![Page 1: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/1.jpg)
Optimal Control
Lecture
Prof. Daniela Iacoviello
Department of Computer, Control, and Management Engineering Antonio Ruberti
Sapienza University of Rome
![Page 2: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/2.jpg)
21/11/2016 Controllo nei sistemi biologici
Lecture 1
Pagina 2
Prof. Daniela Iacoviello
Department of computer, control and management
Engineering Antonio Ruberti
Office: A219 Via Ariosto 25
http://www.dis.uniroma1.it/
Prof.Daniela Iacoviello- Optimal Control
![Page 3: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/3.jpg)
Examples taken from Bruni et al. 2005
Prof.Daniela Iacoviello- Optimal Control
![Page 4: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/4.jpg)
Example 1 (exercise 3.3)
Prof.Daniela Iacoviello- Optimal Control
Consider a vehicle of mass m=1 moving along a straight street.
Assume null initial position and velocity.
Indicate with u the force acting on the vehicle having the same
direction of the street.
The aim is to transfer the vehicle to the maximum
distance with final null velocity in a fixed time interval [0, tf],
with a fixed
energy:
ft
Kdttu0
2 0)(
![Page 5: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/5.jpg)
Prof.D.Iacoviello
Optimal Control
Indicate with:
x1(t) the position of the vehicle
x2(t) the velocity of the vehicle
Cost Index
ft
f
Kdttu
txxx
tutx
txtx
0
2
221
2
21
0)(
0)(0)0()0(
)()(
)()(
ft
f dttxtxxJ0
21 )()()(
![Page 6: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/6.jpg)
Prof.D.Iacoviello
Optimal Control
The problem is not convex since the isoperimetric constraint is not
linear
only necessary conditions
Define the Hamiltonian in the normal case:
)()()()()()(),,1,,( 22212 tututtxttxuxH
0)(
)()(20
)(1)(
0)(
)()(
)()(
1
2
12
1
2
21
ft
ttu
tt
t
tutx
txtx
![Page 7: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/7.jpg)
Prof.D.Iacoviello
Optimal Control
)()(20
0)(
)(
)(1)(
0)(
)()(
)()(
2
*
1
12
1
2
21
ttu
txt
tt
t
tutx
txtx
T
ff
u doesn’t have discontinuity points
Ctt
t
)(
0)(
2
1
can’t be zero
22
)()( 2 Ctt
tu
Substitute in the state
equation
![Page 8: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/8.jpg)
Prof.D.Iacoviello
Optimal Control
)()(
)()(
2
21
tutx
txtx
22
)()( 2 Ctt
tu
tCt
tx
tCt
tx
24)(
412)(
3
2
23
1
0)(2 ftx
2
ftC
2
3
0
2
2
0
2
2
4844
1)(
f
t
f
f
tt
dtt
tttdttuK
ff
![Page 9: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/9.jpg)
Prof.D.Iacoviello
Optimal Control
2
3
0
2
2
0
2
2
4844
1)(
f
t
f
f
tt
dtt
tttdttuK
ff
KK
tt ff
34
tt
tu
ttt
tx
ttt
tx
f
f
f
22
1)(
44)(
812)(
3
2
23
1
Kttt
txfff
f3224
)(
3
1
![Page 10: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/10.jpg)
Prof.D.Iacoviello
Optimal Control
No abnormal extremum exists.
In fact: define the Hamiltonian in the abnormal case:
)()()()()(),,1,,( 2221 tututtxtuxH
0)(
)()(20
)()(
0)(
)()(
)()(
1
2
12
1
2
21
ft
ttu
tt
t
tutx
txtx
Ct
t
)(
0)(
2
1
0C
2)(
Ctu
![Page 11: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/11.jpg)
Prof.D.Iacoviello
Optimal Control
2)(
Ctu
tC
tx2
)(2
That is not compatible with 0)(2 ftx
![Page 12: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/12.jpg)
Prof.Daniela Iacoviello- Optimal Control
Consider a vehicle of mass m=1 moving along a straight street.
Assume null initial position and velocity.
Indicate with u the force acting on the vehicle having the same
direction of the street.
Assume NOT fixed the final instant.
The aim is to perform an automatic coupling of a
second vehicle moving with a uniform move
(with velocity v) along the same
street,
assuming as cost index:
ft
f dttutuJ
0
2 )(2
1),(
Example 2- same as example 1 with a change
![Page 13: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/13.jpg)
Prof.D.Iacoviello
Optimal Control
The final instant is not fixed, therefore the problem is not convex.
)()()()()(2
1),,1,,( 221
2 tuttxttuuxH
)()(0
)()(
0)(
2
12
1
ttu
tt
t
0)(
)(),(
2
1
vtx
rvttxttx
f
ff
ff
21
212
11
)(
)(
)(
CtCtu
CtCt
Ct
![Page 14: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/14.jpg)
Prof.D.Iacoviello
Optimal Control
Integrate the state equations
with null initial conditions
vt
H
txt
T
ft
T
ff
f1
*
2
1
*
,,)(
)(
tCt
Ctx
tC
tCtx
2
2
12
2
2
3
11
2)(
26)(
From the transversality conditions and taking into account the
stationarity of the problem:
01 )( Hvt f 2212 CvC
![Page 15: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/15.jpg)
Prof.D.Iacoviello
Optimal Control
0)(
)(),(
2
1
vtx
rvttxttx
f
ff
ff
2212 CvC
vtCv
tC
rvtt
Cv
tC
f
f
f
ff
2
2
22
2
2
3
22
4
212
2
3
1
2
29
2
3
23
r
vC
r
vC
v
rt f
![Page 16: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/16.jpg)
Prof.Daniela Iacoviello- Optimal Control
Consider a vehicle of mass m=1 moving along a straight street.
Assume null initial position and velocity.
Indicate with u the force acting on the vehicle having the same
direction of the street.
Assume NOT fixed the final instant.
The aim is to perform an automatic coupling of a second
vehicle moving with a uniform move (with velocity v) along the
same street,
assuming fixed the energy for the control
and minimizing the cost index: ff ttJ )(
Example 3- same as example 2 with a change
0)(
0
2 Kdttu
ft
![Page 17: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/17.jpg)
Prof.D.Iacoviello
Optimal Control
Define the Hamiltonian in the normal case:
)()()()()(1),,1,,( 2221 tututtxtuxH
)()(20
)()(
0)(
)()(
)()(
2
12
1
2
21
ttu
tt
t
tutx
txtx
212
11
)(
)(
CtCt
Ct
)0()0(1 221 uuCvC 22
)( 21 CtCtu
![Page 18: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/18.jpg)
Prof.D.Iacoviello
Optimal Control
tCtC
tx
tCtC
tx
24)(
412)(
22
12
223
11
vtCt
v
C
v
rvttCt
v
C
v
ff
f
ff
244
1
4124
1
2
2
2222
2
22
2
3222
)0()0(1 221 uuCvC
v
C
vC
4
122
1
![Page 19: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/19.jpg)
Prof.D.Iacoviello
Optimal Control
2
2
2
22
222
2
3222
0
2
444
1
124
1
)(
fff
t
tCtC
v
C
v
t
v
C
v
dttuK
f
vtCt
v
C
v
rvttCt
v
C
v
ff
f
ff
244
1
4124
1
2
2
2222
2
22
2
3222
vtCt
v
C
v
ff
244
1 2
2
2222
![Page 20: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/20.jpg)
f
ff
t
rvttC 1282
2
2
2
22
222
2
3222
0
2
444
1
124
1
)(
fff
t
tCtC
v
C
v
t
v
C
v
dttuK
f
32232212324212 fffff Ktrvtrvtrvtrvt
![Page 21: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/21.jpg)
If 1r 85.3ft
26.0
83.1
74.0
2
1
C
C
If 1r 1ft
vtCt
v
C
v
ff
244
1 2
2
2222
f
ff
t
rvttC 1282
The two equations
don’t have solution
1231 KvrAssume
![Page 22: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/22.jpg)
Prof.D.Iacoviello
Optimal Control
Define the Hamiltonian in the abnormal case:
Substituting into the admissibility equations and the isoperimetric
Constraint:
)()()()()(),,1,,( 2221 tututtxtuxH
)0()0( 221 uuCvC v
CC
4
22
1
Ktv
Ct
Ct
v
C
vtC
tv
C
rvttC
tv
C
fff
ff
fff
2
3
32
2
223
42
42
22
2
22
223
2
22
164192
216
448
ft
vC 42
K
vt f
3
4 2
![Page 23: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/23.jpg)
Ktv
Ct
Ct
v
C
vtC
tv
C
rvttC
tv
C
fff
ff
fff
2
3
32
2
223
42
42
22
2
22
223
2
22
164192
216
448
r
vK
9
4 3
Abnormal extremal exist only if
the parameters satisfy this relation.
![Page 24: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/24.jpg)
If no solution exists 1r
If
there exist infinite
number of abnormal
solutions
1r 1ft 1221 CC
1231 KvrFor the choice
![Page 25: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/25.jpg)
Example 4
Consider a simple model of phamacodynamic:
x1= quantity of drug in the gastrintestinal section
x2= quantity of drug in the blood
r=reference value of the quantity of drug in the blood
u1 flow rate of the drug by oral administration
u2 flow rate of the drug by administered intravenously
Assume
![Page 26: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/26.jpg)
Minimize:
* * *
Solution of the state equation:
![Page 27: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/27.jpg)
Obviously:
x2=x’2 +x’’2
intravenous Gastrintestinal
![Page 28: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/28.jpg)
Since:
Define:
It can be defined the following tracking problem:
0,0,
0,00,
''2
''2
QTTtQtx
QTtQtx
It could be verified that:
![Page 29: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/29.jpg)
The problem is convex and the Lagrangian is strictly convex
necessary and sufficient conditions and,
if the solution exists, it is unique
By using the theory of optimal tracking:
![Page 30: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/30.jpg)
where:
![Page 31: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/31.jpg)
where
If the admissibility condition is satisfied then the couple
Is the unique optimal solution of the subproblem
0)(2 tuo
![Page 32: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/32.jpg)
To determine Q>0 it is possible to solve another
quadratic problem:
where are suitable affine functions depending
on the Problem; the cost index is strictly convex:
If a solution exists
it is unique and is given by
the condition:
![Page 33: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/33.jpg)
Exercise
Consider the system:
The aim is to have:
with a bounded control
minimizing
)()( tutx 0)0( x
1,1)( ff ttx 1,0)( tu
ft
f dttxtutuJ0
)()(3,
The final instant is not fixed the problem is not convex
The Hamiltonian is:
)()()()(3)(,1),(),( tuttxtuttutxH
![Page 34: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/34.jpg)
Ctt )(*The costate equation yields
The minimum condition is:
The problem may be analyzed for four different cases:
and the transverality conditions will be particularized:
1,0)(1)()(1)( **** ttttu
Ctsigntu 112
1)(*
![Page 35: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/35.jpg)
1) . Three alternatives are possible:
1Ca) . From Ctsigntu 11
2
1)(*
01 Ct
0)(* tu 0)(* tx Not possible
b) . There exists a discontinuity instant
for the control with
1,2 C 1,0t
Ct 1
![Page 36: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/36.jpg)
2C
Not possible
c) . 2C 1,0,01 tforCt
1)(* tu ttx )(*acceptable
In case 1) we have obtained one normal extremum:
1)(* tu ttx )(*1ft
2,)( CCtt
![Page 37: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/37.jpg)
2) . The transversality condition yields:
1C
01)1(* C
0)(* tu 0)(* tx Not possible
3) . The transversality condition yields:
0** ft
H
1)(,1 ff txt
From the stationarity of the problem:
4C 312
1)(* tsigntu
![Page 38: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/38.jpg)
AS far as the value of is concerned two alternatives are possible:
i) . From
*ft
3,1* ft 312
1)(* tsigntu
ttx )(*
Not possible
1)(* tu
1)( *** ff ttx
ii) 3* ft
with 3)( * ftx Not possible
![Page 39: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/39.jpg)
4) . . The transversality condition yields:
0)( *** Ctt ff
1)(,1 ff txt
0)(3 **** ft
txHf
From 312
1)(* tsigntu
There exists an instant of discontinuity
for the control such that:
** ,01 ff ttt
![Page 40: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/40.jpg)
From the transversality condition: 13)( ** ff ttx
With: 1)( tt
In case 4) we have obtained one normal extremum:
The value of the cost index evaluated at point of case 4) is bigger than the
value of the cost index evaluated at the solution of point found in case 1).
1)(* tu ttx )(*1ft
2,)( CCttThe normal extremum is:
![Page 41: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/41.jpg)
As far as the abnormal solution is concerned, all the solutions with
must be rejected .
For
0)( ** ft
1)( * ftx 0)(* Ct
0)( * ftxNot acceptable:
1ftNot possible with the
condition
1ftIf one obtains
with:
1)(* tu ttx )(*1ft
0,)( CCt
![Page 42: Linear Optimal control - uniroma1.itiacoviel/Materiale/Lezione-applicazione_201… · Prof.Daniela Iacoviello- Optimal Control . Examples taken from Bruni et al. 2005 Prof.Daniela](https://reader030.vdocuments.net/reader030/viewer/2022041006/5eac2b43971c7d48d25c6f13/html5/thumbnails/42.jpg)
1)(* tu ttx )(*1ft
The unique possible solution is the extremum
That is both a normal and abnormal solution.