AACIMP 2010 Summer School lecture by Viktor Ivanenko. "Applied Mathematics" stream. "On the Models of Uncertainty in Decision and Control Problems" course. Part 1. More info at http://summerschool.ssa.org.ua

1. Uncertainty Problem in Control &
Decision Theory
I. Introduction
Control Theory: 1. Dynamics; 2. Uncertainty
Decision Theory: 1. Uncertainty; 2. Dynamics
Output
Controller Plant
Control System
Consiquences
Decision Decision
Maker Situation
Decision System

2. Two examples DP
Example 1. Action (Decision)
L H, N
S B, N L H, B, G, N
R G, N S H, B, G, N
R H, B, G, N
Student 1: B>H, B>G, H>G
Student 2: G>B, G>H, H>B
Uncertainty Problem in Control &
Decision Theory

3. Two examples DP
Example 2. Experiment (Observation)
Experiment Observation
H E S
H C11 C12 C13
E C21 C22 C23
S C31 C32 C33
C11> C12, C11> C13
C22> C21, C22> C23
C33> C31, C33> C32
Example 1 є Class of non-parametric DSituations.
Example 2 є Class of parametric DSituations.
Uncertainty Problem in Control &
Decision Theory

4. II. Mathematic Model of DSituation.
U - the set of decisions u U ; ( A)
C - the set of consequences c C
( ) - the multivalued mapping
:U C, (u) C, u U
- the set of parameters
g ( , ) - some mapping g :( U) C
U , C, ( ) ,
Л U , C , Cu C , u U ;
Л - Lottery Scheme (non-parametric DS)
M ( , U , C, g ( , )), g ( , u) C, ,u U
М - Matrix Scheme (parametric DS)
Uncertainty Problem in Control &
Decision Theory

5. Л М !?????
1) 2)
М Л, ,U , C , g ( , ) ; Л М ; (U , C , ( ));
( М ) Л
(u) g ( , u) : , u U; C : (u) (u), u U
: ZМ ( , , , ) ZЛ ( , , , ) g ( , u) (u), , u U
М Л
: Л М
( Л ) Л
T1. Class of DS whose schemes representable in matrix form coinsides
with the class of DS whose schemes representable in a lottery form.
(V.I. Ivanenko, B. Munier, 2000)
(V.I. Ivanenko, V. Mikhalevich, 2007)
Uncertainty Problem in Control &
Decision Theory

6. Uncertainty, necessary condition ( ) but not
sufficient!
P in Z М MМ ( Z М , P)
Data on the uncertainty Q in Z Л MЛ ( Z Л , Q)
Q Qu , u U
Strict
certainty
Strict
uncertainty
Stochastic
uncertainty
Uncertainty Problem in Control &
Decision Theory

7. For Stochastic Uncertainty:
MЛ U , (C , ), u ,u U MЛ MМ
MМ ( , , ),U , (C, ), g ( , ) MМ MЛ
(M Л ) МЛ
T.2. Class of DS whose mathematic models representable in matrix
form coinsides with the class of DS whose mathematic models
representable in a lottery form.
(V. Ivanenko, V. Mikhalevich, 2007)
M DS { M , P} M ,
L( , ) - utility function, L( , u ) R1 C
M ( ,U , L, P).
P - some regularity of uncertainty
Uncertainty Problem in Control &
Decision Theory

8. III. Mathematic Model of Decision Maker
~
1. c C C c - the Binary Relation on C
o
2. First Optimization Problem c C
~
3. u U U
4. Second Optimization Problem uo U
№3 is the essence of Decision Making under Uncertainty.
Uncertainty Problem in Control &
Decision Theory

9. Strict certainty:
g :U C; co C
1
g :C U ; co uo U
Strict uncertainty:
The choice of u is not unique!
M - the Set of DS
~
C U
- Projector or Criterion Choice Rule (CCR)
- the set of all possible projectors
~ ~
C, U,
Uncertainty Problem in Control &
Decision Theory

10. IV. General Decision Problem
Definition. CCR is any mapping ( ) Z
define on , ,
*
and associate to any Z some real function LZ ( ) define on U .
Class of all CCR denote by o( ) ( ) all CCR’s that
satisfied to the next three conditions:
C1. If Zi ( ,U i , Li ) ( ) (i 1,2), U1 U 2 , L1 ( , u ) L2 ( , u ) at all
u U1 , , then L* (u ) L* (u ), at all u U1.
Z1 Z2
C2. If Z ( ,U , L) ( ) u1, u2 U , then from the inequality L( , u1 ) L( , u2 )
*
at all , follows LZ (u1 ) L* (u2 ), and from a, b R, a 0,
Z
*
L( , u1 ) a L( , u2 ) b at all , follows Lz (u1 ) a L* (u2 ) b.
z
C3. If Z ( ), u1 , u 2 , u3 U and L( , u1 ) L( , u2 ) 2 L( , u3 ),
then L* (u1 ) L* (u2 )
z z 2 L* (u3 ).
Z
Uncertainty Problem in Control &
Decision Theory

11. E.Borel A.Kolmogorov
Random
60 in Broad Sense Random Stochastic
Events Events
30
XX
Statistically frequency unstable events.
0011000000 111111 000000000000 }f (k) the set of points
f(1)=f(0)=1/2 f(1)=1/4 f(0)=3/4
V. Ivanenko, B. Munier, I.Zorich (2000)
Uncertainty Problem in Control &
Decision Theory

12. PF ( ) { p (2 [0,1]) : p( ) 1,
p( A B) p( A) p( B A) A, B }
p( ) - the set of all closed (in some topology) subsets p PF ( ) - the
regularities of uncertainty.
: p( ) ( ) Z
if p P( ), Z ( p), Z ( )
( Z ) L* ( ), then p
Z
P
L* ( ) sup L( , u ) p(d )
Z u U
p P
T.3. p( ) o( ) ( Z , P) S
General DP
(V. Ivanenko, V. Labkovsky, 1986,2005)
L* ( )
Z
(V. Ivanenko, B. Munier, 2000)
Uncertainty Problem in Control &
Decision Theory