Table of Contents Author Guidelines Submit a Manuscript
Advances in Fuzzy Systems
Volume 2008 (2008), Article ID 421525, 15 pages
http://dx.doi.org/10.1155/2008/421525
Research Article

On Controllability and Observability of Fuzzy Dynamical Matrix Lyapunov Systems

Department of Applied Mathematics, Acharya Nagarjuna University, Nuzvid Campus, Nuzvid 521 201, Andhra Pradesh, India

Received 27 September 2007; Accepted 1 January 2008

Academic Editor: Hao Ying

Copyright © 2008 M. S. N. Murty and G. Suresh Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We provide a way to combine matrix Lyapunov systems with fuzzy rules to form a new fuzzy system called fuzzy dynamical matrix Lyapunov system, which can be regarded as a new approach to intelligent control. First, we study the controllability property of the fuzzy dynamical matrix Lyapunov system and provide a sufficient condition for its controllability with the use of fuzzy rule base. The significance of our result is that given a deterministic system and a fuzzy state with rule base, we can determine the rule base for the control. Further, we discuss the concept of observability and give a sufficient condition for the system to be observable. The advantage of our result is that we can determine the rule base for the initial value without solving the system.

1. Introduction

The importance of control theory in applied mathematics and its occurrence in several problems such as mechanics, electromagnetic theory, thermodynamics, and artificial satellites are well known. In general, fuzzy systems are mainly classified into three categories, namely pure fuzzy systems, T-S fuzzy systems, and fuzzy logic systems, using fuzzifiers and defuzzifiers. In this paper, we use fuzzy matrix Lyapunov system to describe fuzzy logic system. The purpose of this paper is to provide sufficient conditions for controllability and observability of first-order fuzzy matrix Lyapunov system modeled by 𝑋(𝑡)=𝐴(𝑡)𝑋(𝑡)+𝑋(𝑡)𝐵(𝑡)+𝐹(𝑡)𝑈(𝑡),𝑋(0)=𝑋0,𝑡>0,(1)𝑌(𝑡)=𝐶(𝑡)𝑋(𝑡)+𝐷(𝑡)𝑈(𝑡),(2)𝑈(𝑡) where 𝑛×𝑛 is an 𝑌(𝑡) fuzzy input matrix called fuzzy control and 𝑛×𝑛 is an 𝐴(𝑡),𝐵(𝑡),𝐹(𝑡),𝐶(𝑡) fuzzy output matrix. Here 𝐷(𝑡), and 𝑛×𝑛 are matrices of order 𝑡, whose elements are continuous functions of 𝐽=[0,𝑇]𝑅0x000a0(𝑇>0) on 𝑈(𝑡).

The problem of controllability and observability for a system of ordinary differential equations was studied by Barnett and Cameron [1] and for matrix Lyapunov systems by Murty et al.[2]. Fuzzy control usually decomposes a complex system into several subsystems according to the human expert's understanding of the system and uses a simple control law to emulate the human control strategy.There exist two major types of fuzzy controllers, namely Mamdani fuzzy controllers and Takagi-Sugeno (TS) fuzzy controllers. They mainly differ in the consequence of fuzzy rules: the former uses fuzzy sets whereas the latter employs (linear) functions. Takagi and Sugeno [3, 4] propose a type of fuzzy model in which the consequent part of the rules is defined not by the membership function but by a crisp analytical function. More and more interest appears to shift towards TS fuzzy controllers in recent years, as evidenced by the increasing number of papers in this direction and due to their applications in real world problems (e.g., [512]).

Recently, the controllability and observability criteria for fuzzy dynamical control systems were discussed by Ding and Kandel [13, 14]. In this paper, by converting the fuzzy matrix Lyapunov system into a Kronecker product system we obtain sufficient conditions for controllability and observability of the system (1) satisfying (2).

The paper is well organized as follows. In Section 2, we present some basic definitions and results relating to fuzzy sets [13] and Kronecker product of matrices. Further, we obtain a unique solution of the system (1), when 𝐵(𝑡)=0,0x000a0𝑋,0x000a0𝑈 is a crisp continuous matrix. In Section 3, we generate a fuzzy dynamical Lyapunov system, and also obtain its solution set. In Section 4, we present a sufficient condition for the controllability of the system and illustrate the results by suitable examples. In Section 5, we obtain a sufficient condition for the observability of the fuzzy dynamical Lyapunov system, and the theorem is highlighted by a suitable example. Finally, in Section 6, we present some conclusions and future works.

This paper extends some of the results of Ding and Kandel [13, 14] developed for system of fuzzy differential equations to fuzzy matrix Lyapunov systems and includes their results as a particular case, when 𝑌, and 𝑛 are column vectors of order 𝑃𝑘(𝑅𝑛).

2. Preliminaries

In this section, we present some definitions and results relating to fuzzy sets [13] and Kronecker product of matrices.

Let 𝑅𝑛 denote the family of all nonempty compact convex subsets of 𝑃𝑘(𝑅𝑛). Define the addition and scalar multiplication in 𝑃𝑘(𝑅𝑛) as usual. Radstrom [15] states that 𝛼,𝛽𝑅 is a commutative semigroup under addition, which satisfies the cancellation law. Moreover, if 𝐴,𝐵𝑃𝑘(𝑅𝑛) and 𝛼(𝐴+𝐵)=𝛼𝐴+𝛼𝐵,𝛼(𝛽𝐴)=(𝛼𝛽)𝐴,1𝐴=𝐴,(3), then 𝛼,𝛽0 and if (𝛼+𝛽)𝐴=𝛼𝐴+𝛽𝐴, then 𝐴. The distance between 𝐵 and 𝑑(𝐴,𝐵)=inf𝜖𝐴𝑁(𝐵,𝜖),0x000a0𝐵𝑁(𝐴,𝜖),(4) is defined by the Hausdorff metric 𝑁(𝐴,𝜖)=𝑥𝑅𝑛𝑥𝑦<𝜖,0x000a0for0x000a0some0x000a0𝑦𝐴.(5) where 𝐹𝐽𝑃𝑘(𝑅𝑛)

Definition 1. A set-valued function 𝑢𝑅𝑛 is said to be measurable if it satisfies any one of the following equivalent conditions: (1)for all 𝑡𝑑𝐹(𝑡)(𝑢)=inf𝑣𝐹(𝑡)𝑢𝑣, Gr𝐹={(𝑡,𝑢)𝐽×𝑅𝑛𝑢𝐹(𝑡)}Σ×𝛽(𝑅𝑛) is measurable,(2)Σ,𝛽(𝑅𝑛), where 𝜎 are Borel 𝐽-field of 𝑅𝑛 and {𝑓𝑛()}𝑛1, respectively (Graph measurability),(3)there exists a sequence 𝐹(𝑡)={𝑓𝑛()}𝑛1 of measurable functions such that 𝑡𝐽, for all 𝑆1𝐹 (Castaing's representation).

We denote by 𝐹() the set of all selections of 𝐿1𝑅𝑛(𝐽) that belong to the Lebesgue Bochner space 𝑆1𝐹=𝑓()𝐿1𝑅𝑛(𝐽)𝑓(𝑡)𝐹(𝑡)0x000a0a.e..(6), that is, (𝐴)𝐽𝐹(𝑡)𝑑𝑡=𝐽𝑓(𝑡)𝑑𝑡,𝑓()𝑆1𝐹.(7) We present the Aumann's integral as follows: 𝐹𝐽𝑃𝑘(𝑅𝑛)

We say that 𝐽𝑅 is integrably bounded if it is measurable and there exists a function 𝐿1𝑅𝑛(𝐽),, 𝑢(𝑡) such that 𝑢𝐹(𝑡), 𝐹. From [16], we know that if 𝐽𝐹(𝑡)𝑑𝑡 is a closed valued measurable multifunction, then 𝑅𝑛 is convex in 𝐹. Furthermore, if 𝐽𝐹(𝑡)𝑑𝑡 is integrably bounded, then 𝑅𝑛 is compact in 𝐸𝑛=𝑢𝑅𝑛0,1/𝑢0x000a0satise(i)-(iv)below,(8).

Let 𝑢 where (i)𝑥0𝑅𝑛 is normal, that is, there exists an 𝑢(𝑥0)=1 such that 𝑢;(ii)𝑥,𝑦𝑅𝑛 is fuzzy convex, that is, for 0𝜆1 and 𝑢𝜆𝑥+(1𝜆)𝑦min𝑢(𝑥),𝑢(𝑦);(9), 𝑢(iii)[𝑢]0={𝑥𝑅𝑛/𝑢(𝑥)>0} is upper semicontinuous;(iv)0<𝛼1 is compact. For 𝛼, the [𝑢]𝛼={𝑥𝑅𝑛/𝑢(𝑥)𝛼}-level set is denoted and defined by [𝑢]𝛼𝑃𝑘(𝑅𝑛). Then, from (i)–(iv) it follows that 0𝛼1 for all 𝐷𝐸𝑛×𝐸𝑛[0,).

Define 𝑑𝐷(𝑢,𝑣)=sup[𝑢]𝛼,[𝑣]𝛼/𝛼[0,1],(10) by 𝑑 where 𝑃𝑘(𝑅𝑛) is the Hausdorff metric defined in 𝐷. It is easy to show that 𝐸𝑛 is a metric in (𝐸𝑛,𝐷) and using results of [15], we see that 𝐷 is a complete metric space, but not locally compact. Moreover, the distance 𝐷(𝑢+𝑤,𝑣+𝑤)=𝐷(𝑢,𝑣),𝑢,𝑣,𝑤𝐸𝑛,||𝜆||𝐷(𝜆𝑢,𝜆𝑣)=𝐷(𝑢,𝑣),𝑢,𝑣𝐸𝑛,0x000a0𝜆𝑅,𝐷(𝑢+𝑤,𝑣+𝑧)𝐷(𝑢,𝑣)+𝐷(𝑤,𝑧),𝑢,𝑣,𝑤,𝑧𝐸𝑛.(11) verifies that (𝐸𝑛,𝐷) We note that 𝑢𝐸𝑛 is not a vector space. But it can be imbedded isomorphically as a cone in a Banach space [15].

Regarding fundamentals of differentiability and integrability of fuzzy functions, we refer to Kaleva [17] and Lakshmikantham and Mohapatra [18].

In the sequel, we need the following representation theorem.

Theorem 1 (see [19]). If [𝑢]𝛼𝑃𝑘(𝑅𝑛), then (1) 0𝛼1, for all [𝑢]𝛼2[𝑢]𝛼1; (2) 0𝛼1𝛼21, for all {𝛼𝑘}; (3) if 𝛼>0 is a nondecreasing sequence converging to [𝑢]𝛼=𝑘1[𝑢]𝛼𝑘, then {𝐴𝛼0𝛼1}. Conversely, if 𝑅𝑛 is a family of subsets of 𝑢𝐸𝑛 satisfying (1)–(3), then there exists a [𝑢]𝛼=𝐴𝛼 such that 0<𝛼1 for [𝑢]0=0𝛼1𝐴𝛼𝐴0 and 𝐹𝐽𝐸𝑛.

A fuzzy set-valued mapping 𝐹0(𝑡) is called fuzzy integrably bounded if 𝐹𝐽𝐸𝑛 is integrably bounded.

Definition 2. Let 𝐹 be a fuzzy integrably bounded mapping. The fuzzy integral of 𝐽 over 𝐽𝐹(𝑡)𝑑𝑡 denoted by 𝐽𝐹(𝑡)𝑑𝑡𝛼=(𝐴)𝐽𝐹𝛼(𝑡)𝑑𝑡,0<𝛼1.(12) is defined level-set-wise by 𝐹𝐽×𝐸𝑛𝐸𝑛
Let 𝑢=𝐹(𝑡,𝑢),𝑢(0)=𝑢0.(13),and consider the fuzzy differential equation 𝑢𝐽𝐸𝑛

Definition 3. A mapping 𝑢(𝑡)=𝑢0+𝑡0𝐹𝑠,𝑢(𝑠)𝑑𝑠,𝑡𝐽.(14) is a fuzzy weak solution to (13) if it is continuous and satisfies the integral equation 𝐹 If 𝐴𝐶𝑚×𝑛 is continuous, then this weak solution also satisfies (13) and we call it fuzzy strong solution to (13).

Now, we present some properties and rules for Kronecker products and basic results related to matrix Lyapunov systems.

Definition 4 (see [2]). Let 𝐵𝐶𝑝×𝑞 and 𝐴.Then the Kronecker product of 𝐵 and 𝐴𝐵 written 𝑎𝐴𝐵=11𝐵𝑎12𝐵𝑎1𝑛𝐵𝑎21𝐵𝑎22𝐵𝑎2𝑛𝐵𝑎...𝑚1𝐵𝑎𝑚2𝐵𝑎𝑚𝑛𝐵(15) is defined to be the partitioned matrix 𝑚𝑝×𝑛𝑞 which is an 𝐶𝑚𝑝×𝑛𝑞 matrix and is in 𝐴=[𝑎𝑖𝑗]𝐶𝑚×𝑛.

Definition 5 (see [2]). Let 𝐴𝐴=Vec𝐴=.1𝐴.2𝐴.𝑛,where0x000a0𝐴.𝑗=𝑎1𝑗𝑎2𝑗𝑎𝑚𝑗1𝑗𝑛.(16); one denotes (𝐴𝐵)=𝐴𝐵 The Kronecker product has the following properties and rules [2].
(1)𝐴 (𝐴 denotes transpose of (𝐴𝐵)1=𝐴1𝐵1).(2)𝐴𝐵)(𝐶𝐷)=(𝐴𝐶𝐵𝐷).(3)The mixed product rule(𝐴𝐵=𝐴𝐵. This rule holds, provided the dimension of the matrices is such that the various expressions exist. (4)𝐴(𝑡).(5)If 𝐵(𝑡) and (𝐴𝐵)=𝐴𝐵+𝐴𝐵0x000a0(=𝑑/𝑑𝑡) are matrices, thenVec(𝐴𝑌𝐵)=(𝐵𝐴)Vec𝑌.(6)𝐴.(7)If 𝐵 and 𝑛×𝑛 are matrices both of order Vec(𝐴𝑋)=(𝐼𝑛𝐴)Vec𝑋, then (i)Vec(𝑋𝐴)=(𝐴𝐼𝑛)Vec𝑋,(ii)𝑋𝐼(𝑡)=𝐺(𝑡)𝑋(𝑡)+𝑛𝑋𝐹(𝑡)𝑈(𝑡),𝑋(0)=0𝐼,(17)𝑌(𝑡)=𝑛𝐼𝐶(𝑡)𝑋(𝑡)+𝑛𝐷(𝑡)𝑈(𝑡),(18).

Now by applying the Vec operator to the matrix Lyapunov system (1) satisfying (2) and using the above properties, we have 𝐺(𝑡)=(𝐵𝐼𝑛)+(𝐼𝑛𝐴)𝑛2×𝑛2 where 𝑋=Vec𝑋(𝑡) is an 𝑈=Vec𝑈(𝑡) matrix and 𝑛2, 𝑋𝑋(𝑡)=𝐺(𝑡)𝑋(𝑡),𝑋(0)=0.(19) are column matrices of order 𝜙(𝑡).

The corresponding linear homogeneous system of (17) is 𝜓(𝑡)

Lemma 1. Let 𝑋(𝑡)=𝐴(𝑡)𝑋(𝑡),𝑋(0)=𝐼𝑛,(20)[𝑋(𝑡)]=𝐵(𝑡)𝑋(𝑡),𝑋(0)=𝐼𝑛,(21) and 𝜓(𝑡)𝜙(𝑡) be the fundamental matrices for the systems 𝑋𝑋(𝑡)=(𝜓(𝑡)𝜙(𝑡))0𝜓(𝑡)𝜙(𝑡)=𝜓(𝑡)+𝜙(𝑡)𝜓(𝑡)𝜙=𝐵(𝑡)+=𝐵(𝑡)𝜓(𝑡)𝜙(𝑡)𝜓(𝑡)𝐴(𝑡)𝜙(𝑡)(𝑡)𝐼𝑛+𝐼𝜓(𝑡)𝜙(𝑡)𝑛=𝐵𝐴(𝑡)𝜓(𝑡)𝜙(𝑡)(𝑡)𝐼𝑛+𝐼𝑛.𝐴(𝑡)𝜓(𝑡)𝜙(𝑡)=𝐺(𝑡)𝜓(𝑡)𝜙(𝑡)(22) respectively. Then the matrix 𝜓(0)𝜙(0)=𝐼𝑛𝐼𝑛=𝐼𝑛2 is a fundamental matrix of (19) and the solution of (19) is 𝜓(𝑡)𝜙(𝑡).

Proof. Consider 𝑋𝑋(𝑡)=(𝜓(𝑡)𝜙(𝑡))0 Also 𝜙(𝑡).
Hence, 𝜓(𝑡) is a fundamental matrix of (19). Clearly, 𝑋𝑋(𝑡)=𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝐹(𝑠)𝑈(𝑠)𝑑𝑠.(23) is a solution of (19).

Theorem 2. Let 𝑋𝑋(𝑡)=(𝜓(𝑡)𝜙(𝑡))0+𝑋(𝑡) and 𝑋(𝑡) be the fundamental matrices for the systems (20) and (21). Then the unique solution of the initial value problem (17) is given by 𝑋(𝑡)=𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝐹(𝑠)𝑈(𝑠)𝑑𝑠.(24)

Proof. First we show that the solution of (17) is of the form 𝑢(𝑡), where 𝑤(𝑡)=𝑢(𝑡)𝑋(𝑡) is a particular solution of (17) and is given by 𝑤
Let 𝑋𝑤=(𝜓(𝑡)𝜙(𝑡))0 be any other solution of (17), write 𝑋𝑢(𝑡)=(𝜓(𝑡)𝜙(𝑡))0+𝑋(𝑡), then 𝑋(𝑡)=(𝜓(𝑡)𝜙(𝑡))𝑣(𝑡) satisfies (19), hence 𝑣(𝑡), 𝑋(𝑡)=𝜓(𝑡)𝜙(𝑡)𝑣𝑣(𝑡)+𝜓(𝑡)𝜙(𝑡)𝐼(𝑡)𝐺(𝑡)𝑋(𝑡)+𝑛𝑣𝐹(𝑡)𝑈(𝑡)=𝐺(𝑡)𝜓(𝑡)𝜙(𝑡)𝑣(𝑡)+𝜓(𝑡)𝜙(𝑡)𝑣(𝑡)𝜓(𝑡)𝜙(𝑡)=𝐼(𝑡)𝑛𝐹(𝑡)𝑈(𝑡)𝑣=𝜓(𝑡)1(𝑡)𝜙1𝐼(𝑡)𝑛=𝐹(𝑡)𝑈(𝑡)𝑣(𝑡)𝑡0𝜓1(𝑠)𝜙1𝐼(𝑠)𝑛𝐹(𝑠)𝑈(𝑠)𝑑𝑠.(25).
Consider the vector 𝜙(𝑡)𝜙1(𝑠)=𝜙(𝑡𝑠), where 𝜓(𝑡)𝜓1(𝑠)=𝜓(𝑡𝑠) is an arbitrary vector to be determined so as to satisfy (17), 𝑢𝑖(𝑡)𝐸1 Hence, the desired expression follows immediately by noting the fact that 0x000a0𝑡𝐽 and 0x000a0𝑖=1,2,,𝑛2.

3. Formation of Fuzzy Dynamical Lyapunov Systems

Let 𝑢𝑈(𝑡)=1(𝑡),𝑢2(𝑡),,𝑢𝑛2(𝑡)=𝑢1(𝑡)×𝑢2(𝑡)××𝑢𝑛2=𝑢(𝑡)𝛼1(𝑡),𝑢𝛼2(𝑡),,𝑢𝛼𝑛2=(𝑡)𝛼[0,1]𝑢1(𝑡),𝑢2(𝑡),,𝑢𝑛2(𝑡)𝑢𝑖(𝑡)𝑢𝛼𝑖,(𝑡),0x000a0𝛼[0,1](26), 𝑢𝛼𝑖(𝑡), 𝛼, and define 𝑢𝑖(𝑡) where 𝑈(𝑡) is the 𝑈(𝑡)𝐸𝑛2-level set of 𝑈(𝑡). From the above definition of 𝑋𝐼(𝑡)=𝐺(𝑡)𝑋(𝑡)+𝑛𝑋𝐹(𝑡)𝑈(𝑡),𝑋(0)=0𝐼,(27)𝑌(𝑡)=𝑛𝐼𝐶(𝑡)𝑋(𝑡)+𝑛𝐷(𝑡)𝑈(𝑡)(28) and Theorem 1, it can be easily seen that 𝑈(𝑡).

Now by using the fuzzy control 𝐸𝑛2, we show that the following system 𝑈𝛼=𝑢1(𝑡)×𝑢2(𝑡)××𝑢𝑛2(𝑡)𝑅𝑛2 determines a fuzzy system.

Assume that 𝑇 is continuous in 𝑋𝐼(𝑡)𝐺(𝑡)𝑋(𝑡)+𝑛𝑈𝐹(𝑡)𝛼𝑋(𝑡),𝑡[0,𝑇],(29)𝑋(0)=0.(30). The set 𝑋𝛼 is a convex and compact set in [𝑋(𝑡)]𝛼𝑃𝑘(𝑅𝑛2). For any positive number 0𝛼1, consider the following differential inclusions: 𝑡[0,𝑇]𝑋𝛼 Let 𝐶[[0,𝑇],𝑅𝑛2] be the solution of (29) satisfying (30).

Claim (i). 𝑈𝛼(𝑡), for every 𝑋𝛼, 𝐾=max𝑡[0,𝑇]𝜙(𝑡).
First, we prove that 𝐿=max𝑡[0,𝑇]𝜓(𝑡) is nonempty, compact, and convex in 𝑈𝑀=max{𝑢(𝑡)𝑢(𝑡)𝛼(𝑡),0x000a0𝑡[0,𝑇]}. Since 𝑁=max𝑡[0,𝑇] has measurable selection, we have that 𝐹(𝑡) is nonempty.
Let 𝑋𝑋𝛼, 𝑈𝑢(𝑡)𝛼(𝑡), 𝑋𝑋(𝑡)=𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝐹(𝑠)𝑢(𝑠)𝑑𝑠.(31), 𝑋𝑋(𝑡)𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝑋𝐹(𝑠)𝑢(𝑠)𝑑𝑠𝜓(𝑡)𝜙(𝑡)0+𝑡0𝑋𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝐹(𝑠)𝑢(𝑠)𝑑𝑠𝐾𝐿0+𝐾𝐿𝑁𝑀𝑇.(32).
If for any 𝑡1,𝑡2[0,𝑇], then there is a selection 𝑋(𝑡1)𝑋(𝑡2)=𝜓(𝑡1)𝜙(𝑡1)𝑋0+𝑡10𝜓(𝑡1𝑠)𝜙(𝑡1𝐼𝑠)𝑛𝐹(𝑠)𝑢(𝑠)𝑑𝑠𝜓(𝑡2)𝜙(𝑡2)𝑋0𝑡20𝜓(𝑡2𝑠)𝜙(𝑡2𝐼𝑠)𝑛𝐹(𝑠)𝑢(𝑠)𝑑𝑠.(33) such that 𝑋(𝑡1)𝑋(𝑡2)𝜓(𝑡1)𝜙(𝑡1)𝜓(𝑡2)𝜙(𝑡2)𝑋0+𝑡1𝑡2𝜓(𝑡1𝑠)𝜙(𝑡1𝐼𝑠)𝑛+𝐹(𝑠)𝑢(𝑠)𝑑𝑠𝑡20𝜓(𝑡1𝑠)𝜙(𝑡1𝑠)0x000a0𝜓(𝑡2𝑠)𝜙(𝑡2𝐼𝑠)𝑛𝐹(𝑠)𝑢(𝑠)𝑑𝑠𝜓(𝑡1)𝜙(𝑡1)𝜓(𝑡2)𝜙(𝑡2)𝑋0||𝑡+𝐾𝐿𝑁𝑀1𝑡2||+𝑀𝑁𝑇0𝜓(𝑡1𝑠)𝜙(𝑡1𝑠)0x000a0𝜓(𝑡2𝑠)𝜙(𝑡2𝑠)𝑑𝑠.(34) Then 𝜙(𝑡) Thus 𝜓(𝑡) is bounded.
For any [0,𝑇], 𝑋 Therefore 𝑋𝛼 Since 𝑋𝛼 and 𝑋𝑘𝑋𝛼 are both uniformly continuous on 𝑋𝑘𝑋, 𝑋𝑘 is equicontinuous. Thus, 𝑢𝑘𝑈𝛼(𝑡) is relatively compact. If 𝑋𝑘𝑋(𝑡)=𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝑢𝐹(𝑠)𝑘(𝑠)𝑑𝑠.(35) is closed, then it is compact.
Let 𝑢𝑘𝑈𝛼(𝑡) and {𝑢𝑘𝑗}. For each {𝑢𝑘}, there is a 𝑈𝑢𝛼(𝑡) such that 𝜆𝑗>0 Since 𝜆𝑗=1 is closed, then there exists a subsequence 𝜆𝑗𝑢𝑘𝑗 of 𝑢 converging weakly to 𝜆𝑗𝑋𝑘𝑗=𝜆(𝑡)𝑗𝑋𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝜆𝐹(𝑠)𝑗𝑢𝑘𝑗(𝑠)𝑑𝑠.(36). From Mazur's theorem [20], there exists a sequence of numbers 𝑗, 𝑋𝑋(𝑡)=𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝐹(𝑠)𝑢(𝑠)𝑑𝑠.(37) such that 𝑋𝑋(𝑡)𝛼 converges strongly to 𝑋𝛼.
Thus, from (35) we have 𝑋1 From Fatou's lemma, taking the limit as 𝑋2𝑋𝛼 on both sides of (36), we have 𝑢1,𝑢2𝑈𝛼(𝑡) Thus, 𝑋1𝑋(𝑡)=𝐺(𝑡)1𝐼(𝑡)+𝑛𝑢𝐹(𝑡)1𝑋(𝑡),2𝑋(𝑡)=𝐺(𝑡)2𝐼(𝑡)+𝑛𝑢𝐹(𝑡)2(𝑡).(38), and hence 𝑋𝑋=𝜆1𝑋(𝑡)+(1𝜆)2(𝑡) is closed.
Let 0𝜆1, 𝑋𝑋=𝜆1𝑋(𝑡)+(1𝜆)2𝑋(𝑡)=𝜆𝐺(𝑡)1𝐼(𝑡)+𝑛𝑢𝐹(𝑡)1𝑋(𝑡)+(1𝜆)𝐺(𝑡)2𝐼(𝑡)+𝑛𝑢𝐹(𝑡)2𝜆𝑋(𝑡)=𝐺(𝑡)1𝑋(𝑡)+(1𝜆)2+𝐼(𝑡)𝑛𝐹(𝑡)𝜆𝑢1(𝑡)+(1𝜆)𝑢2.(𝑡)(39), then there exist 𝑈𝛼(𝑡) such that 𝜆𝑢1(𝑡)+(1𝜆)𝑢2𝑈(𝑡)𝛼(𝑡) Let 𝑋𝐼(𝑡)𝐺(𝑡)𝑋(𝑡)+𝑛𝑈𝐹(𝑡)𝛼(𝑡),(40), 𝑋𝑋𝛼, then 𝑋𝛼 Since 𝑋𝛼 is convex, 𝐶[[0,𝑇],𝑅𝑛2], we have [𝑋(𝑡)]𝛼 that is 𝑅𝑛2. Thus 𝑡[0,𝑇] is convex. Therefore, [𝑋(𝑡)]𝛼 is nonempty, compact, and convex in 𝑅𝑛2. Thus, from Arzela-Ascoli theorem, we know that [𝑋(𝑡)]𝛼𝑃𝑘(𝑅𝑛2) is compact in 𝑡[0,𝑇] for every [𝑋(𝑡)]𝛼2[𝑋(𝑡)]𝛼1. Also it is obvious that 0𝛼1𝛼21 is convex in 0𝛼1𝛼21. Thus, we have 𝑈𝛼2𝑈(𝑡)𝛼1(𝑡), for every 𝑈𝛼2(𝑡)=𝑢𝛼21(𝑡)×𝑢𝛼22(𝑡)××𝑢𝛼2𝑛2(𝑡)𝑢𝛼11(𝑡)×𝑢𝛼12(𝑡)××𝑢𝛼1𝑛2=(𝑡)𝑈𝛼1(𝑡).(41). Hence the claim.

Claim (ii). 𝑆1𝑈𝛼2(𝑡)𝑆1𝑈𝛼1(𝑡), for all 𝑋𝐼(𝑡)𝐺(𝑡)𝑋+𝑛𝑈𝐹(𝑡)𝛼2𝐼(𝑡)𝐺(𝑡)𝑋+𝑛𝑈𝐹(𝑡)𝛼1(𝑡).(42).
Let 𝑋𝐼(𝑡)𝐺(𝑡)𝑋+𝑛𝑈𝐹(𝑡)𝛼2𝑋(𝑡),𝑡[0,𝑇],(43)𝐼(𝑡)𝐺(𝑡)𝑋+𝑛𝑈𝐹(𝑡)𝛼1(𝑡),𝑡[0,𝑇].(44). Since 𝑋𝛼2, we have 𝑋𝛼1 Thus, we have the selection inclusion 𝑋𝑋(𝑡)𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝑆𝐹(𝑠)1𝑈𝛼2(𝑠)𝑋𝑑𝑠𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝑆𝐹(𝑠)1𝑈𝛼1(𝑠)𝑑𝑠.(45) and the following inclusion: 𝑋𝛼2𝑋𝛼1 Consider the differential inclusions 𝑋𝛼2𝑋(𝑡)𝛼1(𝑡){𝛼𝑘} Let 𝛼>0 and 𝑋𝛼(𝑡)=𝑘1𝑋𝛼𝑘(𝑡) be the solution sets of (43) and (44), respectively. Clearly, the solution of (43) satisfies the following inclusion: 𝑈𝛼𝑘(𝑡)=𝑢𝛼𝑘1×𝑢𝛼𝑘2××𝑢𝛼𝑘𝑛2 Thus 𝑈𝛼(𝑡)=𝑢𝛼1×𝑢𝛼2××𝑢𝛼𝑛2, and hence 𝑋𝐼(𝑡)𝐺(𝑡)𝑋+𝑛𝑈𝐹(𝑡)𝛼𝑘𝑋(𝑡),(46)𝐼(𝑡)𝐺(𝑡)𝑋+𝑛𝑈𝐹(𝑡)𝛼(𝑡).(47). Hence the claim.

Claim (iii). If 𝑋𝛼𝑘 is a nondecreasing sequence converging to 𝑋𝛼, then 𝑢𝑖(𝑡).
Let 𝑢𝛼𝑖=𝑘1𝑢𝛼𝑘𝑖.(48), 𝑈𝛼(𝑡)=𝑢𝛼1×𝑢𝛼2××𝑢𝛼𝑛2=𝑘1𝑢𝛼𝑘1×𝑘1𝑢𝛼𝑘2××𝑘1𝑢𝛼𝑘𝑛2=𝑘1𝑢𝛼𝑘1×𝑢𝛼𝑘2××𝑢𝛼𝑘𝑛2=𝑘1𝑈𝛼𝑘(𝑡)(49) and consider the inclusions 𝑆1𝑈𝛼(𝑡)=𝑆1𝑘1𝑈𝛼𝑘(𝑡)𝑋𝐼(𝑡)𝐺(𝑡)𝑋+𝑛𝑈𝐹(𝑡)𝛼𝐼(𝑡)=𝐺(𝑡)𝑋+𝑛𝐹(𝑡)𝑘1𝑈𝛼𝑘𝐼(𝑡)𝐺(𝑡)𝑋+𝑛𝑈𝐹(𝑡)𝛼𝑘(𝑡),𝑘=1,2,.(50) Let 𝑋𝛼𝑋𝛼𝑘 and 𝑘=1,2, be the solution sets of (46) and (47), respectively. Since 𝑋𝛼𝑘1𝑋𝛼𝑘.(51) is a fuzzy set and from Theorem 1, we have 𝑋 Consider 𝑋𝐼(𝑡)𝐺(𝑡)𝑋+𝑛𝑈𝐹(𝑡)𝛼𝑘(𝑡),𝑘1.(52) and then 𝑋𝑋(𝑡)𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝑆𝐹(𝑠)1𝑈𝛼𝑘(𝑡)𝑑𝑠.(53). Therefore 𝑋𝑋(𝑡)𝜓(𝑡)𝜙(𝑡)0+𝑘1𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝑆𝐹(𝑠)1𝑈𝛼𝑘𝑋𝑑𝑠𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝑆𝐹(𝑠)1𝑘1𝑈𝛼𝑘=𝑋𝑑𝑠𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝑆𝐹(𝑠)1𝑈𝛼𝑑𝑠.(54) Thus, we have 𝑋𝑋𝛼, 𝑘1𝑋𝛼𝑘𝑋𝛼.(55), which implies that 𝑋𝛼=𝑘1𝑋𝛼𝑘,(56) Let 𝑋𝛼(𝑡)=𝑘1𝑋𝛼𝑘(𝑡).(57) be the solution set to the inclusion 𝑋(𝑡)𝐸𝑛2 Then, [0,𝑇] It follows that 𝑋𝛼(𝑡) This implies that 𝑋𝐼(𝑡)=𝐺(𝑡)𝑋(𝑡)+𝑛𝑋𝐹(𝑡)𝑈(𝑡),𝑋(0)=0𝐼,(58)𝑌(𝑡)=𝑛𝐼𝐶(𝑡)𝑋(𝑡)+𝑛𝐷(𝑡)𝑈(𝑡).(59). Therefore, 𝑋𝑋(𝑡)𝜓(𝑡)𝜙(𝑡)0+𝑡0𝐼𝜓(𝑡𝑠)𝜙(𝑡𝑠)𝑛𝐹(𝑠)𝑈(𝑠)𝑑𝑠.(60) From (51) and (55), we have 𝑈(𝑡)=𝑢1(𝑡)×𝑢2(𝑡)××𝑢𝑖(𝑡)××𝑢𝑛2(𝑡),(61) and hence, 𝑢𝑘(𝑡)𝑅1 From Claims (i)(iii) and applying Theorem 1, there exists 𝑘𝑖, on 𝑖 such that 𝐸1 is a solution set to the differential inclusions (29) and (30). Hence, the system (27), (28) is a fuzzy dynamical Lyapunov system, and it can be expressed as 𝑋𝑋(0)=0𝑋𝑓 The solution set of the fuzzy dynamical system (58), (59) is given by 𝑡1>0

Remark 1. Consider a special case. If the input is in the form 𝑈(𝑡) where 0𝑡𝑡1, 𝑋(𝑡1𝑋)=𝑓 are crisp numbers, then the 𝐹th component of the solution set of (27) is a fuzzy set in 𝑇0𝐹𝑑𝑡=𝑇𝐹.

Proof. The proof follows along similar lines as in the above discussion.

4. Controllability of Fuzzy Dynamical Lyapunov Systems

In this section, we discuss the concept of controllability of the fuzzy system (58) satisfying (59).

Definition 6. The fuzzy system (58), (59) is said to be completely controllable if for any initial state [𝐹]𝛼 and any given final state 𝛼 there exists a finite time 𝐹 and a control 𝑇0𝐹𝑑𝑡𝛼=𝑇0[𝐹]𝛼𝑑𝑡=𝑇[𝐹]𝛼.(62), 𝑇0𝐹𝑑𝑡=𝑇𝐹, such that 𝑃,0x000a0𝑄.

Lemma 2. If (𝑡) is a fuzzy set, then [0,𝑇].

Proof. Let 𝑇0(𝑡)𝑃𝑑𝑡=𝑇0(𝑡)𝑄𝑑𝑡(63) be the 𝑃=𝑄-level set of 𝛼. Since 𝑇0(𝑡)[𝑃]𝛼𝑑𝑡=𝑇0(𝑡)𝑃𝑑𝑡𝛼=𝑇0(𝑡)𝑄𝑑𝑡𝛼=𝑇0(𝑡)[𝑄]𝛼𝑑𝑡.(64) From the definition of fuzzy set, we have 𝑃𝑄.

Lemma 3. Let 𝛼[0,1] be two fuzzy sets and let [𝑃]𝛼[𝑄]𝛼 be a nonzero continuous function on 𝑃,𝑄𝐸1, satisfying 𝑃𝛼=[𝑃min(𝛼),𝑃max(𝛼)] then 𝑄𝛼=[𝑄min(𝛼),𝑄max(𝛼)].

Proof. For each 𝑃min(𝛼)𝑄min(𝛼)-level, we have 𝑃max(𝛼)𝑄max(𝛼) Suppose that 𝑇0(𝑡)𝑃min(𝛼)𝑑𝑡𝑇0(𝑡)𝑄min(𝛼)𝑑𝑡.(65), then for some 𝑇0(𝑡)𝑃max(𝛼)𝑑𝑡𝑇0(𝑡)𝑄max(𝛼)𝑑𝑡.(66), we have 𝑇0𝑃(𝑡)min(𝛼),𝑃max(𝛼)𝑑𝑡𝑇0𝑄(𝑡)min(𝛼),𝑄max(𝛼)𝑑𝑡.(67). Without loss of generality, we assume that 𝑇0(𝑡)𝑃𝛼𝑑𝑡𝑇0(𝑡)𝑄𝛼𝑑𝑡,(68). Let 𝑃=𝑄 and 𝑢,𝑣𝐸1. Then, we have either (i) 𝑘𝑅1 or (ii) [𝑢]𝛼 holds.
If (i) holds, then 𝛼
If (ii) holds, then 𝑢 Thus, in both cases (i) and (ii), we have 𝑢 This implies that 𝑣 which is a contradiction to (64). Hence [𝑢+𝑣]𝛼=[𝑢]𝛼+[𝑣]𝛼=𝑎+𝑏𝑎[𝑢]𝛼,0x000a0𝑏[𝑣]𝛼,(69).

Definition 7 (see [20]). Let 𝑢, 𝑣, and let [𝑢𝑣]𝛼=[𝑢]𝛼[𝑣]𝛼=𝑎𝑏𝑎[𝑢]𝛼,0x000a0𝑏[𝑣]𝛼,(70) be the [𝑘𝑢]𝛼=𝑘[𝑢]𝛼=𝑘𝑎𝑎[𝑢]𝛼.(71)-level set of 𝑥,𝑦𝐸𝑛2. One defines the sum of 𝑥=𝑥1×𝑥2××𝑥𝑛2 and 𝑦=𝑦1×𝑦2××𝑦𝑛2,0x000a0𝑥𝑖,𝑦𝑖𝐸1,0x000a0𝑖=1,2,,𝑛2 by 𝑦=𝑧+𝑥 the difference between 𝑧=𝑦𝑥 and [𝑧]𝛼=[𝑦𝑥]𝛼=[𝑦]𝛼[𝑥]𝛼=[𝑦1]𝛼[𝑥1]𝛼[𝑦𝑛2]𝛼[𝑥𝑛2]𝛼.(72) by 𝑦=𝑤𝑥 and the scalar product by 𝑤=𝑦+𝑥

Definition 8 (see [20]). Let [𝑤]𝛼=[𝑦+𝑥]𝛼=[𝑦]𝛼+[𝑥]𝛼=[𝑦1]𝛼+[𝑥1]𝛼[𝑦𝑛2]𝛼+[𝑥𝑛2]𝛼.(73) and 𝑐𝐶=11𝑐12𝑐1𝑛2𝑐21𝑐22𝑐2𝑛2𝑐𝑛21𝑐𝑛22𝑐𝑛2𝑛2(74), 𝑛2×𝑛2. If 𝑝=𝑝1×𝑝2××𝑝𝑛2, then 𝑝𝑖𝐸1 which is defined by 𝑖=1,2,,𝑛2 If 𝐸𝑛2, then [𝑝𝑖]𝛼 which is defined by 𝛼

Definition 9 (see [20]). Let 𝑝𝑖 be an 𝐶𝑝 matrix, 𝐶, let 𝑝, [𝐶𝑝]𝛼=𝐶[𝑝]𝛼=𝑐11𝑐12𝑐1𝑛2𝑐21𝑐22𝑐2𝑛2𝑐𝑛21𝑐𝑛22𝑐𝑛2𝑛2[𝑝1]𝛼[𝑝2]𝛼[𝑝𝑛2]𝛼=𝑐11[𝑝1]𝛼++𝑐1𝑛2[𝑝𝑛2]𝛼𝑐21[𝑝1]𝛼++𝑐2𝑛2[𝑝𝑛2]𝛼𝑐𝑛21[𝑝1]𝛼++𝑐𝑛2𝑛2[𝑝𝑛2]𝛼.(75), be a fuzzy set in 𝐶𝑝, and let 𝐸𝑛2 be 𝑛2×𝑛2-level sets of 𝑊(0,𝑇)=𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛×𝐼𝐹(𝑡)𝑛𝐹(𝑡)𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑑𝑡(76). Define the product 𝑈(𝑡) of 𝑋𝑋(0)=0 and 𝑋𝑋(𝑇)=𝑓=(𝑥𝑓1,𝑥𝑓2,,𝑥𝑓𝑛2) as 𝑅IF𝑥10x000a0is0x000a0in0x000a0𝑥𝑓1,,𝑥𝑛20x000a0is0x000a0in0x000a0𝑥𝑓𝑛2,0x000a0THEN0x000a0𝑢10x000a0is0x000a0in0x000a0𝑢1,,𝑢𝑛20x000a0is0x000a0in0x000a0𝑢𝑛2,(77)

All these definitions yield the following lemma.

Lemma 4. 𝑢1(𝑡),𝑢2(𝑡),,𝑢𝑖(𝑡),,𝑢𝑛2=1(𝑡)𝑇𝐼𝑛𝐹(𝑡)1𝜓(𝑇𝑡)𝜙(𝑇𝑡)1×𝑥1(𝑇),𝑥2(𝑇),,𝑥𝑓𝑖,,𝑥𝑛2𝐼(𝑇)𝑛𝐹(𝑡)𝜓(𝑇𝑡)𝜙(𝑇𝑡)×𝑊1𝑋(0,𝑇)𝜓(𝑇)𝜙(𝑇)0,𝑖=1,2,,𝑛2.(78) is a fuzzy set in 𝑊(0,𝑇).

Proof. The proof is similar to proof of Lemma 3.1 [13].

Theorem 3. The fuzzy system (58),(59) is completely controllable if the 𝑊1(0,𝑇) symmetric controllability matrix 𝑊1𝑋(0,𝑇)(𝜓(𝑇)𝜙(𝑇))0 is nonsingular. Furthermore,the fuzzy control 𝑋𝜓(𝑇)𝜙(𝑇)0=𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛×𝐼𝐹(𝑡)𝑛𝐹(𝑡)𝜓(𝑇𝑡)𝜙(𝑇𝑡)×𝑊1𝑋(0,𝑇)𝜓(𝑇)𝜙(𝑇)0𝑑𝑡.(79) which transfers the state of the system from 𝑈(𝑡) to a fuzzy state 𝑋𝑋(𝑇)=𝑓=𝑋𝜓(𝑇)𝜙(𝑇)0+𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛𝐹(𝑡)𝑈(𝑡)𝑑𝑡.(80) can be determined by the following fuzzy rule base: 𝑋 where 𝑈(𝑡)

Proof. Suppose that the symmetric controllability matrix 𝑋𝑓 is nonsingular. Therefore 𝑋𝑓=1𝑇𝑇0𝑋𝑓=1𝑑𝑡𝑇𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛×𝐼𝐹(𝑡)𝑛𝐹(𝑡)1𝜓(𝑇𝑡)𝜙(𝑇𝑡)1𝑋𝑓𝑑𝑡.(81) exists. Multiplying 1𝑇𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛×𝐼𝐹(𝑡)𝑛𝐹(𝑡)1𝜓(𝑇𝑡)𝜙(𝑇𝑡)1𝑋𝑓=𝑋𝑑𝑡𝜓(𝑇)𝜙(𝑇)0+𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛𝐹(𝑡)𝑈(𝑡)𝑑𝑡.(82) on both sides of (76), we have 1𝑇𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛×𝐼𝐹(𝑡)𝑛𝐹(𝑡)1𝜓(𝑇𝑡)𝜙(𝑇𝑡)1𝑋𝑓=𝑑𝑡𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛×𝐼𝐹(𝑡)𝑛𝐹(𝑡)𝜓(𝑇𝑡)𝜙(𝑇𝑡)×𝑊1𝑋(0,𝑇)𝜓(𝑇)𝜙(𝑇)0+𝑑𝑡𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛𝐹(𝑡)𝑈(𝑡)𝑑𝑡.(83) Now our problem is to find the control 𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛=𝐹(𝑡)𝑈(𝑡)𝑑𝑡𝑇0𝐼𝜓(𝑇𝑡)𝜙(𝑇𝑡)𝑛×1𝐹(𝑡)𝑇𝐼𝑛𝐹(𝑡)1𝜓(𝑇𝑡)𝜙(𝑇𝑡)1𝑋𝑓𝐼𝑛𝐹(𝑡)𝜓(𝑇𝑡)𝜙(𝑇𝑡)×𝑊1𝑋(0,𝑇)𝜓(𝑇)𝜙(𝑇)0𝑑𝑡.(84) such that 1𝑈(𝑡)=𝑇𝐼𝑛𝐹(𝑡)1𝜓(𝑇𝑡)𝜙(𝑇𝑡)1𝑋𝑓𝐼𝑛𝐹(𝑡)𝜓(𝑇𝑡)𝜙(𝑇𝑡)×𝑊1𝑋(0,𝑇)𝜓(𝑇)𝜙(𝑇)0.(85) Since 𝑋𝑋(𝑇)=𝑓=(𝑥1(𝑇),𝑥2(𝑇),,𝑥𝑛2(𝑇)) is fuzzy and from Lemma 4, 𝑈(𝑡)=(𝑢1,𝑢2,,𝑢𝑛2) must be fuzzy, otherwise the fuzzy left side of (80) cannot be equal to the crisp right side. By Lemma 2, 𝑋(𝑇)=(𝑥1(𝑇),𝑥2(𝑇),,𝑥𝑓𝑖,,𝑥𝑛2(𝑇)) can be written as 𝑈(𝑡) From (80) and (81), we have 𝑈(𝑡)=(𝑢1,𝑢2,,𝑢𝑖,,𝑢𝑛2) Combining (79) and (82), we have 𝑖 It follows that 𝑈(𝑡) By using Lemma 3, we get 𝐸1 Now we have two special cases for (85). First, let 𝑢𝑖(𝑡) be a crisp point, then we will get a corresponding control 𝑢𝑖(𝑡), satisfying (85).
Second, let 𝜇𝑥𝑓𝑖(𝑥𝑖(𝑇)), then the corresponding control 𝑥𝑖(𝑇) will take the form 𝑥𝑓𝑖 in wich the 𝑈th component of 𝑊(0,𝑇) is a fuzzy set in ,𝑒𝐴(𝑡)=0110,𝐵(𝑡)=1001𝐹(𝑡)=𝑡00𝑒𝑡,𝜋,𝐶(𝑡)=0110𝐷(𝑡)=0000,𝑇=2,𝑋0=.1111(86). Obviously, 𝑋𝑓=(𝑥𝑓1,𝑥𝑓2,𝑥𝑓3,𝑥𝑓4)𝐸4 is in 𝑋𝑓𝛼=𝑥𝑓1𝛼𝑥𝑓2𝛼𝑥𝑓3𝛼𝑥𝑓4𝛼=[𝛼1,1𝛼][𝛼1,1𝛼]0.1(𝛼1),0.1(1𝛼)0.1(𝛼1),0.1(1𝛼).(87), the grade of the membership can be determined by 𝑥1=0.5, the grade of the membership of 𝑥2=0.25 in 𝑥3=0.05. Thus, based on the above discussion, we have a fuzzy rule base for the control 𝑥4=0.025,and is given by (77) and (78).

Remark 2. The nonsingularity of the symmetric controllability matrix 𝑥𝑓1 in Theorem 3 is only a sufficient condition but not necessary because the fuzzy rule base cannot guarantee the nonsingularity of the controllability matrix.

Example 1. Consider the fuzzy dynamical matrix Lyapunov system (1) satisfying (2) with 𝑥𝑓2 Let 𝑥𝑓3, where 𝑥𝑓4 We select the points 𝑒𝜙(𝑡)=cos𝑡sin𝑡sin𝑡cos𝑡,𝜓(𝑡)=𝑡00𝑒𝑡.(88), 𝑒𝜓(𝑡)𝜙(𝑡)=𝑡cos𝑡𝑒𝑡𝑒sin𝑡00𝑡sin𝑡𝑒𝑡cos𝑡0000𝑒𝑡cos𝑡𝑒𝑡sin𝑡00𝑒𝑡sin𝑡𝑒𝑡cos𝑡.(89), 𝐼𝜓(𝜃)𝜙(𝜃)𝑛𝐼𝐹(𝑡)𝑛𝐹(𝑡)𝜓(𝜃)𝜙(𝜃)=𝑒𝜃𝑒cos𝜃sin𝜃00sin𝜃cos𝜃0000cos𝜃sin𝜃00sin𝜃cos𝜃𝑡1000010000100001×𝑒𝑡𝑒1000010000100001𝜃=𝑒cos𝜃sin𝜃00sin𝜃cos𝜃0000cos𝜃sin𝜃00sin𝜃cos𝜃𝜋0000𝑒𝜋0000𝑒𝜋0000𝑒𝜋,(90), and 𝜃=𝜋/2𝑡 which are in 𝑊𝜋0,2=0𝜋/2𝑒𝜋0000𝑒𝜋0000𝑒𝜋0000𝑒𝜋𝜋𝑑𝑡=2𝑒𝜋1000010000100001.(91), 𝑈, 𝛼, and 𝑈𝛼=(𝑡)2𝑒𝑡𝜋𝑒1000010000100001𝜃×cos𝜃sin𝜃00sin𝜃cos𝜃0000cos𝜃sin𝜃00sin𝜃cos𝜃[𝛼1,1𝛼][𝛼1,1𝛼]0.1(𝛼1),0.1(1𝛼)0.1(𝛼1),0.1(1𝛼)𝑒𝑡1000010000100001×𝑒𝜃cos𝜃sin𝜃00sin𝜃cos𝜃0000cos𝜃sin𝜃00sin𝜃cos𝜃2𝑒𝜋𝜋1000010000100001×𝑒𝜋/2𝜋cos2𝜋sin2𝜋00sin2𝜋cos2𝜋0000cos2𝜋sin2𝜋00sin2𝜋cos21111.(92) with grades of the membership are 0.5, 0.75, 0.5, and 0.75, respectively. The fundamental matrices of (20), (21) are 𝛼 Now the fundamental matrix of (19) is 𝑈𝛼(𝑡)=2𝑒𝜋/2𝜋2(sin𝑡+cos𝑡)[𝛼1,1𝛼](sin𝑡cos𝑡)[𝛼1,1𝛼](sin𝑡+cos𝑡)0.1(𝛼1),0.1(1𝛼)(sin𝑡cos𝑡)0.1(𝛼1),0.1(1𝛼)𝜋cos𝑡sin𝑡sin𝑡+cos𝑡cos𝑡sin𝑡sin𝑡+cos𝑡(93) Consider (0.5,0.25,0.05,0.025) where 𝑈(𝑡)=𝑢1𝑢2𝑢3𝑢4=2𝑒𝜋/2𝜋20.5(sin𝑡+cos𝑡)0.75(sin𝑡cos𝑡)0.5(sin𝑡+cos𝑡)0.75(sin𝑡cos𝑡)𝜋.cos𝑡sin𝑡sin𝑡+cos𝑡cos𝑡sin𝑡sin𝑡+cos𝑡(94). Therefore, 1𝐴(𝑡)=00𝑡+111𝑡+1,𝐵(𝑡)=𝑒𝑡+1𝑡,1𝑡+101𝐹(𝑡)=02𝑡0(2𝑡)𝑒𝑡1,𝐶(𝑡)=0𝑒𝑡𝑒𝑡0,𝐷(𝑡)=0000,𝑇=1,𝑋0=.1100(95) Clearly, it is nonsingular.
Thus, from Theorem 3, the input 𝑋𝑓=(𝑥𝑓1,𝑥𝑓2,𝑥𝑓3,𝑥𝑓4)𝐸4 can be chosen by the following 𝑋𝑓𝛼=𝑥𝑓1𝛼𝑥𝑓2𝛼𝑥𝑓3𝛼𝑥𝑓4𝛼=[0.5𝛼+0.5,1][0.8𝛼+0.2,1][𝛼1,1𝛼]0.2(𝛼1),0.2(1𝛼).(96)-level sets: 𝑥1=0.75Hence, the 𝑥2=0.8-level sets of fuzzy control are given by 𝑥3=0.75 and the corresponding control function to the point 𝑥4=0.1 is 𝑥𝑓1

Example 2. Consider the fuzzy dynamical matrix Lyapunov system (1) satisfying (2) with 𝑥𝑓2 Let 𝑥𝑓3, where 𝑥𝑓4 We select the points 01𝜙(𝑡)=𝑡+10𝑡+1,𝜓(𝑡)=𝑡+1𝑡0𝑒𝑡.(97), 𝜓(𝑡)𝜙(𝑡)=(𝑡+1)2𝑡0𝑡(𝑡+1)0010𝑡+100𝑒𝑡𝑒(𝑡+1)0000𝑡𝑡+1.(98), 𝐼𝜓(1𝑡)𝜙(1𝑡)𝑛𝐼𝐹(𝑡)𝑛𝐹(𝑡)×𝜓(1𝑡)𝜙(1𝑡)=2𝑡26𝑡+50(1𝑡)𝑒1𝑡002𝑡2𝑒6𝑡+52(𝑡1)0(1𝑡)𝑒𝑡1(1𝑡)𝑒1𝑡0𝑒2(1𝑡)00(1𝑡)𝑒𝑡1.01(99), and 𝑊𝜋0,2=102𝑡26𝑡+50(1𝑡)𝑒1𝑡002𝑡2𝑒6𝑡+52(𝑡1)0(1𝑡)𝑒𝑡1(1𝑡)𝑒1𝑡0𝑒2(1𝑡)00(1𝑡)𝑒𝑡1=801𝑑𝑡303010213𝑒2012𝑒1𝑒102120012𝑒1.01(100) which are in 𝑈, 𝛼, 𝑈𝛼1(𝑡)=02𝑡𝑡1𝑒2𝑡𝑡100𝑒1𝑡02𝑡𝑡12𝑡00𝑒𝑡109𝑒0001[0.5𝛼+0.5,1][0.8𝛼+0.2,1][𝛼1,1𝛼]0.2(𝛼1),0.2(1𝛼)216𝑒(2𝑡)4𝑒209𝑒7216𝑒(1𝑡)+2(8𝑒9)𝑒1𝑡4𝑒20,7(101), and (0.75,0.8,0.75,0.1) with grades of the membership being 0.5, 0.75, 0.25, and 0.5, respectively. The fundamental matrices of (20), (21) are 𝑈(𝑡)=𝑢1𝑢2𝑢3