ontheoptimalcontrol problemforfuzzy differential systems

Viet nam national university-HoChiMinh City University of Science Tran Thanh Tung On the optimal control Problem for fuzzy differential systems Ph.D. Thesis in Mathematics HoChiMinh City - 2008 Viet nam national university-HoChiMinh City University of Science Tran Thanh Tung On the optimal control Problem for fuzzy differential systems Major: Theory on Optimization and Systems Code: 1.01.06 Ph.D. Thesis in Mathematics SCIENTIFIC SUPERVISORS: 1. Assoc.Prof. Ph.D. Nguyen Dinh Phu 2. Prof. D.Sc. Do Cong Khanh HoChiMinh City - 2008 CONFIRMATION I confirm that all the results of this thesis come from my works under the su- pervision of Professors Nguyen Dinh Phu and Do Cong Khanh. They have never been published by other authors. Hochiminh City, December 2008 The author Tran Thanh Tung 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 . . . . . . . . . . . . . . . . . . . 44 3.4.2 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 83 PREFACE 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 the collection 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 intro duced by Kaleva [39, 40] and Nieto [63]. The lo cal 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 by Lakshmikantham, 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  (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. [...]... 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. .. 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... 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... 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... 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 control systems of fuzzy type In Chapter 3, we introduce new concept of set control differential systems and study their existence of solutions,... 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... differential systems set control differential systems Us set of all admissible set controls En the set of all fuzzy numbers on Rn θ D0 [u, v] (E n , D0 ) DH f (t) F DS Uf F CDS (Ω, A, P ) f.r.v the zero element of En the distance between u and v in En a complete metric space derivative of fuzzy mapping f : I → En fuzzy differential systems set of all admissible fuzzy controls fuzzy control differential systems. .. estimation of solutions of differential systems, is following Lemma 1.2 [72, p 83] Suppose that u(t), a(t) are nonnegative functions defined on [t0 , ∞) and t u(t) ≤ C + a(s)u(s)ds, f or all t ≥ t0 ≥ 0, t0 where C is any nonnegative number Then the following inequality holds u(t) ≤ Ce t t0 a(s)ds , t ≥ t0 1.4.2 Existence of solutions of differential systems The differential systems are of the form x = f (t,... ∈ E1 × × E1 = (E1 )p (p times) , t ∈ [0, T ] = I ⊂ R+ and f : I × Rn × Rp → Rn 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 DH X = F (t, X(t), U (t)), X(t0 ) = X0 , (0.5) where F : I × Kc (Rn ) × Kc... sheaf-solutions of these control differential systems d Fuzzy control stochastic differential systems are of the form X (t) = F (t, X(t), U (t)), t ∈ I = [t0 , T ] ⊂ R+ , X(t0 ) = X 0 ∈ Lm 2 (0.7) where F is a mapping: F : I × Lm × Lp → Lm , states X(t) and controls U (t) are 2 2 2 m-dimensional fuzzy random vectors These are systems of combination of fuzziness and randomness and they were firstly introduced in . 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 . . . . . of Science Tran Thanh Tung On the optimal control Problem for fuzzy differential systems Major: Theory on Optimization and Systems Code: 1.01.06 Ph.D. Thesis in Mathematics SCIENTIFIC SUPERVISORS: 1 . . . . . . . . . . . . 25 2.2.3 Control differential systems of fuzzy type . . . . . . . . . . . . 26 Chapter 3 : SET CONTROL DIFFERENTIAL SYSTEMS 34 3.1 EXISTENCE OF SOLUTIONS . . . . . . . .