Thạc Sĩ Ontheoptimalcontrol Problemforfuzzy differential systems

ii
Contents
Contents . ii
Preface iv
Author's heartfelt thanks xii
Notation . xiii
Chapter 1 : PRELIMINARIES 1
1.1 METRIC SPACE (K c (R n ), D), SET-VALUED MAPPING . 1
1.1.1 Metric space (K c (R n ), D) . 1
1.1.2 Set valued mapping 4
1.2 METRIC SPACE (E n , D 0 ) AND FUZZY MAPPING 7
1.2.1 Metric space (E n , D 0 ) . 7
1.2.2 Fuzzy mapping . 10
1.3 METRIC SPACE (L m
2
, ρ) AND FUZZY STOCHASTIC PROCESSES . 13
1.3.1 Metric space L m
2
13
1.3.2 Fuzzy stochastic processes . 14
1.4 SOME KNOWN RESULTS 18
1.4.1 A result on integrable functions and Gronwall inequality 18
1.4.2 Existence of solutions of differential systems . 18
1.4.3 Existence of solutions of differential systems related to fuzziness
and randomness . 20
Chapter 2 : CONTROL DIFFERENTIAL SYSTEMS OF FUZZY
TYPE 23
2.1 CLASSICAL CONTROL DIFFERENTIAL SYSTEMS . 23
2.1.1 A summary of classical control differential systems . 23
2.1.2 The sheaf-solutions . 24
2.2 SOME CONTROL DIFFERENTIAL SYSTEMS RELATED TO FUZZI-
NESS 24
2.2.1 Mamdani models 25
2.2.2 Takagi-Sugeno models . 25
2.2.3 Control differential systems of fuzzy type 26
Chapter 3 : SET CONTROL DIFFERENTIAL SYSTEMS 34
3.1 EXISTENCE OF SOLUTIONS 34
3.1.1 Set control differential systems 34
3.1.2 Existence of solutions . 35
3.2 COMPARISON OF SOLUTIONS . 36
3.2.1 Using maximal solutions of a scalar differential equations . 36
3.2.2 Solutions depend on initial conditions and controls . 39
3.3 COMPARISON OF SHEAF-SOLUTIONS 41
3.3.1 Sheaf-solutions . 41
3.3.2 Comparison and estimation 42
3.4 STABILITY OF SOLUTIONS AND SHEAF-SOLUTIONS . 44
3.4.1 Stability of solutions of SDS . 443.4.2 Stability of solutions of SCDS . 45
Chapter 4 : FUZZY CONTROL DIFFERENTIAL SYSTEMS
AND SOME OPTIMAL FUZZY CONTROL PROBLEMS 51
4.1 QUASI-CONTROLLABILITY AND OBSERVABILITY OF FCDS 51
4.1.1 Quasi-controllability of FCDS . 51
4.1.2 Observability of FCDS . 53
4.2 EXISTENCE OF SOLUTIONS 54
4.2.1 Fuzzy control differential systems 54
4.2.2 Existence of solutions . 57
4.3 COMPARISON OF SOLUTIONS . 57
4.3.1 Using maximal solutions of a scalar differential equations . 57
4.3.2 Solutions depend on initial conditions and controls . 58
4.4 COMPARISON OF SHEAF-SOLUTIONS 60
4.4.1 Sheaf-solutions . 60
4.4.2 Comparison and estimation 60
4.5 STABILITY OF SOLUTIONS AND SHEAF-SOLUTIONS . 62
4.5.1 Stability of solutions of FDS . 62
4.5.2 Stability of solutions of FCDS 63
4.6 SOME OPTIMAL FUZZY CONTROL PROBLEMS 64
4.6.1 Optimal fuzzy control of a poisoning -pest model 64
4.6.2 Optimal fuzzy control of a population of growth model of a species 66
4.6.3 Optimal fuzzy control of the spread of technological innovation 67
Chapter 5 : FUZZY CONTROL STOCHASTIC DIFFER-
ENTIAL SYSTEMS 70
5.1 EXISTENCE OF SOLUTIONS 70
5.1.1 Fuzzy control stochastic differential systems . 70
5.1.2 Existence of solutions . 71
5.2 COMPARISON OF SOLUTIONS . 74
5.2.1 Using maximal solutions of a scalar differential equations . 74
5.2.2 Solutions depend on initials and controls . 75
5.3 COMPARISON OF SHEAF-SOLUTIONS 77
5.3.1 Sheaf-solutions . 77
5.3.2 Comparison and estimation 78
THE MAIN RESULTS OF THE THESIS AND OPEN PROBLEMS 80
THE AUTHOR'S PAPERS RELATED TO THE THESIS 81
REFERENCES 83PREFACE
Many real world phenomena can be described by mathematical models. The con-
trol systems governed by the differential equations (the classical control problems) have
attracted much attention since 1950’s. It is known that in [2], [4], [70]-[74] and [83], the
optimal control problems were researched by Pontryagin and his colleagues in 1956 and
Bellman in 1957, the controllability of linear control systems was studied by Kalman
in 1960 and the stability was investigated in 1960’s by many authors. The control and
optimal control problems have been applied effectively in many fields: economy, tech-
nology, environment . of the real world. We use mathematical models to describe the
real world phenomena. However, an exact description of any real world phenomena
is virtually impossible and we need to accept this fact and adjust to it. To specify
imprecise or vague notions, Zadeh introduced the concept of fuzzy set theory in 1965
[113]. Many results on fuzzy logic, fuzzy control, fuzzy decision making have been
applied in effectively in many fields: economy, technology . Some models of fuzzy
controls such as Mamdani models, Takagi-Sugeno models, have been applied and ones
use fuzzy logic to deal with these models. And, many fields of fuzzy mathematics
such as fuzzy topology, fuzzy norm spaces, fuzzy differential systems, fuzzy program-
ming . have been studied. Some fields as fuzzy differential systems, fuzzy stochastic
differential systems (which use Ito derivative, Ito integral), fuzzy stochastic differential
systems in the sense of Feng [22], set differential systems, have been developed. In
last decades, ones have generalized set, fuzzy, stochastic differential systems from the
ordinary differential equations. Some known differential systems related to fuzziness
and randomness, are following.
1. The set differential systems (SDS) are of the form
D H X = F(t, X), X(t 0 ) = X 0 ∈ K c (R n ), (0.1)
where F : R + × K c (R n ) → K c (R n ), t ∈ [t 0 , T], X ∈ K c (R n ), where K c (R n ) is thecollection of all nonempty, compact, convex subsets of R n .
Brandao Lopes Pinto, De Blasi, and Iervolino first formulated the SDS. They gave
some preliminary resuls of existence, unique and extremal solutions of SDS [51]. Re-
cently, the study of SDS in a semilinear metric space has gained much attention. Laksh-
mikantham, Leela and Vatsala studied the comparison of solutions and local existence
of solutions of SDS [55]. The existence of Euler solutions and flow invariance were
given by Gnana Bhaskar and Lakshmikantham [30]. Gnana Bhaskar, Lakshmikantham
and Vasundhara Devi studied nonlinear variation of parameters formula for SDS in a
metric space [32] . The stability criteria as in the original Lyapunov results for ordi-
nary differential equations and the boundedness were obtained by Gnana Bhaskar and
Vasundhara Devi [33, 34]. Set differential systems were first introduced by Brandao
Lopes Pinto, De Blasi, and Iervolino in 1970, but the main results have been obtained
since 2003 [30], [32]-[34], [51] and [55].
2. The fuzzy differential systems (FDS) are of the form
D H x = f(t, x(t)), x(t 0 ) = x 0 ∈ E n , (0.2)
where f : I = [t 0 , T] × E n → E n , and t ∈ I, x(t) ∈ E n , where E n is set of all fuzzy sets
on R n .
The FDS, firstly, were introduced in 1978 in a conference in Japan. Since the work
of Kaleva [39], FDS have attracted much attention. The existence result on solutions
introduced by Kaleva [39, 40] and Nieto [63]. The local existence paralleled to Peano’s
theorem was presented in [40, 41]. The global existence was given by Lakshmikantham
and Mohapatra in [58]. The FDS with the fuzzy initials was studied by Seikkala
[91]. The comparison results on solutions were presented by Lakshmikantham and
Mohapatra in [58]. Friedman, Ma and Kandel studied numerical solutions of FDS
[27]. Hight order FDS were studied by Buckly, Feuring, Nieto . [7], [8], [29] and
[115]. The relation between set and fuzzy differential systems was investigated byLakshmikantham, Gnana Bhaska .in [31], [50], [55], [56] and [59]. Many authors have
studied and got important results in FDS [1], [5], [38], [52], [54], [64]-[69], [92]-[94],
[102]-[104] and [114].
3. The fuzzy stochastic differential systems (FSDS) are of the form
X 0 (t) = F (t, X(t)), t ∈ I = [t 0 , T] ⊂ R + , (0.3)
with the initial value X(t 0 ) = X 0 ∈ L m
2
, where L m
2
is metric space of all m-dimensional
fuzzy random vectors.
Feng, first introduced a kind of fuzzy stochastic differential systems (FSDS) in 2000
[22] and gave simple existence and comparison results on the solutions. In the case
m = 1, (0.3) is a fuzzy stochastic differential equation. In [22], Feng obtained an ex-
istence result on solutions, a simple comparison result on solutions of FSDS and some
properties of the linear fuzzy stochastic differential systems were given in [22], [24]. In
[98], we provided some more comparison results on solutions and on sheaf-solutions of
FSDS. The stability of solutions and sheaf-solutions of FSDS was presented in [98].
Besides, recently, some concepts of differential systems related to fizziness and ran-
domness such as stochastic differential systems of Ito type [14], in sense of Fei [18],
fuzzy stochastic differential systems in Ito type [44], have been developed.
The real world phenomena are copious. In practice, we are often faced with random
experiments whose outcomes are not exact but are expressed in inexact linguistic vari-
ables of the time parameter. The fuzzy random variables and fuzzy stochastic processes
are the combination of randomness and fuzziness. Recently, research on the combina-
tion of randomness and fuzziness has attracted much attention of common scientists in
both theory and application. Fuzzy random variables have been investigated by many
authors, for example, Hop [35], [36], Kwakernaak [47], [48], Puri and Ralescu [84],
[85]. Many models of this combination have been applied effectively in the real life.In [35], Hop presented a model to measure the superiority and inferiority of fuzzy and
fuzzy stochastic variables. Then these new measures were used to convert the fuzzy
(stochastic) linear programming into the corresponding deterministic linear program.
In [36], a model to measure attainment values of fuzzy stochastic variables and then
these measures were used to convert the fuzzy (stochastic) linear programming into
the corresponding deterministic linear program as the case in [35]. In [61], Luhandjula
surveyed the fuzzy stochastic linear programming and its future research directions. In
[10], the optimal tracking design for stochastic fuzzy systems was studied. The stop-
ping problems have been studied by Yoshida and others authors such as: a stopping
game in a stochastic and fuzzy environment [107], a multiobjective fuzzy stopping in
a stochastic and fuzzy environment [111], fuzzy stopping problems in continuous-time
fuzzy stochastic systems [112], optimal stopping problems in a stochastic and fuzzy
system [110], the optimal stopping models in a stochastic and fuzzy environment [109],
the multicriteria optimal stopping of a stochastic and fuzzy system [108].
The aim of the thesis
The study of fields related to fuzziness and randomness is of promise, especially in
control theory. Motivated by the attraction of the fuzzy and random fields in theory
and applications, we study on control differential systems related to fuzziness and
randomness. In this thesis, based on the results on set, fuzzy, stochastic differential
systems (0.1)-(0.3), we study control systems of fuzzy type, set, fuzzy and stochastic
differential systems (0.4)-(0.7). The linear fuzzy control differential systems have been
investigated by some authors and in this thesis we provide the general form of fuzzy
control differential systems. We introduce some new trends in control theory such as:
set control differential systems, fuzzy control stochastic differential systems. The aim
of the thesis is to research on the existence, comparison and stability of solutions and
sheaf-solutions of these control differential systems. We go into the details.a. Control differential systems of fuzzy type are of the form
¦
x(t) = f(t, x(t), u(t)), (0.4)
where x(0) = x 0 ∈ R n , x(t) ∈ R n , u(t) ∈ [0, 1] × [0, 1] × . × [0, 1] ⊂ R p and we consider
u ∈ E 1 × . × E 1 = (E 1 ) p (p times) , t ∈ [0, T] = I ⊂ R + and f : I × R n × R p → R n .
These are hybrid systems whose state x(t) and function f are crisp, but control u is
fuzzy. We study on comparison and estimation results of solutions and sheaf-solutions
of these control systems.
b. Set control differential systems are of the form
D H X = F(t, X(t), U(t)), X(t 0 ) = X 0 , (0.5)
where F : I × K c (R n ) × K c (R p ) → K c (R n ), t ∈ I ⊂ R + , state X(t) ∈ K c (R n ) and
control U(t) ∈ K c (R p ).
In these control differential systems, states X(t) and controls U(t) are sets and f are
set valued mappings. Set control differential systems are the general forms of classical
control systems and firstly introduced in our papers [75], [79]-[81]. We investigate the
existence, comparison and stability of solutions and sheaf-solutions of these control
differential systems.
c. Fuzzy control differential systems are of the form
D H x = f(t, x(t), u(t)), x(t 0 ) = x 0 ∈ E n , (0.6)
where f : R + × E n × E p → E n , t ∈ R + , states x(t) ∈ E n and fuzzy controls u(t) ∈ E p .
Here, states x(t) and controls u(t) are fuzzy sets and f are fuzzy mappings. Some
results concerned linear fuzzy control differential systems (4.1) such as observability,
quasi-controllability have been researched in [12], [13], [25]. We provide the general
form fuzzy control differential systems (0.6) which are general form of linear fuzzy
control differential systems [12], [13], [25] and study the existence, comparison and
stability of solutions and sheaf-solutions of these control differential systems.d. Fuzzy control stochastic differential systems are of the form
X 0 (t) = F (t, X(t), U(t)), t ∈ I = [t 0 , T] ⊂ R + , X(t 0 ) = X 0 ∈ L m
2
(0.7)
where F is a mapping: F : I × L m
2
× L
p
2
→ L m
2
, states X(t) and controls U(t) are
m-dimensional fuzzy random vectors. These are systems of combination of fuzziness
and randomness and they were firstly introduced in our paper [97].
Up to now, fuzzy mathematics has been developed and it is not complete. Some
concepts such as partial derivatives of fuzzy mappings are not defined. So, a nonlinear
fuzzy differential system can not be approximated by a linear fuzzy differential system
as in classical ones. Because of this difficulty, many parts of control differential systems
related to fuzziness and randomness, are not fully investigated. In control theory, there
are some parts such as: observability, controllability, stability and optimal control
problems. However, set, fuzzy and fuzzy control stochastic differential systems have
been recently investigated in our papers [75], [78]- [82] and [97], some parts in these
new trends (of control theory) have not been fully studied. The study on control theory
related to fuzziness and randomness is of promise and needs more time.
We now give a description of the contents.
The thesis consists of five chapters
The Chapter 1 is devoted to some basic concepts and notations related to set,
fuzzy and stochastic fields. We recall concepts and some properties of the space of all
nonempty, compact, convex subsets of R n , set-valued mappings. Fuzzy sets, the spaces
of all fuzzy sets, fuzzy-valued mappings, fuzzy random variables, fuzzy random valued
mappings are shortly introduced. At last, we introduce some known results which will
be used in the proofs of some theorems in this thesis.
In Chapter 2, control differential systems whose variables, functions are not fuzzy,
but controls are fuzzy, are called control systems of fuzzy type. Our main results in this
chapter are comparisons and estimations of solutions and of sheaf-solutions in controlsystems of fuzzy type.
In Chapter 3, we introduce new concept of set control differential systems and study
their existence of solutions, the dependence of solutions and sheaf-solutions on initials,
right hand sides and controls. Some estimations and the stability of solutions and
sheaf-solutions are also studied.
In Chapter 4, firstly, we recall some results on observability, quasi-controllability
presented in [12], [13] and [25], then we provide the general form of fuzzy control
differential systems. We study the existence of solutions and comparison results on
solutions and on sheaf-solutions of these general control systems. The stability of
solutions and sheaf-solutions of these general control systems and some simple optimal
control problems related to fuzziness are studied.
In Chapter 5, we introduce the new concept of fuzzy control stochastic differential
systems. We investigate the existence and comparison results on solutions and on
sheaf-solutions of these fuzzy control stochastic differential systems. Some estimations
of solutions and sheaf-solutions of these fuzzy control stochastic differential systems
are provided.
Notice that, for the thesis is not too long, some details in the proofs of theorems in
Chapters 4 and 5 of the same techniques with the ones in Chapter 3, are omitted.
The main results obtained in the thesis are the existence, comparison and stability
of solutions, sheaf-solutions of set, fuzzy, control differential systems and fuzzy control
stochastic differential systems. These are the first and important results in researching
control theory related to fuzziness. The restriction of the thesis is the optimal control
problem related to fuzziness which has not been studied. It is still open problem and
of promise.
The results in this thesis come from the papers listed by [75], [76]-[82], [97] and
[99]. Most of the them were reported at the regular conferences of University of ScienceVNU-HCM in 2004 and 2006; ”International conference on differential equations and
applications” 2004-HoChiMinh City and ”Seminar on fuzzy systems, neural networks
and applications” 11/2006 Viet Nam Institute of Mathematics.AUTHOR'S HEARTFELT THANKS
In writing the thesis, the author is indebted to many people and universities.
ã The author would like first to thank Assoc.Prof. Nguyen Dinh Phu, his supervisor,
who has provided from the beginning a continuous guidance and detailed instructions
for his work and Prof. Do Cong Khanh, his co-supervisor, who has given many helps
and moral support.
ã The author is immensely grateful to Prof. Phan Quoc Khanh for continuous helps
and giving a good example on researching.
ã The author thanks the University of Science, Viet Nam National University-Ho Chi
Minh City, for all facilities and working conditions during his Ph.D studies.
ã The author thanks the Leaders of Tay Nguyen University and Faculty of Pedagogy
and his colleagues in Department of Mathematics for moral support and helps.
ã The author is especially grateful to author’s family for the continuous encourag