In this section, we first recall some basic definitions and lemmas on time scales which are used in what follows.
Let be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators and the graininess are defined, respectively, by
A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum m, then , otherwise . If has a right-scattered minimum m, then , otherwise .
Definition 2.1 ([51])
A function is called regulated provided its right-hand side limits exist (finite) at all right-hand side points in and its left-hand side limits exist (finite) at all left-hand side points in .
Definition 2.2 ([51])
A function is called rd-continuous provided it is continuous at right-dense point in and its left-hand side limits exist (finite) at left-dense points in . The set of rd-continuous function will be denoted by .
Definition 2.3 ([51])
Assume and . Then we define to be the number (if it exists) with the property that given any there exists a neighborhood U of t (i.e., for some ) such that
for all . We call the delta (or Hilger) derivative of f at t. The set of functions that is a differentiable and whose derivative is rd-continuous is denoted by .
If f is continuous, then f is rd-continuous. If f is rd-continuous, then f is regulated. If f is delta differentiable at t, then f is continuous at t.
Lemma 2.1 ([51])
Let f be regulated, then there exists a function F which is delta differentiable with region of differentiation D such that for all .
Definition 2.4 ([51])
Assume that is a regulated function. Any function F as in Lemma 2.1 is called a Δ-antiderivative of f. We define the indefinite integral of a regulated function f by
where C is an arbitrary constant and F is a Δ-antiderivative of f. We define the Cauchy integral by for all .
A function is called an antiderivative of provided for all .
Lemma 2.2 ([51])
If , and , then
-
(i)
,
-
(ii)
if for all , then ,
-
(iii)
if on , then .
A function is called regressive if for all . The set of all regressive and rd-continuous functions will be denoted by . We define the set of all positively regressive elements of ℛ by . If p is a regressive function, then the generalized exponential function is defined by for all , with the cylinder transformation
Let be two regressive functions, we define
If , then .
The generalized exponential function has the following properties.
Lemma 2.3 ([51])
Assume that are two regressive functions, then
-
(i)
;
-
(ii)
;
-
(iii)
;
-
(iv)
;
-
(v)
;
-
(vi)
for all ;
-
(vii)
.
Lemma 2.4 ([51])
Assume that are delta differentiable at . Then
Lemma 2.5 ([52])
For each , let N be a neighborhood of t. Then, for , define to mean that, given , there exists a right neighborhood of t such that
where . If t is right-scattered and is continuous at t, this reduces to .
Next, we introduce the Banach space which is suitable for system (1.1)-(1.3).
Let be an open bounded domain in with smooth boundary ∂ Ω. Let be the set consisting of all the vector function which is rd-continuous with respect to and continuous with respect to . For every and , we define the set . Then is a Banach space with the norm , where . Let consist of all functions which map into and is rd-continuous with respect to and continuous with respect to . For every and , we define the set . Then is a Banach space equipped with the norm , where , , .
In order to achieve the global robust exponential synchronization, the following system (2.1)-(2.3) is the controlled slave system corresponding to the master system (1.1)-(1.3):
(2.1)
subject to the following initial conditions
(2.2)
and Dirichlet boundary conditions
(2.3)
where () and () are error functions. () is a constant error weighting coefficient. , , , .
From (1.1)-(1.3) and (2.1)-(2.3), we obtain the error system (2.4)-(2.6) as follows:
(2.4)
subject to the following initial conditions
(2.5)
and Dirichlet boundary conditions
(2.6)
The following definition is significant to study the global robust exponential synchronization of coupled neural networks (1.1)-(1.3) and (2.1)-(2.3).
Definition 2.5 Let and be the solution vectors of system (1.1)-(1.3) and its controlled slave system (2.1)-(2.3), respectively. is the error vector. Then the coupled systems (1.1)-(1.3) and (2.1)-(2.3) are said to be globally exponentially synchronized if there exists a controlled input vector and a positive constant and such that
where α is called the degree of exponential synchronization on time scales.