Advances in Fuzzy Systems
Volume 2008 (2008), Article ID 421525, 15 pages
doi:10.1155/2008/421525
Research Article

On Controllability and Observability of Fuzzy Dynamical Matrix Lyapunov Systems

M. S. N. Murty and G. Suresh Kumar

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 ) = 𝑋 0 , 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 ] ⊂ 𝑅 0 x 0 0 0 a 0 ( 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., [5–12]).

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 , 0 x 0 0 0 a 0 𝑋 , 0 x 0 0 0 a 0 𝑈 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 𝑃 𝑘 ( 𝑅 𝑛 ) .

2. Preliminaries

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

Let 𝑅 𝑛 denote the family of all nonempty compact convex subsets of 𝑃 𝑘 ( 𝑅 𝑛 ) . Define the addition and scalar multiplication in 𝑃 𝑘 ( 𝑅 𝑛 ) as usual. Radstrom [15] states that 𝛼 , 𝛽 ∈ 𝑅 is a commutative semigroup under addition, which satisfies the cancellation law. Moreover, if 𝐴 , 𝐵 ∈ 𝑃 𝑘 ( 𝑅 𝑛 ) and 𝛼 ( 𝐴 + 𝐵 ) = 𝛼 𝐴 + 𝛼 𝐵 , 𝛼 ( 𝛽 𝐴 ) = ( 𝛼 𝛽 ) 𝐴 , 1 ⋅ 𝐴 = 𝐴 , ( 3 ) , then 𝛼 , 𝛽 ≥ 0 and if ( 𝛼 + 𝛽 ) 𝐴 = 𝛼 𝐴 + 𝛽 𝐴 , then 𝐴 . The distance between 𝐵 and 𝑑 ( 𝐴 , 𝐵 ) = i n f 𝜖 ∶ 𝐴 ⊂ 𝑁 ( 𝐵 , 𝜖 ) , 0 x 0 0 0 a 0 𝐵 ⊂ 𝑁 ( 𝐴 , 𝜖 ) , ( 4 ) is defined by the Hausdorff metric 𝑁 ( 𝐴 , 𝜖 ) = 𝑥 ∈ 𝑅 𝑛 ∶ ‖ ‖ ‖ ‖ 𝑥 − 𝑦 < 𝜖 , 0 x 0 0 0 a 0 f o r 0 x 0 0 0 a 0 s o m e 0 x 0 0 0 a 0 𝑦 ∈ 𝐴 . ( 5 ) where 𝐹 ∶ 𝐽 → 𝑃 𝑘 ( 𝑅 𝑛 )

Definition 1. A set-valued function 𝑢 ∈ 𝑅 𝑛 is said to be measurable if it satisfies any one of the following equivalent conditions: (1)for all 𝑡 → 𝑑 𝐹 ( 𝑡 ) ( 𝑢 ) = i n f 𝑣 ∈ 𝐹 ( 𝑡 ) ‖ 𝑢 − 𝑣 ‖ , G r 𝐹 = { ( 𝑡 , 𝑢 ) ∈ 𝐽 × 𝑅 𝑛 ∶ 𝑢 ∈ 𝐹 ( 𝑡 ) } ∈ Σ × 𝛽 ( 𝑅 𝑛 ) is measurable,(2) Σ , 𝛽 ( 𝑅 𝑛 ) , where 𝜎 are Borel 𝐽 -field of 𝑅 𝑛 and { 𝑓 𝑛 ( ⋅ ) } 𝑛 ≥ 1 , respectively (Graph measurability),(3)there exists a sequence 𝐹 ( 𝑡 ) = { 𝑓 𝑛 ( ⋅ ) } 𝑛 ≥ 1 of measurable functions such that 𝑡 ∈ 𝐽 , for all 𝑆 1 𝐹 (Castaing's representation).

We denote by 𝐹 ( ⋅ ) the set of all selections of 𝐿 1 𝑅 𝑛 ( 𝐽 ) that belong to the Lebesgue Bochner space 𝑆 1 𝐹 = 𝑓 ( ⋅ ) ∈ 𝐿 1 𝑅 𝑛 ( 𝐽 ) ∶ 𝑓 ( 𝑡 ) ∈ 𝐹 ( 𝑡 ) 0 x 0 0 0 a 0 a . e . . ( 6 ) , that is, ( 𝐴 ) 𝐽 𝐹 ( 𝑡 ) 𝑑 𝑡 = 𝐽 𝑓 ( 𝑡 ) 𝑑 𝑡 , 𝑓 ( ⋅ ) ∈ 𝑆 1 𝐹 . ( 7 ) We present the Aumann's integral as follows: 𝐹 ∶ 𝐽 → 𝑃 𝑘 ( 𝑅 𝑛 )

We say that ℎ ∶ 𝐽 → 𝑅 is integrably bounded if it is measurable and there exists a function ℎ ∈ 𝐿 1 𝑅 𝑛 ( 𝐽 ) , , ‖ 𝑢 ‖ ≤ ℎ ( 𝑡 ) such that 𝑢 ∈ 𝐹 ( 𝑡 ) , 𝐹 . From [16], we know that if ∫ 𝐽 𝐹 ( 𝑡 ) 𝑑 𝑡 is a closed valued measurable multifunction, then 𝑅 𝑛 is convex in 𝐹 . Furthermore, if ∫ 𝐽 𝐹 ( 𝑡 ) 𝑑 𝑡 is integrably bounded, then 𝑅 𝑛 is compact in 𝐸 𝑛 = 𝑢 ∶ 𝑅 𝑛 → 0 , 1 / 𝑢 0 x 0 0 0 a 0 s a t i s fi e ( i ) - ( i v ) b e l o w , ( 8 ) .

Let 𝑢 where (i) 𝑥 0 ∈ 𝑅 𝑛 is normal, that is, there exists an 𝑢 ( 𝑥 0 ) = 1 such that 𝑢 ;(ii) 𝑥 , 𝑦 ∈ 𝑅 𝑛 is fuzzy convex, that is, for 0 ≤ 𝜆 ≤ 1 and 𝑢 𝜆 𝑥 + ( 1 − 𝜆 ) 𝑦 ≥ m i n 𝑢 ( 𝑥 ) , 𝑢 ( 𝑦 ) ; ( 9 ) , 𝑢 (iii) [ 𝑢 ] 0 = { 𝑥 ∈ 𝑅 𝑛 / 𝑢 ( 𝑥 ) > 0 } is upper semicontinuous;(iv) 0 < 𝛼 ≤ 1 is compact. For 𝛼 , the [ 𝑢 ] 𝛼 = { 𝑥 ∈ 𝑅 𝑛 / 𝑢 ( 𝑥 ) ≥ 𝛼 } -level set is denoted and defined by [ 𝑢 ] 𝛼 ∈ 𝑃 𝑘 ( 𝑅 𝑛 ) . Then, from (i)–(iv) it follows that 0 ≤ 𝛼 ≤ 1 for all 𝐷 ∶ 𝐸 𝑛 × 𝐸 𝑛 → [ 0 , ∞ ) .

Define 𝑑 𝐷 ( 𝑢 , 𝑣 ) = s u p [ 𝑢 ] 𝛼 , [ 𝑣 ] 𝛼 / 𝛼 ∈ [ 0 , 1 ] , ( 1 0 ) by 𝑑 where 𝑃 𝑘 ( 𝑅 𝑛 ) is the Hausdorff metric defined in 𝐷 . It is easy to show that 𝐸 𝑛 is a metric in ( 𝐸 𝑛 , 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 , 𝑤 ∈ 𝐸 𝑛 , | | 𝜆 | | 𝐷 ( 𝜆 𝑢 , 𝜆 𝑣 ) = 𝐷 ( 𝑢 , 𝑣 ) , 𝑢 , 𝑣 ∈ 𝐸 𝑛 , 0 x 0 0 0 a 0 𝜆 ∈ 𝑅 , 𝐷 ( 𝑢 + 𝑤 , 𝑣 + 𝑧 ) ≤ 𝐷 ( 𝑢 , 𝑣 ) + 𝐷 ( 𝑤 , 𝑧 ) , 𝑢 , 𝑣 , 𝑤 , 𝑧 ∈ 𝐸 𝑛 . ( 1 1 ) verifies that ( 𝐸 𝑛 , 𝐷 ) We note that 𝑢 ∈ 𝐸 𝑛 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 ] 𝛼 ∈ 𝑃 𝑘 ( 𝑅 𝑛 ) , then (1) 0 ≤ 𝛼 ≤ 1 , for all [ 𝑢 ] 𝛼 2 ⊂ [ 𝑢 ] 𝛼 1 ; (2) 0 ≤ 𝛼 1 ≤ 𝛼 2 ≤ 1 , for all { 𝛼 𝑘 } ; (3) if 𝛼 > 0 is a nondecreasing sequence converging to [ 𝑢 ] 𝛼 = ⋂ 𝑘 ≥ 1 [ 𝑢 ] 𝛼 𝑘 , then { 𝐴 𝛼 ∶ 0 ≤ 𝛼 ≤ 1 } . Conversely, if 𝑅 𝑛 is a family of subsets of 𝑢 ∈ 𝐸 𝑛 satisfying (1)–(3), then there exists a [ 𝑢 ] 𝛼 = 𝐴 𝛼 such that 0 < 𝛼 ≤ 1 for [ 𝑢 ] 0 = ⋃ 0 ≤ 𝛼 ≤ 1 𝐴 𝛼 ⊂ 𝐴 0 and 𝐹 ∶ 𝐽 → 𝐸 𝑛 .

A fuzzy set-valued mapping 𝐹 0 ( 𝑡 ) is called fuzzy integrably bounded if 𝐹 ∶ 𝐽 → 𝐸 𝑛 is integrably bounded.

Definition 2. Let 𝐹 be a fuzzy integrably bounded mapping. The fuzzy integral of 𝐽 over ∫ 𝐽 𝐹 ( 𝑡 ) 𝑑 𝑡 denoted by 𝐽 𝐹 ( 𝑡 ) 𝑑 𝑡 𝛼 = ( 𝐴 ) 𝐽 𝐹 𝛼 ( 𝑡 ) 𝑑 𝑡 , 0 < 𝛼 ≤ 1 . ( 1 2 ) is defined level-set-wise by 𝐹 ∶ 𝐽 × 𝐸 𝑛 → 𝐸 𝑛
Let 𝑢 = 𝐹 ( 𝑡 , 𝑢 ) , 𝑢 ( 0 ) = 𝑢 0 . ( 1 3 ) ,and consider the fuzzy differential equation 𝑢 ∶ 𝐽 → 𝐸 𝑛

Definition 3. A mapping 𝑢 ( 𝑡 ) = 𝑢 0 + 𝑡 0 𝐹 𝑠 , 𝑢 ( 𝑠 ) 𝑑 𝑠 , ∀ 𝑡 ∈ 𝐽 . ( 1 4 ) is a fuzzy weak solution to (13) if it is continuous and satisfies the integral equation 𝐹 If 𝐴 ∈ 𝐶 𝑚 × 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 𝐵 ∈ 𝐶 𝑝 × 𝑞 and 𝐴 .Then the Kronecker product of 𝐵 and 𝐴 ⊗ 𝐵 written ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑎 𝐴 ⊗ 𝐵 = 1 1 𝐵 𝑎 1 2 𝐵 ⋯ 𝑎 1 𝑛 𝐵 𝑎 2 1 𝐵 𝑎 2 2 𝐵 ⋯ 𝑎 2 𝑛 𝐵 𝑎 . . ⋯ . 𝑚 1 𝐵 𝑎 𝑚 2 𝐵 ⋯ 𝑎 𝑚 𝑛 𝐵 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ( 1 5 ) is defined to be the partitioned matrix 𝑚 𝑝 × 𝑛 𝑞 which is an 𝐶 𝑚 𝑝 × 𝑛 𝑞 matrix and is in 𝐴 = [ 𝑎 𝑖 𝑗 ] ∈ 𝐶 𝑚 × 𝑛 .

Definition 5 (see [2]). Let ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝐴 𝐴 = V e c 𝐴 = . 1 𝐴 . 2 ⋮ 𝐴 . 𝑛 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , w h e r e 0 x 0 0 0 a 0 𝐴 . 𝑗 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑎 1 𝑗 𝑎 2 𝑗 ⋮ 𝑎 𝑚 𝑗 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 1 ≤ 𝑗 ≤ 𝑛 . ( 1 6 ) ; one denotes ( 𝐴 ⊗ 𝐵 ) ∗ = 𝐴 ∗ ⊗ 𝐵 ∗ The Kronecker product has the following properties and rules [2].
(1) 𝐴 ∗ ( 𝐴 denotes transpose of ( 𝐴 ⊗ 𝐵 ) − 1 = 𝐴 − 1 ⊗ 𝐵 − 1 ).(2) 𝐴 ⊗ 𝐵 ) ( 𝐶 ⊗ 𝐷 ) = ( 𝐴 𝐶 ⊗ 𝐵 𝐷 ) .(3)The mixed product rule( ‖ 𝐴 ⊗ 𝐵 ‖ = ‖ 𝐴 ‖ ‖ 𝐵 ‖ . This rule holds, provided the dimension of the matrices is such that the various expressions exist. (4) 𝐴 ( 𝑡 ) .(5)If 𝐵 ( 𝑡 ) and ( 𝐴 ⊗ 𝐵 ) = 𝐴 ⊗ 𝐵 + 𝐴 ⊗ 𝐵 0 x 0 0 0 a 0 ( = 𝑑 / 𝑑 𝑡 ) are matrices, then V e c ( 𝐴 𝑌 𝐵 ) = ( 𝐵 ∗ ⊗ 𝐴 ) V e c 𝑌 .(6) 𝐴 .(7)If 𝐵 and 𝑛 × 𝑛 are matrices both of order V e c ( 𝐴 𝑋 ) = ( 𝐼 𝑛 ⊗ 𝐴 ) V e c 𝑋 , then (i) V e c ( 𝑋 𝐴 ) = ( 𝐴 ∗ ⊗ 𝐼 𝑛 ) V e c 𝑋 ,(ii) 𝑋 ′ 𝐼 ( 𝑡 ) = 𝐺 ( 𝑡 ) 𝑋 ( 𝑡 ) + 𝑛 𝑋 ⊗ 𝐹 ( 𝑡 ) 𝑈 ( 𝑡 ) , 𝑋 ( 0 ) = 0 𝐼 , ( 1 7 ) 𝑌 ( 𝑡 ) = 𝑛 𝐼 ⊗ 𝐶 ( 𝑡 ) 𝑋 ( 𝑡 ) + 𝑛 ⊗ 𝐷 ( 𝑡 ) 𝑈 ( 𝑡 ) , ( 1 8 ) .

Now by applying the Vec operator to the matrix Lyapunov system (1) satisfying (2) and using the above properties, we have 𝐺 ( 𝑡 ) = ( 𝐵 ∗ ⊗ 𝐼 𝑛 ) + ( 𝐼 𝑛 ⊗ 𝐴 ) 𝑛 2 × 𝑛 2 where 𝑋 = V e c 𝑋 ( 𝑡 ) is an 𝑈 = V e c 𝑈 ( 𝑡 ) matrix and 𝑛 2 , 𝑋 ′ 𝑋 ( 𝑡 ) = 𝐺 ( 𝑡 ) 𝑋 ( 𝑡 ) , 𝑋 ( 0 ) = 0 . ( 1 9 ) 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 ) = 𝐼 𝑛 , ( 2 0 ) [ 𝑋 ∗ ( 𝑡 ) ] = 𝐵 ∗ ( 𝑡 ) 𝑋 ∗ ( 𝑡 ) , 𝑋 ( 0 ) = 𝐼 𝑛 , ( 2 1 ) and 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) be the fundamental matrices for the systems 𝑋 𝑋 ( 𝑡 ) = ( 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) ) 0 𝜓 ( 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 ) ⊗ 𝐼 𝑛 + 𝐼 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 𝑛 = 𝐵 ⊗ 𝐴 ( 𝑡 ) 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) ∗ ( 𝑡 ) ⊗ 𝐼 𝑛 + 𝐼 𝑛 . ⊗ 𝐴 ( 𝑡 ) 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) = 𝐺 ( 𝑡 ) 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) ( 2 2 ) respectively. Then the matrix 𝜓 ( 0 ) ⊗ 𝜙 ( 0 ) = 𝐼 𝑛 ⊗ 𝐼 𝑛 = 𝐼 𝑛 2 is a fundamental matrix of (19) and the solution of (19) is 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) .

Proof. Consider 𝑋 𝑋 ( 𝑡 ) = ( 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) ) 0 Also 𝜙 ( 𝑡 ) .
Hence, 𝜓 ( 𝑡 ) is a fundamental matrix of (19). Clearly, 𝑋 𝑋 ( 𝑡 ) = 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 0 + 𝑡 0 𝐼 𝜓 ( 𝑡 − 𝑠 ) ⊗ 𝜙 ( 𝑡 − 𝑠 ) 𝑛 ⊗ 𝐹 ( 𝑠 ) 𝑈 ( 𝑠 ) 𝑑 𝑠 . ( 2 3 ) is a solution of (19).

Theorem 2. Let 𝑋 𝑋 ( 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 ) = 𝑡 0 𝐼 𝜓 ( 𝑡 − 𝑠 ) ⊗ 𝜙 ( 𝑡 − 𝑠 ) 𝑛 ⊗ 𝐹 ( 𝑠 ) 𝑈 ( 𝑠 ) 𝑑 𝑠 . ( 2 4 )

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 𝑋 𝑤 = ( 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) ) 0 be any other solution of (17), write 𝑋 𝑢 ( 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 ) 𝑣 𝑣 ( 𝑡 ) + 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 𝐼 ( 𝑡 ) ⇒ 𝐺 ( 𝑡 ) 𝑋 ( 𝑡 ) + 𝑛 ⊗ 𝑣 𝐹 ( 𝑡 ) 𝑈 ( 𝑡 ) = 𝐺 ( 𝑡 ) 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 𝑣 ( 𝑡 ) + 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) ⇒ 𝑣 ( 𝑡 ) 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) = 𝐼 ( 𝑡 ) 𝑛 ⊗ 𝐹 ( 𝑡 ) 𝑈 ( 𝑡 ) ⇒ 𝑣 = 𝜓 ( 𝑡 ) − 1 ( 𝑡 ) ⊗ 𝜙 − 1 𝐼 ( 𝑡 ) 𝑛 ⊗ = 𝐹 ( 𝑡 ) 𝑈 ( 𝑡 ) ⇒ 𝑣 ( 𝑡 ) 𝑡 0 𝜓 − 1 ( 𝑠 ) ⊗ 𝜙 − 1 𝐼 ( 𝑠 ) 𝑛 ⊗ 𝐹 ( 𝑠 ) 𝑈 ( 𝑠 ) 𝑑 𝑠 . ( 2 5 ) .
Consider the vector 𝜙 ( 𝑡 ) 𝜙 − 1 ( 𝑠 ) = 𝜙 ( 𝑡 − 𝑠 ) , where 𝜓 ( 𝑡 ) 𝜓 − 1 ( 𝑠 ) = 𝜓 ( 𝑡 − 𝑠 ) is an arbitrary vector to be determined so as to satisfy (17), 𝑢 𝑖 ( 𝑡 ) ∈ 𝐸 1 Hence, the desired expression follows immediately by noting the fact that 0 x 0 0 0 a 0 𝑡 ∈ 𝐽 and 0 x 0 0 0 a 0 𝑖 = 1 , 2 , … , 𝑛 2 .

3. Formation of Fuzzy Dynamical Lyapunov Systems

Let 𝑢 𝑈 ( 𝑡 ) = 1 ( 𝑡 ) , 𝑢 2 ( 𝑡 ) , … , 𝑢 𝑛 2 ( 𝑡 ) = 𝑢 1 ( 𝑡 ) × 𝑢 2 ( 𝑡 ) × ⋯ × 𝑢 𝑛 2 = 𝑢 ( 𝑡 ) 𝛼 1 ( 𝑡 ) , 𝑢 𝛼 2 ( 𝑡 ) , … , 𝑢 𝛼 𝑛 2 = ( 𝑡 ) ∶ 𝛼 ∈ [ 0 , 1 ] 𝑢 1 ( 𝑡 ) , 𝑢 2 ( 𝑡 ) , … , 𝑢 𝑛 2 ( 𝑡 ) ∶ 𝑢 𝑖 ( 𝑡 ) ∈ 𝑢 𝛼 𝑖 , ( 𝑡 ) , 0 x 0 0 0 a 0 𝛼 ∈ [ 0 , 1 ] ( 2 6 ) , 𝑢 𝛼 𝑖 ( 𝑡 ) , 𝛼 , and define 𝑢 𝑖 ( 𝑡 ) where 𝑈 ( 𝑡 ) is the 𝑈 ( 𝑡 ) ∈ 𝐸 𝑛 2 -level set of 𝑈 ( 𝑡 ) . From the above definition of 𝑋 ′ 𝐼 ( 𝑡 ) = 𝐺 ( 𝑡 ) 𝑋 ( 𝑡 ) + 𝑛 𝑋 ⊗ 𝐹 ( 𝑡 ) 𝑈 ( 𝑡 ) , 𝑋 ( 0 ) = 0 𝐼 , ( 2 7 ) 𝑌 ( 𝑡 ) = 𝑛 𝐼 ⊗ 𝐶 ( 𝑡 ) 𝑋 ( 𝑡 ) + 𝑛 ⊗ 𝐷 ( 𝑡 ) 𝑈 ( 𝑡 ) ( 2 8 ) and Theorem 1, it can be easily seen that 𝑈 ( 𝑡 ) .

Now by using the fuzzy control 𝐸 𝑛 2 , we show that the following system 𝑈 𝛼 = 𝑢 1 ( 𝑡 ) × 𝑢 2 ( 𝑡 ) × ⋯ × 𝑢 𝑛 2 ( 𝑡 ) 𝑅 𝑛 2 determines a fuzzy system.

Assume that 𝑇 is continuous in 𝑋 ′ 𝐼 ( 𝑡 ) ∈ 𝐺 ( 𝑡 ) 𝑋 ( 𝑡 ) + 𝑛 𝑈 ⊗ 𝐹 ( 𝑡 ) 𝛼 𝑋 ( 𝑡 ) , 𝑡 ∈ [ 0 , 𝑇 ] , ( 2 9 ) 𝑋 ( 0 ) = 0 . ( 3 0 ) . The set 𝑋 𝛼 is a convex and compact set in [ 𝑋 ( 𝑡 ) ] 𝛼 ∈ 𝑃 𝑘 ( 𝑅 𝑛 2 ) . For any positive number 0 ≤ 𝛼 ≤ 1 , consider the following differential inclusions: 𝑡 ∈ [ 0 , 𝑇 ] 𝑋 𝛼 Let 𝐶 [ [ 0 , 𝑇 ] , 𝑅 𝑛 2 ] be the solution of (29) satisfying (30).

Claim (i). 𝑈 𝛼 ( 𝑡 ) , for every 𝑋 𝛼 , 𝐾 = m a x 𝑡 ∈ [ 0 , 𝑇 ] ‖ 𝜙 ( 𝑡 ) ‖ .
First, we prove that 𝐿 = m a x 𝑡 ∈ [ 0 , 𝑇 ] ‖ 𝜓 ( 𝑡 ) ‖ is nonempty, compact, and convex in 𝑈 𝑀 = m a x { ‖ 𝑢 ( 𝑡 ) ‖ ∶ 𝑢 ( 𝑡 ) ∈ 𝛼 ( 𝑡 ) , 0 x 0 0 0 a 0 𝑡 ∈ [ 0 , 𝑇 ] } . Since 𝑁 = m a x 𝑡 ∈ [ 0 , 𝑇 ] has measurable selection, we have that ‖ 𝐹 ( 𝑡 ) ‖ is nonempty.
Let 𝑋 𝑋 ∈ 𝛼 , 𝑈 𝑢 ( 𝑡 ) ∈ 𝛼 ( 𝑡 ) , 𝑋 𝑋 ( 𝑡 ) = 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 0 + 𝑡 0 𝐼 𝜓 ( 𝑡 − 𝑠 ) ⊗ 𝜙 ( 𝑡 − 𝑠 ) 𝑛 ⊗ 𝐹 ( 𝑠 ) 𝑢 ( 𝑠 ) 𝑑 𝑠 . ( 3 1 ) , ‖ ‖ ‖ ‖ ≤ ‖ ‖ 𝑋 𝑋 ( 𝑡 ) 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 0 ‖ ‖ + 𝑡 0 ‖ ‖ 𝐼 𝜓 ( 𝑡 − 𝑠 ) ⊗ 𝜙 ( 𝑡 − 𝑠 ) 𝑛 ‖ ‖ ≤ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ 𝑋 ⊗ 𝐹 ( 𝑠 ) 𝑢 ( 𝑠 ) 𝑑 𝑠 𝜓 ( 𝑡 ) 𝜙 ( 𝑡 ) 0 ‖ ‖ + 𝑡 0 ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ 𝑋 𝜓 ( 𝑡 − 𝑠 ) 𝜙 ( 𝑡 − 𝑠 ) 𝐹 ( 𝑠 ) 𝑢 ( 𝑠 ) 𝑑 𝑠 ≤ 𝐾 𝐿 0 ‖ ‖ + 𝐾 𝐿 𝑁 𝑀 𝑇 . ( 3 2 ) .
If for any 𝑡 1 , 𝑡 2 ∈ [ 0 , 𝑇 ] , then there is a selection 𝑋 ( 𝑡 1 ) − 𝑋 ( 𝑡 2 ) = 𝜓 ( 𝑡 1 ) ⊗ 𝜙 ( 𝑡 1 ) 𝑋 0 + 𝑡 1 0 𝜓 ( 𝑡 1 − 𝑠 ) ⊗ 𝜙 ( 𝑡 1 𝐼 − 𝑠 ) 𝑛 − ⊗ 𝐹 ( 𝑠 ) 𝑢 ( 𝑠 ) 𝑑 𝑠 𝜓 ( 𝑡 2 ) ⊗ 𝜙 ( 𝑡 2 ) 𝑋 0 − 𝑡 2 0 𝜓 ( 𝑡 2 − 𝑠 ) ⊗ 𝜙 ( 𝑡 2 𝐼 − 𝑠 ) 𝑛 ⊗ 𝐹 ( 𝑠 ) 𝑢 ( 𝑠 ) 𝑑 𝑠 . ( 3 3 ) such that ‖ ‖ 𝑋 ( 𝑡 1 ) − 𝑋 ( 𝑡 2 ) ‖ ‖ ≤ ‖ ‖ 𝜓 ( 𝑡 1 ) ⊗ 𝜙 ( 𝑡 1 ) − 𝜓 ( 𝑡 2 ) ⊗ 𝜙 ( 𝑡 2 ) ‖ ‖ ‖ ‖ 𝑋 0 ‖ ‖ + 𝑡 1 𝑡 2 ‖ ‖ 𝜓 ( 𝑡 1 − 𝑠 ) ⊗ 𝜙 ( 𝑡 1 𝐼 − 𝑠 ) 𝑛 ‖ ‖ + ⊗ 𝐹 ( 𝑠 ) 𝑢 ( 𝑠 ) 𝑑 𝑠 𝑡 2 0 ‖ ‖ 𝜓 ( 𝑡 1 − 𝑠 ) ⊗ 𝜙 ( 𝑡 1 − 𝑠 ) 0 x 0 0 0 a 0 − 𝜓 ( 𝑡 2 − 𝑠 ) ⊗ 𝜙 ( 𝑡 2 𝐼 − 𝑠 ) 𝑛 ‖ ‖ ≤ ‖ ‖ ⊗ 𝐹 ( 𝑠 ) 𝑢 ( 𝑠 ) 𝑑 𝑠 𝜓 ( 𝑡 1 ) ⊗ 𝜙 ( 𝑡 1 ) − 𝜓 ( 𝑡 2 ) ⊗ 𝜙 ( 𝑡 2 ) ‖ ‖ ‖ ‖ 𝑋 0 ‖ ‖ | | 𝑡 + 𝐾 𝐿 𝑁 𝑀 1 − 𝑡 2 | | + 𝑀 𝑁 𝑇 0 ‖ ‖ 𝜓 ( 𝑡 1 − 𝑠 ) ⊗ 𝜙 ( 𝑡 1 − 𝑠 ) 0 x 0 0 0 a 0 − 𝜓 ( 𝑡 2 − 𝑠 ) ⊗ 𝜙 ( 𝑡 2 ‖ ‖ − 𝑠 ) 𝑑 𝑠 . ( 3 4 ) Then 𝜙 ( 𝑡 ) Thus 𝜓 ( 𝑡 ) is bounded.
For any [ 0 , 𝑇 ] , 𝑋 Therefore 𝑋 𝛼 Since 𝑋 𝛼 and 𝑋 𝑘 ∈ 𝑋 𝛼 are both uniformly continuous on 𝑋 𝑘 → 𝑋 , 𝑋 𝑘 is equicontinuous. Thus, 𝑢 𝑘 ∈ 𝑈 𝛼 ( 𝑡 ) is relatively compact. If 𝑋 𝑘 𝑋 ( 𝑡 ) = 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 0 + 𝑡 0 𝐼 𝜓 ( 𝑡 − 𝑠 ) ⊗ 𝜙 ( 𝑡 − 𝑠 ) 𝑛 𝑢 ⊗ 𝐹 ( 𝑠 ) 𝑘 ( 𝑠 ) 𝑑 𝑠 . ( 3 5 ) is closed, then it is compact.
Let 𝑢 𝑘 ∈ 𝑈 𝛼 ( 𝑡 ) and { 𝑢 𝑘 𝑗 } . For each { 𝑢 𝑘 } , there is a 𝑈 𝑢 ∈ 𝛼 ( 𝑡 ) such that 𝜆 𝑗 > 0 Since ∑ 𝜆 𝑗 = 1 is closed, then there exists a subsequence ∑ 𝜆 𝑗 𝑢 𝑘 𝑗 of 𝑢 converging weakly to 𝜆 𝑗 𝑋 𝑘 𝑗 = 𝜆 ( 𝑡 ) 𝑗 𝑋 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 0 + 𝑡 0 𝐼 𝜓 ( 𝑡 − 𝑠 ) ⊗ 𝜙 ( 𝑡 − 𝑠 ) 𝑛 𝜆 ⊗ 𝐹 ( 𝑠 ) 𝑗 𝑢 𝑘 𝑗 ( 𝑠 ) 𝑑 𝑠 . ( 3 6 ) . From Mazur's theorem [20], there exists a sequence of numbers 𝑗 → ∞ , 𝑋 𝑋 ( 𝑡 ) = 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 0 + 𝑡 0 𝐼 𝜓 ( 𝑡 − 𝑠 ) ⊗ 𝜙 ( 𝑡 − 𝑠 ) 𝑛 ⊗ 𝐹 ( 𝑠 ) 𝑢 ( 𝑠 ) 𝑑 𝑠 . ( 3 7 ) such that 𝑋 𝑋 ( 𝑡 ) ∈ 𝛼 converges strongly to 𝑋 𝛼 .
Thus, from (35) we have 𝑋 1 From Fatou's lemma, taking the limit as 𝑋 2 ∈ 𝑋 𝛼 on both sides of (36), we have 𝑢 1 , 𝑢 2 ∈ 𝑈 𝛼 ( 𝑡 ) Thus, 𝑋 ′ 1 𝑋 ( 𝑡 ) = 𝐺 ( 𝑡 ) 1 𝐼 ( 𝑡 ) + 𝑛 𝑢 ⊗ 𝐹 ( 𝑡 ) 1 𝑋 ( 𝑡 ) , ′ 2 𝑋 ( 𝑡 ) = 𝐺 ( 𝑡 ) 2 𝐼 ( 𝑡 ) + 𝑛 𝑢 ⊗ 𝐹 ( 𝑡 ) 2 ( 𝑡 ) . ( 3 8 ) , and hence 𝑋 𝑋 = 𝜆 1 𝑋 ( 𝑡 ) + ( 1 − 𝜆 ) 2 ( 𝑡 ) is closed.
Let 0 ≤ 𝜆 ≤ 1 , 𝑋 ′ 𝑋 = 𝜆 ′ 1 𝑋 ( 𝑡 ) + ( 1 − 𝜆 ) ′ 2 𝑋 ( 𝑡 ) = 𝜆 𝐺 ( 𝑡 ) 1 𝐼 ( 𝑡 ) + 𝑛 𝑢 ⊗ 𝐹 ( 𝑡 ) 1 𝑋 ( 𝑡 ) + ( 1 − 𝜆 ) 𝐺 ( 𝑡 ) 2 𝐼 ( 𝑡 ) + 𝑛 𝑢 ⊗ 𝐹 ( 𝑡 ) 2 𝜆 𝑋 ( 𝑡 ) = 𝐺 ( 𝑡 ) 1 𝑋 ( 𝑡 ) + ( 1 − 𝜆 ) 2 + 𝐼 ( 𝑡 ) 𝑛 ⊗ 𝐹 ( 𝑡 ) 𝜆 𝑢 1 ( 𝑡 ) + ( 1 − 𝜆 ) 𝑢 2 . ( 𝑡 ) ( 3 9 ) , then there exist 𝑈 𝛼 ( 𝑡 ) such that 𝜆 𝑢 1 ( 𝑡 ) + ( 1 − 𝜆 ) 𝑢 2 𝑈 ( 𝑡 ) ∈ 𝛼 ( 𝑡 ) Let 𝑋 ′ 𝐼 ( 𝑡 ) ∈ 𝐺 ( 𝑡 ) 𝑋 ( 𝑡 ) + 𝑛 𝑈 ⊗ 𝐹 ( 𝑡 ) 𝛼 ( 𝑡 ) , ( 4 0 ) , 𝑋 𝑋 ∈ 𝛼 , then 𝑋 𝛼 Since 𝑋 𝛼 is convex, 𝐶 [ [ 0 , 𝑇 ] , 𝑅 𝑛 2 ] , we have [ 𝑋 ( 𝑡 ) ] 𝛼 that is 𝑅 𝑛 2 . Thus 𝑡 ∈ [ 0 , 𝑇 ] is convex. Therefore, [ 𝑋 ( 𝑡 ) ] 𝛼 is nonempty, compact, and convex in 𝑅 𝑛 2 . Thus, from Arzela-Ascoli theorem, we know that [ 𝑋 ( 𝑡 ) ] 𝛼 ∈ 𝑃 𝑘 ( 𝑅 𝑛 2 ) is compact in 𝑡 ∈ [ 0 , 𝑇 ] for every [ 𝑋 ( 𝑡 ) ] 𝛼 2 ⊂ [ 𝑋 ( 𝑡 ) ] 𝛼 1 . Also it is obvious that 0 ≤ 𝛼 1 ≤ 𝛼 2 ≤ 1 is convex in 0 ≤ 𝛼 1 ≤ 𝛼 2 ≤ 1 . Thus, we have 𝑈 𝛼 2 𝑈 ( 𝑡 ) ⊂ 𝛼 1 ( 𝑡 ) , for every 𝑈 𝛼 2 ( 𝑡 ) = 𝑢 𝛼 2 1 ( 𝑡 ) × 𝑢 𝛼 2 2 ( 𝑡 ) × ⋯ × 𝑢 𝛼 2 𝑛 2 ( 𝑡 ) ⊂ 𝑢 𝛼 1 1 ( 𝑡 ) × 𝑢 𝛼 1 2 ( 𝑡 ) × ⋯ × 𝑢 𝛼 1 𝑛 2 = ( 𝑡 ) 𝑈 𝛼 1 ( 𝑡 ) . ( 4 1 ) . Hence the claim.

Claim (ii). 𝑆 1 𝑈 𝛼 2 ( 𝑡 ) ⊂ 𝑆 1 𝑈 𝛼 1 ( 𝑡 ) , for all 𝑋 ′ 𝐼 ( 𝑡 ) ∈ 𝐺 ( 𝑡 ) 𝑋 + 𝑛 𝑈 ⊗ 𝐹 ( 𝑡 ) 𝛼 2 𝐼 ( 𝑡 ) ⊂ 𝐺 ( 𝑡 ) 𝑋 + 𝑛 𝑈 ⊗ 𝐹 ( 𝑡 ) 𝛼 1 ( 𝑡 ) . ( 4 2 ) .
Let 𝑋 ′ 𝐼 ( 𝑡 ) ∈ 𝐺 ( 𝑡 ) 𝑋 + 𝑛 𝑈 ⊗ 𝐹 ( 𝑡 ) 𝛼 2 𝑋 ( 𝑡 ) , 𝑡 ∈ [ 0 , 𝑇 ] , ( 4 3 ) ′ 𝐼 ( 𝑡 ) ∈ 𝐺 ( 𝑡 ) 𝑋 + 𝑛 𝑈 ⊗ 𝐹 ( 𝑡 ) 𝛼 1 ( 𝑡 ) , 𝑡 ∈ [ 0 , 𝑇 ] . ( 4 4 ) . Since 𝑋 𝛼 2 , we have 𝑋 𝛼 1 Thus, we have the selection inclusion 𝑋 𝑋 ( 𝑡 ) ∈ 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 0 + 𝑡 0 𝐼 𝜓 ( 𝑡 − 𝑠 ) ⊗ 𝜙 ( 𝑡 − 𝑠 ) 𝑛 𝑆 ⊗ 𝐹 ( 𝑠 ) 1 𝑈 𝛼 2 ( 𝑠 ) ⊂ 𝑋 𝑑 𝑠 𝜓 ( 𝑡 ) ⊗ 𝜙 ( 𝑡 ) 0 + 𝑡 0 𝐼 𝜓 ( 𝑡 −