields:
1− ftt 1)( =tu
( ) )(1)(1)( ttt −=−−−=
Since we have for all
and over this time interval there are no switchings
1)1( =−ft 1− ftt
1)(1
=−− tt fet
Pagina 3331/10/2018Prof.Daniela Iacoviello- Optimal Control
Therefore:
For the switching time
Homework: find the switching time
=
fs
s
tttif
ttiftu
0
01)(*
1−= fs tt
st ft
Optimal solution: we should reinvest all
the output
(and therefore consume nothing)
up to time ts and afterwards we should
consume everything
(and therefore reinvest nothing)
Bang-bang control
Pagina 3431/10/2018Prof.Daniela Iacoviello- Optimal Control
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