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Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.Bài toán điều khiển cho phương trình tiến hóa phân thứ trên không gian các số mờ tương quan tuyến tính.MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ——————–o0o——————— HOANG THI PHUONG THAO CONTROL PROBLEM FOR FRACTIONAL DIFFERENTIAL EVOLUTION EQUATIONS IN THE SPACE OF LINEAR.
MINISTRY OF EDUCATION AND TRAINING HANOI NATIONAL UNIVERSITY OF EDUCATION ——————–o0o——————— HOANG THI PHUONG THAO CONTROL PROBLEM FOR FRACTIONAL DIFFERENTIAL EVOLUTION EQUATIONS IN THE SPACE OF LINEARLY CORRELATED FUZZY-VALUED FUNCTIONS Speciality: Differential and Integral Equations Code: 9460103 SUMMARY OF DOCTORAL DISSERTATION IN MATHEMATICS HA NOI-2022 The dissertation was written on the basis of the author’s research works carried at Hanoi National University of Education Supervisor: Assoc.Prof Nguyen Thi Kim Son Prof Nguyen Nhu Thang Referee 1: Professor Cung The Anh Ha Noi national university of education Referee 2: Associate Professor Do Duc Thuan HaNoi University of science and technology Referee 3: Associate Professor Nguyen Van Tuyen Hanoi Pedagogical University This dissertation is presented to the examining committee at Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Vietnam At the time of , 2020 Full-text of the dissertation is publicly available and can be accessed at: - The National Library of Vietnam - The Library of Hanoi National University of Education INTRODUCTION Motivation and overview of the problems In fact, many physical processes occurring in nature and engineering contain quantities of uncertainty We can mention such as the amount of fuel input of machines, technical processes such as image processing, input adjustment of electronic machines Therefore, when modeling these processes for research purposes, uncertainties play an important role This has been recognized by engineers and scientists A number of uncertainty theories have been studied and have many practical applications, such as fuzzy theory, set value theory and more recently neutrosophic theory Fuzzy analysis is a branch of applied mathematics, studied and applied in many fields such as control engineering, image and signal processing, biomedical engineering Fuzzy analysis starts from the The basic concept of fuzzy sets was introduced by Zadeh in 1960 However, it was not until the 1970s that fuzzy set theory and its applications really came to life And up to now, fuzzy theory has developed and achieved many remarkable achievements The theory of calculus plays an important role in many areas of pure and applied mathematics The basic method for solving various types of differential and integral equations is a combination of integral and fractional derivatives and the corresponding integral transformations Equation of secondary evolution has wide applications in interpretation and modeling in applied mathematics and physics including fluid mechanics, electrical circuits, diffusion, etc Controllability is one of the basic properties of a control system However, very little work has been done in the controllability area of fuzzy differential equations The space of linearly correlated fuzzy numbers is a newly studied space It is constructed by fixing an arbitrary fuzzy number A Using a linear dependency map ψA : R2 → RF(A) , we get operators with form qA+r where (q, r) ∈ R2 Interestingly, RF(A) can be embedded into RF as a completely linear subspace when A is non-symmetric fuzzy numbers However, RF(A) cannot become a linear space when A is a symmetric fuzzy number This means that the algebraic structure of the space RF(A) depends on the symmetry of the fuzzy number A The special feature of the space RF(A) in the case of A is a non-symmetric fuzzy number that also allows us to convert a corresponding program in the space of fuzzy numbers to a corresponding system of equations in the space of real numbers The biggest difficulty encountered when working with the space RF(A) occurs where A is a symmetric fuzzy number Then the mapping ψA is not a isomorphism To deal with this problem, Shen introduced the LC-difference in the space of linearly correlated fuzzy numbers Coincidentally, the LC-difference and the gH difference are equal for interval numbers It is worth mentioning that LC-difference is adaptable regardless of whether the basic fuzzy number is symmetric or non-symmetric, and this difference always exists in the space of linearly correlated fuzzy numbers Based on the LCdifference, the LC-derivative was also proposed for the linearly correlated fuzzy number-valued function Incidentally, the LC-differentiability, the Fr´echet differentiability are equivalent if the basic fuzzy number is non-symmetric The Fr´echet derivative has been defined with the help of the related theory in functional analysis Compared with the Fr´echet derivative, the LC-derivative is introduced by using the derivative of the representation functions of linearly correlated fuzzy number-valued functions In addition, LC-derivative is not only suitable for linearly correlated fuzzy numbervalued functions with the non-symmetric basic fuzzy number, but also it may be employed to deal with the differentiability of linearly correlated fuzzy number-valued functions with the symmetric fuzzy number Therefore we have chosen the topic “Control problem for fractional differential evolution equations in the space of linearly correlated fuzzy-valued functions” Literature review Fractional fuzzy differential equations Fuzzy fractional differential equations (FFDEs) were first introduced by Agarwal et al ignoring the role of fuzzy derivative In other words, they incorporated fuzzy uncertainty into dynamical systems to introduce a new class of FFDEs Then, they presented the fuzzy solution in one special case of nonlinearity but no fuzzy derivative notion was introduced However, the development of FFDEs has strongly been associated with the appearance of different types of fuzzy derivatives FFDEs based on Riemann-Liouville Hukuhara differentiability was investigated by Allahviranloo et al Which is a direct generalization of the fractional Riemann-Liouville derivative using the Hukuhara difference Mazandarani and Kamyad established the Caputo type fuzzy fractional derivative for FFDEs However, fuzzy derivatives based on Hukuhara difference have some drawbacks due to the increasing of diameter of uncertainty in time That is why the generalizations of Hukuhara derivative are recognized as the active research direction and it forms a significant development Follow up the generalizations of Bede and Gal on the differentiability of fuzzy number valued functions, the generalized Hukuhara derivatives of interval valued functions and fuzzy-valued functions were introduced The generalized Caputo fractional derivative, Riemann-Liouville fractional derivative and Caputo-Katugampola fractional derivative were introduced with various applications Recently, new notion of granular fuzzy fractional derivatives with application to FFDEs was established by Najariyan and Zhao The granular fuzzy fractional derivatives based on horizontal membership functions and granular difference have several advantages over previous derivatives since it can be calculated directly via inverse transformation of level sets An extension application of granular fuzzy fractional derivatives to linear quadratic regulator problem governed by neutrosophic fractional partial differential equations was elaborated Recently, Esmi et al has introduced the concept of Fr´echet derivative for linear correlated fuzzy-valued functions They also presented a practical method to calculate the Fr´echet derivative via appropriate standard functions when the fuzzy processes are autocorrected Pedro et al employed Fr´echet derivative and Riemann integral to study the integration and derivative theory for interactive fuzzy processes They established some fundamental calculus for further studying fuzzy differential equations Control problem and stability problem for fuzzy evolutionary equation Stability, controllability, observability can be considered as some inherent characteristics of dynamical systems Although the issue of complete controllability of dynamic systems has been investigated before, there still have many open questions remain In existing literature, there is only a few work investigating the controllable problem for fuzzy dynamical systems Some works can be mentioned in this field such as the controllable problem for nonlinear fuzzy neutral functional differential equations discussed by Balasubramaniam or the approximately controllable problem for fuzzy differential equations driven by Liu process Narayanamoorthy and Sowmiya introduced the concept of approximate controllability for nonlocal problem to nonlinear impulsive neutral fuzzy stochastic differential equations In recent work, Jeong et.al presented an interesting result on exact controllability for fuzzy differential equations by using extremal solutions However, it should be mentioned that most of these results either have some difficult in techniques or demand strict assumptions because there has a lack of fuzzy analysis tools and there has been very little work done in the area of the controllability of abstract differential equations such fuzzy evolution equations or fuzzy fractional evolution equations By differential inclusions, P Diamond studied Lyapunov stability of fuzzy differential equations and the periodicity of the fuzzy solution set The asymptotic equilibrium for fuzzy evolution equations and the stability properties of the trivial fuzzy solution of the perturbed semilinear fuzzy evolution equations were investigated under the concept of Hukuhara derivative Using a generalization of Kharitonov theorem in the context of fuzzy sets, stability of fuzzy linear dynamical systems has been studied With a Lyapunov function presented, stability analysis of fuzzy differential equations under the concept of strongly generalized Hukuhara derivative was investigated Practical stabilities of fuzzy differential equations with the second type of Hukuhara derivative has been considered Research topics The Cauchy problem of fuzzy differential equations under fractional derivative Fr´echet Caputo ( C Dp x F 0+ ˆ(t) = fˆ(t, xˆ(t)), xˆ(0) = x0 t ∈ I = [0, b] (1) The Cauchy problem of fuzzy differential equations under fractional derivatives of linear correlated Caputo fuzzy-valued function ( C Dp x LC 0+ ˆ(t) = gˆ(t, xˆ(t)), xˆ(0) = xˆ0 t ∈ I = [0, b], (2) Complete controllability of fuzzy fractional evolutions equation under Fr´echet Caputo derivative ( C Dp x F 0+ ˆ(t) = Bx(t) ⊕A fˆ(t, xˆ(t)) ⊕A Eu(t), t ∈ I = [0, b], xˆ(0) = xˆ0 , (3) Stability analysis of fractional nonlinear systems under fractional derivatives of linear correlated Caputo ( C Dp x LC 0+ ˆ(t) = a ⊙A xˆ(t) ⊕A fˆ(t, xˆ(t)), xˆ(0) = xˆ0 t ∈ I = [0, ∞) (4) Purpose –Objectives and Research scope Purpose The thesis focuses on the control problem for fractional evolutionary equations The main objective is to establish control functions so that the solution of the problem satisfies some given properties Objectives The fuzzy fractional evolutionary equations Research scope We will study the following problems: (P1): The fractional Caputo Fr´echet derivative and the Caputo fractional linear correlated derivative of linearly correlated fuzzy value functions include the definition and properties of these two types of derivatives (P2): Existence and uniqueness of solutions of the Cauchy problem for the fractional fuzzy evolutionary equations in the space of linearly correlated fuzzy numbers (P3): Controllability for a class of fuzzy evolutionary equations under the derivative Fr´echet Caputo in the space of linearly correlated fuzzy numbers (P4): State feedback control for fractional differential equation system in the space of linearly correlated fuzzy numbers Research Methods Theory of functional analysis, metric space, using contraction mapping principle, ArzelaAscoli theorem, Karanoselski theorem Theory of semigroup of fuzzy-valued mappings, Laplace transform, Mittag-Leffler function in the space of linearly correlated fuzzy numbers Theory of set analysis, multivalued analysis, fuzzy analysis, fractional fuzzy analysis Theory of fuzzy control for fuzzy evolutionary equations The structure and results of the dissertation In addition to the sections Acknowledgment, Introduction, List of symbols, References, List of scientific works related to the thesis, Conclusion, Index, the thesis is divided into three chapters as follows: Chapter The space of linearly correlated fuzzy numbers RF(A) Chapter The Cauchy problem of fuzzy differential equations under fractional derivatives of linear correlated fuzzy-valued function Chapter Complete controllability of fuzzy fractional evolutions equation under Fr´echet derivative in linear correlated fuzzy spaces Chapter State feedback control for fractional differential equation system in the space of linearly correlated fuzzy numbers The results obtained in Chapters 2, and are the answers for problems (P1), (P2), (P3), (P4) Chapter and Chapter are based on the papers [CT1], [CT2] in the List of Publications which were published The results of Chapter 4, which have been submitted for publication, are in the work [CT3] in the List of Publications Chapter PRELIMINARIES In this chapter, the basic knowledge about fuzzy sets and analytical operations is detailed in Section 1.1 This knowledge is extracted from two monographs by Bede and Lakshmikantham and Mohapatra To prepare the mathematical basis in Chapter and Chapter 4, the basic knowledge about the space of linearly correlated fuzzy numbers includes the construction and operations of linearly correlated fuzzy numbers, the measured metrics described in Section 1.2 In addition, the integral calculations, integer derivatives of functions receiving linearly correlated fuzzy values including definitions and properties are also shown in Section 1.3 1.1 The space of fuzzy numbers RF 1.2 The space of linearly correlated fuzzy numbers RF (A) 1.2.1 The space Rnsy F (A) 1.2.2 The space Rsy F (A) 1.3 The linearly correlated fuzzy functions 1.3.1 The definitions and examples 1.3.2 The derivative Fr´echet 1.3.3 The linearly correlated derivative 1.3.4 Integral of the linearly correlated fuzzy function Chapter The Cauchy problem of fuzzy differential equations under fractional derivatives of linear correlated fuzzy-valued function In this chapter, we will introduce some types of Fr´echet fractional derivative defined on the class of linear correlated fuzzy-valued functions Namely, we introduce Fr´echet Caputo fractional derivative and fractional derivative Caputo of of linear correlated fuzzy-valued function Finally, we study the existence of a solution of the Cauchy problem of fuzzy differential equations under Fr´echet Caputo fractional derivative and fractional derivatives Caputo of linear correlated fuzzy-valued function 2.1 Fractional derivatives of linear correlated fuzzy-valued function 2.1.1 Fr´echet Caputo fractional derivatives Definition 2.1 Let A ∈ RF be non-symmetric fuzzy number and fˆ(t) = q(t)A + r(t) Then, the Riemann-Liouville fractional integral of order p ∈ (0, 1] of the function fˆ is defined by RL p ˆ F I0+ f (t) = ψA I0p+ q(t), I0p+ r(t) , t ∈ J + Theorem 2.1 Let A1 and A2 be fuzzy numbers with respective α − level sets [A1 ]α = [a− 1α , a1α ], + [A2 ]α = [a− 2α , a2α ] Assume that there exist qi , ri ∈ L (J, R) ∩ C(J, R), i = 1, 2, such that fˆ(t) = ψA1 (q1 (t), r1 (t)) = ψA2 (q2 (t), r2 (t)) for all t ∈ J and q1 , q2 not change of sign on J Then we have ψA1 (I0p+ q1 (t), I0p+ r1 (t)) = ψA2 (I0p+ q2 (t), I0p+ r2 (t)), t ∈ J (2.1) Definition 2.2 Let A ∈ Rsy e be a linear F be symmetric fuzzy number and f : J ⊂ R → RF(A) correlated fuzzy-valued function which is given by fˆ(t) = ψ e(q(t), r(t)) If q(t) does not change A the sign in J then the Riemann-Liouville (RL) fractional integral of order p ∈ (0, 1] of the function fˆ(t) is defined by RL p ˆ F I0+ f (t) = ψA I0p+ q(t), I0p+ r(t) , t ∈ J Definition 2.3 Let A be a non-symmetric fuzzy number and fˆ : J ⊂ R → RF(A) be a function such that fˆ(t) = q(t)A + r(t) for each t ∈ J For p ∈ (0, 1], the Fr´echet Caputo fractional derivative of order p of function fˆ is defined by C p ˆ F D0+ f (t) 1−p ′ ′ = ψA I01−p + q (t), I0+ r (t) , t ∈ J (2.2) In the case A is a symmetric fuzzy number, for fˆ(t) = q(t)A + r(t), t ∈ J and q ′ (·) not change the sign in J, we define the Fr´echet Caputo fractional derivative of order p of function fˆ by formula (2.2) Proposition 2.1 Let A ∈ Rnsy a fˆ : J → Rnsy be Fr´echet differentiable Then F v` F(A) RL p C p ˆ F I0+ F D0+ f (t) Z = t ′ fˆF (s)ds 2.1.2 The fractional derivatives Caputo for linearly correlated fuzzy number-valued function Definition 2.4 Let fˆ : J → Rnsy with fˆ(t) = q(t)A + r(t), q, r : J → R then the RiemannF(A) Liouville-fractional LC integral of order p ∈ (0, 1] of fˆ(t) is defined by RL p ˆ LC I0+ f (t) = ψA (I0p+ q(t), I0p+ r(t)) Definition 2.5 Let fˆ : J → Rsy with the canonical form fˆ(t) = ψˆA ([qe(t), e r(t)]≡A ) then F(A) Riemann-Liouville-fractional LC integral of order p ∈ (0, 1] of fˆ(t) is defined by RL p ˆ LC I0+ f (t) = ψˆA ([I0p+ qe(t), I0p+ e r(t)]≡A ) Definition 2.6 Let fˆ : J → Rnsy with fˆ(t) = q(t)A + r(t), q, r : J → R The Caputo F(A) fractional LC derivative of order p ∈ (0, 1] of fˆ(t) is defined by p ˆ C LC D0+ f (t) p ˆ′ = RL F I0+ fLC (t) = ψA (D0p+ q(t), D0p+ r(t)) where D0p+ q(t), D0p+ r(t) are Caputo fractional derivative of order p ∈ (0, 1] of the real function q(t), r(t) Definition 2.7 Let fˆ : J → sRF(A) with fˆ(t) = q(t)A + r(t), q, r : J → R Then Caputo fractional LC derivative of order p ∈ (0, 1] of fˆ(t) is defined by p ˆ C LC D0+ f (t) = ψˆA ([|D0p+ qe(t)|, D0p+ e rc (t)]≡A ) where ( |D0p+ qe(t)| = D0p+ qe(t), q ′ (t) ≥ ∀ t ∈ J −D0p+ qe(t)quadq ′ (t) < ∀ t ∈ J ( and rc (t) D0p+ e r(t), D0p+ e = q ′ (t) ≥ ∀ r(t), 2D0p+ qe(t)x∗ + D0p+ e q ′ (t) < ∀ ˆ = ˆ : J → Rnsy with fˆ(t) = ψA (q ˆ(t), r ˆ(t)), gˆ(t) = ψA (qgˆ(t), rgˆ(t)), h(t) Lemma 2.1 Let fˆ, gˆ, h f f F(A) ψA (qhˆ (t), rhˆ (t)) with qfˆ, rfˆ, qgˆ, rgˆ, qhˆ , rhˆ are real functions on J Then ˆ (i) if fˆ(t) ⊕A gˆ(t) = h(t) then C D p fˆ(t) ⊕ C D p g A LC 0+ ˆ(t) LC 0+ p ˆ =C LC D0+ h(t), ˆ (ii) if fˆ(t) ⊟A gˆ(t) = h(t) then C D p fˆ(t) ⊟ C D p g A LC 0+ ˆ(t) LC 0+ p ˆ =C LC D0+ h(t) ˆ : J → sRF(A) with the canonical form fˆ(t) = ψˆA ([qeˆ(t), e Lemma 2.2 Let fˆ, gˆ, h rfˆ(t)]≡A ), gˆ(t) = f ˆ = ψˆA ([qeˆ (t), e ψˆA ([qegˆ(t), e rgˆ(t)]≡ ), h(t) rˆ (t)]≡ ) with qeˆ, e r ˆ, qegˆ, e rgˆ, qeˆ , e rˆ are real functions on J h A h f A f h h and function qfˆ, qgˆ have the same monotonicity on J Then ˆ ˆ A gˆ(t) = h(t) (i) If fˆ(t)⊕ then p C D p fˆ(t)⊕ ˆ AC ˆ(t) LC 0+ LC D0+ g p ˆ =C LC D0+ h(t), ˆ A gˆ(t) = h(t) ˆ (ii) if fˆ(t)⊟ then C D p fˆ(t)⊟ ˆ A C Dp+ gˆ(t) LC LC 0+ p ˆ =C LC D0+ h(t) Proposition 2.2 Let A ∈ RF and fˆ : J → RF(A) be differentiable Then p ˆ RL p C LC I0+ LC D0+ f (t) Z = t ′ fˆLC (s)ds 2.2 The Cauchy problem of fuzzy differential equations under 2.2.1 Problem formulation Consider following fuzzy initial value problem (FIVP) for fractional differential equation ( C Dp x F 0+ ˆ(t) = fˆ(t, xˆ(t)), xˆ(0) = xˆ0 , t ∈ J, (2.3) p where C ˆ(·) is the Fr´echet Caputo fractional derivative of order p ∈ (0, 1] of function xˆ(·), F D0+ x nsy ˆ → xˆ0 ∈ RF(A) with A is a non-symmetric fuzzy number and the function f : J × C J, R F(A) Rnsy is continuous F(A) Lemma 2.3 If the function xˆ ∈ C J, Rnsy F(A) satisfies problem (2.3) then it satisfies the fol- lowing integral equation p ˆ ˆ(t)), xˆ(t) = xˆ0 +A RL F I0+ f (t, x Definition 2.8 If xˆ ∈ C I, Rnsy F(A) t ∈ J (2.4) satisfies the integral equation (2.4) then xˆ is the integral solution of the problem (2.3) 2.2.2 Solvency of problem Next, we give conditions to ensure the existence and uniqueness of the integral solution of the Cauchy problem (2.3) Firstly, we define the supremum metric ρ and the weak metric dm on C I, Rnsy F(A) as follows: ρ(β, γ) = sup dψA (β(s), γ(s)), s∈I dm (β, γ) = sup sm dψA (β(s), γ(s)) , s∈I where β, γ ∈ C(I, Rnsy ) F(A) Theorem 2.2 Assume that the fuzzy-valued function fˆ is continuous and satisfies the Lipschitz condition w.r.t the last argument, i.e., there exists a constant L > such that dA (fˆ(t, xˆ(t)), fˆ(t, yˆ(t))) ≤ LdA (ˆ x(t), yˆ(t)) for all xˆ, yˆ ∈ C J, RF(A) and t ∈ J Then, the fuzzy initial value problem (2.3) has a unique integral solution defined on J Example 2.4 To demonstrate the effective of theoretical result, let us consider following fuzzy fractional differential equations ( CD e p ˆ(t) F 0+ x = β(t)ˆ x(t) + η(t), xˆ(0) =A t ∈ [0, 5], (2.5) ep where C echet Caputo fractional derivative of order p ∈ (0, 1], A ∈ RF is F D0+ denotes for the Fr´ a non-symmetric fuzzy number and β, η : [0, 5] → R are differentiable functions Let us denote fˆ(t, xˆ(t)) = β(t)ˆ x(t) + η(t) for each t ∈ [0, 5] Then, we can see that The function fˆ satisfies Lipschitz condition with the constant L = max β(t) [0,5] Since x ˆ(t) ∈ RF(A) , there exists a pair (qxˆ (t), rxˆ (t)) ∈ R2 such that xˆ(t) = Aqxˆ (t) + rxˆ (t) for each t ∈ [0, 5] Thus, the function fˆ(t, xˆ(t)) can be rewritten as fˆ(t, xˆ(t)) = β(t) (Aqxˆ (t) + rxˆ (t)) + η(t) = Aβ(t)qxˆ (t) + β(t)rxˆ (t) + η(t) Next, denote qfˆ(t) = β(t)qxˆ (t), rfˆ(t) = β(t)rxˆ (t) + η(t) Then, for each t ∈ [0, 5] and xˆ ∈ C [0, 5], RF(A) , we have dA fˆ(t, xˆ(t)), fˆ(t, yˆ(t)) = β(t)qxˆ (t) − β(t)qyˆ(t) + β(t)rxˆ (t) − β(t)ryˆ(t) ≤ max β(t) |qxˆ (t) − qyˆ(t)| + |ru (t) − ryˆ(t)| t∈[0,5] ≤ max β(t)dA (ˆ x(t), yˆ(t)) t∈[0,5] Here, we can see that the function fˆ satisfies all assumptions of Theorem 2.2 Hence, the problem (2.5) has a unique integral solution defined on [0, 5] 2.3 The Cauchy problem of fuzzy differential equations under the fractional derivatives Caputo for linearly correlated fuzzy number-valued function 2.3.1 Problem formulation Consider following fuzzy initial value problem (FIVP) for fractional differential equation ( C Dp x LC 0+ ˆ(t) = gˆ(t, xˆ(t)), xˆ(0) = xˆ0 , t ∈ I = [0, b], (2.6) p where C ˆ(t) is fractional derivatives Caputo for linearly correlated fuzzy number-valued LC D0+ g function p ∈ (0, 1], u0 ∈ RF(A) with A ∈ RF the function gˆ : J × C J, RF(A) → RF(A) is continuous Consider the problem (2.6)in the space Rns F(A) Lemma 2.4 If the function u ∈ C J, Rnsy F(A) is a solution of Cauchy problem (2.6) then it satisfies the following integral equation p ˆ(t, xˆ(t)), xˆ(t) = xˆ0 ⊕A RL LC I0+ g Definition 2.9 If xˆ ∈ C I, Rnsy F(A) t ∈ I (2.7) satisfies the integral equation (2.7) then xˆ is the integral solution of the problem (2.6) We consider the system (2.6) in the space Rnsy F(A) Lemma 2.5 Assume that A is a fuzzy number symmetric with respect to If the Cauchy sy problem (2.6) has a solution xˆ ∈ C I, RF(A) if the diameter of the level set does not increase above I then it satisfies the integral equation p ˆ A RL xˆ0 = xˆ(t)⊕ ˆ(t, x(t)), LC I0+ g t ∈ I Definition 2.10 The function xˆ ∈ C I, Rsy F(A) is called 1) integral solution type of the problem (2.6 ) if it satisfies the integral equation p ˆ A RL xˆ(t) = xˆ0 ⊕ ˆ(t, xˆˆ(t)) F I 0+ g (2.8) 2) integral solution type of the problem (2.6 ) if it satisfies the integral equation p ˆ A RL ˆ(t, xˆ(t)) xˆ0 = xˆ(t)⊕ F I0+ g (2.9) 2.3.2 Solvency of problem We hypothesize the function fˆ as follows Theorem 2.3 Assume that the function fˆ satisfies the Lipschitz condition that there exists a constant L > such that dψA (ˆ g (t, xˆ(t)), gˆ(t, yˆ(t))) ≤ LdψA (ˆ x(t), yˆ(t)) for all xˆ, yˆ ∈ C I, Rns F(A) and t ∈ I Then, the initial value problem (2.6) has a unique integral solution defined on C(I, Rns F(A) ) 10 Chapter Complete controllability of fuzzy fractional evolutions equation under Fr´ echet derivative in linear correlated fuzzy spaces In this chapter, we study the control problem for the evolution equation under Fr´echet−Caputo differentiability ( C Dp x F 0+ ˆ(t) = B xˆ(t) ⊕A fˆ(t, xˆ(t)) ⊕A Eu(t), t ∈ I = [0, b], (3.1) xˆ(0) = xˆ0 , where C D p µ(t) F 0+ is Fr´echet Caputo of function xˆ : J → Rnsy FA , B is the generator of a C0 − semigroup {T (t)}t≥0 on RF(A) , nsy The fuzzy-valued function u ∈ L J, Rnsy F(A) is called control input where L J, Rnsy F(A) is space of integrable function in J nsy nsy The linear operator E and the function fˆ : J × RF(A) → RF(A) satisfy some given conditions mentioned later In this chapter, we study the complete controllability for a class of fuzzy fractional evolution equations Some sufficient conditions of the controllability for the fuzzy fractional evolution equations are considered in both not unique or an unique of control variable Technics of fuzzy semigroups and noncompactness measure are used in new linear conrelated fuzzy spaces Some illustrated examples are given 3.0.1 Laplace transformation for functions in RF (A) Definition 3.1 Let fˆ : [0, +∞) → Rnsy Assume that the function fˆ(t) is given by fˆ(t) = F(A) ψA (q(t), r(t)), where q(t), r(t) are continuous real-valued functions Then, the fuzzy Laplace transform of the function fˆ(t) is defined by L[fˆ(t)](s) = ψA (L[q(t)](s), L[r(t)](s)) , where L[q(t)](s), L[r(t)](s) are classical Laplace transforms of real functions of q, r : [0, +∞) → R respectively Proposition 3.1 Let fˆ : [0, +∞) → Rnsy , fˆ(t) = ψA (q(t), r(t)), where q, r : [0, +∞) → F(A) ′ R, s ∈ R Assume that fˆ (.) is an Frechet derivative of fˆ Then ′ L[fˆ (t)](s) = sL[fˆ(t)](s) ⊖A fˆ(0) Proposition 3.2 Let fˆ, g : [0, +∞) → Rnsy be continuous fuzzy-valued functions Suppose F(A) that c1 , c2 are real constants, then L[c1 ⊙A fˆ(t) ⊖A c2 ⊙A gˆ(t)](s) = c1 ⊙A L[fˆ(t)](s) ⊕A c2 ⊙A L[ˆg (t)](s), s ∈ R Remark 3.1 Let fˆ : [0, +∞) → Rnsy be continuous fuzzy functions Then F(A) L[λ ⊙A fˆ(t)](s) = λ ⊙A L[fˆ(t)](s) for λ ≥ 0, s ∈ R 11 3.0.2 The C0 − semigroup of fuzzy-valued mappings Definition 3.2 A mapping T : Rnsy → Rnsy is called a bounded linear mapping if F(A) F(A) (i) T (ˆ u1 ⊕A uˆ2 ) = T (ˆ u1 ) ⊕A T (ˆ u2 ) and T (λ ⊙A uˆ1 ) = λ ⊙A T (ˆ u1 ) for all uˆ1 , uˆ2 ∈ Rnsy and F(A) λ ∈ R (ii) There exists K > such that dψA (T (ˆ u1 ), ˆ0) ≤ KdψA (ˆ u1 , ˆ0) for all uˆ1 ∈ Rnsy F(A) Definition 3.3 For each bounded operator T : Rnsy → Rnsy , denote F(A) F(A) ∥T ∥op := sup{dψA (T (ˆ u), ˆ0)|ˆ u ∈ Rnsy , d (ˆ u, ˆ0) ≤ 1} F(A) ψA Then, we have (i) dψA (T (ˆ u), ˆ0) ≤ ∥T ∥op dψA (ˆ u, ˆ0) for all uˆ ∈ Rnsy F(A) (ii) dψA (T (ˆ u1 ), T (ˆ u2 )) ≤ ∥T ∥op dψA (ˆ u1 , uˆ2 ) for all uˆ1 , uˆ2 ∈ Rnsy F(A) Definition 3.4 A family {T (t)}t≥0 of bounded linear mappings on Rnsy is called a strongly F(A) nsy continuous semigroup (or C0 − semigroup) on RF(A) if it satisfies (i) T (0) = IdA , where IdA is the identity mapping on Rnsy , F(A) (ii) T (t1 + t2 ) = T (t1 )T (t2 ) ∀t1 , t2 ≥ (iii) ∀ˆ x ∈ Rnsy , the orbit maps ξxˆ : [0, ∞) → Rnsy , given by ξxˆ (t) = T (t)ˆ x, are continuous F(A) F(A) Definition 3.5 A strongly continuous semigroup {T (t)}t≥0 on Rnsy is called a contraction if F(A) for each t ≥ 0, ∥T (t)∥op ≤ Proposition 3.3 If {T (t)}t≥0 is a strongly continuous semigroup on Rnsy then F(A) lim ∥T (t) ⊖A IdA ∥op = t→0 For p ∈ (0, 1) and t ∈ (0, ∞), we define the probability density functions on (0, ∞) as follows Sp (t)z = ∞ Z ϕp (θ)T (t θ)zdθ, p Tp (t)z = p ∞ Z θϕp (θ)T (tp θ)zdθ where ∞ 1X Γ(np + 1) ψp (θ) = (−1)n−1 θ−pn−1 sin(nπp), θ ∈ [0; +∞), π n! n=1 1 ϕp (θ) = θ−1− p ψp (θ− p ), θ ∈ (0; +∞), n ∈ N p Proposition 3.4 For each t ≥ 0, the mappings Sp (t) and Tp (t) satisfy the following assertions: (i) Sp (t) and Tp (t) are bounded linear mappings, specifically for all xˆ ∈ Rnsy , we have F(A) ∥Sp (t)ˆ x∥A ≤ K1 ∥ˆ x∥A and ∥Tp (t)ˆ x∥ A ≤ 12 pK1 ∥ˆ x∥A , Γ(p + 1) (ii) The families {Sp (t)}t≥0 and {Tp (t)}t≥0 are strongly continuous, specifically, for each xˆ ∈ Rnsy , we have F(A) dA (Sp (t)ˆ x, Sp (t1 )ˆ x) → dA (Tp (t)ˆ x, Tp (t1 )ˆ x) → and as t → t1 Definition 3.6 Let {T (t)}t≥0 be a strongly continuous semigroup on Rnsy Then, we define F(A) nsy nsy the infinitesimal generator B : D(B) ⊂ RF(A) → RF(A) of semigroup {T (t)}t≥0 by Bx := lim h→0+ where D(B) = xˆ ∈ Rnsy F(A) (T (h)ˆ x ⊖A xˆ), h xˆ ∈ D(B), x ⊖A xˆ) exists : lim (T (h)ˆ + h→0 h For xˆ ∈ D(B), denote Mλ xˆ = λˆ x −A Bx and ρ(B) := {λ ∈ C : Mλ xˆ is bijective and M−1 λ ∈ nsy L(RF(A) )} Then for all λ ∈ ρ(B), the resolve operator is nsy R(λ, B) := M−1 λ ∈ L(RF(A) ) 3.1 Complete controllability 3.1.1 Formula of solution Lemma 3.1 Assume that the function xˆ : J × Rnsy → Rnsy is Fr´echet differentiable on J F(A) F(A) and satisfies the problem (3.1) Then, for each t ∈ J, we have Z xˆ(t) = Sp (t)ˆ x0 +A t (t − s)p−1 Tp (t − s)(fˆ(s, xˆ(s)) +A Euxˆ (s))ds, ≤ t ≤ b (3.2) Definition 3.7 A continuous function : J ì C J, Rnsy F(A) → Rnsy satisfying the integral F(A) equation (3.2) is said to be an integral fuzzy solution of the problem (3.1) Definition 3.8 The problem (3.1) is completely controllableon I if for each initial state xˆ0 nsy and the final state xˆ1 , there exists an input control u ∈ L I, RF(A) such that xˆ(b) = xˆ1 , where xˆ(·) is an integral solution of (3.1) We define the operator P : C(I, Rnsy ) → C(I, Rnsy ) by F(A) F(A) P[ˆ x](t) = Sp (t)ˆ x0 ⊕A Z t (t − ς)p−1 Tp (t − ς)fˆ(ς, xˆ(ς))dς Z0 t ⊕A (t − ς)p−1 Tp (t − ς)Eux (ς)dς, where input control uxˆ (.) " uxˆ (t) = Π−1 xˆ1 ⊖A Z Sp (b)ˆ x0 ⊖A !# b (b − ς)p−1 Tp (b − ς)fˆ(ς, xˆ(ς))dς 3.1.2 complete controllability with an unique control variable Consider assumptions (B),(F1), (F2), (C), (P) 13 (t) (B) The operator B generates a strong continuous semigroup {T (t)}t≥0 in the space Rnsy F(A) Hence there exists K1 such that ∥T (t)∥op ≤ K1 (F1) For each t ∈ I, the operator fˆ(t, ·) : C(I, Rnsy ) → Rnsy is continuous, fˆ(t, ˆ0) = ˆ0 and F(A) F(A) for each xˆ ∈ Rnsy , fˆ(., xˆ) : I → Rnsy is strongly measurable F(A) F(A) x ∈ C(I, Rnsy ) : ρ(ˆ x, ˆ0) ≤ r} with r > there is a constant afˆ > (F2) For xˆ, x ∈ Br := {ˆ F(A) such that dψA (fˆ(t, xˆ(t)), fˆ(t, x(t))) ≤ af dψA (ˆ x, x) (C) ∥E∥op ≤ K2 with K2 > (P) The operator Π : L1 I, Rnsy F(A) → Rnsy and F(A) b Z (b − ς)p−1 ⊙A Tp (b − ς)Eu(ς)dς, Π(u) = has an inverse operator and there exists a constant K3 > such that ∥Π−1 ∥op ≤ K3 for all u ∈ L1 I, Rnsy F(A) Theorem 3.1 If the hypotheses (B), (F1), (F2), (C), (P) hold then the fuzzy fractional evolution equation(3.1) is completely controllable provided that K1 bp afˆ Γ(p + 1) where ξ = (1 + ξ) ≤ (3.3) K1 K2 K3 bp Furthermore, the control input is unique Γ(p + 1) 3.1.3 Complete controllability with not unique control variable We consider some conditions following: (F3) There exist p ∈ (0, p) and gfˆ(·) ∈ L p (J, R+ ) such that dA (fˆ(t, xˆ(t)), fˆ(t, x(t))) ≤ gfˆ(t)H(ˆ x, x) for all xˆ, x ∈ C(J, Rnsy ) and t ∈ J F(A) (F4) For each bounded set Ω ⊂ Rnsy , there is a p1 ∈ (0, p) and X : J × J → R+ such that F(A) X(t, ·) ∈ L p1 ([0, t), R+ ) and β(Tp (t − s)fˆ(s, Ω)) ≤ X(t, s)β(Ω) and β(Tp (t − ς)fˆ(ς, Ω)) ≤ X(t, ς)β(Ω) ∥X(t, ·)∥ L p (I,R+ ) < η p−1 b η p−p , where β(·) is the measure of noncompactness on Rnsy F(A) 1−p We recall that the operator P : C(J, Rnsy ) → C(J, Rnsy ) by F(A) F(A) for all t ∈ [0, b], η = P[ˆ x](t) = Sp (t)ˆ x0 ⊕A t Z (t − ς)p−1 ⊙A Tp (t − ς)fˆ(ς, xˆ(ς))dς 14 t Z (t − ς)p−1 ⊙A Tp (t − ς)Euxˆ (ς)dς, ⊕A where the control input uxˆ (t) is given by " uxˆ (t) = Π−1 xˆ1 ⊖A Sp (b)ˆ x0 ⊕A !# b Z (b − ς)p−1 ⊙A Tp (b − ς)fˆ(ς, xˆ(ς))dς (t) Now, we will prove that the operator P has at least one fixed point x ∈ C J, RF(A) by the virtue of Krasnoselskii’s fixed point theorem To this, we represent P as a sum of two following operators t Z P1 [ˆ x](t) = Sp (t)ˆ x0 +A (t − s)p−1 Tp (t − s)Euxˆ (s)ds Z t P2 [µ](t) = (t − s)p−1 Tp (t − s)fˆ(s, xˆ(s))ds (3.4) For r > 0, Ωr = {µ ∈ C J, Rnsy F(A) : H(ˆ x, ˆ0) ≤ r} Lemma 3.2 Under assumption that the hypotheses (B), (F1),(F3),(F4) and (P) hold , the operator P2 defined by (3.4) is continuous on Ωr Lemma 3.3 Under assumption that the hypotheses (F1),(F3),(F4) and (P) hold then P2 is a compact Theorem 3.2 If the hypotheses (B, ((F1)), ((F3)), ((F4)) (C) and (P) are fulfilled, then the problem (3.1) is completely controllable on J = [0, b], provided that [(σ − 1)Γ(p + 1) + σK1 K2 K3 bp ] ≥ 0, K1 K4 p v`a K4 = where σ = Γ(p + 1) − p1 p − p1 1−p1 bp−p1 ∥gfˆ∥ L p (I,R+ ) (3.5) 3.1.4 Example Example 3.1 This example illustrates the solvability of the following fuzzy fractional finite time delay system under the Fr´echet−Caputo differentiability −t C D1/3 xˆ(t) = e xˆ(t) + 2t + xˆ(t) + u A A F 0+ 3 π (3.6) µ(0) = where A is a given non-symmetric fuzzy number, t ∈ I = [0, 1], < s < t We can see that 2t + fˆ(t, xˆ(s)) = xˆ(t) Firstly, the B = e−t/3 IdA generates a semigroup {T (t)}t≥0 and ∥T (t)∥op ≤ Therefore K1 = is the number satisfying hypothesis (F1) For t ∈ [0, 1], we have ∥fˆ(t, xˆ(t))∥A ≤ 2t + ∥ˆ x∥A ≤ ∥ˆ x∥ A 15 and 2t + dA (ˆ x(t), x(t)) ≤ dA (ˆ dA (fˆ(t, xˆ(t)), fˆ(t, x(t)) = x(t), x(t)), for all xˆ, x ∈ C(J, Rnsy ) It implies that (F2) is satisfied with afˆ = Moreover, one gets F(A) E = IdA then π ∥E∥op ≤ And so (C) are satisfied with K2 = For u ∈ L1 (J, Rnsy F (A)), the linear operator Π is given by Πu = π t Z (t − s) Tp (t − s)u(s)ds , we have ∥Πu∥A ≤ H(u, ˆ0) By applying Theorem 3.1, we can Γ( ) conclude that the fractional differential system (3.6) is completely controllable on I = [0, 1] Let K3 = ∈ 0, Example 3.2 This example is devoted to investigate the controllability of the following fuzzy fractional differential system t e− 2t + 1/3 C F D0+ xˆ1 (t) = xˆ1 (t) +A xˆ2 (t) +A u1 (t) 3 π (3.7) t e− C D1/3 xˆ2 (t) +A u2 (t), ˆ2 (t) = (−1)ˆ x1 (t) +A F 0+ x π that satisfies the initial conditions ( xˆ1 (0) = xˆ01 xˆ2 (0) = xˆ02 , where A is a given non-symmetric fuzzy number, t ∈ I = [0, 1] and the input controls u1 , u2 : [0, 1] → Rnsy are continuous fuzzy-valued functions Then, by applying Theorem 3.2, we can F(A) conclude that the fractional differential system (3.7) is completely controllable on I provided √ π that there exists a constant K3 ∈ 0, 5Γ( ) such that [(σ − 1)Γ(p + 1) + σK1 K2 K3 ] ≥ √ where K1 = 2, K2 = π r √ 0.88 and σ ≈ , K = 729 Γ 16 108 1− √ e + √1 162 ≈ 0.44 Chapter State feedback control for fractional differential equation system in the space of linearly correlated fuzzy numbers In this chapter, we study the stability and attractiveness of the equilibrium point of the fuzzy evolutionary equation according to the Caputo linear correlation derivative in the space RF(A) We provide conditions to ensure that the given problem has attractive equilibrium and asymptotically stable (Theorem 4.1, Theorem 4.2, Theorem 4.3, Theorem 4.4, Theorem 4.5) In case the equilibrium point is not attractive or asymptotically stable, we design the control variable u(t) so that when u(t) is introduced into the problem, its equilibrium is attractive or stable approach (Theorem 4.6, Theorem 4.7) The content of this chapter is extracted from article in the list of scientific works related to the thesis 4.1 Problem formulation We stipulate that if xˆ(t), fˆ(t, xˆ(t)) ∈ RF(A) , there exists q, r, hfˆ, gfˆ : I → R such that xˆ(t) = q(t)A + r(t), fˆ(t, xˆ(t)) = hfˆ(t)A + gfˆ(t) Definition 4.1 The constant xˆe is an equilibrium of system (4) if and only if fˆ(t, xe ) = ˆ0 We consider the system (4) in the space Rnsy F(A) be a fuzzy function that takes a linear correlation value Definition 4.2 Let xˆ : I → Rnsy F (A) We say that xˆ is a solution of the system (4) if xˆ is differentiable on I and satisfies the system (4) Note that the problem (4) always has a trivial solution xˆ = ˆ0 Definition 4.3 The trivial solution xˆ = ˆ0 of the system (4) is stable if for every ϵ > there exists δ = δ(ϵ) such that ∥ˆ x(t0 )∥A < δ , the solution xˆ(t) of the system (4) satisfies ∥ˆ x(t)∥A < ϵ for all t ≥ attractive if exist δ > such that ∥ˆ x(t0 )∥A < δ, the solution xˆ(t) of the system (4) satisfy lim xˆ(t) = ˆ0 t→+∞ asymptotically stable if it is stable and attractive Lemma 4.1 xˆ(t) = ψA (q(t), r(t)) is a solution of problem (4) if and only if q(.), r(.) : I → R are fractional differentiable on I and satisfy following two independent problems ( D0p+ q(t) = aq(t) + bhfˆ(t), q(0) = q0 , t ∈ I, (4.1) t ∈ I (4.2) and ( D0p+ r(t) = ar(t) + bgfˆ(t), r(0) = r0 , 17 We consider the following problem (4) in the space sRF(A) and we rewrite problem (4) as the new problem follow ( C Dp x LC 0+ ˆ(t) ˆ A xˆ(t)⊕ ˆ A fˆ(t, xˆ(t)), = a⊙ t∈I (4.3) xˆ(0) = xˆ0 p where C ˆ(t) is the Caputo fractional LC derivative of order p ∈ (0, 1) of state xˆ : I → LC D0+ x sy sy ˆ ˆ ˆ ˆ(t) = qe(t)A + e r(t), fˆ(t, xˆ(t)) = e hfˆ(t)A + e gfˆ(t) RF(A) , f : I × Rsy F(A) → RF(A) satisfies f (t, 0) = 0, x are the canonical form of xˆ(t), fˆ(t, xˆ(t)), respectively, a < Definition 4.4 Let xˆ : I → Rsy be a linearly correlated fuzzy number-valued function with F(A) the canonical form xˆ(t) = qe(t)A + e r(t) We say that x ˆ is a differentiable solution of problem (4.3) if qe, e r are differentiable and xˆ satisfies the system (4.3), x ˆ is a continuous solution of problem (4.3) if qe, e r are both left and right differentiable and xˆ satisfies the system (4.3) Definition 4.5 The trivial solution xˆ = ˆ0 of the system (4.3) is stable if for every ϵ > there exists δ = δ(ϵ) such that ∥ˆ x(t0 )∥Aˆ < δ , the solution xˆ(t) of the system (4) satisfies ∥ˆ x(t)∥Aˆ < ϵ for all t ≥ attractive if exist δ > such that ∥ˆ x(t0 )∥Aˆ < δ, the solution xˆ(t) of the system (4) satisfy ˆ lim xˆ(t) = t→+∞ asymptotically stable if it is stable and attractive We can prove that problem (4.3) always has a differentiable solution xˆ = ˆ0 And we call the solution xˆ = ˆ0 is the trivial differentiable solution of (4.3) Lemma 4.2 Let qe(.) be non-decreasing function on the I0 ⊂ I and ∈ I0 xˆ(t) = qe(t)A + e r(t) is the differentiable solution of system (4.3) on the interval I0 if and only if qe(t) is non-negative and qe(t) and e r(t) satisfy the following system p e D0+ qe(t) = −aqe(t) + hfˆ(t), D0p+ e r(t) = ae r(t) + 2aqe(t)x∗ + e gfˆ(t), qe(0) = qe0 , e r(0) = e r0 , t ∈ I0 (4.4) Lemma 4.3 Let qe(.) be a decreasing function on the I0 ⊂ I and ∈ I0 xˆ(t) = qe(t)A + e r(t) is the differentiable solution of system (4.3) on the interval I0 if and only if qe(t) is non-negative and qe(t), e r(t) satisfy the following system p e D0+ qe(t) = aqe(t) − hfˆ(t), D0p+ e r(t) = ae r(t) + 2e hfˆ(t)x∗ + e gfˆ(t), qe(0) = qe0 , e r(0) = e r0 , t ∈ I (4.5) 4.2 The asymptotic stability of the equilibrium point We consider this following assumption Assumption (H) If fˆ(t, φ(t)) = ψA (h(t, φ(t)), g(t, φ(t))) and φ(t) = ψA (qφ (t), rφ (t)) then there exist l > such that |h(t, φ(t))| ≤ l|qφ (t)| v`a |g(t, φ(t))| ≤ l|rφ (t)| for all t ∈ I 18