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Contraction Mapping Theory and Approach to LMI-Based Stability Criteria of T-S Fuzzy Impulsive Time-Delays Integrodifferential Equations
In this paper, Banach fixed point theorem is employed to derive LMI-based exponential stability of impulsive Takagi-Sugeno (T-S) fuzzy integrodifferential equations, originated from Cohen-Grossberg Neural Networks (CGNNs). As far as we know, Banach fixed point theorem is rarely employed to derive LMI criteria for T-S fuzzy CGNNs, and this inspires our present work. It is worth mentioning that the conditions on the behavior functions are weaker than those of existing results, and the formulated contraction mapping and fixed point technique are different from those of previous literature. Even a corollary of our main result improves one of existing main results due to extending linear function to nonlinear function. Besides, the LMI-based criteria are programmable for computer MATLAB LMI toolbox. Moreover, an analytical table and a numerical example are presented to illustrate the advantage, feasibility, and effectiveness of the proposed methods.
In this paper, we consider a class of integrodifferential equations, which is originated from Cohen-Grossberg Neural Networks (CGNNs). CGNNs model was proposed originally by Cohen-Grossberg in 1983 . Since then, the stability analysis of CGNNs has attracted extensive attentions owing to its wide applications. Lyapunov function method is a common technique which derived stability criteria of various time-delays dynamic systems [2–13]. However, any method has its limitations. Fixed point technique may be one of common replacements. Brouwer fixed point theorem and Schauder fixed point theorem were always applied to stability analysis of various CGNNs models [14–17]. Of course, Banach fixed point theorem was also employed to derive the stability criteria of various CGNNs models [17–22]. But for most of existing literature, either using the same inverse function translates CGNNs model into another model similar as follows (see, e.g., [17, 19–21]), or the stability criteria are too complex [22, Theorem 3.1], which cannot be programmable for computer software while practical engineering is often involved in large-scale computing. Of course, both methods and results of [17, 19–22] are good and referential. But, in order to innovate, we have to find another way. In , a LMI-based stability criterion was given for the following CGNNs:
Theorem A (see [18, Theorem 4]). If (H1)–(H4) are satisfied and there exists a positive constant such that the following LMI condition holds, then System (2) is globally exponentially stable, where the four conditions are proposed as follows.(H1)For any , there exist constants , , , and such that (H2)For any , is differentiable, and there exists a constant such that (H3)There exist nonnegative constants such that (H4)For , there exist and a constant such that
Remark 1. Denoting , we know from [18, (H4)] that can only be a linear function on . So it is the main objective of this paper to make up the deficiency. Below, we try to make the condition of better so that may be a nonlinear function on .
All the good results and methods in existing literature, particularly in [17, 19–22], inspired our current work. Of course, we cannot imitate (1) again, and the condition [18, (H4)] should be replaced by a new suitable condition. Below, we shall propose a new condition which is better than [18, (H4)]. Due to the changes of conditions, we have to formulate new contraction mapping to derive new LMI-based exponential stability criterion for Takagi-Sugeno (T-S) fuzzy impulsive CGNNs with discrete and distributed delays. This is one of main innovations of this paper. In addition, we shall admit weaker condition on the behavior functions than that of existing literature (e.g., [17, 19–22]) published from 2007 to 2016. Moreover, a corollary of our main result is better than [18, Theorem 3.1] due to the above reasons.
|Conditions [19, ()], [22, ()], and [17, (H3)] are similar to .|
For convenience’s sake, we introduce the following standard notations.(i) 0 (<0) is a positive (negative) definite matrix; that is, 0 (<0) for any .(ii) 0 (⩽0) is a semipositive (seminegative) definite matrix; that is, 0 (⩽0) for any .(iii) implies that for all with , , and .(iv) means matrix is a semipositive (seminegative) definite matrix.(v) means matrix is a positive (negative) definite matrix.(vi) denotes the largest and smallest eigenvalue of matrix , respectively.(vii)Denote for any matrix .(viii) for any vector .(ix) (⩾) implies that (⩾) , , and (>) implies that (>) , for any vectors and .(x) is identity matrix with compatible dimension.
Cohen-Grossberg Neural Networks (CGNNs) with discrete and distributed delays have been investigated in many papers [23–28]. Consider the following impulsive CGNNs with discrete and distributed delays,with the initial condition where with being the state variables of the th neuron at time . The neuron active functions for any given . is -dimension diagonal matrix with representing an amplification function, and with being an appropriately behavior function. , , and are the connection weight matrices with representing the strengths of connectivity between cells and at time . Vector functions , and with activation functions telling how the th neuron reacts to the input. is discrete delay, and is distributed time-delay. is a positive constant with denotes the impulsive function, and the fixed impulsive moments satisfy with and stand for the right-hand and left-hand limit of at time , respectively. Further, suppose that In addition, we assume that , where is the space of all the continuous functions defined on .
Since practice has shown that fuzzy logic theory is an efficient approach to deal with the analysis and synthesis problems for complex nonlinear system, the fuzzy model is far more important than stochastic model [29–31]. Among various kinds of fuzzy methods, Takagi-Sugeno (T-S) fuzzy models provide a successful method to describe certain complex nonlinear systems using some local linear subsystems.
Below, we describe the T-S fuzzy mathematical model with time-delay as follows.
Fuzzy Rule . IF is is THENwhere is the premise variable, is the fuzzy set that is characterized by membership function, is the number of the IF-THEN rules, and is the number of the premise variables. By way of a standard fuzzy inference method, System (11) is inferred as follows:where , , and is the membership function of the system with respect to the fuzzy rule . can be regarded as the normalized weight of each IF-THEN rule, satisfying and .
Throughout this paper, we assume that and the following.(A1) all are Lipschitz continuous functions with Lipschitz constants , respectively. In addition, there are positive constants such that (A2)There is positive definite diagonal matrix such that , and is Lipschitz continuous with Lipschitz constant , for all , , where is -dimension diagonal matrix.(A3)For each , there exists a constant such that the function is Lipschitz continuous with the Lipschitz constant .
Remark 3. There are a large number of functions satisfying (A2) and (A3).
For example, we denote Obviously, is continuous for all so that exists, and Let , and then the constant Next, let and then Hence, and it is easy to prove that the above function is differentiable, and for all So we can set such that In addition, we can prove that is also Lipschitz continuous. Indeed, Below, we only need to prove is Lipschitz continuous. Firstly, we claim that is differentiable for all . In fact, we can get by L’Hospital’s rule and Hence, is differentiable for all , and . Further, Lagrangian differential mean value theorem derives Owing to we can conclude that the continuous function is bounded in all , which implies that is a positive constant, and hence both and are Lipschitz continuous for all This has proved that and defined as (15) satisfy (A2) and (A3).
Remark 4. Let ; we know that condition (A3) admits that is a nonlinear function on while [18, (H4)] derives that can only be a linear function on .
Definition 6. Impulsive fuzzy CGNNs (12) with initial condition (9) are said to be globally exponentially stable if, for any initial condition , there exists a pair of positive constants and such that where the norm
Lemma 7 (contraction mapping theorem, see ). Letting be a contraction operator on a complete metric space , then there exists a unique point for which .
3. Main Results
If (A1)–(A3) hold, we can derive the following main result.
Proof. First of all, we need to formulate integral equations equivalent to (12).
Denote, for convenience, , and then . Thereby, we have And hence .
Next, we claim that System (12) with initial condition (9) is equivalent to with initial condition (9).
On one hand, we can prove that the solution of System (27) with initial condition (9) is that of System (12) with initial condition (9).
Indeed, we get by (27) Differentiating both sides of (28) on results in that for , which generates the first equation of System (12).
Further, let in (27) produce , and let in (27) come to , which derives the second equation of (12).
Thus, we have proved the above claim.
On the other hand, we claim that the solution of System (12) with initial condition (9) is that of System (27) with initial condition (9).
In fact, multiplying both sides of the first equation of System (12) with results in for all Moreover, integrating from to gives which yields, after letting , for all
Throughout this paper, we assume that is a sufficiently small positive number. Now, taking in (32) reaches which yields by letting Combining (32) and the above equation comes to for all . Thereby, we have Synthesizing the above analysis derives System (27). Hence, we have proved that each solution of (12) with initial condition (9) is that of (27) with initial condition (9). Now, we have proved that System (12) with initial condition (9) is really equivalent to integral equations (27) with initial condition (9).
To apply the contraction mapping theorem, we firstly define the complete metric space as follows.
Let be the space consisting of functions , satisfying the following:(a) is continuous on .(b), for .(c) and exists, for all .(d) as , where is a positive constant, satisfying .It is not difficult to verify that the product space is a complete metric space if it is equipped with the following metric: where and
Hence, we define the mapping as follows: and for all
Below, we are to prove that is contraction mapping from into .
We may firstly prove . So we need to verify that satisfies conditions (a)–(d) for all
Indeed, for is continuous on owing to (38), and satisfies condition (a). Further, the definition of implies that satisfies condition (b). Besides, we can derive from (38) that which comes to a conclusion that satisfies condition (c). Remark, the above convergence is under the metric of the metric space . Below, all the convergence is in this sense. No longer repeat.
Throughout this section, we assume that is a sufficiently small positive real number.
Below, we need to prove that satisfies condition (d).
Indeed, for , we have where Owing to condition (d), for as So, for any given , there exists sufficiently large such that where the fixed vector When , we have It is obvious that Besides, which together with arbitrariness implies that Now we have actually proved that as
Similarly, we have Similarly, we use the methods employed in (44)–(47), obtaining as
Besides, Similarly, we use the methods employed in (44)–(47), obtaining as
Next, It is clear that Besides,