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Optimal Control
Prof. Daniela Iacoviello
Lecture 1
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Systems Identification and Optimal Control
Optimal control:
Period: September - December
Credits: 6
Professor: Daniela Iacoviello
System Identification:
Period: February- May
Credits: 6
Professor: Stefano Battilotti
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Systems Identification and Optimal Control
Optimal control:
Period: September - December
Credits: 6
Professor: Daniela Iacoviello
System Identification:
Period: February- May
Credits: 6
Professor: Stefano Battilotti
Pagina 326/09/2019Prof.Daniela Iacoviello- Optimal Control
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Prof. Daniela Iacoviello
Department of computer, control and management
engineering Antonio Ruberti
Office: A219 Via Ariosto 25
http://www.dis.uniroma1.it/~iacoviel
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26/09/2019 Pagina 5Prof.Daniela Iacoviello- Optimal Control Pagina 526/09/2019
Mailing list:
Send an e-mail to: [email protected]
Subject: Optimal Control 2019
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Grading
Project+ oral exam
Example of project (1-3 Students):
-Read a paper on an optimal control problem
-Study: background, motivations, model, optimal control, solution, results
-Simulations
-Conclusions
-References
You must give me, before the date of the exam (about a week):
-A .doc document
-A power point presentation
-Matlab simulation files
Oral exam: Discussion of the project AND on the topics of the lectures
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Some projects proposed
Application Of Optimal Control To Malaria: Strategies And Simulations
Performance Compare Between Lqr And Pid Control Of Dc Motor
Optimal Low-thrust Leo (Low-earth Orbit) To Geo (Geosynchronous-earth Orbit)
Circular Orbit Transfer
Controllo Ottimo Di Una Turbina Eolica A Velocità Variabile Attraverso Il Metodo
dell'inseguimento Ottimo A Regime Permanente
Optimalcontrol In Dielectrophoresis
On The Design Of P.I.D. Controllers Using Optimal Linear Regulator Theory
Rocket Railroad Car
Optimal Control Of Quadrotor Altitude Using Linear Quadratic Regulator
Optimal Control Of An Inverted Pendulum
Glucose Optimal Control System In Diabetes Treatment ………
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Optimal Control Of Shell And Tube Heat Exchanger
Optimal Control Analysis Of A Mathematical Model For Unemployment
Time optimal control of an automatic Cableway
Glucose Optimal Control System In Diabetes Treatment
Optimal Control Of Shell And Tube Heat Exchanger
Optimal Control Analysis Of A Mathematical Model For Unemployment
Time Optimal Control Of An Automatic Cableway
Optimal Control Project On Arduino Managed Module For Automatic Ventilation Of
Vehicle Interiors
Optimal Control For A Suspension Of A Quarter Car Model
………
Pagina 826/09/2019
Some projects proposed
Prof.Daniela Iacoviello- Optimal Control
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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
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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 1026/09/2019Prof.Daniela Iacoviello- Optimal Control
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References
B.D.O.Anderson, J.B.Moore, Optimal control, Prentice Hall, 1989
E.R.Pinch, Optimal control and the calculus of variations, Oxford science publications,
1993
C.Bruni, G. Di Pillo, Metodi Variazionali per il controllo ottimo, Masson , 1993
A.Calogero, Notes on optimal control theory, 2017
L. Evans, An introduction to mathematical optimal control theory, 1983
H.Kwakernaak , R.Sivan, Linear Optimal Control Systems, Wiley Interscience, 1972
D. E. Kirk, "Optimal Control Theory: An Introduction, New York, NY: Dover, 2004
D. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise
Introduction", Princeton University Press, 2011
How, Jonathan, Principles of optimal control, Spring 2008. (MIT OpenCourseWare:
Massachusetts Institute of Technology). License: Creative Commons BY-NC-SA.
……………………………………..
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References
B.D.O.Anderson, J.B.Moore, Optimal control, Prentice Hall, 1989
E.R.Pinch, Optimal control and the calculus of variations, Oxford science publications,
1993
C.Bruni, G. Di Pillo, Metodi Variazionali per il controllo ottimo, Masson , 1993
A.Calogero, Notes on optimal control theory, 2017
L. Evans, An introduction to mathematical optimal control theory, 1983
H.Kwakernaak , R.Sivan, Linear Optimal Control Systems, Wiley Interscience, 1972
D. E. Kirk, "Optimal Control Theory: An Introduction, New York, NY: Dover, 2004
D. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise
Introduction", Princeton University Press, 2011
How, Jonathan, Principles of optimal control, Spring 2008. (MIT OpenCourseWare:
Massachusetts Institute of Technology). License: Creative Commons BY-NC-SA.
……………………………………..
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Course outline
• Introduction to optimal control
• Calculus of variations
• Calculus of variations and optimal control
• The maximum principle
• The Hamilton Jacobi Bellman equation
• The linear quadratic regulator
• The minimum time problem
• The linear quadratic gaussian problem
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Course outline
• Introduction to optimal control
• Calculus of variations
• Calculus of variations and optimal control
• The maximum principle
• The Hamilton Jacobi Bellman equation
• The linear quadratic regulator
• The minimum time problem
• The linear quadratic gaussian problem
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Background
First course on linear systems
(free evolution, transition matrix, gramian matrix,…)
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Notations
nRtx )(
pRtu )(
RRRRf pn → ::
kC
State variable
Control variable
Function (function with k-th derivative continuous a.e.)
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Introduction
Optimal control is one particular branch of modern control that sets
out to provide analytical designs of a special appealing type.
The system, that is the end result of an optimal design, is supposed to
be the best possible system of a particular type
A cost index is introduced
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Introduction
Linear optimal control is a special sort of optimal control:
✓ the plant that is controlled is assumed linear
✓ the controller is constrained to be linear
Linear controllers are achieved by working with quadratic cost
indices
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Introduction
Advantages of linear optimal control
✓ Linear optimal control may be applied to nonlinear systems
✓ Nearly all linear optimal control problems have computational
solutions
✓ The computational procedures required for linear optimal design
may often be carried over to nonlinear optimal problems
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History
Pagina 2026/09/2019
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VARIATIONAL METHODS
❑ Early Greeks – Max. area/perimeter
❑ Hero of Alexandria – Equal angles of incidence /reflection
❑ Fermat - least time principle (Early 17th Century)
❑ Newton and Leibniz – Calculus (Mid 17th Century)
❑ Johann Bernoulli - brachistochrone problem (1696)
❑ Euler - calculus of variations (1744)
❑ Joseph-Louis Lagrange – Euler-Lagrange Equations (??)
❑ William Hamilton – Hamilton's Principle (1835)
❑ Raleigh-Ritz method – VA for linear eigenvalue problems (late 19th Century)
❑ Quantum Mechanics - Computational methods – (early 20th Century)
❑ Morse & Feshbach – technology of variational methods (1953)
❑ Solid State Physics, Chemistry, Engineering – (mid-late 20th Century)
❑ Personal computers – new computational power (1980’s)
❑ Technology of variational methods essentially lost (1980-2000)
❑ D. Anderson – VA for perturbations of solitons (1979)
❑ Malomed, Kaup – VA for solitary wave solutions (1994 – present)
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26/09/2019 Pagina 22Prof.Daniela Iacoviello- Optimal Control
History
In 1696 Bernoulli posed the
Brachistochrone problem to his contemporaries:
“it seems to be the first problem which explicitely
dealt with optimally controlling the path or the behaviour of a dynamical
system»
Suppose a particle of mass M moves from A to B along smooth wire
joining the two fixed points (not on the same vertical line),under the
influence of gravity.
What shape the wire should be in order that the bead, when released
from rest at A should slide to B in the minimum time?
E.R.Pinch, Optimal control and calculus of variations, Oxford Science Publications, 1993
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Example 1 (Evans 1983)
Reproductive strategies in social insects
Let us consider the model describing how social insects (for example
bees) interact:
represents the number of workers at time t
represents the number of queens
represents the fraction of colony effort
devoted to increasing work force
lenght of the season
)(tw
)(tq
)(tu
T
Motivations
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0)0(
)()()()()(
ww
twtutsbtwtw
=
+−= Evolution of the
worker population
Death rate
Known rate at which each worker
contributes to the bee economy
( )
0)0(
)()()(1)()(
twtstuctqtq
=
−+−= Evolution of the
Population
of queens
Constraint for the control:1)(0 tu
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The bees goal is to find the control that maximizes the number of
queens at time T:
The solution is a bang- bang control
( ) )()( TqtuJ =
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Example 2 (Evans 1983…and everywhere!) A moon lander
Aim: bring a spacecraft to a soft landing on the lunar surface, using
the least amount of fuel
represents the height at time t
represents the velocity =
represents the mass of spacecraft
represents thrust at time t
We assume
)(th
)(tv
)(tm
)(th
)(tu
1)(0 tu
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Motivations
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Consider the Newton’s law:
We want to minimize the amount of fuel
that is maximize the amount remaining once we have
landed
where is the first time in which
ugmthm +−=)(
0)(0)()()(
)()(
)(
)()(
−=
=
+−=
tmthtkutm
tvth
tm
tugtv
)())(( muJ =
0)(0)( == vh
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Analysis of linear control systems
Essential components of a control system
✓ The plant
✓ One or more sensors
✓ The controller
Controller Plant
Sensor
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Analysis of linear control systems
Feedback:
the actual operation of the control system is compared to the desired
operation and the input to the plant is adjusted on the basis of this
comparison.
Feedback control systems are able to operate satisfactorily despite
adverse conditions, such as disturbances and variations in plant
properties
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Definitions
Consider a function
and
denotes the Eucledian norm
A point is a local minimum of over
If
RRf n →::
nRD
•
Dx * f nRD
)()(
0
*
*
xfxf
xxsatisfyingDxallforthatsuch
−
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Definitions
Consider a function
and
denotes the Eucledian norm
A point is a strict local minimum of over
If
RRf n →::
nRD
•
Dx *f nRD
*)()(
0
*
*
xxxfxf
xxsatisfyingDxallforthatsuch
−
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Definitions
Consider a function
and
denotes the Eucledian norm
A point is a global minimum of over
If
RRf n →::
nRD
•
Dx * f nRD
)()( * xfxf
Dxallfor
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Definitions
The notions of a local/strict/global maximum are defined similarly
If a point is either a maximum or a minimum
is called an extremum
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Unconstrained optimization - first order necessary conditions
All points sufficiently near in are in
Assume and its local minimum.
Let be an arbitrary vector.
Being in the unconstrained case:
Let’s consider:
0 is a minimum of g
x
nR
DnR
1Cf *x
)(:)( * += xfg
0
*
toenoughcloseR
Dx
+
*x
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First order Taylor expansion of g around
Proof: assume
For these values of
If we restrict to have the opposite sign of
Contraddiction
0=
0)(
lim),()0(')0()(0
=++=→
ooggg
0)0(' =g
26/09/2019 Pagina 35Prof.Daniela Iacoviello- Optimal Control
Unconstrained optimization - first order necessary conditions
0)0(' g
)0(')(
0
gofor
thatsoenoughsmall
)0(')0(')0()( gggg +−
)0('g
)0(')0(')0()( gggg +− 0)0()( − gg
0)0(' =g
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δ is arbitrary
First order necessary condition for optimality
( )fofgradienttheis
fffwherexfgT
xx n
1:)()(' * =+=
0)()0(' * == xfg
0)( * = xf
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Unconstrained optimization - first order necessary conditions
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A point satisfying this condition is a stationary point*x
Unconstrained optimization - first order necessary conditions
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Assume and its local minimum. Let be
an arbitrary vector.
Second order Taylor expansion of g around
Since
nR2Cf *x
0=
0)(
lim),()0(''2
1)0(')0()(
2
2
0
22 =+++=→
oogggg
0)0(' =g
0)0('' g
Unconstrained optimization – second order conditions
26/09/2019 Pagina 38Prof.Daniela Iacoviello- Optimal Control
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Proof: suppose
For these values of
Contraddiction
0)0('' g
22 )0(''2
1)(
0
gofor
thatsoenoughsmall
0)0()( − gg
26/09/2019 Pagina 39Prof.Daniela Iacoviello- Optimal Control
Unconstrained optimization – second order conditions
0)0('' g
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By differentiating both sides with respect to
Second order necessary condition for optimality
=
+=
n
i
ix xfgi
1
* )()('
)()()0(''
)()(''
*2
1,
*
1,
*
xfxfg
xfg
T
j
n
ji
ixx
j
n
ji
ixx
ji
ji
==
+=
=
=
=
nnn
n
xxxx
xxxx
ff
ff
f
1
111
2
Hessian matrix
0)( *2 xf
26/09/2019 Pagina 40
Unconstrained optimization – second order conditions
Prof.Daniela Iacoviello- Optimal Control
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Remark:
The second order condition distinguishes minima from maxima:
At a local maximum the Hessian must be negative semidefinite
At a local minimum the Hessian must be positive semidefinite
26/09/2019 Pagina 41
Unconstrained optimization – second order conditions
Prof.Daniela Iacoviello- Optimal Control
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Unconstrained optimization- second order conditions
Let and
is a strict local minimum of f
0)( * = xf2Cf 0)( *2 xf
*x
26/09/2019 Pagina 42Prof.Daniela Iacoviello- Optimal Control
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Global minimum – Existence result
Weierstrass Theorem
Let f be a continuous function and D a compact set
there exist a global minimum of f over D
26/09/2019 Pagina 43Prof.Daniela Iacoviello- Optimal Control
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26/09/2019 Pagina 44Prof.Daniela Iacoviello- Optimal Control
nRd A vector is a feasible direction at if
for small enough
*x Ddx +*
0
Feasible directions
x*
Not feasiblefeasible
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26/09/2019 Pagina 45Prof.Daniela Iacoviello- Optimal Control
Feasible directions
If not all direction d are feasible, then the condition is no
longer necessary for optimality
If is a local minimum for every feasible
direction d
If is a local minimum is true if
which together with
implies for every
feasible direction d satisfying
0)( * = xf
*x 0)( * dxf
*x 0)0('' g 0)0(' =g
dxfg = )()0(' * dxfdg T = )()0('' *2
0)( *2 dxfd T
0)( * = dxf
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Procedure for finding a global minimum
26/09/2019 Pagina 46Prof.Daniela Iacoviello- Optimal Control
0)( * = xf
0)( * = xf
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Convex set
26/09/2019 Pagina 47Prof.Daniela Iacoviello- Optimal Control
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Convexity
26/09/2019 Pagina 48Prof.Daniela Iacoviello- Optimal Control
0)( * = xf
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Constrained optimization
Let
Equality constraints
Inequality constraints
Regularity condition:
1, CfRD n
1,:,0)( ChRRhxh p →=
1,:,0)( CgRRgxg q →
( )
a
a
x
a
qg
qpx
ghrank
dimensionwith
ofconstraintactivethearegwhere
,
a
*
+=
26/09/2019 Pagina 49Prof.Daniela Iacoviello- Optimal Control
It is a technical
assumption
needed
to rule out
degenerate situations
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Constrained optimization
Lagrangian function
If the stationary point is called normal
and we can assume
( ) )()()(,,, 00 xgxhxfxL TT ++=
*x00
10 =
26/09/2019 Pagina 50Prof.Daniela Iacoviello- Optimal Control
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From now on and therefore the Lagrangian is
If there are only equality constraints the are called
Lagrange multipliers
The inequality multipliers are called Kuhn – Tucker multipliers
i
10 =
( ) )()()(,, xgxhxfxL TT ++=
26/09/2019 Pagina 51
Constrained optimization
Prof.Daniela Iacoviello- Optimal Control
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First order necessary conditions for constrained optimality:
Let and
The necessary conditions for to be a constrained local minimum are
If the functions f and g are convex and the functions h are linear these
conditions are necessary and sufficient
i
ixg
x
L
i
ii
T
T
x
=
=
0
,0)(
0
*
*
Dx * 1,, Cghf
*x
26/09/2019 Pagina 52Prof.Daniela Iacoviello- Optimal Control
Constrained optimization
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First order necessary conditions for constrained optimality:
Let and
The necessary conditions for to be a constrained local minimum are
Dx *
*x
26/09/2019 Pagina 53Prof.Daniela Iacoviello- Optimal Control
Constrained optimization- a simple problem
Proof (sketch):
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26/09/2019 Pagina 54Prof.Daniela Iacoviello- Optimal Control
Constrained optimization- a simple problem
Let us consider the function:
Let us now characterize the tangent space
Note that 0 is a minimum of
Consider the first order Taylor expansion:
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26/09/2019 Pagina 55Prof.Daniela Iacoviello- Optimal Control
Constrained optimization- a simple problem
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Consider the admissible set
Minimize the objective function
26/09/2019 Pagina 56Prof.Daniela Iacoviello- Optimal Control
Example from Bruni et al. 2005
Necessary conditions:
Lagrangian function:
✓ The existence theorem of Weierstrass can’t be applied
✓ g is not convex
only necessary conditions can be applied
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26/09/2019 Pagina 57Prof.Daniela Iacoviello- Optimal Control
Since:
The point P is the only possible candidate.
It can be shown that IT IS the optimal solution
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Second order sufficient conditions for constrained optimality:
Let and and assume the conditions
is a strict constrained local minimum if
ixgx
Liii
T
T
x
==
0,0)(0 *
*
Dx * 2,, Cghf
*x
pidx
xdhthatsuch
x
L
x
i
x
T ,...,1,0)(
0**
2
2
==
26/09/2019 Pagina 58Prof.Daniela Iacoviello- Optimal Control
Constrained optimization
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Definition:
26/09/2019 Pagina 59Prof.Daniela Iacoviello- Optimal Control
Constrained optimization
2
2
0
j
i
Tj
i
hL
xx
h
x
Bordered Hessian
A point in which and is called
a non-degenerate critical point of the constrained problem.
*x 0L =
2
2
det 0
0
j
i
Tj
i
hL
xx
h
x
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Theorem:
26/09/2019 Pagina 60Prof.Daniela Iacoviello- Optimal Control
Constrained optimization
A necessary and sufficient condition for the non-degenerate critical point
to minimize the cost subjects to the constraints
is that for all non-zero tangent vector
E.R.Pinch, Optimal control and calculus of variations, Oxford Science Publications, 1993
*x
f *( ) 0, 1,2,...,jh x j m= =
2
20T L
x
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Let us consider the problem of minimizing function
with one constraint
The previous theorem implies that we should look for and
such that:
If the normal to the constraint curve
at has its
normal parallel to
the level surface of that passes through has the
same normal direction as the constraint curve at
In R2 this means that the two curves touch at
26/09/2019 Pagina 61Prof.Daniela Iacoviello- Optimal Control
A geometrical interpretation1 2( , )f x x
1 2( , )h x x c=
1 2( , )x x x=
( ) 0f h at x + = f h at x = −
0f 0 0h
1 2( , )h x x c= 1 2( , )x x x=
( )f x
1 2( , )f x x1 2( , )x x x=
1 2( , )x x x=
1 2( , )x x x=
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Minimize the function
subject to
Introduce a Lagrange multiplier and consider:
There are three unknowns and we need another equation:
and
26/09/2019 Pagina 62Prof.Daniela Iacoviello- Optimal Control
A geometrical interpretation - Example
2 21 2 1 2( , ) 1f x x x x= − −
21 2 2 1( , ) 1 0h x x x x= − + =
( )2 2 21 2 2 1( ) 1 ( 1 ) 0f h x x x x + = − − + − + =
1 1 22 0 2 0x x x − + = − + =
22 11 0x x− + =
1 20, 1, 2x x = = = 1 21 , 1/ 2, 1
2x x = = =
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26/09/2019 Pagina 63Prof.Daniela Iacoviello- Optimal Control
A geometrical interpretation - Example
1 20, 1, 2x x = = =
1 21 , 1/ 2, 1
2x x = = =
The minimum is at
You can find it evaluating
in the two points
Note that the level curves of f
are centered in the origin O
and f decreases
as you move away from the
origin.
1 20, 1x x= =
2 21 2 1 2( , ) 1f x x x x= − −
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RVJ →:
VA
26/09/2019 Pagina 64Prof.Daniela Iacoviello- Optimal Control
•
Functional
Vector space V,
We need to equipe the function space V with a norm :
it is a real valued function on V
- positive definite
- homogeneous
- satisfies the triangle inequality
DISTANCE or METRIC
00 yify
VyRallforyy = ,
zyzy ++
zyzyd −=),(
Function spaces
RVJ →:
VA
•
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26/09/2019 Pagina 65Prof.Daniela Iacoviello- Optimal Control
)(max0
xyybxa
=On the space a common norm is
where is the Eucledian norm
Function spaces
)],,([0 nRbaC
•
)],,([1 nRbaC
)('max)(max1
xyxyybxabxa
+=
On the space a common norm is
and so on.....
The k-norm can be used on klRbaC nl ),],,([
pb
a
p
pdxxyy
/1
)(
=
0-norm
1-norm
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Function spaces
26/09/2019 Pagina 66Prof.Daniela Iacoviello- Optimal Control
Extrema of J with respect to the 0- norm are called strong extrema
Extrema of J with respect to the 1- norm are called weak extrema
If a curve is a strong minimum then it is a weak one.The converse is not true
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Function spaces
Functional
Vector space V,
is a local minimum of J over A if there exists an
RVJ →:
VA
Az *
)()(
0
*
*
zJzJ
zzsatisfyingAzallforthatsuch
−
26/09/2019 Pagina 67Prof.Daniela Iacoviello- Optimal Control
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• RVz + ,,
RVJz
→:)()()()( oJzJzJ z ++=+
0* =z
J
26/09/2019 Pagina 68Prof.Daniela Iacoviello- Optimal Control
Function spaces
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• RVz + ,,
RVJz
→:)()()()( oJzJzJ z ++=+
26/09/2019 Pagina 69Prof.Daniela Iacoviello- Optimal Control
Function spaces
(see the proof of the necessary condition for optimality of a function)
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• RVz + ,,
RVJz
→:)()()()( oJzJzJ z ++=+
26/09/2019 Pagina 70Prof.Daniela Iacoviello- Optimal Control
Function spaces
(see the proof of the necessary condition for optimality of a function)
Question: how can we compute the first variation of a functional?
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• RVJz
→:2
and
)()()()()( 222 oJJzJzJ zz +++=+
0)(*
2 z
J
Az *VA
Function spaces
26/09/2019 Pagina 71Prof.Daniela Iacoviello- Optimal Control
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The Weierstrass Theorem is still valid
If J is a convex functional and is a convex set a local minimum is
automatically a global one and the first order condition are necessary
and sufficient condition for a minimum
VA
Function spaces
26/09/2019 Pagina 72Prof.Daniela Iacoviello- Optimal Control
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•VA
Function spaces
26/09/2019 Pagina 73Prof.Daniela Iacoviello- Optimal Control